Graph-Grammar Based Longest-Edge Refinement Algorithm for Three-Dimensional Optimally p Refined Meshes with Tetrahedral Elements.

Finite element method is a popular way of solving engineering problems in geoengineering. Three-dimensional grids employed for approximation the formation layers are often constructed from tetrahedral finite elements. The refinement algorithms that avoids hanging nodes are desired in order to avoid constrained approximation on broken edges and faces. We present a new mesh refinement algorithm for such the tetrahedral grids, with the following features (1) it is a two-level algorithm, refining the elements’ faces first, followed by the refinement of the elements’ interiors; (2) for the face refinements it employs the graph-grammar based version of the longest-edge refinement algorithm to avoid the hanging nodes; and (3) it allows for nearly perfect parallel execution of the second stage, refining the element interiors. We describe the algorithm using the graph-grammar based formalism. We verify the properties of the algorithm, by breaking 5,000 tetrahedral elements, and checking their angles and proportions. On the generated meshes without hanging nodes we span the polynomial basis functions of the optimal order, selected via metaheuristic optimization algorithm. We use them for the projection based interpolation of formation layers.