A priori error analysis of shape derivatives of linear functionals in structural topology optimization

Abstract

We provide theoretical analyses and numerical comparisons of boundary-based and volumetric shape derivative expressions of linear objective functionals encountered in topology optimization of linear elastic structures. The two expressions yield identical results if the domain is smooth and the governing equation is solved exactly; however, the finite element approximation of the expressions for less regular domains yield different results. We first review the two expressions to show that the volumetric shape derivative places weaker regularity requirements, which, unlike the requirements for boundary-based shape derivatives, are satisfied in most finite element approximations. We then analyze the error in the degree-k polynomial finite element approximations of the two expressions; we show that, for sufficiently regular problems, the boundary-based and volumetric shape derivatives provide kth and 2k-th order accurate approximations, respectively, of the true shape derivative. We finally assess, through numerical examples, the practical implications of using the volumetric vs boundary-based shape derivatives in topology optimization problems; we demonstrate that methods based on the volumetric shape derivative yield more robust solutions to topology optimization problems.