Geometric-structure Preserving Methods for Surface Evolution in Curvature Flows with Minimal Deformation Formulations

In this article, we design novel weak formulations and parametric finite element methods for computing surface evolution under mean curvature flow and surface diffusion while preserving essential geometric structures such as surface area decrease and volume conservation enclosed by the surface. The proposed methods incorporate tangential motion that minimizes deformation energy under the constraint of normal velocity, ensuring minimal mesh distortion from the initial surface. Additionally, they employ a global constant multiplier to preserve the geometric structures in mean curvature flow and surface diffusion. Specifically, for mean curvature flow, the proposed method preserves the decrease of surface area; for surface diffusion, it preserves both the decrease of surface area and the conservation of the volume enclosed by the surface. Extensive numerical examples are presented to illustrate the convergence of the proposed methods, their geometric-structure-preserving properties, and the improvement in mesh quality of the computed surfaces.