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An inertial minimal-deformation-rate framework for shape optimization

Falai Chen, Buyang Li, Jiajie Li, Rong Tang

TL;DR

The paper tackles slow progress in PDE-constrained shape optimization and poor mesh quality during geometry evolution by introducing a second-order inertial gradient flow coupled with Minimal Deformation Rate (MDR) mesh motion, enhanced with surface-diffusion regularization inside the Barrett–Garcke–Nürnberg (BGN) framework. This inertial MDR approach accelerates convergence to lower objectives while preserving mesh quality and avoiding remeshing, and it is extended to Willmore-driven hole filling to achieve high-order smooth reconstructions from non-ideal initial data. The authors provide continuous and fully discrete schemes, prove energy-dissipation properties, and demonstrate strong numerical results across shape reconstruction, Stokes drag minimization, eigenvalue optimization, and hole filling. The work offers a robust, scalable framework for PDE-guided geometry optimization with practical benefits for mesh integrity and convergence speed, and lays the groundwork for future theoretical convergence analyses and broader geometric-flow applications.

Abstract

We propose a robust numerical framework for PDE-constrained shape optimization and Willmore-driven surface hole filling. To address two central challenges -- slow progress in flat energy landscapes, which can trigger premature stagnation at suboptimal configurations, and mesh deterioration during geometric evolution -- we couple a second-order inertial flow with a minimal-deformation-rate (MDR) mesh motion strategy. This coupling accelerates convergence while preserving mesh quality and thus avoids remeshing. To further enhance robustness for non-smooth or non-convex initial geometries, we incorporate surface-diffusion regularization within the Barrett-Garcke-N"urnberg (BGN) framework. Moreover, we extend the inertial MDR methodology to Willmore-type surface hole filling, enabling high-order smooth reconstructions even from incompatible initial data. Numerical experiments demonstrate markedly faster convergence to lower original objective values, together with consistently superior mesh preservation throughout the evolution.

An inertial minimal-deformation-rate framework for shape optimization

TL;DR

The paper tackles slow progress in PDE-constrained shape optimization and poor mesh quality during geometry evolution by introducing a second-order inertial gradient flow coupled with Minimal Deformation Rate (MDR) mesh motion, enhanced with surface-diffusion regularization inside the Barrett–Garcke–Nürnberg (BGN) framework. This inertial MDR approach accelerates convergence to lower objectives while preserving mesh quality and avoiding remeshing, and it is extended to Willmore-driven hole filling to achieve high-order smooth reconstructions from non-ideal initial data. The authors provide continuous and fully discrete schemes, prove energy-dissipation properties, and demonstrate strong numerical results across shape reconstruction, Stokes drag minimization, eigenvalue optimization, and hole filling. The work offers a robust, scalable framework for PDE-guided geometry optimization with practical benefits for mesh integrity and convergence speed, and lays the groundwork for future theoretical convergence analyses and broader geometric-flow applications.

Abstract

We propose a robust numerical framework for PDE-constrained shape optimization and Willmore-driven surface hole filling. To address two central challenges -- slow progress in flat energy landscapes, which can trigger premature stagnation at suboptimal configurations, and mesh deterioration during geometric evolution -- we couple a second-order inertial flow with a minimal-deformation-rate (MDR) mesh motion strategy. This coupling accelerates convergence while preserving mesh quality and thus avoids remeshing. To further enhance robustness for non-smooth or non-convex initial geometries, we incorporate surface-diffusion regularization within the Barrett-Garcke-N"urnberg (BGN) framework. Moreover, we extend the inertial MDR methodology to Willmore-type surface hole filling, enabling high-order smooth reconstructions even from incompatible initial data. Numerical experiments demonstrate markedly faster convergence to lower original objective values, together with consistently superior mesh preservation throughout the evolution.
Paper Structure (21 sections, 2 theorems, 71 equations, 12 figures, 1 table, 1 algorithm)

This paper contains 21 sections, 2 theorems, 71 equations, 12 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Representative models of shape optimization
  3. Shape reconstruction
  4. Drag minimization in Stokes flow
  5. Eigenvalue minimization
  6. Surface hole filling driven by Willmore flow
  7. Inertial Gradient Flow for PDE-constrained Shape Optimization
  8. ${\bf H}^1(\varOmega(t))$ gradient flow and domain evolution
  9. Inertial ${\bf H}^1(\varOmega(t))$ gradient flow
  10. Linear time discretization of the inertial term
  11. Fully discretized linear numerical scheme
  12. Inertial MDR Framework for PDE--Constrained Shape Optimization
  13. Minimal Deformation Rate Framework
  14. Inertial MDR Framework regularized by surface diffusion
  15. Inertial MDR method for Willmore-driven surface hole filling
  16. ...and 6 more sections

Key Result

Lemma 3.1

Let $\varOmega(t)$ be a smoothly evolving domain with velocity field $\tilde{\bm w} \in {\bf W}^{1,\infty}(\varOmega(t))$. Then, for any sufficiently regular function $f:\varOmega(t)\times [0,T]\to \mathbb{R}$, it holds that where $\partial_t^{\bullet} f := \partial_t f + \tilde{\bm w} \cdot \nabla f$ denotes the material derivative of $f$ along $\tilde{\bm w}$.

Figures (12)

  • Figure 1: Optimal shapes produced by the inertial $H^1$ gradient flow (left) and the classical $H^1$ gradient flow (middle), together with the corresponding original objective and mechanical energy convergence histories (right), for Example 1 (Case 1).
  • Figure 2: Mesh evolution for Example 1 (Case 2): initial mesh (left), intermediate stage (middle left), final optimized mesh via the inertial BGN--MDR flow (middle right), and the BGN--harmonic extension (right).
  • Figure 3: Example 1 (Case 3): Row 1 shows initial mesh (left), optimized mesh obtained by the $H^1$ gradient flow (middle), and optimized mesh obtained by the inertial MDR method (right); Row 2 shows optimized mesh obtained by inertial BGN--MDR method (left) and convergence histories of original objective (right).
  • Figure 4: Example 2 (Case 1): initial mesh (left), intermediate mesh (middle), and optimized mesh (right) obtained by the inertial BGN--MDR method.
  • Figure 5: Example 2 (Case 1): velocity field around the optimized obstacle (left), and convergence histories of the original objective (middle) and mechanical energy (right).
  • ...and 7 more figures

Theorems & Definitions (3)

  • Lemma 3.1: Walker2015
  • Lemma 3.2: Energy dissipation for the inertial gradient flow
  • proof