On performance bounds for topology optimization

TL;DR

This paper explores how close a given topology optimized design is to the global optimum by developing a computational framework based on Lagrange duality, addressing a key challenge in non-convex topology optimization problems.

Contribution

It introduces a novel method using Lagrange duality to estimate performance bounds and quantify optimality gaps in topology optimization.

Findings

The framework provides bounds on how far a design is from the global optimum.

Numerical examples demonstrate the method's effectiveness in practical design problems.

The approach helps identify suboptimal solutions in complex topology optimization tasks.

Abstract

Topology optimization has matured to become a powerful engineering design tool that is capable of designing extraordinary structures and materials taking into account various physical phenomena. Despite the method's great advancements in recent years, several unanswered questions remain. This paper takes a step towards answering one of the larger questions, namely: How far from the global optimum is a given topology optimized design? Typically this is a hard question to answer, as almost all interesting topology optimization problems are non-convex. Unfortunately, this non-convexity implies that local minima may plague the design space, resulting in optimizers ending up in suboptimal designs. In this work, we investigate performance bounds for topology optimization via a computational framework that utilizes Lagrange duality theory. This approach provides a viable measure of how \say{close} a given design is to the global optimum for a subset of optimization formulations. The method's capabilities are exemplified via several numerical examples, including the design of mode converters and resonating plates.