Analytical relationships for imposing minimum length scale in the robust Topology Optimization formulation

TL;DR

This paper derives analytical formulas linking minimum length scale parameters to the robust topology optimization framework, simplifying parameter selection and ensuring consistent control over solid and void phase sizes.

Contribution

It provides explicit analytical expressions relating minimum length scale to density filter and projection parameters, validated within a density-based topology optimization framework.

Findings

Analytical expressions for minimum length scale parameters

Validation of formulas on density-based topology optimization

Guidelines for involving eroded, intermediate, and dilated designs in optimization

Abstract

The robust topology optimization formulation that introduces the eroded and dilated versions of the design has gained increasing popularity in recent years, mainly because of its ability to produce designs satisfying a minimum length scale. Despite its success in various topology optimization fields, the robust formulation presents some drawbacks. This paper addresses one in particular, which concerns the imposition of the minimum length scale. In the density framework, the minimum size of the solid and void phases must be imposed implicitly through the parameters that define the density filter and the smoothed Heaviside projection. Finding these parameters can be time consuming and cumbersome, hindering a general code implementation of the robust formulation. Motivated by this issue, in this article we provide analytical expressions that explicitly relate the minimum length scale and the parameters that define it. The expressions are validated on a density-based framework. To facilitate the reproduction of results, MATLAB codes are provided. As a side finding, this paper shows that to obtain simultaneous control over the minimum size of the solid and void phases, it is necessary to involve the 3 fields (eroded, intermediate and dilated) in the topology optimization problem. Therefore, for the compliance minimization problem subject to a volume restriction, the intermediate and dilated designs can be excluded from the objective function, but the volume restriction has to be applied to the dilated design in order to involve all 3 designs in the formulation.