## Finite element approximation for optimal shape design: theory and applications |

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Page 55

The optimal shape design problem is stated as follows f Find a* G U&d such that (

P) 1 J(a*) < J(a) Va G .

Proof. We shall verify all assumptions of

Lemma ...

The optimal shape design problem is stated as follows f Find a* G U&d such that (

P) 1 J(a*) < J(a) Va G .

**Theorem**3.1. There exists at least one solution to (P).Proof. We shall verify all assumptions of

**Theorem**2.1. This will be done inLemma ...

Page 67

Problem (Pfc) is now equivalent to We shall prove (

a solution a* to (P). Due to the equivalence of (Pj,) and (P) the existence of a

solution a*h to (Pfc) follows. Moreover, we shall give in

...

Problem (Pfc) is now equivalent to We shall prove (

**Theorem**4.1) the existence ofa solution a* to (P). Due to the equivalence of (Pj,) and (P) the existence of a

solution a*h to (Pfc) follows. Moreover, we shall give in

**Theorem**4.2 an answer to...

Page 68

corresponding state inequality (P(a^)h). Then there exist subsequences {a*h.} C (

«*> (ahj )} c and elements a* G C/ad, u G K(a*) such that a*h. =fc a* in [0, 1] , as j

...

**Theorem**4.2. Let a*h G U£d be a solution to (Ph) and uj,(ajj) the soiu- tion to thecorresponding state inequality (P(a^)h). Then there exist subsequences {a*h.} C (

«*> (ahj )} c and elements a* G C/ad, u G K(a*) such that a*h. =fc a* in [0, 1] , as j

...

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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 | |

10 other sections not shown

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### Common terms and phrases

adjoint algorithm Appendix applied approximation boundary conditions boundary value problem Chapter constraints contact problems convergence convex set cost functional defined denote design sensitivity analysis differentiable discrete domain elastic exist a subsequence fi(a Figure finite element fixed follows formula ft(a G K(a G R2 G U&d G V(a given gradient Green's formula Haslinger Haug Hilbert space Hlavacek initial iterations Komkov Lagrange multipliers least one solution Lemma liminf limsup linear Lipschitz Lipschitz continuous lower semicontinuous mapping material derivative method minimize Moreover Necas Neittaanmaki nodal nodes nonlinear programming nonsmooth Numerical results obtain optimal control optimal pair optimal shape design parameter Pironneau Proof prove respect results for Example satisfying sequence sequential quadratic programming shape design problems shape optimization Sokolowski solves P(a stresses subgradient sufficiently small Tc(a Theorem triangulation variational inequality vector Zolesio