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optimal control problem; elliptic, linear symmetric operator; unique solution of stationary variational inequalities; convex set; principle of virtual power; unilateral constraints; bending; cylindrical shell; thickness function; obstacle
The aim of the present paper is to study problems of optimal design in mechanics, whose variational form are inequalities expressing the principle of virtual power in its inequality form. We consider an optimal control problem in whixh the state of the system (involving an elliptic, linear symmetric operator, the coefficients of which are chosen as the design - control variables) is defined as the (unique) solution of stationary variational inequalities. The existence result proved in Section 1 is applied in Section 2 to the optimal design of an elastic cylindrical shell subject to unilateral constraints. We assume that the bending of the shell is limited by a rigid obstacle. The role of the design variable is played by the thickness of the shell.
[1] R. A. Adams: Sobolev Spaces. Academic Press, New York, San Francisco, London 1975, MR 0450957 | Zbl 0314.46030
[2] H. Attouch: Convergence des solution d'inéquations variationnelles avec obstacle. Proceedings of the International Meeting on Recent Methods in Nonlinear analysis. (Rome, May 1978) ed. by E. De Giorgi - E. Magenes - U. Mosco.
[3] V. Barbu: Optimal control of variational inequalities. Pitman Advanced Publishing Program, Boston. London, Melbourne 1984. MR 0742624 | Zbl 0574.49005
[4] I. Boccardo C. Dolcetta: Stabilita delle soluzioni di disequazioni variazionali ellittiche e paraboliche quasi-lineari. Ann. Universeta Ferrara, 24 (1978), 99-111.
[5] J. Céa: Optimisation, Théorie et Algorithmes. Dunod Paris, 1971. MR 0298892
[6] G. Duvaut J. L. Lions: Inequalities in mechanics and physics. Berlin, Springer Verlag 1975. MR 0521262
[7] R. Glowinski: Numerical Methods for Nonlinear Variational Problems. Springer Verlag 1984. MR 0737005 | Zbl 0536.65054
[8] I. Hlaváček I. Bock J. Lovíšek: Optimal Control of a Variational Inequality with Applications to Structural Analysis. II. Local Optimization of the Stress in a Beam. III. Optimal Design of an Elastic Plate. Appl. Math. Optimization 13: 117-136/1985. DOI 10.1007/BF01442202 | MR 0794174
[9] D. Kinderlehrer G. Stampacchia: An introduction to variational inequalities and their applications. Academic Press, 1980. MR 0567696
[10] V. G. Litvinov: Optimal control of elliptic boundary value problems with applications to mechanics. Moskva "Nauka" 1987, (in Russian).
[11] M. Bernadou J. M. Boisserie: The finite element method in thin shell. Theory: Application to arch Dam simulations. Birkhäuser Boston 1982. MR 0663553
[12] J. Nečas I. Hlaváček: Mathematical theory of elastic and elasto-plastic bodies: An introduction. Elsevier Scientific Publishing Company, Amsterdam 1981. MR 0600655
[13] U. Mosco: Convergence of convex sets of solutions of variational inequalities. Advances of Math. 3 (1969), 510-585. DOI 10.1016/0001-8708(69)90009-7 | MR 0298508
[14] K. Ohtake J. T. Oden N. Kikuchi: Analysis of certain unilateral problems in von Karman plate theory by a penalty method - PART 1. A variational principle with penalty. Computer Methods in Applied Mechanics and Engineering 24 (1980), 117-213, North Holland Publishing Company.
[15] P. D. Panagiotopoulos: Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy functions. Birkhäuser-Verlag, Boston-Basel-Stutgart, 1985. MR 0896909 | Zbl 0579.73014
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