Introduction to Design Optimization

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1 Introduction to Design Optimization First Edition Krishnan Suresh i

2 Dedicated to my family. They mean the world to me. ii

3 Origins of this Text Preface Like many other textbooks, this text has evolved from teaching a formal course, Optimum Design of Mechanical Elements and Systems, at the University of Wisconsin, Madison. It has undergone several informal revisions over the last decade. Target Audience The primary audience for this text include senior undergraduate students, first year graduate students and practicing engineers. Given this wide audience, the text starts from 1D numerical optimization, and culiminates in the fascinating world of 3D shape and topology optimization. No prior background on mathematical or computer programming is assumed. Topics Not Covered Optimization is too broad to be able to cover in a single text. Consequently, there are several topics that have been omitted; these include stochastic/nature-inspired global optimization and discrete optimization. These two topics, amongst others, typically serve as students projects. Additional Resources The MATLAB code accompanying this text is an integral part of student learning, and can be downloaded from the author s website at The MATLAB code is object-oriented; it teaches the basic concepts of data encapsulation, inheritance and code resuse. Through exercises, the reader is encouraged to extend the classes presented to address other optimization problems. While MATLAB is an excellent numerical analysis package, it does not offer much when it comes to 3D geometry modeling. To create and analyze 3D geometry, one must use computer-aided design packages such as SolidWorks. SolidLab, presented in this text, was developed by the author s research group to serve as an interface between SolidWorks and MATLAB, thereby providing access to both software packages. It is only supported on Windows. Underneath the hood, SolidLab runs compiled Python code to communicate with SolidWorks. One can, for example, modify feature dimensions of SolidWorks models, query mass properties, carry out a finite element analysis, and optimize, all from within the comfort of MATLAB. Acknowledgements Textbook writing inevitably takes time away from family, and I would like to acknowledge the support of my dear wife, Vanitha, for her constant encouragement to finish this text. Meanwhile, our two sons, Sanjay and Arjun, have smartly learnt to work around my busy schedule and deadlines. Several graduate students from my research lab have contributed both directly and indirectly to this textbook. In particular, Josh Danczyk was instrumental in developing SolidLab, while Amir M. Mirzendehdel helped proof-read this text. iii

4 I would also like to thank the support of National Science Foundation and University of Wisconsin Graduate School for their research support; it has greatly helped shape this text. Krishnan Suresh Madison, WI January Department of Mechanical Engineering University of Wisconsin, Madison, iv

5 Table of Contents 1 OPTIMIZATION A DESIGN OPTIMIZATION EXAMPLE OVERVIEW OF TEXT MODELING STANDARD FORMULATION SAMPLING ALGORITHM FEASIBLE AND INFEASIBLE REGIONS EXERCISES MATLAB PROGRAMMING BASICS MATLAB SCRIPT FILES LINEAR ALGEBRA COMPLEX NUMBERS PLOTS SYMBOLIC OPERATIONS USER-DEFINED FUNCTIONS VARIABLE ARGUMENTS MODULES SAMPLING ALGORITHM POLYNOMIAL CLASS EXTENDING THE POLYNOMIAL CLASS EXERCISES MATHEMATICS OF OPTIMIZATION LOCAL VERSUS GLOBAL MINIMUM v

6 3.2 TAYLOR SERIES FIRST-ORDER OPTIMALITY CRITERIA SECOND-ORDER OPTIMALITY GRADIENT IDENTITIES QUADRATIC AND CONVEX FUNCTIONS ILLUSTRATIVE EXAMPLES EXERCISES NUMERICAL OPTIMIZATION ONE-VARIABLE MINIMIZATION UNIMODAL FUNCTIONS ONE-DIMENSIONAL METHODS: OVERVIEW Trisection Golden Section Quadratic Interpolation Bisection Newton-Raphson MATLAB OPTIMIZATION TOOLBOX: 1D fminbnd fminsearch fminunc Using Matlab s fsolve MULTIVARIABLE MINIMIZATION LINE-SEARCH SEARCH DIRECTIONS EIGEN-DIRECTIONS CONJUGATE-DIRECTIONS POWELL S METHOD vi

7 4.11 GRADIENT METHODS Steepest Descent Method Linear Conjugate Gradient Method Non-Linear Conjugate Gradient Method NEWTON-RAPHSON METHOD MATLAB OPTIMIZATION TOOLBOX: N-D fminsearch fminpowell Using Matlab s fminunc SUMMARY OF METHODS EXERCISES OPTIMIZATION: A PHYSICAL PERSPECTIVE EQUILIBRIUM OF SPRING SYSTEMS: TWO STRATEGIES Conventions and Terminology Force Balance Strategy Residual Computation for Arbitrary Spring Systems Potential Energy Strategy Algorithm for Potential Energy Computation SPRING MODELING IN MATLAB EQUILIBRIUM OF TRUSS SYSTEMS Small Displacement Force Balance Small Displacement Potential Energy Assembly Algorithm for Truss Systems TRUSS MODELING IN MATLAB EXERCISES EQUALITY CONSTRAINTS PHYSICAL INTERPRETATION OF CONSTRAINTS vii

8 6.2 OPTIMALITY CRITERIA Single Equality Constraint Quadratic Problems Multiple Constraints PATHOLOGICAL CASES Linearly Dependent Constraints Linearly Dependent Gradients Vanishing Gradients TWO INTERPRETATIONS OF LAGRANGIAN MULTIPLIERS Sensitivity of Constraints Reaction Forces in a Structural System LAGRANGIAN FUNCTION AUGMENTED LAGRANGIAN METHOD MATLAB ROUTINE: FMINCON EXERCISES INEQUALITY CONSTRAINTS OPTIMALITY CRITERIA MULTIPLE CONSTRAINTS ACTIVE-SET METHOD MATLAB ROUTINE: FMINCON EXERCISES LEAST SQUARES LEAST SQUARES PROBLEMS LINEAR LEAST SQUARES PROBLEM NONLINEAR LEAST SQUARES PROBLEM EXERCISES viii

9 9 TRUSS OPTIMIZATION DISPLACEMENT MINIMIZATION OF TRUSS SYSTEMS STRESS CONSTRAINTS ON TRUSS SYSTEMS BUCKLING CONSTRAINTS ON TRUSS SYSTEMS INDETERMINATE TRUSS SYSTEMS TRUSS2D OPTIMIZATION: MATLAB IMPLEMENTATION SCALING OPTIMIZATION VARIABLES SCALING OBJECTIVES AND CONSTRAINTS TRUSS OPTIMIZATION: OBJECTIVE GRADIENT EXERCISES GRADIENT COMPUTATION FINITE DIFFERENCE IN 1D FINITE DIFFERENCE IN N-D HIGHER ORDER DERIVATIVES COMPLEX VARIABLE APPROACH AUTOMATIC DIFFERENTIATION EXERCISES FUNCTIONAL MINIMIZATION MOTIVATION FORCE BALANCE APPROACH MINIMIZATION OF POTENTIAL ENERGY APPROXIMATE SOLUTIONS PIECEWISE LINEAR SOLUTION NUMERICAL INTEGRATION MATLAB IMPLEMENTATION BEAM BENDING PROBLEM ix

10 Potential Energy Formulation Finite Element Approximation EXERCISES LINEAR ELASTICITY AND FINITE ELEMENT ANALYSIS PLANE STRESS PROBLEMS Potential Energy of Plane Stress Problems PLANE STRAIN PROBLEMS AXISYMMETRIC PROBLEMS FINITE ELEMENT ANALYSIS MESH GENERATION SHAPE FUNCTIONS Triangles Quadrilaterals VARIABLE TRANSFORMATION Triangles Quadrilaterals ELEMENT STIFFNESS MATRIX AND FORCE VECTOR GAUSSIAN INTEGRATION ASSEMBLY AND SOLUTION HIGHER ORDER APPROXIMATION D LINEAR ELASTICITY EXERCISES FINITE ELEMENT IMPLEMENTATION BOUNDARY REPRESENTATION BREP2D CLASS IN MATLAB TRIMESHER CLASS IN MATLAB TRIELASTICITY CLASS IN MATLAB x

11 13.5 QUADMESHER CLASS IN MATLAB QUADELASTICITY CLASS IN MATLAB ADDITIONAL EXAMPLES Plate with a Hole Problem A Plane Strain Problem An Axisymmetric Problem EXERCISES SHAPE OPTIMIZATION SHAPE PARAMETERS PARAMETRIC STUDIES SHAPE OPTIMIZATION PROBLEMS SCALING FOR NUMERICAL ROBUSTNESS DIRECT FINITE DIFFERENCE SENSITIVITY OF COMPLIANCE INDIRECT FINITE DIFFERENCE SENSITIVITY OF COMPLIANCE SHAPE OPTIMIZATION GEOMETRIC CONSTRAINTS AVOIDING ADDITIONAL SOLVES EXERCISES FINITE ELEMENT ANALYSIS AND SHAPE OPTIMIZATION IN 3D OVERVIEW GEOMETRY-RELATED METHODS Opening a SolidWorks Model Querying Mass Properties Querying All Dimensions and Values Querying and Modifying Specific Dimension Values FINITE ELEMENT STUDIES xi

12 Running Finite Element Analysis via SolidLab Suppressing Features PARAMETRIC STUDIES SHAPE OPTIMIZATION EXERCISES TOPOLOGY OPTIMIZATION OVERVIEW SIMP Formulation Optimality Criteria Algorithm Implementation Numerical Examples PARETO: A TOPOLOGICAL LEVEL-SET METHOD Background Topological Sensitivity Algorithm Implementation Numerical Examples TOPOLOGY OPTIMIZATION ON THE CLOUD Overview of CTO Topology Optimization Problems Supported Compliance Minimization Stress Minimization Pareto Optimal Designs Draw Constraints EXERCISES xii

APPLIED OPTIMIZATION WITH MATLAB PROGRAMMING

APPLIED OPTIMIZATION WITH MATLAB PROGRAMMING Second Edition P. Venkataraman Rochester Institute of Technology WILEY JOHN WILEY & SONS, INC. CONTENTS PREFACE xiii 1 Introduction 1 1.1. Optimization Fundamentals