An inertial minimal-deformation-rate framework for shape optimization

TL;DR

This paper introduces a robust PDE-constrained shape optimization framework combining inertial flow and minimal-deformation-rate mesh strategies, improving convergence speed and mesh quality, and extending to Willmore surface hole filling.

Contribution

It presents a novel inertial minimal-deformation-rate approach that accelerates convergence and maintains mesh quality in shape optimization and surface hole filling tasks.

Findings

Faster convergence to lower objective values.

Superior mesh preservation during evolution.

Effective extension to Willmore surface hole filling.

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.