Optimization of Elliptic Systems,9780387272351

Optimization of Elliptic Systems

by ; ;
ISBN13: 9780387272351
ISBN10: 0387272356
Format: Hardcover
Pub. Date: 2006-01-31
Publisher(s): Springer Verlag
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Summary

"This monograph provides a comprehensive and accessible introduction to the optimization of elliptic systems. This area of mathematical research, which has many important applications in science and technology, has experienced an impressive development during the last two decades. This book also aims to address some of the pressing unsolved questions in the field."--BOOK JACKET.

Table of Contents

Preface v
A Brief Reader's Guide xi
Chapter 1. Introductory Topics 1(24)
1.1 Some General Notions
1(2)
1.2 Motivating Examples
3(22)
1.2.1 Cost Functionals
3(3)
1.2.2 Partial Differential Equations Setting
6(8)
1.2.3 Applications
14(6)
1.2.4 Variable Domains
20(5)
Chapter 2. Existence 25(58)
2.1 A General Situation
25(8)
2.2 Special Existence and Uniqueness Results
33(11)
2.2.1 Second-Order Problems
33(6)
2.2.2 Fourth-Order Problems
39(5)
2.3 Variable Domains
44(39)
2.3.1 Some Examples
45(8)
2.3.2 General Dirichlet Problems
53(11)
2.3.3 Neumann and Mixed Boundary Conditions
64(11)
2.3.4 Partial Extensions
75(8)
Chapter 3. Optimality Conditions 83(110)
3.1 Abstract Approaches
83(19)
3.1.1 The Convex Case. Subdifferential Calculus
83(5)
3.1.2 The Convex Case. Mathematical Programming
88(4)
3.1.3 The Differentiable Case
92(10)
3.1.3.1 Singular Control Problems
96(6)
3.2 Penalization
102(41)
3.2.1 The Standard Approach
103(10)
3.2.2 Penalization of the State Equation
113(14)
3.2.3 Semilinear Equations and Exact Penalization
127(16)
3.3 Control of Variational Inequalities
143(25)
3.3.1 An Abstract Result
143(5)
3.3.2 Semilinear Variational Inequalities
148(11)
3.3.3 State Constraints and Penalization of the Equation
159(9)
3.4 Thickness Optimization for Plates
168(25)
3.4.1 Simply Supported Plates
168(7)
3.4.2 Clamped Plates and Control Variational Methods
175(18)
3.4.2.1 Penalization and Variational Inequalities
183(10)
Chapter 4. Discretization 193(56)
4.1 Finite Element Approximation of Elliptic Equations
193(13)
4.1.1 The Finite Element Method
194(4)
4.1.2 Error Estimates for the FE Equations
198(8)
4.2 Error Estimates in the Finite Element Discretization of Control Problems
206(14)
4.3 Semidiscretization
220(6)
4.4 Optimal Control Problems Governed by Elliptic Variational Inequalities
226(12)
4.4.1 The Ritz-Galerkin Approximation
228(10)
4.5 Error Estimates in the Discretization of Control Problems with Nonlinear State Equation
238(11)
Chapter 5. Unknown Domains 249(92)
5.1 Free Boundary Problems
249(29)
5.1.1 The Dam Problem
250(4)
5.1.2 Free Boundary Problems and Optimal Design
254(4)
5.1.3 Shape Optimization of Systems with Free Boundaries
258(9)
5.1.4 Controllability of the Coincidence Set
267(11)
5.2 Direct Approaches
278(45)
5.2.1 An Algorithm of Céa, Gioan, and Michel
278(6)
5.2.2 Characteristic Functions
284(13)
5.2.2.1 Structural Material Optimization
285(9)
5.2.2.2 Bang-Bang Controls and Characteristic Functions
294(3)
5.2.3 Controllability and Fictitious Domains Approaches
297(26)
5.2.3.1 Boundary Controllability
297(13)
5.2.3.2 Distributed Controls
310(13)
5.3 Domain Variations
323(18)
5.3.1 Classical Approaches
323(11)
5.3.2 Topological Asymptotics
334(7)
Chapter 6. Optimization of Curved Mechanical Systems 341(96)
6.1 Kirchhoff—Love Arches
342(29)
6.1.1 Application of the Control Variational Method
343(8)
6.1.2 Optimization of Nonsmooth Arches
351(18)
6.1.3 Variational Inequalities for Arches
369(2)
6.2 General Three-Dimensional Curved Rods
371(35)
6.2.1 A Local Frame Under Low Differentiability Assumptions
372(2)
6.2.2 A Generalized Naghdi-Type Model
374(15)
6.2.3 Shape Optimization
389(17)
6.3 Applications to Shells
406(31)
6.3.1 A Generalized Naghdi Shell Model
406(7)
6.3.2 Proof of Coercivity
413(9)
6.3.3 Shell Optimal Design
422(15)
Appendix 1. Convex Mappings and Monotone Operators 437(14)
Appendix 2. Elliptic Equations and Variational Inequalities 451(8)
Appendix 3. Domain Convergence 459(16)
Bibliography 475(20)
Index 495(10)
Notation 505

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