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.

翻译：拓扑优化已发展成为一种强大的工程设计工具，能够综合考虑多种物理现象，设计出卓越的结构与材料。尽管该方法近年来取得了巨大进展，但仍存在若干未解问题。本文旨在探讨其中一个核心问题：给定的拓扑优化设计距离全局最优解有多远？这通常是一个难以回答的问题，因为几乎所有有意义的拓扑优化问题都是非凸的。遗憾的是，这种非凸性意味着设计空间可能存在大量局部极小值，导致优化器最终陷入次优设计。本研究通过基于拉格朗日对偶理论的计算框架，系统探究拓扑优化的性能界。该方法为特定优化问题子集提供了一种有效度量，能够评估给定设计距离全局最优解的"接近"程度。我们通过多个数值算例（包括模式转换器与谐振板的设计）展示了该方法的实际应用能力。