Topology optimization concerning the mass distribution via filtered gradient flows on the Wasserstein space

In this article, we formulate topology optimization problems concerning the mass distribution as minimization problems for functionals on the Wasserstein space. We relax optimization problems regarding non-convex objective functions on the Wasserstein space by using the Neumann heat semigroup and prove the existence of minimizers of relaxed problems. Furthermore, we introduce the filtered Wasserstein gradient flow and derive the error estimate between the original Wasserstein gradient flow and the filtered one in terms of the Wasserstein distance. We also construct a candidate for the optimal mass distribution for a given fixed total mass and simultaneously obtain the shape of the material by the numerical calculation of filtered Wasserstein gradient flows.

💡 Research Summary

In this paper the authors develop a novel framework for topology optimization of material mass distribution by formulating the design problem as a minimization problem on the Wasserstein space of probability measures. Traditional topology optimization distinguishes between shape optimization (boundary control) and topology optimization (control of the material’s distribution). While shape optimization has been successfully studied using infinite‑dimensional Riemannian manifolds, topology optimization requires a space that can accommodate creation and disappearance of holes. The authors choose the space (\mathcal{P}(D)) of Borel probability measures on a bounded convex domain (D\subset\mathbb{R}^d) equipped with the (L^2) Wasserstein distance (W_2). Otto’s calculus provides a formal Riemannian structure: tangent vectors are identified with elements of the weighted Sobolev space (\dot H^{-1}_\mu(D)) and the Wasserstein gradient (\nabla_W J