Altair's Student Guides - CAE and Design Optimisation - Basics

CAE and Design Optimization – Basics

Contents

Contents Introduction ......................................................................................................2 About This Series ...........................................................................................2 About This Book .............................................................................................2 Supporting Material ........................................................................................3 Engineering Design Practice ...............................................................................4 Characteristics of Different Sectors ..................................................................4 CAE And The Design Cycle ..............................................................................5 The Impact of Optimization on CAE .................................................................6 Summary: How Engineers Should Design.........................................................8 Optimization Theory ........................................................................................ 10 What is an Optimum Design? ........................................................................ 10 Analysis and Design...................................................................................... 11 Finding An Optimum..................................................................................... 13 The Optimization Model ................................................................................ 16 Workable Implementations ........................................................................... 19 Summary ..................................................................................................... 19 FEA Essentials ................................................................................................. 21 Why use Numerical Methods at all? ............................................................... 21 What is Finite Element Analysis? ................................................................... 22 Choosing a Numerical Model ......................................................................... 24 The Role of Physical Testing ......................................................................... 25 Quick Summary of Analysis Terminology........................................................ 26 What are Elements? ..................................................................................... 30 Steps in FE Modeling .................................................................................... 31 Guidelines on Element Choice ....................................................................... 34 OptiStruct ....................................................................................................... 35 Before We Start ........................................................................................... 35 Techniques to Design Optimum Products ....................................................... 39 Putting it all together.................................................................................... 47 Summary ..................................................................................................... 49 Laminates ....................................................................................................... 51 The Miracle Material – Plastics....................................................................... 51 Reinforced Plastics: One Step Ahead ............................................................. 51 Data Required for Stress Analysis .................................................................. 53 Finite Element Approaches............................................................................ 57 Design Optimization Issues ........................................................................... 59 Glossary And References.................................................................................. 63

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Introduction

CAE and Design Optimization – Basics

Introduction About This Series To make the most of this series you should be an engineering student, in your third or final year of Mechanical Engineering. You should have access to licenses of HyperWorks, to the Altair website, and to an instructor who can guide you through your chosen projects or assignments. Each book in this series is completely self-contained. References to other volumes are only for your interest and further reading. You need not be familiar with the Finite Element Method, with 3D Modeling or with Finite Element Modeling. Depending on the volumes you choose to read, however, you do need to be familiar with one or more of the relevant engineering subjects: Design of Machine Elements, Strength of Materials, Kinematics of Machinery, Dynamics of Machinery, Probability and Statistics, Manufacturing Technology and Introduction to Programming. A course on Operations Research or Linear Programming is useful but not essential.

About This Book Finite Element Analysis has traditionally been used as a design-verification method. Recent developments in Mathematics, Software and Mechanics have led to a dramatic change: Computer Aided Engineering (CAE) is now widely deployed at the concept-design stage itself. This volume presents these changes in the engineering industry and introduces you to the theory necessary to effectively apply optimization techniques using OptiStruct. You should read Chapters 2 and 3 in their entirety. Chapter 4 can be skipped if you are familiar with the Finite Element Method, but is essential reading if you’re not. Chapter 5 is best read once, then referred to again when you are working on your assignment. Chapter 6 is essential if you want to work on a project addressing laminated composites, but can be safely omitted if you’re working with other materials. The various references cited in the book will probably be most useful after you have worked through your project and are interpreting the results.

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Introduction

Supporting Material Your instructor will have the Student Projects and Student Projects Summaries that accompany these volumes – they should certainly be made use of. Further reading and references are indicated both in this book and in the Projects themselves. If you find the material interesting, you should also look up the HyperWorks On-line Help System. The Altair website, www.altair.com, is also likely to be of interest to you, both for an insight into the evolving technology and to help you present your project better.

"Mach 2 travel feels no different." a passenger commented on an early Concorde flight. "Yes," Sir George replied. "That was the difficult bit." Sir George Edwards

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Engineering Design Practice

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Engineering Design Practice It’s rare to meet a mechanical engineer who hasn’t, at one time or another, been fascinated by automobiles and aircraft. With attractive looks, complex designs, exciting performance, and a long history of colorful personalities, they embody everything you ever wanted to create. Who’d ever want to design anything else? Lots of people, it turns out. While the automotive and aerospace sectors continue to define technology and trends in mechanical engineering even today, these are not the only sectors that make widespread use of design software. Neither are they the only segments where serious sums of money hinge on the efforts of the Design Engineer. The section below outlines the bases for evaluating designs in various engineering sectors.

Characteristics of Different Sectors First, let’s specify what we mean by the word “Design”. As engineers, we usually ignore aesthetics, leaving that to Industrial Designers. One of the tasks of an engineering designer is to come up with a design that is functionally “satisfying”. That’s very often very hard to do, since there are so many ways to define “satisfaction”. To understand how optimization fits into engineering design practice, it’s instructive to look at the different ways in which “satisfaction” is measured by some industry segments, Cars, motorcycles, trucks, buses and other roadtransport vehicles are often grouped in one sector. Appearance, ride quality, safety and fuel economy are the most important factors in design – apart from cost, of course. Plastics and Steel are the most commonly used materials. Most often, aesthetics and cost dictate the visible areas (interiors and body), with the other parts being designed for efficiency in function. Performance and cost, in most cases, determine satisfaction levels. 4

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Engineering Design Practice

Aircraft and space vehicles are often clubbed in the “aerospace” sector. Cost is rarely an issue, but performance is of paramount importance. Safety regulations in passenger aircraft are among the most stringent in the engineering world. Combat aircraft are subject to harsh environments. Spacecraft can gain useful life with every gram of weight shaved off. Advanced materials – ceramics, composites, honeycombs, and exotic alloys – tend to be widely used, along with advanced manufacturing techniques. Ranging from the cardboard boxes that toothpaste tubes are sold in to the containers of cars that are loaded on to cargo ships, “packaging” is a multi-billion dollar industry. Not very surprising, in fact, when you stop to consider that products like toothpastes and soft-drinks are literally sold in billions. Packaging materials are often “outside” standard engineering technology – Styrofoam and paper are rarely treated as load-bearing materials in engineering courses! Products such as cell phones, stereos, watches, washing machines, etc. rarely cause loss of life if they fail. Sometimes called “consumer goods”, these products are usually designed for elegance and cost. Product life is sometimes measured in weeks, translating into extreme time pressures on designers. Plastics are used very widely. One thing that’s common to all sectors is the fact that designers are always under pressure to create better products in less time and at a lower price. And this, of course, is why optimization plays such an important role in product design.

CAE And The Design Cycle It’s long been recognized that the designer is almost always under pressure to meet time targets, performance targets and cost targets. The figure below shows the “typical” design cycle:

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CAE and Design Optimization – Basics

The design cycle almost always originates with a drawing – a sketch to illustrate a concept - and almost always ends with a drawing – the manufacturing drawing. This is the biggest problem – how to translate the sketch into an acceptable, manufacturable design. A typical design cycle involves numerous trade-offs: appearance vs. function, cost vs. ease of manufacture, etc. Every trade-off changes the design, and changes are inevitable. One of the rules written by Kelly Johnson, legendary head of Lockheed’s Skunk Works, demands that “A very simple drawing and drawing release system with great flexibility for making changes must be provided”. The widespread use of 3D CAD software has made it easier for engineers to re-create manufacturing drawings when the design changes. But CAE1 is often viewed as a visit to the dentist: put off as long as possible, and usually painful. The reason for this is easy to find. Since CAE has traditionally been used to verify the preliminary design, analysts usually bear bad news: that the design has failed the verification and must therefore be changed. If the analyst carries good news, it’s often ignored since it’s too late in the design cycle to implement the changes! Wouldn’t it be great if the concept-designer had a tool that could help suggest designs that are least likely to get rejected by subsequent CAE?

The Impact of Optimization on CAE Relatively recent advances in mechanics and software have provided just these capabilities: software can suggest the design that is best suited to the conditions you specify. In other words, an “optimum” design.

1

Short for Computer Aided Engineering. Usually taken to mean simulation of performance under operating conditions.

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Engineering Design Practice

Design vs. Analysis As opposed to analysis software, optimization tools are for designers. Put forward your definition of a “satisfactory” design, and the optimization software will suggest to you shapes and sizes that are most likely to pass the subsequent analyst’s verification. This means a tremendous change in the way we can now view CAE: the visit to the dentist will not be as unpleasant! Since our design has been “certified” by the optimization tool, we can approach the traditional “verification” stage with a much higher level of confidence. In the rest of this book you will learn how OptiStruct brings these capabilities to the designer.

Other Designer Issues This section summarizes some design approaches that are not addressed by linear optimization – which is the focus of this book. If you think the below are relevant to your problem or are of interest to you, you should read the other volumes of this series. Design-Of-Experiments Statistics teaches us that in a “Normal Distribution”, a large part of the population lies within 6 standarddeviations of the mean. Engineering industries refer to this as “six-sigma” quality – less than 3.4 failures per million parts produced. To achieve these levels of quality, designers setup numerical experiments to account for the statistical variations in various design parameters – the elasticity modulus of bars of steel, the rigidity of supports, the variations in loads, and so on. Designers also use non-linear equations to represent material behaviors. As a result, optimization tools are often classified as linear-optimization software, such as OptiStruct, and design-of-experiment tools such as HyperStudy that interface with non-linear solvers like Radioss. Multi-Disciplinary-Optimization As mechanical engineers, we study the difference between mechanisms and structures, and develop different sets of equations to design these. We study equations that govern the flow of heat, equations that govern the flow 7

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of fluids, equations that govern the stress-distribution, …each as a different subject, with little if any interaction between the equations. How would you optimize a product that has some parts that move rigidly (i.e. act as mechanisms) and some that flex (i.e. behave as structures), and that has to withstand stresses and also be aerodynamic? MotionSolve, used in conjunction with OptiStruct, allows designers to find optimal solutions to some of these problems, while HyperStudy addresses others. Process Optimization A product that’s been designed and verified still stands the risk of rejection: by the process engineers, if it’s too expensive to manufacture. How can you design the manufacturing process to be most “satisfactory” – in other words, how can you optimize the manufacturing process? HyperForm is used to simulate the sheet-metal forming process, and, together with HyperStudy, can be used to arrive at optimal process designs.

Summary: How Engineers Should Design All these tools, then, change the way you should address the design cycle. Rather than use CAE to verify (and in all probability reject!) suggested designs, the preferred approach is to use software to suggest a design that is much more likely to work. In many cases, the “redesign” cycle is all but eliminated.

Computer software continues to redefine the way products are designed. But that does not eliminate the need for engineering judgment. In fact, it increases the burden on the engineer, who now has to act both as the investigator and as an impartial and knowledgeable judge. 8

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Engineering Design Practice

First, you phrase the design requirements in as realistic a way as possible. You may even want to try different statements of the different requirements. Then you ask the software to come up with its suggestions. Finally you sit in judgment: which of the statements was the most realistic, and which of the suggestions do you want to adopt?

Engineering problems are under-defined, there are many solutions, good, bad and indifferent. The art is to arrive at a good solution. This is a creative activity, involving imagination, intuition and deliberate choice. Sir Ove Arup

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Optimization Theory

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Optimization Theory What is an Optimum Design? It’s evident from the previous chapter that as a designer, you should search for an optimum design. What is not so clear is how exactly we can recognize the “optimum” design. The dictionary definition is a good place to start. An “optimum”, says the dictionary, is “the greatest degree or best result obtained or obtainable under specific conditions”. It’s the phrase “specific conditions” that gives you your design freedom. As a designer, you define the conditions that allow you to evaluate your design alternatives. In engineering terms, this means you draw up mathematical equations that quantify the performance of a design. The statement “good ride quality” would translate, for instance, into a specification of the maximum values of the components of acceleration that the passenger’s seat can experience. The quantitative parameter that you use to evaluate a design is called the objective. Of course, you may well have multiple objectives. For instance, it’s very likely a car designer would simultaneously want excellent safety and low cost. Unfortunately, in many cases, the objectives are contradictory, making it increasingly difficult for the designer to reach the best compromise2. A working design almost always involves a compromise of some sort or the other. To make things harder for you, few designers have the luxury of infinite resources in the pursuit of their objectives. Whether the resources are the money you can afford to spend on materials, the amount of fuel the spacecraft can carry or the maximum drag coefficient permitted for a sports car, there are usually limits you have to work between. These limits, or constraints give rise to the subject named constrained optimization. A solution that satisfies the constraints is called a feasible solution, while one that does not is called an infeasible solution. It’s important to realize that not all design is done from scratch. In several cases, we have to start from existing designs and improve them to the best extent possible. This could be for various reasons, ranging from the 2

MOO, or multi-objective optimization, is covered in CAE And Design Optimization Advanced. 10

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necessity to liquidate existing inventory to the modification of a manufactured design that has failed a test. If you’re starting from scratch, you can list the objectives and constraints and search for the best solution. If you’re working on modifying an existing design things are usually a little harder since you have less flexibility to change things. Mechanical Engineers face one further requirement. Most components you design have to assemble with other components. They need to fit together. This means you have to work with a package space within which your component needs to fit3, and assembly points that cannot be varied since they’re decided by other components. In mathematics, the package space is referred to as the design space or the optimization domain. Finally, you may not be allowed to change every possible parameter. For example, the material you can work with may be restricted by factors beyond your control: working with sheet steel limits you to commercially available thickness. The parameters that you have the freedom to vary are called design variables. The dependence of the objective on the design variables is expressed as an equation, called the objective function. The statement of the design optimization problem then, consists of the package space, the design variables, the constraints and the objectives. If you have any of these wrong, it’s pretty likely your design proposals will be useless!

Analysis and Design Coming up with a concept involves a synthesis of ideas to suggest different alternatives or proposals. Evaluating the performance of each proposed design involves analysis of function of that particular proposal. As a designer, which should you focus on? The nice part of using Design Optimization as a part of CAE is that you can simultaneously do both, instead of doing them one after the other. As we 3

CAD users will be familiar with the use of “envelopes” in 3D modeling 11

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saw in the previous Chapter, the separation of conceptual-design and design-verification into distinct steps was one of the main reasons analysis is frowned upon even though it’s essential for good product design. In the conventional design process, the designer would have to rely on experience or insight to come up with proposals. The analysis tool is then used to evaluate each proposal, with the designer using these analysis results or responses to choose the “best”. Optimization changes this. The designer outlines the constraints, and leaves it to the optimization tool to come up with proposals. The optimizer uses the analysis tool to decide how to change the initial design to arrive at a better one. In qualitative terms, an analysis problem has only one correct answer4. Design, of course, has no single “correct” answer. There are always a variety of options that can satisfy the same requirements, which is why it is extremely important to search for an optimum design. This is the reason a good analyst often does not make a good designer! For the designer, then, analysis and optimization are very much complementary functions. They are equally important parts of design optimization: a design optimization model consists of an analysis model and an optimization model. These are related and dependent but distinct areas, so we will take some care to understand which parts of the design problem will be defined in the analysis model and which in the optimization model. This chapter outlines the background of optimization, while the next outlines the basics of one of today’s most popular analysis methods, Finite Element Analysis. Other analysis methods can also be used, of course, as in multidisciplinary optimization or non-linear optimization. These are covered in the other volumes of this series.

4

At least in linear analyses, where uniqueness-of-solution is an important mathematical proof.

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Optimization Theory

The world of optimization is a hard one to live in. It’s a little like being asked to search for a black cat in a dark room. You know it’s in there somewhere, but have to feel your way forwards, backtracking and changing direction frequently since the cat changes its position every time you move5. In the world of linear equations, at least we’re assured that there’s a cat in the room, and that there’s only one cat to look for. In many real world problems, we cannot always count on this, as we’ll see. Since a person who analyses is called an analyst, perhaps a person who seeks to optimize should be called an optimist! Our objective, then, is to find a better design than the one we are starting with. In some cases it will be the best while in other cases it may not.

Finding An Optimum Since we will be happy to find a better solution even if it’s not the best, we are looking for an optimum solution, not necessarily the optimum solution. Why are we emphasizing this statement? In optimization theory, by convention, we search for the minimum of the objective function. This is not a limitation since maximization of an objective is equivalent to minimizing its reciprocal6. A function that has only one minimum within the optimization domain is called a convex function. It’s useful to recall the basics of differential calculus. In calculus, a minimum (as well as any other “turning point”) of a curve is characterized by a zero slope (or first derivative). If the objective function is a quadratic function of the design variables, we are then guaranteed a global minimum. This is because a second order curve has only one turning point – and therefore only one minimum in the design space. 5

In mathematical terms, this behaviour is a characteristic of implicit equations. The “knowns” and “unknowns” cannot be neatly separated into the right-hand-side and left-hand-side. 6 Sometimes maximization of x is addressed as minimization of the negative value of x, i.e. -x 13

Optimization Theory

CAE and Design Optimization – Basics

A higher order curve may7 have multiple turning points within the design space. If it does, then we may have multiple minima8. The turning point at which the objective function has the least value is the global minimum, while the other minima are called local minima. A real life problem may well have hundreds, if not thousands of design variables. And the objective function may well be a non-convex function, with multiple local minima within the design space. How does optimization software arrive at a better solution within a reasonable time? How does it interface with or use analysis software?

The Mathematics of Optimization Note that this book is restricted to “linear” problems only –where there is a linear correlation between inputs and responses. For non-linear problems, an entirely different approach is used, as described in CAE And Design Optimization - Advanced. To recap, to define a problem in design optimization you must specify the design space, the design variables, the constraints, and the objectives. The corresponding mathematical statement is:

Minimize

f(x) = f(x1, x2, x3, …. xn)

Subject to

gj(x) ≤ 0,

j = 1, … m

xiL ≤ xi ≤ xiU

7

A higher order curve has more than one turning point, but some may lie outside the design space. 8 Recall your calculus: a turning point can be a maximum, an inflection point or a minimum. 14

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Optimization Theory

where f(x) is the objective function, g(x) are the constraint functions, and x is a vector of design variables.

An Example We may be asked to design a light-weight bracket that has to fit in a 300 mm x 300 mm x 600 mm volume. We want the bracket to be made of steel, to carry a load of 100 Kg. The maximum permissible deflection of the bracket is 0.1mm, and the maximum permissible stress is 20 Kg/mm2. We are allowed to use sheet-steel that can be 1 mm, 2mm or 4 mm thick. In this case, our design space would be the 300 mm x 300 mm x 600 mm volume. The objective would be to minimize the mass. The constraints would be the permissible stress and deflection. The design variables would be the thickness of the steel, and the layout of the steel (i.e. how the sheet should “flow”– where material should be located - within the design space). To solve a problem like this, the optimizer would start with an initial configuration or proposal. It would ask the analysis software to evaluate the mass, stress and deformations of this configuration – the values calculated by the analysis package and tracked by the optimizer are called responses. The optimizer would evaluate the sensitivity of the responses to the various design variables, and decide which to change and by how much. When the design variables change, the responses change too. If the steel thickness changes, the mass of the bracket changes. The displacement would probably change too, as would the stress. So the optimizer would again need to ask the analysis package to evaluate the responses. This iterative procedure would continue until the optimizer concludes it has found the best possible design for the given constraints and variables.

Evaluating Sensitivity Evaluating the sensitivity of the responses to changes in design variables is, obviously, a very key part of the optimization process. If we define the response as a function of the deformation u by the equation

g=

T

u

then the sensitivity of the responses to the design variables is given by the equation 15

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CAE and Design Optimization – Basics

∂g ∂ T u+ = ∂x ∂x

T

∂u ∂x

Some design problems have more constraints than design variables, while others have more design variables than constraints. Different algorithms are used by OptiStruct for each case, in order to efficiently arrive at the optimum solution.

The Optimization Model Asking the analysis package to evaluate the responses each time a variable is changed can be very expensive in terms of computer time. OptiStruct takes a different approach: the optimizer builds an approximate model, and does most of its work within this approximate model, turning back to the analysis software only when essential. This makes the optimization much faster. It also has another implication. The analysis model itself is an approximation of the physical behavior of the product. Since the optimization model is an approximation too, the responses evaluated by the optimizer are unlikely to be very precise. They are twice removed from the physical product. This means that as a designer you must subject the final proposal of every optimized design to a verification-analysis. There are various techniques the optimization model uses to reduce computer-time and still get an accurate solution. Most of these are programmed into OptiStruct. As a designer you can control these methods, but doing that requires a good understanding of the mathematics. That’s not the intent of this book. Here, we want to develop a good understanding of better product design. The following sections summarize some of the more important techniques. Remember that the intention is not to be mathematically rigorous. Rather, the intention is to provide you with an overview of the workings of the optimization model. This should help you phrase your design problems correctly.

Managing Local and Global Minima Even if the analysis-model is linear, the optimization problem is frequently not. Take, for instance, the deflection of a cantilever beam that has a rectangular cross-section. The deflection equation is 16

CAE and Design Optimization – Basics

δ=

Optimization Theory

wL3 3EI

The analysis model is linear since the equilibrium equation is a linear function of the state variable δ. If the Elasticity Modulus (E) were a function of the deflection, as in a plastic analysis, the analysis model would be nonlinear. Suppose we want to choose an optimum depth for the cross-section. The Moment of Inertia is given by

bd 3 I= 12 which is not a linear function of the design variable – d. Depending on the objective function chosen, the optimizer might have to search for the minimum of a non-convex function. One of the parameters that determine whether the optimizer finds a global or a local minimum is the starting point of the search – the initial configuration or proposal. Another is the move size, which is the step that the optimizer takes in the direction dictated by the Gradient Search algorithm. If the step is too large, the optimizer may overshoot the optimum, which means it will have to reverse its direction in the next iteration. If it’s too small, the optimizer may take too long to locate the optimum. What does this mean to you as a designer? First, you can vary the move size if the optimizer doesn’t converge. Second, an intelligent choice of the initial configuration and design variables can significantly affect the design suggested by the optimizer.

Convergence and Iteration Control As the optimizer searches through the design space, it needs to check whether proposals it comes up with are indeed optimal or not. Mathematical formulations of the optimization problem usually include Lagrange Multipliers, which can be interpreted as a method to find extrema 17

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of a bounded surface. The objective function, of course, can be viewed as a surface, with the constraints as boundaries. An interesting interpretation of the Lagrange Multipliers is provided by S.Jensen9:

the constraint function g(P) can be thought of as "competing" with the desired function f(P) to "pull" the point P to its minimum or maximum. The Lagrange multiplier λ can be thought of as a measure of how hard g(P) has to pull in order to make those "forces" balance out on the constraint surface The Karush-Kuhn-Tucker conditions, also called the Kuhn-Tucker conditions, are a necessary condition for the solution of an optimization problem to be optimal. The KKT conditions are often used not to find the solution but to obtain information about the solution. This is useful to us, since we are looking for a design solution, not a mathematically precise solution. From our design perspective, it is important to understand that the search is an iterative procedure. First, we can instruct the optimizer how long to search, by telling it the maximum number of iterations. Further, we can tell it how fine the search should be. If the difference between two successive proposals is less than a convergence tolerance, the optimizer can be asked to conclude that this is acceptable to us from a design perspective.

Gradient Search Methods Most engineers are familiar with Newton’s method to find the roots of a polynomial. As shown in the figure, this method uses the slope of the curve to guess at which direction the initial guess should be adjusted in – to increase or decrease it. In practice, the gradient is often computed using a finite difference method. 9

See http://www.slimy.com/~Esteuard/professional.html for an excellent introduction.

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The gradient search method, also called the method of steepest descent, is one of the many methods used by the optimizer to move from the initial configuration to the final solution. Non-linear optimization normally uses other methods, as described in CAE And Design Optimization - Advanced.

Workable Implementations Very often, an exact answer is worthless if it comes too late. An approximate answer that is available in time is much more useful. In order to speed up the optimization process, the optimization model uses Constraint Screening, Constraint Linking and Constraint Deletion. The first, constraint screening, is a technique used to identify which of the constraints are critical to the current iteration. In an effort to reduce the number of variables, the optimizer uses one or more criteria to choose a subset of all variables for each iteration. This subset is likely to change from one iteration to another as the optimizer moves through the design space. Constraint linking is when you can use factors such as symmetry to reduce the number of constraints that need to be considered. Suppose you want all beams in a structure to use the same cross-section because it makes the purchase process easier. In this case, it makes sense to link all of them together, thereby reducing the load on the optimizer. As the optimizer searches through the design space, the current configuration may violate only 2 of 3 constraints. In this case, the third constraint is not important for the iteration. It can be marked inactive and ignored in other words, the constraint can be deleted for this iteration.

Summary Part of the challenge of optimizing a product is that designers are not always able to clearly define their design problem or state their definition of “optimum”. Don’t let this deter you too much. Even if you don’t arrive at the “best” design, any improvement over your current proposal is better. Optimization technology is fairly robust today. Most of the methods outlined above are implemented intelligently by the software. You can, however, make things easier for the optimizer and reduce your

Design Variables Things that can vary – thickness, density, etc.

Responses Things calculated by the analysis model, and of interest to the Optimizer. Mass, deflection, stress, etc.

Constraints Limits on responses or design variables.

Objective

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Value that measures quality of your design. Mass, frequency, center of gravity, etc.

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design time by intelligent choices in both phases of design optimization: the Optimization Model and the Analysis Model. You can also, of course, set the optimizer an impossible task if the statement of your problem is itself wrong.

Therefore O students study mathematics and do not build without foundations. Leonardo Da Vinci

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FEA Essentials

FEA Essentials As we’ve seen, design optimization relies on CAE to calculate the response of the product. Computer Aided Engineering, unfortunately, is a catch-all phrase that’s not very well defined. It can mean just about anything. And often does. In more general usage, CAE, sometimes also called “Simulation”, is the use of numerical analysis to study various behaviors of products. How they react to forces, what drag or lift they experience in a fluid, how they react to different thermal conditions, the forces generated as they experience accelerations, and so on. The other volumes of this set10 address some of these aspects. In the context of our study, however, we’ll focus mainly on the Finite Element Method. Without going into the mathematics of the method, we’ll look at those aspects that let us understand how it fits into our goal: using optimization to enhance product design. Building models for analysis involves making approximations to the initial geometry to omit irrelevant details, specifying conditions on the boundary or at initial time, specifying solver options, choosing output options, and so on. It can help, therefore, if you are familiar not just with the terms used but also with some of the background. This can help you make intelligent decisions when you prepare your models for optimization. This chapter is not intended to be rigorous – numerous textbooks are available that do that job admirably well. Rather, this is a quick summary of some of the salient aspects of Finite Element usage and theory.

Why use Numerical Methods at all? Most engineering problems can be solved using one of two methods: analytical or numerical11. An “analytical solution” is a mathematical expression that gives the values of the desired unknown quantity or quantities at any location in the body, and at any instant of time. But analytical solutions can usually be obtained only for relatively simple 10

See CAE And Design Optimization – Advanced and CAE For Multi Body Dynamics. A third method, often neglected by engineering beginners, is Physical Testing, or “Test”. 11

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problems: as the geometries or mechanics or both become complicated, the effort of finding an analytical solution is often so high that the solution cannot be found at an acceptable cost or in an acceptable time. For “complicated” problems, numerical methods provide approximate solutions that are usually of adequate accuracy. One way of setting up these numerical solutions is to discretize the original body. This means we break the original geometry into several smaller geometries. We first solve the equations governing the mechanics over these smaller bodies, then piece the results back together to get the complete solution. The two most widely used methods are Finite Element Analysis and Finite Difference Methods. The latter are used mainly for problems in Computational Fluid Dynamics (CFD), while the former is used in a wide range of applications.

What is Finite Element Analysis? Finite Element Analysis (FEA) simulates a physical part or assembly’s behavior by dividing the geometry of the part into a number of elements of standard shapes, applying loads and constraints, then calculating variables of interest – deflections, stresses, temperatures, pressures, etc. The behavior of an individual element is usually described by a relatively simple set of equations. Just as the set of elements would be joined together to build the whole structure, the equations describing the behaviors of the individual elements are joined into a set of equations that describe the behavior of the whole structure. One way of looking at it is to recall the approach you studied in Engineering Mechanics12. There, you drew free body diagrams of each member in the structure, wrote equations that related the unknown forces in each member, then wrote equations that had to be satisfied for the forces between members if equilibrium is to be satisfied. Solving these equations gave you the forces in each member. Elements themselves are defined by specifying the nodes, which are the vertices of the element. Just as 4 corners define a rectangle, the nodes define the shape of an element.

12

A more complete discussion is presented in A Designer’s Guide To Finite Element

Analysis. 22

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FEA Essentials

When you choose an element to represent a part of the product, you are also specifying the parameters that define the behavior across the element. For instance, in a stress analysis, if you know the 6 components of deformation13 at any point, you can calculate the strain from this by taking the first spatial derivative. And once you know the strain, you can use the material properties to calculate the stress. For the Finite Element Method, every node has these parameters associated with it, just as in a trussstructure every member has forces associated with its end-points. From the values at the nodes, you can interpolate for the values between the nodes. Suppose you were asked to digitize a surface, using a Coordinate Measuring Machine. Unless your surface were absolutely flat, you would not space the measurement points evenly. Since you have to interpolate between measured values, you would naturally choose to have more measurement points at areas where the surface curves sharply. In maths, we say these areas have a high derivative, or rate-of-change. In a similar fashion, for an FE analysis you would create smaller elements (which means more nodes) at areas where you expect the stress to be high14. The choice of the sizes of elements depends on many things - the anticipated stress levels of a certain area, the detail wanted in the results, the stability of a solution algorithm, the available computational power, and so on. A Finite Element program takes the elements you have defined, lists the equations for each unknown value, puts them together as a matrix equation, then solves all these for the values of the unknown parameters. The equilibrium equation is of the form

[K ]{u} = { f } Since it’s analogous to the equations of spring-deflection, K is often called the Stiffness Matrix, u is called the deformation vector, and f is called the load vector. K is a square matrix, with one row (and column) for each 13

The 6 components are the translations along the 3 axes, and rotations about the 3 axes 14 A high stress means a high strain, from Hooke’s Law. Strain is the first derivative of deformation. Hence a high stress area is one where the deformation has a high derivative. And this, of course, means the rate-of-change of deformation is high in areas of high stress. 23

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unknown variable in the problem-definition. If, for instance, you have used 100 nodes in your model, and each node has 6 unknowns15, your stiffness matrix would be 600 x 600. u and f are each column-matrices. In our example, each has 1 column and 600 rows. A computer is required because of the large number of calculations needed to analyze a part or assembly. It is not uncommon for a model to have more than 1,00,000 unknowns (called degrees of freedom). The power and low cost of modern computers has made Finite Element Analysis available to many disciplines and companies. Finally, remember that most Finite Element Analysis models are applicable only to “structures” – they cannot be applied to “mechanisms”. Components such as the shackles that hold up the leaf-springs of a truck chassis require different treatment. These are not treated in this volume – refer to CAE For Multi Body Dynamics for that.

What are Finite Difference Methods? In some areas, mainly in fluid flow, analysts often prefer to use a different mathematical approach than FEA. In this approach, there are no elements – the discrete points are referred to as grid points or grids. Some analysis programs call for “structured” grids – the numbering of and positioning of grid points must follow specific patterns. Other analysis programs are less stringent in their requirements – unstructured grids or blocks are supported.

Choosing a Numerical Model As a designer, you need to anticipate the behavior of the product you’re designing. You will need to guess at the conditions it is likely to be exposed to, and then to predict how it will respond to these conditions. In some situations, the behavior is independent of time – these are called steady state problems. In others, the solution varies with time – these are called transient problems. In some situations, the response of the body to stimuli is linear. That is, there is a linear correlation between input and output. Such a model is, obviously, called a linear problem. Other situations are non-linear because there’s no linear dependence between stimulus and response. 15

The 6 components of deformation are the translations along 3 axes and the rotations about the 3 axes

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It’s important to remember that the product you’re analyzing does not know whether it is “linear” or not. You, as the analyst, can choose to model it as linear or as non-linear, depending on which is more likely to give you useful results. Since we are designers, not mathematicians, we are not interested in results that are “exactly correct”. We are willing to settle for “approximately correct” provided we get the results in time and at a cost we can afford. As you know from your courses on Linear Algebra and Differential Equations, linear equations are far easier to solve than non-linear equations. Therefore, we very often choose to model behaviors as linear even if a non-linear model is more precise. We tend to choose non-linear models only if there’s no linear model that’s even reasonably accurate. Non-linear models are of several types – the materials used, the geometry involved, or conditions on the boundary can cause the “non-linear” nature. Examples of material non-linearity are plastic deformation, melting and solidification – the stiffness of the body changes as the material properties change. In other problems, the stiffness changes as the body deforms even if the material’s properties do not change – take for example the reduced rigidity of a plastic bottle as it is crushed. Examples of boundary nonlinearities are contact and thermal radiation. In the former, the stiffness of the part or assembly changes as sections come into contact with each other. In the latter, the heat lost is proportional to the 4th power of temperature. Some models, such as those required to simulate the behavior of a car when it crashes, can involve several of these types of “non-linearities”.

The Role of Physical Testing Remember that the the terms in the preceding paragraphs describe the mathematical model of the physical behaviour. It’s your responsibility to exercise your engineering judgement to ensure that the model you’ve chosen does a reasonably good job of capturing the physical behavior. In order to make it easier for the design engineer to verify this, in actual engineering design, results of physical tests are used to verify that the numerical model is capable of reproducing conditions obtained under test conditions. Then why not just test a physical model? Why simulate it at all using a numerical or analytical model? 25

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For two reasons. First, constructing test models is expensive and time consuming. In many cases, there’s no way to reduce the time for the test. This is starkly different from computer-methods. In the computer-world, a faster computer produces results faster. This same time-compression effect cannot be obtained in most tests! Second, tests themselves are not very easily controllable. As a result, most engineering practice requries that the analysis model be validated against a test result. That is, the model is used to simulate performance of the product under conditions similar to an existing test. If the model is capable of doing this, then we assume it is capable of reproducing behaviour under different conditions too. As a result, we can dispense with the physical test for further studies.

Quick Summary of Analysis Terminology Linear, Static This model is used when the response of the body is linear, and there’s no variation with time. In stress analysis, this model is appropriate when operating within the elastic region (i.e. the stress-strain curve is linear) and when the deformations are small16 and when the loads do not vary with time. This model is used widely since it’s quick to solve and relatively easy to interpret the results. Very often, even if a non-linear model is more realistic, a linear model is used to investigate likely behavior. Once the options have been narrowed, a full non-linear analysis can be used. The equilibrium equation is

[K ]{u} = { f } where K, u and f are functions of x, y and z only – they are independent of t.

16

26

Remember that large deformations need not mean inelastic behaviour.

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Linear, Transient In stress analysis, this model is appropriate when operating within the elastic region (i.e. the stress-strain curve is linear) and when the deformations are small but when the external conditions do vary with time. Transient problems themselves are often subdivided into further classes, depending on whether the load varies with time in a periodic fashion or not. If the load is periodic, as for instance if the excitation source is a rotating unbalance, it is called “Harmonic” excitation. This is usually easier to solve than when the load is non-periodic. It’s important to note that Finite Element mathematics is applicable only to spatial discretization. A Finite Difference method is usually used to step the solution forward in time, from the initial time to the final time. To setup the problem for analysis, then, the values at the boundary are specified at the initial time (often referred to as “t = 0”). Time-variant solutions can also be calculated by representing the solution17 as a weighted sum of the “mode shapes”. In the equilibrium equation

 ∂ 2u  [M ] 2  + [C ] ∂u  + [K ]{u} = { f }  ∂t   ∂t  K is a function of x, y and z only – it is independent of t. F and u, however, vary with t. M represents the mass, and C the damping.

Normal Modes Sometimes our design problem is not just to calculate stresses or deformations. We may be interested in identifying the resonance frequencies of the system. In vehicle design, avoidance of resonance enhances ride comfort by cutting out unwanted rattles. When designing a loudspeaker or a megaphone, on the other hand, you may want resonance to occur. In cases like these, we need to solve the “eigenvalue18” problem and evaluate the natural frequencies of the body. 17

Look up the use of Rayleigh’s method or Dunkerley’s method. Also look up the Rayleigh-Ritz method for a rough idea as to how this works. 18 Refer to a course or a text on Dynamics of Machinery for more details 27

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The equilibrium equation is

 ∂ 2u  [M ] 2  + [K ]{u} = {0}  ∂t  where K and u are functions of x, y and z only – they are independent of t. The solutions to this equation are pairs of natural frequencies and the corresponding “mode shapes”.

Random Response In some situations, we cannot specify the exact value of the loads as a function of time, but can specify the total energy in these loads. An example would be the forces experienced by a plane when its engines are firing. We know the total energy being transferred from the jet engines to the frame, but cannot claim that we know the loads precisely as functions of time. In cases like these, the loads are characterized by Probability Density Functions, and the behavior is called stochastic. The designer’s goal then is to predict a probability of safety. The several ways to evaluate these responses is beyond the scope of this book.

Inertia Relief Setting up a Finite Element model for static analysis requires that the structure be supported adequately. Some structures, like aircraft in steady flight, are not supported explicitly but are still best represented by staticanalysis models. Inertia Relief is an approach used to model such problems.

Frequency Response In many designs where vibration is important, and correlation with testresults is essential, designers have to characterize the response of the structure as a function of frequency-of-excitation instead of as a function of time. In these cases a Fourier Transform converts the equilibrium equation from the “time domain” to the “frequency domain”.

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In the equilibrium equation, the variables are expressed as functions of ω rather than time. This is called the frequency domain. Of course, the Inverse Fourier Transform can convert the solution back to the time domain.

Linear Buckling Designers sometimes have to take into account the fact that even if stresses are less than permissible values, the structure may fail if it buckles – like a tall column in compression. The equilibrium equation is similar to that of Normal Modes, but the results are interpreted as a “buckling load factor”. Buckling load factors are often important in the design of aerospace structures, where the quest for a minimal weight and the use of advanced materials leads to the frequent use of thin-walled designs.

Non-linear – Gap / Contact The terms “gap” and “contact” are often used to mean the same thing – an opening in the body that may close or widen under the influence of external factors. Clearly, if a gap closes or opens, the stiffness of the body changes. Since the gap opens or closes depending on the deformation, this means the stiffness depends on the deformation. In other words, in the equilibrium equation

[K ]{u} = { f } K is a function of u, making the equation non-linear. The force, f, too can be a function of u.

Component Mode Synthesis When working with large models, resource constraints sometimes force the analyst to break the problem into smaller parts. In static analysis, this approach is called sub-structuring. When used in dynamic analyses, it is called Component Mode Synthesis. It’s sometimes impossible to treat the product purely as a structure or purely as a mechanism. Consider, for example, the feed mechanism for a highspeed packing machine. The rates of acceleration that the mechanism experiences may be quite high. High enough that the deflection of the levers is large enough, perhaps, for the feed mechanism to jam because of misalignment. 29

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Designing such a product requires that the equations of rigid-bodymechanics be coupled with the equations of structural deformation. Component Mode Synthesis also provides a way to do this.

What are Elements? An element is a shape for which the Finite Element program can write out the equations relating the unknown and known quantities. An element is defined by its nodes – the unknowns at each node are called the degrees of freedom. Shapes that are accepted in most finite element programmes are triangles, quadrilaterals, lines, tetrahedra, pentahedra and hexahedra. The sizes of and the number of elements usually have a bearing on the accuracy of the solution. As problems become more complex (advancing in complexity from linear-statics to nonlinear-dynamics), the requirements on shapes and sizes of elements become increasingly stringent. These requirements are often referred to as mesh-specifications, and these are usually strongly analysis-program dependent. In most analyses, the more the number of elements, the better the results. However, the computer time and disk-space required to solve the equations also goes up. Most analysts have to settle for a quality of results that they can afford, given the available computer resources. Fortunately for us, optimization methods are less dependent on mesh quality. As we saw in the previous chapter, the optimization algorithm makes some simplifications and assumptions, so it’s not important that a mesh be “perfect”. Only that it be “adequate”. In your assignments, you will learn how to judge whether your mesh is sufficiently fine.

Element Types Choosing the element type is an important part of any Finite Element analysis. Elements are categorized based on their shape or topology, the number of nodes needed to define them, and the mechanics or behavior they represent. Element types are usually solver dependent – they vary based on the solver used. The elements listed below are specific to OptiStruct, but are available in almost every commercially available analysis package. 30

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Categorization based on Mechanics Beams and Bars (or rods or trusses) are represented by one-dimensional elements – lines or curves – but can lie in 3D space. Plain Strain, Plane Stress and Axi-symmetric elements are two-dimensional shapes that can be used only if the entire model lies in one plane only. Plates and Shells represent surfaces that are two-dimensional in the sense that they have no volume, but lie in 3D space. Solid Elements represent volumes. Categorization based on Topology Standard 2D Elements (plane strain, plane stress, axi-symmetric, plate and shell) are either triangular or quadrilateral. Standard 3D elements are either tetrahedral, pentahedral, or hexahedral. A pyramid with a rectangular base is a pentahedron, as is a wedge. However the two are different element types: the pyramid has 5 nodes while the wedge has 6. Not all solvers support pentahedral elements, and some support only one of the two pentahedra. In most stress-analysis problems, quadrilateral and hexahedral elements are preferred over triangular and tetrahedral elements. For reasons that you can find in the references listed at the end of this volume, they give much better results: more accurate and less CPU intensive. 1D elements are all (topologically) curves – either straight lines or arcs, depending on the number of nodes. Typical applications are as beams, bars, rods, pipes, springs, cold- or hot-runners, and axi-symmetric shells. Categorization based on Order The variation of the unknown quantity between nodes is assumed (by the analysis code) to be linear, or quadratic, or cubic, etc. Linear and parabolic elements are the most common. Linear elements have two nodes along each edge, while parabolic elements have three nodes along each edge. Further refinements do exist – for instance, parabolic quadrilateral elements can have either 8 or 9 nodes.

Steps in FE Modeling Geometry Preparation While it is possible to build a model directly using elements and nodes, this is not often done today. The geometry that defines the area to be analyzed 31

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(also called the “domain”) is usually created first using a CAD program, and elements are created to encompass that boundary or represent the volume. CAD designers create models for manufacture. As many details are included as possible. For a numerical analysis, we often choose to ignore aspects that we think will not significantly affect the solution. For instance, a single hole of 1 mm radius in a plate that is 2 meters wide can probably be ignored safely when calculating the deformation of the plate. Therefore the first task that most analysts are faced with is that of preparing the geometry for analysis. This involves tasks like removal of features, extraction of mid-surfaces, extrapolation of surfaces, etc. Further, the CAD world has an abundance of data exchange formats, since most CAD applications use proprietary data storage formats. A transfer of data from the CAD package to the FE preprocessor sometimes results in a loss of accuracy – gaps are introduced during the import process, for example. Also, CAD assembly models are sometimes made up of parts that were created in different CAD applications. Therefore a cleaning-up of the geometry is often required. This involves filling gaps, eliminating small edges or surfaces that will mislead the automatic-mesh-generation routines, eliminating dangling faces, and so on.

Mesh Creation Once the geometry is more or less ready for discretization, you then start to subdivide the geometry into elements or grid points. The collection of elements is usually referred to as a mesh. Meshes that consist of triangular or quadrilateral elements can often be generated automatically, while tetrahedral or hexahedral meshes usually require considerable manual intervention.

Mesh Editing Once a mesh has been created, the analyst checks if it meets the specifications – several measures of quality are checked, depending on the analysis requirements. Usually, some editing of the mesh is required. Depending on the complexity of the mesh, this can be done either semiautomatically or manually.

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Preparing for Analysis Once the mesh is ready, additional data is specified – the properties of the materials used, the thickness or cross-sectional properties of shell or beam elements, the conditions on the boundaries (restraints, loads or excitations), initial conditions, data for the specific solution algorithm to be employed, kind of output required for text and graphics records, and so on. Once this is done, the data is turned over to the solution program for the next phase – solving. Data is often written out in the form of a text file, which is referred to as a deck. Each line of text in the deck is commonly referred to as a card. A card image is the format followed by the analysis program to interpret the text on the line. The procedure of building the Finite Element Model is sometimes referred to as FEM – short for Finite Element Modeling. Some books, however, use FEM to refer to the Finite Element Method.

Solving The model created in the earlier steps is now taken up for solution – the computer program reads the data, calculates matrix entries, solves the matrix equations and writes data out for interpretation. This task is CPU-intensive, and is often called processing19. Most of the time, very little interaction from the user is required. In some cases, the analyst periodically monitors results to check that they are indeed on the right track. If the solution seems to be evolving in an unexpected direction, the analyst can stop the solver and modify the model, thereby saving valuable time.

Post-Processing After the program has evaluated the results, the analyst examines and interprets the data – looking for errors or improvements in design. As with pre-processing, this calls for substantial interaction from the analyst.

19

Hence the term pre-processing for the preceding steps, and post-processing for the subsequent steps. 33

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Guidelines on Element Choice Learning which element to choose is a little like learning driving. Guidelines exist, but can’t be applied blindly. You need to adapt them to specific situations. Remember this warning! If your product has a region that is long and thin, you can probably model it using beam elements. If this region is connected to the rest of the structure by pin-joints, then you should use truss elements. Regions that are like plates are best modeled using shell elements. Any areas that don’t fall in the earlier categories should be modeled using solid elements. If you have different element types in your model, there are rules that govern the assemblage. For several models, we choose to use just one element type to avoid these complications.

Engineering is the art of modeling materials we do not wholly understand, into shapes we cannot precisely analyse so as to withstand forces we cannot properly assess, in such a way that the public has no reason to suspect the extent of our ignorance. Dr.A.R.Dykes

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OptiStruct

OptiStruct Before We Start The previous chapters outlined the reasons we want optimization to be a part of the product design cycle, and introduced both the Optimization Model and the Analysis Model. The procedure, to summarize, is as follows: 1. As the designer, you decide the design variables, the constraints and the objectives. You also choose the design space, the loads and the restraints – usually dictated by other components in the assembly. It’s a good idea to list these as a design-specification document. 2. Next, you prepare the FE model. To do this, you i. inherit the product definition as a CAD model. If necessary, you must modify it to omit unnecessary details. Note that this step is optional. It makes sense if you are trying to improve an existing design, or if it is easier to build the design-space in a CAD modeler. You may choose to define the design space within the FE pre-processor itself if you’re working on a new concept with a geometrically simple design space. ii. mesh the product-geometry or the design space, depending on which you are starting with. The design space can, but need not, span the entire product. For instance your design may not allow you to change mounting points. In this case, the restraint-areas will not be a part of the design space, although they will be a part of the analysis model. iii. specify material data for the elements – Modulus of Elasticity, etc. iv. specify element properties – the thickness of shell elements or the cross-section for beam elements 35

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v. define the forces acting on the body vi. specify the restraints on the body – where and how it’s supported vii. choose the type of analysis you want to perform – linear-static, modal, etc. 3. Before running the optimizer, you should check that the Analysis Model is adequate. A good way to do this is to run the analysis for meshes of different element-sizes. If the reported results (deformation, stress, frequency, etc., depending on your interest) do not vary with the mesh, it’s reasonable to conclude that it’s adequate. 4. Once the FE model is ready, you prepare the Optimization Model. This means you specify i. the design variables. Remember that different parts of the design space can have different variables. You may have the freedom to place cutouts in one region, but only to vary the thickness in others. ii. the responses that the optimization model needs from the analysis model. The optimizer will use these to evaluate sensitivities. iii. the design constraints. iv. the objective function. 5. Now you are ready to perform the optimization. 6. After the optimization is done, you review the results to check that the optimization has proceeded in line with your design requirement. You may have to revise or restate the optimization model to better reflect the statement of the design requirements. 7. When you are satisfied with the design configuration proposed by the optimizer, you take this geometry back to your CAD modeler for further CAD-related work such as drawing generation, etc. 36

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OptiStruct

In the subsequent sections we’ll review the specific methods OptiStruct uses20.

Terminology OptiStruct includes both an FE solver21 and an Optimizer. In other words, it can be used to solve the Analysis Model and the Optimization Model. The models themselves are created using HyperMesh, which is the preprocessor. HyperMesh is used to define both the Analysis Model and the Optimization Model. The table below lists the key terms used by HyperMesh and correlates them with the Analysis and Optimization Models. Analysis Model

Collector

A way to group related items together. For instance all elements that have the same thickness would be in the same collector.

Load

External forces acting on the boundary. Includes concentrated forces, moments, pressures, gravity, etc.

SPC

Short for Single Point Constraint. Refers to restraints applied to the analysis model at locations where the body is supported22.

Subcase

Combination of SPCs and Loads. Since they represent values on the boundary, these are often clubbed together as Boundary Conditions. A subcase is sometimes called a Load Case.

Card

Some data in the analysis model, such as the material properties, cannot be displayed graphically. Such data is entered as a card image by typing in text or numerical values.

Optimization Model

Response

Any quantity calculated by the Analysis Model, and of interest to the Optimization Model. This could include

20

Some of these features are unique to OptiStruct. See A Designer’s Guide To CAE 22 Do not confuse these with design constraints, which are applicable to the optimization model. 21

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interest to the Optimization Model. This could include mass of the model, volume, deformation, stress, frequencies, etc.The Analysis Model calculates a lot of things, not all of which are relevant to the design problem. Any quantity or variable that you want to use as a design-constraint or as an objective must be identified as responses.

Design Space

One or more component collectors that contain entities that can be altered as part of the design effort. All other component collectors are non-design areas.

Desvar

Short for Design Variable.

Discrete DVs

Design variables that can vary only in predefined steps rather than continuously.

Dconstraint

Design Constraint. Limits on the values selected responses can take. For instance the maximum permissible stress, or a range of frequencies to avoid.

Objective

The goal of the optimization. This must be a quantity that has been tagged as a response. An objective can either be minimized or maximized. MultiObjective Optimization is not covered in this book. We restrict our attention to a single objective function23.

Minmax

This means you want to minimize the maximum value of the objective function.

Maxmin

Used when you want to maximize the minimum value of the objective function.

Opti Control

Optimization Control. Parameters to control the optimization algorithms

23

While MOO is covered in CAE And Design Optimization - Advanced, note that you can define an equation that clubs a series of responses together to form a single response, which can then be used in single-objective optimizations. This is similar to the use of an average-mark in exams.

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OptiStruct

Techniques to Design Optimum Products Using optimization at the design stage itself is a good way to reduce the chances that the design will fail. Thus far, this has been our main motivation for modifying the CAE cycle to include optimization. But your design could be rejected for other reasons too. Having a design criticized on the grounds that it is too hard or too expensive to manufacture is no fun for the designer. It is even worse if your proposal is impossible to manufacture. Mechanical engineering designs can be manufactured in a wide variety of ways: they can be cast, machined, molded, drawn, blanked, forged, welded, and so on. As we saw in the earlier chapter, materials used can range from steel to paper. You may be limited to working with predefined sizes and shapes, as for instance with standard-section beams or metal sheets of standard thickness. With all these complications, how can you work towards designs that are not only likely to pass verification, but also likely to be approved as manufacture-able? In this section, we’ll first run over the techniques OptiStruct offers, and then see how to put these together in a meaningful way for product design.

Types of Analyses OptiStruct’s Finite Element solver can address several different types of linear analyses: static, inertia relief, normal modes, linear buckling, transient, and frequency response24. Component mode synthesis and multibody dynamics are also supported. Non-linear analysis is restricted to simulation of gaps. From an optimization perspective, this means that we can obtain responses for any one of, or any combination of these analyses. All we need to do is tell the Optimization Model which responses of the Analysis Model it should track.

24

Transient analysis and frequency response analysis can be performed using either direct-integration or modal-superposition. 39

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Types of Responses Responses are quantities that are calculated by the Finite Element solver and are important for the optimizer. They can be any of, or any combination of, mass, volume25, center of gravity, moments of inertia, compliance, natural frequency, deformation, von Mises stress, strain, and buckling load factor. For transient and frequency response analyses, velocity and acceleration can be identified as responses. For laminates26, the failure index can be a response.

Types of Objectives The objective can be any of the responses, but some care needs to be exercised. Choosing an objective that’s insensitive to the design variables makes it harder for the optimizer to reach a minimum. Some guidelines on appropriate choices for the objective function are included in the assignments that accompany this book. A more detailed description for each optimization method is available in the on-line documentation.

Types of Constraints Constraints are of two types; first, design constraints, which are any responses that can be obtained from the Finite Element solver. Second, manufacturing constraints. These are guidelines we give the optimizer to reflect our preferred manufacturing method. It would certainly be nice if the optimizer could tell us which of the available materials is best for the design. Engineering designers, unfortunately, know that the factors that govern material choice are often impossible to quantify. And we have been clear that the objective function should be clearly quantifiable. Therefore the choice of material is the responsibility of the designer, as is the choice of the manufacturing method. Having made this choice, we use manufacturing constraints to restrict the set of available designs to those that can be manufactured. The various manufacturing constraints available, and their relevance, are summarized below, with examples to illustrate the difference between design-optimization with and without these options.

25

For a model that has only a single material, mass and volume are equivalent. But for products with multiple materials, there’s a significant difference. 26 See the next chapter for details 40

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OptiStruct

Member Size: Extremely thin or extremely large members can pose problems in metal-casting or injection molding. Size control avoids designs like the one shown on the left, producing “cleaner” designs, as shown on the right.

Draw Direction: Cast or molded components must be ejected from the cavity, which means undercuts must be avoided, and a positive draft must be provided. Split molds require drafts on either direction of the parting surface. Draw direction constraints yield designs such as the one on the right, which is far better than the one on the left. Extrusion: Materials like Aluminum are often extruded. The design on the right, obtained using extrusion constraints, can be manufactured in an extrusion die. The one on the left cannot.

Pattern Repetition: Components like airplane wings have different sections (ribs) that are linked together by spars, covered with a skin and 41

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then subjected to a load. The different ribs must match topologically so that the spar can connect all of them, as shown on the right. Pattern repetition constraints enforce this behavior. The wing sections on the left are worse from a manufacturing perspective.

Pattern Grouping: Symmetry affects the aesthetics of a product. Repetitive patterns often look nicer. OptiStruct’s pattern grouping constraints enforce several types of symmetry: planar, linear, cyclical, radial and circular. The figure on the right uses radial grouping, while the one on the left doesn’t. It’s easy to decide which the better looking design is! Laminated Composites require different manufacturing methods. Some of these are covered in the next Chapter.

Types of Optimization There are six distinct approaches that OptiStruct can take to arrive at the optimum design. These can be used either singly, or in sequence, or simultaneously. In order to decide which to use when, you obviously need to understand the methods themselves. As with our review of the Finite Element method, we will focus on a qualitative effort to grasp the spirit of the methods.

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Topology Topology is a well-established branch of mathematics, dealing with continuous transformations27. From one point of view, it is the study of the relationships between the edges, faces, vertices and volumes of 3D objects, independent of the dimensions. A circle and an ellipse, for example, are topologically identical, though their dimensional characteristics are obviously different. Several theorems in mathematics require that the topology of the domain remain the same. From a stress analysis perspective, vertices are particularly troublesome. Stress is unbounded or singular at a vertex, so introducing new vertices can change the scenario drastically. Obviously, restricting a designer to a single topology is unacceptable: even a simple cutout changes the topology of a design, but cutouts are an essential part of design! OptiStruct’s Topology Optimization approach provides an excellent way to allow the design configuration to change, neatly sidestepping the mathematical problems caused by changes in topology. There are two approaches used by OptiStruct. These are called the density method and the homogenization method, but for the purpose of our qualitative discussion we will not distinguish between them. Think of the difference between a steel sieve and a steel plate, both of the same size. The plate is solid metal, while the sieve has a mesh of holes. As the holes in the sieve get larger and larger, the sieve not only gets lighter, it also gets weaker. Making the holes smaller results in a heavier, stronger and stiffer sieve. From a mathematical perspective, we can treat the solid plate as a sieve that has holes of zero diameters. We can also calculate an “equivalent density” for the sieve. When the diameter of the holes is zero, the equivalent density is the density of steel. As the holes get larger and larger, the “equivalent density” approaches zero. In the Topology Optimization approach, OptiStruct treats each element as a “sieve”. It treats the “equivalent density” of each element as a design variable. The equivalent density is normalized so that 1 is equivalent to no holes or 100% material, while 0 is equivalent to no material in the element. Now given the constraints and the objective, OptiStruct can calculate what equivalent density to assign to each element in order to yield the best design. 27

Interested students should look up the litho-cuts of M.C.Escher for some remarkable topology-related art. 43

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In most cases, you cannot take the same “sieve” approach in your design. Manufacturing constraints will limit the sizes of holes as well as the thickness of material between holes28. Therefore, you take this “equivalent density” plot and exercise your design discretion to decide where to omit material completely (i.e. create holes) and where to retain material.

The density plots closely follow the flow of forces in the domain. As a result, this approach provides a very intuitive understanding of the package space:

where material is wasted, and where it’s most effective. The original-design and the topology-optimized design are shown above. The improvement is clearly visible in the performance-evaluation shown below. The technique is applicable to shell elements, to one-dimensional elements like bars, and to solid elements. In the assignments that accompany this book, we will restrict our attention to shell and solid elements only. Topography Every mechanical engineer knows of the fact that the further a fiber is from the neutral axis, the better it is at resisting bending forces. A plate with ribs can be much lighter than, but just as strong as, a thicker plate. This effect has long been used by designers to strengthen thin plates by providing them with beads or swages. The presence of beads stiffens the plate by moving the fibers away from the neutral surface of the plate. In effect, this alters the topography of the plate. Applicable only to shell elements, OptiStruct’s Topography Optimization makes use of the change in stiffness with depressions in the plate. With an 28

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Sometimes referred to as a ligament.

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intelligent use of manufacturing constraints, this can yield structures that are very efficient, and outside the realm of experience-based design.

Asked to stiffen a water tank, most designers would suggest “regular” stiffening, as shown on the right. The water tank shown on the left is unlikely to be suggested by any “experience”, but is the optimal shape and is eminently manufacturable! Remember that beads can also be interpreted as ribs, which means topography optimization can also be used to decide where to add ribs! Size Suppose you are assigned the task of increasing the strength of a pressure vessel that has already been manufactured because subsequent testing has identified weaknesses. Ribs are ruled out because of assembly requirements. You obviously cannot form beads on the vessel. The only alternative you have is to weld reinforcing plates over selected areas, thereby increasing the effective thickness, and consequently the strength. Using reinforcement plates indiscriminately is not a sensible answer. Not only is it expensive, it may alter the dynamic response of the vessel. How can you address this?

Size Optimization is perfectly suited for such requirements. Scenarios where the design parameters are limited to the thickness of plates, or the cross-section of beams are best addressed by this method. Changes in topology or topography are not relevant here.

Free-size Optimization also treats the thickness of shell elements as design variables, but with a slight difference. Unlike size optimization, which treats elements in groups, it allows the thickness to vary for each element individually. To achieve the same effect using size optimization, you would have to define one collector for each element.

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Ideally suited for machined components or laminates, it provides an effect somewhat similar to topology optimization. In the latter the “equivalent density” is allowed to vary continuously, while in Free-Size, the element thickness can vary continuously. If you specify a minimum thickness of 0, Free-Size Optimization can even provide for cutouts. If you have a thick structure that must be modeled with 3D elements, topology optimization will do the job. If you model a plate with layers of hexahedral elements, the equivalent-density approach allows you to remove elements, thereby achieving a variation in thickness. Free-size is ideal if you have a thin structure that would be prohibitively expensive, from a CPU-time point of view, to model using 3D elements. Free-size also provides a powerful approach for the optimization of laminates, as addressed in the next Chapter. Shape Stress is a very local phenomenon. It often dies away very rapidly. Unfortunately, it also peaks surprisingly rapidly. Any designer who has been faced with a cracked product would rue the fact that adequate fillets were not used. Unlike other types of optimization, here you are dealing with the very definition of the external boundary of the design space. In some sense, this is a little like CAD modeling, treating the dimensions of selected feature as design variables. While Topography, Topology and Size Optimization tell you about overall behavior, Shape Optimization helps you pay attention to detail. The external boundaries of the analysis model are modified by OptiStruct to improve performance. This can dramatically increase your power as a designer, allowing you to step beyond the limits defined by the Finite Element method, giving you capabilities that have traditionally been seen as “pure-CAD”. The figure on the left shows the initial design, with the shapeoptimized design on the right. As you can see, the optimizer has changed the fillet radius significantly. Shape Optimization requires you to specify the perturbation of the boundary. This is not an easy task if the product has more potentially-modifiable boundaries than the designer can specify with reasonable effort. 46

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Free-shape Optimization gets around this. OptiStruct automatically determines the allowable movements and includes these in the Optimization Model.

Putting it all together Remember that the optimizer cannot choose the materials, the loads or the restraints. These come from you. The optimizer can work with the variables you provide to it – these are either element properties, “density” as in topology optimization, or bead-parameters in topography optimization. An intelligent choice of the design variables and constraints helps. It’s not essential, but can help the optimizer find the solution faster if you restrain yourself from taking the brute force approach and leaving all decisions to the optimizer. However a wrong choice of design variables and constraints can completely destroy your design effort, since it may result in no feasible solution, or, even worse, a wrong solution. As you will see in the assignment problems, we frequently track the objective function even as the solution is in progress, to check that it’s proceeding in the right direction! Remember to check that the Finite Element model is adequate. You can help speed up things by organizing elements intelligently. Which of the optimization methods you use depends to some extent on your problem, and on the time you have available. In an ideal scenario, you would first use topology optimization to get an idea as to the best path for load to flow in your design space. Remember that “best” is defined by your objective: weight, frequency, etc. Then you would look at the manufacturing process you can choose from. If you can cast the component, you would set appropriate manufacturing constraints (symmetry, draw direction, grouping, etc.) and run topology optimization again. When you’re satisfied with this, shape optimization can help choose details like fillet radii. If restricted to using sheet-metal, you would lay out a configuration that can be guided by the results of the earlier topology optimization. You would then use topology optimization again on this to identify areas where you can have cutouts, topography optimization to stiffen the product and size optimization to choose the sheet thickness.

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An excellent example of usage of OptiStruct can be found at www.altairindia.com/edu/chassisdesign. This discusses how design optimization was used to arrive at a better design for the chassis of an SUV.

Controlling the Optimizer There are several parameters that control the performance of the optimizer. Some, such as design variable linking and manufacturing constraints, relate to the algorithms used by the software. Others relate to the computer resources available to you, such as the RAM on your computer. OptiStruct makes an effort to choose these intelligently. While you can set these in HyperMesh, choosing appropriate values requires a deeper understanding of the mathematics and algorithms than has been presented in this book.

List of Optimization Results An OptiStruct analysis-and-optimization creates several files, all created in the same directory where your model resides. In general, the best way to view and interpret results is using the postprocessor, HyperView. If things go wrong, though, it sometimes helps to look at other ways to review the information logged by the optimizer. The files all have the same name as the model, but have different suffixes, as summarized below:

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modelname.hm

The model you create using HyperMesh. This is a binary file containing the geometry, analysis model and optimization model.

modelname.fem

This is an intermediate file. It contains the analysis and optimization models only, without any geometry. It is created by HyperMesh and read by OptiStruct. It’s a text file and can be interpreted using the format-definitions listed in the OptiStruct On-line Help.

modelname.out

This is a text file created by OptiStruct. The contents depend on the instructions you specify in HyperMesh when creating the model.

modelname.spcf

This file is created only if you explicitly tell OptiStruct to record all reactions. It is a useful

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check to ensure that you have applied loads correctly.

modelname.stat

This is a text file created by OptiStruct, containing statistics on CPU usage.

modelname.h3d

People who don’t have access to HyperWorks licenses but want to view results of analyses use HyperView Player, freely downloadable from www.altair.com. The Player reads this binary file that is created by OptiStruct.

modelname.html

This is a quick summary of analysis and optimization results. Viewable using any webbrowser.

modelname.mvw

This is a text file, intended for use by HyperView. You will use this file to view stresses, displacements, density, convergence history, constraint violation, etc.

modelname.res

This is a binary file containing the results of the analysis and optimization. It’s readable only by HyperMesh and HyperView.

modelname_frames.html

If you have Macromedia’s Flash installed on your computer, you can use this file to view the results using HyperView Player.

modelname_oss

OSSmooth is a module that helps you take your optimized design back to a CAD model in either STL or IGES format. This file controls the performance of OSSmooth.

There are several other files29, most of which you can ignore in the normal course of events.

Summary You’re now well equipped to use Design Optimization as an integral part of the design process30. Remember that while this changes the way CAE has

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Detailed in the on-line help documentation If you want to learn about design using laminated composites, you should read the next Chapter before going on to the assignments.

30

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traditionally been viewed, it puts some burden on you to think of your problem as one of design rather than as a problem in analysis. Don’t be discouraged if your optimization does not yield results at first. Learning from failure is an integral part of design. Use your understanding of the Optimization Model to revisit the design variables, constraints and objective. Verify that you have indeed used your knowledge of manufacturing technology and machine design properly, and you will be rewarded with safe, elegant, and effective designs.

Some Examples of Usage Worldwide, Conferences include papers presented by designers highlighting applications where they have successfully used the very same technologies that are available to you. www.altair-india.com.edu is periodically updated to reflect the latest publications, so browsing through these is well worth your time.

Aeroplanes are not designed by science, but by art in spite of some pretence and humbug to the contrary. I do not mean to suggest that engineering can do without science, on the contrary, it stands on scientific foundations, but there is a big gap between scientific research and the engineering product which has to be bridged by the art of the engineer. British Engineer to the Royal Aeronautical Society, 1922. Quoted by Walter G Vincenti in 'What Engineers Know and How They Know It'.

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Laminates The Miracle Material – Plastics First used extensively in packaging products like PVC pipes and polythene bags, plastics are today the “visible” part of almost any product, from expensive automobiles to cheap MP3 players. The reason for this is easy to find: plastics are an industrial designer’s delight because of the ease with which they can be shaped into almost any form you can think of. With metals, manufacturing planners are likely to reject shapes that it’s hard to hammer the metal into. Plastics, on the other hand can be formed, extruded, molded, sintered and even machined with a fraction of the effort it takes to work metals. Most plastics are also much lighter than metals. Unfortunately, from a stress-analysis perspective, while few engineering students have trouble finding design equations that govern the behavior of steel, plastic is another story altogether. While the material is present everywhere you look, calculating the stresses in plastics is a bit of an art. Another disadvantage of plastics, from a designer’s point of view, is just as easy to find. As the name shows, they are not “elastic”. They cannot carry loads as well as most metals can, and certainly not as well as steels. As with many other milestones in engineering history, Nature showed the way out of this. Wood has long been used as a structural material. Its fibers give it strength. As carpenters know, it is easier to saw a piece of wood “along” the grain than “against”.

Reinforced Plastics: One Step Ahead The use of fibers to reinforce plastics, then, led to the development of Fiber Reinforced Plastics, also called Composites. While preserving their weight advantage over steel, they can match or even exceed the strength of even special-alloys. If we look at the strength-to-weight ratio, it’s almost no contest. Kevlar, for example, is used to build bullet-proof vests that are light enough to be worn, yet strong enough to stop speeding bullets.

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The development of composites is a science all by itself. Composite materials come in many forms, including MMCs – metal matrix composites. Our focus, however, is restricted to FRPs – fiber reinforced plastics.

Laminated Composites Laminates are one form of composite materials, in which the reinforcing fibers are laid out in thin mats or “laminae”. The lamina or mat is also often referred to as a ply. Remember that fibers are like cables. They are good at resisting tension, but poorer at handling bending and compression. To take advantage of this behavior, the mats are laid out so that the fibers lie along the direction in which you expect tensile forces to act. If the mats are oriented along one direction only, the laminate is called a uniaxial laminate. It has good strength in one direction, but much poorer resistance to forces applied in any other directions. Most laminates, however, have multiple layers of mats. Also, a lamina may consist of fibers woven together to form a mat. The designer’s job is to determine how many layers to use in which part of the product, and how to orient each layer. Now something’s needed to fill the gaps between the fibers, and to hold the layers together. This is achieved using a binder. The binder, which is usually an epoxy, is sometimes referred to as the matrix. Laminates are widely used in applications that demand excellent strength and low weight. They are also sometimes used in non-load-bearing situations, such as the cowls of motorcycles, but these stylistic applications are not in the scope of our study. Aircraft, spacecraft, and sports equipment are excellent examples of sectors where laminates are very widely used for their excellent strength-to-weight ratio. Most high-performance bicycles, for instance the ones ridden in races such as the Tour de France or the Olympics, have frames constructed from laminated composites. Aircraft like the Stealth Bomber also make extensive use of these advanced materials. You will sometimes hear them referred to as CFRPs and GFRPs. These stand for Carbon Fiber Reinforced Plastics and Glass Fiber Reinforced Plastics. Numerous fibers can be used for reinforcement. As with plastics such as Kevlar, trade names are widely used to refer to the materials.

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As shown in the picture31 of the Airbus, developments in composites technology are driven by the fact that the fraction of composites used in aircraft is increasing steadily!

Data Required for Stress Analysis Stress analysis entails two distinct steps. The first is to estimate the stiffness of the structure, from which we can calculate the stresses or resonant frequencies. Once this is done, we check if the stress is within permissible limits using an appropriate failure theory. There are two main differences you should keep in mind when moving from the design of steel components to the design of laminated composites. First, steels are isotropic. That is, their properties are the same in any direction. Plies are different. They resist tension well in the direction of the fiber, but not in other directions. Such behavior is called orthotropic. Orthotropic materials can be characterized by specifying the properties along orthogonal axes. If there are a number of plies stacked up, each oriented at different angles, the material may be better characterized as anisotropic – the properties vary in all directions, not just along orthogonal axes. 31

The image is by Phillipe Cognard, www.specialchem.com 53

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Second, Steels are nicer for the designer since the Elasticity Modulus and Poisson’s Ratio are largely independent of heat treatment, rolling, etc. The permissible stress is the property that varies. In fact, the Modulus and Poisson’s Ratio of steel do not even vary much with alloying elements. This means the first step in stress-analysis, estimating the stiffness, can be done using standard properties. This is not the case for laminates. Unfortunately for the designer, as with all plastics, the properties of the laminates are strongly dependent on processing conditions or manufacturing conditions. Accordingly, it’s essential that properties be obtained from the manufacturer. It is a foolhardy designer who relies on “standard” properties for a plastic or a laminate! The next question that the designer faces is whether the laws of elastic stress analysis are applicable at all. Plastics are known to creep at lower temperatures than metals, they behave differently under fatigue loads, and most important of all they’re not elastic. In other words, the behavior of plastics, and therefore of laminates, is best described using non-linear relations between stress, strain and deformation. Remember what we discussed earlier – linear models of behavior are often chosen even if a non-linear model is more accurate. Designers of plastic components often use the same approach. Even if the material behavior is best modeled as non-linear, linear models are used at the preliminary design stage to narrow down the choices. If the component is critical enough, however, you may need to use non-linear analyses right from the start32. To sum up, there are two classes of data we require to design laminates. First, the material constants that characterize the stiffness. Second, the failure criteria we can use to estimate if calculated stress is within permissible limits.

Material Constants Strain is the first derivative of deformation – that’s a definition that’s independent of the material properties. The stress-strain relationship, however, is material dependent. The 3D equivalent of Hooke’s Law relates 32

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This is covered in the volume on CAE And Design Optimization – Advanced.

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the six components of stress with the six components of strain. In the equation,

σ 11   c11 σ 12   c 21    σ 13   c 31  = σ 22   c 41 σ 23   c 51    σ 33   c 61

c12 c 22 c32 c 42 c52 c 62

c13 c 23 c33 c 43 c53 c 63

c14 c 24 c 34 c 44 c 54 c 64

c15 c 25 c35 c 45 c55 c 65

c16  ε 11  c 26  ε 12  c 36  ε 13    c 46  ε 22  c 56  ε 23    c 66  ε 33 

the constants Cij represent material behavior, and the matrix C is called the Constitutive Matrix. Since the stress and strain components are symmetric (that is, σij = σji), C is symmetric too. This means that the general relationship between stress and strain in 3D requires 21 constants to fully characterize the behavior of the material. An anisotropic material, then, requires 21 constants to be specified for stress-analysis. For an isotropic material, it can be shown that two constants are enough. All others can be derived from these two. Most often, we specify the Modulus of Elasticity (E) and the Poisson’s Ratio (ν). For an orthotropic material in 3D, 9 constants are required, while for an orthotropic material in 2D (as in a ply) 4 constants are required. Usually, we specify the Elasticity Modulus along the two principal directions, and the Shear Modulus and Poisson’s ratio in the “12” direction: E1, E2, G12 and ν12. In this notation, the subscripts refer to the ply’s coordinate system. The “1” direction is along the direction of the fiber, the “2” direction is in the plane of the fiber but perpendicular to the fiber. The “3” direction can be obtained by taking the cross-product of the “1” and “2” vectors.

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2

Y 1 - Fiber Direction

X

Failure Theories From a design perspective, there are a number of different ways in which a component can “fail” – it may buckle, it may deform too much and interfere with another part, it may deform permanently, it may rupture, and so on. In this section, we will restrict our attention to stress as a measure of failure. You are familiar with the use of the von Mises condition, the Tresca condition, etc. to characterize the failure of steel under the permissiblestress mode of design. We will look for a similar method to quantify laminates: how can we determine whether the stresses within the laminated composite are within safe limits or not? Failure criteria, historically, have been derived from experimental observations, not from fundamental laws. One extreme view is that “most

Failure Criteria are meaningless curves passed through unrelated data points”33! Experimental Mechanics is the field in which engineers subject materials to various conditions, then attempt to define boundaries on acceptable environments. These are called Failure Theories. For ductile materials some theories extend till yield (or permanent deformation, which is the elastic limit) while others extend till rupture or fracture. Composites are no different. There are several different failure theories that have been propounded to allow the designer to decide whether or not the stresses are within safe limits. Since composites are unlike ductile materials, they have different failure modes. A composite, for example, can fail by “delamination” – when the binder fails and a layer peels off, like the bark on 33

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J. Hart-Smith, cited in Mechanics of Composite Materials, R. Jones

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a tree. Since the failure can occur either within the fiber or within the binder, characterizing failure is far more difficult. Just as there are no “standard” properties that you can use for a composite, there are no “standard” failure theories you can apply. Most failure theories are valid only for a select set of materials or within a select range of conditions. Extensive testing is often resorted to, with numerical analysis being used as a guideline. Some commonly used theories are the Hill theory, the Hoffman theory, the NASA Larc02 theory, the Tsai-Hill theory, and the Tsai-Wu theory, all of which apply to failure of plies. The Strain Invariant Failure Theory is often used to estimate failure of the matrix.

Finite Element Approaches The sections above outline the theory. How can we use this theory to frame and solve actual problems? From an analysis perspective the question, of course, is how we can determine the stress components from the given loads. Numerical analysis has proved to be very useful in dealing with the complexity posed by laminated composites. The Finite Element Method is more or less the standard numerical analysis method used in engineering design. As we’ve seen earlier, FEA consists of defining the Finite Element model, running the analysis to create the matrix equations and solve them, and then interpreting the results.

Choosing Elements Most laminated composites are thin, which means they are best modeled using shell elements. If the component is not thin, as for example in the hub of the propeller, shear stress plays a more prominent part than bending stress. In this case, solid elements are better used than shells. From this perspective, the guidelines for laminated composites are no different than the general guidelines we saw earlier. The main difference from FE models of isotropic materials lies in the way the material data is specified.

The ABD Matrix The mechanics of composite materials is not very well understood. Several approaches have been propounded, but as the science is still evolving, most commercially available software takes an approximate approach. 57

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Rather than investigate the behavior of each ply and the matrix separately, an “average’ approach is taken. In this approach, we assume the stiffness of the laminate can be separated into bending, extensional, and coupled bending-extensional stiffness. Each of these is represented by a 3 x 3 matrix. To calculate the strains in a laminate, given the forces and moments, the equation used is

 Nx   A11  Ny   A21     Nxy   A31  = Mx   B11 My   B 21    Mxy   B 31

A12 A22 A32 B 21 B 22 B 22

A13 A23 A33 B 31 B 32 B 33

B11 B12 B13  εox  B 21 B 22 B 23  ε 0 y  B 31 B 32 B 33  γ 0 xy    D11 D12 D13   Kx  D 21 D 22 D 23  Ky    D 31 D 32 D 33   Kxy 

where N and M are the Force and Moment components, A represents the extensional-stiffness, D represents the bending stiffness, and B represents the coupled bending-extension stiffness. The 6 x 6 matrix is often referred to as the ABD matrix. In this formulation, the entries in the ABD matrix depend on the thickness of each ply, the orientation of each ply, the distance of each ply from the neutral surface of the laminate, and the material properties of the plies and the binder. If the ABD matrix is available, the stresses and strains can be calculated from the forces. In other words, the problem can now be solved.

Material Data Specifications For the Finite Element solver to do its work, then, the data required is now clear. First we need to supply the material properties (Modulus of Elasticity, Shear Modulus, etc.) separately for the binder and for each ply. Next, we need to specify the “stackup sequence”. That is, we need to describe the order in which the laminae are stacked, the thickness of each 58

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lamina, and the orientation of each lamina with respect to some reference coordinate system. Finally, for each element, we need to specify the coordinate system with reference to which laminae orientations are specified. Given this data, the Finite Element Solver calculates the ABD matrix for each element and proceeds with the solution. Remember that the analyst does not directly specify the ABD matrix itself: it is derived from the data of the constituents of the composite. This is very convenient. Why? Remember that we make frequent use of experimentally measured data. It would be impossible for an experimental scientist to generate data for every single possible combination of plies! This is because a fabricator buys the lamina and binder, then lays up the laminae as required. There are literally infinite ways in which the same laminae can be stacked up and oriented. With the approach described above, the analyst only needs to ask the material supplier for the properties of each constituent. Then, when building the model, the analyst specifies the specific stackup-sequence used.

Interpreting Results Having solved the problem, how do you use the results for your design? Unlike a ductile material where a single equivalent stress can be compared to the permissible stress of the material, the composite designer is often faced with the necessity to examine the laminate layer by layer. You’ll see how to do this in the project that addresses optimization of composites. Fortunately, the HyperWorks approach makes it fairly easy to present the data in the form that’s most convenient.

Design Optimization Issues What we’ve seen so far is the theory of laminate behavior, and the analysis approach. Our interest however is not in analysis. We are interested in design optimization. Given the understanding we have of optimization theory, and of how OptiStruct works, we’ll spend the rest of this Chapter understanding the recommended method to use OptiStruct for the design of laminates.

What we do not Optimize It’s important to realize that there are some parts of a laminate that are omitted from our design-approach. Take, for example, an I-section beam 59

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constructed using laminates. The figure shows the junction in the detailed view. The approach we will follow ignores these areas.

Further, we will restrict our attention to shell elements.

The Poverty of Abundance Let’s restate the design problem before we proceed further. You know the loads you want the component to carry. You have the design space that the component can occupy. You have available a set of plies of predefined thickness, and a binder. You also know your design objective – stress, deformation, frequency, buckling load factor, etc. The problem is to determine how many layers to use, in what sequence to stack them, and what orientation each ply should have. An engineer familiar with the rate at which complexity rises when faced with permutations and combinations can be forgiven for giving up. Given all this complexity, is it even worth it trying to come up with an optimal design? Why not just settle for the design-thenverify approach, even if the accepted design is sub-optimal? It’s like a starving man who is offered a choice of hundreds of dishes: faced with the impossibility of making a “best” choice, he settles for any available food. There are just too many choices! The main motivation to cope with this abundance, of course, lies in the reason we use composites in the first place. Their strength-to-weight ratio 60

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makes a compelling case in the design of weight sensitive equipment. A reduction of even a few percentage points can make the difference between a product that’s “good enough” and a product that’s “outstanding”. With OptiStruct, it’s surprisingly easy to specify an initial configuration and ask the software to come up with a suggestion that’s better.

Discrete Optimization When optimizing laminates, one of the challenges lies in the fact that the laminae can have finite thickness only. Further, the designer should have the freedom to cut a layer off at any point. In other words, some of the design variables can vary in steps, rather than varying continuously. As you will recall from your calculus courses, gradients of a discontinuous function are not defined at the discontinuity. Gradient Search Methods, therefore, need to be modified to work effectively. The mathematics used to solve such problems is called Discrete Optimization. Topology optimization, as we’ve seen, uses a density-based approach. Freesize, on the other hand, uses the element’s thickness as the design variable. Note that this is not the same as size optimization. Using free-size, we can obtain results similar to those of a topology-optimization, but with a more even spread of material through the design space. Problems where bucklingstiffness is the main objective would be better off with topology optimization, while problems where stress is a constraint would yield better results with free-size optimization. Free-size optimization changes the thickness of the laminae during the optimization cycle. Remember that the A matrix represents the extensional or in-plane behavior. Consequently, it’s largely independent of the sequence in which the plies are laid up. (The D matrix is sequence sensitive, since bending stiffness varies with distance from the neutral surface). For most shell-like structures, the extensional stress dominates the behavior, so Freesize works quite well. This makes things much easier for the designer, since you only need to specify which plies you want to choose from – you do not need to specify the number of layers of each ply. OptiStruct will add or remove layers of plies as required!

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Laminates

CAE and Design Optimization – Basics

It’s important to note that laminate optimization problems often have a number of local minima, so the optimization-for-design precept holds good: you are looking for a better design, not necessarily for the best design.

Putting it all together The flow of steps in OptiStruct is as follows: 0. Ensure the FE model is complete. In addition to the usual definition of the elements, loads and restraints, use HyperLaminate to define the properties of the matrix and plies, and to specify the stacking sequence. Each combination of plies+orientation+thickness+stacking-sequence is saved as a PCOMP or a PCOMPG. Then use the “Composites” command to specify the relevant coordinate system and PCOMP / PCOMPG for the elements in the model. 1. Choose the Design Variables. 2. Specify the Responses that the optimizer needs from the Finite Element solver. 3. Specify the constraints and the objective. 4. Run the optimizer. This is pretty much the same as running any other OptiStruct optimization, except for the fact that you are dealing with discrete optimization.

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CAE and Design Optimization – Basics

Glossary, Tips and References

Glossary And References CAD

Computer Aided Design. Usually means creation of 3D geometric models of parts and assemblies.

CAE

Computer Aided Engineering. Said to have been coined by Dr.Jason Lemon in 1980, and meant to consist of CAD + FE Modeling + FE Analysis + Design. Today includes Multi-Body Dynamics (MBD). Often separated into MCAE (for Mechanical CAE, or the analysis of structures) and FCAE (for Fluid CAE, or the analysis of flow of heat and liquids / gases).

CFD

Computational Fluid Dynamics. The use of computers to solve various forms of Navier-Stokes equations to analyze the flow of fluids.

DOF

See Degree of Freedom

Degree of Freedom

Variable we want the FE analysis to solve for. In an FE model every node / grid has at least 1 dof, often more. For a stress-analysis problem, each node can have upto 6 dofs – 3 rotations and 3 translations. Other variables such as temperature at the nodes, pressure, etc. are also included in some models.

Bandwidth

The stiffness matrix of a typical FE model has zeroes in most entries except for a band about the diagonal. The bandwidth measures this “spread” of non-zeroes in the matrix. A smaller bandwidth means faster computation.

Core Memory

High-speed memory, usually RAM. Lower-speed storage, such as disk, is used to provide virtual memory to augment available core memory.

Design Space

In a topology optimization, this space contains the elements whose density can be varied by the optimizer.

Packaging

The art of laying out components in 3D such that best use is made of available space, keeping in mind required clearances between components.

Response Surface

In the absence of a continuous function relating the objective to design variables, numerical experiments can be used to generate a table of objective-function values vs. design-variable values. A surface fitted through this table of points, called the Response Surface, is then used to find optimal locations.

Sensitivity

Reflects the rate of change of the objective function with changes in the design variables. A zero sensitivity is an indicator of a badly

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Glossary, Tips and References

CAE and Design Optimization – Basics the design variables. A zero sensitivity is an indicator of a badly phrased problem: If the objective is independent of the design variable, the optimizer is lost!

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Convex Function

A function that has only one minimum in the domain. This minimum is the “global minimum”.

Convergence

An iterative process can be terminated when it has “converged” to a solution – that is, when subsequent iterations do not change the solution. OptiStruct offers a choice between soft and hard convergence. The former is faster while the latter is more precise.

MDO

Multi-disciplinary Optimization. Used, for example, when your product needs to be designed for optimal performance as a mechanism and as a structure.

Compliance

Reflects the reciprocal of the stiffness. Maximizing the compliance is the same as minimizing the stiffness.

Stiffness

Usually referred to as the “stiffness matrix” in FE models, relates the applied loads to the deformation of the structure. The matrix is square, with “n” rows and columns. “n” is the number of unknowns (dofs) in the FE model.

Stochastic

Something that involves chance or probability, but with an overall and measurable trend or direction – this makes it possible to predict the behavior. Engineers frequently encounter stochastic processes and stochastic variables.

Robust Design

A design method to reduce sensitivity of the design to inherent unpredictability of design parameters.

Isotropic

Material whose properties are independent of direction. Applies to most metals. 2 elasticity constants are required to fully specify the material for stress analysis. The Modulus of Elasticity and the Poisson’s Ratio are most frequently used. In OptiStruct, these materials are of type MAT1.

Orthotropic

Material whose properties vary along principal or orthogonal directions. Applies to many fibrous materials, and to composites that have 2 ply directions. Upto 9 elasticity constants are required to fully specify the material for stress analysis. In OptiStruct, these materials are of type MAT8 for shell elements.

Anisotropic

Material whose properties vary with direction, but not necessarily along orthogonal directions. Several fused or sintered materials are anisotropic. 21 elasticity constants are required to fully specify the material for stress analysis. In OptiStruct, these materials are of type MAT2 for shell elements. MAT9 should be used for solid elements.

CAE and Design Optimization – Basics

Glossary, Tips and References

Supported Responses Responses can be used either as constraints or as an objective. Choices can be quite complex, particularly when multiple optimization-techniques are used simultaneously. Exercise your discretion when assigning responses as objectives. For instance, a topography optimization is unlikely to have much affect on mass, since no material is being added or removed. The responses that OptiStruct supports include: compliance, frequency, compliance index, volume, mass, volume fraction, mass fraction, moments of inertia, center of gravity, displacements, velocities, accelerations, buckling factor, stresses, strains, composite failure, forces, synthetic responses, external (user-defined) functions.

It’s a bad idea to work with a response that you do not understand from an engineering and a mathematical point of view. It’s a good idea to search for a tutorial problem (part of the on-line documentation) that uses the responses you plan on using.

Errors It’s frustrating, and often confusing, to have the solver reject your model because it contains errors. As a general practice, use the “check” option before running an analysis or optimization. It can save you considerable time. Reading the output file (modelname.out) is usually the best way to figure out what went wrong. Common errors include: • • • •

•

putting elements in a wrong collector. For instance, shell elements in a PSOLID collector, or mass elements in a PSHELL collector forgetting to create loads and / or restraints creating restraints and loads in the same load-collector neglecting to specify density when using gravity loads or calculating dynamic response mixing up units

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Glossary, Tips and References

CAE and Design Optimization – Basics

References Engineering Optimization: Theory and Practice, 3rd Edition, Singiresu S. Rao, Engineering Optimization: Methods and Applications, 2nd Edition, A. Ravindran, K. M. Ragsdell, G. V. Reklaitis Optimization Methods for Engineering Design, R.L.Fox, Addison Wesley Arora, J., Introduction to Optimum Design (McGraw-Hill, 1989). Bendsoe, M.P., and Sigmund, O., Topology Optimization - Theory, Methods and Applications (Springer, 2003). Haftka, R.T., and Guerdal, Z., Elements of Structural Optimization (Kluwer Academic Publishers, 1996). Rozvany, G.I.N., Topology Optimization in Structural Mechanics (Springer, 1997).

Other Resources www.altair-india.com/edu, which is periodically updated, contains case studies of actual usage. It also carries tips on software usage.

OptiStruct and the FE Model The table below shows the names of some of the data types OptiStruct and HyperMesh use, along with the relations between them. Use this as a guideline to remember when you need to specify which data. Normally, you would proceed down the table: first the mat collector, then the component collector, then elements (grids or nodes are implicit), and then the load collector. While load collectors can contain both loads and restraints, it’s a good practice to keep them in separate load-collectors so that you can organize them into sub-cases. Data Entity Mat& Component Property% Grid Elem

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Refers to Mat Coordsys Grids,

Data 1

Data 2

Data 3

Data 4

Matid CompName PropId Gridid Elemid

E ElemType+

Nu Mat Id

etc. Prop Data#

X Propid

Y Grid Id 1

Z Grid Id 2

CAE and Design Optimization – Basics Load@ Restraints! SubCase$

Notes: &

Props Grids/ Elems Grids Loads, Spcs

Glossary, Tips and References etc. Loadid

Gridid

Value

Spcid SubCaseid

Gridid

Dofs

Values

A Mat needs a card image. Use Mat1 for linear isotropic, Mat8 for orthotropic shells, Mat9 for linear anisotropic.

+

The component collector is either a PSOLID or a PSHELL. Composites use a PCOMP or PCOMPG.

#

Data depends on the element type. For a solid, there’s nothing. For a shell, there’s the thickness. For a composite, thickness is derived from the PCOMP or PCOMPG data.

%

A Prop collector is needed only for 1D elements like beams or special elements such as springs, connectors, etc.

@

Forces and Moments need no card image. Loads such as gravity, which cannot be depicted graphically, require card images.

!

Restraints normally will not require a card image. Remember that nonzero displacements may be specified, in which case you will need to enter values. Restraints on non-existent dofs are ignored (for instance, specifying restraints on all the rotational dofs of a solid element).

$

Sub-Case definitions are followed by a set of cards, each with a keyword (Load, Spc, etc.) followed by the relevant id. These can be viewed in the “fem” file, not using the card editor.

Common Material Properties Be careful in using these properties. Some properties vary widely with alloying elements or processing parameters, so treat these as indicative. It’s probably safe to use them in exploratory design efforts, but not in designs that will be manufactured. For those, you should look for values from the material supplier. Also remember to check the units in your model – they must be consistent! Material

Modulus of

Poisson’s Ratio

Density

Sample Permissible 67

Glossary, Tips and References

CAE and Design Optimization – Basics

of Elasticity

Units

N/m2

Steel

200x109

Aluminum

Ratio

Permissible Stress

Kg/m3

N/m2

0.29

7800

250x106

69 x109

0.33

2700

110 x106

Wood34

13 x109

0.029

480

50 x106

Cast Iron

190 x109

0.21

7150

170 x106

ABS Plastics

2.3 x109

40x106

Epoxy

1790

Sample properties for a Carbon Fiber Composite material: E1 181 GPa E2 10.3 GPa G12 7.17 GPa ν12 0.28 Density 1.60 gm/cm3

Consistent Units Mixing up units is one of the most common errors. It’s also the least forgivable if committed by an engineer who is allowed to use the SI system of units. While FPS can be challenging, the SI system is very straightforward. The table below lists some common properties of Steel in consistent units. Mass

Length

Time

Force

Stress

Density

Young’s Modulus

Acceleration due to Gravity

kg kg g g ton

m cm cm mm mm

s s s s s

N 1.0e-02 N dyne 1.0e-06 N N

Pa

7.83e+03 7.83e-03 7.83e+00 7.83e-03 7.83e-09

2.07e+11 2.07e+09 2.07e+12 2.07e+11 2.07e+05

9.806 9.806e+02 9.806e+02 9.806e+03 9.806e+03

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In compression

dy/cm² Pa MPa

CAE and Design Optimization – Basics lbfs2/in slug kg

in ft mm

s s s

lbf lbf mN

Glossary, Tips and References psi psf 1.0e+03 Pa

7.33e-04 1.52e+01 7.83e-06

3.00e+07 4.32e+09 2.07e+08

386 32.17 9.806e+02

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