Structural optimization considering smallest magnitude eigenvalues: a smooth approximation

An issue that frequently arises in structural optimization problems considering eigenvalues is the non differentiability of repeated eigenvalues. In order to overcome this difficulty, several schemes were already presented in the literature. However, these approaches generally have other disadvantages such as the inclusion of additional constraints, the inaccuracy of representation of smallest/largest eigenvalues, the significant increase in the computational effort required and incompatibility with finite differences schemes. In this paper a smooth p-norm approximation for the smallest magnitude eigenvalue is employed. The resulting approximation is differentiable, converges to the exact value as p is increased and is very simple to use (it is also compatible with finite difference schemes). Although the use of smooth approximations for maximum/minimum operators is a classical approach, for some reason it was not extensively studied in the context of structural optimization considering eigenvalues. Three examples concerning topology optimization for the maximization of the first natural vibration frequency of plane stress structures are presented in order to show the effectiveness of the proposed approach.