Efficient Mesh Optimization Using the Gradient Flow of the Mean Volume

TL;DR

This paper introduces a gradient flow-based mesh optimization method using mean volume, which effectively regularizes hybrid meshes and improves finite element quality through a fast, theoretically grounded smoothing process.

Contribution

It presents a novel gradient flow approach for mesh optimization based on mean volume, applicable to various polyhedral elements, enhancing the GETMe method.

Findings

Gradient flow regularizes hybrid mesh elements to regular polyhedra.

The method converges quickly and improves mesh quality.

Theoretical and experimental analysis confirms effectiveness.

Abstract

The signed volume function for polyhedra can be generalized to a mean volume function for volume elements by averaging over the triangulations of the underlying polyhedron. If we consider these up to translation and scaling, the resulting quotient space is diffeomorphic to a sphere. The mean volume function restricted to this sphere is a quality measure for volume elements. We show that, the gradient ascent of this map regularizes the building blocks of hybrid meshes consisting of tetrahedra, hexahedra, prisms, pyramids and octahedra, that is, the optimization process converges to regular polyhedra. We show that the (normalized) gradient flow of the mean volume yields a fast and efficient optimization scheme for the finite element method known as the geometric element transformation method (GETMe). Furthermore, we shed some light on the dynamics of this method and the resulting smoothing procedure both theoretically and experimentally.