An inertial minimal-deformation-rate framework for shape optimization

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.

💡 Research Summary

The paper introduces a novel numerical framework that simultaneously tackles two pervasive difficulties in PDE‑constrained shape optimization and Willmore‑driven surface hole filling: (1) stagnation on flat regions of the objective landscape, and (2) mesh degradation during large geometric deformations. The authors combine a second‑order inertial flow with a Minimal‑Deformation‑Rate (MDR) mesh‑motion strategy, and further augment the method with surface‑diffusion regularization implemented via the Barrett‑Garcke‑Nürnberg (BGN) discretization.

Inertial flow. Starting from the classic H¹‑gradient descent, an inertial term ε₀>0 and a time‑dependent damping η(t)>0 are added, yielding the continuous variational equation
ε₀⟨∂ₜₜX, v⟩ + η(t)⟨∂ₜX, v⟩ = −dJ(Γ(t); v) ∀ v∈H(t).
The associated mechanical energy H(t)=J(Γ(t)) + (ε₀/2)‖∂ₜX‖²_L² satisfies dH/dt≤0, guaranteeing energy dissipation while allowing temporary non‑monotonicity of J. A simple central‑difference time discretization leads to a linear system at each step, preserving computational efficiency.

MDR mesh motion. The normal component of the boundary velocity is prescribed by the inertial flow (˜w·n). The bulk velocity w is then obtained by minimizing the instantaneous deformation‑rate energy ½∫_Ω|ε(w)|² subject to w·n|_Γ = ˜w·n. This constrained quadratic problem yields a linear variational formulation that can be solved with a single linear solve per time step. By directly minimizing the strain‑rate, the method maintains high mesh quality without the need for remeshing, and it generalizes naturally to three dimensions.

Surface‑diffusion regularization. To improve robustness for non‑smooth or non‑convex initial domains, a surface‑diffusion term α∆_ΓH n is added to the normal velocity (α is dynamically set to the L²‑norm of ˜w). This term corresponds to the H⁻¹‑gradient flow of the surface area and smooths geometric singularities while preserving volume. The fourth‑order diffusion is discretized using the BGN framework, which also generates an artificial tangential velocity that further protects mesh quality.

Willmore‑driven hole filling. For surface reconstruction with G¹ continuity, the authors formulate a Willmore‑energy (∫_Γ H²) minimization problem. They apply the same inertial‑MDR coupling, imposing clamped boundary positions and prescribed conormals. The resulting second‑order flow avoids the need for auxiliary geometric evolution equations, reduces long‑time error accumulation, and remains stable even with incompatible or highly irregular initial data.

Numerical experiments. The authors test the methodology on three representative PDE‑constrained shape‑optimization problems (Poisson reconstruction, Stokes drag minimization, Laplace eigenvalue minimization) and on Willmore‑driven hole filling. Results show:

• 2–3× faster convergence compared with pure first‑order flows.
• Lower final objective values, indicating that the inertial momentum helps escape shallow minima.
• Mesh quality metrics (minimum angle, element distortion) remain essentially unchanged throughout the simulations, eliminating the need for remeshing.
• In the hole‑filling case, the reconstructed patches exhibit smooth curvature and G¹ continuity despite starting from highly irregular initial patches.

Conclusions and future work. By integrating inertial dynamics, minimal deformation‑rate mesh motion, and surface‑diffusion regularization, the paper delivers a robust, efficient, and mesh‑preserving algorithm for a broad class of shape‑optimization problems. Future directions include adaptive time‑stepping, multi‑physics extensions, and real‑time interactive design tools.