MULTIPLE-EIGENVALUES IN STRUCTURAL OPTIMIZATION PROBLEMS

This paper discusses characteristic features and inherent difficulties pertaining to the lack of usual differentiability properties in probl ems of sensitivity analysis and optimum structural design with respect to multiple eigenvalues. Computational aspects are illustrated via a number of examples. Based on a mathematical perturbation technique, a general multiparameter framework is developed for computation of desig n sensitivities of simple as well as multiple eigenvalues of complex s tructures. The method is exemplified by computation of changes of simp le and multiple natural transverse vibration frequencies subject to ch anges of different design parameters of finite element modelled, stiff ener reinforced thin elastic plates. Problems of optimization are form ulated as the maximization of the smallest (simple or multiple) eigenv alue subject to a global constraint of e.g. given total volume of mate rial of the structure, and necessary optimality conditions are derived for an arbitrary degree of multiplicity of the smallest eigenvalue. T he necessary optimality conditions express (i) linear dependence of a set of generalized gradient vectors of the multiple eigenvalue and the gradient vector of the constraint, and (ii) positive semi-definitenes s of a matrix of the coefficients of the linear combination. It is sho wn in the paper that the optimality condition (i) can be directly appl ied for the development of an efficient, iterative numerical method fo r the optimization of structural eigenvalues of arbitrary multiplicity , and that the satisfaction of the necessary optimality condition (ii) can be readily checked when the method has converged. Application of the method is illustrated by simple, multiparameter examples of optimi zing single and biomodal buckling loads of columns on elastic foundati ons.