## Finite Element Approximation for Optimal Shape Design: Theory and ApplicationsExplains how to speed the optimal shape design process using a computer. Outlines the problems inherent in optimal shape design and discusses methods of their solution. Concentrates on finite element approximation and describes numerical realization of optimization techniques. Treats optimal design problems via the optimal control theory when the state systems are governed by variational inequalities. Provides useful background information, followed by numerous approaches to optimal shape design, all supported by illustrative examples. Appendices provide algorithms and numerous examples and their calculations are included. |

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Page vii

In Chapter 7 we

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 ...

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 ...

Page 51

As a consequence of the restriction to small separations one may

stiffness of the bodies to be unchanged by the design process . The problem may

be stated as ( P ) min sup 1 * ( x ) a where \ * is the solution stress to the contact ...

As a consequence of the restriction to small separations one may

**consider**thestiffness of the bodies to be unchanged by the design process . The problem may

be stated as ( P ) min sup 1 * ( x ) a where \ * is the solution stress to the contact ...

Page 90

This fact will also be utilized in Chapter 7 , where we

and in the FE - grid optimization ( Chapter 11 ) . Let us

function J3 ( an ) defined in Section 5 . 2 : J3 ( an ) = 1 $ ( an , un ( an ) ) = 1 Jun (

an ) ...

This fact will also be utilized in Chapter 7 , where we

**consider**contact problems ,and in the FE - grid optimization ( Chapter 11 ) . Let us

**consider**the criterionfunction J3 ( an ) defined in Section 5 . 2 : J3 ( an ) = 1 $ ( an , un ( an ) ) = 1 Jun (

an ) ...

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### Contents

Preliminaries | 1 |

Abstract setting of optimal shape design problem and | 28 |

Optimal shape design of systems governed by a unilateral | 53 |

Copyright | |

14 other sections not shown

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### Common terms and phrases

algorithm Appendix applied approach approximation associated assume body boundary bounded called Chapter closed compute Consequently consider constant constraints contains continuous convergence convex corresponding cost functional defined definition denote depend differentiable direction discrete displacement domain elasticity element equivalent Example exists field Figure Finally Find fixed follows force formula function give given hand Haslinger holds initial iterations Lemma linear mapping material derivative matrix means method minimize Moreover moving multipliers Neittaanmäki nodes nonlinear numerical Numerical results obtain optimal shape design parameters positive present programming Proof prove reads refer relation Remark respect results for Example satisfying sequence shape design problems smooth solution solving space Step stress structural subgradient subset sufficiently suppose Table term Theorem triangulation unilateral unique vector write Zolesio