Variational analysis of unbounded and discontinuous generalized eigenvalue functions with application to topology optimization

Abstract

The maximum (or minimum) generalized eigenvalue of symmetric positive semidefinite matrices that depend on optimization variables often appears as objective or constraint functions in structural topology optimization when we consider robustness, vibration, and buckling. It can be an unbounded or discontinuous function where matrices become singular (where a topological change of the structural design occurs). Based on variational analysis, we redefine the maximum (and minimum) generalized eigenvalue function as an extended real-valued function and propose a real-valued continuous approximation of it. Then, we show that the proposed approximation epi-converges to the original redefined function, which justifies solving problems with the approximation instead. We consider two specific topology optimization problems: robust compliance optimization and eigenfrequency optimization and conduct simple numerical experiments.