Optimal Shape Design for Elliptic Systems

1. Front Matter

   Pages i-xii

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2. Elliptic Partial Differential Equations

   • Olivier Pironneau

   Pages 1-15

3. Problem Statement

   • Olivier Pironneau

   Pages 16-29

4. Existence of Solutions

   • Olivier Pironneau

   Pages 30-44

5. Optimization Methods

   • Olivier Pironneau

   Pages 45-67

6. Design Problems Solved by Standard Optimal Control Theory

   • Olivier Pironneau

   Pages 68-80

7. Optimality Conditions

   • Olivier Pironneau

   Pages 81-98

8. Discretization with Finite Elements

   • Olivier Pironneau

   Pages 99-120

9. Other Methods

   • Olivier Pironneau

   Pages 121-142

10. Two Industrial Examples

    • Olivier Pironneau

    Pages 143-162

11. Back Matter

    Pages 163-168

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The study of optimal shape design can be arrived at by asking the following question: "What is the best shape for a physical system?" This book is an applications-oriented study of such physical systems; in particular, those which can be described by an elliptic partial differential equation and where the shape is found by the minimum of a single criterion function. There are many problems of this type in high-technology industries. In fact, most numerical simulations of physical systems are solved not to gain better understanding of the phenomena but to obtain better control and design. Problems of this type are described in Chapter 2. Traditionally, optimal shape design has been treated as a branch of the calculus of variations and more specifically of optimal control. This subject interfaces with no less than four fields: optimization, optimal control, partial differential equations (PDEs), and their numerical solutions-this is the most difficult aspect of the subject. Each of these fields is reviewed briefly: PDEs (Chapter 1), optimization (Chapter 4), optimal control (Chapter 5), and numerical methods (Chapters 1 and 4).

• Design
• Diskretisation
• Elliptische Differentialgleichung
• Konstruktion
• Optimale Regelung
• boundary element method
• calculus of variations
• differential equation
• fields
• finite element method
• numerical method
• partial differential equation
• simulation
• solution
• system
• fluid- and aerodynamics

• Centre Scientifique et Polytechnique, Departement de Mathematiques, Universite Paris-Nord, Villetaneuse, France

  Olivier Pironneau

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