## 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 consider the shape optimization of the contact surface of a two -

dimensional elastic

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 . Theproblem is to redesign the contact surface in such a way that the total potential

energy ...

Page 50

... strain tensor , and

surface traction applied to

the surface of

chosen as ...

... strain tensor , and

**body**force in**body**s c = 1 , 2 ) , respectively , and Pi issurface traction applied to

**body**N . One may identify rį and r ? as the regions onthe surface of

**body**127 and**body**522 where contact may occur . These arechosen as ...

Page 160

12 we see the numerical results obtained for Examples 7 . 1 - 7 . 6 . 1 )

triangulation of initial

initial

the optimal ...

12 we see the numerical results obtained for Examples 7 . 1 - 7 . 6 . 1 )

triangulation of initial

**body**before deformation 2 ) scaled displacement field forinitial

**body**at nodal points 3 ) decrease of E versus iteration 4 ) triangulation ofthe optimal ...

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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