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

Fumiya Okazaki, Takayuki Yamada

Abstract

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.

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

Abstract

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.
Paper Structure (9 sections, 8 theorems, 106 equations, 72 figures, 1 algorithm)

This paper contains 9 sections, 8 theorems, 106 equations, 72 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Formulation of mass optimization
  4. Optimization for heat conduction
  5. Optimization for mean compliance of linear elastic structures
  6. Numerical algorithm
  7. Numerical results
  8. Heat conduction
  9. Mean compliance of linear elastic structures

Key Result

Lemma 3.1

If $J_{\delta}$ is proper lower semi continuous with respect to the metric $\mathcal{W}_2$ on $\mathcal{P}(\overline{D})$, then $J_{\delta}$ admits a minimizer on $\mathcal{P}(\overline{D})$.

Figures (72)

  • Figure : $(\delta,\eta)=(10^{-2},10^{-2})$
  • Figure : $(\delta,\eta)=(10^{-3},10^{-2})$
  • Figure : $(\delta,\eta)=(10^{-4},10^{-2})$
  • Figure : $(\delta,\eta)=(10^{-2},10^{-2})$
  • Figure : $(\delta,\eta)=(10^{-3},10^{-2})$
  • ...and 67 more figures

Theorems & Definitions (19)

  • Definition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • Remark 3.5
  • proof
  • ...and 9 more