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

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Page 7

Since the set M is weakly compact, there exist a subsequence {unt}jij and an

element uo G M such that unk - ' uo- By assumption, the functional F is weakly

liminf ...

Since the set M is weakly compact, there exist a subsequence {unt}jij and an

element uo G M such that unk - ' uo- By assumption, the functional F is weakly

**lower semicontinuous**on the set M. This and (1.4) together imply (1.5) F(u0) <liminf ...

Page 222

V - » (-00, +00] a convex,

U will be denoted by || ||y and || \\u, respectively. We shall deal with the following

optimization problem (P) minimize J(u,y) , where u and y are related by (10.1) ...

V - » (-00, +00] a convex,

**lower semicontinuous**, proper function. The norms in V,U will be denoted by || ||y and || \\u, respectively. We shall deal with the following

optimization problem (P) minimize J(u,y) , where u and y are related by (10.1) ...

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... uh G £^ad (approximations of au); — <ph- Vh -* (- oo,+oo] convex,

Vfc') (approximations of / and B); — Jh ' Uh x Vh -* R1 convex,

... uh G £^ad (approximations of au); — <ph- Vh -* (- oo,+oo] convex,

**lower****semicontinuous**, proper functions (approximations of <p); — fh G Vh\ Bh G L(Uh,Vfc') (approximations of / and B); — Jh ' Uh x Vh -* R1 convex,

**lower****semicontinuous**, ...### What people are saying - Write a review

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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