Wasserstein crossover for evolutionary algorithm-based topology optimization

Abstract

Evolutionary algorithms (EAs) are promising approaches for non-differentiable or strongly multimodal topology optimization problems, but they often suffer from the curse of dimensionality, generally leading to low-resolution optimized results. This limitation stems in part from the difficulty of producing effective offspring through traditional crossover operators, which struggle to recombine complex parent design features in a meaningful way. In this study, we propose a novel crossover operator for topology optimization, termed Wasserstein crossover, and develop a corresponding EA-based optimization framework. Our method leverages a morphing technique based on the Wasserstein distance -- a distance metric between probability distributions derived from the optimal transport theory. Its key idea is to treat material distributions as probability distributions and generate offspring as Wasserstein barycenters, enabling smooth and interpretable interpolation between parent designs while preserving their structural features. The proposed framework incorporates Wasserstein crossover into an EA under a multifidelity design scheme, where low-fidelity optimized initial designs evolve through iterations of Wasserstein crossover and selection based on high-fidelity evaluation. We apply the proposed framework to three topology optimization problems: maximum stress minimization in two- and three-dimensional structural mechanics, and turbulent heat transfer in two-dimensional thermofluids. The results demonstrate that candidate solutions evolve iteratively toward high-performance designs through Wasserstein crossover, highlighting its potential as an effective crossover operator and validating the usefulness of the proposed framework for solving intractable topology optimization problems.