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).

1. Elliptic Partial Differential Equations.- 1.1 Introduction.- 1.2 Green's Formula.- 1.3 Sobolev Spaces.- 1.4 Linear Elliptic PDE of Order 2.- 1.5 Numerical Solutions of Linear Elliptic Equations of Order 2.- 1.6 Other Elliptic Equations.- 1.7 Continuous Dependence on the Boundary.- 2. Problem Statement.- 2.1 Introduction.- 2.2 Definition.- 2.3 Examples.- 2.4 Principles of Solution.- 2.5 Future of Optimal Design Applications in Industry.- 2.6 Historical Background and References.- 3. Existence of Solutions.- 3.1 Introduction.- 3.2 Dirichlet Conditions.- 3.3 Neumann Boundary Conditions.- 3.4 Conclusion.- 4. Optimization Methods.- 4.1 Orientation.- 4.2 Problem Statement.- 4.3 Gradients.- 4.4 Method of Steepest Descent.- 4.5 Newton Method.- 4.6 Conjugate Gradient Method.- 4.7 Optimization with Equality Constraints.- 4.8 Optimization with Inequality Constraints.- 5. Design Problems Solved by Standard Optimal Control Theory.- 5.1 Introduction.- 5.2 Optimization of a Thin Wing.- 5.3 Optimization of an Almost Straight Nozzle.- 5.4 Thickness Optimization Problem.- 6. Optimality Conditions.- 6.1 Introduction.- 6.2 Distributed Observation on a Fixed Domain.- 6.3 Other Cases with Linear PDE.- 7. Discretization with Finite Elements.- 7.1 Introduction.- 7.2 Neumann Problem.- 7.3 Dirichlet Conditions.- 7.4 Other Problems.- 7.5 Convergence.- 8. Other Methods.- 8.1 Introduction.- 8.2 Method of Mappings.- 8.3 Finite Difference Discretization.- 8.4 Method of Characteristic Functions.- 8.5 Discretization by the Boundary Element Method.- 9. Two Industrial Examples.- 9.1 Introduction.- 9.2 Optimization of Electromagnets.- 9.3 Optimization of Airfoils.- 9.4 Conclusion.- References.