We shall be dealing with the eigenvalue optimization problem for an anisotropic plate. The plate is partly unilaterally supported on its boundary and subjected to longitudinal forces causing its buckling. The state problem has then the form of an eigenvalue variational inequality expressing the deflection of the plate and the maximal possible value of the acting forces keeping its stability which corresponds to the first eigenvalue. The demand of the maximal first eigenvalue with respect to variable thicknesses of the plate means to solve the optimal design problem with eigenvalue variational inequality as the state problem. The existence of a solution in the framework of the general theory will be examined. The necessary optimality conditions will be derived. The convergence of the finite elements approximation will be verified.
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The eigenvalue optimization problem for anisotropic plates has been dealt with. The variable thickness of a plate plays the role of a design variable. The state problem arises considering free vibrations of a plate. The demand of the lowest first eigenfrequency means the maximal first eigenvalue of the elliptic eigenvalue problem. The continuity and differentiability properties of the first eigenvalue have been examined. The existence theorem for the optimization problem has been stated and verified. The finite elements approximation has been analyzed. The shifted penalization and the method of nonsmooth optimization can be used in order to obtain numerical results.
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