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Space Mapping for PDE Constrained Shape Optimization

Sebastian Blauth

TL;DR

This work introduces a Riemannian space-mapping framework for PDE-constrained shape optimization, leveraging Steklov-Poincaré-type metrics to define a gradient-driven deformation on the shape manifold. It develops an aggressive space mapping (ASM) strategy in both a manifold and a volume-based formulation, enabling coarse-model optimizations to drive fine-model solutions without requiring fine-model derivatives. Numerical experiments on a semi-linear transmission problem and a Navier–Stokes flow distribution problem demonstrate dramatic reductions in fine-model evaluations (often only a handful of PDE solves) while maintaining high accuracy, including successful coupling with commercial solvers for the fine problem. The approach offers a practical, scalable pathway for industrial PDE-constrained shape optimization, with open-source software support and clear directions for further analysis and applications.

Abstract

The space mapping technique is used to efficiently solve complex optimization problems. It combines the accuracy of fine model simulations with the speed of coarse model optimizations to approximate the solution of the fine model optimization problem. In this paper, we propose novel space mapping methods for solving shape optimization problems constrained by partial differential equations (PDEs). We present the methods in a Riemannian setting based on Steklov-Poincaré-type metrics and discuss their numerical discretization and implementation. We investigate the numerical performance of the space mapping methods on several model problems. Our numerical results highlight the methods' great efficiency for solving complex shape optimization problems.

Space Mapping for PDE Constrained Shape Optimization

TL;DR

This work introduces a Riemannian space-mapping framework for PDE-constrained shape optimization, leveraging Steklov-Poincaré-type metrics to define a gradient-driven deformation on the shape manifold. It develops an aggressive space mapping (ASM) strategy in both a manifold and a volume-based formulation, enabling coarse-model optimizations to drive fine-model solutions without requiring fine-model derivatives. Numerical experiments on a semi-linear transmission problem and a Navier–Stokes flow distribution problem demonstrate dramatic reductions in fine-model evaluations (often only a handful of PDE solves) while maintaining high accuracy, including successful coupling with commercial solvers for the fine problem. The approach offers a practical, scalable pathway for industrial PDE-constrained shape optimization, with open-source software support and clear directions for further analysis and applications.

Abstract

The space mapping technique is used to efficiently solve complex optimization problems. It combines the accuracy of fine model simulations with the speed of coarse model optimizations to approximate the solution of the fine model optimization problem. In this paper, we propose novel space mapping methods for solving shape optimization problems constrained by partial differential equations (PDEs). We present the methods in a Riemannian setting based on Steklov-Poincaré-type metrics and discuss their numerical discretization and implementation. We investigate the numerical performance of the space mapping methods on several model problems. Our numerical results highlight the methods' great efficiency for solving complex shape optimization problems.
Paper Structure (15 sections, 3 theorems, 76 equations, 9 figures)

This paper contains 15 sections, 3 theorems, 76 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Space Mapping Technique
  4. Shape Calculus
  5. Steklov-Poincaré-Type Metrics for Shape Optimization
  6. Basics of Riemannian Geometry
  7. Space Mapping Methods for Shape Optimization
  8. A Riemannian Framework for Space Mapping Shape Optimization
  9. Volume-Based Formulation of the Space Mapping Methods for Shape Optimization
  10. Numerical Solution of Shape Optimization Problems with Space Mapping Methods
  11. Implementation Details
  12. A Semi-Linear Transmission Problem
  13. Uniform Flow Distribution
  14. Coupling with Commercial Solvers
  15. Conclusion and Outlook

Key Result

Proposition 2.8

\newlabelprop:volume0 The reduced cost functional $J$ corresponding to problem eq:interface_strong is shape differentiable with shape derivative where $u \in H^1_0(D)$ is the solution of eq:interface_weak, $D\mathcal{V}$ denotes the Jacobian of $\mathcal{V}$, and the adjoint state $p \in H^1_0(D)$ solves

Figures (9)

  • Figure 1: Sketch of the problem setup for $d=2$.
  • Figure 1: Initial inclusion $\Omega$ (orange) and reference ellipse (outlined in gray).
  • Figure 2: History of the ASM method for problem \ref{['eq:semi_linear_transmission']}.
  • Figure 3: Evolution of the inclusion $\Omega$ over the course of the ASM method. The interior domain $\Omega$ is depicted in orange, the outer domain $D\setminus \hbox{$\Omega$}$ in blue, and the reference ellipse is outlined in gray.
  • Figure 4: Reference geometry for the uniform flow distribution.
  • ...and 4 more figures

Theorems & Definitions (17)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Example 2.5
  • Definition 2.6
  • Remark 2.7
  • Proposition 2.8
  • Theorem 2.9: structure theorem
  • Proposition 2.10
  • ...and 7 more