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

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Page 23

Then the following

be such that yn - y in IT1 (ft) . Let a„ 4 a in [0, 1] (a„,a G U&d). Then (1.64) (7>(Vn),

0an - (%), 0a G C°°(ft) . Proof. Let us write / (yn(otn) + an)~t(an)dx2- (y(a) + ...

Then the following

**lemma**holds:**Lemma**1.2. Let yn, y G H1^) (Q D Q(a) Va G U^)be such that yn - y in IT1 (ft) . Let a„ 4 a in [0, 1] (a„,a G U&d). Then (1.64) (7>(Vn),

0an - (%), 0a G C°°(ft) . Proof. Let us write / (yn(otn) + an)~t(an)dx2- (y(a) + ...

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The rest of the proof now follows from the next

.

a„ G U&d. Then there exist a subsequence of {(a„, un)} (again denoted by {(a„, ...

The rest of the proof now follows from the next

**lemma**. □ Let ejt = 1 (for simplicity).

**Lemma**3.4. (verification of A(i)) Let u„ = u(a„) G V(ttn) be solutions of (P(a„)ek-\),a„ G U&d. Then there exist a subsequence of {(a„, un)} (again denoted by {(a„, ...

Page 63

again) such that Vj - » v in ff1(ft) and fj|n*j G K(akj ) (see the proof of

From the definition of the state equation (P(akj )ek. ) we have (3.55) (u*,^),^ + j-{H

^kMau = (Mofik> VV e Bl(Ci) . Setting <p = vj - ukj in (3.55), we obtain (3 56) (u*' ...

again) such that Vj - » v in ff1(ft) and fj|n*j G K(akj ) (see the proof of

**Lemma**3.1).From the definition of the state equation (P(akj )ek. ) we have (3.55) (u*,^),^ + j-{H

^kMau = (Mofik> VV e Bl(Ci) . Setting <p = vj - ukj in (3.55), we obtain (3 56) (u*' ...

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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