Optimum design of thin plates via frequency optimization using BEM

Abstract

The thickness optimization is used to regulate the dynamic response of a thin plate of arbitrary geometry subjected to any type of admissible boundary conditions. The optimization problem consists in establishing the thickness variation law for which the fundamental frequency is maximized, minimized or forced to reach a prescribed value. Beside the equality constraint of constant volume, the thickness variation is subjected also to inequality constraints resulting from serviceability requirements (upper and lower thickness bounds) as well as to a nonlinear inequality constraint which ensures that the optimum solution remains within the limits of Kirchhoff plate theory. The evaluation of the objective function requires the solution of the dynamic bending problem of a plate with variable thickness which is solved using the analog equation method in conjunction with the boundary element method. A nonlinear optimization problem is formulated, and the optimum solution is obtained through the sequential quadratic programming algorithm. The thickness is approximated using integrated radial basis functions which approximate accurately not only the thickness function but also its first and second derivatives involved in the plate equation and in the constraints. Several plate optimization problems have been studied giving realistic and meaningful optimum designs without violating the validity of the thin plate theory.