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 91
... respect to the x1- coordinate of nodal point N ; = ( x1 ( i ) , x2 ( i ) ) is given by ( 5.38 ) dJ4 ( an ) : = with 1 2 Jou suppi a AjVu Vur dx - -Jupp , or ( fi ) un dz Jouppe . θφί Οφί Jx1 ax 2 A ; Əyi θφί Əx2 მ x1 i = 1 , ... , ñ ...
... respect to the x1- coordinate of nodal point N ; = ( x1 ( i ) , x2 ( i ) ) is given by ( 5.38 ) dJ4 ( an ) : = with 1 2 Jou suppi a AjVu Vur dx - -Jupp , or ( fi ) un dz Jouppe . θφί Οφί Jx1 ax 2 A ; Əyi θφί Əx2 მ x1 i = 1 , ... , ñ ...
Page 116
... respect to t gives for t = 0 ( 6.22 ) ( AVzh , V4h ) 0 , n + ( żh , ❤h ) 1 , n + ( div Vzn , 4h ) 0,5 % = ( AVun , V❤h ) 0 , 2 + ( ún , Oh ) 1 , n + ( div Vuh , h ) 0 , n - ( div ( fv ) , h ) 0 , " where un Є Khan ) solves ( P ( an ) ...
... respect to t gives for t = 0 ( 6.22 ) ( AVzh , V4h ) 0 , n + ( żh , ❤h ) 1 , n + ( div Vzn , 4h ) 0,5 % = ( AVun , V❤h ) 0 , 2 + ( ún , Oh ) 1 , n + ( div Vuh , h ) 0 , n - ( div ( fv ) , h ) 0 , " where un Є Khan ) solves ( P ( an ) ...
Page 302
... respect to design variables . The aim of this appendix is to present general results concerning the differentiability of a projection in a Hilbert space onto its closed convex subset . For more detailed analysis see Mignot ( 1976 ) and ...
... respect to design variables . The aim of this appendix is to present general results concerning the differentiability of a projection in a Hilbert space onto its closed convex subset . For more detailed analysis see Mignot ( 1976 ) and ...
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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adjoint algorithm Appendix applied approximation boundary value problem C₁ Céa compute constraints contact problems convex convex set cost functional defined denote design sensitivity analysis differentiable discrete domain elastic exist a subsequence Figure Find finite element follows formula given Gm(a H¹(Î Haslinger Haug Hlaváček I₁ Ir(an ITERATION jEJk Komkov Lagrange multipliers least one solution Lemma lim inf lim sup linear Lipschitz Lipschitz continuous lower semicontinuous mapping material derivative matrix method minimization Nečas Neittaanmäki nodes nonlinear nonlinear programming nonsmooth Numerical results obtain optimal control optimal design optimal pair optimal shape design parameter Pironneau Proof results for Example Section sensitivity analysis sequence shape design problems Shape optimization Sokolowski solves P(a subgradient subset T₁ Theorem triangulation un(an unilateral boundary value variational inequality vector w₁ Zolesio г₁