An introduction to structural optimization

by Christensen, Peter W.; Klarbring, Anders

• ISBN13: 9781402086656
• ISBN10: 1402086652
• Format: Hardcover
• Copyright: 2008-12-04
• Publisher: Springer Verlag

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
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