文档介绍：This is page i
Printer: Opaque this
Variational Methods for Structural Optimization
This is page ii
Printer: Opaque this
This is page i
Printer: Opaque this
Variational Methods for Structural
Optimization
Andrej Cherkaev
Department of Mathematics
The University of Utah
Salt Lake City, UT 84112-0090
iv
Insert the copyright information, etc.
This is page v
Printer: Opaque this
Contents
List of Figures xiii
Preface xv
I Preliminaries 1
1 Relaxation of One-Dimensional Variational Problems 3
An Optimal Design by Means posites . . . . . . . . 3
Stability of Minimizers and the Weierstrass Test . . . . . . 7
Necessary and Sufficient Conditions . . . . . . . . . 7
Variational Methods: Weierstrass Test . . . . . . . . 10
............................ 14
Nonconvex Variational Problems . . . . . . . . . . . 14
ConvexEnvelope.................... 16
Minimal Extension and Minimizing Sequences . . . . 19
Examples: Solutions to Nonconvex Problems . . . . 24
Null-Lagrangians and Convexity . . . . . . . . . . . 27
Duality......................... 29
................... 32
2 posites 35
Conductivity of Inhomogeneous Media . . . . . . . . . . . . 35
Equations for Conductivity . . . . . . . . . . . . . . 35
Continuity Conditions in Inhomogeneous Materials . 39
Energy, Variational Principles . . . . . . . . . . . . . 42
........................... 45
Homogenization and Effective Tensor . . . . . . . . . 46
vi Contents
Effective Properties of Laminates . . . . . . . . . . . 51
Effective Medium Theory: Coated Circles . . . . . . 55
................... 57
3 Bounds and G-Closures 59
Effective Tensors: Variational Approach . . . . . . . . . . . 59
Calculation of Effective Tensors . . . . . . . . . . . . 59
Wiener Bounds . . . . . . . . . .