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 92
... contains 12 modules . Procedure 5.1 . ( for solving shape design problems ) This procedure contains 12 modules inter - connected as described in Figure 5.6 . I ( a * , x ( a * ) 92.
... contains 12 modules . Procedure 5.1 . ( for solving shape design problems ) This procedure contains 12 modules inter - connected as described in Figure 5.6 . I ( a * , x ( a * ) 92.
Page 95
... contains 128 or 512 elements . The dimension of the optimization problem is 9 or 17 , i.e. D ( h ) = 8 or D ( h ) = 16 . = The state problem contains 56 or 224 unknowns ; i.e. n ( h ) = 56 or n ( h ) = 224. Furthermore , in Examples 5.1 ...
... contains 128 or 512 elements . The dimension of the optimization problem is 9 or 17 , i.e. D ( h ) = 8 or D ( h ) = 16 . = The state problem contains 56 or 224 unknowns ; i.e. n ( h ) = 56 or n ( h ) = 224. Furthermore , in Examples 5.1 ...
Page 206
... contains all the piecewise linear functions from Uad . ad A family of triangulations { T ( h , an ) } , an € Uha , h E ( 0,1 ) of N ( an ) will satisfy the same assumptions introduced in Chapter 4. By h ( an ) , or shortly Nh , we ...
... contains all the piecewise linear functions from Uad . ad A family of triangulations { T ( h , an ) } , an € Uha , h E ( 0,1 ) of N ( an ) will satisfy the same assumptions introduced in Chapter 4. By h ( an ) , or shortly Nh , we ...
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