(RAO_00.PDF)
Applied Numerical Methods for Engineers and Scientists Singiresu S. Rao University of Miami, Coral Gables, Florida
©2002 Prentice Hall (http://vig.prenhall.com/), Prentice-Hall Inc. (USA). 高立圖書有限公司 (http://www.gau-lih.com.tw), Tel: (02) 2290-0318 (～1000 pages, ～NT1000)
Contents 1.
Introduction to Numerical Methods
2.
Solution of Nonlinear Equations
3.
Solution of Simultaneous Linear Algebraic Equations
4.
Solution of Matrix Eigenvalue Problem
5.
Curve Fitting and Interpolation
6.
Statistic Methods
7.
Numerical Differentiation
8.
Numerical Integration
9.
Ordinary Differential Equations: Initial-Value Problems
10. Ordinary Differential Equations: Boundary-Value Problems 11. Partial Differential Equations 12. Optimization 13. Finite-Element Method
Preface The use of numerical methods for the analysis, simulation, and design of engineering processes and systems has been increasing at a rapid rate in recent years. The availability of cheap high-speed computing power makes the numerical solution of even complex engineering problems economically feasible. In the face of ever increasing demands on engineering profession to perform better, the students who learn numerical methods in preparing to face the challenges of 21st century should learn not only the theory behind the methods, but also acquire skills to implement the methods for computer solution. In addition, the students should be aware of the many commercial software systems available and their use in the solution of engineering problems. Although a student may not learn all the numerical methods described in this book and use all the software systems available in any one course, he or she should be in a position to intelligently select and use suitable numerical methods and software systems as the need arises in practice. The use of numerical methods in engineering can be considered partly science and partly art. Thus, a cookbook-type procedure will not be effective in learning the methods. A student should solve a problem using different approaches and variety of software systems and experiment with the various parameters of the problem. The different results obtained through this procedure will form an experience base for selecting a suitable method and interpreting the results for a new problem. It is always desirable to compare and verify the results with other available solutions based on engineering judgment and
intuition. This book is intended for course on numerical methods at the junior and senior level as well as at the beginning graduate level. The book also serves as reference for numerical methods in engineering. Fortran and C programs, along with illustrative examples, are given in each chapter to implement many of the numerical methods discussed in that chapter. The use of commercial numerical softwares − MatLab, Maple, and MathCAD − in the solution of practical problems is demonstrated in every chapter. Even when a program from a software package is used, we need to understand the basic principles, purpose, and limitations of the program. Often, in many engineering applications, an available standard program cannot be used directly; we need to adapt and modify it. This invariably requires a sound knowledge of the numerical method as well as some computational experience with the method. The book is aimed at presenting numerical methods along with their practical applications in a manner that helps students achieve the goals just outlined.
Organization Applied Numerical Methods for Engineers is organized into 13 chapters and 6 appendices. Chapter 1 presents an overview of numerical methods, iterative processes, numerical errors, software available for numerical methods, programming languages, and the various aspects of computer program development. The methods of solving nonlinear equations are given in Chapter 2. The solution of sets of linear algebraic equations is
presented in Chapter 3. Both direct and iterative methods are considered. The matrix eigenvalue problem is the tropic of Chapter 4. Chapter 5 deals with the methods of curve fitting and interpolation. The probabilistic and statistical methods are considered in Chapter 6. The numerical differentiation and numerical integration are the tropics of Chapter 7 and 8, respectively. The numerical solution of ordinary differential equations is considered in Chapter 9 and 10. While Chapter 9 presents the methods of solving initial-value problems, Chapter 10 deals with the solution of boundary-value problems. The numerical solution of partial differential equations is considered in Chapter 11. The optimization and the finite-element methods are presented in Chapter 12 and 13, respectively. Appendices A and B provide the basics of Fortran and C languages while Appendices C, D, and E summarize the basics of Maple, MatLab, and MathCAD, respectively. A review of matrix algebra is given ion Appendix F. Finally, Appendix G presents tables of statistical distributions. The material of the book provides flexible options for different types of numerical methods courses. A junior and senior level course may cover the basic techniques of Chapters 1, 2, 3, and 5 to 9. A first-level graduate course can cover Chapters 4, 10, 11, 12, and 13 as well. The prerequisites for using the text are elementary calculus, basic concepts of linear algebra, and an introduction to differential equations. Each tropic for Applied Numerical Methods for Engineers is self-contained. In derivations and developments, steps needed for continuity of understanding have been included to aid the reader at the introductory level. Representative engineering
applications are given at the beginning of each chapter so that the reader can appreciate the practical use and application of the numerical methods presented in that chapter. Many sample problems are solved by using several methods, and the results are compared, discussed, and general conclusions are drawn. Most of the algorithms described in the book are implemented in the form of Fortran and C codes and are made available at the Web site of the book. The use of different commercial software systems, as well as the programs available at the Web site of the book, is illustrated in each chapter.
Features The specific features of the book include 1. A variety of engineering applications at the beginning of each chapter to illustrate the practicality of the methods considered in that chapter. 2. The presentation of the material in a simple and user-friendly form. Illustrative examples follow the presentation of the tropics. 3. A discussion of convergence rate, error, relative performance, and recommendations for the numerical methods. 4. Review questions to help students in reviewing and testing their understanding of the text material. These include multiple choice questions, questions with brief answers, true-false questions, questions involving matching of related descriptions, and fill-in-the-blank type questions. Answers to review questions can be found at the
Web site of the book. 5. A summary of important algorithms in the instructor’s manual. 6. Over 700 problems, with solutions in the instructor’s manual. 7. The inclusion of several open ended, project type and design problems at the end of chapters. 8. Fortran and C programs for many of the methods presented in the book can be found at the Web site of the book. 9. The inclusion of example and problems based on the use of MatLab, Maple, and MathCAD in every chapter. 10. References to lead the reader to specialized and advanced literature. 11. Brief biographical information and photographs of scientists and mathematicians who contributed to the development of numerical methods, found at the Web site of the book.
Wed site of the book The Fortran and C programs used in the book, answers to problems, solutions to review questions, and brief biographical information of scientists can be found at the web site of the book: http://www.prenhall.com/rao. Note that the programs and techniques presented in the book and at the web site are intended for use by students in learning the material.
Although the material has been tested, no warranty is implied as to their accuracy. I would appreciate receiving any errors found in the book.
Acknowledgments I would like to express my appreciation to the students who used the notes that led to the present text. I would like to thank Mr. Qiang Fu, Mr. Lingtao Cao and Ms. Qing Liu, graduate students at the University of Miami, for their help in solving some of the examples and problems using MatLab, Maple, and MathCAD. I wish to thank my family, wife Kamala, daughters Sridevi and Shobha, and grand daughter Siriveena, for their numerous intangible contributions to this work. In particular, I dedicate this book to my wife, Kamala, for providing me the inspiration and support in completing this book.
Singiresu S. Rao University of Miami Coral Gables, FL
Singiresu S. Rao Professor & Chairman Mechanical and Aerospace Engineering University of Miami (http://www.miami.edu/UMH/CDA/UMH_Main/1,1770,6561-1,00.html)
P.O. Box 248294 Coral Gables, FL 33124-0620 (305) 284-3308
[email protected] Education: (http://www.miami.edu/UMH/CDA/UMH_Main/1,1770,6561-1;6644-2;34956-3,00.html) ¾ B.E.(Hons) Andhra University, Waltair, India, 1965 ¾ M.Tech. Indian Institute of Technology, Kanpur, India, 1968 ¾ Ph.D. Case Western Reserve University, Cleveland, Ohio, 1971
Teaching: ¾ Mechanical Design, ¾ Optimization and Reliability in Mechanical and Structural Design, ¾ Finite Element methods, ¾ Structural Dynamics.
Research: ¾ Engineering Optimization, ¾ Uncertainty Modeling in Analysis and Design, ¾ Fuzzy approaches, ¾ Design for Manufacturability, ¾ Structural Control.
Publications: Books Published:
¾ S.S. Rao, Engineering Optimization Theory and Practice, 3rd Edition, 903 pages, John Wiley, New York, 1996. ¾ S.S. Rao, Mechanical Vibrations, 4th Edition, 920 pages, Addison-Wesley, Reading, Mass., 1995 ¾ S.S. Rao, Reliability-Based-Design, 569 pages, McGraw-Hill, Inc., New York, 1992 ¾ S.S. Rao, Applied Numerical Methods for Engineers and Scientists, 1060 pages, Prentice Hall, Upper SaddleRiver, NJ, 2002 Books Edited: ¾ H.-S. Tzou and S.S. Rao (eds), "Intelligent Structures and Vibrations, "Proceedings of the 1995 Design Engineering Technical Conferences, American Society of Mechanical Engineers, New York, Sept 1995, Vols 3, Part C (DE-Vol. 84-3) Encyclopedia Edited: ¾ S. Braun, D. J. Ewins and S.S. Rao(Eds), "Encyclopedia of Vibration", Four Volumes, Academic Press Ltd. London, 2000(in print) ¾ S. Braun, E. Ewing, and S.S. Rao(Editors), "Encyclopedia of Vibration", Volumes I, II, and III, Academic Press Ltd. London, 2002
Technical Papers published (In Transactions of ASME and AIAA Journals): ¾ E.L. Mulkay and S.S. Rao, "Fuzzy Heuristics for Sequential Linear Programming," Journal of Mechanical Design, Vol. 119, 1997(in press). ¾ Li Chen and S.S. Rao, "A Fuzzy Approach for the Manipulation of Uncertainties in Determination of Optimum Machining Conditions Under Multiple Criterian," Journal of Manufacturing Science and Engineering, Vol. 119, No. 2, pp. 186-192,1997 ¾ K. Y. Yoon and S.S. Rao, "Dynamic Load Analysis of Spur Gears Using a New Tooth Profile," Journal of Mechanical Design, Vol. 118, pp. 1-6. March 1996 ¾ S.S. Rao and L. Berke, "Analysis of Uncertain Structural Systems Using Interval Analysis," AIAA Journal, Vol. 35, No. 4, pp. 727-735, April 1997 ¾ M. Sunar and S.S. Rao, "Thermopiezoelectric Control Design and Actuator Placement," AIAA Journal, Vol. 35, No. 3, pp. 534-539, March 1997 ¾ M. Sunar and S.S. Rao, "Distributed Modeling and Actuator Location for Piezoelectric Control System," AIAA Journal, Vol. 34, No. 10, pp. 2209-2211, October 1996 ¾ S.S. Rao and Li Chen, "Generalized Hybrid Method for Fuzzy Multiobjective Optimization of Engineering Systems," AIAA Journal, Vol. 34, No. 8, pp. 1709-1717, 1996
Experience: ¾ 1971-1982 Assistant Professor (1971-1977) and Professor (1978-1982), Department of Mechanical Engineering, Indian Institute of Technology, Kanpur, India ¾ 1981-1985 Professor, Department of Mechanical Engineering, San Diego State University, San Diego, California. ¾ 1985-1998 Associate Professor (1985) and Professor (1988-1998), School of Mechanical Engineering, Purdue University, West Lafayette, Indiana. ¾ 1998-Present Professor and Chairman, Department of Mechanical Engineering, University of Miami, Coral Gables, Florida. ¾ 1980-1981 Visiting Scientist, Multidisciplinary Analysis and Optimization Branch, NASA Langley Research Center, Hampton, Virginia ¾ 1986 AFOSR Summer Faculty Program, Flight Dynamics Laboratory, Wright Patterson Air Force Base, Ohio
Honors: ¾ US Government Fellowship, for studies and training in optimization and reliability in engineering design, 1969-1971 ¾ Vepa Krishnamurthi Memorial Gold Medal, for University First Rank among students of all branches of engineering in each of the five years of the B. E. degree program 1965 ¾ Lazarus Prize for University First Rank among students of Mechanical Engineering branch in B.S.M.E. 1965 ¾ Award for outstanding original contribution in engineering by Telugu Association of North America (TANA), St. Louis, Missouri, 1987. ¾ First Prize in James F. Lincoln Graduat Design Contest, for the paper, "Automated Optimum Design of Airplane Wing Structures", 1971 ¾ Award for outstanding performance in teaching and personal dedication, Graduate student Association, San Diego State University, 1984
Language: English, Telugu, Hindi
Dr. Singiresu S. Rao's career spans the last 35 years. He has been working as professor, chairman and graduate program director of Department of Mechanical Engineering at University of Miami since 1998. Prior to he worked as professor of mechanical engineering, Purdue University for 13 years. He served as major professor for 23 MS (Thesis) and 20 Ph.D. students and published 140 technical papers. He published several pioneering papers in the area of game theory, probabilistic design, fuzzy optimization and interval methods. He introduced the subjects of reliability-based design and design for manufacturability into mechanical engineering curriculum at Purdue University and popularized the use of probabilistic design in automobile industry through Singiresu S. Rao, Ph.D several distance education courses. He served as paper chairman, conference chairman and associate ASME Fellow – 2000 review editor for ASME Design Automation Committee during 1985-1988. He is the author of five popular and American Society well-respected textbooks, which are being used in of Mechanical Engineering hundreds of universities throughout the world. http://www.asme.org/Governance/Honors/Fellows/Fellows_Listing.cfm?FellowshipYear=2000
Biographies of Scientists and Mathematicians: (1571 - 1630) Johannes Kepler
(Chapter 8)
(1638 - 1675) James Gregory
(Chapter 5)
(1643 - 1727) Isaac Newton
(Chapters 2, 5, 7, and 8)
(1648 - 1715) Joseph Raphson
(Chapter 2)
(1682 - 1716) Roger Cotes
(Chapter 8)
(1685 - 1731) Brook Taylor
(Chapters 2, 7, 8, and 9)
(1692 - 1770) James Stirling
(Chapters 5)
(1698 - 1746) Colin Maclaurin
(Chapters 2, 7, 8, and 9)
(1704 - 1752) Gabriel Cramer
(Chapter 3)
(1707 - 1783) Leonhard Euler
(Chapter 9)
(1710 - 1761) Thomas Simpson
(Chapter 8)
(1735 - 1796) Alexandre-Théophile Vandermonde (Chapters 5) (1736 - 1813) Joseph-Louis Lagrange
(Chapters 5 and 13)
(1752 - 1833) Adrien-Marie Legendre
(Chapter 8)
(1777 - 1855) Johann Carl Friedrich Gauss
(Chapters 3, 6, and 8)
(1781 - 1840) Simeon-Denis Poisson
(Chapters 11 and 13)
(1784 - 1846) Friedrich Wilhelm Bessel
(Chapters 5)
(1789 - 1857) Augustin Louis Cauchy
(Chapters 9)
(1804 - 1851) Carl Gustav Jacob Jacobi
(Chapters 3 and 4)
(1815 - 1864) George Boole
(Chapter 8)
(1819 - 1892) John Couch Adams
(Chapter 9)
(1819 - 1912) Francis Bashforth
(Chapter 9)
(1821 - 1894) Pafnuty Lvovich Chebyshev
(Chapters 5 and 8)
(1821 - 1895) Arthur Cayley
(Chapters 3 and 4, and Appendix F)
(1821 - 1896) Philipp Ludwig von Seidel
(Chapter 3)
(1822 - 1901) Charles Hermite
(Chapters 5 and 13)
(1834 - 1886) Edmond Nicolas Laguerre
(Chapter 8)
(1838 - 1922) Marie Ennemond Camille Jordan
(Chapter 3)
(1856 - 1927) Carl David Tolme Runge
(Chapter 9)
(1859 - 1929) Karl Heun
(Chapter 9)
(1862 - 1943) David Hilbert
(Chapter 3)
(1867 - 1944) Martin Wilhelm Kutta
(Chapter 9)
(1872 - 1952) Forest Ray Moulton
(Chapter 9)
(1875 - 1918) Andre-Louis Cholesky
(Chapter 3)
(1880 - 1963) Leonard Bairstow
(Chapter 2)
(1881 - 1953) Lewis Fry Richardson
(Chapter 7, 8, and 9)
(1888 - 1970) Richard Vynne Southwell
(Chapter 3)
(1889 - 1961) Eric Harold Neville
(Chapter 5)
(1890 - 1971) William Edwin Milne
(Chapter 9)
(1895 - 1967) Alexander Craig Aitken
(Chapter 2, ???)
(1909 - 2003) Werner Romberg
(Chapter 8)
(1915 - 1998) Richard Wesley Hamming
(Chapter 9)
(1933 - ????) John C. Butcher
(Chapter 9)
(Bookmarks in the file bio.htm)
Kepler (1571-1630)
Gregory (1638-1675)
Newton (1643-1727)
Raphson (1648-1715)
Cotes (1682-1716)
Taylor (1685-1731)
Stirling (1692-1770)
Maclaurin (1698-1746)
Cramer (1704-1752)
Euler (1707-1783)
Simpson (1710-1761)
Lagrange (1736-1813)
Legendre (1752-1833)
Gauss (1777-1855)
Poisson (1781-1840)
Bassel (1784-1846)
Cauchy (1789-1857)
Vandermonde (1735-1796)
Jacobi (1804-1851)
Boole (1815-1864)
Adams (1819-1892)
Bashforth (1819-1912)
Chebyshev (1821-1894)
Cayley (1821-1895)
Seidel (1821-1896)
Hermite (1822-1901)
Laguerre (1834-1886)
Jordan (1838-1922)
Runge (1856-1927)
Heun (1859-1929)
Hilbert (1862-1943)
Kutta (1867-1944)
Moulton (1872-1952)
Cholesky (1875-1918)
Bairstow (1880-1963)
Richardson (1881-1953)
Southwell (1888-1970)
Neville (1889-1961)
Milne (1890-1971)
Aitken (1895-1967)
Romberg (1909-2003)
Hamming (1915-1998)
Butcher (1933-????)
David. E. Muller (chap. 2, in 1956) Muller, D. E. (1956), “A method for solving algebraic equations using an automatic computer,” Mathematical Tables and Other Aids to Computation, 10, 208-215. Thomas Algorithm (chap. 3, in 1949) Thomas, L. H.: Elliptic problems in linear difference equations over a network. Watson Sc Comput. Lab. Rept., Columbia University, New York, 1949. Crout’s Method (chap. 3) Doolittle’s Method (chap. 3) E. Fehlberg (chap. 9, in 1969) E. Fehlberg, "Classical fifth-, sixth-, seventh-, and eighth-order Runge–Kutta formulas with stepsize control" NASA Techn. Rep. , 287 (Abstract in: Computing (1969), 93–106) E. Fehlberg, "Low-order classical Runge–Kutta formulas with stepsize control and their application to some heat transfer problems" NASA Techn. Rep. , 315 (Abstract in: Computing (1969), 61–71) R. H. Merson (chap. 9, in 1957) R.H. Merson, "An operational method for the study of integration processes" , Proc. Symp. Data Processing , Weapons Res. Establ. Salisbury , Salisbury (1957) pp. 110–125 S. Gill (chap. 9) Ralston’s Method (chap. 9) (Ralston in 1962, Ralston and Rabinowitz in 1978)
Reference-1
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2 9 16 23 30
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4 11(7) 18 25(8) 11 (假)
5 12 19 26 2(補)
6 13 20 27 3
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8 15 22 29 5
17 6 7 8(9) 9 10 11 12 18 13 14 15 16 17 18 http://www.esoe.ntu.edu.tw/courses/50529120.htm
