Surface and hypersurface meshing techniques for space-time finite element methods

TL;DR

This paper presents a novel method for constructing 2D surface and 3D hypersurface meshes in space-time domains, enabling improved finite element analysis through flexible mesh generation across time slabs.

Contribution

The paper introduces a general, detailed approach for meshing space-time domains using unstructured simplicial meshes, applicable to 2D surfaces in 3D and 3D hypersurfaces in 4D.

Findings

Method successfully generates meshes for complex space-time domains.

Numerical experiments validate the method's accuracy and flexibility.

Applicable to various space-time finite element applications.

Abstract

A general method is introduced for constructing two-dimensional (2D) surface meshes embedded in three-dimensional (3D) space time, and 3D hypersurface meshes embedded in four-dimensional (4D) space time. In particular, we begin by dividing the space-time domain into time slabs. Each time slab is equipped with an initial plane (hyperplane), in conjunction with an unstructured simplicial surface (hypersurface) mesh that covers the initial plane. We then obtain the vertices of the terminating plane (hyperplane) of the time slab from the vertices of the initial plane using a space-time trajectory-tracking approach. Next, these vertices are used to create an unstructured simplicial mesh on the terminating plane (hyperplane). Thereafter, the initial and terminating boundary vertices are stitched together to form simplicial meshes on the intermediate surfaces or sides of the time slab. After describing this new mesh-generation method in rigorous detail, we provide the results of multiple numerical experiments which demonstrate its validity and flexibility.