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**Finite element approximation for optimal shape design: Theory and applications.**
*(English)*
Zbl 0713.73062

Chichester etc.: John Wiley & Sons. xii, 335 p. (1988).

The main aim of this book is to investigate optimal design problems via optimal control theory where the states of systems are governed by variational inequalities. The most characteristic property of variational inequalities is the fact that their solution does not depend smoothly, in general, on the control, i.e. it is not possible to differentiate the solution of the state problem with respect to variation of the boundary. Hence shape sensitivity analysis is a crucial question, especially in the design of solution procedures, as the objective functional may not be smooth.

The contents of this book are organized as follows:

Chapter 1 contains some fundamental results for subsequent reference. The aim of Chapter 2 is, firstly, to formulate and to give results concerning the existence of a solution to the problem of the optimal shape design in an abstract setting, secondly, to describe a regularization technique applicable to state problems (possibly non-smooth), and finally, to present a general framework of approximation for these problems. In addition, several examples of the technical applications of optimal shape design are presented.

Chapters 3-4 consider optimal shape design problems governed by the scalar Dirichlet-Signorini boundary value problem. We show how to regularize the problem by utilizing the method of penalization and we prove that the corresponding optimal designs associated with the penalized problems are close (in an appropriate sense) to an optimal design for the original problem. We give the discretization by the finite element method for both approaches and prove the convergence of the approximation as well as the relation between these two approaches. Chapter 5 is devoted to the numerical realization of problems treated in Chapter 4. The design problem leads to a nonlinear programming problem in which function evaluation leads in turn to the solving of nonlinear algebraic systems. We present various techniques for carrying out design sensitivity analysis. Furthermore, we give a procedure for solving optimal shape design problems by applying nonlinear programming algorithms. Several numerical examples are presented.

In Chapter 6 the “boundary flux” cost functional is considered. We present two different approaches together with numerical results. In Chapter 7 we consider the shape optimization of the contact surface of a two-dimensional elastic body unilaterally supported by a rigid foundation. The problem is to redesign the contact surface in such a way that the total potential energy of the system in the equilibrium state will be minimal. We consider separately the following two cases: 1) an elastic body on a rigid frictionless foundation, 2) a model with friction between the body and the support taken into account. We present the formulation of the problem in a continuous setting as well as in a discrete setting (in matrix form). When the discretization has been done, our discrete design formulation leads to a nonconvex but smooth minimization problem with linear constraints. The evaluation of the cost functional involves the nonlinear state problem.

Chapter 8 is devoted to the so-called ‘punch’ problem. We consider the problem of finding the optimal design for a two-dimensional perfectly plastic body (a punch) supported by a rigid frictionless foundation. It is assumed that the material of the body is elastic - perfectly plastic, obeying Henky’s law. Variational formulation in terms of stresses is utilized.

In Chapter 9 we consider a “packaging problem”. The state of the system is governed by a free boundary value problem with additional state constraints given at the interior of the domain. Various formulations of the problem together with numerical examples are presented.

Chapter 10 deals with optimal control problems where the constraints are both on the state and the control. Several numerical examples together with sensitivity analysis are also given. Chapter 11 is devoted to optimal grid design for FEM. Chapter 12 contains concluding remarks on the references and software available on optimal shape design and related topics.

We have included five appendices in this book. Appendix I is devoted to the numerical algorithms for solving unilateral boundary value problems. We make a comparison of efficiency between different algorithms. In Appendix II the differentiation of stiffness and mass matrices and force vectors with respect to the discrete design variables is discussed. In Appendix III we present algorithms for the minimization of nonsmooth functions over a convex set. Appendix IV briefly describes the sequential quadratic programming (SQP) algorithm frequently used in numerical tests. Finally, Appendix V contains results concerning the differentiability of the projection operator onto a closed, convex subset of a Hilbert space. These results play a crucial role in sensitivity analysis.

The contents of this book are organized as follows:

Chapter 1 contains some fundamental results for subsequent reference. The aim of Chapter 2 is, firstly, to formulate and to give results concerning the existence of a solution to the problem of the optimal shape design in an abstract setting, secondly, to describe a regularization technique applicable to state problems (possibly non-smooth), and finally, to present a general framework of approximation for these problems. In addition, several examples of the technical applications of optimal shape design are presented.

Chapters 3-4 consider optimal shape design problems governed by the scalar Dirichlet-Signorini boundary value problem. We show how to regularize the problem by utilizing the method of penalization and we prove that the corresponding optimal designs associated with the penalized problems are close (in an appropriate sense) to an optimal design for the original problem. We give the discretization by the finite element method for both approaches and prove the convergence of the approximation as well as the relation between these two approaches. Chapter 5 is devoted to the numerical realization of problems treated in Chapter 4. The design problem leads to a nonlinear programming problem in which function evaluation leads in turn to the solving of nonlinear algebraic systems. We present various techniques for carrying out design sensitivity analysis. Furthermore, we give a procedure for solving optimal shape design problems by applying nonlinear programming algorithms. Several numerical examples are presented.

In Chapter 6 the “boundary flux” cost functional is considered. We present two different approaches together with numerical results. In Chapter 7 we consider the shape optimization of the contact surface of a two-dimensional elastic body unilaterally supported by a rigid foundation. The problem is to redesign the contact surface in such a way that the total potential energy of the system in the equilibrium state will be minimal. We consider separately the following two cases: 1) an elastic body on a rigid frictionless foundation, 2) a model with friction between the body and the support taken into account. We present the formulation of the problem in a continuous setting as well as in a discrete setting (in matrix form). When the discretization has been done, our discrete design formulation leads to a nonconvex but smooth minimization problem with linear constraints. The evaluation of the cost functional involves the nonlinear state problem.

Chapter 8 is devoted to the so-called ‘punch’ problem. We consider the problem of finding the optimal design for a two-dimensional perfectly plastic body (a punch) supported by a rigid frictionless foundation. It is assumed that the material of the body is elastic - perfectly plastic, obeying Henky’s law. Variational formulation in terms of stresses is utilized.

In Chapter 9 we consider a “packaging problem”. The state of the system is governed by a free boundary value problem with additional state constraints given at the interior of the domain. Various formulations of the problem together with numerical examples are presented.

Chapter 10 deals with optimal control problems where the constraints are both on the state and the control. Several numerical examples together with sensitivity analysis are also given. Chapter 11 is devoted to optimal grid design for FEM. Chapter 12 contains concluding remarks on the references and software available on optimal shape design and related topics.

We have included five appendices in this book. Appendix I is devoted to the numerical algorithms for solving unilateral boundary value problems. We make a comparison of efficiency between different algorithms. In Appendix II the differentiation of stiffness and mass matrices and force vectors with respect to the discrete design variables is discussed. In Appendix III we present algorithms for the minimization of nonsmooth functions over a convex set. Appendix IV briefly describes the sequential quadratic programming (SQP) algorithm frequently used in numerical tests. Finally, Appendix V contains results concerning the differentiability of the projection operator onto a closed, convex subset of a Hilbert space. These results play a crucial role in sensitivity analysis.

### MSC:

74P99 | Optimization problems in solid mechanics |

74S05 | Finite element methods applied to problems in solid mechanics |

49J40 | Variational inequalities |

74M05 | Control, switches and devices (“smart materials”) in solid mechanics |

65K10 | Numerical optimization and variational techniques |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74P10 | Optimization of other properties in solid mechanics |

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

49N60 | Regularity of solutions in optimal control |