Sokołowski, Jan Nonsmooth optimal design problems for the Kirchhoff plate with unilateral conditions. (English) Zbl 0789.73052 Kybernetika 29, No. 3, 284-290 (1993). Summary: The form of directional derivative of the metric projection in the Sobolev space \(H^ 2_ 0(\Omega)\) onto the convex set \(K=\{f\in H^ 2_ 0(\Omega)\mid f\geq \psi\}\) is derived in [M. Rao and the author, Numer. Funct. Anal. Optimization 14, 125-143 (1993)]. In the present paper, this result is used to obtain the first order optimality conditions for a class of nonsmooth optimal design problems for the Kirchhoff plate with an obstacle. Cited in 1 Document MSC: 74P99 Optimization problems in solid mechanics 74K20 Plates 49K20 Optimality conditions for problems involving partial differential equations 49J40 Variational inequalities Keywords:directional derivative; metric projection; Sobolev space; convex set; first order optimality conditions; obstacle × Cite Format Result Cite Review PDF Full Text: EuDML Link References: [1] M. P. Bendsøe N. Olhoff, J. Sokolowski: Sensitivity analysis of problems of elasticity with unilateral constraints. J. Struct. Mech. 13 (1985), 201-222. [2] M. P. Bendsøe, J. Sokolowski: Design sensitivity analysis of elastic-plastic analysis problem. Mech. Structures Mach. 16(1988), 81-102. [3] M. P. Bendsøe, J. Sokolowski: Sensitivity analysis and optimal design of elastic plates with unilateral point supports. Mech. Structures Mach. 15 (1987), 383-393. [4] G. Duvaut, J. L. Lions: Inequalities in Mechanics and Physics. (Grundlehren der mathematischen Wissenschaften 219.) Springer-Verlag, Berlin 1976. · Zbl 0331.35002 [5] A. Haraux: How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities. J. Math. Soc. Japan 29 (1977), 615-631. · Zbl 0387.46022 · doi:10.2969/jmsj/02940615 [6] E. J. Haug, J. Cea (eds.): Optimization of Distributed Parameter Structures. Sijthoff and Noordhoff, Alpen aan den Rijn, The Netherlands 1981. · Zbl 0511.00034 [7] I. Hlaváček I. Bock, J. Lovíšek: Optimal control of a variational inequality with applications to structural analysis. Part III. Optimal design of an elastic plate. Appl. Math. Optim. IS (1985), 117-136. · Zbl 0582.73081 [8] A. M. Khludnev, J. Sokolowski: book in preparation. · Zbl 1067.74056 [9] F. Mignot: Controle dans les inequations variationelles elliptiques. J. Funct. Anal. 22 (1976), 25-39. · Zbl 0364.49003 · doi:10.1016/0022-1236(76)90017-3 [10] A. Myslinski, J. Sokolowski: Nondifferentiable optimization problems for elliptic systems. SIAM J. Control Optim. 23 (1985), 632-648. · Zbl 0571.49010 · doi:10.1137/0323040 [11] M. Rao, J. Sokolowski: Sensitivity analysis of Kirchhoff plate with obstacle. Rapport de Recherche No. 771 (1988), INRIA, Rocquencourt, France. [12] M. Rao, J. Sokolowski: Shape sensitivity analysis of state constrained optimal control problems for distributed parameter systems. (Lecture Notes in Control and Information Sciences 114.) Springer-Verlag, Berlin 1989, pp. 236-245. · Zbl 0702.49015 [13] M. Rao, J. Sokolowski: Differential stability of solutions to parametric optimization problems. Math. Methods Appl. Sci. 14 (1991), 281-294. · Zbl 0749.90077 · doi:10.1002/mma.1670140405 [14] M. Rao, J. Sokolowski: Sensitivity analysis of unilateral problems in \(H_0^2(\Omega)\) and applications. Numer. Funct. Anal. Optim. 14 (1993), 1-2, 125-143. [15] M. Rao, J. Sokolowski: Sensitivity analysis of unilateral problems in \(H_0^2(\Omega)\) and applications. Emerging Applications in Free Boundary Problems (J.M. Chadam and II. Rasmussen, Pitman Research Notes Math. Ser., No. 280), Longman 1993. [16] J. Sokolowski: Sensitivity analysis of control constrained optimal control problems for distributed parameter systems. SIAM J. Control Optim. 25 (1987), 1542-1556. · Zbl 0647.49019 · doi:10.1137/0325085 [17] J. Sokolowski: Shape sensitivity analysis of boundary optimal control problems for parabolic systems. SIAM J. Control Optim. 26 (1988), 763-787. · Zbl 0663.49012 · doi:10.1137/0326045 [18] J. Sokolowski: Sensitivity analysis of shape estimation problems. In the volume dedicated to Jean Cea, to appear in special issue of Mechanics of Structures and Machines. · Zbl 0516.73108 [19] J. Sokolowski, J. P. Zolesio: Introduction to Shape Optimization. Shape sensitivity analysis. (Springer Series in Computational Mathematics 16.) Springer-Verlag, New York 1992. · Zbl 0761.73003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.