Penalization in Density-Based Topology Optimization: A Mathematical Perspective

Abstract

Most real-life engineering optimization problems are non-convex by nature. In a topology optimization context, this non-convexity is even exacerbated by the extra restrictions imposed during the optimization process to enforce mesh-independent black/white manufacturable solutions. Such restrictions include intermediate density penalization, as well as external regulation techniques imposed to tackle some numerical instabilities such as checkerboarding and mesh dependence, in addition to various design constraints. This non-convexity gives rise to the problem of local minima, where the converged solution is greatly affected by the algorithmic parameters as well as the initial guess. To overcome this non-convexity, it's often advised to use continuation methods, that is to introduce non-convexification gradually between iterations. In this article, we present a comprehensive treatment of the sources of nonconvexity in density-based topology optimization problems, with a special emphasis on linear elastic compliance minimization. This is in an attempt to better bridge the gap between the mathematical and the engineering/physical aspects of the topic.

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