Shape optimization by the homogenization method由均匀化方法的形状优化

作者简介

This book provides an introduction to the theory and numerical developments of the homogenization method. It's main features are: a comprehensive presentation of homogenization theory; an introduction to the theory of two-phase composite materials; a detailed treatment of structural optimization by using homogenization; a complete discussion of the resulting numerical algorithms with many documented test problems. It will be of interest to researchers, engineers, and advanced graduate students in applied mathematics, mechanical engineering, and structural optimization.

书籍目录

PrefaceNotation1 Homogenization  1.1  Introduction to Periodic Homogenization　　1.1.1  A Model Problem in Conductivity　　1.1.2  Two-scale Asymptotic Expansions　　1.1.3  Variational Characterizations and Estimates of the Effective Tensor　　1.1.4  Generalization to the Elasticity System  1.2  Definition of H-convergence　　1.2.1  Some Results on Weak Convergence　　1.2.2  Problem Statement　　1.2.3  The One-dimensional Case　　1.2.4  Main Results  1.3  Proofs and Further Results　　1.3.1  Tartar's Method　　1.3.2  G-convergence　　1.3.3  Homogenization of Eigenvalue Problems　　1.3.4  A Justification of Periodic Homogenization　　1.3.5  Homogenization of Laminated Structures　　1.3.6  Corrector Results  1.4  Generalization to the Elasticity System　　1.4.1  Problem Statement　　1.4.2  H-convergence　　1.4.3  Lamination Formulas2  The Mathematical Modeling of Composite Materials  2.1  Homogenized Properties of Composite Materials　　2.1.1  Modeling of Composite Materials　　2.1.2  The G-closure Problem  2.2  Conductivity　　2.2.1  Laminated Composites　　2.2.2  Hashin-Shtrikman Bounds　　2.2.3  G-closure of Two Isotropic Phases  2.3  Elasticity　　2.3.1  Laminated Composites　　2.3.2  Hashin-Shtrikman Energy Bounds　　2.3.3  Toward G-closure　　2.3.4  An Explicit Optimal Bound for　　　　Shape Optimization3  Optimal Design in Conductivity  3.1  Setting of Optimal Shape Design　　3.1.1  Definition of a Model Problem　　3.1.2  A first Mathematical Analysis　　3.1.3  Multiple State Equations　　3.1.4  Shape Optimization as a Degeneracy Limit　　3.1.5  Counterexample to the Existence of Optimal Designs  3.2  Relaxation by the Homogenization Method　　3.2.1  Existence of Generalized Designs　　3.2.2  Optimality Conditions　　3.2.3  Multiple State Equations　　3.2.4  Gradient of the Objective Function　　3.2.5  Self-adjoint Problems　　3.2.6  Counterexample to the Uniqueness of Optimal Designs4  Optimal Design in Elasticity  4.1  Two-phase Optimal Design　　4.1.1  The Original Problem　　4.1.2  Counterexample to the Existence of　　　　Optimal Designs　　4.1.3  Relaxed Formulation of the Problem　　4.1.4  Compliance Optimization    ……5 Numerical AlgorithmsIndex

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