Revisiting Non-Convexity in Topology Optimization of Compliance Minimization Problems

TL;DR

This paper analyzes the sources of non-convexity in topology optimization for compliance minimization, emphasizing the dominant role of density penalization and advocating continuation methods to find global optima.

Contribution

It provides a comprehensive mathematical analysis of non-convexity sources, especially highlighting the effects of penalization and filtering in density-based topology optimization.

Findings

Penalization of intermediate densities is the main cause of non-convexity.

Filtering techniques can have penalization-like effects influencing convexity.

Continuation methods are effective in overcoming local minima.

Abstract

Purpose: This is an attempt to better bridge the gap between the mathematical and the engineering/physical aspects of the topic. We trace the different sources of non-convexification in the context of topology optimization problems starting from domain discretization, passing through penalization for discreteness and effects of filtering methods, and end with a note on continuation methods. Design/Methodology/Approach: Starting from the global optimum of the compliance minimization problem, we employ analytical tools to investigate how intermediate density penalization affects the convexity of the problem, the potential penalization-like effects of various filtering techniques, how continuation methods can be used to approach the global optimum, and how the initial guess has some weight in determining the final optimum. Findings: The non-convexification effects of the penalization of intermediate density elements simply overshadows any other type of non-convexification introduced into the problem, mainly due to its severity and locality. Continuation methods are strongly recommended to overcome the problem of local minima, albeit its step and convergence criteria are left to the user depending on the type of application. Originality/Value: In this article, we present a comprehensive treatment of the sources of non-convexity in density-based topology optimization problems, with a focus on linear elastic compliance minimization. We put special emphasis on the potential penalization-like effects of various filtering techniques through a detailed mathematical treatment.