Generation of unstructured meshes in 2-D, 3-D, and spherical geometries with embedded high resolution sub-regions

We present 2-D, 3-D, and spherical mesh generators for the Finite Element Method (FEM) using triangular and tetrahedral elements. The mesh nodes are treated as if they were linked by virtual springs that obey Hooke's law. Given the desired length for the springs, the FEM is used to solve for the optimal nodal positions for the static equilibrium of this spring system. A 'guide-mesh' approach allows the user to create embedded high resolution sub-regions within a coarser mesh. The method converges rapidly. For example, in 3-D, the algorithm is able to refine a specific region within an unstructured tetrahedral spherical shell so that the edge-length factor $l_{0r}/l_{0c} = 1/33$ within a few iterations, where $l_{0r}$ and $l_{0c}$ are the desired spring length for elements inside the refined and coarse regions respectively. One use for this type of mesh is to model regional problems as a fine region within a global mesh that has no fictitious boundaries, at only a small additional computational cost. The algorithm also includes routines to locally improve the quality of the mesh and to avoid badly shaped 'slivers-like' tetrahedra.