PhD Thesis by Iasef Md Rian

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Fractal-Based Computational Morphogenesis of Architectural Structures

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Iasef Md Rian A thesis presented for the degree of Doctor of Philosophy Department of Architecture and Design Politecnico di Torino Turin, Italy March, 2015 Guided by: Mario Sassone (Politecnico di Torino, Italy) Shuichi Asayama (Tokyo Denki University, Japan)

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April 9, 2015

Ph.D. in ’Architecture and Building Design’ XXVII Cycle (2012 - 2013 - 2014) Politecnico di Torino Viale Mittioli 39 Turin-10125, Italy Ph.D. Candidate: Iasef Md Rian Matriculation No: 189951 Tutors: Mario Sassone (Politecnico di Torino, Italy) Shuichi Asayama (Tokyo Denki University) c 2015 Iasef Md Rian

All Rights Reserved

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To

my father, my inspiration

maa, my strength

and

my wife, my life

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Abstract This research is an interdisciplinary research that encompasses mathematics, architecture and civil engineering, more precisely, the fractal geometry of mathematics, computational design of architecture and structural design of civil engineering. Geometry is a fundamental tool for articulating design ideas into building forms, it is an instrument that makes structures withstand against unfavorable forces. To shape architectural structures in innovative ways without compromising the interest of aesthetics, architects and engineers apply different geometric systems and exploit them to extract the most useful essence related to form creations and development. Since the past, available knowledge of geometric concepts and their development, including the development of construction and material technologies, in different times gave birth of architecture of corresponding forms. In the recent years, the quest for new forms drives architects and designers to interact with mathematics, which is a source of increasingly new geometric shapes. Today, mathematics delivers complex geometric systems, especially the nonlinear geometry, to the contemporary architects to design some unimaginable architecture and also enriches the art of structural forms. However, in this venture, there are few geometric notions that have not been exploited yet enough by architects and designers, and remain as mathematical entity. Fractal geometry is one such example, which we see more often in nature than any other geometric forms. This form system, abundantly appears in nature, shows beauty and complexity, and moreover, displays the functional, biological and structural properties, which is a source of design ideas and solutions for architects and engineers. This thesis has attempted to explore the mathematical concept of fractal geometry by understanding its usability in the practical field, especially for architecture with the target of developing new structural forms and evaluate their structural efficiency. It is a relatively unexplored area of research with regard to the understanding of the mechanical characteristics of fractal forms and its application in designing architectural structures. A fractal is a unique concept that deals with the objects that are not integer dimensional, like 2-dimensional plane, there dimensional sphere, but deals with the fractional dimensional object. Thus a fractal object has its own dimension, known as fractal dimension. Dimension has an important impact on structural behavior, and hence, this new concept of fractal dimension also must have some impact on the strength and mechanical behaviors of structures that are fractal in shape. So far, there

ii are no known investigations in finding relation between fractal dimension and structural strength. Therefore, an important focus of this research is dedicated to this untouched area of investigation. The recent practice of computational morphogenesis has brought a new opportunities in the field of architecture and civil engineering. Computational morphogenesis allows architects and engineers to articulate the mathematicsbased structural forms and to solve some problems in the fields of architecture and civil engineering. In this research, computational morophogenesis has been used as a technical tool for the investigation, experiment and applications.

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Acknowledgment It is a pleasure for me to express my gratitude to all of them who have helped me or have been an important part of my Ph.D. research. It was 2005 when I was introduced with the term ‘fractal’ for the first time by Professor Jin-Ho Park, my Masters supervisor from Inha University, South Korea. I thank him for giving me such a complex and hard to understand gift. This term ‘fractal’ drove me to a new world of geometric form system. It fascinated me such immensely that I spent years on exploring it. I thank the Department of Architecture and Design (DAD) of Poliecnico di Torino that gave an excellent opportunity to me for doing full time research, and elevated my fascination turning it into a serious investigation in scholastic manner. First of all, I am grateful to my supervisor Mario Sassone for taking me out from the blind fascination towards fractals, and insisting me to look at fractals from scientific lenses with the rational and engineering perspectives without compromising designing aptitude. It was he who helped me to frame a clear road map to follow the design-oriented scientific approach for applying the concept of fractal geometry in the field of architecture. His consistent guidance and critical assessments show his quality of an excellent mentorship. I am equally thankful to Professor Shuchi Asayama of Tokyo Denki University, Tokyo. His intense and active involvement during my six months of an abroad research program had engineered the fundamental framework for the theory part of my research. His contribution to develop my mathematical aptitude related to the theory of fractal geometry has been enormous, which became the backbone of my research. I am indebted to him also for his availability in online conversation whenever I required his guidance. I am lucky to have him as my co-advisor who is without a doubt an amazing mentor. I must mention my senior Tomas Mendez’s contribution to my research work. I consider him as my amateur supervisor for the first two years. His excellent support for developing my technical and computational skills is unforgettable. My other seniors, Antonio Spinelli and Andrea Rosada were great sources of support and discussion. I am highly thankful to my colleagues, especially Shaghayegh Razabzadeh, Han Xiao Fein, Shanshan Li, Marco, Matteo, Yang, Asma Mehan, Daniele, Cristina, Lucia for their supports and advices. Their company at the university coffee bar boosted me up when I have been stressed and frustrated. Some of my juniors’ involvement in my experimental work and physical pro-

ii totyping is undeniable. Riccardo Manitta, Giulia Mazza, Elisa Pitassi, Leonaro, Bruno and Gabriele, thank you so much. Dialogues and discussion with Professor Daniel Lordick from the Technical University Dresden and Dr. Alberto Pugnale from the University of Melbourne have been potentially useful. I thank Mansi Bapna for proofchecking some of the important drafts prepared for the journal publications. I would like to convey my special thanks to the Politecnico di Torinonos’ PhD scholarship grant. Without their financial support, doing my research would not be possible staying in Italy. Apart from financial support, universities, special grant such as attending conferences, meeting some experts have appeared as an excellent support. I would like to thank Professor Antonio de Rossi and Professor Giovanni Durbiano for their administrative supports and important tips. Nevertheless, being a married and family person, it was a hard three year long journey leaving my dear family behind in my home country India for Italy. This journey of research was truly difficult realizing that this journey, mainly made to suffer my wife who always showed her joy in my work and kept encouraging me by hiding her pain of loneliness. I may never pay her back for her immense self-less sacrifice. My heartiest thank to my wife for her love and support. There is no proper word for me to thank my dear mother who always keeps praying for my success. My brother, sister and their spouses, my in-laws and my other family members are my constant supports and well-wishers. The main credit of my PhD research goes to my father. His inspiration has been the consistent strength for me. He was the person who was counting days to see me as a qualified Ph.D. holder. My entire effort has been to make him happy who devoted his life to promoting our education, even in his hard times. But, unfortunately, six months back, he could not wait any more for the final day of my Ph.D., and passed away in October 2014. It was an unacceptable loss. During this concluding phase of my Ph.D., although, my journey became emotionally harder and painful after my father’s death, yet his teachings and his dream have not let me stay weak, and made me stronger and determined. His inspiring forces and blessings have always been surrounding me. It is impossible to return him even a drop of what he gave me. I dedicate my thesis to my father, who has been my inspiration and strength, to my mother and to my lovely wife.

Contents I

Introduction

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1 Introduction 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Geometry-Aided Architecture . . . . . . . . . . . . . 1.1.2 A Brief Idea about Fractal Geometry . . . . . . . . . 1.1.3 Relation between Fractals and Architecture . . . . . 1.1.4 Motivation of the Research . . . . . . . . . . . . . . 1.2 Statement of The Problem . . . . . . . . . . . . . . . . . . . 1.3 Hypotheses: Fractal Properties and their Structural Merits 1.4 Chapter Outlines . . . . . . . . . . . . . . . . . . . . . . . .

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2 Literature Review 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Fractal Geometry as a Mathematical Concept . . 2.1.2 Methods for Generating Fractals . . . . . . . . . 2.1.3 Fractals in Architecture . . . . . . . . . . . . . . 2.1.4 Fractals in Civil Engineering . . . . . . . . . . . 2.1.5 Fractal Structures in Pure and Applied Sciences 2.2 Research Gap and Final Objective . . . . . . . . . . . . 2.2.1 Remaining Research Gap and Questions . . . . . 2.2.2 Research Objective . . . . . . . . . . . . . . . . .

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II

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Theory, Formulations and Tools

3 Fractal Mathematics 3.1 Introduction . . . . . . . . . . . . . . 3.2 A Brief History of Fractal Geometry 3.3 Definition of Fractal Geometry . . . 3.3.1 Common Definition . . . . . iii

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CONTENTS

3.4 3.5 3.6 3.7

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3.3.2 Hausdorff Definition . . . . . . . . . . . . . . . . . . . . . The Hausdorff Metric Space and Hausdorff Measure . . . . . . . Topological Dimension vs. Fractal Dimension . . . . . . . . . . . The Contraction Mapping Theorem . . . . . . . . . . . . . . . . 3.6.1 The Hutchinson Operator . . . . . . . . . . . . . . . . . . Iterated Function System . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Example1: Sierpinski Triangle . . . . . . . . . . . . . . . 3.7.2 Example2: Fractal Fern . . . . . . . . . . . . . . . . . . . Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Affine Transformation and Self-Affine Sets . . . . . . . . . The Concept of Pre-Fractal and Random Fractal . . . . . . . . . Different Measures of Fractal Dimensions . . . . . . . . . . . . . 3.10.1 Hausdorff Dimension of Self-Similar Sets . . . . . . . . . . 3.10.2 Minkowski-Bouligand Dimension (Box-Counting Dimension) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Construction Methods of Fractals . . . . . . . . . . . . . . . . . . 3.11.1 Iterated Function System: IFS . . . . . . . . . . . . . . . 3.11.2 Lindenmayer Rewriting System: L-System . . . . . . . . . Automatic Fractal Constructions: IFS Code . . . . . . . . . . . . Fractals: From Mathematical Theory to Engineering Applications 3.13.1 Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . 3.13.2 Fractal Scale: From Macro to Micro . . . . . . . . . . . . 3.13.3 Finite Fractals for Finite Element Analyses . . . . . . . . 3.13.4 Self-Similarity and Structural Module System . . . . . . . 3.13.5 Fractal Dimension and Structural Response . . . . . . .

4 Fractal-Based Computational Morphogenesis 4.1 Introduction: Computational Morphogensis in Architecture 4.2 Generative Design and ‘Generative Tool Box’ . . . . . . . . 4.2.1 The Concept of Generative Design . . . . . . . . . . 4.2.2 Generative Design based on IFS Rule . . . . . . . . 4.2.3 IFS-based ‘Generative Tool Box’ . . . . . . . . . . . 4.3 Finite element analyses (FEA): ‘FEA Tool Box’ . . . . . . . 4.4 Optimal Form Finding: ‘Optimization Tool Box’ . . . . . . 4.4.1 Structural Optimization . . . . . . . . . . . . . . . . 4.4.2 ‘Optimization Tool Box’ . . . . . . . . . . . . . . . . 4.5 Computational Morphogenesis of Fractal-Based Structures . 4.6 Fractal-Aided Computational Morphogenesis . . . . . . . .

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CONTENTS

III

v

Experiments

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5 Experiment 1: Fractal-Based Hierarchical Truss 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Self-similar Repetition of Structural Members . . . . 5.1.2 Recent Development: Hierarchical Fractal Structures 5.2 Geometric Modeling . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Mathematical Expression and IFS . . . . . . . . . . 5.2.2 Hausdorff Dimension Calculation . . . . . . . . . . . 5.3 Structural Analyses . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Structural Modeling and Preprocessing . . . . . . . 5.3.2 Finite Element Analysis and Results . . . . . . . . . 5.4 Structural Optimization . . . . . . . . . . . . . . . . . . . . 5.5 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Experiment 2: Branching Structures 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 6.2 Geometric Modeling . . . . . . . . . . . . . . . . . 6.2.1 Mathematical Expression . . . . . . . . . . 6.2.2 IFS Function and Coding . . . . . . . . . . 6.2.3 Hausdorff Dimension Calculation . . . . . . 6.3 Architectural Design . . . . . . . . . . . . . . . . . 6.4 Structural Analyses . . . . . . . . . . . . . . . . . . 6.4.1 Finite Element Modeling and Preprocessing 6.4.2 Finite Element Analyses . . . . . . . . . . . 6.4.3 Results and Discussion . . . . . . . . . . . . 6.5 Form Finding and Structural Optimization . . . . 6.5.1 Finding Optimal Form . . . . . . . . . . . . 6.5.2 Optimization of the Structural Form . . . .

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IV

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Applications

7 Application 1: Fractal-Based Generative Design of the alisa Pavilion’ 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Design Objective: Recreating a Forest . . . . . . . 7.1.2 Geometric Modeling . . . . . . . . . . . . . . . . . 7.1.3 Parametric Modeling . . . . . . . . . . . . . . . . . 7.1.4 Architectural Design . . . . . . . . . . . . . . . . . 7.1.5 Structural Analyses . . . . . . . . . . . . . . . . . 7.1.6 Practical Construction . . . . . . . . . . . . . . . .

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CONTENTS 7.1.7

Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . 122

8 Application 2: Fractal-Based Trusses Design 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Trusses and Self-Similarity Features . . . . . . . . . . . 8.2.1 The History of Truss Designs . . . . . . . . . . . 8.2.2 Self-Similarity Features of Traditional Trusses . . 8.3 Geometric Modeling of Fractals for Truss Design . . . . 8.3.1 Sierpinski Triangle . . . . . . . . . . . . . . . . . 8.3.2 Pinwheel Fractal . . . . . . . . . . . . . . . . . . 8.3.3 Asayama and Mae’s Fractal . . . . . . . . . . . . 8.3.4 Fan Fractal . . . . . . . . . . . . . . . . . . . . . 8.3.5 Baltimore Fractal . . . . . . . . . . . . . . . . . . 8.3.6 Comparison of the Hausdorff Dimensions . . . . 8.4 Architectural Design of the Fractal-Based Trusses . . . . 8.4.1 Design Considerations and Truss Designs . . . . 8.4.2 Statically Determinacy of Fractal-Based Trusses 8.5 Structural Analyses of Fractal-Based Trusses . . . . . . 8.5.1 Finite Element Analyses . . . . . . . . . . . . . . 8.5.2 Results and Discussion . . . . . . . . . . . . . . . 8.6 Structural Optimization . . . . . . . . . . . . . . . . . . 8.6.1 Design Optimization . . . . . . . . . . . . . . . . 8.6.2 Cross Section Optimization . . . . . . . . . . . . 8.7 Final Remarks . . . . . . . . . . . . . . . . . . . . . . .

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9 Application 2: Fractal-Based Grid Shell Design 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Midpoint Displacement Method . . . . . . . . . . . . . . . . 9.2.1 Archimedes’ Method . . . . . . . . . . . . . . . . . . . . 9.2.2 Generalization to Blancmange function: Takagi-Landsberg Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Takagi-Landsberg Curve Construction: Vector-Based Midpoint Displacement Method . . . . . . . . . . . . . . . . 9.2.4 Hausdorff Dimension of Takagi-Landsberg Curve . . . . . 9.2.5 IFS Coding . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Application 2a: Takagi Arch Structure . . . . . . . . . . . . . . . 9.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Geometric Modeling . . . . . . . . . . . . . . . . . . . . . 9.3.3 Architectural Design . . . . . . . . . . . . . . . . . . . . . 9.3.4 Structural Analyses . . . . . . . . . . . . . . . . . . . . . 9.4 Application 2c: Fractal-Based Grid Shell Structure . . . . . . . .

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CONTENTS 9.4.1 9.4.2 9.4.3 9.4.4 9.4.5 9.4.6 9.4.7

vii Introduction . . . . . . . . . . . . . . . Geometric Modeling . . . . . . . . . . . Hausdorff Dimension of Takagi Surface . Architectural Design . . . . . . . . . . . Structural Analyses . . . . . . . . . . . Computational Form Finding . . . . . . Physical Prototype Construction . . . .

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10 Application 3 - Folded Roof Design using the Concept of Random Fractals 187 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 10.2 Random Midpoint Displacement Method . . . . . . . . . . . . . . 188 10.2.1 Fractal Dimension Calculation . . . . . . . . . . . . . . . 189 10.3 Diamond Square Algorithm Method . . . . . . . . . . . . . . . . 191 10.3.1 Fractal Dimension Calculation . . . . . . . . . . . . . . . 194 10.4 Fractal-Based Folded-Plate Shell Structure . . . . . . . . . . . . 196 10.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 196 10.4.2 Geometric Modeling . . . . . . . . . . . . . . . . . . . . . 199 10.4.3 Structural Analyses . . . . . . . . . . . . . . . . . . . . . 201 10.4.4 Computational Form Finding and Structural Optimization 202 10.5 Canopy Structure with Crinkled Roof . . . . . . . . . . . . . . . 204 10.5.1 Introduction: Design Concept . . . . . . . . . . . . . . . . 204 10.5.2 Geometric Modeling of a Branching Column . . . . . . . 205 10.5.3 Geometric Modeling of a Crinkled Roof . . . . . . . . . . 205 10.5.4 Architectural Design . . . . . . . . . . . . . . . . . . . . . 207 10.5.5 Structural Modeling . . . . . . . . . . . . . . . . . . . . . 209 10.5.6 Structural Analyses . . . . . . . . . . . . . . . . . . . . . 212 10.5.7 Computational Morphogenesis and Form Finding . . . . . 216 10.5.8 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . 217

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Conclusion

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11 Conclusion 221 11.1 Introduction: Complexity, Fractals and Architecture . . . . . . . 221 11.2 Summary of the Results . . . . . . . . . . . . . . . . . . . . . . . 223 11.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

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CONTENTS

Part I

Introduction

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Chapter 1

Introduction ... clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. —Benoit Mandelbrot [46]

1.1 1.1.1

Introduction Geometry-Aided Architecture

Geometry and architecture are closely associated with each other. Since the ancient time, architects have employed the principles of geometry in an applied way. The mathematical methods, especially the geometric processes, serve as tools for designing buildings and cities. Several historical examples demonstrate the interest in applied architectural geometry. Vitruvius was one of the foremost thinkers and mathematicians who insisted on the importance of geometry and mathematics as a tool for the architect. In his ten books, he teaches the work of the mathematicians Pythagoras, Archytas and Eratosthenes as a base for the representation of the architectural design (ichnographia, orthographia, scaenographia) [84]. These mathematical methods provide rules for proportion, harmony and order. In his opinion, ‘summum templum architecturae’, which means, the architect’s multidisciplinary profession incorporating large knowledge in different domains is the most excellent of all sciences. 3

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Chapter 1. Introduction

Some of D¨ urer’s most famous drawings are teaching the principles of the central perspective based on Alberti’s method, which became a common architect’s tool for the representation of designs. Besides, D¨ urer published a very important book back in 1950s, titled Four Books on Measurement (De Symmetria... and Underweysung der Messung) [16] introducing the variations of Platonic solids by studies on Archimedes’s semi-regular solids. Knowing the work of Vitruvius, he talks about Euclid’s Elements, Ptolemy and Apollonius, who taught him several methods of curve design. Finally, in the third book, the principles of geometry are applied to architecture and civil engineering. In the 18th century, the French mathematician Gaspar Monge founded the principles of descriptive geometry, which is a unique branch of geometry that allows the representation of three-dimensional objects in two dimensions, by using a specific set of procedures. After a couple of decades of non-disclosure for military use, Monge’s method of descriptive geometry became available to architects and civil engineers around 1850. Until today, his method is used as a standard for technical drawings for engineering, architecture, design and even in art. [45] In the second half of the 20th century, the development of computers has revolutionized the geometrical processes. Computer has made easy for exploring new geometric shapes by transofrming mathematical functions into graphical models. Mathematicians have started using computers as a new operating tool for exploiting the mathematics of geometries, especially non-classical, nonlinear and complex geometries. Computer based geometric modeling of simple to highly complex shapes allowed architects to think about new designs beyond the conventional boundary that were restricted by the regular geometric systems. Computer aided design and the articulations of modern mathematical functions as digital figures has offered numerous opportunities to architects and designers to play with new, complex, unimaginable and unnoticeable geometries. The gap between mathematics and architecture including the civil engineering has become closer that forms a modern version of the interdisciplinary design process devised by geometry. If we only focus on the historical evolution of architecture as structural objects, then we find that the geometry has not only contributed to the aesthetics and planning of architectural buildings and cities, in fact, it has pivotal contributions in designing structural shapes by means of its stability and strength. It acts as a tool to serve for the functional needs in buildings. This association between geometric contributions to design structures depends on the development of science and technology since the ancient times. In the ancient time, the Platonic and the Euclidean geometries were the instrument for designing structures. The Egyptian Pyramids and Greek Pantheon are the perfect examples of such fact. The great example is a semi-circular curve or

Chapter 1. Introduction

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Figure 1.1: Geometries used in ancient architectural structures.

arc which is one of the oldest and simplest geometric shapes, but the powerful one which contributed an uncountable number of structures from the ancient age to today’s contemporary era. This simple arc had been used for designing an arch, one of the most fundamental structural elements as a compressive member, amazingly significant, especially for the times when the knowledge of using tensile materials for the construction purpose was not existing, and only masonry materials such as bricks or stones had to be used for grand structures. This simple curve and the arch-based compression technique were used in many different ways to design simple to complex arch systems, and also to its three-dimensional expansions as simple as well as complex vault designs. Later, in the 17th century, with the development of mathematics and physics, architects were exposed to a new curve system known as catenary curve, an idealized hanging chain or cable assumes under its own weight when supported only at its ends. Robert Hook, a notable physicist, announced that the arches which are semi-circular in shape are not the best from structural point of view; instead, the catenary is the optimal shape for the arch structures that carry self-weights. Similarly, parabolic curve, a non-Euclidean shape, was also found to be the perfect shape for an arch that can take the uniformly projected loads. Based on these new findings, architects and engineers designed a score of new structures. These findings had revolutionized the trend of architecture, especially in the late 19th century and in the beginning of the 20th century. During this period we find a number of beautiful structures that are designed based on these non-Euclidean curves, among them Gaudi’s buildings are the pioneering examples of such structures. Later, in the 20th century, reinforced concrete technology had changed the paradigm of the trend of architectural design and structures. During this period, we see a number of new shapes, mainly paraboloid structures. In the late 20th century, architect shows huge interest in complex and irregular designs. Thanks

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Chapter 1. Introduction

Figure 1.2: Geometries used in modern architectural structures.

to the development of computers, mathematicians have become able to exploit new and complex shapes that were not possible to generate by hand. Computer aided design software has helped architects to model architectural shapes using the functions of complex geometries. Structural innovations and simulative testing allowed architects and engineers to get liberty of designing complex and free-form structures. In the contemporary period, further advancement of computer aided design software, parametric design tools, computer aided manufacturing technology, innovative construction system and advanced materials, open a new frontier for architects and engineers to construct almost all possible forms of structures. More importantly, the recent practice of computational optimization process aids architects and engineers to find the optimal shape from their generative designs. All these progressions and advancement in the field of architecture and construction have been extensively contributed and shaped by mathematics and geometric systems.

Figure 1.3: Is highly delicate shape of cauliflower applicable for designing architectural structures?

During this architectural geometric evolution, in the 1970s, there was an introduction of a new geometric system, known as fractal geometry. The concept of fractal geometry was not the new concept in the 1970s, it was practiced even. But, in this decade, it was re-introduced and re-systematized as a new

Chapter 1. Introduction

7

geometric system and categorized as a unique branch of mathematics. Like the development of previous geometries and their applications in the innovations of architectural and structural shapes, this latest geometric system fractal may also posses the capability to contribute in designing new but efficient forms for architectural structures. Based on this premise, this dissertation deals with the topic which explores the possibility and feasibility of applying the concept of fractal geometry in designing architectural structures with a strategy for obtaining a new and efficient structural system as well as innovative aesthetic appearance.

1.1.2

A Brief Idea about Fractal Geometry

Fractal geometry is a unique branch of mathematics that was born from the early studies of Cantor, Koch, Menger, and Julia, and re-systematized and reconceptualized in the 1970s by a 20th century pioneer mathematician Benoit Mandelbrot. He was the first who coined the term ’fractal’ from the Latin word f ractus (means irregular or fragmented). This unique branch of mathematics studies abstract configurations characterized by the patterns of self-similarity and recursive growth. Fractal objects show the properties of being exactly or nearly the same at different range of visual scales. At their most basic, fractals are a visual expression of a repeating pattern or formula that starts out simple and gets progressively more complex. From the point of view of pure science, a fractal object is a mathematical set that has a fractional dimension, so that they are intermediate objects between one and two dimensional figures (as lines and surfaces) or two and three dimensional (as surfaces and solids) forms. The basic properties of fractal shapes are: • self-similar repetition • reflection of whole in a part • fragmented edges at any zooming scale • fractional dimension The domain of fractal geometry applications covers almost all scientific disciplines, ranging from mathematics, pure and applied sciences, biology, medicine, including engineering and architecture. It is a powerful theoretical tool for the analysis, interpretation and description of natural and human phenomena, where continuous geometry cannot be applied. Fractal geometry, in fact, is a geometry of discontinuity. The detailed definition and mathematical concept of fractal geometry has been described in the Chapter 3.

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Chapter 1. Introduction

Figure 1.4: Koch curve, a mathematical fractal figure showing the repetition of full into its parts and never ending roughness.

Figure 1.5: Dimensions of different geometric objects. Objects having fractaional dimensions are fractal objects.

1.1.3

Relation between Fractals and Architecture

In architecture, we notice some design features which are self-similar and the repetition of whole shape into its parts, especially in Indian Hindu temples (Figure 2.2a), Gothic churches, African villages layouts ( (Figure 2.2b), and so on. This means, some properties of fractal geometry already appeared in these centuries old architecture. Mostly, these features were used as decorative features (such as in Hindu temples and Gothic churches) to produce complexity, yet beautiful and aesthetically rich, and sometimes as a planning scheme such the layouts of some African villages. After the modern movement in the early 20th century, the ornamentations and decorative features, including fractal-like features, were started to disappear. Few architects, like F. L. Wright maintained self-similar repetitions concept for some of their buildings’ layout designs in a simplified manner. The Chiesa Cattolica Parrocchiale Santa Teresa is an unnoticed built example of fractal architecture that was built in 1958 in Turin, Italy (Figure 2.2c). However, in the late 20th century, there was a beginning of new movement, which was marked as ’complexity’ and ’Deconstructivism’ in architecture. During this period, a number of architects started rediscovering

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9

complexity in different way where some abstract forms of self-similar repetitions have been reincarnated giving surprisingly complex impressions. With the development of computer technology and construction advancement, these architects were able to design radically complex designs. Modern mathematics aided them to achieve such complexity in their designs. In the recent years, we observe a new trend for finding innovative forms for architecture. Within this frame of form-finding design trend, fractal geometry as a mathematical tool seems to be highly useful and relevant, which can offer new but highly complex forms by using simple initial shape and simple algorithmic rules.

1.1.4

Motivation of the Research

Today, the form findings for architectural design are not only driven by the aesthetic interest, but also and mainly driven by the functional, structural, environmental and contextual adaptability or reasonability. Today, computational morphogenesis is one of the widely practiced methods that are used for finding or modifying forms as an aspiration to express or respond to contextual processes. This process of forming finding or modifying are mainly dealing with the shapes that are originated from some conventional geometric systems. In this context, a great scope of using fractal geometry is clearly visible. Before the invention of computers, fractal shapes were extremely difficult to represent by the hand-made process because of its high complex shapes, although resulted by following some simple rules. That was the reason, architects and designers used to avoid such shapes in architecture and in other designs as it would be painfully time consuming and expensive to recreate such figures and models. However, after the invention of computers in the mid 20th century, it became easy to model fractal shapes with few lines of a simple algorithm using recursive logic. Yet, because of the mathematical difficulty of the fractal concept, designers, as the non-expert mathematicians, still have not adopted fractal shapes for their designs. As a result, mathematicians invest their full interests of the fractal concept within its theory level, and do not pay much attention for its practical applicability, especially within the context of architectural applications. On the other hand, designers pay more attentions to other geometric systems that are relatively easy to understand and handle. As a result, a big gap has emerged between the mathematics of fractal geometry dealt by mathematicians and its possibility in architectural applications dealt by architects and engineers. Hence, the increasing gap between the potential scope of applying fractal concept in architecture and the lack of its serious attention among the architectural community has deeply motivated me to do a research on this particular subject. So, this research represented in this dissertation aims to bridge this gap by using the contemporary tool of computational morphogenesis.

10

1.2

Chapter 1. Introduction

Statement of The Problem

Since the 1970s, when Benoit Mandelbrot systematically developed the concept of fractal geometry as a new system of form, this unique branch of mathematics has started influencing different research fields that include science, engineering, medicine and art. These are the fields where the notion of fractal geometry has surpassed from its theory level of mathematics to practical application level. Although a very few, but there are recent architectural examples that are directly influenced by the concept of fractal geometry as an art because of it’s unique beauty of complexity; these are the fa¸cade designs of the Federation Square Building in Melbourne (Figure 1.2.a) and the Grand Egyptian Museum in Cairo (Figure 1.2.b).

Figure 1.6: (a) Federation Square building inspired by Pinwheel tiling by LAB Architects (1997); (b) Grand Egyptian Museum inspired by Sierpinski triangle by Heneghan Peng and ARUP (2015).

But, unfortunately, there are almost no built examples where fractal geometry has been deliberately applied for the structural purpose in construction. We notice some historical structures where repetitions of self-similar units were assembled to obtain high strength. Because, apparently, repeating the selfsimilar shapes in a particular arrangement can offer high stiffness. Based on this principle, Mandelbrot [46] claimed that, the Eiffel Tower is an important example where a self-similar repetition has been wisely applied. As a result, the Eiffel Tower is notably a weight-efficient structure by virtue of its architecture. By this statement, Mandelbrot gave a clue that the concept of fractal geometry developed in the field of mathematics can effectively contribute in the field of construction, too. Hence, there is a threshold of new research about the theory of fractal geometry for the practical applications in architecture and civil engineering. However, so far, there are no sufficient explorations about the structural properties of fractal shapes and their practical applicability for designing structures in architecture. There is a lack of rigorous and significant investigations about finding the structural merits of fractal shapes for designing the efficient and lightweight structures as a form-finding and problem solving

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11

strategies. After the above discussion, it can be summarized that there is a need for • bridging the gap between the mathematical theory of fractal geometry and its different possible applications in designing architectural structures. • exploring the structural rationality of fractal properties if applied in the field of architecture and construction. • develop a process of form finding using the concept of fractal geometry.

1.3

Hypotheses: Fractal Properties and their Structural Merits

Fractal has the following three assets that can significantly contribute to the field of architecture and construction: • The property of self-similar repetitions of fractal geometry is an outstanding asset which can be potentially useful for designing hierarchical structures whose members are replaced by submeneber, and the submembers are further replaced by their own subsubmebers, thus saving huge amount of weight by maintaining a high strength. • The property of complex appearance of fractal shapes, which follows simple production rule, is an excellent key tool that has potential to create innovative designs of complex and intricate designs of architectural structures or a radically new assemblage of structural members that contribute to offer unique structural behavior. • The property of fractional dimensionality of a fractal object makes it lie in between two successive integer dimensional objects by means of geometric shapes. Accordingly, it can be hypothesized that a fractal object as a structural element may exhibit the strength which is in between the strengths of two successive integer-dimensional structural objects, thus a unique mechanical behavior is expected from a fractal object. As an example, a three-dimensional solid as a structural member is stronger than a two dimensional structural surfaces, and a fractal object in between them, may exhibit the strength in between them. This unique concept of fractal dimensionality can be useful for designing structural shapes where the intermediate strength is sufficient, and by doing so, the weight of a structure can be significantly saved. Alternatively, fractal objects may show completely different structural attitude which is unknown and may

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surprisingly offer new results. In this sense, this area of research regarding the exploitation of dimensions and its consequent structural impacts can unfold new and unpredictable outcomes. The above hypotheses are supported by some existing examples as references. We notice the abundance of the features self-similar repetitions in nature as well as in man-made structures. This feature is an outcome of the strategy for functional and mechanical reasons. In nature, trees are the most common of such examples. Its self-similar branches are a strategic outcome for some functional, biological and mechanical reasons [37]. Pollack [61] clarifies that, ”Trees are organisms that stand by themselves, so their shape has an inherent structural rationality”. Its fractal-like branching arrangement is a result of tree’s optimization of shape for structural demand [50]. The structural responsibility of branches is to spread the leaves to obtain maximum surface area for absorbing maximum sunlight during the whole day, and then take the loads of leaves and fruits, including their own weights, and finally pass the loads concentrating towards the main trunk. Thus, self-similar branching configuration acts as a network of cantilevers supported by successive parent branches [81]. This concept of tree’s structural tactic and optimization aided by fractal-like geometric configurations has inspired architects and engineers to develop their designs since centuries [63]. Similarly, in man-made world, especially in engineering, we see the use of self-similar repetition as a tactic to obtain high stiffness and save weight. Hierarchical truss structures are such examples where we notice that the original assemblage of truss members is repeated in its each member which is an similar patterned assemblage of its own submenbers, and each submemeber follow the same rule. Well known lattice structures, for example, Eiffel Tower was designed using this principle where the property of self-similar repetitions facilitate to strengthen them and offer mechanical efficiency. (Figure 1.7) These natural and man-made examples provide an immediate hint that the fractal geometry can be potentially useful if employed in the real world applications, specifically in the field of construction. These examples as existing references support the above mentioned hypotheses and suggest that the fractal geometry can display its architectural and structural merits if applied in the practical fields of design. Hence, based on the hypotheses, a systematic investigation about the mathematical properties of fractal geometry and its possible applicability in designing architectural structures is the central purpose of this dissertation.

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13

Figure 1.7: Hierarchical truss system of Eiffel Tower that shows fractal property of self-similar repetition at smaller scales.

1.4

Chapter Outlines

Part I deals with the introductory part of this research laying a background of the topic. Chapter 1 has discussed about the brief background and motivation followed by the statement of problem and a quick hypothesis of the research. Chapter 2 discusses about the literature review to address and investigate related to the statement of the problem. It follows two coordinates; the first coordinate is the fractal analysis of architecture and civil engineering, and the second coordinate is the fractal-based designs in architecture and civil engineering. It mainly addresses the current state of research development related to address the above research problem. This chapter is ended with identifying some questions that are not addressed significantly in the latest investigations and published papers that are mentioned in the section of the literature review. Based on the research questions and remaining gaps, this dissertation develops an objective to find their answers, and accordingly makes a precise plan and methodology to achieve the answers which are the main specific target of this dissertation. Part II explains the theory and formulations of fractal geometry, set up detailed methods, and develop tools for the experiment and applications. This part acts as a vehicle for the methodical investigation, experiments and applications for the next parts. Chapter 4 explains in details about the technical definitions and mathematical explanations of fractal geometry to dig out and understand the theoretical

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properties of fractals, and identifies its limitations for practical applications. Finally, it filters the properties of fractals that are useful for the practical applications in designing structures. In this chapter, the techniques and algorithms of modeling the fractals are discussed. Chapter 5 develops a tool of computational morphology using the concept of fractal geometry, and transform the mathematical functions, developed according to design goal or structural problem, into digital models. For this a ‘Generative Tool Box’ has been developed. For structural analyses of fractalbased model, an ‘FEA (Finite Element Analyses) Tool Box’ has been developed. In the end to find the optimal design of fractal-based structures, an ‘Optimization Tool Box’ has been created. Part III discusses about the experiments checking the validation and effectiveness of the developed tools and testify the hypotheses. Chapter 6 is the first experiment on hierarchical trusses that is developed using the fractal concept. The tools are applied here to testify the hypothesis about the efficiency and lightness of fractal-based hierarchical trusses. Chapter 7 is the second experiment which evaluates the efficiency of the toolboxes and check the claim about the potency of the fractal concept for transforming natures’ structure into man-made structures in a more efficient way. Part IV is all about the applications of fractal-base computational morphogenesis process. Chapter 8 is the application of fractal concept in designing planar roof truss structures. The objective of this application is to find the ability of fractals for offering new configurations of truss topology and to evaluate their unique structural behavior. The Iterated Function System has been used in this chapter for generating the fractal shapes. Chapter 9 proposes a new design of fractal-based grid-shell structures. This chapter is an original research in a sense that the fractal concept has been extended to three-dimensional space in designing space structures. This chapter is significant here, because the notion of fractal dimension has been interpreted with the mechanical properties of the structure. In this chapter, the Midpoint Displacement Method has been used for generating the fractal shapes. Chapter 10 talks about a unique application of fractal geometry. Here, the principle of fractal geometry shows its versatility in designing structural forms by not limiting itself within the self-similar properties, but go beyond to its stochastic or random property. This chapter is also an original application of

Chapter 1. Introduction

15

fractal geometry where the factor of fractal dimension act as a design variable, which is completely a unique and a new design variable in the field of computational design of architecture. It stretches the possibility of applying fractal concept in a diverse way utilizing its wider possible mathematical features. Part V concludes the research with the discussion of results and findings, and ended with the possible future scopes.

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Chapter 2

Literature Review 2.1

Introduction

This chapter reviews the literatures from different fields, specifically mathematics, architecture, civil engineering and pure science, which deal with the concept of fractal geometry and its applications. It encompasses old and primary sources, including the most recent and state of the art investigations. The literatures are confined within the selected sources that lead to build a research framework and supporting references for this dissertation.

2.1.1

Fractal Geometry as a Mathematical Concept

The most canonical reference at where the concept of fractal geometry is systematically defined and described, and from where a new branch of mathematics evolved, is Benoit Mandelbrot’s Fractal Geometry of Nature [46]. It was a pathbreaking book first published in 1982, redefined the concept of geometric explanation about the shapes and patterns of natural phenomena as well as many irregular and complex shapes or schemes that were not explained by the available geometric systems before the 1970s. Later, M. F. Barnsley [2] expanded the theory of Mandelbrot’s fractal geometry using the old similar theories given by Julia, Koch, etc. In addition, Barnsley proposed his own theory of contraction mapping and developed a new technique the Iterated Function System for generating fractals. He also explained in details the process of calculating the fractal dimensions using the Hausdorff method and related it to his contraction mapping theory. Falconer, in 2003, summarized the works of Mandelbrot and Barnsley, and made his book on fractal geometry easier for the readers who are not expert in mathematics, but having fair mathematical knowledge. In the 17

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late 20th century, a number of mathematicians investigated and enriched this latest branch of mathematics. Soon after the development of fractal geometry as a new branch of mathematics, this idea has been extensively used in many disciplines ranging from science [83] [77] to engineering [14], and medicine [42] to arts [5]. Its immediate applications were frequently seen in the field of image processing [34] and simulations of natural phenomena [24]. Mandelbrot’s further developments had advanced these specific fields. In the later phases, he dedicated his works mainly in studying the randomness from the perspective of fractal geometry [47] [48].

2.1.2

Methods for Generating Fractals

Fractals are highly complex by their appearances, but they can be produced by using very simple rules. Mandelbrot used the earlier methods borrowed from the methods developed by Sierpinski, Koch, Cantor, Menger and Julia. Thanks to computers, it made possible for Mandelbrot to model the mathematical functions of their mathematical functions and rules. One of the famous graphical realizations was Julia set (Figure 2.1a). Based on this, Mandelbrot proposed a new set based on complex number and made a graphical model which was surprisingly new, beautiful but highly complex (Figure 2.1b). However, to model fractals, more easily with simple rules, Barnsley’s Iterated Function System method is very handy. For this, a simple algorithm is enough to produce highly complex fractal shapes. Till now, this process is the most popular method that can not only produce self-similar or self-affine fractals, but also produce random fractals and simulate nature. In this dissertation, the Iterated Function has been used as a main tool of experiments and applications. This method has been described in details in the Chapter 3. Much before the development of fractal geometry, in 1968, Aristid Lindermayer [40] proposed a new method, which was based on rewriting grammar system, to simulate natural trees and plants. His method became more popular in botany, because it helps to model a wide range of different trees and plants [62]. However, this method has become useful for making fractal shapes, such as Koch curves, dragon curves, and so on. Apart from these two popular methods, some methods were also developed among them Finite Subdivision Rule is useful producing fractals by subdividing a shape [10]. These above methods are mainly useful for producing self-similar fractals. For generating the random and approximate fractals, stochastic rules are appropriate, and some of the well known methods are Levy flight [46], percolation clusters, self avoiding walks [28] and Brownian motion [49] methods. For making natural terrain and fractal landscapes, the midpoint displacement method based

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19

Figure 2.1: (a) Graphical expression of Julia set, (b) Graphical representation of Mandelbrot set, (c) Koch curve.

on the factor Gaussian randomness proposed by Fourier and his colleagues [24] is very useful. This method has been described in details in Chapter 10.

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2.1.3

Chapter 2. Literature Review

Fractals in Architecture

Fractal as a Tool for Analyses of Architecture In the field of urban design and planning, fractal geometry is one of the advanced and highly sophisticated approaches to analyze several aspects of urban morphology, urban topology, growth patterns, communication, traffic networking system, and so on. Many examples can be found in the classical book of Michael Batty’s Fractal Cities [3] which discusses about the analyses of cities for several different aspects. Fractal geometry has been used to evaluate the relationship between urban road patterns and population [80] and in analyzing the satellite-detected urban heat island effect [87]. Mustapha Ben-Hamouche used fractal geometry as a tool to explore how succession law shaped the urban morphology in Muslim countries. He found that the successive iterations of subdivision of properties over the course of decades or even centuries, gave a fractal character to the cities [4]. As per Ji Zhu and his team’s recent works, fractal geometry is also a very useful instrument to assist remote sensing system for making expert classification of land use or land cover types of any even or uneven terrains [90]. Besides, GIS and fractal analysis are also applied together for the studying of urban morphology [51].

Figure 2.2: (a) Hindu Temple, (b) African village, (c) Chappel of Chiesa Cattolica Parrocchiale Santa Teresa, Turin.

In the field of architectural design, the concept of fractal geometry has been mostly used for analyzing the complexity, richness and self-similarity of building forms and their design elements from visual perspective. These fractal-based architectural analyses include some historical buildings (Figure 2.2) (Figure 2.3)that shows some extend of fractal characteristics [82], [65], [72] as well as some modern buildings [8], [59], [27]. A long list of papers was published by an urban theorist Nikos Salingaros who did extensive studies to analyze the fractal characters of a series of architectural examples in terms of their building forms and urban patterns from theoretical, philosophical, psychological and functional

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21

Figure 2.3: Fractal analyses of Kandariya Mahadev temple. [65]

perspectives [70], [71]. In this context, Charles Jencks insisted that the concept of fractal geometry is a tool of a new paradigm in architecture after post-modern period, and gave some philosophical insight about the fractal architecture and its aesthetic impact relating to the harmony with nature [30]. Fractal as a Tool for Architectural Designs and Patterns Fractal geometry has been applied in architecture since centuries as a building geometry [82] [69] and design patterns [8]. Early fractal patterns can be traced to ancient Maya settlements. In Europe, fractals were found in the early 12th century buildings. The floors of the cathedral of Anagni in Anagni built in 1104 and the Sistine chapel in Rome built in 1481 are adorned with dozens of mosaics in a form of a Sierpinski gasket fractal (Figure 2.4).

Figure 2.4: Floor of Anagni cathedral, Anagni (1104); (b) Mosaic pattern in Sistine Chappel , Rome (1481); (c) Elevation of Ca’ d’Oro, Venice (1440).

Fractals have been applied to many elevation structures to exclusively ad-

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dress power and balance. Some very excellent examples of classical architecture can be seen in many parts of the Europe, in the Middle East and Asia, which have effects of fractal elevations, for example, Ca’ d’Oro in Venice (Figure 2.4c). More vital evidence shows that fractals exist in Gothic cathedrals in general. The pointed arch, an impression of elevation, appear in entrance, at windows and the costal arch with many scales and details [41]. In the Middle East, fractal patterns have been adopted widely in designing stucco, a typically Persian art form for the decoration of dome interiors. In Asia, architectures with fractal structures have also been found in Humayun’s Mausoleum, Shiva Shrine in India and the Sacred Stupa Pha That Luang in Laos. Fractals have been used to study Hindu temples. In China, some mosques in the west were more likely to incorporate such domes which are fractals. Besides geographical localities, in recent times, the concept of fractals has been extended to many well known architectures including Frank Lloyd Wright’s several buildings, such as Robie House, Falling Water, Palmer House (Figure 2.5a) and Marion County Civic Building (Figure 2.5b), which demonstrate that fractals have universal appeal and are visually satisfying because they are able to provide a sense of scale at different levels. Fractals have inspired many renowned modern designers such as Zaha Hadid, Daniel Liebeskind, Frank Gehry and others with many notable fractal architectures [68]. Carl Bovill noticed that architects and designers started to adopt fractals as a design form and tool in the 1980s [8]. Bovill was one of the foremost researcher, who used the mathematical theory of fractal geometry to analyze old fractal buildings, and systematically showed the potency of fractal geometry as a tool for designing modern and complex buildings maintaining harmony with the surroundings yet bring surprise.

Figure 2.5: F. L. Wright’s fractal design in his architecture, (a) Plan of Palmer House (built in 1901); (b) Fa¸ cade design of Marin County Civic building (built in 1958).

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Before the 1970s, the application of fractal geometry was unintentional without having a mathematical concept and theory of fractal geometry. That is why, many of the old examples may not be called fractal from true mathematical definition, but from a design point of view, they are approximate fractals. Therefore, after the 1980s, when fractal geometry as a mathematical concept appeared in science, few designers attempted to use fractals from more systematic and mathematical way. In this context, Yessios was probably among the first utilizing the mathematical formulation of fractal geometry in architectural design [89]. They developed a computer program to aid architecture using fractal generators with the target of modeling architectural forms. Later, in the 1990s, Durmisevic and Ciftcioglu applied fractal tree as an a design tool for architecture design [17]. The recent development of computers facilitated architects to design some complex designs which were time consuming in the past. Consequently, computerbased algorithmic architecture has become a new design trend that brings high complexity and intricacy in design with the use of simple mathematical function and a few lines of algorithmic codes. This recent development of computational design in architecture allowed architect to use fractal geometry more mathematical way. One of the finest examples of using the pure mathematical features of fractal geometry in architecture is the fa¸cade design of the Federation Square Building in Melbourne designed by Lab Architecture Studio built in 1997 [26]. But, unfortunately, because of the mathematical difficulty to exploit and adopt the theory of fractal geometry by architects who are usually not expert in advanced mathematics, the applications are not much although there is a high interest of using fractals in architecture. However, in the field of research and in academia, there are some attempts to develop and use fractal geometry as a tool for computational design of architectural shapes at early design phases [18] [11].

2.1.4

Fractals in Civil Engineering

Fractals for Analyzing Structural Mechanics In the field of purely structural mechanics, a set of noteworthy research works has been done, which are based on fractal-like patterns. A.Y.T. Leung is one of the leading researchers who did a number of research works based on the Fractal Finite Element method dealing with different mechanical aspects and diagnosis of structures. Among them, by using by Fractal Finite Element method, parametric study on two-dimensional crack problems, addressing the exterior problems of acoustics [38], assessing thermal stress and separately for unbounded problems [39] is significant works. In this specific topic interesting applications have been developed. A very recent research has been done to assess the drag force on branches of trees caused by wind. The computational model was based

24

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on the fluid flow over the fractal patterned plate and allowed the evaluation of the drag forces and near wakes. [32]. Mistakeidis and Panagouli’s investigation about strength evaluation of retrofit shear wall elements with interfaces of fractal geometry has been a scientific use of fractals in the field of structural mechanics [54]. Fractal as a Tool for Structural Designs The application of fractal geometry is noticed in old buildings, many years before the development of fractal geometry as an established mathematical concept in the 1970s. In the past, architects and engineers had excellent technical aptitudes, they applied the features of self-similar repetitions in building structures to obtain strength, but reduce mass [46]. This logic became a common principle in the post-medieval and modern constructions. The colossal and monumental structures, like Pyramid and Colosseum, were no more built as massive masses, but as lattice structures by reducing heavy masses without compromising strengths. Schroeder believes, that the principle of branching points play the key role to provide strength both in nature and man-made structures. He argues, Think of the Eiffel Tower in Paris, designed by Gustave Eiffel. If instead of its spidery construction, it had been designed as a solid pyramid, it would have consumed a lot of iron, without much added strength. Rather, Eiffel used trusses,that is, structural frames whose members exploit the rigidity of the triangle [ .....] However, the individual members of the largest trusses, which in turn are made from members that are trusses again. This self-similar construction guarantees high resilience at low weight. The structures of Gothic cathedrals, oo, betray great faith in this principle of achieving maximum strength with minimum mass. And Buckminster Fuller [.....] and his skeletal domes popularized the fact that strength lies not in mass but in branch points. [73] Based on this fact and observation, and for exploring the deeper understanding about the structural behavior of fractal structures, apart from saving masses, in the recent years, Epstein and Adeeb [19] carried out a systematic study taking an example of a planar truss which was designed based on the Sierpinski gasket. Their investigation found a striking feature of stiffness pattern which shows self-similarity by their force distribution behavior. However, if the Sierpinski truss is compared with the same size of the complete planar truss, according to Kishimoto and Natori [35], the Sierpinski truss is slightly weaker, yet useful for the real-world application in construction. In other such research,

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25

Asayama and Mae [1] did a unique study about the innovation of new truss design based on fractal geometry (Figure 2.6), and claimed that the fractal-based new truss design is more efficient than the conventional king-post truss. In their experiment, they found a fractal-based truss, which is stiff enough against different loads and which appreciably consumes less weight. They also applied the principle of fractal geometry in designing fractal layered arch structures and found it stronger than conventional smooth arches against a heavy wind [44].

Figure 2.6: Asayama-Mae’s fractal trusses with different base angles, and their deformations under vertical load. [1]

Figure 2.7: (a) Folded shell design using Fractal’s rule of Iterated Function System [78], (b) Fractalbased reticulated shell structures design by Vyzantiadou, et. al. [85]

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In terms of developing a new design as a venture of finding new structural forms, a systematic and scientific study of fractal geometry was attempted by the IBOIS research group at EPFL (Ecole Polytechnique Federale de Lausanne) in Switzerland [78] (Figure 2.7a). Their study was significant in a sense that their engineering approach showed the prospect of the applicability of fractal geometry in the practical field of construction. In this frame, besides, ”Fractal Pre-Structured Building for (Temporary) Housing”, a research paper written by P. Fiamma and G. Carra [23] of Department of Civil Engineering in University at Pisa is an updated study which talked about structural and compositional property of fractal geometry and its application to the temporary housing units with the base of purely mathematical fractals. In 2007, a detailed mathematical and structural analysis for the application of fractal geometry in designing of grid or reticulated shell structures has been done by Vyzantiadou, et. al. from Department of Architecture and Civil Engineering in Aristotle University, Greece [85] (Figure 2.7b). However, until now, special attention of unfolding the structural logic in the geometry of fractal pattern exhibited in nature (apart from tree branches), and the implementation of that geometry-based structural logic (biomimicry) for constructing architecture and structures has very little been paid.

2.1.5

Fractal Structures in Pure and Applied Sciences

In the applied science, the observations of Mandelbrot and Schroeder about the fractal’s merit of saving masses without affecting strength, have been reconsidered and testified in a scientific way by physical demonstrations, interestingly beyond the normal range of scales. In 2012, an outstanding feat has been achieved on the potency of fractal geometry and its practical applicability in construction. Robert Farr and his team have found a remarkable result after experimenting on a small-scale fractal truss column developed by them in the laboratory which claimed that fractal can produce the ’lightest and strongest’ structure [20]. A step further, in the field of material science, Professor Julia Greer and her team have recently developed a ’world’s lightest’ truss, named as ’nano-lattice’ using the concept of fractal geometry (Figure 2.8) [52]. According to their experimental results, ’nano-lattice’ is as light as a feather but features outrageous mechanical strength. They hope that this fractal nanostructure might one day be used in structural engineering. In this context, similar investigation has been carried out by Professor Kenneth Cheung and Professor Gershenfeld at the MIT. They have developed a system that can construct a large structure by assembling small ultra-light tiniest pieces using the principle of fractal geometry [12]. They claim that, in the field of construction, its application can offer a large space-frame structure that can be exceptionally light in

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weight and highly strong.

Figure 2.8: A fractal nanotruss made in Greer’s lab. Credit - Lucas Meza, Greer lab, Caltech. (URL: http://www.jrgreer.caltech.edu/home.php)

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2.2

Research Gap and Final Objective

2.2.1

Remaining Research Gap and Questions

This particular area of research about the applications of fractal geometry in architecture and civil engineering is recently an emerging area, and only a few studies have been done so far on this particular field of research. Hence, there exist multiple research scopes and questions which are necessary to be addressed for the further advancement of this specific research field. Some of the important research questions are: Fractals as a Tool for Form Finding Process So far, the applications of fractal geometry in the previous studies followed a straight-forward approach. The fractal concept has been directly applied as a venture or trial in designing structures to explore their shapes, observe their mechanical behaviors and find efficiencies. What is lacking in this strategy is the structural reasonableness for choosing a fractal concept. Therefore, why not change the perspective about the approach of applying fractal notion in designing or improving structures? What if we follow the inverted approach in which first a problem of a structure is identified, then find solutions in the properties of fractal geometry, and accordingly the concept of fractal is applied for responding to the problem? The Interpretation of the Fractal Dimension One of the most fundamental properties of fractal shapes is their fractional dimensionality in the Hausdorff metric space. In fact the fractal dimension can be changed continuously from integer to fractal while correspondingly changing a surface from smooth to rough. Based on this concept, any conventional smooth shape of a structure can be transformed into a complex and fractal shape. This is independent from the number of iterations that are performed. But, how does the Hausdorff dimension or fractal dimension affect the design outcome and mechanical behavior of a structure? Is it possible to set up a generalized but a reasonable relationship between fractal dimension and structural strength? How to interpret the fractal dimension of the structure with its strength? Flexibility of the Concept of Fractal Geometry Until now, only the property of self-similarity has been considered for applications, especially in the field of architecture and construction. But, there is another very important concept of fractal geometry, which is randomness. So,

Chapter 2. Literature Review

29

the question is, how flexible is the concept of fractal geometry if we talk about its structural applications. How can the property of strict ‘self-similarity’ be overcome by ‘randomness’ while still obtaining feasibility?

2.2.2

Research Objective

The proposed research encompasses two main domains of mathematics and architecture, more precisely the areas of fractal geometry and the computational morphogenesis of structures. The main aim of this interdisciplinary research is to explore and develop the practical and efficient applications of fractal geometry in the field of architecture and construction, and react to the above research questions. Structural Rationality of Fractal-Based Shapes and Optimization The first objective of the proposed research is to explore the structural significance of fractal shapes. For this, a problem-solving approach will be developed to apply the concept of fractal geometry for designing the shapes of the structures or their elements. This approach can be considered as the response to the first research question of changing the perspective of the research approach. In this strategic approach, instead of straight-forwardly applying the principle of fractal geometry in designing structural forms as a venture or trial, the proposed research will investigate about the merits of fractal features and fractal dimensions of fractal-based shapes which can solve some structural problems and optimize structures. Fractal-Based Light-Weight Structure Designs One of the key objectives of the proposed research is to develop very light-weight and high-strength structures in the field of construction using the concept of fractal geometry. This particular objective is directly inspired by the findings of previous researches done by the teams of Robert Farr, Julia Greer and Kenneth Cheung in the field of physics and material science. Fractal Dimension and Its Influence on Structural Behavior The proposed research will exploit the fractal dimension of a structure by transforming it from integer to fractional, and then assess its structural impact. Hence, it will establish a generalized relationship between fractal dimension and structural stiffness.

30

Chapter 2. Literature Review

Random Fractals and Their Structural Applications In the proposed research, the concept of random fractals, generated by stochastic rules, will be applied for getting a variety of rather unpredictable shapes, which open a design space. The task is to find the most suitable and optimized shape for a structure within this space of countless random shapes, so that it can improve the mechanical efficiency of the structure while reflecting on feasibility.

Part II

Theory, Formulations and Tools

31

Chapter 3

Fractal Mathematics 3.1

Introduction

Architecture is shaped and assembled by various geometries. From the structural point of view, geometric configurations are designed to provide the structural stability and mechanical strength to the buildings. Since history, architects and engineers have been using the knowledge of Euclidean geometry for designing different trends of architectural structures. Before the 1970s, there were no clear mathematical expressions about the other forms of geometries which widely exist in the nature, and we can see these unexplained form system in the nature. The geometries of nature like tree, cloud, lightning, river basin and mountain could not be explained by any mathematical expression and could not be expressed by Euclidean geometry. But, in the late 1970s, thanks to Benoit Mandelbrot who introduced the concept of fractals as a new form of geometric expression that can explain the form system of nature mathematically by some simple algorithmic definitions. Using the computer, this algorithmic formula can reproduce the nature’s fractal shapes easily and rapidly, which we might never be able to draw by hand or manually in CAD. Nature’s shapes, many of which are fractals, are the reasons and outcomes of their structural and functional necessities. Apart from functional rationality, these shapes offer structural stability and mechanical strength to the natural objects so that they can exist and survive in the nature against some unwanted forces. Therefore, it hints that the previously unknown form system which was later defined as fractals have some structural and mechanical potency which can be applied in designing architectural structures. But, before applying this form system of fractals to the real world applications, we need to understand its mathematical notion and explanations properly. Although, fractals can explain 33

34

Chapter 3. Fractal Mathematics

the geometry of nature, but its origin, definitions and measures lie in the pure mathematical ground that does not include nature at first hand. Its mathematical notion was born in the 17th century and later nurtured by some early mathematical concepts and theorems, credits goes to some notable mathematicians like Gottfried Leibniz, Helge von Koch, Waclaw Sierpi´ nski, Gaston Julia and Felix Hausdorff. In this chapter, the very essence of fractal mathematics has been explained briefly with the target of its real world engineering applications. In this chapter, it firstly explains the concept of metric space, specifically the Hausdorff metric space and Hausdorff measure as the ground where the fractals lie. Then, it describes the Hausdorff dimension system as a measure and indicator of fractals. Fractional Hausdorff dimension confirms a set or an object as a fractal. In this part, the Hausdorff dimensions of some sets have been calculated with some geometric examples and illustrations. For the calculation of Hausdorff dimension, an essential concept contraction mapping theory has been explained. Later, the Iterated Function System (IFS) has been described to construct fractals followed by the concepts of other measures of fractal dimensions such as Box Counting Dimension, etc. The mathematical concept of random fractals is also discussed in this chapter which is closer to the mathematical explanation of nature’s fractals or approximate fractals. Finally, the constraints and the extents of the notion of mathematical fractals are addressed for the finite parameters of real world applications. The final paragraph of this chapter has discussed about the threshold for connecting the theory of pure mathematical fractals to the applied finite iterated fractals for real world applications.

3.2

A Brief History of Fractal Geometry

Although the earliest history of the origin of fractal-like concept is traced from 17th century, we see more specific development on fractals from the very beginning of the 19th century. Since 1900s, the chronology of the development of fractals can be distinguished roughly as follows. Around 1900, strange counterexamples were found by famous mathematicians: • Cantor’s set consisting of continuum many separate points (Cantor) • Plane-filling curves (Peano, Hilbert) • Curves without intersection points with positive area (Osgood) • Nowhere differentiable functions (Weierstra, von Koch) • Graphs where each point is a branch point (Sierpinski)

Chapter 3. Fractal Mathematics

35

Starting 1920, important theory was created but not yet publicly recognized: • Hausdorff and Besicovich developed the concepts of fractional dimension and fractional measure. • Wiener and Kolmogorov founded the theory of Brownian motion. • In the 1940s Levy discovered more general self-similar random processes which now carry his name. After 1975, fractals became popular, and a wave of simple applications followed: • Benoit Mandelbrot, who considers himself a student of Levy, created the word fractal and wrote his book Les objets fractals: forme, hasard et dimension which in English became Fractal Geometry of Nature. • Thousands of papers throughout science, in engineering and the social sciences were inspired by Mandelbrot’s book. • Computer graphics provided impressive visualizations of fractals. • In probability theory, new self-similar constructions were developed • Branching Brownian motion, Aldous’ tree, Le Gall’s Brownian snake etc. Since 2000, new important theory is being worked out. • Lawler, Schramm and Werner developed a theory of con formally invariant random fractal curves. • S. Smirnov proved conformal invariance of two-dimensional percolation. • Analysis on fractals (which started 1982 with two letters in J. Physique) develops into an own mathematical area.

3.3

Definition of Fractal Geometry

A non-strict and loose denition of a fractal is a shape or an image that contains the copies of itself at innitely dierent range of scales; which means it is selfsimilar at every magnifying scale. Nevertheless, its mathematical definition is more precise termed by pure mathematical notions. According to Robert L. Devaney’s definition [15], A fractal is a subset of Rn (a n-dimensional metric space) which is self-similar and whose fractal dimension exceeds its topological dimension.

36

Chapter 3. Fractal Mathematics

To understand Devany’s definition of fractals, first we need to understand the mathematical concepts of self-similarity and magnification followed by the concept of ‘topological dimension’ and the meaning of ‘fractal dimension’. In this chapter, therefore, the definitions and properties of a fractal are explained from the purely mathematical background, but limited within the relevance of the needs of this research for real world engineering applications. In short, in this chapter, the explanation of mathematical background of fractal is targeted to lay the foundation for architectural and structural design-oriented research works and applications, but not go into the depth of pure mathematics which is more relevant to the interest of pure science disciplines. Barnsley’s seminal book Fractal Everywhere [2] and Kenneth Falconer’s important book Fractal Geometry: Mathematical Foundations and Applications [?] are very useful references for understanding the in depth explanations of fractal mathematics.

Figure 3.1: The formation of Sierpinski triangle (or gasket)

For developing the knowledge of the basic essence of fractal mathematics useful for this design-oriented research thesis, the following mathematical concepts are important understand: 1. The Hausdorff Metric Space and Hausdorff Measure 2. Topological Dimension vs. Fractal Dimension 3. The Contraction Mapping Theorem 4. Iterated Function System (IFS) 5. Different Measures of Fractal Dimensions

3.3.1

Common Definition

The purpose of this study is to design a new shape of shell structure based on fractal geometry followed by exploiting the smooth quadric surface of a paraboloid using Iterated Function System (IFS), and then briefly investigate its structural characteristics. Fractals are categorized as a new family of geometry that is different than the Euclidean and other regular but non-Euclidean geometries. Apart from the striking features

Chapter 3. Fractal Mathematics

3.3.2

37

Hausdorff Definition

The purpose of this study is to design a new shape of shell structure based on fractal geometry followed by exploiting the smooth quadric surface of a paraboloid using Iterated Function System (IFS), and then briefly investigate its structural characteristics. Fractals are categorized as a new family of geometry that is different than the Euclidean and other regular but non-Euclidean geometries. Apart from the striking features

3.4

The Hausdorff Metric Space and Hausdorff Measure

Conceptually, Hausdorff metric space is the space where fractals live. Here, the Hausdorff metric is denoted by H(X) for some metric space X. Definition 1: If (X, d) is a complete metric space, then, excluding the empty set, Hausdorff metric H(X) is the space whose points are the compact subsets of X. This means, as an example, if we are working in H(R), then [0, 1] and [-1, 5] are each different point, i.e., each different compact subsets in H(R). Now, we have to define distance from a point to a set so that we can later properly define distance in the Hausdorff space. Definition 2: If (X, d) is a complete metric space, x ∈ X, and B ∈ H(X), then the distance d from x to B is defined as d(x, B) = min{d(x, b) : b ∈ B}

(3.1)

Now we need to define distance in the Hausdorff metric. Definition 3: If (X, d) is a complete metric space and A, B H(X), then the distance d from A to B is d(A, B) = min{d(a, B) : a ∈ A}

(3.2)

Intuitively, the Hausdorff distance h is the same as putting an “epsilon net” around one set so we can capture the other. This means if there are two sets, A and B, and we are finding the distance from A to B then we must find the point b in B closest to A. Then, as shown in ’Figure 3.2’, we find the smallest radius possible that, when one end is fixed at b, will inscribe A. This radius is the distance from A to B and is sometimes known as ‘epsilon net’.

38

Chapter 3. Fractal Mathematics

It can be noted that this distance is not commutative; d(A, B) is not necessarily the same as d(B, A). We thus come to the definition of the Hausdorff distance:

Figure 3.2: Epsilon Net from A to B where d(A,B) = epsilon1; epsilon2 Net from B to A where d(B,A) = epsilon2

Definition 3: If (X, d) is a complete metric space and A,B H(X), then the Hausdorff distance, h, is defined as h(A, B) = max{d(A, B), d(A, B)}

(3.3)

It means, if we draw an epsilon net from A to B and from B to A and the largest of the two is h(A, B). Theorem 1: If X is a complete metric space then h is a metric for Hausdorff metric H(X). According to the ‘theorem 1’, h is a metric for H(X) where (H(X), h) is a perfect metric space. Because, this is a metric space, therefore it will be possible to define distances between fractals and convergence of shapes in R2 . Therefore, the Hausdorff metric space, (H(X), h), is a metric space where we can truly define and understand fractals. It is commonly understood that many fractals, such as the Sierpinski triangle, are the limit points of iterated functions. Which means, what we see with our naked eyes to a fractal figure, for example Sierpinski triangle, is actually not the final figure (which has to be undergone through a

Chapter 3. Fractal Mathematics

39

limitless infinite iterations), rather it is a capture of the iteration process at a finite number of iterations. But, it does not make sense to speak of an entire shape being a limit point in a metric space such as (R2 , d). How would we decide whether an iterated point is within a certain distance of the limit point? The Hausdorff metric space is a space where we can treat entire compact sets as points which simplifies definitions and equations easily.

3.5

Topological Dimension vs. Fractal Dimension

The Sierpinski triangle is self-similar which contains 3n identically-sized triangles within it. By zooming any one of them up to 2n times of its original size we can find the original Sierpinski triangle. From this example we can define the term affine self-similar. Definition 4: A set is considered affine self-similar if it can be divided into some integer m matching subsets. The matching subsets must be able to be magnified by some positive constant n to produce the original set. However, to understand Devany’s definition [15] of a fractal, we must still define the topological dimension. Definition 5: A set has a topological dimension of zero if all its points have arbitrarily small boundary ovals such that the boundary ovals do not intersect the set. Here, “boundary ovals” represents small circles or ellipses circumscribing points. For instance, in ‘Figure 3.3’, the set depicted by the points has a topological dimension of zero because around each point an ellipse can be placed that are individual and isolated and does not intersect any of the other points. Hence, the general definition of topological dimension is: Definition 6: A set has topological dimension m if at each point an arbitrarily small boundary oval can be created which intersects the original set in a topological dimension of (m−1) and m must be the smallest nonnegative integer for which this is true. The definition means to say that if the boundary ovals around the points in some set K intersect K so that the intersection has a topological dimension of 0, the topological dimension of K must be 1; because K − 1 = 0, i.e.,K = 1. This is graphically illustrated in the top figure in ‘Figure 3.4’. For another object,

40

Chapter 3. Fractal Mathematics

Figure 3.3: An example of zero topological dimensions

for example a filled in circle, any boundary oval around any point in that circle will produce an intersection with a topological dimension greater than one. If we consider a boundary oval around a point in the intersection we get a new set with a topological dimension of 0. This implies that the intersection of the circle with the original boundary oval had a topological dimension of 1 which means the filled in circle had a topological dimension of 2. Similarly, a sphere has a topological dimension of 3. This is shown in the bottom in ‘Figure 3.3’. It is noticeable that, when a boundary oval intersects a line (topological dimension 1) then intersection has two separate points which are non-connected points and which can be independently enclosed by another boundary ovals that never overlaps to each other (‘Figure 3.4’, Top). But, when a boundary oval intersects a surface (topological dimension 2), then we find that the intersection is a collection of consecutive connected points and this intersection is a curve whose topological dimension is 1 (Figure 3.4, Bottom). From the above explanation we find that the topological dimension of a set can be an iterative process. The Sierpinski triangle therefore has a topological dimension of 1 because no matter where a boundary oval is created, the boundary oval intersects the Sierpinski triangle at non-connected points. This statement makes sense, because the Sierpinski triangle is composed entirely of straight lines which means at infinite level of iterations, Sierpinski triangle completely losses its all surface areas (means losses its topological dimension of 2). Therefore, when a boundary oval intersects any part of the triangle, we notice that at each intersection there is no two consecutive points. In other way, no two consecutive points of the ellipse (the shape of a boundary oval) will intersect any lines of the Sierpinski triangle because of the curvature of the ellipse

41

Chapter 3. Fractal Mathematics

Figure 3.4: Examples of 0, 1, 2 and 3 topological dimensions

so the intersection of the Sierpinski triangle with a boundary oval will produce non-connected points. Now, we need to understand the definition of fractal dimension and how is it different than topological dimension. Definition 7: If a set K can be broken into some integer m identical parts and those parts can each be magnied by some factor R to yield K then K has a fractal dimension D of D = log m/ log R

(3.4)

The above formulation of fractal dimension is a simple outline of the Hausdorff dimension. The notion of Hausdorff dimension and its calculation has been explained later in paragraph ‘Fractal Dimensions’ by using the tools of Contraction Mapping Theorem and Iterated Function System (IFS). However, the above equation 3.4 is also popularly known as Self-similar dimension. From the above definition, in the case of Sierpinski triangle, for any n ∈ N , the triangle can be broken into 3n identical pieces, each of which can be magnified by a factor of 2n to yield the Sierpinski triangle. This means it has a fractal dimension of D = log 3n / log 2n = log 3/ log 2 = 1.584962501.

42

Chapter 3. Fractal Mathematics

Hence, it can be concluded from the above result that the Sierpinski triangle is, according to Devany’s definition, a fractal because it is a subset of R2 , it is affine self-similar, and it has a fractal dimension 1.584962 which is greater than its topological dimension 1.

Figure 3.5: Stages of the construction of Koch curve.

Similarly, in the case of Koch curve we can calculate its fractal dimension (Figure 3.5). In Koch curve one line is split into three equal segments and the middle segment is replaced by two same length segments arranged 60o to each other. Thus from one line it produces an arrangement of 4 self-similar line segments and this operation iteratively continues for each line segment. Here, at any step of iteration we find two any line segments intersect each other and their intersection is a point whose topological dimension is 0. Therefore, Koch curve has topological dimension 1. Therefore, for any n ∈ N , the line can be broken into 4n identical segments, each of which can be magnified by a factor of 3n to produce the Koch curve. This means it has a fractal dimension of vD = log 4n /log3n = log4/log3 = 1.2619. So what does a fractal dimension of 1.58496 mean when normally we deal with dimensions that are integers? We know a line is one dimensional. Intuitively if we were placed on the line, we could either move forward or backward but that is it. We would only have choices in one dimension. A filled in circle is two dimensional with the same idea. We could move up, down, left, right, or

Chapter 3. Fractal Mathematics

43

some combination which involves two dimensions. With the Sierpinski Triangle, however, at each point there are more choices than just back and forth, however, not as many choices as if we were in an infinite plane. This is the reason, its dimension is stuck between 1 and 2.

3.6

The Contraction Mapping Theorem

The Sierpinski triangle is actually a limit point of an iterated map of a function. Odd as it is to think of it this way, it is true. How can an entire shape be considered a point? Remember that we are working in the Hausdorff metric space so compact sets are actually considered points. Not only is it a limit point of the iterated map of this function, but it is the limit point no matter what shape you start o with or where in R2 this shape is located. How can we show the Sierpinski triangle is the limit point of any initial point’s orbit under this special function? We can do this by the contraction mapping theorem. Definition 8: If f (X) → X is a function, where X is a metric space and 0 < λ < 1, such that d(f (a), f (b) ≤ λ(d(a, b))

(3.5)

for all a, b ∈ X, then f function is a λ-contraction of X. Any contraction is continuous. If equality holds, i.e. if d(f (a), f (b)) = λ(d(a, b))

(3.6)

then f transforms sets into geometrically similar sets, and in that case f is called as a contracting similarity. Theorem 2 (The Contraction Mapping Theorem): If X is a complete metric space and f (X) → X is a λ − contractionof Xthenf orallx ∈ X, the orbit of x under f converges to some unique fixed point x◦. This is, therefore, a very powerful theorem especially for constructing fractals. In understanding the proof of this theorem in details, one can go into the details of Carol Schumacher’s Closer and Closer book [74]. The convergence action can be generally explained by the example of Sierpinski triangle. For example, if we consider the left self-similar triangle only, then we will find that at infinite iteration left triangles are being contracted and finally converges to the fixed point of the left vertex of the main triangle. It is important to know that the convergence point must be a fixed point.

44

Chapter 3. Fractal Mathematics

3.6.1

The Hutchinson Operator

It is proven in Barnsley’s Fractals Everywhere that Hausdorff metric space (H(X), h) is a complete metric space if (X, d) is also a complete metric space. Therefore, we can apply the contraction mapping theorem to k-contractions within H(X). There is a method which iterates functions on points in the Hausdorff metric space. A specific function exists which, when iterated at an infinite number of times, generates end-less self-similar copies. We call this function system as Iterated Function System (IFS) discussed later in the paragraph of ‘Iterated Function System’. However, the newly generated self-similar copies, which are k-contracted, are collected by the Hutchinson operator to a single set that is a fractal. Definition 9: Let X be a metric space and let A be a subset of X. Let (f0 , f1 , ..., fn ) be a finite number of affine transformations. The Hutchinson Operator of (f0 , f1 , ..., fn ) on A is f0 (A) ∪ f1 (A) ∪ . . . ∪ fn (A). This definition basically means that if there is a group of functions where each function changes an image in R2 , the Hutchinson operator of these functions creates their resulting images simultaneously. There are many choices that are possible for the individual affine transformations and lead to different interesting fractals. In the case of the Sierpinski triangle, however, we will be working with the three affine transformations. The affine transformation methods and their examples are explained at bellow in the chapter ‘Iterated Function System’.

3.7

Iterated Function System

There are many fractals which are made up of parts that are, in some way, similar to the whole. For example, the middle third Cantor set is the union of two similar copies of itself, and the Koch curve is an assembly of four self-similar copies. These self-similarities are not only the properties of the fractals, but they also may actually be used to define them. Iterated function systems (IFS) do this in a unified way and, moreover, often lead to a simple way of finding dimensions as shown in the chapter ‘Different Measures of Fractal Dimension’. IFS is also a powerful tool to generate fractals. In this thesis, IFS has been used as the main tool for producing fractal forms and for measuring their Hausdorff dimensions. The history of IFS is not so much old. IFSs were conceived in their present form by John E. Hutchinson in 1981 [29] and popularized by Michael Barnsley’s book Fractals Everywhere [2]. IFSs provide models for certain plants, leaves, and ferns, by virtue

45

Chapter 3. Fractal Mathematics

of the self-similarity which often occurs in branching structures in nature. - Michael Barnsley, et. al. [2] According to Barnsleys explanation, a finite family of contractions (f1 , f2 , ..., fm ), with m ≥ 2, is called an iterated function system or IFS. We call a non-empty compact subset ∆ of D an attractor (or invariant set) for the IFS if ∆=

m [

fi (∆)

(3.7)

i=1

The fundamental property of an iterated function system is that it determines a unique attractor, which is usually a fractal. For a simple example, take ∆ to be the middle third Cantor set. Let f1 , f2 : R → R be given by 1 x 3

(3.8)

2 1 x+ 3 3

(3.9)

f1 (x) =

f2 (x) =

Then f1 (∆) and f2 (∆) are just the left and right ’halves’ of ∆, so that ∆ = f1 (∆) ∪ f2 (∆); thus ∆ is an attractor of the IFS consisting of the contractions (f1 , f2 ), the two mappings, which represent the basic self-similarities of the Cantor set. The fundamental property of an IFS is it has a unique attractor which is non-empty compact, i.e. closed and bounded. This means, for example, that the middle third Cantor set is completely specified as the attractor of the mappings (f1 , f2 ) given above. Theorem 3: Consider the iterated function system given by the contractions (f1 , f2 , ..., fm ) on X ⊂ Rn so that d(fi (a), fi (b)) ≤ λi (d(a, b)); (a, b) ∈ X(3.10) i.e., |fi (a) − fi (b)| ≤ λi |a − b|; (a, b) ∈ X(3.11) with λi < 1 for each i. Then there is a unique attractor ∆, i.e. a non-empty compact set such that ∆=

m [ i=1

fi (∆)

(3.12)

46

Chapter 3. Fractal Mathematics

Moreover, if we define a transformation f on the class f of non-empty compact sets by f (E) =

m [

fi (E)

(3.13)

i=1

for E ∈ f , and write f n for the nth iterate of f (so f 0 (E) = E and f n (E) = f (f n−1 (E)) for n ≥ 1), then ∆=

∞ \

f n (E)

(3.14)

n=0

for every set E ∈ f such that fi (E) ⊂ E for all i.

3.7.1

Example1: Sierpinski Triangle

Sierpinski triangle shown in Figure 3.6 are generated by means of the contraction mapping repeatedly. The figures show the convergent sequence of them set in 2- dimensional space. Based on the above theorem of IFS, we get the following geometric properties: ∆n = f1 (∆n−1 ) ∪ f2 (∆n−1 ) ∪ f3 (∆n−1 )

(3.15)

∆0 ⊃ ∆1 ⊃ ∆2 ⊃ ∆3 ⊃ ..... ⊃ ∆n ⊃ .....

(3.16)

where,

Hence, the attractor S is, ∆=

∞ \

∆i

(3.17)

i=1

which forms a perfect self-similar set of Sierpinski triangle.

Figure 3.6: Convergent sequence of the Sierpinski triangle by using contraction mapping and IFS.

47

Chapter 3. Fractal Mathematics

3.7.2

Example2: Fractal Fern

There are different approaches to construct a fractal fern using IFS. One of the simple approaches is the union of two different functions derived from one initial function, and this process of union is continued at each new self-similar functions. ‘ Figure 3.7’ shows the construction of a fern using union two different affine transformation functions.

Figure 3.7: Construction of fern usning IFS that is made with two functions.

3.8

Attractor

Definition10: From equation 3.12, if X is a complete metric space and F = (f1 , f2 , .., ., fm ) is a collection of contraction mappings of X to itself, then F is said to be a Iterated Function System. Therefore, for such an F there is a unique compact set ∆ such that ∆=

m [

fi (∆)

(3.18)

i=1

then, ∆ is called the attractor for F .

3.8.1

Affine Transformation and Self-Affine Sets

Self-affine sets form an important class of sets, which include self-similar sets as a particular case. An affine transformation f : Rn → Rn is a transformation of

48

Chapter 3. Fractal Mathematics

the form f (x) = T (x) + b

(3.19)

n

where T is a linear transformation on R (representable by a n × n matrix) and b is a vector in Rn . Thus an affine transformation f can be a combination of a translation, rotation, dilation and, perhaps, a reflection. Two-Dimensional Affine Transformation A two-dimensional affine transformation on the Euclidean plane is        x a b x e T = + y c d y f

(3.20)

This consists of a linear transformation performed by the matrix containing a, b, c and d and then a translation by the matrix [e, f ]. However, unlike similarities, affine transformations contract with differing ratios in different directions. The contractive and rotational affine transformation of (x, y) on XY plane is        x Γ cos θ − sin θ x e T =λ + (3.21) yΓ sinθ cos θ y f where, θ is the angle of rotation, λ is the contractivity of (x, y) and e and f are translation/shifting of x Γ and yΓ. If an IFS consists of affine contractions (f1 , f2 , .., fm on Rn , the attractor F guaranteed by ’Theorem 3’ is termed as a self-affine set. Example 4.2: In Sierpinski triangle, there are three affine transformations. Let, the vertices of an equilateral triangle are (0, 0), (x, 0), (x/2, py). For equilateral triangle the common angle is 60o and the height y = (3/2).x.. In Sierpinski triangle the contractivity is 1/2.  ∆1a = f1 (∆0 ) = T



x1a y1a

=

 1 cos 0o · 2 sin0o

− sin 0o cos 0o





x y

 +

0 0

 (3.22)

the affine transformation of the second self-similar triangle is,  ∆1b = f2 (∆0 ) = T

x1b y1b



 1 cos 0o = · 2 sin0o

− sin 0o cos 0o



x y



 +

x/2 0

 (3.23)

and the affine transformation of the third self-similar triangle is,  ∆1c = f3 (∆0 ) = T

x1c y1c



 1 cos 60o = · 2 sin60o

− sin 60o cos 60o



x y



 +

x/4 y/2

 (3.24)

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Chapter 3. Fractal Mathematics

Figure 3.8: Contraction mapping and affine transformation of the Sierpinski triangle. Angle is 60o and contractivity is 1/2.

’Figure 3.9’ shows different process of two-dimensional affine transformation. using the same process, Barnsle’s fern has been generated shown in ’Figure 3.10’ Three-Dimensional Affine Transformation Barnsley’s method of affine transformations are also able to produce three dimensional fractals. It operates in three-dimensional space. Let, λ is the contractivity, θ, α and β are the angles of rotation on XY, XZ and YZ planes respectively. Let, e, f and j are the displacement on YZ plane. Then, there rotational transformations in the 3-dimensional space are, on XY plane, 

  x1 cosα  y1  = sinα z (θ) = f3 (∆0 ) = z1 0

  0 x0 0  y0  1 z0

(3.25)

  0 sinβ x1 1 0   y1  0 cosβ z1

(3.26)

−sinα cosα 0

on XZ plane, 

  x2 cos β  y2  =  0 y (θ) = f3 (∆0 ) = z2 −sinβ on YZ plane, 

  x2 1    y 0 (θ) = f (∆ ) = = 3 z 3 0 z3 0

0 cosθ sinθ

  0 x2 −sinθ  y2  cosθ z2

(3.27)

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Figure 3.9: Different modes of two-dimensional affine transformations.

Therefore, the final affine transformation on YZ plane,     x2 cosα −sinα 0 cos β T  y3  = λ·sinα cosα 0  0 z3 0 0 1 −sinβ

is

 0 sinβ 1 1 0  0 0 cosβ 0

    0 x e −sinθ  y + f  cosθ z j (3.28) Three-dimensional affine transformation has been applied in this dissertation 0 cosθ sinθ

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Figure 3.10: Bensley’s fern generation using his IFS based on two dimensional affine transformations.

for modeling three-dimensional fractal structures, such as branching tree column in Chapter 6.

3.9

The Concept of Pre-Fractal and Random Fractal

The transformation S introduced in ‘Theorem 3’ is the key to computing the attractor of an IFS. In fact, the sequence of iterates fi (E) converges to the attractor F for any initial set E in f , in the sense that d(f i (E), F ) → 0. This follows d(f (E), F ) = d(f (E), f (F )) ≤ λd(E, F ), so that d(f i (E), F )λi d(E, F ), where λ = max(1≤i≤m) λi < 1.T husthefi (E) provide increasingly good approximations to F. If F is a fractal, these approximations are sometimes called pre-fractals for F.

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Figure 3.11: Construction of the attractor F for contractions S1 and S2 which map the large ellipse E onto the ellipses S1 (E) and S2 (E). The sets S n (E) = ∪(1,2) s1 (E), ...., Sn (E) give increasingly good approximations to F .

There are many IFS fractals that are not self-similar, but randomized at each stage by certain defined range. Random fractals can be produced by different tactics using IFS. In practice, IFS fractals are almost always drawn by using a probabilistic technique. Each of the similitudes is assigned a probability. With each iteration, the transformation to be used is then a random selection, with the chances that a particular one will be used depending on the probabilities assigned. This sequence is still attracted by the same attractor and will hence converge to the same fractal as would be obtained by using the non-random method. An illustration of the above theory can be obtained using the first image of a Sierpinski gasket. The three smaller boxes in the image correspond to the three affine transformations used to create the fractal. Following values have been used in affine transformation function for the three attractors. a 50 50 50

b 0 0 0

c 0 0 0

d 50 50 50

e 0 128 64

f 0 0 80

Probability 1/3 1/3 1/3

Suppose we modify the attractors now and see what happens to the image. The first change we make is to change the translation factor of one transformation. That is, we shift it by a bit. As can be seen, the resulting attractor is still easily recognizable as a Sierpinski gasket. Nevertheless, it is different to the

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Figure 3.12: Left- affine transformation of Sierpinski triangle, Centre – randomized affine transformation by shifting upper right triangle, Right – randomized affine transformation by rotating upper left triangle by 100o .

original as it is skewed. Starting again with the original factors of the Sierpinski fractal, the second modification was to change the dilation ratio of the bottom attractor. The resulting fractal now takes on a very different appearance. The most remarkable change occurs when the leftmost attractor was rotated through about 100 degrees in the clockwise direction. Whilst such ‘random fractals’ do not have the self-similarity of their nonrandom counterparts, their non-uniform appearance is often rather closer to natural phenomena such as coastlines, topographical surfaces or cloud boundaries. Indeed, random fractal constructions are the basis of many impressive computer-drawn landscapes or skyscapes. Random fractals can also be applied in designing complex architectures. In this dissertation, the concept of random fractals has been applied in designing structural forms and and evaluate their structural behavior in the Chapter 10.

3.10

Different Measures of Fractal Dimensions

According professor Barnsley’s definition, ’Fractal dimension of a set is a number which tells how densely the set occupies the metric space which it lies.’ [2] Apart from Barnseley’s conceptual definition, there exist a number of different measures and definitions of fractal dimension. Most of them are equivalent. The most celebrated and well accepted one is the Hausdorff (or Hausdorff -

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Besicovitch) dimension. But, the most popularly used fractal dimension is the Box-Counting Dimension. In mathematics, there are several equivalent definitions termed as a Box-Counting Dimension. Some of them are the capacity dimension (the definition given by Kolmogorov) and the Minkowski–Bouligand dimension. Although in many cases the Hausdorff Dimension equals to the BoxCounting Dimension, but in general the Hausdorff is used mainly in theoretical settings and is too subtle sometimes for practitioners. Other well-known methods for estimating fractal dimension are correlation dimensions (GrassbergerProcaccia, Takens estimators) and information (or entropy) dimensions.

3.10.1

Hausdorff Dimension of Self-Similar Sets

One of the advantages of using the iterated function system is that the dimension of the attractor is most of the time relatively easy to calculate or measure in terms of the defining contractions. In this section I discuss the case where f1 , f2 , .., fm : Rn → Rn , are similarities, i.e. with d(fi (a), fi (b)) ≤ λi (d(a, b)); (a, b) ∈ X(3.29) i.e., |fi (a) − fi (b)| ≤ λi |a − b|; (a, b) ∈ X(3.30) where 0 < λi < 1 (lambdai is called the contractivity ratio of fi ). Thus, each fi transforms the subsets of Rn into geometrically similar sets. The attractor of such a collection of similarities is called a (strictly) self-similar set, because it forms a union of a number of smaller similar copies of itself. Standard examples of self-similar sets are the middle third Cantor set, the Sierpinski triangle and the von Koch curve. However, under certain conditions, a self-similar set F has Hausdorff dimension equal to the value of D satisfying, m X (λi )D = 1

(3.31)

i=1

If, λi has the same value which is seen in strictly self-similar fractals, and if 0 < λ < 1, then, from equation 3.31 we get, λD + λD + λD + .....λD = 1; m times, m >1

(3.32)

m · λD = 1

(3.33)

so, i.e., λD =

1 m

(3.34)

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Therefore, D=

3.10.2

log (1/m) log(λ)

(3.35)

Minkowski-Bouligand Dimension (Box-Counting Dimension)

Box-counting dimension is one of the most widely used fractal dimensions. Its relatively easy empirical estimation is the reason of its wide popularity. The box-counting dimension is simple only if one operates on one- or two-dimensional data. Although, the simplicity of the “box-counting” idea is very appealing, but its time complexity is exponential with respect to dimension of space. That means, the amount of computation grows exponentially with the number of space coordinates. Box-counting dimension of a set S in a in a metric space (X, d) is defined by, Definition 12: The lower box-counting dimension is defined by Dimlower (X) = lim inf r→0

log N (r) log(1/r)

(3.36)

Similarly, the upper box-counting dimension is defined by Dimupper (X) = lim sup r→0

log N (r) log(1/r)

(3.37)

If Dimlower (X) = Dimupper (X), then the common value is called the BoxCounting Dimension. Thus, the Box-Counting Dimension is defined by DimBOX (X) = lim sup r→0

log N (r) log(1/r)

(3.38)

if such a limit exists. N (r) is the number of s-mesh cubes that intersect X. Equivalently, one can also determine N (r) as the smallest number of closed sets of diameter at most r (closed balls or cubes) that cover X or as the largest number of disjoint balls of radius r with centers in X. The upper box dimension is sometimes known as the entropy dimension, Kolmogorov dimension, Kolmogorov capacity or upper Minkowski dimension, while the lower box dimension is also known as the lower Minkowski dimension. However, the upper and lower box dimensions are strongly related to the more popular Hausdorff dimension. The box dimensions and the Hausdorff dimension are related by the inequality,

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i.e., DimHaus (X) ≤ DimLow (X) ≤ DimU p (X)

(3.39)

Very simple methods are popularly used for estimating the box-counting dimension. In this process, the numbers of closed boxes that cover any part of an image are counted and its side length is measured at each different scale, and then their log-log graph is plotted to get the box-counting dimension easily and quickly. The process is demonstrated at bellow. Example A typical process to evaluate the box-counting dimension of an image, starts with overlaying a squares grid on the image where the size of each grid box, let S, determines the scale of the grid. Then the boxes, that covers any mark or line of the image within the grid, are counted; let it is N . After repeating the same process on the same image by changing the box size, fractal dimension D of that image can be obtained by transforming the results of S and N into the log–log graph. The slope of the resulting line of the log–log graph determines the fractal dimension of the image.

Figure 3.13: Left- affine transformation of Sierpinski triangle, Centre – randomized affine transformation by shifting upper right triangle, Right – randomized affine transformation by rotating upper left triangle by 100o .

Box counting operation of a fractal figure. Extreme left—fractal figure; (a) grid size 24; (b) grid size 12; (c) grid size 6; and (d) grid size 3. Step 1 2 3 4

Grid size (s) 24 12 6 3

Marked boxes (N ) 140 420 1226 3497

Log(S) 1.380 1.079 0.778 0.477

Log(N ) 2.146 2.623 3.088 3.544

For example, by taking the box sizes or grid sizes 24, 12, 6 and 3 of a squares grid for a fractal image, covered boxes are counted (Figure 3.13). Then by taking the log of grid sizes (S) on the X-axis and log of marked boxes (N )

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Figure 3.14: Log-log graph whose slope is the value of fractal dimension.

on the Y-axis following results have been obtained as shown in ’Figure 3.14’. Slope of the resulting line in log–log graph shows the fractal dimension of the image is 1.5478. The resulted points lied on a smooth straight line in the graph indicate the uniformity of roughness of the image in different scales.

3.11

Construction Methods of Fractals

Mathematical fractal shapes are infinitely detailed and complex. Even finitely iterated fractal shapes are hard to construct by manual process. Therefore, computer is the best solution to automatically construct fractals. However, there are different geometric rules to that govern the automatic productions of fractals. In this chapter these rules are briefly discussed with few demonstrations.

3.11.1

Iterated Function System: IFS

The Iterated Function System (IFS) is the most popular and simple rule which is based on the concept of recursion system. In the previous section, IFS has been explained in details by using Barnsley’s method. IFS is very useful for generating perfectly self-similar as well self-affine fractal shapes. In this thesis, IFS has been as the main tool for the construction of fractals.

3.11.2

Lindenmayer Rewriting System: L-System

The Lindenmayer System, also known as L-System, is mainly based on the parallel rewriting system and a form of formal grammar. This system was developed in 1968 by Aristid Lindenmayer, a Hungarian theoretical biologist and

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botanist [40]. This technique is widely used for simulating the natural fractals, especially in botany to generate complex branching systems. It is also useful for generating self-similar fractals. L-System are also known as parametric LSystem, defined as a tuple G, which is a ordered list of elements in the following way, G = (V, A, C, P )

(3.40)

where, V is a variable which is a set of alphabets or symbols containing elements that can be replaced. A is an axiom or initiator that is a string of symbols from V defining the initial state of the system. C is a constant that guides for directions and P is a production rule. Example: Tree Branching The alphabets can be replaced by some symbols or geometric shapes to produce some fractal shapes. One of the common such example is the construction of branching tree as follows: variables : 0, 1 ; where ’0’ means draw a line segment ending in a leaf and ’1’ means draw a new line segment. constants : [, ] ; ’[’ means push position and angle, turn 45o left. ’]’ means push position and angle, turn 45o right. axiom : 0 constants : [, ] ; (1 → 11), (0 → 1[0]0) The shape is built by recursively feeding the axiom through the production rules. Each character of the input string is checked against the rule list to determine which character or string to replace it with in the output string. In this example, a ’1’ in the input string becomes ’11’ in the output string, while ’[’ remains the same. Applying this to the axiom of ’0’, we get: axiom : 0, 1 0 1st recursion : 1[0]0 2nd recursion : 11[1[0]0]1[0]0 3rd recursion : 1111[11[1[0]0]1[0]0]11[1[0]0]1[0]0 Applying the graphical rules listed above to the earlier recursion, we get a fractal branching shown in Figure 6.2.

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Figure 3.15: Construction of fractal branching using L-System.

Based on the simple L-System rule, different forms of composite L-Systems are developed, such as stochastic, context sensitive and parametric grammars. For further understanding of these advanced L-Systems, one can read the book The Algorithmic Beauty of Plants by Przemyslaw Prusinkiewicz and Aristid Lindenmayer [62] Apart from these well known methods, there are also othe methods which are able to generate self-similar and approximat fractals. Among them, the Finite Subdivison Rule. 1 is useful for genarting fractals that are applicable in the field of architectural and structural designs. In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. It uses a recursive topological algorithm for refining tilings and they are similar to the process of cell division. The uniqeness of this process is the slight changing of shape at each new iteartion, thus control the elegancy of fractal shapes. This process has not popular yet in the filed of design, however, it has a great potential if applied in designig structures where subdibvisions are demanded from aesthetic as well as mechanical point of views. Nevertheless, in this dissertation, only the Iterated Function System and the Midpoint Displacement Method has been used as mathematical tools and rulebased algorithms to generate fractal shapes for designing architectural structures. The Midpoint Displacement Method has been explained in details in the Chapter 9 and Chapter 10.

1 The

detailed description about this method can be found in [10].

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Figure 3.16: Constructing conformal finite subdivision rules. [10]

3.12

Automatic Fractal Constructions: IFS Code

Based on the mathematical description of contraction mapping and affine transformation, the Iterated Function System (IFS) can be coded as a scripting language for automatic generation in computer. Manually, it is impossible to produce fractals especially after high number of iterations. By scripting it for computer aided automatic generations, fractal figures can be easily made parametric so that the output shapes can be modified by controlling the initial input values. The IFS works step by step as follows, The pseudo code for the above IFS shown in flowchart, is presented at bellow that can be used in python scripting language.

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Figure 3.17: Flowchart of IFS method

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Figure 3.18: IFS pseudocode

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3.13

Fractals: From Mathematical Theory to Engineering Applications

3.13.1

Metric Space

Fractals are lying in a metric space, more precisely in Hausdorff metric space. In engineering application we also work in metric space. So, from a mathematical point of view, fractals and architectural as well as engineering forms lie in metric space. Accordingly, fractal geometry is feasible and real if applied for architectural and engineering applications as geometric shapes and structural topologies.

3.13.2

Fractal Scale: From Macro to Micro

Size matters in structure. In other sense, the scale of discrete structural members decides the strength of a structure. Using fractal geometry we can control size by exploiting the scalability of the structure. Euclidean geometry is probably unable to bridge the gap between mega scale to nano scale. But, in the concept of fractal geometry, theoretically, one shape can undergoes from infinitely minimum scale to infinitely maximum scale. Thus, it can generate a massive range of scales and at each scale it is self-similar to other scales. This concept can be used for designing mega space-frame hierarchical structures where smallest member can be used and recursively repeated as a strategy to obtain light-weight structure. In the field of civil engineering, we see some such application to design hierarchical trusses to save matrails [21]. Moreover, the smallest practical member can go further smaller scale, even to micro and nano scales as a contracted copy of the smallest members. Farr and Mao [20], Cheung and Gershenfeld [12] and Julia Greer and her team [52] have shown that contraction principle of fractal geometry can be applied even in nano and micro scale to produce ultra-light hierarchical structure by maintain a high strength. Their experimental lattice trusses are the copy of normal scale lattice structure. So, in future, we can use the finding of ultra-light nano lattice structure to construct mega structure in architecture by largely scaling up.

3.13.3

Finite Fractals for Finite Element Analyses

Mathematical fractals are infinite. Therefore, the mathematical fractal figures which we see with our naked eyes are in actual endlessly iterated infinite shape. So the figure we see in certain scale is a result of a limit point of the infinite iteration. For example, the Sierpinski gasket is a collection of many self-similar triangles and we see the figure of a finitely iterated Sierpinski gasket fractal only

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Figure 3.19: Possibility to construct mega scale lattice structures from nano scale hierarchical trusses using the concept of fractal geometry.

for the graphical representation. But in actual, at infinite level, we cannot see the gasket as we see usually for illustration. At infinite level, it tends to loss all its surface area, and it tends to become a collection of lines only, which means at infinite level it tends to become a one dimensional. But, if we take fractals for applying in engineering applications and for other applications in a practical world, then we cannot take the infinite stage of fractals. Rather, we have to stop somewhere and only take a limit point fractal, i.e., finite fractal. For the purpose of structural analysis, Finite Element Analysis (FEA) is one the easiest and fastest computational tool. So, if we select FEA as structural analysis strategy, then finite iterated fractal is fit for FEA. Hence, in this thesis, for the actual engineering structure, the number of iterations is finite and we can call it as Finite Iterated Modeling of mathematical fractals. The number of iteration depends on the overall size of the structure as well as the feasible size of the smallest element, if we consider the human scale, not micro and nano scale.

3.13.4

Self-Similarity and Structural Module System

The finite fractal has the foremost quality of self-similarity, which means each shape is repeated in the next iterations, and sometimes, each smaller part is identical to its original shape. This quality is very useful for engineering structures where the module system offers cost and time effective efficiencies. For example, to construct a truss having self-similar fractal properties may need only one size of the element or only one type of module combined with selfsimilar elements. Thus, it will be efficient to quickly assemble the elements because of self-similarity. Besides, every smaller module may have similar structural and mechanical behaviors same as the main truss because of their self-similarities and the property of the replication of the whole configuration in its smaller parts or modules. The topology of each smaller module is similar to the topology of main structure. Stress are uniformly distributed and forces follow self-similar pattern in

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the self-similar configuration of truss.

3.13.5

Fractal Dimension and Structural Response

Fractals are quantified by fractal dimension which tells how densely the set occupies the metric space where it lies [2]. Accordingly, fractal dimension can be applied for measuring the density of engineering structure, especially trusses, where it can tell how densely the nodes occupy the structure. Higher the fractal dimension of a structure, higher is the number of structural nodes in it. For a defined topological pattern, the number of nodes is generally related to the stiffness of the structure. Higher the number of the nodes, the higher is the stiffness of the structure. However, this relation is not always true, because the stiffness is primarily dependent on the structural topology and boundary conditions, including other factors like load amounts and their directions, material property, etc. Yet, in some cases where these factors are kept constant, the variable number of nodes influences the stiffness noticeably. Few researchers have found a unique self-similar stiffness behaviour of fractal structures because of its selfsimilar geometry and topology as well as its corresponding fractal dimension. The Hausdorff dimension of a fractal truss changes with the changing of their base angles for a defined topology of a truss having fractal properties [1] [64]. Hausdorff fractal dimension is mainly measured by contraction values. Therefore, in a self-similar fractal truss, the Hausdorff fractal dimension is determined by the contraction, i.e. the scaled value of the original shape. Interestingly, with the changing of this factor of contraction in a fractal truss, we see not only the changing of its fractal dimension, but also the changing of its stiffness value as well as the stress distribution pattern. We will demonstrate this relationship in the chapter of ’Fractal Dimension and Structural Stiffness’. Whereas, the Hausdorff fractal dimension is appropriate for measuring the fractal density of a self-similar pure fractal shape, the box counting dimension is appropriate for measuring the density of any non-self-similar and approximate fractal shapes. Therefore, any structure which is not a fractal, can also be measured by its density using the box counting method.

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Chapter 4

Fractal-Based Computational Morphogenesis 4.1

Introduction: Computational Morphogensis in Architecture

In architecture, computational morphogenesis or digital morphogenesis is understood as a set of methods that employ digital media not as representational tools for visualization but as generative tools for the derivation of form and its transformation often in an aspiration to express performance-based contextual processes in built form [36]. In this inclusive understanding, Stanislav Roudavski argues, that the‘computational morphogenesis in architecture bears a largely analogous or metaphoric relationship to the processes of morphogenesis in nature, sharing with it the reliance on gradual development but not necessarily adopting or referring to the actual mechanisms of growth or adaptation’ [67]. Context-sensitive goal-oriented form-finding is linked to the recent discourse on computational morphogenesis in architecture. Among the benefits of natureinspired forms, their advocates list the potential for structural benefits derived from redundancy and differentiation and the capability to sustain multiple simultaneous functions [86]. From the structural point of view, computational morphogenesis is a word that is generally used for expressing those techniques or ways of thought by which the configuration or the system itself of the structures is generated mainly 67

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through the usage of the computers, which is realized on the firm foundation of both FEM (Finite Element Method) as a tool of numerical analysis and various kinds of method based on relatively newly developed algorithms for structural optimization [58]. Recently, it has been practiced by considerable number of users such as structural engineers or engineering architects for the structural design of the actual buildings as well as the proposal for the architectural competitions. In this dissertation, the aim of the computational morphogenesis of fractalbased structures is to find a rational shape which defines an ideal internal stress state. It sets forth a kind of computational method to generate fractal-based structurally optimal structures, which is a synthesis of mathematics-aided design modeling, finite element analyses and structural optimization. Computational morphogenesis of fractal-based structures consists of three major operations – a. Generative modeling of fractals for designing structural shapes, b. Finite element analyses and c. Structural optimization as a process of form-finding in architecture. To perform the computational morphogenesis that includes all these three operations of a fractal-based structure at a time interactively on a same digital dais, a main tool box, named here as the ’Computational Morphogenesis Tool Box’ has been developed that is combination of (a) ‘Generative Tool Box’, (b) ‘FEM Analyses Tool Box’ and (c) ‘Optimization Tool Box’.

4.2 4.2.1

Generative Design and ‘Generative Tool Box’ The Concept of Generative Design

In the recent decades, the computer simulations have become a powerful tool for studying the structural performance of the building. It is possible to fully understand the complex interaction between design features, structural system, architectural function and occupant’s necessity by only resorting to simulation. Moreover using simulation tools lead to engage in design practice based on feedback loops between design decision and structural behavior including environmental impacts. Usually, these simulations are based on scenario-by-scenario scheme in which these scenarios are introduced and formatted by designers so that they can test and evaluate to make design decisions. Nevertheless, this process is repetitive and highly time-consuming. In this process, only a few scenarios can be evaluated among large possible choices. That is the reason, this method is not accurate because the designers are only able to pick some selected scenarios to simulate and evaluate. Therefore, the ideal method should be such that can generate and determine different design alternatives and evaluate them automatically.

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69

In contrast to scenario-by-scenario method, in the recent years a new approach has been arisen which are capable to evaluate different design alternatives based on previously defined goals, in which this method named as goal-oriented design [9]. The idea of goal-oriented design in the fields of architecture and civil engineering was borrowed from the field of Artificial intelligence. This idea of goal-oriented design is based on the automatic search and optimization technique, such as Genetic Algorithms (GA), which transfers the theory of natural evolution to the field of optimization, imitating in a virtual environment a fast forward analogy of the mechanism of evaluation in nature. Therefore, the Generative design can be expressed as an algorithmic or rulebased approach of design in which various possible design solutions can be generated. Rules in generative design include parameters and variables. There are numerous rules mainly based on mathematical formulations that govern the output shapes, and it is designer’s choice and aptitude to choose a specific rule for specific design objective and goal.

4.2.2

Generative Design based on IFS Rule

In this dissertation, the Iterated Function System (IFS) has been used as the rule for generative design of fractal-based structures. The IFS is one of the easiest and powerful mathematical rules that can produce fractal shapes using a simple and short algorithm, and that is the obvious reason behind choosing the IFS for this study.

4.2.3

IFS-based ‘Generative Tool Box’

In this research, two main strategic methods have been used for the computational morphogenesis of fractal-based structures. The first method is a generative method in which a ‘Generative Tool Box’ has been developed that serves to the second method which is an objective-oriented structural optimization and form-finding method. The key task of the ‘Generative Tool Box’ is to transform the mathematical functions of fractal geometry into digital figures and then to finite element parametric models for form finding process. Because, fractals are the results of self-similar or self-affine recursions, hence, the generative operation follows a recursive algorithm aided by the Iterated Function System (IFS). All the mathematical functions and their corresponding algorithms have been translated into a scripting code. Some variables will be defined which will control the size, shape and generation of a resulted fractal model, thus it will make it as a parametric model. The above fractal-based recursive codes are graphically expressed as geometric models on a digital interface. McNeel’s Rhinoceros 3D has been used as the

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Figure 4.1: IFS pseudocode

NURBS based modeling software and visual interface. These digital simulations of fractal shapes are sensitive to the initial geometric and procedural conditions which act as parameters and variables for generating various alternative shapes. The values and domains of the parameters and variables are set based on the design target. To obtain various alternative possible shapes out of which the optimal one will be searched, the digital model has been transformed into parametric model. For this transformation, a parametric software Grasshopper, which is a Rhinoceros 3D plug-in, has been used. In Grasshopper, there is a GhPython component which is a platform for inserting codes developed in python language having input options for parameters and variables such as iteration number, size factor,

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71

shape factor, etc. The fractal-based recursive codes will be developed inside GhPython by keeping the variables as shape controllers.

4.3

Finite element analyses (FEA): ‘FEA Tool Box’

In this dissertation, the objective of developing generative model is structuraloriented. For this reason, the digital model has to be passed through the structural evaluation process. Generative Tool Box produces the models which are fractals and they are theoretically infinite entity. In practical, only the finitely iterated models are valid for the real world applications. Therefore, the generative model is required to be transformed into the model which is valid for the structural analysis. For this operation, ’FEA Tool Box’ has been developed which transforms the finitely iterated model into finite element model for the structural calculation by a Finite Element Analysis solver. This scheme is processed through three steps. The first step is ’preprocessing’ step. In this step, the digital model is processed for the structural analyses. In this step, first, a finitely iterated model is discretized by exploding the continuous shapes into individual pieces as finite number of elements. Then, the duplicate and overlapped pieces are removed. Then, it is transformed into finite element model by assigning some structural properties, load conditions and mechanical constrains, thus prepare the model for finite element analyses. In the second step, FEM solver, analyze the model and give different results. Thus, the ’FEA Tool Box’ evaluates the structural property of the fractal model. But, for finding the best design, this evaluation has to be done on each alternative shapes that are produced by the ’Generative Tool Box’. Therefore, an interactive connection is required so that each variation of the model can be automatically evaluated by FEA Tool Box. Since, this whole method is done in a parametric environment of Grasshopper, an interactive tool is needed for the structural analysis so that it can give instant FEA feedback with the changing of the parameters and variable values of geometric model. There are many methods to interact one software with another by passing the data from one to another and getting instant feedbacks. In this study, Karamba , an instant finite element solver and a plug-in for Grasshopper, has been used because it gives the opportunity to visualize the action of interaction and feedback outcomes. GeometryGym has also been used for this purpose. Unlike Karamba which is an independent FEM solver embedded in the parametric environment of Grasshopper, the GeometryGym is a readymade connector which sends the preprocessing data to a commercial FEM solver such Oyasis GSA, Autodesk Robot or SAP, and receives instant analytical results

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back to the Grasshopper environment.

Figure 4.2: Schematic diagram representing the scheme of ’FEA Tool Box’. Top - Karamba, Bottom - GeometryGym

4.4 4.4.1

Optimal Form Finding: ‘Optimization Tool Box’ Structural Optimization

Computer-based structural optimization, an emerging area in the field of architecture, mainly in civil engineering from the last few decades, has facilitated architects and engineers to find the optimal shapes for design. It helps during the early design stage and even in the final stage of design process. The purpose beyond optimization in architectural or structural system is to declare the best state within the model under a set of constraints, implied or expressed.. In architecture, the structural optimization process has been used as a form-finding tool to pick most efficient structural form if its strength, stability and efficiency are the predefined objectives. This process starts with a model which is approximately fitted for its purpose. Then adjustments are made to the geometry and the form to move the overall performance closer to the optimized goal. Additionally, optimization introduced an architectural

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structure as a design problem, and to solve this problem the design process is defined as a solver. In the past, before introduction of computer aided design and simulation tools, many attempts had been made to define the optimal form for the geometry of structures. For instance, the use of dynamic analogue (physical) models to compute the optimal form and structural system was one of the earliest methods of structural form finding. In the history of structural optimization, Antonio Gaudi, Heinz Isler and Frei Otto are the best known for using the physical expression of mechanical behavior of hanging models as process of form finding for structures. In the recent years, the advancement of Computer-Aided Design (CAD), Computer-Aided Engineering (CAE), Computer-Aided Manufacturing (CAM) and optimization software allow designers to solve very complex problem. However, it is significant to have a good formulation to solve a design problem; it is therefore the designer has to consider the four elements of the formulation which are: Design Variables: The mathematical expression or any quantities that controls the shape of design for example, the height, width or length of the building. These variables are controlled by designer. Design Objectives: Designers set design objectives in order to achieve some conceptual goal. These objectives are represented as mathematical functions which designers try to maximize or optimize. Parameters: Parameters are quantities that affect the design objective but are considered fixed so they cannot be changed by the designers Constrains or Domains: The limitations or allowable domains of the design space in which characteristically happen by reason of finiteness of source or technologically limitation of some design variables. There is two different type of optimization – single-objective and multiobjective optimizations. Single-Objective Optimization: When the design problem optimizes a single objective, which mostly maximize or minimize. Therefore the optimization model is scalar. Multi-objective Optimization: When the problem has more than one objective to optimize, the optimization model will have vector objective instead

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of scalar. In the optimization process, not only definition of different variables, objectives, parameters and constraints are important, but also selecting a proper algorithm, like genetic algorithms, as searching engine has a significant role in defining the optimal design solutions. Searching algorithms are various, yet Genetic Algorithms is mostly used one. This is because they are used both as a form generation tools and as a optimization tool.

4.4.2

‘Optimization Tool Box’

In this step, an ’Optimization Tool Box’ has been developed which can act as a problem-responsive, context-sensitive or goal-oriented form-finding device. For this, first, a structural problem or design goal has been defined as a design objective that can possibly be solved or improved by shape modifications, and then find possible solutions underlying in the properties of fractal geometry. To achieve the design objective, the ’Generative Tool Box’ has been used as a tool to produce different variations of the initial model. The variables of fractal functions and their parameters are determined based on the structural demands and allowable feasibility. ’Generative Tool Box’ accordingly generates the flexible and parametric fractal-based structural model which are reactive to the variables defined as designed parameters. Fitness functions as an aspiration is prepared for the solution, and finally the parametric model is optimized to satisfy the fitness functions. To perform the optimization, Grasshopper0 s own genetic algorithm-based single-objective optimizer component Galapagos and multi-objective optimizer component Octopus have been used. The ’Optimization Tool Box’ results the final fractal-based optimal structures based on the initial conditions (’structural problem’ or ’design objective’), initial parameters and defined constrains.

Figure 4.3: Schematic diagram representing the scheme of computational morphogenesis of fractalbased structures using Tool Boxes

Chapter 4. Fractal-Based Computational Morphogenesis

4.5

75

Computational Morphogenesis of Fractal-Based Structures

Computational morphogenesis of a structure, as it is mentioned before, is always based on some contexts or demands. Therefore, before the morphogenesis, a target is defined or a structural problem is identified to solve. It can be a target of getting stiffest structure. Or, it can be a target of reducing weight by maintaining the strength. It can have multiple targets. However, the selection of targets is such that it can be possible to solve by applying the properties or geometric variables facilitated by fractal geometry. For example, if we make an intention to design a large span roof supported by least number of columns, then one can select the fractal’s property of self-similar branching inspired by nature’s tree structure where the branches are spread to cover large area above and standed on a single trunk. Similarly, as another example, if a target is to obtain high stiff but light-weight structural frame structure such as truss beams, then the idea of fractal’s self-similar repetition as hierarchical system can be chosen as the suitable solution. In this case, at each iteration, a solid beam member will be replaced by a scaled version of the whole truss beam’s replica; it can be continued for many repetitions. Thus, at each iteration, a replica truss replaces the solid beam member, and saves high amount of weight. This example is demonstrated in ‘Chapter 5’ in details. In this thesis, mainly, the maximum vertical displacement of the fractal-based structures has been considered as a suitable structural parameter to evaluate its local mechanical behavior. In some cases, the total strain energy has been considered as a fitness function that represents, on the contrary, a global measure of the mechanical behavior of structures. Domain is an important factor for the optimization. The choice of an appropriate domain for the design variables has been another fundamental task which is defined by the demands from architectural point of view. It is well known that the functioning of the morphogenetic process largely depends on the definition of the range of variability of each variable. Higher the numbers of variables and larger the range of domains, heavier is the process to find the solution. Therefore, for the obtaining the quick solutions, the choice of variables and their limits of parameters should be precisely defined. ’Generative Tool Box’ turns the mathematical function of a fractal into the finite element model as a new structure. Therefore, the initial function is the main controller which produces the basic fractal shape which is parametric. For the computational morphogenesis, first an affine transformation function is defined based on the suitable selection of the basic fractal shape. Then this function is made target-sensitive or fitness-oriented. For this, the variables of the functions are decided that are linked with the fitness target. These

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variables can be size, angle, iteration number, etc. For example, in the case of branching structures, the length of initial column, number of branches, scale factor (contractivity) of each branch, branching angles and iteration number are made variables. These variables control the whole shape of the branching columns. Based on the target, these variables can be changed to get the most efficient but wide span free form roof. This is elaborately demonstrated in the Chapter 6.

4.6

Fractal-Aided Computational Morphogenesis

Computational morphogenesis is seeded by the design variables. Commonly, the variables are associated with the Euclidean geometry. as a result, the adjustment of the variables’ values adjust the overall model offering limited diversity and variety maintaining the overall impression of the shape. In this dissertation, the term ’fractal-based computational morphogenesis’ is slightly different than the common computational morphogenesis process. Fractal-based morphogenesis implies the process that is controlled by some unique variables. One such unique variable in the fractal-based computational morphogenesis is the iteration number, i.e., the generation number. This variable affects drastically the overall shape of a model. Even the common geometric variables are also sensitive to this unique variable. A slight change of a geometric variable in Euclidean morphogenesis process possibly do not affect the final shape so much, but in Fractal-based morphogenesis process its impact is enormous and sometimes unpredictable with the changing of iteration variable. Besides, there is another unique variable that act distinctively in fractal-based morphogenesis process is the fractal dimension. This variable (fractal dimension) further alter and deform the main model and changes the texture and noise of its surface, and making it floating in between two successive integer dimensional shapes. As a result, this unique process seeded by two exceptional variables (iteration number and fractal dimension) generates huge variations of the original model, expands massive varieties of design possibilities, and thus allows the automatic optimizers to find the optimal and perhaps more accurate model from this immense collection of alternative possibilities. In this sense, fractal-based computational morphogenesis is truly unconventional where the factors of fractal geometry play the key roles as unique variables for providing the optimal shape. Hence, this fractal-based morphogenesis process can be considered as an improved form finding process by means of its unique variables such as iteration number, fractal dimension, etc.

Part III

Experiments

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Chapter 5

Experiment 1: Fractal-Based Hierarchical Truss ... Trusses can be made enormously lighter than cylindrical beams of identical strength. Benoit Mandelbrot, p131-132 [46]

5.1 5.1.1

Introduction Self-similar Repetition of Structural Members

The feature of self-similar repetition is not a new concept even in the field of construction. In the language of civil engineering, this type of configuration is called as hierarchical configuration, and the structures having this configuration are called hierarchical structures. Many of the lattice structures, such as trusses, are the examples of hierarchical structures. The fundamental reason for adopting hierarchical arrangement of members and their sub-members in a structure is to achieve high strength but light weight. The pioneer of using this principle of developing incredibly light weight but high stiff structure was Alexander Graham Bell. Using this concept of self-similar repetition as hierarchical assemblage, he developed several light-weight lattice structures, famously 79

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Chapter 5. Experiment 1: Fractal-Based Hierarchical Truss

the tetrahedral. Few of them also resemble with the Sierpinski triangle. Amazingly, he attempted to make his hierarchical structures so light so that he could literally fly the structure as a kite. This experiment was a part of his investigation about the aerodynamic lightness if apply in man-made objects. This experiment was one of the earliest attempts to develop flight. In this sense, he was probably the foremost engineer who applied the basic concept of self-similar repetitions in the field of engineering to obtain ultra lightness.

Figure 5.1: Figure: (Top left) – One of Alexander Graham Bell’s tetra kites made of silk and wood; Bottom left - Alexander Graham Bell’s ’Siamese Twin’ kites, from Alexander Graham Bell, ’Aerial Locomotion, With a Few Notes of Progress in the Construction of the Aerodrome’, National Geographic Magazine (Jan., 1907), 1-33; (Top right) - Bell’s ’Cygnet II’, February 25, 1909. Bulletins, from January 4, 1909 to April 12, 1909, Alexander Graham Bell Family Papers at the Library of Congress, 1862-1939, Manuscript Division, Library of Congress; (Bottom right) - Alexander Graham Bell’s Tower, from ’Dr. Bell’s Tetrahedral Tower,’ National Geographic Magazine (Oct., 1907), 672-675.

After Graham Bell’s brilliant innovations, one of the finest and ceremonial examples of such structures is the Eiffel Tower, whose every member is composed of sub-members and each sub-member is further composed of its own sub-members and so on, maintianing the same configuration. Thus, by its hierarchical features, it shows a character of recursive self-similar repetitions, which is a fundamental property of fractal geometry. In this context, with reference to the design of the Eiffel tower, Benoit Mandelbrot [46] claimed in his seminal

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81

book The Fractal Geometry of Nature, My claim is that (well before Koch, Peano, and Sierpinski), the tower that Gustave Eiffel built in Paris deliberately incorporates the idea of a fractal curve full of branch points. . . . However, the A’s and the tower are not made up of solid beams, but of colossal trusses. A truss is a rigid assemblage of interconnected submembers, which one cannot deform without deforming at least one submember. Trusses can be made enormously lighter than cylindrical beams of identical strength. And Eiffel knew that trusses whose ’members’ are themselves subtrusses are even lighter.. – [46] This cliam clarifies the strength lies not in mass, but in cleverly designed geometric shapes and in the case of Eiffel tower, it lies in branching points. Fractal geomtery can be a tool that aid to develop such branching point in many hierarchical scales. Later, this example of self-similar repetition of hierarchical configuration has been adopted as a fractal configuration after the development of the concept of fractal geometry in the 1970s. With this new concept, architects and engineers have got a scope to refine the shape of hierarchical trusses by following the grammar of fractal geometry to obtain a perfect self-similar structure. In the concept of hierarchical structure, the range of scale is limited; while, in the concept of fractal geometry, the range of scale is theoretically infinite, and in the finite level, the scale can range from smallest of nano scale to largest of mega scale. This high range of the scaling factor of fractal geometry provides an opportunity and liberty to engineers and scientists to develop hierarchical structures at its tiniest (magnifying) level which will allow further savings of weight without compromising the strength.

5.1.2

Recent Development: Hierarchical Fractal Structures

Based on the concept of limitless scaling factor of fractal geometry and the idea of developing hierarchical features at smallest scale, some scientists and engineers have recently attempted to develop some materials and structures that are extremely light but highly stiff at small, micro and nano levels. In 2012, Robert Farr and his team have found a remarkable result after experimenting on a small-scale fractal-based hierarchical truss column which was developed by replacing a solid cylinder, claimed that the application of fractal concept can produce the ’lightest and strongest’ structure (Figure 5.3a) [20]. A step further, in the field of material science, Professor Julia Greer and her team have recently developed a ’world’s lightest’ truss, named as ’nano-lattice’ using the concept of

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Figure 5.2: Hierarchical truss composition in Eiffel Tower.

fractal geometry and hierarchical configuration (Figure 5.3b) [52]. According to their experimental results, ’nano-lattice’ is as light as feather yet features outrageous mechanical strength. They hope that this fractal nanostructure might one day be used in structural engineering. In this context, similar investigation

Chapter 5. Experiment 1: Fractal-Based Hierarchical Truss

83

has been carried out by Professor Kenneth Cheung and Professor Gershenfeld at the MIT. They have developed a system that can construct a large structure by assembling small ultra-light tiniest pieces using the principle of fractal geometry (Figure 5.3c) [12]. They claim that, in the field of construction, its application can offer a large space-frame structure that can be exceptionally light in weight and highly strong.

Figure 5.3: (a) Fractal truss column [20]; (b) Nano-truss based on hierarchical system of arrangement [52]; (c) Structural lattice assembled by ultra-light tiniest pieces [12].

5.2

Geometric Modeling

In this chapter, the concept of fractal-based hierarchical structure has been applied for developing a truss beam as an experiment to analyze its structural behavior with the changing of scaling factors. The configuration of this truss beam is inspired by the truss model developed at micro scale by Robert Farr and his team [20]. Here, the beam is considered as a planar truss beam.

5.2.1

Mathematical Expression and IFS

Hierarchical trusses are the assemblage of the self-similar members that are further assembled by their own sub-members in a similar fashion. In the case hierarchical truss beam, each beam member is replaced by self-similar truss whose sub-members are further replaced by its corresponding truss beams. This geometric shape follows a rule that can be constructed by a generative process such as the Iterated Function System. Here, the Barnsley’s contraction mapping method has been applied to produce such beam. In the beginning, a single line has been taken as a beam B0 , and initial shape. In the first iteration, B0 is transformed into a truss beam B1 which is an assemblage of 25 self-similar copies (b1 , b2 , b3 , . . . ., b25 ) of B0 . There are 4 groups of different self-similar copies that are produced by the contractions of λ1 , λ2 , λ3 andλ4 . In the next steps, this process of repetition is continued using the same transformation rules. Thus,

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it results a fractal figure which is an attractor. It is an intersection set of all the identical subsets of B0 , as shown in F igure 5.5. The attractor B can be expressed as, ∞ \

B=

Bi

(5.1)

i=1

where, B1 = b1 ∪ b2 ∪ b3 ..... ∪ b25

(5.2)

B1 = f1 (B0 ) ∪ f2 (B0 ) ∪ f3 (B0 )........ ∪ f25 (B0 )

(5.3)

and at nth iteration, Bn = f1 (Bn−1 ) ∪ f2 (Bn−1 )........ ∪ f25 (Bn−1 )

(5.4)

In short, Bn =

25 [

fi (Bn−1 )

(5.5)

i=1

If B1 , B2 , . . . .., Bn , . . . are contraction sets of B0 , that are contracted by using the contractivity factor λi and transformed by using an affine transformation function fi , such that, B0 ⊃ B1 ⊃ B2 ⊃ B3 ⊃ ..... ⊃ Bn ⊃ .....

(5.6)

then, they form a perfect self-similar fractal set. In the case of constructing the fractal hierarchical truss beam, the following function of IFS is applied that consists 25 different affine transformations (f1 , f2 , f3 , ......, f25 ) of the main triangle.      cos θ − sin θ x δx fi = λ i + sinθ cos θ y δy i = 1 to 25(5.7) In this geometric configuration of truss beam, there are four groups of selfsimilar copies based on 4 different contractions, which are, 1 7

(5.8)

1 1 · 7 cosθ

(5.9)

λ1 = λ2 =

Chapter 5. Experiment 1: Fractal-Based Hierarchical Truss

λ3 = r λ4 =

2 · sinθ 7

2 1 1 p ( · tanθ)2 + ( )2 = · 4 · tan2 θ + 1 7 7 7

85

(5.10)

(5.11)

Figure 5.4: IFS variables and their values.

The above IFS function has been transformed into a scripting code for making a parametric digital model of the structure where θ, L and the iteration

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number n are made variables. The code results a graphic model shown in the F igure 5.5.

Figure 5.5: Convergent seqquence of the fractal-based truss beam.

Chapter 5. Experiment 1: Fractal-Based Hierarchical Truss

5.2.2

87

Hausdorff Dimension Calculation

Based on the Barnsley’s contraction mapping theory, the Hausdorff dimension of a fractal, is linked to the contractivity factors by the following relation: 25 X

λ2 i = 1

(5.12)

i=1

where λi is a contractivity factor of transformation fi and k is the number of transformations [2]. Therefore, in the case of fractal-based hierarchical truss beam: 10λ1 D + 4λ2 D + 6λ3 D + 5λ4 D = 1

109

(5.13)

D  p D  D   D 1 1 1 2 1 tanθ +5 4tan2 θ + 1 = 1 (5.14) +4 +6 7 7 cosθ 7 7

The aboveequation 5.14 shows that the Hausdorff dimension of the shape of fractal-based hierarchical truss beam is the function of angle θ. The value of Hausdorff dimension D can be obtained by the Newton-Raphson method. In the experimental model, the angle is taken 20o , and therefore the calculated Hausdorff dimension of the shape of the beam truss is approximately 1.565.

5.3 5.3.1

Structural Analyses Structural Modeling and Preprocessing

The above digital model is transformed into a finite element model for the FEM analysis by taking the following considerations: • The length of the truss beam is 10 meter. • Only the finitely iterated fractals are applicable in real world constructions. Therefore, here from initial model to 2nd iterated model has been taken. The high iterated model has not been taken to avoid heavy calculation by FEM solver that would take painful long time. • The overlapping lines are removed, and the remaining lines are considered as beam elements.

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• Each beam element is hollow steel tube having the circular cross-section of 200 cm diameter and 10 mm thickness for initial model, 25 cm diameter and 8 mm thickness for the 1st iterated model, and 5 cm diameter and 4 mm thickness for the 2nd iterated model. • The overlapping points are removed, and the remaining points are considered as structural nodes. • Each node is a welded joint connecting the corresponding beams. • The overlapping of beams near the joints is ignored. • One end of the main structure is horizontally as well as vertically restrained, while the other end has been vertically retrained only. • Self-weight and the vertically applied distributed point loads of 10 KN are considered for the analyses.

5.3.2

Finite Element Analysis and Results

After preprocessing and applying the loads, the FEM solver calculates the maximum displacements of the structures, shown in ’Figure 5.6’. Here, the allowable displacement has been set as = beam length/500 = 10000 mm/500 = 20 mm. After the finite element analyses, it has been found that the loss of mass of the solid beam B0 , when transforming into the first iterated truss beam B1 , does not affect significantly to the overall strength of beam structure. This fact is the same when the first iterated truss beam B1 transforms into the second iterated beam B2 . Until the allowable maximum deformation under a certain practical load, the beam can be iterated for several generations, thus reduce as much mas as possible. The analytical results, thus, indicate the advantage of using fractal concept for reducing mass as much possible, as hypothesized by Mandelbrot in the beginning [46] and studied some examples by researchers from applied sciences [20] [52]. In this case, the number of iteration plays the main role to tactically reduce the mass from the main solid beam. However, apart from this variable, there are other geometric variables too, especially in this demonstration, which can further help to reduce the mass by maintaining enough strength. In the following section, the optimal form of this hierarchical truss has been discussed.

Chapter 5. Experiment 1: Fractal-Based Hierarchical Truss

Figure 5.6: Finite Element Aanalyses of different iterated models of the fractal truss beam.

89

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5.4

Chapter 5. Experiment 1: Fractal-Based Hierarchical Truss

Structural Optimization

The fractal-based hierarchical truss, designed in this section as a case study for experiment, has been made parametric where iteration number is a key variable. However, apart from iteration number, the cross-sectional sizes of the members and sub-members are a significant factor for obtaining an optimal design of the truss beam. From the above FEM analyses, it is observed that the 2nd iterated model is the lightest among initial and 1st iterated truss beam and yet its maximum displacement is about 6 mm which is within the allowable maximum deformation of 20 mm. This means, the model with further iterations is possible that will be lighter than 2nd iterated model within the allowable deformation of 20 mm. For avoiding the high calculation and for the limited memory and speed of the personal computer, this trial has not been attempted. Instead, the crosssectional size can be optimized in the 2nd iterated model that can deform within the allowable limit of 20 mm, and save further weight of the structure. For doing the optimization operation, only the second iterated model is chosen and preprocessed for the FEM analyses using ’preprocessing tool’, where the angle and cross-sectional diameter as the design variable is made only variable. The domain of the angle is set from 30o to 60o to avoid the large overlappings of the sub-members and the domain of the cross-sectional diameter has been set from 1.0 cm to 5.0 cm with the constant thickness of 4.0 mm. Design Variables

Domain

Angle, θ Cross-section Diameter

30o to 60o 1.0 cm to 10.0 cm

The load condition remains the same. ’FEM Solver Tool’ Karamba, calculates the preprocessed finite element model of the structure, and gives instant feedback with the changing of the value of diameter. It works interactively in the Grasshopper’s parametric environment and produces the digital model in the Rhinoceros3D. This tool box is connected to the ’Optimization Tool Box’, Galapagos, which helps to find the optimal shape of the structure. In this case, the allowable maximum deformation of 20 mm has been defined as the fitness function. The optimization process has been stopped at 100th generation, and the optimal truss beam design has been found shown in F igure 5.8 whose end angle is 39.2o . The optimized cross-sectional diameter of the truss beam has been found 2.1 cm. The weight of this optimal beam truss is 225 kg and the maximum deformation touches to the allowable deformation of 20 mm.

Chapter 5. Experiment 1: Fractal-Based Hierarchical Truss

91

Figure 5.7: Schematic diagram of the computational searching using Galapagos.

Figure 5.8: Schematic diagram of the computational searching using Galapagos.

5.5

Final Remarks

The computational morphogenesis and optimization of the fractal-based truss beam shows that the high iterated design is structurally efficient and optimal. This experiment confirms Mandelbrot’s claim about merit of fractal geometry for developing light-weight and high-strength lattice structures. It also ensures and gives a confidence to explore further applicability in designing efficient and innovative structural shapes that can offer novel and inventive design possibilities in architecture. In this optimization process, the cross-sectional sizes of all the members are uniform. However, the ideal process of optimization to find the optimal size of the cross-section of each member is based on its minimum

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capacity to carry internal distributing force within the allowable limit. In this case, the process will result different sizes of all the members and they will not be uniform any more. This approach of optimization has been adopted in the Chapter 8.

Chapter 6

Experiment 2: Branching Structures 6.1

Introduction

Tree, nature’s one of the common examples of self-similar fractals, has been used in designing branching supports even before the development of the concept of fractal geometry.1 Gaudi was one of those forefront architects who understood the mechanical advantage of self-similar branching forms of trees as a structural support to carry a wide canopy, thus providing a large free space underside (Figure 6.1a). Later, in the mid 20th century, Frei Otto, used the same principle but by sophisticating the form of tree-structure experimenting with some hanging models (Figure 6.1b). Even, at the time of Otto, the concept of well defined fractal geometry did not exist mathematically. However, after the 1970s, a list of branching structures began to appear, some of which were designed by intentional application of fractal geometry idea, such as the tree columns in Stuttgart Airport Terminal in Stuttgart (Figure 6.1c) and The Tote in Mumbai (Figure 6.1d), and some of which were designed by straight-forwardly replicating the abstract form of tree structures. Intentional application of fractal geometry allows architects to computationally design a parametric model of branching structure whose shape can be changeable by controlling the design variables, such as branching angles, branching numbers and scaling factors. This computational model of fractal-based tree is helpful for the optimization process that 1 A part of this chapter has been published as a journal paper in the Frontiers of Architectural Research in 2014: Iasef Md Rian and Mario Sassone, ‘Tree-Inspired Dendriforms and Fractal-Like Branching Structures in Architecture: A Brief Historical Overview’, Frontiers of Architectural Research, 2014, 3(3), 298-323.

93

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Chapter 6. Experiment 2: Branching Structures

can result the optimized shape of branching structure.

Figure 6.1: (a) Tree Columns in the model of Sagrada Familia, Spain by Antonio Gaudi (b) Frei Otts hanging models of branching systems (Nerdinger, 2005, p67), (c) Tree column in Stuttgart Airport terminal, and (d) The Tote, Mumbai.

Trees and forests have inspired architects and engineers to design structures in architecture, and there is long chronological history about the structures of branching columns as tree-like structures supporting roofs inspired by the structural appearance of natural trees [63]. As a case study, in this section, a branching structure has been designed to support a wide-span roof so that we can obtain a large-span indoor space, similar as the canopy structure in the Stuttgart Airport Terminal. In this experiment, the structural efficiency of such structures will be analyzed, and then a computational search will be performed to find an optimal form of the structure.

6.2

Geometric Modeling

There are different methods to simulate branching trees. Among them, the Lindermayer System (L-System) is popular in the field of botany where the simulation of natural trees and plants are the main objectives. However, here, the Barnsley’s Iterated Function System, which is based on the contraction mapping method, has been used to generate an abstract form of a tree structure targeting for the realization and practical application in the field of architecture and construction.

6.2.1

Mathematical Expression

Branching tree is an example of a perfectly self-similar fractal which is a union of self-similar branching sets that lie in the Metric space, more precisely, in the Hausdorff Metric space. Based on the contraction mapping theory introduced by M. F. Barnsley [2], a fractal tree is an attractor when it is the resulting figure at the limit state obtained from a set of affine transformations fi (i = 1 to k),

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Chapter 6. Experiment 2: Branching Structures

applied infinite times. As an example, a fractal tree with three transformations f (k = 3) has been constructed in the two-dimensional space shown in ‘Figure 6.2’

Figure 6.2: Fractal as a union of perfectly self-similar subsets after affine transformation.

As a perfectly self-similar fractal, here in example, the fractal tree is a unique non-empty compact set. The resulting figure has been obtained by the union of all self-similar subsets, and it can be represented as, ∆=

∞ [

∆i

(6.1)

i=1

i.e., ∆ = ∆1 ∪ ∆2 ∪ ∆3 ........ ∪ ∆n ∪ ∆0

(6.2)

∆1 = f1 (∆0 ) ∪ f2 (∆0 ) ∪ f3 (∆0 ) ∪ ∆0

(6.3)

where,

and at nth iteration, ∆n = f1 (∆n−1 ) ∪ f2 (∆n−1 ) ∪ f3 (∆n−1 )∆0

(6.4)

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Chapter 6. Experiment 2: Branching Structures

If ∆1 , ∆2 , . . . .., ∆n , . . . are contraction sets of ∆0 , that are contracted by using the contractivity factor λi and transformed by using an affine transformation function fi , such that, ∆0 ⊂ ∆1 ⊂ ∆2 ⊂ ∆3 ⊂ ..... ⊂ ∆n ⊂ .....

(6.5)

then, they form a perfect self-similar fractal set. This IFS method is the most common method to produce branching tree structure. One can notice that, in this IFS process, we always will get a fractal tree whose terminal points are not in a line but form an approximate curved outline. That means, the end points in these fractal trees, if generated in threedimensional space, do not lie in a plane. Therefore, this type of tree structures is not applicable for making canopy structure which has flat slab. For making flat slab canopy structures, we require the branching tree structures whose endpoints of the terminal branches will lie on a flat planar space. In this regard, the above mathematical formulation of the self-similar union as well as the IFS process has to be modified by changing the affine transformations strategy. In this chapter, a modified method has been proposed that will generate self-similar fractal branching with its terminal ends lying on a plane. This proposed structure is an experiment of fractal-based computational morphogenesis for analyzing its functional and structural potencies and finding an efficient shape as a case study of such canopy structures. In this method, unlike the ‘Figure 6.2’, it does not repeat the trunk after contractions and affine transformations. Instead, it repeats its branches at each iteration, and the first branch is generated from the trunk. It means, it follows two steps in the beginning. First step is the creation of the first group of branches B0 from the trunk T . In the second, this group of branches B0 is repeated using a new set of affine transformations f1 , f2 , f3 and f4 . Using the IFS process, the second step is continued iteratively for the next steps. The first step produces the first branch B0 which can be represented as, B0 = b1 ∪ b2 ∪ b3 ∪ b4 = f1 (T ) ∪ f2 (T ) ∪ f3 (T ) ∪ f4 (T )

(6.6)

And thus the first tree is born which is, ∆0 = B 0 ∪ T

(6.7)

It follows the following affine transformation,  cos α fi = λi  sinα 0

− sin α cos α 0

 0 cos β 0  0 1 −sinβ

    0 sin β x δx 1 0   y  +  δy  0 cos β z δz

(6.8)

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Figure 6.3: First affine transformation as the first step transforming trunk T to the first tree ∆0 .

Figure 6.4: IFS variables and their values.

Starting with this first tree ∆0 , the IFS produces the proposed fractal tree which is an attractor at nth iteration, and mathematically expressed as, ∆=

n [ i

(6.9)

i=1

i.e., ∆ = ∆1 ∪ ∆2 ∪ ∆3 ........ ∪ ∆n ∪ ∆0

(6.10)

∆1 = f1 (B0 ) ∪ f2 (B0 ) ∪ f3 (B0 ) ∪ f4 (B0 ) ∪ T

(6.11)

where,

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∆2 = f1 (B1 ) ∪ f2 (B1 ) ∪ f3 (B1 ) ∪ f4 (B1 ) ∪ T

(6.12)

and at nth iteration, ∆n = f1 (Bn−1 ) ∪ f2 (Bn−1 ) ∪ f3 (Bn−1 ) ∪ f4 (Bn−1 )∆0

(6.13)

Here, B1 , B2 , . . . .., Bn , . . . are contraction sets of B0 , that are scaled by using the contractivity factor λi and transformed by using an affine transformation function fi , such that, ∆0 ⊂ ∆1 ⊂ ∆2 ⊂ ∆3 ⊂ ..... ⊂ ∆n ⊂ .....

(6.14)

then, they form a perfect self-similar fractal set.

Figure 6.5: Convergent sequence of fractal branching tree in three-dimensional space.

6.2.2

IFS Function and Coding

In this operation, the affine transformations of the branches do not include rotations, but only contractions and displacements. Therefore, the IFS function can be represented as follows:     δx x  fi = λi y +  δy  1 to 4. (6.15)   z δz In the above IFS function, there are several variables which control the whole appearance of the branching structure. In the ‘Figure 6.4’, the trunk height is set 6 meters and its IFS variables are set as follows, Here in F igure 6.6 the variables such as the contractivity (λi ), horizontal angles (α1 ) and the vertical angles (β1 ) are the same as in ‘Figure 6.4’. Li

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Figure 6.6: Inputs for the affine transformations for the second and remaining steps.

is the length of each member of the first branching group B0 , i.e., Li = λi .H, where H is the initial height of the trunk which is 6.0 meter for the proposed tree structure. Therefore, for generating the figure of the branching tree by transforming the above mathematical functions, we have the following input variables:

Figure 6.7: Design Variables for generating branching column

Using the above inputs and the IFS functions, an IFS code has been developed as follows:

6.2.3

Hausdorff Dimension Calculation

According to Barnesley’s method, the Hausdorff dimension D of the fractal branching is depending on the contractivity factors based on the following relation, 4 X

λ2 i = 1

(6.16)

i=1

where, λi is a contractivity factor of transformation fi of the branch Bi . For proposed branching structure, λ1 = λ2 = λ3 = λ4 = 0.6

(6.17)

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Figure 6.8: IFS pseudo code for generating branching column.

Using Newton’s method, we get, D ≈ 2.714

(6.18)

Chapter 6. Experiment 2: Branching Structures

6.3

101

Architectural Design

The above generative fractal model is infinite mathematically. But, for architectural applications, this model should be finite, and for this reason, only finitely iterated model is considered. The stability of a single branching structure is highly sensitive to asymmetric load. Hence, four units of branching structures are composed to make a canopy structure in this experiment. Here, the following design considerations of are taken into account for the real-world architectural applications: • Only the 4th iterated model of each branching structure has been considered. • The branching structures are made of tubular steel. • The cross-sectional sizes: Radius 16 cm and thickness 8 mm for the trunk, radius 12 cm and thickness 6 mm for the first order branches, radius 8 cm and thickness 4 mm for the second order branches, and the radius 6 cm and thickness 4 mm for the terminal branches. • The bottom of the trunk of each branching structures is fixed to ground. • Each branching structure is attached with a flat canopy at the top and connected by the tip points of each end branches. The roof is a single layer planar grid structure made of steel tubes having 6 cm diameter of cross-section. This planar grid roof is useful for fixing the end branches and prevents them from bending. • The four branching units are arranged such a way so that there canopy roofs are attached to each other. • At the top, glass panels are fixed on the roof.

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Figure 6.9: Architectural model of the propsed branching canopy structure

6.4

Structural Analyses

Branching structures exhibit a particularly close relationship between the forces flows and their shapes, both in their overall appearance and in the nature of the structure itself. It is a functional combination between the roof construction and supporting structures. The advantage of the tree-like branching system is to have short distances from the loading points to the supports. Branching structures are usually referred to as tree-like structures. However, their action cannot be compared with that of a natural tree. While the branches of a tree are under bending stress, bending forces are systematically avoided in technically constructed tree-like structures [57]. The inner structure of the tree-like columns represents a type of framework that is unique in the construction industry. It is not a truss with a triangular structure, which would brace the structure, allows articulation joints between the truss elements, and prevents bending even under alternating loads. In the tree-like column, therefore, the individual elements must be rigidly connected at the joints. A tree-like column is particularly well suited for only the one main load scheme for which it is optimized. All other loading conditions will cause bending stresses within the structure. However, in this experiment, the branching structure is connected with the planar roof grid

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such a way so that its terminal branches are prevented from bending.

6.4.1

Finite Element Modeling and Preprocessing

For the finite element analyses, the geometric model of the overall branching structure has been transformed into a finite element model by the following way: • All the overlapping line segments are removed. • Each remaining line is a beam element. • All the overlapping points are discarded. • Each remaining point is a structural node. • All the elements are connected as welded joints at the nodes. • The bottom of each trunk is vertically and horizontally restrained at the ground. • Only the canopy weight is considered as applied load. For this, the nodal loads(100 N for each node) at all terminal points are considered.

6.4.2

Finite Element Analyses

The finite element analyses is targeted for checking the maximum nodal displacements, global buckling effect and axial stresses. The ’FEA Tool Box’ has analyzed the structure and given the results shown in ‘Figure 6.10’.

6.4.3

Results and Discussion

After the analyses, it is found that the maximum displacement of the structure along vertical axis is 25 mm which is allowable and safe (Figure 6.10, Top). Its stress output shows that most of the branches are experiencing tension, and bending stress (Figure 6.10, Middle). Its buckling calculations results the buckling load factor of the structure is 14.94 which fair enough to to carry almost 14 time additional load than the load is presently applied which is 100N at each node of terminal branches (Figure 6.10, Bottom). These results confirm the structural feasibility of the fractal-based branching canopy structure if applied in the practically. However, it can be quickly realized that the structure has numerous design possibilities. The branching angles,there number of tiers, scaling factor, the cross section of the memebers, all of these influence the overall design of the structure. Hence, it is quite impossible to find the optimal design if check all variations separately. Therefore, an automatic system is needed to

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Figure 6.10: FEA Resultsunder the defined load case. Top - Deformation under, Middle - Axial stresses, Bottom - Global Buckling effect.

find the optimal structure with regard to the design and structural objectives, which are called fitness functions. In the following section, a computational searching process has been performed to find the most efficient design of the branching canopy structure using the ‘Optimization Tool Box’.

Chapter 6. Experiment 2: Branching Structures

6.5 6.5.1

105

Form Finding and Structural Optimization Finding Optimal Form

The above generative model of the branching structure has been transformed into a parametric model so that it can provide a set of different design variations. The target was to find the optimal shape. To do this, some of the key geometric inputs are considered as the key design variables and set some reasonable domains:

Figure 6.11: Design variables and their domains

The parametric model of the branching canopy structure has been changed to different forms with the changing of the key design variables, and their corresponding maximum displacements have been recorded to analyze their structural behavior. For the structural analyses, an interactive FEM solver Karamba has been used that promptly gives feedbacks with the changing the value of design variables. ‘Figure 6.12’ shows that how the changing of vertical angle impacts on the maximum displacement and the canopy area of the structure. Contractivity also influences on the maximum displacements and the canopy area of the structure shown in ‘Figure 6.13’.

Figure 6.12: Left - Graph representing the relation between vertical angle and maximum displacements; Right - Graph representing the vertical angle and canopy area. (Contractivity is constant, which is 0.6)

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Figure 6.13: Left - Graph representing the relation between contractivity and maximum displacements; Right - Graph representing the contractivity and canopy area. (Vertical angle is constant, which is 60o )

From the above relationships, we noticed the separate impacts of the design variables on the canopy structure. But, if we consider both the design variables together and if we fix the maximum allowable displacement and the minimum covering areas, then we can find a suitable shape out of the multiple variations of the structure. However, it is not easy to find the optimal shape from the above resulted graphs. Therefore, a computational searching method is required to find the optimal design of the proposed canopy structure.

6.5.2

Optimization of the Structural Form

To automatically find the most optimal form for these targets, a multi-objective optimization method is required to be performed. In this case, the active variables are verical angle, contractivity and the iteration number. The target is to minimize the displacement and to maximize the anopy area. For the optimization, the domains of the variables of the geometric model have been set (shown in ‘Figure 6.14’) such a way so that it can be reasonable in terms of allowable maximum height and functional canopy area of the structure. For the optimization process, the cross-sectional sizes of all the memebers have been kept as they were. They will be further optimized later in the second stage of optimization process. The allowable deformation has been set at 20 mm. Here, the optimization process has been performed in a single task by making a single function that includes both the objectives as follows, f = maximize(A + δmax ); so that, δmax ≤ 20(6.19) where, A is the total roof area and δmax is the maximum displacement.

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Table 6.1: Design variables of the branching structure.

Branching structure Design variables Domains Contractivity (scale), λ 0.4 to 0.8 Vertical angle, β 30o to 70o Iteration number, n 2 to 4

After 100th generations, the following optimal shape has been obtained whose vertical angle is 53.9o , and the contractivity factor is 0.57. The maximum displacement of this optimal model is 15 mm, and its total canopy area is 231.35 sqmt and total weight is 9080.59 kg.

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Figure 6.14: Structural form finding of the branching canopy structure.

Part IV

Applications

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Chapter 7

Application 1: Fractal-Based Generative Design of the ‘Monalisa Pavilion’ 7.1

Introduction

This chapter applies the concept of fractal geometry as a generative design tool to design a pavilion structure as the first application of this research, mainly focusing on creating a new type of architectural form and not a structural form at this time.1 Therefore, no structural analyses have been performed to evaluate its structural behavior and to find structure-based optimal design. Here, finding suitable form is completely based on the design objective, functional needs as well as on the sense of design aptitude of designers and the agreement among them who involve in this early stage of design process. The intention is to demonstrate the ability of the fractal-based generative design tool, which is ‘Generative Tool Box’ developed in the ‘Chapter 4’, to produce a new type of architectural designs, and translate a design idea into a visual form and then to a architectural model. Besides, this application is a also intended to 1 The research of this Chapter has been presented in the proceedingS of the International Scientific Conference on Hardwood Processing: Guido Callegari, Iasef Md Rian, Antonio Spinelli, Gaetano Castro and Roberto Zanuttini. ‘Monalisa wood Pavillon: optimization and parametric design by using poplar plywood engineered components’. ISCHP 2013, Florence, Italy. October 2013.

111

112 Chapter 7. Application 1: Fractal-Based Generative Design of the ‘Monalisa Pavilion’ see how mathematics based computer supported generative design technique or such machine like ’Generative Tool’ catalyzes the design process without compromising architectural quality, instead enhancing it and expanding the possibilities of innovations. For the application, an excellent opportunity came during a summer workshop project named ‘Monalisa Pavilion’ held in 2012 at the Politecnico di Torino in Italy led by Guido Callegary and sponsored by the regional companies who manufacture poplar plywoods. The objective of this workshop was to produce an architectural showpiece which had to be developed using advanced digital-aided design and fabrication technologies, and make the pavilion design contextual to the contemporary trend of architectural complexity. The commercial aim of the workshop was to produce an architectural showpiece that will serve to create a manifesto on poplar and to explore the uses of this wood in architecture.

7.1.1

Design Objective: Recreating a Forest

The main objective of the workshop was to promote poplar plywood and to exhibit its versatility in an innovative way for creative purposes in the industrial sector. The intention was to showcase the unique quality of poplar plywood, which is smooth in color, lightweight, flexible and easy to cut. To achieve this challenging task, design ideas were communicated through sketches. The concept development exercise started with a thread, ‘cannot an architectural piece be a visual narration of a story of the production of construction material itself, if its intention is to emphasize the material’s usefulness and versatility?’ Poplar sprouts from its seed, and then gradually grows into a young plant and finally a full-grown woody tree. Poplar trees altogether exist in a plantation like a family, provide shading to others and finally they become useful for making plywood needed for manufacturing furniture, interiors and architectural objects. The growth and age of a poplar tree is recognized by the increasing of its height, trunk radius and number of its branches. All these elements, which mark poplar’s growth, were symbolically taken as design vocabularies to narrate the design concept. Therefore, the idea was to recreate a poplar plantation as a human designed pavilion that symbolizes the usefulness of poplar as a shelter and as furnitures for human needs. However, this whole conceptual idea had to be transformed into a design model.

7.1.2

Geometric Modeling

To narrate the story of poplar’s growth and usefulness, and to recreate forest, we needed a geometry which underlies in the complex appearance in nature. The shape of a tree and forest can not be represented by Euclidean geometry or any other conventional geometries. Fractal geometry, in this context, is the

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Figure 7.1: Replication of poplar tree growth and its representation as architectural design elements.

perfect geometric system that can help to simulate forest. Natural forest is full of richness and detailed from large scale to tiny insect scale. So, for the simplification, an abstract form of forest was decided to be modeled which can satisfay the architectural objective and functional needs. The first task was to decide the overall shape and plan of the pavilion. Then the second task was to model trees of different ages and place them creatively so that it can offer a new appearance. In ‘Chapter 6’, the generation of tree structure was started with a straight line as the initial shape that produced straight branches later, as a result the ‘Generative Tool Box’ resulted a tree structure which looks like man-made nature-inspired structure, and did not display natural look. In this application, because of the demand of design conceptual, we had to model trees which could display natural appearance. That is the reason, we started the generative modeling process with a curve T0 as the original geometric seed which was symbolically represents the budding of poplar tree as a baby plant. Later this curve becomes larger and represents as a trunk when the tree becomes adult. In the next step, i.e., symbolically at the next age of growth, two branches b1 and b2 were developed that are scaled replicas of the first curve T0 . As age grows, the iteration number increases and, in mathematical language, it converges towards an attractor, which is in pavilion design, appears as a full tree T . Therefore, a single fully-grown tree T is an attractor which is a union of all self-affine subsets,

114 Chapter 7. Application 1: Fractal-Based Generative Design of the ‘Monalisa Pavilion’ that are branches, obtained after anisotropic affine transformation to get natural appearance. As discussed in the ‘Chapter 3’, self-affine transformation is a process which is differ from the self-similar transformation process be means of different scaling along x and y axes. It means, in this case, the trunk T0 is not exactly repeated in branches, instead, the trunk is deformed a little by controlling the mid point of the curve and then scaled to make first branch using lambda1 and the second branch using different scale lambda2 . For different affine transformations fi n (i is number different transformations at each iteration, and n is the number iterations), the resulting figure that represent an adult tree, is a fractal in the form of an attractor T , can be mathematically expressed as, T =

∞ [ i

(7.1)

i=1

i.e., T = T1 ∪ T2 ∪ T........ ∪ Tn ∪ T0

(7.2)

T1 = b1 ∪ b2

(7.3)

T1 = f1 1 (T0 ) ∪ f2 2 (T0 )

(7.4)

Tn = f1 n (Tn−1 ) ∪ f2 n (Tn−1 )

(7.5)

where,

i.e.,

and at nth iteration,

The above mathematical processed was transformed into scripting codes for generating the digital model of one tree unit. Making tree as one unit was the first step of the final modeling of the structure. In the second step, this unit, whose growth can be easily controlled by the number of iteration, is replicated in different scales with different growth members, and arranged along the overall shape of the pavilion that was designed for the functional purpose. The arrangement of the tree or plant units were arranged based on the growth phenomena of the tree and thus geometrically based on their increasing iteration numbers. Because of the curve feature, the branches of adult tree units touched to the opposite side adult trees, thus created a skeletal forested canopy with complex network of branches. The another important feature of the overall design was to install sitting benches which symbolized the plywood furnitures showing the usefulness of poplar tree. This part was modeled by simple a linear array algorithm. Some pieces which represented

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Figure 7.2: Top - Branching tree as an attractor by following affine anisotropic transformation process. Bottom - Resulted tree models that shows their age growth.

sprout of young poplar tree in fact were intended to be used as a seat with backrest.

7.1.3

Parametric Modeling

The digital modeling was done entirely in the Grasshopper parametric environment and visualized in Rhinoceros3D. During this process, the geometric variables, such as the size of the overall functional space, number of branches and their vertical as well as horizontal angles, scaling factors, number of units followed by the main spatial shape and the gap between the units, were made changeable, and thus the whole geometric model has been transformed into a parametric model, so that by controlling these variables can deliver several design possibilities, out of which, the most satisfied one was selected. The advantage of working with parametric software turned out very useful especially in the final stage, when the client asked for cost reduction.Some design variables suing simple parametric slider components in Grasshopper allowed us to reduce

116 Chapter 7. Application 1: Fractal-Based Generative Design of the ‘Monalisa Pavilion’ the number and size of tree units and bench elements to achieve low-cost by maintaining functionality, comfort and aesthetic appearance.

7.1.4

Architectural Design

Figure 7.3: Top view of the Monalisa Pavilion.

The above digital model has been transformed into an architectural model by assigning the assembling features, sizes and materials to all components. To show the versatility and flexibility of the poplar-made plywood, we attempted to stretch its material and mechanical potency by making each tree branches a single strip coming from the ground. In this sense, in the tallest tree, the last branch had to be the longest strip coming through different curve paths which showed the excellent bending capacity of poplar plywood in both direction at a time. Architecturally, the exploitation of the mechanical as well as aesthetic qualities of poplar plywood had to went through some unique challenges, and that was the reason we adopted an innovative assembling. This way, the overall all design offered a sense of continuity yet display high complexity resembling with nature’s forest. All the pieces are made by softwood white finish poplar plywood with 2 cm thickness. One important element, a triangular wedge, was at the branching points for keeping two adjacent strips apart from each other so that they can spread out as branches (Figure 7.4). the branching angles were obtained by controllng the angles of wedges. The primary intention was to use plywood as much as possible for the real fabrication. Some architectural expressions of the pavilion design has been shown in ‘Figure 7.5’.

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Figure 7.4: Creating branches by inserting curved triangular wedges for making physical model and prototypes.

7.1.5

Structural Analyses

Because this pavilion is a free standing structure supported tightly at the ground, hence no critical structural analyses was performed except at the branching point where the plywood strips faced different directional bending and large deformations. For assessing the bending and internal stress behavior of plywood strips, especially long ones, finite element analyses was performed using a commercial FEM solver program ANSYS. An elastic orthogonal material was selected by considering the fiber directions of all sheets perpendiculars to each other; because, plywood is a set of different ply sheets placed orthogonally with different angles of fiber directions (Figure 7.6, Top Rright). Elastic modulus and Poisson’s ratios of different ply sheets having different fiber directions were taken from company’s poplar plywood specifications. Because the ply wood was subjected to face large deformation in bending to form curves, a nonlinear analysis was done. After nonlinear analysis, ANSYS resulted in a failure spot near the curve of plywood during bending (Figure 7.6, Bottom Right). Therefore, after simulative analysis, we faced two other problems. First problem was how to deal with the weakest point in large-space bending. Second, how to assemble or joint different pieces of plywood to produce a single complete strip for making one ‘tree’ unit, because one standard plywood board is not large enough. The solution of assembling different strips and make a bigger tree is shown in ‘Figure 7.7’

7.1.6

Practical Construction

As soon as the final design has been selected from several possibilities of the pavilion, a small scale (1:20) physical model was then made (Figure 7.8) to assess its physical behavior: every piece was cut in a CNC cutter in FabLab Torino, and all pieces were assembled to get the final physical outcome. No serious problem appeared because of the smaller scale, except the heavy exposure of too many metal bolts used for jointing pieces. But, in making real scale physical prototype, we faced two main challenges. The first challenge was to

118 Chapter 7. Application 1: Fractal-Based Generative Design of the ‘Monalisa Pavilion’

Figure 7.5: Architectural representations of the Fractal Forest Pavilion.

create one long but complete strip by jointing two different pieces of poplar plywood boards. This was done through ‘loose tongued joints’ (Figure 7.9a). Four

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Figure 7.6: Left – Nonlinear nodal solution of stress intensity in ANSYS after applying loads on two adjacent ply-sheets for large deflection. Right Top – Ply-sheets placed with orthogonal fiber directions. Left Bottom – The weakest (failure) point ‘MX’ during large deflection in bending..

Figure 7.7: Assembling of strips to get a large tree unit.

such strips were jointed together by screwing tightly with brass screws, so that the colors of poplar and brass screws could merge to avoid the heavy exposure of metals (Figure 7.9b) However, the joints had to be made strong enough to carry the load of the upper part of the strips, i.e., the group of branches which acted as cantilevers. For physical testing of internal stresses and bending behavior, students jointed two larger curve-shaped strips in the University workshop building and then pulled the tails of each strip manually for getting large deflection bending in order to obtain the curved branching configuration. At certain level of stress a crack appeared at the bent area due to torsion effect. Therefore, jointing plywood pieces was not the only critical aspect since ensuring the crack-less

120 Chapter 7. Application 1: Fractal-Based Generative Design of the ‘Monalisa Pavilion’

Figure 7.8: (a) Small scale (1:20) physical model.

Figure 7.9: (a) ‘Loose tongued joint’ for making longer strip (source: internet); (b) Two plyboards are joined by ‘loose tongued joints’ to make strips and four such strips were screwed by brass screws.

large deflection of bending was the another main challenge. Accordingly, in real scale production of one ‘tree’ unit, screws were placed diagonally at turning position but not perpendicular to the tangent of bending curve so that it would not only ensure crack-less bending but also offer a relaxation space in torsion (Figure 7.10, Left). One of the most important elements was the wedges for generating branches. A triangular wooden wedge was produced on the basis of the dimensions provided by the Rhinoceros model of a branch (Figure 7.4a) and then it was inserted between two strips, thus the wedge forced the strips to deform outwards in opposite directions, forming the branch. The sides of the wedges were gently curvilinear so as to give branches a natural shape (Figure 7.4). The next step of the workshop event was to start to realizing all individual components for making a real scale prototype of two ‘tree’ units. After cutting all pieces in a CNC cutter, all the components were assembled and connected to form two different ‘tree’ unit prototypes. Two units were then placed facing each other to observe their structural stability as well as their aesthetic appearances (Figure 7.11). The final and last challenge was yet to be faced: realization and installation of the complete structure in real scale. After a long period of designing, modeling, structural testing and making prototype units, final

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Figure 7.10: Left- Weakest part in large deflection bending of strips when screws were perpendicular to the tangent. Right – Screws not-perpendicular to tangent to offer relaxation in torsion while bending a strip.

production of all units was done in an industrial workshop building by the industrial participants and supervised by WoodLab team from Politecnico di Torino. Then all assembled units were transported to the exhibition center in Milan two days before the event of ‘MadeExpo2012’ exhibition. It took one full day to fabricate and install the final real scale pavilion structure on the event site, and on the first day of exhibition it was finally ready for inauguration.

Figure 7.11: Real scale ‘tree’ units as test displayed in the university workshop ground.

122 Chapter 7. Application 1: Fractal-Based Generative Design of the ‘Monalisa Pavilion’

Figure 7.12: : Design to production process in factory workshop building.

7.1.7

Final Remarks

This workshop was a perfect platform to practically apply the concept of fractal geometry in developing architectural design. Mathematics based generative modeling of fractal figures as an abstract replica of forest, which is highly complex by its appearance, showed the potency of fractals as a design tool for real world architectural applications. Moreover, this workshop event, had confirmed the validity, merit and flexibility of the ‘Generative Tool Box’ which easily transformed the complex mathematical expression of fractals into a digital model. However, in this chapter, computational morphogenesis process had only considered the generative process without employing the structural analyses tool to find a optimal shape. Instead, here, it shows the adaptability of the fractalbased generative tool for taking design decision and opt the finest one based on designer’s choice and functional needs. However, in the next chapters, all the steps and tool boxes of computational morphogenesis are applied not only to generate fractal-based structures, but also to find the optimal fractal structures. The next applications are more inclined towards finding structural forms and dealing with assessment of structural behaviors of the structures that are generated by using the principle of fractal geometry.

Chapter 7. Application 1: Fractal-Based Generative Design of the ‘Monalisa Pavilion’

Figure 7.13: Final outcome of Monalisa Pavilion in MadeExpo2012 event, Milan

Figure 7.14: Production progress of real scale ‘tree’ unit in the university workshop.

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124 Chapter 7. Application 1: Fractal-Based Generative Design of the ‘Monalisa Pavilion’

Chapter 8

Application 2: Fractal-Based Trusses Design 8.1

Introduction

The main objective of this chapter is to propose a list of new structural trusses by applying the concept of fractal geometry, focusing on the design of pitched roof trusses.1 The core strategy is to expand the feasibility of new shape of truss structures by using a rule-based design development process supported by fractal geometry. For this purpose, our investigation proposes some fractal-based trusses as the non-conventional new designs, and some fractal-based trusses as the geometric refinement of conventional truss designs. Similar as Asayama and Mae’s approaches [43] [1], the Iterated Function System (IFS) based on Barnsley’s contraction mapping method [2] are used in this study as a generative design tool for the automatic forms generation of fractal-based trusses and for the calculations of their Hausdorff dimensions. The Box Counting dimensions are calculated to measure the detailness of the internal lattices of the fractal-based trusses. The finitely iterated geometric models of all the proposed fractals are transformed into the finite element models of fractal-based trusses for their finite element analyses, and the results have been discussed to assess 1 The research of this Chapter has been published as a journal paper in the International Journal of Space Structures in 2015: Iasef Md Rian and Mario Sassone. ’Fractal-Based Generative Design of Structural Trusses using Iterated Function System’, International Journal of Space Structures, 2015, 29(4), 181-203.

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their structural characteristics.

8.2 8.2.1

Trusses and Self-Similarity Features The History of Truss Designs

The history of the design of structural trusses can be traced back to the construction of pitched roof trusses in the Roman period. In the earlier times, often an overall shape of a triangle was used for constructing a wooden roof frame. Sometimes, one post was added under a pair of two rafters for low-pitched or long-span roof. This post, traditionally known as crown post, was rested on the centre of a tie beam which could take upper loads. In this situation, the rafters and the post acted as compression members (Figure 8.1a) [66]. Sometimes, the beam was made gentle upward curve to prevent sagging from the upper loads transformed by the crown post. The abstract detail of this frame system is shown in ‘Figure 8.1a’. Surviving example of such crown post truss is seen in the 12th century St Clement Church roof at Old Romney in Kent, England. (Figure 8.1b) Later, the Romans invented a new structural system in the same truss design, but by modifying the details and the mechanical principle of the system. Romans did not rest the vertical post on the tie beam. Instead, they attached the tie beam with the post that was anchored at the ridge where two rafters met. Thus, the vertical post became a hanging post, and acted as a tension member (Figure 8.2a) [88]. An example of traditional king post is seen in the 18th century Bolduc House in Ste. Genevi`eve in Missouri (Figure 8.2b). Therefore, a crown post was designed to be in a compression and transfers weight to the tie beam, while a king post was designed to be in tension and supports the tie beam. This technical improvement by means of mechanics of truss system had advanced the history of truss development in the later periods. In the later periods, diagonal members were added to the truss designs so that the diagonals can transfer the vertical and horizontal loads. Apart from transferring the loads, the major role of diagonals were stiffening the truss structure, especially for the long-span trusses. Struts were added further in a regular pattern as posts to provide a robust structure. For long-span structures, more members were added depending on the bearing capacity and required stiffness, and formed a denser lattice configuration. After the Renaissance period, in Europe, different configurations of lattice patterns were seen in long-span truss designs depending on the requirements and applied loads. Since the 19th century, when steel has become popular and affordable in construction, several different varieties of truss systems started to appear and a list of engineers introduced some innovative truss system having unique lattice configurations to

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127

Figure 8.1: (a) A vertical post under compression traditionally known as ’crown post’ standing on a tie beam, (b) An example of crown post in St Clement Church roof, Old Romney, Kent, England (12th century).

Figure 8.2: (a) A vertical post under tension traditionally known as ‘king post’ hanging from the joint of two rafters and hold the tie beam (b) An example of king post in the Bolduc House in Ste. Genevi` eve, Missouri (18th century).

fill the interior of the main frame using the tension diagonals, compression struts and posts for increasing strength, stability and stiffness of the trusses. Today, all of these trusses are considered as the standard trusses widely used in the contemporary constructions. Among them King post truss, Queen post truss, Pratt truss, Fink truss, Howe truss, Fan truss, Fan Fink truss and Baltimore truss are most common for the pitched roof construction.

128

8.2.2

Chapter 8. Application 2: Fractal-Based Trusses Design

Self-Similarity Features of Traditional Trusses

From a geometric point of view, when the internal configuration of a truss is designed by assembling the posts, diagonals and struts within the main frame, the foremost target is always structural, dealing with the compressions and tensions in the members to obtain the maximum stability and high stiffness, and in some cases, it considers the aesthetic appearances too. While the geometric constructions of the internal lattices of conventional trusses basically follows simple geometric rules, one striking feature, clearly recognizable in many examples, is the modular arrangement and repetition of simple geometric shapes at different scales. This repetition is meant for increasing the stiffness of the truss structure, by reducing the members slenderness. Generally, the more the repetitions are nested inside the truss frame, the more is the stiffness of the truss structure. In the ‘Figure 8.3’, it can be noticed that Double Howe truss, Fink truss and Fan Fink truss show some extend of self-similarity features at increasing orders of their compound configurations. In this figure, ’A’ shape (King Post truss) is exactly or approximately copied in the second order configurations of Double Howe truss, Fink truss and Fan Fink truss with different orientations at different positions after its contraction. Similarly, ’B’ shape (Fan truss) is also copied in the third order configuration of the Fan Fink truss. The geometric process of forming the compound configurations of these three conventional trusses derived from the very basic truss shapes such as King Post truss and Fan truss, illustrated in the ‘Figure 8.4’, somehow resembles with the rule-based process that generates self-similar fractal shapes. Unaware with the notion of fractal geometry much before its introduction in 1970s by Mandelbrot, the engineers purposely used self-similar modular units to stiffen the conventional trusses. Accordingly, it is explorable to design rule-based self-similar features in the compound configurations of structural trusses that follow the principle of fractal geometry, and analyze the mechanical properties of these new fractal-based trusses.

Figure 8.3: Some common types of trusses and their ‘self-similar’ features that are formed by the self-similar or self-affine repetitions of ‘A’ and ‘B’.

Chapter 8. Application 2: Fractal-Based Trusses Design

8.3

129

Geometric Modeling of Fractals for Truss Design

Fractal system has two important properties that are useful for structural design in the field of construction. The first property is that the fractal system can be expressed by using a simple algorithmic function. This property is useful for easily programming a fractal model in computer for designing structures. Second, fractal system is an assembly of some self-similar unit elements; hence, they form some hierarchical or modular system in structure. This property can be useful to construct long-span structures where modular units are required for stiffening and strengthening a structure. There are different methods to design a simple algorithmic function that can generate a fractal model. In this paper, Barnsley’s method [2] of fractal construction is adopted which is based on his contraction mapping theorem and the Iterated Function System. Using his method and the principle of fractal geometry, some fractal models are constructed in this study in order to make different geometric configurations of some structural trusses. First two trusses are designed based on the well-known fractals such as Sierpinski triangle and Pinwheel Fractal, whereas other trusses are designed by reorganizing the configurations of traditional trusses using fractal rule. Asayama and Mae’s fractal truss is also included in this family of fractal-based trusses.

8.3.1

Sierpinski Triangle

Sierpinski triangle is a canonical example of fractal geometry. By its construction, generally, a triangle S0 , as an initiator, is subdivided into four identical triangles and then the central sub-triangle is discarded. This process continues to the other remaining three sub-triangles to produce further smaller congruent sub-triangles, and thus continue this process on each remaining smaller sub-triangles ad infinitum. Sierpinski triangle can be generated by using Barnsley’s Iterated Function System (IFS) method based on his theory of contraction mappings and affine transformations. In this method, Sierpinski triangle S is an attractor which is an intersection set of all the infinitely iterated self-similar sets that are S0 , S1 , S2 , S3 ,. . . . . . ,Sn , .... In this method of constructing a Sierpinski triangle, the IFS produces three identical copies (σ1 , σ2 and σ3 ) from initial triangle 0 S00 at the first iteration using the contractivity factor λ as 1/2 and the affine transformation functions as fi , shown in ’Figure 8.4’. Thus, a new shape S1 is generated which is a union of σ1 , σ2 and σ3 , mathematically

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expressed as, S1 = σ1 ∪ σ2 ∪ σ3 = f1 (S0 ) ∪ f2 (S0 ) ∪ f3 (S0 )

(8.1)

Sn = f1 (Sn−1 ) ∪ f2 (Sn−1 ) ∪ f3 (Sn−1 )

(8.2)

S0 ⊃ S1 ⊃ S2 ⊃ S3 ⊃ ..... ⊃ Sn ⊃ .....

(8.3)

such that Hence, the attractor S is, S=

∞ \

Si

(8.4)

i=1

which is a mathematical expression of Sierpinski triangle, a fractal entity. In the case of constructing the Sierpinski triangle, the following function of IFS is applied that consists of k = 3 different affine transformations (f1 , f2 and f3 ) of the original triangle shown in Figure 8.4.       µ 0 cos θ − sin θ x δx fi = λ i 1 + (8.5) 0 µ2 sinθ cos θ y δy

Based on the equation (3.3.1), if the Hausdorff dimension of Sierpinski triangle is D, then λ1 D + λ2 D + λ3 D = 1; where,λ1 = λ2 = λ3 =

1 2

(8.6)

Therefore,  D  D  D  D 1 1 1 1 + + =3 =1 2 2 2 2

(8.7)

So, D=

log (1/3) ≈ 1.585 log (1/2)

(8.8)

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131

Figure 8.4: Top - Contraction mapping and affine transformation of the Sierpinski triangle. Contractivity is 1/2; Bottom - Convergent sequence of Sierpinski triangle.

8.3.2

Pinwheel Fractal

Pinwheel Tiling: An interesting opportunity to create fractal shapes is given by a non periodic tiling of the plane, as the well known Pinwheel Tiling developed by John Conway and Charles Radin in 1994 [13]. Pinwheel Tiling is an assembly of identical triangular tiles yet produces randomness with high visual complexity. It is a non-periodic tiling which means their orientations is infinite and do not follow the same orientations. Conway, first noticed that a right-angled triangle P0 having its width double than its height can be divided in five isometric copies of itself that are 0 t0 s, where each copy is 1/sqrt5 times smaller than P0 . Starting from that, Conway and Radin assembled four copies of the triangle P0 around it such a way so that the union of five isometric triangles becomes a big rightangled triangle P1 (Figure 8.5). Now, around the P1 , its four isometric copies are translated and assembled in the similar fashion so that the union of all triangles of P1 and its copies becomes a larger triangle P2 . This process can be iterated to obtain an infinite increasing sequence of growing triangles where all the new triangular planes are the isometric copies of P0 . In this tiling, one interesting feature appears is the infinitely many orientations of the tiles. This is because of the two angles of the P0 which are arctan(1/2) and arctan(2), and both of these values are both non-commensurable with . Despite this, Radin and Conway [?] confirm that all the vertices of the isometric copies of P0 in the tiling have rational coordinates. In this sense, Pinwheel Tiling is not exactly repetitive. The pinwheel tiling can even be regarded as an iterated subdivision of an initial shape P0 , so that the generating procedure becomes similar to the iterated function system. In this context, Barnsley’s method can be applied. In this

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Figure 8.5: Conway and Radin’s method to make Pinwheel tiling pattern which starts from one small triangle tile and then it iteratively grows outwards with above rule.

case, P0 is an initial triangle and it will be subdivided to produce pinwheel tiling pattern. Using, the contraction mapping process on the main triangle P0 , five isometric copies (t1 , t2 , t3 , t4 and t5 ) are generated which are contraction mappings of P0 itself using the contractivity factor of 1/sqrt5. Then, each of them pass through a unique affine transformation process so that all of them can converge into and accommodated inside the space of main triangle P0 without overlapping but touching with each other (Figure 8.6, Top).

Figure 8.6: Pinwheel Tiling construction using contraction mapping method; Top - Affine transformations of right-angle triangle. Bottom - Convergent sequence of Pinwheel Tiling.

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The output figure at the first iteration can be expressed as, P1 = t1 ∪ t2 ∪ t3 ∪ t4 ∪ t5 = f1 (P0 ) ∪ f2 (P0 ) ∪ f3 (P0 ) ∪ f4 (P0 ) ∪ f5 (P0 ) (8.9) and at nth iteration, Pn = f1 (Pn−1 ) ∪ f2 (Pn−1 ) ∪ f3 (Pn−1 ) ∪ f4 (Pn−1 ) ∪ f5 (Pn−1 )

Pn =

5 [

fi (Pn−1 )

(8.10)

(8.11)

i=1

such that P0 ⊃ P1 ⊃ P2 ⊃ P3 ⊃ ..... ⊃ Pn ⊃ .....

(8.12)

Hence, the Pinwheel Tiling as an attractor C is an intersection set of all the identical subsets of P0 , and can be expressed as, C=

∞ \

Pi

(8.13)

i=1

In the case of constructing the Pinwheel Tiling, the following function of IFS is applied that consists k = 5 different affine transformations (f1 , f2 , f3 , f4 and f5 ) of the main triangle.       µ 0 cos θ − sin θ x δx fi = λ i 1 + (8.14) 0 µ2 sinθ cos θ y δy

According to the Hausdorff dimension method, if the fractal dimension of a Pinwheel Tiling is D, then λ1 D + λ2 D + λ3 D + λ4 D + λ5 D = 1

(8.15)

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where, 1 λ1 = λ2 = λ3 = λ4 = λ5 = √ 5

(8.16)

Therefore,  5

1 √ 5

D =1

(8.17)

So, D=

log(1/5) √ = 2.0 log (1/ 5)

(8.18)

The calculated Hausdorff dimension of this iterated figure of Pinwheel Tiling is 2.0 which is an integer value, and it is equivalent to a two dimensional surface. That is obvious in the subdivision (or tiling) process, because no surface is added or subtracted at each iteration. Pinwheel Fractal : If the central triangle of Pinwheel Tiling is discarded ad infinitum, then the whole triangular surface starts losing its dimension (Hausdorff dimension, D) from 2 to fractional such that 1.0 < D < 2.0. Thus, it becomes no longer a two dimensional surface and transforms into a fractal shape, which is known as a Pinwheel Fractal (Figure ∞ \

P =

Pi

(8.19)

i=1

where, P1 = t1 ∪ t2 ∪ t3 ∪ t4 = f1 (P0 ) ∪ f2 (P0 ) ∪ f3 (P0 ) ∪ f4 (P0 )

(8.20)

and at nth iteration, Pn = f1 (Pn−1 ) ∪ f2 (Pn−1 ) ∪ f3 (Pn−1 ) ∪ f4 (Pn−1 )

Pn =

4 [

fi (Pn−1 )

(8.21)

(8.22)

i=1

such that P0 ⊃ P1 ⊃ P2 ⊃ P3 ⊃ ..... ⊃ Pn ⊃ .....

(8.23)

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135

Figure 8.7: Pinwheel Fractal; Above - Affine transformations; Bottom – Convergent sequence.

If the Hausdorff dimension of Pinwheel Fractal is D, then  4

1 √ 5

D =1

(8.24)

So, D=

8.3.3

log(1/4) √ ≈ 1.7227 log (1/ 5)

(8.25)

Asayama and Mae’s Fractal

Asayama and Mae [1] were the first who proposed a new fractal shape for designing a truss by using IFS method. The first iterated figure of their fractal was formed by the union of three self-similar copies (a1 , a2 and a3 ) of the first triangle ∆0 after its two dimensional affine transformations (f1 , f2 and f3 ) in which the contractions are λ1 , λ2 and λ3 (Figure ∆=

∞ \

∆i

(8.26)

i=1

where, ∆1 = a1 ∪ a2 ∪ a3 = f1 (∆0 ) ∪ f2 (∆0 ) ∪ f3 (∆0 )

(8.27)

and at nth iteration, ∆n = f1 (∆n−1 ) ∪ f2 (∆n−1 ) ∪ f3 (∆n−1 )

(8.28)

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∆n =

4 [

fi (∆n−1 )

(8.29)

i=1

such that ∆0 ⊃ ∆1 ⊃ ∆2 ⊃ ∆3 ⊃ ..... ⊃ ∆n ⊃ .....

(8.30)

Figure 8.8: Pinwheel Fractal; Above - Affine transformations; Bottom – Convergent sequence.

If the Hausdorff dimension ofAsayama-Mae’s Fractal is D, then 

1 √ 5

D

 +

1 √ 5

D +

 D 3 =1 8

(8.31)

Using the Newton-Raphson method, we get D ≈ 1.5953

(8.32)

Chapter 8. Application 2: Fractal-Based Trusses Design

8.3.4

137

Fan Fractal

Fan Fractal is also constructed by following the same process as above based on the IFS method. In this case, F0 is the initial shape, and the nth iterated shape Fn is the union of the two contracted shapes of their previous shape Fn−1 after the two affine transformations f1 and f2 shown in ’Figure ∞ \

F =

Fi

(8.33)

i=1

where, F1 = ϕ1 ∪ ϕ2 = f1 (F0 ) ∪ f2 (F0 )

(8.34)

and at nth iteration, Fn = f1 (Fn−1 ) ∪ f2 (Fn−1 )

Fn =

4 [

fi (Fn−1 )

(8.35)

(8.36)

i=1

such that F0 ⊃ F1 ⊃ F2 ⊃ F3 ⊃ ..... ⊃ Fn ⊃ .....

(8.37)

If the Hausdorff dimension ofAsayama-Mae’s Fractal is D, then √ !D 5 + 4

√ !D 5 =2 4

√ !D 5 =1 4

(8.38)

Therefore, D=

log(1/2) √ ≈ 1.1918 log ( 5/4)

(8.39)

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Figure 8.9: Fan Fractal; Above - Affine transformations; Bottom – Convergent sequence (The dotted lines are the impressions of previous shapes).

8.3.5

Baltimore Fractal

Based on the similar method used for the constructions of previous fractals, the Baltimore Fractal is constructed by the union of the two contractions of previous shape at each new iteration using the affine transformations f1 and f2 (Figure 8.10). Unlike other fractals, the initial shape B0 of the Baltimore Fractal is a triangle having an additional vertical line that connects the vertex to the middle of base line (Figure 8.10a). Similar as the construction of the Fan Fractal, the dotted lines shown in the ’Figure 8.10’ are the impressions of previously iterated shape. The shape of Baltimore Fractal as an attractor B and its corresponding affine transformations can be expressed as, ∞ \

B=

Bi

(8.40)

i=1

where, B1 = bi1 ∪ b2 = f1 (B0 ) ∪ f2 (B0 )

(8.41)

and at nth iteration, Bn = f1 (Bn−1 ) ∪ f2 (Bn−1 )

Bn =

4 [ i=1

fi (Bn−1 )

(8.42)

(8.43)

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139

such that B0 ⊃ B1 ⊃ B2 ⊃ B3 ⊃ ..... ⊃ Bn ⊃ .....

(8.44)

Figure 8.10: Fan Fractal; Above - Affine transformations; Bottom – Convergent sequence (The dotted lines are the impressions of previous shapes).

If the Hausdorff dimension ofAsayama-Mae’s Fractal is D, then  D  D  D 1 1 1 + =2 =1 2 2 2

(8.45)

Therefore, D=

log(1/2) = 2.0 log (1/2)

(8.46)

which means, at infinite iteration, the Baltimore Fractal (without dotted lines) will be transformed into a one dimensional line.

8.3.6

Comparison of the Hausdorff Dimensions

In the previous sections, the Hausdorff dimensions of all five fractal shapes are calculated by using Barnsley’s contraction mapping method [2] considering all the triangles have the same base angles θ1 and θ2 such a way so that θ1 = θ2 = arctan(1/2). It is observed that the estimated Hausdorff dimensions are variable to the base angles θ1 and θ2 in the case of Fan Fractal shape and Asayama-Mae’s Fractal shape. Because, in these two shapes the contractions

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are related to the base angles. Oppositely, in the cases of Sierpinski triangle, Pinwheel Fractal shape and Baltimore Fractal shape, their contractions are not related to the base angles, and hence, the Hausdorff dimensions in these three fractal shapes are constant and invariable to the base angles. The comparative values of Hausdorff dimensions of these five fractal shapes are shown in ’Figure 8.11’.

Figure 8.11: A comparison of the Hausdorff dimensions.

8.4 8.4.1

Architectural Design of the Fractal-Based Trusses Design Considerations and Truss Designs

Five different fractal shapes, which are constructed in the previous section using IFS method, are transformed into the geometric configurations useful for designing pitched roof trusses. A fractal is an infinite entity. Therefore, in the scale of architecture and engineering, only finitely iterated fractal models are applicable. The following considerations are taken into account for transforming the mathematical fractal shapes into the truss designs, • Only the third iterated fractal models are considered as the truss configurations except the Pinwheel Fractal shape which is its second iterated model. These finitely iterated models are chosen by keeping the overall

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141

size of each truss in mind so that the longest member and the shortest member of the truss do not become too long and too short with respect to their cross-sections. • The overall height of each truss is 3 meters; hence, the base is 12 meters wide, because it is considered in the beginning that the length of its base is always 4 times more than its height for each truss. • Only the line segments (i.e., the outer lines of all shaded parts) of the above fractal shapes are considered for designing the truss lattices. • All the individual lines are considered as truss members and their connecting points as truss joints. • Each member is made of steel having a circular hollow cross-section with 6 cm diameter and 3 mm thickness. • Each joint is a steel plate which connects steel tubular members and acts as a hinged joint. The connecting system at the joints is made such a way so that the posts and diagonals, which are attached to the rafters and their junction, act as hanging members; and the other members, which are attached to the tie beam, carry the weight of tie beam (Figure 20b). This connecting system is followed by the similar mechanism as traditional King post truss shown in ‘Figure 2a’. • Two far end joints at the base of each truss are considered as two supports. One support is vertically and horizontally restrained, while another one is only vertically restrained as a roller. Based on the above design considerations, the third iterated shapes of the Sierpinski triangle and the Asayama-Mae’s Fractal are straightforwardly transformed into the designs of Sierpinski Fractal truss and Asayama-Mae’s Fractal truss by only taking the outlines of their shaded parts (Figure 8.11). Two copies of the second iterated shapes of Pinwheel Fractal are attached by placing them opposite to each other along their heights, and then transform this jointed shape into the design of Pinwheel Fractal truss (Figure 8.12). Fan Fractal and Baltimore Fractal shapes are also transformed into line lattices for designing their corresponding trusses, but by adding new line F and a group of lines B respectively at each new iteration such a way so that these new lines replaces of dotted lines that were shown in ’Figure 8.13’ and ’Figure 8.14’. The intention is to obtain truss-like configurations. This new addition in

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Figure 8.12: Fractal-based trusses design by transforming the the third iterated models of Sierpinski Triangle and Asayama-Mae’s Fractal, and the second iterated model of Pinwheel Fractal.

Fan Fractal is marked as red lines and shown in ’Figure 8.13’. Now, this new shape Ft can be expressed as follows: Ft =

∞ \

Fi

(8.47)

i=1

where, F1 = ϕ1 ∪ ϕ2 ∪ F = f1 (F0 ) ∪ f2 (F0 ) ∪ F ; F⊃ Ft (8.48) and at nth iteration, Fn = f1 (Fn−1 ) ∪ f2 (Fn−1 ) ∪ F

Bt =

∞ \ i=1

Bi

(8.49)

(8.50)

Chapter 8. Application 2: Fractal-Based Trusses Design

143

Figure 8.13: Transformation of third iterated Fan Fractal into its correspnding truss design by adding a newline F at each iteration.

where, B1 = b1 ∪ b2 ∪ B = f1 (B0 ) ∪ f2 (B0 ) ∪ B; B⊃ Bt (8.51) and at nth iteration, Bn = f1 (Bn−1 ) ∪ f2 (Bn−1 ) ∪ B

(8.52)

In the ’Figure 8.16 Left’ we have calculated the Box Counting dimension of each line lattice to measure its density inside the main triangular frame. Apparently, the denser the internal lattice of a truss, the stiffer is the truss structure having a reasonable topology. The comparative values of Box Counting dimensions are shown in ’Figure 8.16 Left’ which informs the degree of lattice density among all five fractal-based trusses. It will be interesting to see if there is some relation between the Box Counting dimension and the stiffness of a truss. Based on the above design considerations, we calculated the total weights of all truss structures, and their comparison is shown in ’Figure 8.16 Right’. It is seen that there is a similarity between Box Counting dimension differences of the trusses and their weight differences. This is understandable, because, the denser the lattice, the heavier is the truss. In this sense, Box Counting dimension can be interpreted as visual measure of truss weights.

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Figure 8.14: Transformation of third iterated Baltimore Fractal into its correspnding truss design by adding a newline B at each iteration.

Figure 8.15: (a) An example of the architectural appearance of a fractal-based truss (Fan Fractal truss); (b) An illustration of a typical joint detail.

8.4.2

Statically Determinacy of Fractal-Based Trusses

For the practical applicability of the fractal-based new trusses in construction, it is important to check if they are statically determinate or not as a measure of their internal stabilities. Generally, this property of a truss is checked based on the rule of force equilibrium in which the total number of unknown forces F and the total numbers of equilibrium equations Eq are counted and compared. If, F < Eq , then the truss is a hypostatic, i.e., statically indeterminate; if, F = Eq , then the truss is an isostatic, i.e., statically determinate; and if F > Eq , then the truss is a hyperstatic, i.e., statically indeterminate. It is evaluated in ’Figure 8.17 Left’ that informs whether the fractal-based trusses are statically determinate or indeterminate.

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145

Figure 8.16: Comparison of Box Counting dimensions (left) and total weighs (right) of different fractal-based trusses including conventional compound trusses when all the members have same cross-sectional area.

From the ’Figure 8.17 Left’, we notice that all the fractal-based trusses are statically determinate, except Pinwheel Fractal truss. To transform the Pinwheel Fractal truss into an isostatic truss, some members should be eliminated such a way so that F = Eq . So, the number of eliminated members should be = F − Eq = (2b − 2j + r) = (81 − 2 × 40 + 3) = 4 without compromising the number of joints. Therefore, the four members, which carry no or the lowest axial forces after applied loads, can be chosen for the elimination to make the Pinwheel Fractal truss as an isostatic. As a result, the shape of final isostatic Pinwheel Fractal truss will lose its fractal property of self-similarity, but can be addressed as a fractal-like. Hence, after the structural analyses, if we find some bars are inactive under some load conditions, then we can opt them out which will turn the shape of a fractal-based truss into a non-strict fractal.

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Chapter 8. Application 2: Fractal-Based Trusses Design

Figure 8.17: Evaluation of the statically determinacy of fractal-based trusses.

8.5 8.5.1

Structural Analyses of Fractal-Based Trusses Finite Element Analyses

In order to have a first understanding of the mechanical behavior of the fractalbased trusses, a linear static analysis under various load conditions has been performed. Before the analysis, it is assumed that all members of the truss are perfectly straight, all joints are frictionless and all loads and reaction occur only at the joints. Since all the fractal-based truss models are statically determinate, except the Pinwheel Fractal truss model, the stiffness of members does not influence the distribution of internal axial forces, while it has an effect on the nodal displacements. Further, the self weight of the trusses is not taken into account, being usually much smaller than the other loads, so that a unique kind of cross section is adopted for all the bars that form the trusses. Because the self weight is also an indicator of the amount of steel required for construction, and then of the cost, using the same cross section for all bars allows to evaluate this quantity directly from the total length of bars. As it has already been said before, the trusses are externally constrained with a hinge and a trolley, so that the global behavior is analogous to the corresponding simply supported beam. In that way, the influence of large displacements on the internal stress state

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147

can be neglected and the linear static analysis is reliable. This fact has been confirmed by some non linear analyses, performed taking into account large displacements. Three different load conditions are used in analysis: vertical uniform at the top, vertical uniform at the bottom and lateral load (Figure 8.18). The loads are applied as nodal loads and they are calibrated in order to have the same resultant and then to produce the same reactions when they are applied on different trusses. The total vertical load applied at top chord is 15 KN and at bottom chord is 15 KN, and the total lateral load applied is 15 KN. All members are hollow tubes having 6 cm diameter and 3 mm thickness made of steel, and the Young’s Modulus of each member is 2.05e+011 Pa and Poisson’s Ratio is 0.3.

Figure 8.18: Different load cases; (a) vertical nodal loads applied on two rafters, (b) vertical nodal loads applied on the bottom chord, and (c) lateral nodal loads at one rafter.

8.5.2

Results and Discussion

’Figure 8.19’ shows the deformations of fractal-based trusses under nodal loads applied on the top rafters. The maximum nodal deformations n of all the fractal-based trusses have been evaluated and their stiffnesses k = F/n have been compared in ’Figure 8.20’ where F is the total force applied on the truss. We found that the maximum displaced node is not necessarily the top node. In this case the global behavior of each truss is analogous to the corresponding simply supported beam, because it is externally constrained with a hinge and a trolley. ’Figure 8.20 Left’ shows the differences between stiffness among the fractal-based as well as two conventional compound trusses. ’Figure 8.20 Right’ is the Box Counting dimensions of the entire truss that express the detailness of internal lattices. By comparing these two figures in ’Figure 8.20’, we notice that here is no significant relation between the stiffness and the detailness of a truss. For example, the Box Counting dimension of Baltimore truss is the lowest among others which means it is the least dense truss, but its stiffness is the highest among others. This comparison confirms that the density of internal lattices has no effect on the strength of the truss structures, instead, the strength

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comes from their design topologies. Hence, this structural behavior can be more clearly understood by analyzing the internal forces inside the truss. The ’Figure 8.21a’ and ’Figure 8.21b’ show the internal forces, inside the Sierpinski Fractal truss, due to nodal forces applied on the upper rafters and on the lower chord, respectively. It is noticed in the ’Figure 8.21a’ that the distribution of internal forces shows some extend of self-similarity nature. The presence of branching fractal patterns allows the application of different loads on the boundary of the truss, by stiffening its long straight sides. As for all the other cases, the flow of forces depends on the load condition, so that different parts of the structure are stressed under different loads. In fact, the most important feature of fractal’s self similar subdivision is to reduce the size of the elements and to increase the number of nodes. Depending on the load condition, some elements can be fully unloaded, because the force flow does not involve them.

Figure 8.19: The deformations of fractal-based truss structures under vertical nodal loads applied on the top.

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Figure 8.20: Left - The comparison of stiffness k = F/ n of fractal-based trusses and conventional compound trusses, where F is applied forces and n is maximum nodal deformation; Right – Comparison of Box Counting dimensions of the truss structures.

Figure 8.21: Internal forces in the Sierpinski Fractal truss; (a) Internal forces due to nodal forces applied on the upper rafters; (b) Internal forces due to nodal forces applied on the bottom chord.

If an element is never loaded under any load condition, it can be considered as redundant and the truss can be simplified by removing it. In general, every time a node links three elements two of which are aligned, the third becomes inactive, unless an external load is directly applied on the node. In practical cases, the imposed loads on a truss only act on the nodes on the boundary of the truss. Some variation of the reference fractal pattern, in order to avoid inactive elements, is shown in ’Figure 8.22a’ and ’Figure 8.22b’, from the Sierpinski fractal shape. In ’Figure 8.22a’ is shown the simplified version with bottom load only, while in ’Figure 8.22b’ the simplified version for top load. ’Figure 8.23a’ and ’Figure 8.23b’ show the corresponding modified versions for the Pinwheel fractal truss. ’Figure 8.24’ shows the internal forces due to the combined three load cases ’a’, ’b’ and ’c’ of ’Figure 8.18’ From the figure, we have noticed that almost

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most of the internal members are active under the combined loads, but some members do not experience any force, especially in the Pinwheel Fractal truss, Asayama-Mae’s Fractal truss and Fan Fractal truss. The inactive members can be taken out which may transform the trusses statically indeterminate. But, from aesthetic point of view, and to keep the truss statically determinate, these inactive members can be kept as they are.

Figure 8.22: Reference fractal pattern of Sierpinski Fractal truss to avoid inactive members; (a) when applied vertical forces are on the top rafters; (b) when applied vertical forces are on the bottom chord.

Figure 8.23: Reference fractal pattern of Pinwheel Fractal truss to avoid inactive members; (a) when applied vertical forces are on the top rafters; (b) when applied vertical forces are on the bottom chord.

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Figure 8.24: Axial forces in the fractal-based trusses under the combined three load cases ’a’, ’b’ and ’c’ of ’Figure 8.7’ Green to blue and violet is under compression, and yellow to brown and red is under tension.

8.6

Structural Optimization

In the previous section, the Iterated Function System has been used for generating a new family of truss designs as a generative approach. Analytical results shows their structural behavior under some specified load conditions based on which one can find the most efficient fractal-based truss designs from its family. But, it will not provide the accurate information that which one is the most optimal design. To find the most efficient design from this new truss family, an optimization and computational form finding process is required. Here, the form finding is based on the the optimal sizing of the members. Besides, the iteration number has also an important impact on the strength and weight of the structure. Hence, the computational searching will also take the number of iteration as an important variable into account. The target is to search the fractal-based truss design which has the minimum displacement and the least weight. Here, the searching process will follow two different steps. In the first step, it will optimize the size of each member based on their capacity of carrying

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internal forces. In the second step, it will find the lightest and strongest truss if the iteration number including other design variable such as height is changed.

8.6.1

Design Optimization

In the beginning of the optimization process, cross section of all members have been considered uniform. First, the overall design optimization has been performed by taking the cross-sectional diameter and the height (base angle) as two different variables. The domain of the diameter has been set 1.0 cm to 15.0 cm, the domain of the height has been set from 2 meters to 6 meters. These domains are chosen based on the architectural functionality and apparent structural feasibility. The allowable maximum deformation has been limited at 15 mm.Only a point load of 20 KN at the top has been applied. The target is to obtain an optimal design which can deform to the maximum allowable displacement but reduce the weight. For this,a multi-objective optimizer is required. A ready-made multi-objective optimizer Octopus, which is a tool of ’Optimization Tool Box’ has been used for this purpose. The process has been stopped at 100th generation, and the optimal design has been found which is shown in F igure 8.25. The optimal design has the

Figure 8.25: Optimal design of Sierpinski truss with uniform member cross section.

But, obviously, this optimal design is not final optimized shape. Because, if we consider the sufficient capacity of internal force carrying capacity, then each member will have unique cross section size, that will allow us to further saving of weight. For example, in the ’Figure 8.25Right’, we can see that the internal members are not so much active when load is applied. Hence, these members’

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cross- section can be reduced. Usually, the members under compression are thick, whereas the member under tension are thin. This means, the further optimization can be performed by using the concept of internal forces. This approach has been discussed in the following section.

Figure 8.26: Different maximum displacements of Asayama-Mae’s truss with the different base angles under the point load at top, and their Hausdorff dimensions.

Figure 8.27: Optimized configuration of the Asayama-Mae’s Fractal truss.

In the case Asyama-Mae’s truss it has also many variations of forms based on its base angles, and accordingly each base angle shows different mechanical strength (Figure 8.26. It is difficult to find the structurally optimal by checking different variations scheme by scheme. That is why, automatic optimizer, here it is the ’Optimization Tool Box’ is required to find the best configuration under the defined load case. Therefore, an optimization process has also been performed by taking its base angles as the design variables, and fixing its base length 12 meters and height 4 meters. The optimization algorithm run for 50 generations, and resulted the most optimized truss topology of Asayama-Mae’s

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Fractal truss, shown in ’Figure 8.27’.

8.6.2

Cross Section Optimization

The size optimization of the truss members are dependent on their capacity of carrying the internal forces. Therefore, it is important to know the magnitude of internal forces of each member that will help to find its optimal size of the cross-section. It means, this optimization process is not based on the searching of the best shape from a list of randomly produced variations. Instead, it is a straight-forward approach based on the factor of internal forces. The calculation of the optimal sizes of each memeber based on their internal forces are calculated by the following way: Based on the internal forces and permissible deformation as well as buckling factor, the cross-section of each member of a truss structure can be numerically optimized. Here, a readymade component of optimizing the cross-section available in Karamba has been used which is developed by Clemens Preisinger (Clemens2014) by the following aprroach: Karamba does cross section design by going through the list of cross sections in a group of cross sections called a family. It starts at the first entry and proceeds to the next until a cross section is found that is suffcient for the given cross section forces. The core routine of the cross section optimization is the function ’isSufficient(...)’ which is listed below. It returns ’True’ if the cross section resistance surpasses the acting cross section forces. This is the original C++ listing of the variant for elastic cross section design which makes use of the elastic resisting moments Wy and Wz : In line 9 the buckling length lk is determined. The function ’bucklingLength’ starts with the end-points of the given beam and procedes to its neighbors until nodes with more than two beams attached are detected. The distance between these nodes is taken as the buckling length. In lines 13 and 14 the materials yield stress fy and Young’s Modulus E are determined. The critical buckling load of a beam is given as, Nc r =

π2 · E · I I 2k

(8.53)

This calculation is done in lines 16 to 18 for the two principal axes of the cross section. A variable h acts as a helper, ’M-PI’ represents π. The procedure for calculating the design value of the buckling force Nbrd is taken from Eurocode EN 1993-1-1:2005, paragraph 6.3.1.1, equation (6.47): Nb,Rd =

X · A · fy = X · A · fyd γ · M1

(8.54)

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Karamba assumes cross sections of class 1,2 or 3. This means that their most strained fibre yields before local buckling occurs. When a material is defined, the Karamba further assumes that the given yield stress fy is already reduced by the safety factor M1 which in case of steel is 1.0 in many European countries. Equation 6.49 of EC3 states that X=

Φ+

√

1 ≤ 1.0 Φ2 − λ−2

(8.55)

¯ and The equivalent code can be found in lines 26 and 27. The values for λ Φ are calculated according to the formulas of EC3 section 6.3.1.2: ¯ − 0.2) + λ ¯2] Φ = 0.5 · [1 + α · (λ ¯= λ

r

A · fy Ncr

(8.56)

(8.57)

¯ gets calculated for the principal directions of the cross section In Karamba λ (see lines 19 to 21). Φy and Φz for the principal directions get calculated in lines 22 and 23. The imperfection factors αy (alpha-y) and −z (alpha-z) are taken from the cross section tables. When using cross sections created within a Grassshopper definition then αy = αz = 0.3 is assumed. The larger of Φy and ¯ (see lines 24 and 25) are selected for calculating X Φz and the corresponding λ in lines 26 and 27. In lines 29 to 35 the resisting moment and forces of the cross section are calculated. In case of elastic design the elastic resistance moments Wy and Wz come into play, otherwise their plastic counterparts Wy,pl and Wz,pl . Ay and Az are the equivalent shear areas in y- and z-direction. Inside the loop which comprises lines 45 to 75 the cross section resistance is compared to the cross section forces. This is done for the two endpoints of index zero and one. In order to assess the effect of combined loads the superposition formula in line 58 or 60 is used for axial stress. If the normal force is tensile (N > 0) then N is compared to the plastic resisting force Nrd (see line 29), if compressive then Nb,rd enters the picture. These formulas correspond to that given in equation 6.41 of EC3. The superposition of the effect of shear in y- and z-direction and torsional moments is considered in line 63 along the same lines as for the axial stresses. The fact that shear and normal stress are considered separately constitutes a simplification. The larger of the utilization numbers of shear and normal stress is taken as the resulting cross section utilization (see line 66). By using the above approach, an arbitrary design of a Sierpinski truss has been optimized, shown in Figure 8.29.

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Figure 8.28: C++ code for the cross section optimization. (Clemens2014)

Chapter 8. Application 2: Fractal-Based Trusses Design

Figure 8.29: Cross section optimization of a Sierpinski truss.

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Final Remarks

This chapter has attempted to address the scope of applying the concept of fractal geometry in the field of construction aiming to introduce a new geometric system in designing structural roof trusses. The mathematical formulation called Iterated Function System has been applied to automatically generate the shapes of structural trusses as rule-based configurations. From the full mathematical definition of the analyzed fractal configurations, it has been possible to evaluate their Hausdorff dimensions and their Box Counting dimensions. Based on this strategy, a set of new fractal-based structural trusses have been generated, in part as geometric refinement of more conventional schemes, in part as truly new designs. The definition of the abstract fractal geometry is just the first step of the design: the translation to real structures has required a set of adjustments as the substitution of all the two dimensional generating figures with segments, in some cases the removal of elements from the fractal pattern and in general the arrangements required to obtain statically determinate configurations. The structural analysis shows that the self-similar fractal branching of trusses is suitable to make them able to carry almost distributed loads, on both the upper inclined rafters and on the bottom chord. The internal forces flows largely depend on the applied load condition and they suggest that new shapes can be derived from the fractal patterns. This is particularly interesting from the architectural point of view, because such new solutions can show aesthetical values and innovative appearance that make them suitable for real design applications. The analysis also informs that the lattice density, as a measure of Box Counting dimension, has no significant influence on the strength of truss structures. It is the truss topology by means of geometric configuration that can offer strength. A more complete evaluation of the structural applicability will require the whole design of fractal-based trusses, including optimal sizing of members, design of joints and more refined stability analyses, in order to compare their performances with more traditional solutions. Hence, it would be fruitful to pursue further research about applying the fractals as a new geometric concept in designing and developing innovative structural systems, as well as finding the efficient shapes in a new but geometric way in the domain of architecture and construction.

Chapter 9

Application 2: Fractal-Based Grid Shell Design 9.1

Introduction

In the field of architecture and construction, geometric shapes or the assemblage of different shapes play a pivotal role in providing the strength, stability and safety to structures.1 Apart from functional needs and structural benefits, aesthetic interests also drive architects and engineers to find forms. Based on these combined approaches, for many years, especially in the 20th and 21st centuries, numerous different innovative shell structures were designed, exploiting their structural features and geometric forms. In the recent years, thanks to the advancement of computer and CAD software, several previously unimaginable, complex, free-form as well as highly efficient and optimized vault, shell and other space structures were developed and realized. Most of these designs follow regular and continuous geometric shapes, in the frame of Euclidean geometry, and only in very few recent cases, they assume more intricate shapes, somehow inspired by nonlinear mathematics and non-Euclidean geometries [25]. However, we rarely find any built examples of these space structures, whose shapes are developed based on the mathematical rule of fractal geometry. In academia, 1 The research of this chapter has been presented in the proceeding of the IASS-SLTE 2014 Symposium in 2014: Iasef Md Rian, Mario Sassone and Shuichi Asayama, ’Fractal Shell Design using Iterated Function System’, Proceedings of the International Association for Shell and Spatial Structures (IASS-SLTE2014) Symposium, Brasilia, Brazil,2014.

159

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only a couple of investigations have been done so far where fractal geometry was used for designing folded and free-form shell structures [78] [85]. However, these investigations were limited within the straight-forward application of fractal geometry for generating new shapes only, but do not provide sufficient information about the structural characteristics of such shapes, especially, with regard to the effect of fractal dimension on their structural behaviors. In this chapter, a new type of vault and shell structures are proposed whose shape is a fractal, but derived from a non-fractal paraboloid. Barrel vaults are designed based on the extrusion of a single arch or pairs of arches. Barrel vault made of parabolic arches has always an smooth surface which 2-dimensional. On the other hand, paraboloids are 2-dimensional and smooth quadric surface in Euclidean space. Therefore, from the dimensionality perspective, if we exploit the smoothness of the barrel vault and paraboloid shell’s surface, then we can obtain shapes which are in between 2-dimensional surface and 3-dimensional solid. This way, the barrel vault and paraboloid shell can be transformed into a fractal vault and fractal shell respectively. From the structural point of view, parabolic arch, its vault and paraboloid shell are structurally efficient shape for uniformly projected and gravitational loads, and that is the reason parabola and paraboloid are widely used geometric forms for designing vault and shell structures, especially in the 20th century. But, we are still unfamiliar about its structural behavior if their surfaces are subjected to transform from smooth surface to unsmooth fractal surface. In this chapter, the structural characteristics when parabolic barrel vault and paraboloid shell starts morphing into a fractal shell structure has been investigated. Finally, an structural optimization process has been performed to find the most efficient form.

9.2 9.2.1

The Midpoint Displacement Method Archimedes’ Method

The history of the invention of the Midpoint Displacement Method traces back to the third century BCE. Archimedes (287-212 BC) is credited for developing this method [2], which was based on the Method of Exhaustion. Several ancient Greek philosophers and mathematicians considered the problem of calculating the area bounded by a curve. In the 5th and 4th centuries BCE, Antiphon and Eudoxus developed the idea of finding the area bounded by a curve by inscribing polygons in it, now known as the Method of Exhaustion [76]. The idea is to inscribe infinitely many polygons within the curve, which do not overlap and which cover the entire area, so that the sum of the areas of polygons approaches the area bounded by the curve. The Method of Exhaustion was

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perhaps taken furthest by Archimedes, who used it in the 3rd century BCE to perform calculations of areas and volumes bounded by circles, ellipses, spheres and cylinders.

Figure 9.1: Archimedes’ method for calculating the area of a parabola segment

By the time of Archimedes, the following facts were known concerning an arbitrary parabolic segment S with base AB and vertex P (see ’Figure 9.1’): Fact 1 The tangent line at P is parallel to AB. Fact 2 The straight line through P parallel to the axis of the parabola intersects AB at its midpoint M . Fact 3 Every chord QQ1 parallel to AB is bisected by P M . Fact 4 With the notation in the ’Figure 9.1’, PN N Q2 = PM M A2

(9.1)

For measuring the area of a parabolic segment, Archimedes first drew a line from the midpoint of the baseline to the vertex of the parabolic curve, and then connected the endpoints of the baseline to that point where the line touches on a point on the parabolic curve. Thus he obtained a large triangle ∆AP B. The area of this new triangle T0 is the first approximation of the area the parabolic segment S. In the next step, he repeated the same process in the two gap areas that were left in between the triangle ∆AP B and the parabolic curve inside the segment. He continued this process iteratively on each gap remained from the previous step until he approximately covered all the area of the parabolic segment by a collection of many triangles which collectively forms

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a polygon having a large number of sides (Figure 9.2 Top). In this process, 20 , 21 , 22 , . . . ., 2n number of triangles were created at 1, 2, 3, . . . , n iterations to obtain a polygon having area Pn . According to Archimedes’ propositions, the area of each smaller triangle is one eighth of the area of its parent triangle, i.e., T1 = 81 T0 , and in general, Tn = 18 Tn−1 . The above relation can be shown using the facts (Fact 1 to Fact 4) mentioned above by illustrating in the ’Figure 9.2b’ Applying the principle of ’Fact 2’ to both the original parabolic segment S and the smaller segment with base AP , we obtain two lines parallel to the axis of the parabola, one going through P and intersecting AB at its midpoint M , and one going through P1 and intersecting AP at its midpoint Y . Let M1 be the intersection point of this second parallel line with AB. Then M1 is the midpoint of AM because the triangles ∆Y M1 A and ∆P M A are similar. Finally, let V be the intersection with P M of the line through P1 parallel to AB (so V M M1 P1 is a parallelogram). From ’Figure 9.2b’, based on the ’Fact 4’ and the fact that V P1 = M M1 = 1 AM , we have 2 PV V P12 1 = = 2 PM AM 4

(9.2)

P M = 4P V

(9.3)

V M = 3P V

(9.4)

i.e., and From parallelogram V M M1 P1 , P1 M1 = V M = 3P V

(9.5)

Because, ∆Y M1 A and ∆P M A are similar triangles, hence 1 VM 2

(9.6)

1 P M = 2P V 2

(9.7)

AM1 = Y M1 = Therefore,

Y P1 = P1 M1 − Y M1 = 3P V − 2P V = P V

(9.8)

Hence, based on equation 9.2, Y P1 = P V =

1 PM 4

(9.9)

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Accordingly, area of ∆AP B, T0 =

1 AB · P M 2

(9.10)

Using the equation 9.3, we obtain T0 =

1 1 · (2AM ) · (4P V ) = 8 · ( · AM · P V ) 2 2

(9.11)

And, area of ∆AP1 P , T1 =

1 · AM · Y P1 2

(9.12)

Using the equation 9.8, we obtain, T1 =

1 · AM · P V 2

(9.13)

Therefore, from the equation 9.9 and the equation 9.15, it is proved that, T1 =

1 · T0 8

(9.14)

Figure 9.2: Top - Archimedes Midpoint Displacement Method for dissecting of a parabolic segment into infinitely many triangles; Bottom - The sizes of parent triangle and its child triangle.

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Now, the area of the polygon Pn , can be represented by the summation of the areas of all the triangles. Therefore, Area(Pn ) = T + 21 (

T T T T ) + 22 ( 2 ) + +23 ( 3 ) + ...... + 2n ( n ) 1 8 8 8 8

(9.15)

i.e., 1 1 1 1 1 + 1 + 2 + 3 + ..... + n ) · T (9.16) 0 4 4 4 4 4 The larger the value of n, the more accurate is the area of the parabolic segment. In other words, a sufficiently large number of n can reduce the difference between the area of Pn and the area of the parabolics segment S, thus can provide almost accurate value of the area of S. In modern mathematics, the area S can be represented by, Area(Pn ) = (

S = lim ( n→∞

1 1 1 1 1 + 1 + 2 + 3 + ..... + n ) · T 0 4 4 4 4 4

(9.17)

And, from equation 9.9 h1 =

1 h0 4

(9.18)

i.e., 1 height(n) (9.19) 4 From the above equation 9.17 it is understood that the sequence of Relative displacement factor for the midpoint displacements to construct the parabola are (1/4)0 , (1/4)1 , (1/4)2 , . . . .., (1/4)n for n number of iterations, and therefore, based on the equation 9.19, therelative height factorfor constructiong a patabolais 1/4. height(n+1) =

9.2.2

Generalization to Blancmange function: Takagi-Landsberg Curve

After centuries gap, in 1910, Japanese number theorist Teiji Takagi introduced a new sequence for constructing the parabola. Instead of using the sequence of (1/4)0 , (1/4)1 , (1/4)2 , . . . .., (1/4)n as relativesizef actors, Takagi used the sequence of (1/2)0 , (1/2)1 , (1/2)2 , . . . .., (1/2)n [79]. By doing so, a remarkable change is noticed that the smooth curve of the parabola is transformed into a highly unsmooth curve, later known as Takagi curve. The mathematical formulation of the Archimedes’ parabola and the Takagi curve was expressed by Blancmange function, identified for the first time by Takagi in 1903, and reinvented and generalized by van der Waerden in 1930 [47]. The Blancmange

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function of Archimedes’ parabola as a Cartesian equation on the unit interval is formulated by,

A(x) =

∞ X 1 ( )n s(2n x) 4 n=1

(9.20)

Whereas, the Blancmange function of Takagi curve is,

Blanc(x) =

∞ X 1 ( )n s(2n x) 2 n=1

(9.21)

where s(2n x) is sawteethf unction, in which s(x) = min(n∈z) |x − n| that is, s(x) is the distance from x to the nearest integer. Later, German mathematician George Landsberg apply w as the relativesizef actor which is 1/2 < w < 1 and performed the midpoint displacement by following the same pattern of sequence, i.e., (w)0 , (w)1 , (w)2 , . . . .., (w)n , thus obtain a result of another highly unsmooth curve which is nowhere-differentiable but uniformly continuous function [?], and expressed by

Tw (x) =

∞ X

(w)n s(2n x)

(9.22)

n=1

From the Takagi-Lindsberg function, the periodic version of the TakagiLindsberg curve can also be defined by, Tw (x) = s(x) + wTw (2x)

(9.23)

The equation 9.23 shows that the Takagi curve (for 0 < x < 1) is the attractor of the family of two affine contractions fL and fR , that are,  fL =  fR =

x y

x y



 =



 =

x wTw (2x)



1−x wTw (2x − 1)

if 0 < x < 0.5

(9.24)

 if 0.5 < x < 1

(9.25)

By using the above recursive definition of Takagi-Lindsberg function which is a Cartesian equation, the Takagi curve can be easily constructed through the process of IFS.

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Takagi-Landsberg Curve Construction: Vector-Based Midpoint Displacement Method

The above affine transformation process can also be performed using vector method which is based on the same principle of the Iterated Function System, but by applying the vector-based affine transformation instead of the Cartesian affine transformations. While, in the Cartesian system always two different transformation functions (fL and fR ) are required at each new iteration, in the vector system only one transformation function fm is sufficient, and it commonly operates on each newly born line at every new iteration. As an example, shown in ’Figure 9.3’ the endpoints of a line AB are considered as two vectors v1 at A and v2 at B. A unit vector m1 is generated in the midpoint of the line AB and then displaced vertically (towards the global Z-direction) by a relative size factor of δ. This displaced vector is a new vector m gets a position at C. This way, the only affine transformation fm of vector v1 creates a new vector m. Then, v1 , m and v2 are connected, thus produced two new lines AC and BC. In the next step, the same process of fm is repeated by appointing now A as v1 and C as v2 for AC line, and C as v1 and B as v2 for BC line. Thus, at each step, each newly born line’s endpoints are considered as v1 and v2 and repeat the same process by applying fm only ad infinitum.

Figure 9.3: Midpoint displacement using vector method.

For modeling the Takagi-Landsberg curve, plays the key role for defining the final outcome of the curve. For the Takagi-Landsberg curve, L 1 δ = ( )v where 2 2w

(9.26)

and L is the length of the first line, i.e., the distance between first two vectors v1 and v2 (Figure 9.3 and Figure 9.4). This process is translated into a scripting code which could generate the digital model of the final outcome. The iteration number n and the relativesizef actor w are the key variables as inputs that make the model parametric and control its curve texture. The parameter of w value ranges from 0.25 to 1.0, and changing this value from 0.20 to 1.0 in the code for more than 4 iterations shows the transformation of smooth parabolic

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167

Figure 9.4: Construction of a curve using vector-based Midpoint Displacement Method.

curve (w = 0.25) to unsmooth Takagi-Landsberg (w 6= 0.25) curve (Figure 9.5). As the w value increases, the roughness of the curve also increases.

Figure 9.5: Takagi-Landsberg curve and its Hausdorff dimension with the changing of w value.

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Figure 9.6: Relation between the relative size factor w and the Hausdorff dimension DH of TakagiLandsberg curve.

9.2.4

Hausdorff Dimension of Takagi-Landsberg Curve

Hunt (Hunt1998) estimated the Hausdorff dimension of the fractal curve, which follows the Takagi-Landsberg function, is DH = 2 +

log (w) for 0.5 < w < 1. log (2)

(9.27)

When 0.5 < w < 1, the Hausdorff dimension of the Takagi-Landsberg curve ranges from 1.0 to 2.0 (Figure 9.6). It is important to mention that when its Hausdorff dimension reaches 1.0, then the curve is actually a borderline curve of dimension 1.0, but of logarithmically infinite length [47]. However, if 0 < w < 0.5 and w 6= 0.25, then the Takagi-Landsberg function becomes a Lipschitz function, which implies that the curves are rectifiable, with the fractal dimension saturated at the value 1.0 [22]. ’Figure 9.5’ shows the parabola and different Takagi-Landsberg curves based on w value and their corresponding Hausdorff dimensions.

9.2.5

9.3 9.3.1

IFS Coding

Application 2a: Takagi Arch Structure Introduction

In the field of construction, there are very few examples where the fractal concept is intentionally applied in arch construction. The ancient example where

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we notice the properties of fractal geometry in arch construction is the Pont du Gard bridge, a hallmark of Roman engineering, built in the 1st century AD (Figure 9.7a). In this structure, the repetition of arches in three tiers follows the fractal-like characters, especially on the top two tiers where one arch is replaced by three self-similar arches in the next upper layer, intending for attaining the structural rigidity in long span structure. ’Fractal layered arch’, designed by Mae and Aasayama, is a recent example, where the fractal concept has been intentionally applied for achieving the higher structural stiffness in the steelmade arches, designed after analysing its high structural strength against heavy wind and vertical roof load (Figure 9.7b) [43]. In this section, a different fractal based arch structure will be designed by adopting the Takagi-Landsberg curve. While, the Mae and Asayam’s fractal-basd arch configuration has a tendency of the vertical and horizontal expansion, the Takagi-Landsberg curve has always the tendency of the vertical growth only and it does not occupy the horizontal spaces, which is a kind of functional advantage than that of the Mae and Asayama’s fractal arch.

Figure 9.7: (a) Pont du Gard bridge in Gard, France; (b) Fractal Layered Arch in Tokyo, Japan.

9.3.2

Geometric Modeling

Takagi arch is the direct adoption of the Takagi-Landsberg curve which is parametric. When w 6= 0.25, the curve has a set of downward curve ends as multifoil arches. In this type of configuration, if vertical loads are applied, then the Takagi arch will not be strong enough. Theirfore, to stiffen the arch structure, an addional frame is needed which may act as tensile or compression depending on the applied force and the main arch. The main arch is the type − 2f ractal (i.e., intersection of self-similar sets) obtained at nth iteration. The frame is made from the same Takag-Landsberg curve, but taking its type−1f ractal (i.e., union of self-similar sets) figure obtained after after (n − 1)th iteration. The input design variables for both the main arch and additional frame structures

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are the same.

9.3.3

Architectural Design

The Takagi arch is designed from the 5th iterated figure of Takagi-Landsberg curve. It is further connected by an additional frame which is developed from the 4th iterated figure of Takagi-Landsberg curve. The whole configuration of the Takagi arch including its additional frame is constructed by wooden bars having square cross-section with 2.4 cm each side.

9.3.4

Structural Analyses

We have performed the finite element analyses on this structure by transforming its geometric model into a finite element model. For this, all duplicate line segments and overlapping connecting points have been removed. All the remaining lines and points are considered as the bar elements and the structural nodes as hinged joints respectively. The endpoints, i.e., the feet of the main arch are pinned and restrained at all directions. For the analyses, we have applied uniformly projected vertical load, asymmetric load and horizontal wind load (Figure 9.8). It is well known that a parabola is a very efficient shape against the uniformly projected loads, and that is the reason we see many long-span structures, especially bridges as built examples on which uniformly projected loads are applied.

Figure 9.8: Different load cases. Left - Unformly projected load, Center - Asymmetric load and Right - Wind load.

However, it is not well known that how parabolic arch acts against other applied loads apart from uniform projected loads. More interestingly, it is quite unknown that how parabolic arch performs when its smooth curve turns into non-smooth fractal curve under different applied loads. This query was one of

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the motivations that drove us to analyze the structural characteristics of fractalbased parametric arch structure. We have performed a nonlinear static analysis on the parabolic and Takagi arches by using FEM solver software. ’Figure 9.9’ and ’Figure 9.10’ show the results calculated by the FEM solver.

Figure 9.9: Axial forces due to uniform projected loads; Left – Parabolic arch structure (w = 0.25); Right – Takagi arch structure (w = 0.5).

From the ’Figure 9.9’ we observe that the internal forces are mainly carried by the outer curve in the parabolic arch under uniformly projected loads, and its additional frame does not actively take part to carry the applied loads. But, in Takagi arch (w = 0.5), in fact, the outer curve plays minor role for carrying the applied forces when it is subjected to uniformly projected loads. In this case, the additional frame acts the major role for carrying the uniformly projected loads. It means, in this load condition, when w = 0.25, we do not need any additional frame. But, if w = 0.5, then we need an additional frame structure which should be strong especially in compression. On the other hand, under

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Figure 9.10: Axial forces due to asymmetric (right half) applied loads; Left – Parabolic arch structure (w = 0.25); Right – Takagi arch structure (w = 0.5).

asymmetric load, shown in ’Figure 9.9’, the role of additional frame in the parabolic arch structures is again almost none; while, in the Takagi arch, a part of the additional frame is highly active.

9.4 9.4.1

Application 2c: Fractal-Based Grid Shell Structure Introduction

In the beginning of the 20th century when reinforced concrete technology became prevalent in architecture and construction industries, a list of innovative and freeform shell structures begun to appear. At that time paraboloid became one of the most popular and efficient geometric forms for designing shell and other space structures because of its excellent load distribution capacity

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under uniformly projected loads and self-weight. Even now, paraboloid is one of the most useful structural forms for designing shell structures. The surfaces of most of these shell structures, which are either paraboloid or other regular or Euclidean geometric forms used as domes, vaults, ellipsoid, etc., are smooth except some examples of multi-folded or complex origami-based shell structures. However, mathematically, most of these surfaces are two-dimensional, i.e., integer dimensional. So far, numerous investigations have been done on the structural behavior and the design optimization of these regular shell structures. In this paper, however, we aim to propose a new form for designing a shell structure based on a non-conventional form-system which is fractal geometry and which is irregular in shape and fractional dimensional. In fact, our core interest is to explore the design and structural impacts of the relative size factor w, which is responsible for transforming a two-dimensional paraboloid surface into a fractional-dimensional fractal surface. With this key objective, we plan to design a grid-shell structure using Archimedes’ Midpoint Displacement Method which was modified and generalized by Takagi and Landsberg. The design will be made parametric by keeping the relative size factor w as a design variable.

9.4.2

Geometric Modeling

The basic shape for designing our proposed shell structure is a paraboloid which is the surface counterpart of Archimedes’ parabolic curve. Mandelbrot called this paraboloid as M ountArchimedes and the surface counterpart of the TakagiLandsberg curve as M ountT akagi [47]. Our design model is a generalized parametric surface that is sensitive to the w value, which ranges from 0.20 to 1.00 and which exploits the surface from smooth (w = 0.25) to unsmooth, i.e., fractal (w0.25). We will call this parametric surface as T akagisurf ace. This surface is constructed by using the same process of Midpoint Displacement Method. We have generated this parametric surface from an equilateral triangle as the initial base. Similar as the construction of Takagi-Landsberg curve, vector-based Midpoint Displacement Method has been applied to produce the Takagi surface. The vertices of the base triangle are considered as three vectors v1, v2 and v3 (Figure 9.11a). Midpoint of each side of the triangle has been elevated along global Z-axis by which is a function of relative size factor w, shown in equatione 9.28, and which is the same for constructing Takagi-Landsberg curve. 1 L (9.28) δ = ( )v where 2 2w and L is the length of each side, i.e., the distance between any two vectors. All the elevated midpoints are then connected by lines that give four new triangles (Figure 9.11b). In the next step, the same method of midpoint displacements is repeated on each newly born triangle which is considered as the

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new base and whose vertices are assigned as the new v1, v2 and v3 for the next generations (Figure 9.11c). This process of repetition is continued for multiple generations. After few generations we start getting a parametric surface in a three-dimensional space which is dependent on the w value (Figure 9.12). When w = 0.25, the surface becomes paraboloid, and when w is more than 0.25, the surface become rough, i.e., fractal. ’Figure 9.13’ shows the different outcomes of the surface with the changing of w value. Apart from w value, the shape of the original base also determines the different outcomes of such surface if applied the same process. ’Figure 9.14’ shows the different resulted shapes when the bases are square and pentagon. v 3

v 2 v 1

Figure 9.11: Midpoint Displacement Method applied on an equilateral triangle.

Figure 9.12: The construction of a Takagi surface after a number of iterations when w = 0.5.

Figure 9.13: Transformation of paraboloid’s smooth surface to fractal surface and their fractal dimensions.

9.4.3

Hausdorff Dimension of Takagi Surface

A paraboloid is a 2-dimensional surface. Therefore, the fractal surface comes from the paraboloid’s smooth surface is in between 2 and 3, which means the higher value of fractal dimension will tend to transform the 2-dimensional

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Figure 9.14: Construction of different types of Takagi surfaces using different polygonal bases when w = 0.5.

paraboloid into a 3-dimensional solid. Accordingly, w is the main factor that determines the fractal dimension of the surface (Figure 9.15). Therefore, the fractal dimension is a function of w. According to Felton, et. al. [22], the Hausdorff fractal dimension of the Takagi surface is, DH = 2 +

log (w) for 0.5 < w < 1. log (2)

(9.29)

Figure 9.15: Relative size factor w Vs. Hausdorff dimension DH graph.

9.4.4

Architectural Design

The proposed gridshell structure is designed by transforming the Takagi surface into a structural grid. For the structural analyses based on the changing of w

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value, we transform the geometric model of the Takagi surface into a parametric model by the following ways: • The original shape of the base is an equilateral triangle with each side 4 meters. • Only the finitely iterated models are valid for the real world applications. So, we have taken the 4th iterated model of the Takagi surface. • Only the triangular mesh edges of the Takagi surface are taken as a grid and call it as a T akagigrid (Figure 9.16b). • For the structural reason by means of strengthening the grid structure from large deformation, an extra lattice frame has been added which is also a fractal (after 3 iterations) generated by using the same Midpoint Displacement Method (Figure 9.16a). • All lines of the Takagi grid and the additional frame are wooden bars having square cross-section with each side 2.4 cm. • All these bars are connected at the joints by a wooden sphere having 9 cm diameter. • Three feet of the final structure are vertically and horizontally restrained at the ground. • The gridshell is covered by panels (Figure 9.16d).

Figure 9.16: Fractal-based gridshell design when w = 0.5; (a) Additional frame structure (‘type 1’ fractal) obtained after 3 iterations; (b) Takagi grid obtained after 4 iterations; (c) Fractal gridshell design by combining the Takagi grid and the additional frame structures; (d) Panels fixed on the grid.

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9.4.5

177

Structural Analyses

Similar as parabola, paraboloid is also known as a very efficient and optimized shape under the uniform projected loads in three-dimensional space. In this study, we take a case of three-legged paraboloid shell. However, we are not familiar with the structural behavior of a paraboloid if it experiences non-uniform loads. In this context, we can doubt that a paraboloid may not be the versatile shape for a shell structure against non-uniform or asymmetric loads. One possible way to verify this doubt and to find the possible adaptable shape for a shell structure derived from the paraboloid shape under different load conditions is to transform the paraboloid shape into a parametric shape so that it can morph its own surface using some geometric variables, and then all the new form variations react to different applied forces. For this purpose, we have taken the parametric Takagi gridshell structure designed in the previous section in order to find the better shape for a gridshell structure that can tackle different types of loads. In this sense, the application of fractal geometry can be considered as an exploration for finding a new and verseatile shape for shell structures under varying load conditions.

Figure 9.17: Different load cases (a) uniformly projected loads, (b) asymmetric load and (c) horizontal wind load.

For performing the finite element analyses using FEM solver, we have transformed the geometric model of the Takagi gridshell into a finite element model by removing the overlapping lines and the duplicate connecting points. All the joints are hinged; hence, all the lines act as structural bar elements. Its feet are fixed to the ground such a way so that they are vertically and horizontally restrained. Three different loads are applied - uniformly projected loads (1 KN/Meter), asymmetric load (2 KN) and wind load (1/2 KN) (Figure 9.17). After this preprocessing stage, a nonlinear static analysis has been performed on the parametric finite element model of Takagi gridshell structure which is sensitive to w value. For this purpose, the parametric modeling tool and the FEM solver are connected such a way so that they can act interactively. In this way, FEM solver gives prompt feedback with the changing of w value. Our

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main objective of using the FEM solver is to see the forces flow behavior and the maximum displacements of the variations of gridshell structure under different applied loads. With this arrangement, first, we performed the nonlinear static analyses on the paraboloid gridshell when w = 0.25 and then on the Takagi gridshell when w = 0.5, and see their results to compare the structural behavior between the smooth form and the fractal form of both the gridshells (Figure 9.18 and Figure 9.19). Later, the maximum displacements of the different variations of parametric gridshell structure with respect to w value are recorded under three different applied loads in order to find a relation between the w value of the gridshell and its strength (1/δ) shown in ’Figure 9.20’.

Figure 9.18: High nodal displacements shown as red discs under uniform projected loads; Left Paraboloid gridshell when w = 0.25; Right - Takagi gridshell when w = 0.5.

From the analytical results (Figure 9.18, Figure 9.19 and Figure 9.20), we observe a significant changing of structural behavior with respect to the changing of w values. ’Figure 9.18 Left’ shows the nodal displacements along vertical axis is almost none at the top region in the case of the paraboloid gridshell structure (w = 0.25) when it is subjected to uniformly projected loads. We noticed some large displacements only at the bottom of its legs (red discs) which is due to a slight buckling effect. On the contrary, ’Figure 9.18 Right’ shows that the nodal displacement of the fractal-based gridshell (w = 0.5) along vertical axis is quite significant at the top region and mainly concentrated on the central part. Unlike, paraboloid gridshell, fractal-based gridshell does not show any significant buckling effect. The internal forces flow behavior is also different in both the gridshell structures shown in ’Figure 9.19’. While paraboloid gridshell tackles the uniformly distributed loads by itself and slightly shared by the additional frame structure, the fractal-based gridshell is heavily dependent on its additional frame and almost all the additional members take part

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Figure 9.19: Internal Axial forces; Left - Paraboloid gridshell (w = 0.25); Right - Takagi gridshell (w = 0.5).

in distributing the applied loads. This behavior is nearly similar to their two dimensional counterparts, i.e., parabolic arch and the Takagi arch (Figure 9.9 and (Figure 9.10). ’Figure 9.20’ shows the relation between the w value of the parametric Takagi gridshell structure and its strength which is inversely proportional to the maximum vertical displacement (Figure 9.20, Right) and vertically central nodal displacement (Figure 9.20, Left) under uniformly projected loads. It is noticed that the paraboloid shape (w = 0.25) is highly strong under uniformly projected loads as we expected if the displacements at legs due to buckling are avoided; but surprisingly, the strongest one is found when w = 0.23. When w > 0.25, then the strength of the gridshell structure starts decreasing very gradually. If the maximum displacement due to buckling is considered, then we find that the paraboiloid shape is weak enough until w = 2.8. The shape with w = 2.8 is the strongest one under uniformly projected loads considering buckling effect, and from w > 0.28, it starts loosing its strength again very slowly, but always stronger than paraboloid shape. When the structure is subjected to asymmetric vertical loads, and if we consider the maximum displacements including the major effect of buckling, then we observe, as w gets higher, the strength of the gridshell also gets higher till w = 0.5. After that, the strength is slowly increasing, and after w = 0.8, the strength becomes constant and no significant impact of w value on the strength of the structure is observed. However, if we avoid the major buckling effect, and consider only the displacement of central node, then we notice that the strength sharply gets higher till w = 0.3, and after that the strength gradually decreases. Under wind load, if we consider both the displacements (maximum and central

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Figure 9.20: Relation between relative size factor w and the stiffness 1/δ under uniformly projected loads, asymmetric load and wind load. Here, δ is the nodal displacements along vertical axis for uniformly projected l and asymmetric loads, and along horizont al axis for wind load. Top Displacement of the central node; Bottom - Maximum nodal displacement.

nodal), we notice that the strength gets higher with the increasing of w value. However, no significant change of strength is observed when w increases from 0.8 to 0.1.

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Therefore, from these results shown in ’Figure 9.18’, we can conclude that the gridshell structures having w value in between 0.23 to 0.25 are the best choices as the strongest structure if only uniformly projected loads are applied and if the major buckling that occur at legs are stopped. But, if we consider the no-uniform loads such as vertical asymmetric and horizontal wind loads, then fractal-based gridshell structure (w > 0.25) is better choice than the paraboloid gridshell structure. Accordingly, we may argue that the fractal-based gridshells are structurally stronger than that of the paraboloid gridshell structure if the additional frame structures are added and if the uniformly projected loads are not considered only. To find the most optimized shape from the parametric shape variations of fractal-based gridshells, a single-objective or multi-objective computational search process can be performed if a combination of different applied loads is considered. In this regard, it is important to mention that the strength of the overall fractal-based structure (w > 0.25) is largely influenced by the additional frame structure.

9.4.6

Computational Form Finding

The parametric model of the gridshell structure has been processed through ’Optimization Tool Box’ to find the optimal design with lowest deformation under different load conditions. Although, we may not need an automatic optimizer to find the best shape, because, from the graphs shown in ’Figure 9.20’, it ca easily select which one is the best. Yet, for accuracy, the optimization has been completed by taking only relative size factor w as the only variable whose domain has been set in between 0.2 to 1.0. The target is to find the shape which has the lowest deformation. In this case, only a single-objective optimizer is suitable, so Galapagos has been used as a member of the ’Optimization Tool Box’. The optimization has run for 100 generations, and resulted the outcomes shown in ’Figure 9.21’. After the optimization, it is shown the similar result that we could obtained from the graphs shown in ’Figure 9.20’. As a result, at the one hand it validates the accuracy of the optimization tool Galapagos, on the other hand it confirms that the gridshell with higher dimension is the better choice if we do not only consider the uniformly distributed loads. It is also observed that the additional members other than than the main grid lattice are mainly experience compression for higher dimensional fractal-based gridshell structures. Thant means, they actively involve in stress distribution. While, in the case of paraboloid gridshell (w = 0.25), these additional members are not active, and all the stresses are dealt by the main grid lattice. So, if we opt higher fractal dimensional gridshell structure, then the additional members should be strong enough.

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Figure 9.21: Optimal designs under different load conditions optimized by single-objective optimizer Galapagosasamemberof the0 OptimizationT oolBox0 .

9.4.7

Physical Prototype Construction

After the finite element analyses of the virtual model of the fractal-based gridshell structure, we planned to check its practical feasibility as a structure by making its real scale physical prototype. Another purpose of the prototype construction was to verify the constructability of such complicated structural form. The big advantage of making fractal-based complex structure is its self-similar modules which allowed us to make a complex assembly of one module only,

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and then repeat the same module for the assemblage of the main structure. For the prototype construction, we took the fractal-based gridshell structure whose w = 0.5. In this structure we had three sets of self-similar modules as three legs, and one unique module in the centre (Figure 9.22, Right). For the easy construction and to avoid using too many bars, we took the 3rd iterated model instead of 4th iterated. Besides, we removed some bars from the additional frame. These removed bars were selected based on their less involvement in internal forces distribution, and which occupied some usable internal space (Figure 9.22, Left). Before starting the fabrication process, we again performed the finite element analyses on this new grid shell model to check any large deformation and buckling issues. Here, we considered only the gravitational and wind loads. The analyses had confirmed its strength and stability under these load conditions. Henceforth, we started fabricating the modules first, and then finally assembled them to get the final structure. The whole fabrication process was intended to be done by manual process. The purpose was to present that such complicated shape can be realized without any robotic and computer aided manufacturing tools, and thus representing the versatility of fractal geometry for any type of constructions.

Figure 9.22: Left - Simplified assemblage of additional frame structure; Right - Four structural modules, three of them (1, 2 and 3) are perfectly self-similar.

The whole structure was completely made with wood. The joints are prepared by wooden balls so that each bar member coming from different directions can be easily connected at a specific point (Figure

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Figure 9.23: Manual fabrication of different parts for each structural module.

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Figure 9.24: Fractal-based gridshell structure without covering panels.

185

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Figure 9.25: Final structure after fixing the covering panels.

Chapter 10

Application 3 - Folded Roof Design using the Concept of Random Fractals 10.1

Introduction

In nature, there are plenty of objects that have the impressions of self-similarity, but they are not strictly self-similar or exactly identical although exhibit fractal behavior. 1 This type of fractal is known as statistically self-similar fractal or random fractal, displaying a high visual complexity. Random fractal repeats a pattern stochastically, yet a part of it sometimes gives a near impression of the whole. Coast lines, mountain terrain and trees are such examples. A part of tree branches gives an impression of the whole tree although they are not exactly similar in reality. Random fractals are useful for simulating the accurate replica of natural phenomena. Because of its random characteristics, this type of fractals can be generated by using stochastic rule. There are several methods to generate random fractals, such as Levy flight [46], percolation clusters, self avoiding walks [28] and Brownian motion [49] methods. Among these processes, Random Midpoint Displacement method and the Diamond Square (DS) Algorithm method are based on simple rule that produce very natural 1 The research of this chapter was presented in the proceedings of the 2nd International Conference on Biodigital Architecture Genetics in 2014: Iasef Md Rian, Mario Sassone and Shuichi Asayama, ‘Nature-Inspired Fractal Geometry and Its Applications in Architectural Designs’, in the proceedings of 2nd International Conference on Biodigital Architecture Genetics, Barcelona, Spain; June 2014.

187

188 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals appearances of fractal terrains. These processes can be executed using the Iterated Function Method. In this chapter, the DS Algorithm method has been used for designing nature-inspired random forms for space structures. The DS Algorithm method is the three-dimensional extension of the Random Midpoint Displacement method. Both of these methods are explained in the following sections.

10.2

Random Midpoint Displacement Method

The midpoint displacement method, which was initially popular for generating smooth curves (first used by Archimedes in the 3rd century B.C. to produce parabolic curve) and self-similar unsmooth curves (used by Teiji Takagi and Landsberg in the early 20th century to produce Blancmange or Takagi fractal curve), is useful for generating unsmooth random curves with unpredictable outcome. This can be done by two processes. The first process is based on the Archimedes’s Midpoint Displacement Method where the scaling factor (λ) of displacement with respect to the length of base line or previous line (L) is always constant. In this process, at the first step, the midpoint of a straight line is vertically moved up by a certain height (δ0 ) which is a scaled value (λ) of the base length (L0 ), i.e., δ0 = λ · L0 .Then, this elevated midpoint is connected by the endpoints which produces two new lines. In the second step, the midpoint of each new line is again moved up vertically by the same scaled value (λ) of its new line length (L1 ), and this displacement is δ1 = λ · L1 . In this stage, the midpoints of both the two new lines can be moved up vertically. But, to randomize the curve, the midpoint of the left new line can be vertically moved up, while the midpoint of the right new line can be vertically moved down. In the next steps, this process of left midpoint up and right midpoint down is continued for each newly generated line segments for infinite times. This process shown in the ’Figure 10.1a’ is deterministic yet offer random appearance. To make the outcome of the curve unpredictable and more random, one can flip a coin to determine whether the midpoint of left newly line and the midpoint of right newly line born from each previous line will go upward or downward. The second process is based on the Gaussian Midpoint Displacement Method which is more complex form of simple Midpoint Displacement Method discussed above. In this method, displacement of the midpoint of each base line at each step is vertically displaced by a random value taken from Gaussian distribution (Figure 10.3 Left). This random displacement (σ) at each step is scaled by a factor (λ) which is related to the Hurst exponent (H) that varies in between 0 to 1 in the case of the one-dimensional fractal curve. The relation between the

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Figure 10.1: (a) Deterministic fractal curve with random impression using Archimedes’ Midpoint Displacement method; (b) Unpredictable random fractal curve using Archimedes’ Midpoint Displacement method and flipping a coin; (c) Unpredictable random fractal curve using Gaussian Midpoint Displacement method.

scaling factor (λ) and the Hurst exponent is as follows: λ = 2−H

(10.1)

It means, when the value of Hurst exponent gets higher, the scaling factor λ gets smaller, which means the fluctuations of the values of midpoint displacements get smaller and the fractal curve becomes less noisy. And, when H becomes 0.5, the distribution generated becomes the same as brown noise because of its relation to the Brownian motion [8].

10.2.1

Fractal Dimension Calculation

The primary feature of a random fractal is its roughness or noisiness, and this characteristic is measured by fractal dimension. Higher the roughness or noise of a shape, higher is the fractal dimension of it. There are several methods that can measure the fractal dimensions of random fractals, among which the Box Counting Method is a very simple and useful method [8]. In this method, a square grid having a 2n box size is overlaid on a curve and the number of boxes is counted that contains any part of the curve. In the next steps, the box size of the grid is reduced to 2(n−1) , and the again the number smaller boxes are counted that contains any part of the fractal curve. This process is continued till the box size becomes 2. Then, a log-log graph drawn by taking number of boxes along X-axis and the size of boxes along the Y-axis, and their coefficient gives the value which the fractal dimension of the fractal curve. This dimension is also called as the Box Counting dimension. In this method, the

190 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals fractal dimension is only possible to calculate once the fractal curve is finally generated.

Figure 10.2: Left - Box counting method for a random fractal curve; Right - Log-log graph for calculating the fractal dimension of the curve which is the slope of the trend line. The fractal dimension is 1.385.

In the case of random fractal curves constructed by using Gaussian Midpoint Displacement method, we noticed that the factor of the Hurst exponent H results the roughness or noise of the curve which is a property that can be measured by fractal dimension. Hence, there is a relation between the Hurst exponent H and the fractal dimension D of the resulted fractal curve [2], which is, For the one-dimensional fractal curve, D =2−H

where, 0 < H < 1

(10.2)

Therefore, the scaling factor, λ = 2D−2

(10.3)

It means, in the case of random fractal curves produced by Gaussian Midpoint Displacement Method and using the equation 10.3, we can control its noise by deciding the value of fractal dimension D as an input value at the beginning before the curve construction, and this is a unique advantage for modeling a

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fractal dimension-controlled fractal curve. ‘Figure 10.3(Right)’ shows that how the fractals dimension controls the noise of the fractal curve, keeping the deviation () as constant (−1 ≤ σ ≤ 1). This method is commonly used to generate an outline of a distant mountain range in the field of visual graphics.

Figure 10.3: Left – A Gaussian distribution centered at zero; Right - Fractal noise generated using the different values of fractal dimensions as a parametric input. [8]

The midpoint displacement is also applicable in exploiting a two-dimensional surface following the similar process which exploits a one-dimensional curve. While, the Archimedes’ Midpoint Displacement Method can produce a smooth paraboloid as a two-dimensional counterpart of one-dimensional parabola, the Gaussian Midpoint Displacement method can produce noisy fractal surface or the topology of natural random terrain as a two-dimensional counterpart of onedimensional random fractal curve. The two-dimensional operation of a surface using Gaussian Midpoint Displacement Method for producing a natural terrain is discussed in the ’section 3.2’ for modeling a roof surface.

10.3

Diamond Square Algorithm Method

The Diamond Square (DS) Algorithm is a stochastic algorithm that results not a deterministic fractal surface, but a random surface with fractal behavior. The DS algorithm, which is the three-dimensional counterpart of Gaussian Midpoint Displacement Method discussed in the previous section, considered as a very simple method which results a realistic simulation of natural terrains. The concept of DS Algorithm was first introduced by Fournier, Fussell and Carpenter [24]. Later, it was analyzed by G.S.P. Miller citeMiller1986 and found a little flaw in the algorithm that results visible vertical and horizontal ’creases’ in the model. As a result, Miller proposed a different algorithm based on weighted averaging and control points [53]. Later, in 1997, Paul Martz claimed that

192 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals Miller’s complain about Fournier and his team’s DS Algorithm might be wrong, because Martz reused the same principle of DS Algorithm by using the proper seeding values for random displacement so that the ’creasing’ problem does not appear. In this paper, Matz’s simplified method of DS Algorithm has been adopted for modeling a surface replicating natural terrain in the following way. The DS Algorithm follows the same rule of the vertical displacements of midpoints. What is important and tricky in this algorithm is the pattern of seeding of values which is able to create a surface of natural terrain and whose randomness and surface noise can be easily controlled. This algorithm follows four steps out of which two steps are main key steps. The key steps are ’diamond step’ and ’square step’. (i) Initial Step In the initial step, a large empty two-dimensional square grid is constructed whose array size is equal to 2n+1 , where n is the number of iteration. The four corners of the square grid are assigned with some random height values as the first seed for the operation. In the ‘Figure 10.4’, as a demonstration, we have assumed 5x5 square grids where the corners are A, B, C, and D and their height values are A, B, C and D respectively marked as red dots. (ii) Diamond Step In the ‘diamond step’, the center point (E) of the square which is obtained by intersecting the two diagonals (AC and BD), is vertically displaced by a value which is the average value of the heights of four corners added with a random value that ranges from −∆ to Delta. Thus, the displacement of E is, E=

A+B+C +D + rand∆ 4

(10.4)

This displacement of the center point E transforms the flat square into a pyramid. At this stage, E is a new value, i.e., a new seed, and that is why it is shown in red dot, while the corner heights are shown in blue dots as the previous values in the ‘Figure 10.4’. When there are multiple squares arranged in a grid, then this step of central displacement of each square produces diamonds. (iii) Square Step In this step, when the grid has multiple squares, then the height of the center of each diamond is assigned by a value which is the average heights of its corners with the addition of a random perturbation similar way as in ‘diamond

Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals 193

Figure 10.4: The method DS Algorithm that produces fractal terrain. Red dots are the new vertices while the blue dots represent previous vertices. Top and middle - Plan views; Bottom – Isometric views.

step’. In this step one thing is important to notice that the diamonds, which are at the edges of the square grid, have three corners, while the diamonds inside the array have four corners. Hence, the each of the diamonds that lay at the edges of the grid, are assumed to have four corners where fourth corner is an imaginary corner which has the same height values as the vertex of the pyramid obtained in the diamond step. By assuming so, we calculate the height of the center point of these edged diamonds. Therefore, if we see the ‘Figure 10.4c’, we notice that there are four diamonds resided at the edges of the main grid and each of them has three corners where red dots are their peaks. Therefore, the calculated height values of their peaks, i.e., center points are,

F =

A+B+E+E + rand(∆) 4

(10.5)

G=

B+C +E+E + rand(∆) 4

(10.6)

194 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals

H=

C +D+E+E + rand(∆) 4

(10.7)

D+A+E+E + rand(∆) (10.8) 4 But, in the case of internal diamonds having four existent corners, the height value of the center of each internal diamond are calculated by the average of the heights of its four corners plus a random perturbation. As an example, in the ‘Figure 10.4d’, the value of N which is the center of diamond EJFK is, I=

N=

E+J +F +K + rand(∆) 4

(10.9)

O=

E+K +G+L + rand(∆) 4

(10.10)

P =

E+L+H +M + rand(∆) 4

(10.11)

O=

E+M +I +J + rand(∆) 4

(10.12)

Similarly,

(iv) Recursive Step Next, the ‘diamond step’ is repeated to each square that was created in the ‘square step’, then the ’square step’ is repeated to each diamond that was created in the ‘diamond step’, and continue this process recursively until the grid becomes sufficiently dense and gives an impression of a crinkled surface. The ‘Figure 10.5’ shows the transformation of a flat surface into natural terrainlike or randomly folded crinkled-like surfaces after increasing the number of iterations using the process of the DS Algorithm.

10.3.1

Fractal Dimension Calculation

In the DS Algorithm, the range of random perturbation is key factor that controls the noise of the surface generated. This range is based on the value of roughness coefficient which is referred to the Hurst exponent H. If the Hurst exponent H is 1.0, then the random perturbation is multiplied by a value that ranges from -1.0 and 1.0. At each new iteration, the value of H is reduced by a scaling factor λ such a way so that, λ = 2−H

(10.13)

Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals 195

Figure 10.5: Natural terrain-like crinkled surface generation from a flat surface using DS Algorithm.

Therefore, if the height scale or z-deviation is , then the random value of the displacement in the first iteration is, ∆1 = λ · δ = 2−H · δ

(10.14)

∆n = λn · δ = 2−nH · δ

(10.15)

i.e., at nth iteration,

According to Barnsley, et. al. [60], there is a relation between the Hurst exponent H and the fractal dimension D of the resulting surface produced by the Gaussian random midpoint displacement process, which is, D =3−H

(10.16)

λ = 2D−3

(10.17)

i.e., From the above equation, if H gets larger, the scaling factor λ gets smaller and so the fractal dimension D.

196 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals

10.4

Fractal-Based Folded-Plate Shell Structure

10.4.1

Introduction

The principle of folding as a tool to develop a general structural shape has been known for a long time. Folded structure systems which are analogous to several biological systems such as found at broad leaf-tree leaves, petals and foldable insect wings, are adopted to be employed in a new, technical way. The aerospace and the automotive industry, e.g. apply this principle to create self-supporting wall, mould and slab elements with a high load capacity out of flat and thin semi-manufactured metals. This stands in contrast to the building industry, where the principle of folding has played a secondary role so far. There are two different folding system, that are rigid folding system and kinematic folding system. Both of these systems are increasingly useful in the field of construction in the recent years. However, the rigid folding system has been used from the long past. Especially, in the beginning of the 20th century, when the reinforced technology emerged as new construction technology that facilitated modern architects to play with free form architecture, folding shape also became a high interest among architects and engineers. The unique selfstiffening quality of folds not only influences engineers to develop knowledge about the mechanical benefits out of it, but also inspire architects for its unique design appearance. By principle, folding systems represent one category of plane structural surfaces, alongside with plates and slabs. Their special structural behavior is due to their structural subdivision arrangement in pairs which correlate with each other and so they are connected through a shear connection. The structural characteristics of folding structures depend on the shape of the folding (longitudinal or pyramidal), on their geometrical basic shape, on its material (concrete, timber, metal, synthetics), on the connection of the different folding planes and on the design of the bearings. The characteristics of the folding structures are interactivity related to each other.

Figure 10.6: Structural condition of folding structures. [75]

The structural behavior of folded shapes are unique. The inner load trans-

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fer of a folding structure happens through the twisted plane, either through the structural condition of the plate (load perpendicular to the centre plane) or through the structural condition of the slab (load parallel to the plane) (Figure 10.6) . At first, the external forces are transferred due to the structural condition of the plate to the shorter edge of one folding element. There, the reaction as an axial force is divided between the adjacent elements which results in a strain of the structural condition of the slabs. This leads to the transmission of forces to the bearing.

Figure 10.7: (a) Example of longitudinal folding as a structural shape, (b) Example of composite pyramidal folding as more complex structural shape. [75]

Figure 10.8: Left - Air Force Academy Chapel, Colorado Springs, USA, architect Walter Netsch, year 1962; Right - Interior of the Yokohama port terminal designed by Foreign Office Architects (FOA) in 1995.

There are plenty of structures that are designed by using regular folding system which can be easily described by simple mathematical formulations and modeled by simple geometric operations. In the 20th century, the art of folding, got new dimension when Japanese mathematicians developed the mathematical formulation of foldings, today it is known as Origami. This mathematics-based folding has revolutionized and offered numerous opportunity to designers for finding new and innovative forms by folding. In the late 20th century, after the development of origami science, several free-form complex designs of folded structures have been designed and

198 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals constructed. It added a new trend in architecture, even in other design fields including fashion and product designs, the contemporary period when the art of complexity is its peak of interest. During this time, origami-based irregular and sophisticated forms of folded structures have started to appear.

Figure 10.9: Different forms of folded structures based on their geometries and constructions. [75]

Computer has further widen the possibilities of new designs for folded structures. Simulation based structural analyses and optimization software has allowed designers to choose the optimal folded shape out of different possibilities. Nevertheless, this development has been progressed within the frame of Euclidean, regular and in some cases non-linear geometric systems. As a result, the possibilities to obtain and model different innovative forms of folds are confined within these geometric boundaries. As a consequence, we are unfamiliar with other possible folded forms that lie outside of these geometric arenas, and

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hence, we do not know their structural behavior and possible potencies. Interestingly, in nature, we see a plenty of objects that are folded shapes such as mountains, crinkled paper, etc., but unexplainable by regular and conventional geometric systems. These folds are also irregular, but randomly irregular that do not follow any visible geometric rule. In this stage, the concept of fractal geometry is highly useful which is able to explain and simulate such folded shapes by using some tricky but simple rules. Fractal geometry allow us to investigate about the possibilities of new forms generated by folding in a random way, and provide the opportunity to explore the structural property and potency of this randomly folded shapes if applied in designing structures. In this regard, the stiffness behavior of a crinkled piece of paper gives us a quick idea about the advantage of such shapes. So far, insufficient investigations have been done on finding the structural behavior of random folded structures, perhaps, due to the complex mathematical descriptions of such complicated shapes. In this chapter, a random folded plate shell structure has been designed to evaluate its geometric properties, structural behavior and practical feasibility. The concept of random fractals has been used to mathematically express its forms, and the Diamond Square Algorithm has been applied to generate the model. The model has been structurally analyzed using finite element analyses method. In the end, an optimization process have been performed to find the optimal shape, thus complete the computational morphogenesis of a random folded shell structure.

10.4.2

Geometric Modeling

The random folded-plate shell structure has been designed by adopting the terrain-like surface generated in the previous section using the Diamond Square Algorithm method. The ‘Generative Tool Box’ has been used here to generate the geometric model of the proposed folded shell structure. The input variables have been set such a way so that it can produce a model which has adequate height and enough indoor space with less numbers of supports. The height is controlled by z-deviation factor δ, while the complexity and noise is controlled by the fractal dimension D, i.e., the model produced by the generative tool is a parametric model, whose parameters of the variables are fixed as follows to design the first model by keeping the architectural demands in mind. Only 15 supports are assigned as the supports for the structure. These supports are the 15 vertices that have the lowest heights. Because of the factor of Gaussian randomness in the function of Z-limit value, the same magnitudes of the variables produce countless different form variations some of which are structurally infeasible with the 10 lowest supports, and some are feasible (’Figure 10.10). After some trials one arbitrary model has been selected which are

200 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals Table 10.1

Variables Size, s Z-Limit, δ Fractal dimension, D Iteration number, n

Fixed Values 17 meters 1.5 2.5 4

Figure 10.10: Elevations of some random form variations of the proposed folded-plate shell structure showing the positions of their supports for checking the apparent structural feasibility.

apparently structurally feasible. Later, an optimal form will be chosen by using the computational searching process in the next section.

Figure 10.11: an architectural impression of the random folded-plate fractal shell structure

The selected model of the structure is constructed by using RCC with the thickness of 8 cm. all the 15 supports are RCC pillars with square cross-sections having 30 cm X 30 cm size. It is the initial design. However, for the optimization process, the roof or plate thickness and the number of supports are made parametric so that the optimized thickness and number of supports can be achieved to satisfy a given condition and structural target.

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10.4.3

Structural Analyses

For the structural analyses of the above structure, the geometric model has been translated into a finite element model using the preprocessing tool of the ‘FEA Analyses Tool Box’ by considering the following aspects: • The roof surface is a mesh of triangular grid. • The mesh is composed of triangular shell elements. • All the overlapping shell elements are discarded. • The shell material has been chosen as a conventional concrete C25/30 (defined by the Eurocode), and the shell thickness has been defined equal to 10 cm. • All the supports are concrete column assumed that they are fixed joints restrained vertically and horizontally, even if that is not the real condition, as explained in the book ’From control to Design’. • Regarding the load case, gravity load as well as the self-weight and a live load of 6KN/m2 has been applied. After the analyses by ‘FEA Tool Box’, the maximum displacement is found 14 mm. ‘Figure 10.10’ shows the vertical displacement and principal stress of the structure. The analyses confirm that the structure is feasible and stable. However, this model might not be the best shape with this configuration. In the following section, the optimal shape of the structure has been found by using a computational optimization process.

Figure 10.12: Vertical displacement and principal stress.

202 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals

10.4.4

Computational Form Finding and Structural Optimization

As we see in the modeling of the random folded-plate shell structure that its geometric model is highly sensitive to its variables. Hence, it is impossible to find the optimal shape by using scheme by scheme process from the countless variations of the model. So, an automatic computational searching process is needed to find the optimal shape. The design target is to get an architecturally functional and spacious, and structurally feasible and strong structure. In this situation, the ‘Optimization Tool Box’ has been used for finding the optimal shape of the structure. The optimality of a structure depends on the objective, it can be single objective optimization, or multi-objective optimization, and accordingly a suitable tool is used. In this case, the objectives are multiple.

• The first objective is to obtain the shape whose deformation is the lowest • The second objective is reduce the weight of the structure with the strategy to save materials and cost. This folded shell model has some unique design variables that morph the shape of the initial model very distinctively. Commonly, the optimization of structural shapes are controlled by the variables which act within the Euclidean geometric system. Here, the variables act beyond the conventional geometric system and morph the shape into unconventional ways. Some of the variables are even not typical geometric entities, but they are different such as the number of iterations and the fractal dimension. For the optimzaition, the domains of these variables are constrained as follows to get the reasonable output that can be practically applicable and efficient. Table 10.2

Variables Z-Limit, δ Fractal dimension, D

Fixed Values 1.0 to 1.5 2.0 to 3.0

The operation has been performed in a multi-objective tool Octopus which is a readymade multi-objective optimizer software embedded in Grasshopper by making 50 population size and 100 maximum generations. The optimization gives the results shown in ‘Figure 10.13’. The optimization algorithm has been stopped after a fixed limited number of generations at 100th due to the high computational time necessary to perform all the FEM analyses, because, a better result could be obtained interrupting

Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals 203

Figure 10.13: Optimal shapes at different generations.

the algorithm only when the evaluated performance of at least an individual has reached a defined satisfactory value. The searching algorithm has resulted in a structural form with maximum displacements of 2.7 mm. One important fact has been noticed during optimization is the value of variables. Most of the optimal shapes at increasing generations are found at the highest value of

204 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals fractal dimension (3.0) and at the maximum of the z-limit value (1.5) that are constrained before the optimization.

10.5

Canopy Structure with Crinkled Roof

10.5.1

Introduction: Design Concept

Commonly, most of the existing architectural structures with branching supports have roofs which are flat in shape. From architectural point of view, flat roof is an impression of a man-made object, while the branching structure is an impression of nature. Here, it is aimed to design a canopy roof which is not flat, but irregular inspired by the morphology of natural terrain, so that it can gives an impression of nature similar as the branching structures. The reason behind deliberately mimicking the natural terrain for roof design is its crinkled surface which is apparently a self-stiffened shape. As an example, a flat sheet of a paper is flexible and not stiff, but after crumpling, its crinkled form, which looks similar to natural land terrain, becomes stiffer. Crumpling of a piece of paper is a relevant and ubiquitous example of a stress induced morphological transformation in thin sheets [6]. As a demonstration shown in ‘Figure 10.14’, if we place the flat piece of a thin paper on the top of a bottle, then two sides of the sheet bent down. But, if we crumple the same paper, and after unfolding if we place its crinkled version at the same position on the bottle, then the crinkled paper do not noticeably bent down, instead it stay almost non-deformed, because ridges of crinkles, i.e., its random folds resulted after crumpling act as stiffeners. Even, to some extent, the crinkled paper sheet can carry some load with a negligible deformation. This mechanical feature has motivated us to mimic such surface for the roof design.

Figure 10.14: Left - A piece of flat paper on a bottle; Right – A piece of crinkled paper on a bottle.

So, the main canopy structure will be composed of two main structural components. First is a tree-inspired branching structures which is an example of perfectly self-similar fractal and the second is a natural terrain-inspired crinkled roof which is an example of a random or statistically self-similar fractal. In

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the following sections, both of these structures’ shapes are modeled which are assembled at the end for designing the final canopy structure. There is an attempt by Falk and von Buelow [?] who proposed a non-flat roof for such tree-column supported canopy structure after hanging model experiments (Figure 10.15a). Their roof is a folded plate origami roof supported by optimally designed branching columns (Figure 10.15b). Nevertheless, the folded roof in their proposal does not resemble with any natural objects, because their design intuition might not be mimicking nature. However, we notice a few built examples of canopy structures, such as at ION Orchard in Singapore designed by Benoy in 2009 (Figure 10.15c) and at WestendGate in Frankfurt designed by Heinrich Rohlfing GmbH and Sternwede-Niedermehnen in 2010 (Figure 10.15d), where both the columns and roof form gives natural impressions.

Figure 10.15: (a) Thread hanging model for designing an optimized canopy structure [?], (b) Canopy structure with folded plates origami roof [?], (c) Canopy structure at ION Orchard in Singapore by Benoy (2009), (D) Canopy structure at WestendGate in Frankfurt by Heinrich Rohlfing GmbH, Sternwede-Niedermehnen. (2010)

10.5.2

Geometric Modeling of a Branching Column

Branching support is designed based on the exact process discussed in the ‘Chapter 6’.

10.5.3

Geometric Modeling of a Crinkled Roof

In the proposed canopy structure, the roof form is designed crinkled inspired by the morphology of natural terrain and by the self-stiffness character of a crumpled paper. Each crumpling produces unique pattern of random folds or crinkles. However, physically it is easy to crumple the paper infinite times and then obtain infinite patterns of crinkled surface. But, geometrically, it is difficult to simulate crinkled surface especially by using Euclidean or other regular geometric system and without any algorithmic process. Several methods have been developed that are able to simulate the realistic models of crinkled

206 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals surface which are mainly based on the random fold lines due to crumpling [7] [6] [56]. It is noticeable by a quick glance that the surface of crinkled sheet after crumpling resembles with the surface of fractal-like natural terrain, although the pattern of random fold lines in both the examples are slightly different to each other. While the geometric shape of crumpled paper is mainly related to the fold lines, the geometric shape of natural land terrain is mainly associated with the height maps. However, in the end, the morphologies of both of them are closely similar to each other from a particular scale of view. In this study, the design objective is to replicate nature’s form in roof design. Nakamura and Asayama [55] applied an Algorithm of Land Erosion to model a non-smooth random form of a space structure to replicate the nature’s land terrain topology and assess the mechanical behavior of such nature-inspired unconventional structure. However, in this design, the Diamond Square Algorithm method, which is mainly popular in simulating the natural terrain, has been used for generating the crinkled surface of the canopy roof. The same DS Algorithm scripting code, developed in the previous subsection, has been used for generating the parametric form of the canopy roof by keeping the overall size as 17 meters X 17 meters. All the other variables are set with structurally and functionally reasonable parameters. The main key factor is the Hausdorff dimension of the proposed roof surface. ‘Figure 10.16’ shows the different forms affected by the fractal dimension.

Figure 10.16: Noises of a surface resulted after different fractal dimensions.

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10.5.4

Architectural Design

The final canopy structure is designed by combining the crinkled roof and four self-similar copies of the branching structures developed in the previous sections. The roof is supported by four branching structures. To support the crinkled roof efficiently, some key points on its surface are required as joints where the column ends will be connected. If we insert a set of vertical columns connecting those key points, then these columns will occupy a large indoor space inside. Therefore, to minimize the number of supporting members by keeping the key supporting points under the roof surface same, we will use branching columns. Branching support is useful for its minimal path principle in which the distributed loads are transferred through the branches and flown towards their trunk. The tree supports are modeled here by adopting the three-dimensional branching trees modeled in the previous section using the process of IFS. Usually, in this type of canopy structures, flat roof is connected at the end points of the terminal branches of tree-like supports. However, in this design, since, the roof surface is not flat but crinkled, hence, the endpoints of all the terminal branches do not necessarily touch the surface. In some cases the branches penetrate through the roof surface, while in some cases the last branches are far from the surface and do not touch the surface. Therefore, to cope up with this design issue in modeling, we place the roof surface such a way so that all the end branches either penetrate through the surface or intersect the surface or at least touch the surface. After such strategic intersections, all the branches that overlap the surface are trimmed and splitted. Thus, the roof surface divides the branches into two groups, – one group is above the surface and another group is under the surface (Figure 10.17b). We consider the second group under the surface as the structural supports and the first group above the surface is discarded (Figure 10.17c). In this situation, now all the end points of the end branches are connected to the surface at the intersection points. This has been done parametrically which means, whenever the roof surface randomly changes its topology, automatically we will get the second group of branches as the resultant structural supports. The whole model is made parametric keeping the architectural and structural feasibilities in mind. For example, in the case of branching supports, the vertical angles influence the structure by expanding its topmost area, while the iteration number influences the supporting connections at the roof surface thus increase number of load distribution. Similarly, the contractivity factor (λ) and horizontal angle also influence the aesthetic appearance as well structural behavior of the branching structure. On the other hand, the crinkled roof also influences the structure by changing its values of the fractal dimension and zlimit factor (λ). As a result, the final model has several variables in total which could produce infinite design variations of the main model. For finding a prac-

208 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals

Figure 10.17: (a) Independent branching structure (b) Wrinkled roof, which is placed at the centre of the overall branching configuration at a certain height, overlaps the branches separating some branches above the surface and some are under the surface, (c) the upper branches are trimmed out from the intersection points in between the surface and branches.

tical and efficient form, we have fixed the values of some variables and define the parameters of the remaining variables such a way so that they can generate feasible design possibilities, not beyond the impractical models. The variables and their values and parameters are shown in ‘Table 10.3’ and ‘Table 10.4’,

Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals 209

along which some input values are fixed to avoid the heavy calculations during the computational optimization process. Table 10.3

Branching Structure Design Variables

Values/Domains

Initial height, h0 Contractivity, s Vertical angle, θy Horizontal angle, θx Iteration number, n

5.5 meter (fixed) 0.6 (fixed) 35o to 50o (variable) 45o (fixed) 4 (fixed)

Table 10.4

Crinkled Roof Surface Design Variables

Values/Domains

Size, s Z-Limit, δ Fractal dimension, D Iteration number, n

17 meter (fixed) 1.0 to 2.0 (variable) 2.0 to 3.0 (variable) 4 (fixed)

Based on the above unrestricted variables, different design variations of a single unit (main canopy structure is composed of four units with four branching supports) of the main canopy structure are shown in ‘Figure 10.18’. ‘Figure 10.17Top’ shows different design possibilities due to the variations of branching supports influenced by the changing of vertical angles and contractivity factors, keeping the parameters of roof surface are constant. ‘Figure 10.17Bottom’ shows the different design variations of roof surface influenced by the changing of deviation (δ) and the fractal dimension (D), while the parameters of branching supports are constant. However, to design the first model of the main canopy structure, we have defined the values of different variables mentioned in ‘Table 10.5’ and ‘Table 10.6’, which results an architectural outcome shown in ‘Figure 10.19’

10.5.5

Structural Modeling

The geometric complexity of roof surface is achieved by the heights of its mesh vertices that are present in an array of a square grid pattern. Hence, height of

210 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals

Figure 10.18: Design variations of the pavilion unit; Top - The design variations of branching supports with the changing of vertical angles and contractivity values; Bottom- The design variations of roof surface from flat to random and noisy with the changing of z-limits and fractal dimensions. Table 10.5

Branching Structure Design Variables

Values/Domains

Initial height, h0 Contractivity, s Vertical angle, θy Horizontal angle, θx Iteration number, n

5.5 meter 0.6 45o 45o 4

each vertex plays the main role to offer such complexity of surface. In the case of structural optimization of shell structures or similar smooth or flat roof canopy structures, the height of each vertex are kept variable, and optimization process control each height as a variable to obtain the most optimized shape with regard to some defined fitness functions and structural conditions. Computational morphogenesis for form-finding performed on a NURBS based (or non-NURBS based) grid surface generally results a smooth surface as an optimized form of the roof chosen and prefered by designers. Usually, smooth surface offers better distribution of forces-flows, and that is why computational morphogenesis with respect to structural form-finding indicates the smooth surface as a preferable choice. However, since, in our example, our architectural design goal is to obtain a natural terrain like irregular roof form and structural goal is study

Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals 211

Table 10.6

Crinkled Roof Surface Design Variables

Values/Domains

Size, s Z-Limit, δ Fractal dimension, D Iteration number, n

17 meter 1.5 to 2.0 2.5 4

Figure 10.19: An architectural model of the main nature-inspired fractal-based canopy structure.

the mechanical behavior of fractal-based random surfaces, we will maintain the unsmooth fractal character of the roof surface during the form-finding process. For this, unlike other common examples, we do not take all the heights of the points as the individual variables; instead, we keep only two variables that are deviation or z-limit factor δ and the fractal dimension factor D. These two factors will maintain the fractal characters of the roof surface, and to stick with this idea, we have restricted the parameter in between 1.0 and 2.0 for z-limit factor and 2.0 and 3.0 for fractal dimension factor. The z-limit value ranging from 0.0 to 0.5 usually produces nearly flat and approximately smooth surfaces. For the structural analysis of the fractal-based canopy structure, its geometric model has been transformed into a finite element model by assuming the following consideration: • All the duplicate lines of branching supports are removed. Each line is considered as a beam element. • All beam elements are hollow steel tube. The diameters of all trunks are

212 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals 20 cm, all first tier branches are 15 cm and the remaining branches are 8 cm. The thickness of all the tubes is 5 mm. • All the duplicate connecting points, i.e., overlapped structural nodes are removed. Each connecting node is considered as a structural welded joint. • The bottom of the trunk is vertically and horizontally restrained. • The roof surface is a triangular mesh and all the duplicate mesh units are discarded. Each mesh units are shell elements. • Shell elements are made of cross laminated timber (CLT) and thickness is 5 cm. • The shell and the branching tubes are connected by screws via steel plates at the intersection joints. For investigating the influence of fractal dimension of the roof’s surface on its structural strength, the ‘Generative Tool Box’ and ‘FEA Tool Box’ have been used that instantly give structural feedbacks after finite element analysis with the changing of design variables that are unique such as the fractal dimension. The strategy is to find the appropriate form that provides maximum strength by means lowest displacement under a defined load case.

10.5.6

Structural Analyses

Before the structural analysis of the main canopy structure, we perform a finite element linear static analysis on a smaller prototype of the main roof that is centrally supported by a space truss. This trial is a kind of mock up of a piece of paper which is centrally placed on the top of a bottle shown in ‘Figure 10.14, Top’. As we see in the ‘Figure 10.14’, the flat paper sheet is not stiff enough by its own weight and its two sides bent down; but, after crumpling the paper, its crinkled shape becomes stiffer and do not bent down as before, but act as a kind of cantilever, because the crinkled paper is self-stiffened by its random folds. Therefore, through the finite element analysis, we expect to see the similar mechanical behavior in prototype flat roof and then its crinkled version. In order to confirm this prediction, we modeled three different roof surfaces, one is flat having z-limits, δ = 0 and fractal dimension, D = 2.0, second is crinkled having deviation, δ = 1.0 and fractal dimension, D = 2.5, and the third is further crinkled surface having deviation, δ = 1.5 and fractal dimension, D = 2.9. After performing FEM analyses on these the models using the parametrictoolbox we obtained the results shown in ‘Figure 10.20’. In the ‘Figure 10.20’ we notice that when the roof is flat then its deformation is large, and bents down at its

Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals 213

edges similar as the flat paper sheet. The maximum displacement is 264 mm. Soon after the shape of flat roof is crinkled with fractal dimension 2.5, it does not bend down; instead it stays almost in the same position. The maximum displacement in this case jumps down to 38 mm from the previous 264 mm of flat roof. But, when the crinkled surface is further folded with the fractal dimension of 2.9, then we do not see a significant change. In this condition, the maximum displacement is 32 mm. The similar behavior we also notice when a paper sheet is further crumpled, then the stiffness gets little higher but not very large than the less crumpled paper. The only big difference occurs between flat paper and crinkled paper. These results confirm our hypothesis that the crinkled surface acts as a self-stiffened structure which can be practically useful for designing large-span canopy roof.

Figure 10.20: Deformations of a roof due to self-weight when its flat surface is transformed into crinkled surface.

Based on the FEA results of single unit demonstration shown in‘’Figure 10.20’, we predict that the roof of main canopy structure will act the similar way as the crinkled paper and the roof of prototype unit structure. For this purpose, at present, we only consider the gravity load and mesh load of the roof and check its maximum displacement as a measure of stiffness, as we know the stiffness is directly proportional to acting forces and inversely proportional to the

214 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals displacements. Our curiosity is to see whether the maximum displacement of the roof gets smaller when the fractal dimension as a measure of the crinkliness of the roof surface gets higher; although, the displacement is also affected by the vertical angle of the branching supports. At this moment, we have fixed the branching vertical angle as 45o , horizontal angle as 45o , trunk height as 5.5 meters, contactivity factor as 0.6 and iteration number as 5. For the roof model, we have fixed the size as 17 meters and z-limit as 1.5. So, the only variable is the fractal dimension that ranges between 2.0 to 3.0. The parametrictoolbox has analyzed the structure for different fractal dimensions which are shown in the ‘Figure 10.21’ in which red circles indicate the positions of maximum displacements. From the ‘Figure 10.21’ we notice that when the fractal dimension of the roof surface gets larger, the displacement under gravity and mesh loads gets smaller. When the roof is flat, i.e., fractal dimension is 2.0, which is an integer value, the displacement is relatively very high. As soon as the surface starts getting crinkled (D > 0 and δ > 1), the maximum displacement sharply falls. Although, in the Table we see the value of maximum displacement is getting smaller with the increasing of fractal dimension, it is always not true. Because of the factor of randomness, the repeated input of same fractal dimension results different topology of the roof surface by maintaining the degree of roughness, hence delivers different stiffness. For understanding the influence of fractal dimension on the structural strength of crinkled roof, we repeated the same value of fractal dimension two times, and get two different maximum displacement behaviors shown in ‘Figure 10.22’ In the ‘Figure 10.22’, we observe that in each case, the maximum displacement sharply falls as soon as fractal dimension becomes more than 2.0 to 2.1, and from 2.1 to 2.9, its displacement is significantly different to each other as compared to the maximum displacement of flat surface (D = 2.0), and fluctuates in between 70 mm to 150 mm. These fluctuations are completely different to each other in both trials. It represents that as soon as the roof is crinkled from its flat surface, it becomes self-stiffened due to its random folds. However, the further folding makes the surface more crinkle, but not necessarily increase its stiffness, because stiffness is further depending in the surface topology. Besides, increasing of fractal dimension increases the roughness of the surface, hence increases the weight of the roof. ‘Figure 10.23’ shows the relationship between the fractal dimension and the weight of the roof. This relation is helpful to find an optimized shape of the roof with regard to its minimum weight but maximum stiffness. As we see from the ‘Figure 10.22’ that the roof surface having D = 2.1 is stiff enough as compared to the flat roof, and ‘Figure 10.23’ shows that the roof surface with D = 2.1 is the lightest roof structure among the other roofs having D > 2.1. On the contrary, the roof form should be such

Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals 215

Figure 10.21: The displacements of the roof under the gravity and mesh loads with the changing of fractal dimensions, while z-limit is constant (1.5).

so that it can express the appearance natural terrain, neither nearly flat nor too rough. In this condition, we can select the shape which has fractal dimension in between 2.3 to 2.5. In this range the roofs are lighter and stiffer enough, while maintaining the natural expression by its appearance.

216 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals

Figure 10.22: Fractal dimension vs. maximum displacement graphs for two different trials.

Figure 10.23: Fractal dimension vs. roof weight graphs.

10.5.7

Computational Morphogenesis and Form Finding

Because of the factor of randomness, we obtain an infinite number of form variations of the structure. Even, the same value of fractal dimension results different forms of the roof surface at each new attempt. Hence, it is not easy to find the most suitable form in terms of high strength and less weight, and that is why, a computational optimization method based on automatic search algorithms is required to find the optimal form of the structure. For the automatic form finding, the domains of the variables are set by keeping structural and architectiral feasibility in mind, and also to avoid heavy and time-consuming calculations during optimization process. The ‘Optimization Tool Box’ has been applied for the optimization, and an optimal designed has been found after stopping the process at 100th generation (Figure 10.25).

Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals 217

The parameters are set as follows: Table 10.7

Crinkled Roof Surface Design Variables

Domains

Z-Limit, δ Fractal dimension, D Iteration number, n

1.0 to 2.0 2.0 to 3.0 1 to 3

Figure 10.24: Schematic diagram of the automated computational search using the ‘Optimization Tool Box’

10.5.8

Final Remarks

This paper has attempted to apply the concept of fractal geometry from its theory level of mathematics to applied fields of architecture and construction. While the fractal geometry’s main property of self-similarity are recently studied in applying for architectural designs, it’s another important property of randomness have not been studied so far to find its scope and potency in architecture. This paper mainly focuses on this study to find an opportunity to use the concept of random fractals in designing a wide span crinkled roof for designing a canopy structure inspired by nature’s random forms of land terrain. Mathematical formulations of self-similar and random fractals have been transformed into algorithms for the computational design of the canopy structure. Finite element analyses have been performed to assess the structural behavior

218 Chapter 10. Application 3 - Folded Roof Design using the Concept of Random Fractals

Figure 10.25: Optimal design of the randomly folded fractal-based roof canopy structure after 100th generation of optimization process. Top - Nodal displacements, Bottom - Architectural view.

of the structure, and tried to explore the relation between the fractal dimension and the structural strength as well as another relation between the fractal dimension and the weight of the roof. Based on these two relations, a suitable form can be selected that will give an impression of nature. Because of the unique design variables, related to fractal geometry, there is a huge number of form variations the canopy structure, and from this massive variety, a suitable one is not easy to be found using scheme by scheme process. Therefore, an computer-based optimization process has been run on the parametric model of this canopy structure, and after 100th generations, it has resulted an optimal design.

Part V

Conclusion

219

Chapter 11

Conclusion Fractal architecture could be good for you ! - Yannick Joye [31]

11.1

Introduction: Complexity, Fractals and Architecture

Yannick Joye’s ‘philosophical’ remark (in his own words), mentioned above in the chapter quotation, about the aesthetic importance of fractal geometry in art and architecture with respect to the well-beings of human emotion and environmental psychology [31], is a source of optimism that has driven this study for exploring the position and usability of the notion of fractal geometry in architecture and construction from the ground of practical applications apart from artistic value. In this context, Charles Jencks’ observation about the recent inclination (and probably the necessity) of architects and designers towards the new paradigm of complexity in architecture and urban design [30] has further inspired this study to find the possible structural meaning and potency of fractal concept as a new skeleton to embody the ideas of complexity. The complexity of a design idea, may it be visual or thematic, can be achieved by its physical manifestation in buildings and urban forms. In the recent years, with the development of computers and advanced construction technologies, the domain of the complexity has become wider as well as deeper. Computer has made possible to visualize and model complex forms (even ideas) that were not easy to draw, model and (sometimes even) imagine. It was unimaginable and seemed 221

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to be paradoxical to see a visual form of the Julia set, a mathematical function. Thanks to computer, in the 1960s, Benoit Mandelbrot shows us the graphical impression of the Julia set, the output image is astonishing which might not be imagined by Julia himself ! Therefore, computer-aided mathematics has opened a new world of exploiting old and new geometries dealing with complexity, and thus offered a new opportunity to architect and designers to produce their designs with complex form systems and manifest their unique ideas with the nonlinear design vocabularies. In this contemporary trend of design complexity, Charles Jenks believes that the complexity should not bring painful chaos and confusion, but should bring harmony so that it can tune with the complex appearance of nature that follow some hidden order of geometry [30]. Human beings are a part of the complex system of nature, and that is why, probably, nature’s complex yet orderly appearance soothes our eyes, although, the visual experience and the taste of complex appearances are not universal, and depended on person to person [33]. However, biologically, and generally, as Salingaros claims, fractal art and architecture can reduce our physiological stress and provide visual comfort [71]. It was Mandelbrot, who revealed the hidden order of nature’s complexity using mathematical vocabulary, and termed it as fractal [46]. His systematic development of fractal geometry, which became a new branch of mathematics in the late 20th century, had changed the course of scientific development, and allowed researchers to study nature from a closer distance with the eyes of mathematicians. In the field of architecture, fractal geometry, i.e., ‘nature’s geometry’, has offered designers a new tool to bring complexity in their designs by maintaining the synchronization with nature. Structural innovations and constructional advancement have eased to realize complex designs as built forms. These advancements have allowed contemporary architects to articulate their boundary-less free-form design ideas and digital models into physical forms. In this sense, fractal geometry shows huge opportunity not only to offer order-based complexity in architecture, but also to provide structural trick by its property of self-similar repetitions for the modular structures and construction which are always the high interest among architects and engineers. This dissertation has attempted to utilize this opportunity of using fractal geometry for developing architectural structures with order-based complex designs, thus to explore the structural functions and architectural design of fractal forms.

Chapter 11. Conclusion

11.2

223

Summary of the Results

Based on the core research objective, an extensive research has been done by focusing on finding structural forms using the concept of fractal geometry and explore their efficiencies and opportunities. This study was confined within the exploration of finding structural merits of fractal geometry. The whole experiment and applications were done using the technical theme of computational morphogenesis, which transforms mathematical functions of fractals into generative design for architectural structures, and then finally find the optimal shape The results of this study are as follows: Fractal Mathematics and its Practical Usefulness This dissertation started its investigation by understanding the mathematics of fractal geometry and then extracting its essence from its theoretical landscape to practical ground of architecture and construction. The study of fractal mathematics from the lens of architectural aptitude extracted a list of features that are potentially applicable in the practical field. The concept of fractal geometry increasingly encompasses vast aspects on nonlinear geometries. It has the ability to mathematically explain many intricate exactly self-similar orderly figures, and also explain unpredictable highly complex and random features. By understanding the vastness of the theory of fractal geometry, this study attempted to establish a wider panorama about its practical architectural applications, thus to exhibit the versatility of this unique concept of nonlinear fragmented geometry. An attractor, as a union or an intersection of countless self-similar subsets, described in mathematical language, exists in the practical world in its approximate forms such as tree branches, river networks, etc. Through the study of the theory of fractal geometry, this research has found the resemblance of mathematical entity of fractal geometry with practical world, thus obtain a rational link between fractal mathematics and architecture. The concept of random fractals explains irregular and unsmooth forms, and helps to generate random, nonlinear and unsmooth forms. Through the study of mathematical principle of fractal geometry, it was found that the randomness and the degree of unsmoothness can be controlled by geometric variables. This characteristic shows the versatility of fractal geometry within the frame of formalism. From an architectural point of view, when designers are hunting for new forms, this concept of random fractals shows it immense usefulness. One of the fundamental features of fractal geometry is its fractal dimension that defines the dimensionality of any irregular and unsmooth objects. This concept of fractional dimensionality surpasses its theoretical explanation when

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we see an abundance of unsmooth objects in the practical world. That means, this concept of fractal dimension shows a huge possibility in offering infinite form variations that lie between two successive integer dimensional objects. Mathematics provides us a key that is useful for generating the figures with different fractal dimensions, and hence, from architectural point of view, this feature of non-integer dimensionality exhibits an enormous formal utility. Structurally, the concept of fractal dimension shows effectiveness in term of changing the mechanical behavior by exploiting the regular forms of structures.

Fractal Geometry, a Form Generator for Architecture From the studies of fractal mathematics, it has been found that the fractal geometry can produce infinite variations of forms by changing the magnitudes and configurations of geometric variables, such as initial condition (shape), contractivity factor, affine transformation function and iteration number. Translation of the mathematical function of a fractal, as an attractor, into a scripting code allowed us to generate graphical models with plentiful variations. The codes, in this dissertation, are developed by using the Iterated Function System (IFS), which provided an opportunity to develop some design forms to achieve an architectural target. In this thesis, fractal geometry acted as a tool to recreate nature as a man-made structure. Nature always inspires architects to develop their designs, and we see plentiful examples of nature-inspired architecture. They adopted nature’s forms using different geometric tools, mostly regular and conventional geometric systems. But the mathematical interpretation of recreating nature using nature’s geometry, fractal geometry, could be more straight forward, and probably reasonable. In this dissertation, the mathematical formulations have been used to recreate a forest. ’Fractal Forest’, a design workshop project, was a perfect opportunity where the fractal geometry was applied as a theoretical tool of generative designs. This project was a kind of testimony for the next experimentations and applications in this research. The mathematical function, supported by IFS tool, produced a parametric model of the forest where the mathematical vocabularies were used to generate architectural features, i.e., recreate trees. The parametric model, controlled by fractal features, produced a number of design variations which allowed us to choose the best design out many options. The full-scale physical realization of the digital model of ’Fractal Forest’, guaranteed the potential usefulness of the fractal geometry, as a mathematical tool for architectural designs. This project was also important to testify the toolbox that was developed for generating fractal models and transforming to architectural designs.

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225

Fractals to Develop Light-Weight Structures Fractals are defined by components on hierarchy of different scales. This property has an enormous prospect in the field of construction where the lightness with high strength is always the foremost demand among engineers, especially in contemporary architecture. In the field of construction, this feature is known as hierarchical system, and the structures are called hierarchical structures, mostly seen in lattice trusses. In this system, one member of a structure is replaced by it submembers with a defined lattice arrangement, and each submember is replaced by its own submember maintaining the same assembling configuration for different levels. Mathematically, this system can be explained by fractal geometry, and so, using the concept fractal geometry, the hierarchical systems can be easily modeled, as a result we will get a grammar-based hierarchical structure that follows a particular rule. Based on this idea and to see the potency of fractal mathematics for designing hierarchical trusses, a prototype of truss beam has been designed in a two-dimensional space. Using Bransley’s [2] concept of contraction mapping, an affine transformation function has been developed whose attractor is the proposed truss beam. The generative tool box, developed in this research based on the IFS for translating mathematical function into a graphical form, has been applied to produce a parametric model of the hierarchical truss beam whose design variables are the factors of fractals, such as initial shape and condition, contractivity factor, affine transformation function and the magnitudes of this function’s variable. It is important to notice that the design variables here are unique, because these variables are not related to the Euclidean or other conventional geometric systems, but related to fractals. In this sense, iteration number as a variable is completely unique that produces different hierarchical forms. This was an experiment to check the viability of fractal concept and the credibility of generative tools that were developed in this study. In the first step of the experiment, generative tool successfully produced the parametric model of the truss beam, and generate a design of truss beam that strictly follows the grammar of fractal geometry at each hierarchical level. It has allowed to perfectly replace each member by its submembers maintaining the original assembling configuration developed by IFS function in the beginning. In the second step of this experiment, the potency of fractal geometry has been tested from structural perspectives. For a defined structural condition, increasing levels of hierarchical truss beams have been tested for an allowable maximum displacement. Finite Element Analyses showed that, for a maximum allowable displacement, the higher level of the hierarchical truss is much lighter than the less level of hierarchical truss beam, yet shows sufficient strength. The computational optimization process has helped to find the optimal shape of the hierarchical truss beam, which is having a high level of hierarchy and

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save huge amount of weight. This experiment further assures the merit and rational applicability of fractal geometry in the field of both architecture and construction.

11.3

Future Work

Advanced Assessment of Fractal-Based Structures In this thesis, the core task was to transform the mathematical essence of fractal geometry from its theory level to architectural and structural designs of practical level. Here structure has been used as a measure to find an architectural form. Therefore, in this study the structural analyses and assessments have been restricted and not considered other evaluations, such buckling analyses, stability checking, etc. Hence, in future there is a scope dedicating only for assessing the true and accurate mechanical behaviors of structures that are made using the principle of fractal geometry. Enormity of Fractal Shapes in Architectural Design Unlike other geometric system, fractal geometry’s variables are powerful in a sense that it can produce a vast number of form variations and complexity. A small change in the initial condition and affine transformation, and its consequent outputs in different iterations delivers a huge variety of forms and complexity originating from one. This is an exceptional mathematical seed that has potential to generate far wider design possibilities than that of the Euclidean and other regular geometric systems. Therefore, there is a large scope of investigation related to architectural and structural formalism based on fractal geometry that can contribute to develop a new paradigm in architectural design and construction. Geometric Refinement of Fractal-Like Structures Fractal geometry is a rule-based geometry. In the field architecture and engineering, we observe a number of structures resemble with fractal shapes, but they are not true fractals which follow a strict geometric rule. An investigation can be done in future on comparing the structural behavior of existing fractal-like structures and the same structures if remodeled using the strict rule of fractal geometry. As an example, transmission towers exhibit some extend of self-similar repetitions at each successive tier, but their topological configuration mostly do not follow a precise geometric rule. It is probably because of the no interaction between the designer and the service tower engineers during its design process. In this context, fractal-aided rule-based design of the towers as

Chapter 11. Conclusion

227

a geometric refinement may offer better mechanical output. Hence, in future, there is a potential scope of research based on the refinement of such structures and explore its effect on their strength and stability. Application of Different Methods of Fractal Generation In this thesis, only Bransley’s IFS method and Midpoint Displacement methods have been used for modeling self-similar and random fractals. But there are other different methods that produce fractals. In some cases, the final outcome differs from different methods. For example, L-system and Finite Subdivision Rule produces different types of fractals that are not possible to make using IFS. Similarly, Brownian Motion method, Percolation method and Self-Avoiding Walk method can produce different forms of random fractals. In this sense, as a future work, different other methods can be applied that might result some exceptional structural forms and show unique mechanical behavior. Interpretation of Fractal Dimension on Structural Behavior In this thesis, particular emphasis has been given to deal with the role of fractal dimension, which is usually a theoretical entirety in mathematics as well as in graphics or image analyses, for exploiting the form of structures and assess the strength and force behavior. This attempt is probably an original approach so far, especially in three-dimensional space. In the future, it has a great scope of research investigating about the importance of fractal dimension in terms of its relation with the structural mechanics and behavior. Structural Optimization using Wider Range of Domains One part of future study can be dedicated to the computational optimization of fractal-based structures. Here, in this dissertation, a ready-made component of Galapagos which was developed based on the concept of genetic algorithm has been used for the optimal form finding of fractal-based structures. In the future, a different other optimization process can also be applied to get more comparable results. Because of the limitations of computer memory, the high iterated models have not been tested here, and therefore, their structural behaviors are unknown. During optimization, these highly iterated models have not been considered for avoiding painful long calculations. Besides, the scale factor has also been restricted within the practical level. As a result, the true optimal designs of fractal-based structures have not been obtained. In future, a part of investigations can be dedicated to analyzing the high iterated models of fractal-based structures. High range of scales can also be implemented to check

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the impact of long range variations of a fractal-based structure. In this way, a true optimal design can be found. Practical Applicability and Materiality In the end, the whole research is intended to apply to the real world architecture and construction fields. Therefore, a further exploration about the practical construction methods of such complex yet modular (except random fractals) structures is required. The investigation can consider both methods, i.e., the advanced computer aided rapid production method for small scale structures and manual method for large scale structures. Besides, the selection of construction material for realizing such fractal-based structures can be a tricky task. Therefore, a significant study can also be devoted for the investigation on the materiality of fractal-based architectural as well as engineering structures.

List of Figures 1.1 1.2 1.3 1.4 1.5 1.6

1.7

2.1 2.2 2.3 2.4

2.5

2.6

Geometries used in ancient architectural structures. . . . . . . . Geometries used in modern architectural structures. . . . . . . . Is highly delicate shape of cauliflower applicable for designing architectural structures? . . . . . . . . . . . . . . . . . . . . . . . Koch curve, a mathematical fractal figure showing the repetition of full into its parts and never ending roughness. . . . . . . . . . Dimensions of different geometric objects. Objects having fractaional dimensions are fractal objects. . . . . . . . . . . . . . . . (a) Federation Square building inspired by Pinwheel tiling by LAB Architects (1997); (b) Grand Egyptian Museum inspired by Sierpinski triangle by Heneghan Peng and ARUP (2015). . . . Hierarchical truss system of Eiffel Tower that shows fractal property of self-similar repetition at smaller scales. . . . . . . . . . . . (a) Graphical expression of Julia set, (b) Graphical representation of Mandelbrot set, (c) Koch curve. . . . . . . . . . . . . . . (a) Hindu Temple, (b) African village, (c) Chappel of Chiesa Cattolica Parrocchiale Santa Teresa, Turin. . . . . . . . . . . . . Fractal analyses of Kandariya Mahadev temple. [65] . . . . . . . Floor of Anagni cathedral, Anagni (1104); (b) Mosaic pattern in Sistine Chappel , Rome (1481); (c) Elevation of Ca’ d’Oro, Venice (1440). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. L. Wright’s fractal design in his architecture, (a) Plan of Palmer House (built in 1901); (b) Fa¸cade design of Marin County Civic building (built in 1958). . . . . . . . . . . . . . . . . . . . . Asayama-Mae’s fractal trusses with different base angles, and their deformations under vertical load. [1] . . . . . . . . . . . . . 229

5 6 6 8 8

11 14

19 20 21

21

22 25

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List of Figures

2.7

2.8

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

3.12

3.13

3.14 3.15 3.16 3.17 3.18 3.19

4.1

(a) Folded shell design using Fractal’s rule of Iterated Function System [78], (b) Fractal-based reticulated shell structures design by Vyzantiadou, et. al. [85] . . . . . . . . . . . . . . . . . . . . . 25 A fractal nanotruss made in Greer’s lab. Credit - Lucas Meza, Greer lab, Caltech. (URL: http://www.jrgreer.caltech.edu/home.php) 27 The formation of Sierpinski triangle (or gasket) . . . . . . . . . . Epsilon Net from A to B where d(A,B) = epsilon1; epsilon2 Net from B to A where d(B,A) = epsilon2 . . . . . . . . . . . . . . . An example of zero topological dimensions . . . . . . . . . . . . . Examples of 0, 1, 2 and 3 topological dimensions . . . . . . . . . Stages of the construction of Koch curve. . . . . . . . . . . . . . Convergent sequence of the Sierpinski triangle by using contraction mapping and IFS. . . . . . . . . . . . . . . . . . . . . . . . . Construction of fern usning IFS that is made with two functions. Contraction mapping and affine transformation of the Sierpinski triangle. Angle is 60o and contractivity is 1/2. . . . . . . . . . . Different modes of two-dimensional affine transformations. . . . . Bensley’s fern generation using his IFS based on two dimensional affine transformations. . . . . . . . . . . . . . . . . . . . . . . . . Construction of the attractor F for contractions S1 and S2 which map the large ellipse E onto the ellipses S1 (E) and S2 (E). The sets S n (E) = ∪(1,2) s1 (E), ...., Sn (E) give increasingly good approximations to F . . . . . . . . . . . . . . . . . . . . . . . . . . . Left- affine transformation of Sierpinski triangle, Centre – randomized affine transformation by shifting upper right triangle, Right – randomized affine transformation by rotating upper left triangle by 100o . . . . . . . . . . . . . . . . . . . . . . . . . . . . Left- affine transformation of Sierpinski triangle, Centre – randomized affine transformation by shifting upper right triangle, Right – randomized affine transformation by rotating upper left triangle by 100o . . . . . . . . . . . . . . . . . . . . . . . . . . . . Log-log graph whose slope is the value of fractal dimension. . . Construction of fractal branching using L-System. . . . . . . . . Constructing conformal finite subdivision rules. [10] . . . . . . . Flowchart of IFS method . . . . . . . . . . . . . . . . . . . . . . IFS pseudocode . . . . . . . . . . . . . . . . . . . . . . . . . . . . Possibility to construct mega scale lattice structures from nano scale hierarchical trusses using the concept of fractal geometry. . IFS pseudocode . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36 38 40 41 42 46 47 49 50 51

52

53

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List of Figures

4.2 4.3

5.1

5.2 5.3

5.4 5.5 5.6 5.7 5.8

6.1

6.2 6.3 6.4 6.5

Schematic diagram representing the scheme of ’FEA Tool Box’. Top - Karamba, Bottom - GeometryGym . . . . . . . . . . . . . . Schematic diagram representing the scheme of computational morphogenesis of fractal-based structures using Tool Boxes . . . Figure: (Top left) – One of Alexander Graham Bell’s tetra kites made of silk and wood; Bottom left - Alexander Graham Bell’s ’Siamese Twin’ kites, from Alexander Graham Bell, ’Aerial Locomotion, With a Few Notes of Progress in the Construction of the Aerodrome’, National Geographic Magazine (Jan., 1907), 133; (Top right) - Bell’s ’Cygnet II’, February 25, 1909. Bulletins, from January 4, 1909 to April 12, 1909, Alexander Graham Bell Family Papers at the Library of Congress, 1862-1939, Manuscript Division, Library of Congress; (Bottom right) - Alexander Graham Bell’s Tower, from ’Dr. Bell’s Tetrahedral Tower,’ National Geographic Magazine (Oct., 1907), 672-675. . . . . . . . . . . . . Hierarchical truss composition in Eiffel Tower. . . . . . . . . . . . (a) Fractal truss column [20]; (b) Nano-truss based on hierarchical system of arrangement [52]; (c) Structural lattice assembled by ultra-light tiniest pieces [12]. . . . . . . . . . . . . . . . . . . . IFS variables and their values. . . . . . . . . . . . . . . . . . . . Convergent seqquence of the fractal-based truss beam. . . . . . . Finite Element Aanalyses of different iterated models of the fractal truss beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic diagram of the computational searching using Galapagos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic diagram of the computational searching using Galapagos. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Tree Columns in the model of Sagrada Familia, Spain by Antonio Gaudi (b) Frei Otts hanging models of branching systems (Nerdinger, 2005, p67), (c) Tree column in Stuttgart Airport terminal, and (d) The Tote, Mumbai. . . . . . . . . . . . . . . . . . Fractal as a union of perfectly self-similar subsets after affine transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . First affine transformation as the first step transforming trunk T to the first tree ∆0 . . . . . . . . . . . . . . . . . . . . . . . . . . . IFS variables and their values. . . . . . . . . . . . . . . . . . . . Convergent sequence of fractal branching tree in three-dimensional space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

231

72 74

80 82

83 85 86 89 91 91

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List of Figures

6.6 6.7 6.8 6.9 6.10 6.11 6.12

6.13

6.14

Inputs for the affine transformations for the second and remaining steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Design Variables for generating branching column . . . . . . . . . IFS pseudo code for generating branching column. . . . . . . . . Architectural model of the propsed branching canopy structure . FEA Resultsunder the defined load case. Top - Deformation under, Middle - Axial stresses, Bottom - Global Buckling effect. Design variables and their domains . . . . . . . . . . . . . . . . . Left - Graph representing the relation between vertical angle and maximum displacements; Right - Graph representing the vertical angle and canopy area. (Contractivity is constant, which is 0.6) . Left - Graph representing the relation between contractivity and maximum displacements; Right - Graph representing the contractivity and canopy area. (Vertical angle is constant, which is 60o ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural form finding of the branching canopy structure. . . . .

Replication of poplar tree growth and its representation as architectural design elements. . . . . . . . . . . . . . . . . . . . . . . . 7.2 Top - Branching tree as an attractor by following affine anisotropic transformation process. Bottom - Resulted tree models that shows their age growth. . . . . . . . . . . . . . . . . . . . . . . . 7.3 Top view of the Monalisa Pavilion. . . . . . . . . . . . . . . . . . 7.4 Creating branches by inserting curved triangular wedges for making physical model and prototypes. . . . . . . . . . . . . . . . . . 7.5 Architectural representations of the Fractal Forest Pavilion. . . . 7.6 Left – Nonlinear nodal solution of stress intensity in ANSYS after applying loads on two adjacent ply-sheets for large deflection. Right Top – Ply-sheets placed with orthogonal fiber directions. Left Bottom – The weakest (failure) point ‘MX’ during large deflection in bending.. . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Assembling of strips to get a large tree unit. . . . . . . . . . . . . 7.8 (a) Small scale (1:20) physical model. . . . . . . . . . . . . . . . 7.9 (a) ‘Loose tongued joint’ for making longer strip (source: internet); (b) Two plyboards are joined by ‘loose tongued joints’ to make strips and four such strips were screwed by brass screws. . 7.10 Left- Weakest part in large deflection bending of strips when screws were perpendicular to the tangent. Right – Screws notperpendicular to tangent to offer relaxation in torsion while bending a strip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 99 100 102 104 105

105

106 108

7.1

113

115 116 117 118

119 119 120

120

121

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233

7.11 Real scale ‘tree’ units as test displayed in the university workshop ground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.12 : Design to production process in factory workshop building. . . 122 7.13 Final outcome of Monalisa Pavilion in MadeExpo2012 event, Milan123 7.14 Production progress of real scale ‘tree’ unit in the university workshop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.1

(a) A vertical post under compression traditionally known as ’crown post’ standing on a tie beam, (b) An example of crown post in St Clement Church roof, Old Romney, Kent, England (12th century). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 (a) A vertical post under tension traditionally known as ‘king post’ hanging from the joint of two rafters and hold the tie beam (b) An example of king post in the Bolduc House in Ste. Genevi`eve, Missouri (18th century). . . . . . . . . . . . . . . . . 8.3 Some common types of trusses and their ‘self-similar’ features that are formed by the self-similar or self-affine repetitions of ‘A’ and ‘B’. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Top - Contraction mapping and affine transformation of the Sierpinski triangle. Contractivity is 1/2; Bottom - Convergent sequence of Sierpinski triangle. . . . . . . . . . . . . . . . . . . . . 8.5 Conway and Radin’s method to make Pinwheel tiling pattern which starts from one small triangle tile and then it iteratively grows outwards with above rule. . . . . . . . . . . . . . . . . . . 8.6 Pinwheel Tiling construction using contraction mapping method; Top - Affine transformations of right-angle triangle. Bottom Convergent sequence of Pinwheel Tiling. . . . . . . . . . . . . . . 8.7 Pinwheel Fractal; Above - Affine transformations; Bottom – Convergent sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Pinwheel Fractal; Above - Affine transformations; Bottom – Convergent sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Fan Fractal; Above - Affine transformations; Bottom – Convergent sequence (The dotted lines are the impressions of previous shapes). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.10 Fan Fractal; Above - Affine transformations; Bottom – Convergent sequence (The dotted lines are the impressions of previous shapes). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11 A comparison of the Hausdorff dimensions. . . . . . . . . . . . . 8.12 Fractal-based trusses design by transforming the the third iterated models of Sierpinski Triangle and Asayama-Mae’s Fractal, and the second iterated model of Pinwheel Fractal. . . . . . . . .

127

127

128

131

132

132 135 136

138

139 140

142

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List of Figures

8.13 Transformation of third iterated Fan Fractal into its correspnding truss design by adding a newline Fateachiteration. . . . . . . . . 143 8.14 Transformation of third iterated Baltimore Fractal into its correspnding truss design by adding a newline Bateachiteration. . . . 144 8.15 (a) An example of the architectural appearance of a fractal-based truss (Fan Fractal truss); (b) An illustration of a typical joint detail.144 8.16 Comparison of Box Counting dimensions (left) and total weighs (right) of different fractal-based trusses including conventional compound trusses when all the members have same cross-sectional area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 8.17 Evaluation of the statically determinacy of fractal-based trusses. 146 8.18 Different load cases; (a) vertical nodal loads applied on two rafters, (b) vertical nodal loads applied on the bottom chord, and (c) lateral nodal loads at one rafter. . . . . . . . . . . . . . . . . . . . . 147 8.19 The deformations of fractal-based truss structures under vertical nodal loads applied on the top. . . . . . . . . . . . . . . . . . . . 148 8.20 Left - The comparison of stiffness k = F/ n of fractal-based trusses and conventional compound trusses, where F is applied forces and n is maximum nodal deformation; Right – Comparison of Box Counting dimensions of the truss structures. . . . . . . . . . 149 8.21 Internal forces in the Sierpinski Fractal truss; (a) Internal forces due to nodal forces applied on the upper rafters; (b) Internal forces due to nodal forces applied on the bottom chord. . . . . . 149 8.22 Reference fractal pattern of Sierpinski Fractal truss to avoid inactive members; (a) when applied vertical forces are on the top rafters; (b) when applied vertical forces are on the bottom chord. 150 8.23 Reference fractal pattern of Pinwheel Fractal truss to avoid inactive members; (a) when applied vertical forces are on the top rafters; (b) when applied vertical forces are on the bottom chord. 150 8.24 Axial forces in the fractal-based trusses under the combined three load cases ’a’, ’b’ and ’c’ of ’Figure 8.7’ Green to blue and violet is under compression, and yellow to brown and red is under tension.151 8.25 Optimal design of Sierpinski truss with uniform member cross section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.26 Different maximum displacements of Asayama-Mae’s truss with the different base angles under the point load at top, and their Hausdorff dimensions. . . . . . . . . . . . . . . . . . . . . . . . . 153 8.27 Optimized configuration of the Asayama-Mae’s Fractal truss. . . 153 8.28 C++ code for the cross section optimization. (Clemens2014) . . 156 8.29 Cross section optimization of a Sierpinski truss. . . . . . . . . . . 157

List of Figures

9.1 9.2

9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10

9.11 9.12 9.13 9.14 9.15 9.16

9.17 9.18

9.19

235

Archimedes’ method for calculating the area of a parabola segment161 Top - Archimedes Midpoint Displacement Method for dissecting of a parabolic segment into infinitely many triangles; Bottom The sizes of parent triangle and its child triangle. . . . . . . . . . 163 Midpoint displacement using vector method. . . . . . . . . . . . 166 Construction of a curve using vector-based Midpoint Displacement Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Takagi-Landsberg curve and its Hausdorff dimension with the changing of w value. . . . . . . . . . . . . . . . . . . . . . . . . . 167 Relation between the relative size factor w and the Hausdorff dimension DH of Takagi-Landsberg curve. . . . . . . . . . . . . . 168 (a) Pont du Gard bridge in Gard, France; (b) Fractal Layered Arch in Tokyo, Japan. . . . . . . . . . . . . . . . . . . . . . . . . 169 Different load cases. Left - Unformly projected load, Center Asymmetric load and Right - Wind load. . . . . . . . . . . . . . 170 Axial forces due to uniform projected loads; Left – Parabolic arch structure (w = 0.25); Right – Takagi arch structure (w = 0.5). . 171 Axial forces due to asymmetric (right half) applied loads; Left – Parabolic arch structure (w = 0.25); Right – Takagi arch structure (w = 0.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Midpoint Displacement Method applied on an equilateral triangle.174 The construction of a Takagi surface after a number of iterations when w = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Transformation of paraboloid’s smooth surface to fractal surface and their fractal dimensions. . . . . . . . . . . . . . . . . . . . . 174 Construction of different types of Takagi surfaces using different polygonal bases when w = 0.5. . . . . . . . . . . . . . . . . . . . 175 Relative size factor w Vs. Hausdorff dimension DH graph. . . . . 175 Fractal-based gridshell design when w = 0.5; (a) Additional frame structure (‘type 1’ fractal) obtained after 3 iterations; (b) Takagi grid obtained after 4 iterations; (c) Fractal gridshell design by combining the Takagi grid and the additional frame structures; (d) Panels fixed on the grid. . . . . . . . . . . . . . . . . . . . . . 176 Different load cases (a) uniformly projected loads, (b) asymmetric load and (c) horizontal wind load. . . . . . . . . . . . . . . . . . 177 High nodal displacements shown as red discs under uniform projected loads; Left - Paraboloid gridshell when w = 0.25; Right Takagi gridshell when w = 0.5. . . . . . . . . . . . . . . . . . . . 178 Internal Axial forces; Left - Paraboloid gridshell (w = 0.25); Right - Takagi gridshell (w = 0.5). . . . . . . . . . . . . . . . . . 179

236

List of Figures

9.20 Relation between relative size factor w and the stiffness 1/δ under uniformly projected loads, asymmetric load and wind load. Here, δ is the nodal displacements along vertical axis for uniformly projected l and asymmetric loads, and along horizont al axis for wind load. Top - Displacement of the central node; Bottom Maximum nodal displacement. . . . . . . . . . . . . . . . . . . . 180 9.21 Optimal designs under different load conditions optimized by single-objective optimizer Galapagosasamemberof the0 OptimizationT oolBox0 .182 9.22 Left - Simplified assemblage of additional frame structure; Right - Four structural modules, three of them (1, 2 and 3) are perfectly self-similar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 9.23 Manual fabrication of different parts for each structural module. 184 9.24 Fractal-based gridshell structure without covering panels. . . . . 185 9.25 Final structure after fixing the covering panels. . . . . . . . . . . 186 10.1 (a) Deterministic fractal curve with random impression using Archimedes’ Midpoint Displacement method; (b) Unpredictable random fractal curve using Archimedes’ Midpoint Displacement method and flipping a coin; (c) Unpredictable random fractal curve using Gaussian Midpoint Displacement method. . . . . . . 189 10.2 Left - Box counting method for a random fractal curve; Right Log-log graph for calculating the fractal dimension of the curve which is the slope of the trend line. The fractal dimension is 1.385.190 10.3 Left – A Gaussian distribution centered at zero; Right - Fractal noise generated using the different values of fractal dimensions as a parametric input. [8] . . . . . . . . . . . . . . . . . . . . . . . . 191 10.4 The method DS Algorithm that produces fractal terrain. Red dots are the new vertices while the blue dots represent previous vertices. Top and middle - Plan views; Bottom – Isometric views. 193 10.5 Natural terrain-like crinkled surface generation from a flat surface using DS Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 195 10.6 Structural condition of folding structures. [75] . . . . . . . . . . . 196 10.7 (a) Example of longitudinal folding as a structural shape, (b) Example of composite pyramidal folding as more complex structural shape. [75] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 10.8 Left - Air Force Academy Chapel, Colorado Springs, USA, architect Walter Netsch, year 1962; Right - Interior of the Yokohama port terminal designed by Foreign Office Architects (FOA) in 1995. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 10.9 Different forms of folded structures based on their geometries and constructions. [75] . . . . . . . . . . . . . . . . . . . . . . . . . . 198

List of Figures

10.10Elevations of some random form variations of the proposed foldedplate shell structure showing the positions of their supports for checking the apparent structural feasibility. . . . . . . . . . . . . 10.11an architectural impression of the random folded-plate fractal shell structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.12Vertical displacement and principal stress. . . . . . . . . . . . . . 10.13Optimal shapes at different generations. . . . . . . . . . . . . . . 10.14Left - A piece of flat paper on a bottle; Right – A piece of crinkled paper on a bottle. . . . . . . . . . . . . . . . . . . . . . . . . . . 10.15(a) Thread hanging model for designing an optimized canopy structure [?], (b) Canopy structure with folded plates origami roof [?], (c) Canopy structure at ION Orchard in Singapore by Benoy (2009), (D) Canopy structure at WestendGate in Frankfurt by Heinrich Rohlfing GmbH, Sternwede-Niedermehnen. (2010) 205 10.16Noises of a surface resulted after different fractal dimensions. . . 10.17(a) Independent branching structure (b) Wrinkled roof, which is placed at the centre of the overall branching configuration at a certain height, overlaps the branches separating some branches above the surface and some are under the surface, (c) the upper branches are trimmed out from the intersection points in between the surface and branches. . . . . . . . . . . . . . . . . . . . . . . 10.18Design variations of the pavilion unit; Top - The design variations of branching supports with the changing of vertical angles and contractivity values; Bottom- The design variations of roof surface from flat to random and noisy with the changing of z-limits and fractal dimensions. . . . . . . . . . . . . . . . . . . . . . . . . 10.19An architectural model of the main nature-inspired fractal-based canopy structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.20Deformations of a roof due to self-weight when its flat surface is transformed into crinkled surface. . . . . . . . . . . . . . . . . . . 10.21The displacements of the roof under the gravity and mesh loads with the changing of fractal dimensions, while z-limit is constant (1.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.22Fractal dimension vs. maximum displacement graphs for two different trials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.23Fractal dimension vs. roof weight graphs. . . . . . . . . . . . . . 10.24Schematic diagram of the automated computational search using the ‘Optimization Tool Box’ . . . . . . . . . . . . . . . . . . . . .

237

200 200 201 203 204

206

208

210 211 213

215 216 216 217

238

List of Figures

10.25Optimal design of the randomly folded fractal-based roof canopy structure after 100th generation of optimization process. Top Nodal displacements, Bottom - Architectural view. . . . . . . . . 218

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