Numerical simulations of evolving surfaces, such as mean curvature flow and shape optimization, often suffer from mesh deterioration under large deformations. In this talk, I will present a numerical framework based on the Minimal Deformation Rate (MDR) strategy, whose main purpose is to preserve mesh quality throughout the evolution without remeshing.
First, for curvature-driven flows, including mean curvature flow and surface diffusion on both closed and open surfaces with moving contact lines, we design a new BGN--MDR scheme by exploiting the key observation that the Barrett–Garcke–Nürnberg (BGN) and MDR formulations differ only in one degree of freedom, thereby combining the discrete energy stability of the classical BGN approach with the mesh-quality preservation of MDR.
Second, for PDE-constrained shape optimization and surface hole filling, we couple a second-order inertial flow with MDR, which both accelerates convergence in flat energy landscapes and avoids frequent remeshing during large geometric deformations through a suitable MDR-based mesh-motion strategy.