Bimodal optimal design of vibrating plates using theory and methods of nondifferentiable optimization

This paper is concerned with an optimal design problem of vibrating plates. The optimization problem consists in maximizing the smallest eigenvalue of the elliptic eigenvalue problem describing the free plate vibration. The thickness of the plate is the variable subject to optimization. The volume of the plate is constant and the thickness of the plate is bounded.In this paper, we consider the case where the smallest eigenvalue is multiple. This implies that the optimization problem is nondifferentiable. A necessary optimality condition is formulated. The finite-element method is employed as an approximation method. A nonsmooth optimization method is used to solve this optimization problem. Numerical examples are provided.This work was supported by the Polish Academy of Sciences and the Education Ministry of Japan. Lemarechal's implementation of his method was used for numerical computations.on leave from Systems Research Institute, Warsaw, Poland.