An introduction to structural optimization
by Christensen, Peter W.; Klarbring, AndersFormat: Hardcover
Pub. Date: 2008-12-04
Publisher(s): Springer Verlag
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Summary
Mechanical and structural engineers have always strived to make as efficient use of material as possible, e.g. by making structures as light as possible yet able to carry the loads subjected to them. In the past, the search for more efficient structures was a trial-and-error process. However, in the last two decades computational tools based on optimization theory have been developed that make it possible to find optimal structures more or less automatically. Due to the high cost savings and performance gains that may be achieved, such tools are finding increasing industrial use.
Table of Contents
| Introduction | p. 1 |
| The Basic Idea | p. 1 |
| The Design Process | p. 1 |
| General Mathematical Form of a Structural Optimization Problem | p. 3 |
| Three Types of Structural Optimization Problems | p. 5 |
| Discrete and Distributed Parameter Systems | p. 7 |
| Examples of Optimization of Discrete Parameter Systems | p. 9 |
| Weight Minimization of a Two-Bar Truss Subject to Stress Constraints | p. 9 |
| Weight Minimization of a Two-Bar Truss Subject to Stress and Instability Constraints | p. 12 |
| Weight Minimization of a Two-Bar Truss Subject to Stress and Displacement Constraints | p. 14 |
| Weight Minimization of a Two-Beam Cantilever Subject to a Displacement Constraint | p. 18 |
| Weight Minimization of a Three-Bar Truss Subject to Stress Constraints | p. 21 |
| Weight Minimization of a Three-Bar Truss Subject to a Stiffness Constraint | p. 31 |
| Exercises | p. 33 |
| Basics of Convex Programming | p. 35 |
| Local and Global Optima | p. 35 |
| Convexity | p. 37 |
| KKT Conditions | p. 41 |
| Lagrangian Duality | p. 46 |
| Lagrangian Duality for Convex and Separable Problems | p. 47 |
| Exercises | p. 52 |
| Sequential Explicit, Convex Approximations | p. 57 |
| General Solution Procedure for Nested Problems | p. 57 |
| Sequential Linear Programming (SLP) | p. 58 |
| Sequential Quadratic Programming (SQP) | p. 59 |
| Convex Linearization (CONLIN) | p. 59 |
| The Method of Moving Asymptotes (MMA) | p. 66 |
| Exercises | p. 72 |
| Sizing Stiffness Optimization of a Truss | p. 77 |
| The Simultaneous Formulation of the Problem | p. 77 |
| The Nested Formulation and Some of Its Properties | p. 84 |
| Convexity of the Nested Problem | p. 85 |
| Fully Stressed Designs | p. 87 |
| Minimization of the Volume Under a Compliance Constraint | p. 88 |
| Numerical Solution of the Nested Problem Using MMA | p. 91 |
| Sensitivity Analysis | p. 97 |
| Numerical Methods | p. 97 |
| Analytical Methods | p. 98 |
| Direct Analytical Method | p. 98 |
| Adjoint Analytical Method | p. 99 |
| Analytical Calculation of Pseudo-loads | p. 100 |
| Bars | p. 101 |
| Plane Sheets | p. 104 |
| Exercises | p. 112 |
| Two-Dimensional Shape Optimization | p. 117 |
| Shape Representation | p. 117 |
| Bezier Splines | p. 118 |
| B-Splines | p. 120 |
| Treatment of Geometrical Design Constraints | p. 127 |
| C[superscript 1] Continuity Between Bezier Splines | p. 128 |
| C[superscript 1] Continuity at a Point on a Line of Symmetry | p. 129 |
| A Composite Circular Arc | p. 131 |
| Mesh Generation and Calculation of Nodal Sensitivities | p. 132 |
| B-Spline Surface Meshes | p. 133 |
| Coons Surface Meshes | p. 134 |
| Unstructured Meshes | p. 137 |
| Summary of Sensitivity Analysis for Two-Dimensional Shape Optimization | p. 139 |
| Exercises | p. 143 |
| Stiffness Optimization of Distributed Parameter Systems | p. 147 |
| Calculus of Variations | p. 147 |
| Optimality Conditions and Gateaux Derivatives | p. 149 |
| Handling a Constraint | p. 153 |
| Equilibrium Principles for Distributed Parameter Systems | p. 156 |
| One-Dimensional Elasticity | p. 156 |
| Beam Problem | p. 158 |
| Two-Dimensional Elasticity | p. 159 |
| Abstract Equilibrium Principles | p. 162 |
| The Design Problem | p. 163 |
| Optimality Conditions | p. 166 |
| The Stiffest Rod | p. 168 |
| Beam Stiffness Optimization | p. 170 |
| Exercises | p. 174 |
| Topology Optimization of Distributed Parameter Systems | p. 179 |
| The Variable Thickness Sheet Problem | p. 179 |
| Problem Statement and FE-Discretization | p. 179 |
| The Optimality Criteria (OC) Method | p. 182 |
| Penalization of Intermediate Thickness Values | p. 188 |
| Solid Isotropic Material with Penalization (SIMP) | p. 188 |
| Other Penalizations | p. 190 |
| Well-Posedness and Potential Numerical Problems | p. 190 |
| The Archetype Problem and an Analogy | p. 190 |
| Numerical Instabilities | p. 191 |
| Restriction of the Archetype Problem | p. 193 |
| Bounds on the Design Gradient | p. 194 |
| Filters | p. 195 |
| Relaxation of the Archetype Problem | p. 198 |
| Exercises | p. 200 |
| Answers to Selected Exercises | p. 203 |
| References | p. 207 |
| Index | p. 209 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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