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

TL;DR

This paper develops a variational analysis framework to handle unbounded and discontinuous generalized eigenvalue functions in topology optimization, providing a continuous approximation that ensures stable optimization solutions.

Contribution

It introduces a redefinition and continuous approximation of eigenvalue functions, with theoretical guarantees of convergence, applicable to topology optimization problems.

Findings

The approximation epi-converges to the original eigenvalue function.

Numerical experiments demonstrate the effectiveness of the approach.

The method improves stability in topology optimization involving eigenvalues.

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.