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 28
... convergence of domains . Furthermore , ( 2.2 ) un → u , as n → denotes the convergence of un Є V ( n ) to u Є V ( N ) where Nn and NE Õ . Let us assume that Õ is equipped with an appropriate topology and convergences ( 2.1 ) and ( 2.2 ) ...
... convergence of domains . Furthermore , ( 2.2 ) un → u , as n → denotes the convergence of un Є V ( n ) to u Є V ( N ) where Nn and NE Õ . Let us assume that Õ is equipped with an appropriate topology and convergences ( 2.1 ) and ( 2.2 ) ...
Page 31
... ( convergence in V ( N ) -norm ) . Now we are able to state and prove an important convergence result : Theorem 2.2 . Let ( P ) have at least one solution for any εk > 0 and let assumptions A ( ii ) , A ( j ) – A ( jjj ) be satisfied ...
... ( convergence in V ( N ) -norm ) . Now we are able to state and prove an important convergence result : Theorem 2.2 . Let ( P ) have at least one solution for any εk > 0 and let assumptions A ( ii ) , A ( j ) – A ( jjj ) be satisfied ...
Page 297
... convergence usually slows down . On the other hand in the second method the interior of S may be empty and one can move along the boundary of S without losing any convergence properties . AIII.3.2 . Direction finding Instead of ...
... convergence usually slows down . On the other hand in the second method the interior of S may be empty and one can move along the boundary of S without losing any convergence properties . AIII.3.2 . Direction finding Instead of ...
Contents
Preliminaries | 1 |
Abstract setting of optimal shape design problem and | 28 |
Optimal shape design of systems governed by a unilateral | 53 |
Copyright | |
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algorithm Appendix applied approximation boundary value problem C₁ Céa Computer constraints contact problems convex convex set cost functional defined denote design sensitivity analysis differentiable discrete domain elastic element method exist a subsequence Figure Find finite element finite element method follows formula given Glowinski Gm(a H¹(Î Haslinger Haug Hlaváček Ir(an ITERATION jEJk ji Eli Komkov Lagrange multipliers Lemma lim inf lim sup linear Lipschitz continuous lower semicontinuous matrix minimization Nečas Neittaanmäki nodes nonlinear programming nonsmooth Numerical results obtain optimal control optimal design optimal pair optimal shape design parameter Pironneau Proof results for Example Section sequence shape design problems Shape optimization Sokolowski solves P(a structural design structural optimization subgradient subset T(Un T₁ Theorem triangulation un(an variational inequality vector w₁ Zolesio г₁