Aggressive Tetrahedral Mesh Improvement

Abstract

Summary. We present a tetrahedral mesh improvement schedule that usually creates meshes whose worst tetrahedra have a level of quality substantially better than those produced by any previous method for tetrahedral mesh generation or "mesh clean-up." Our goal is to aggressively optimize the worst tetrahedra, with speed a secondary consideration. Mesh optimization methods often get stuck in bad local optima (poor-quality meshes) because their repertoire of mesh transformations is weak. We employ a broader palette of operations than any previous mesh improvement software. Alongside the best traditional topological and smoothing operations, we introduce a topological transformation that inserts a new vertex (sometimes deleting others at the same time). We describe a schedule for applying and composing these operations that rarely gets stuck in a bad optimum. We demonstrate that all three techniques-smoothing, vertex insertion, and traditional transformations-are substantially more effective than any two alone. Our implementation usually improves meshes so that all dihedral angles are between 31• and 149• , or (with a different objective function) between 23• and 136• .

Optimizing Voronoi Diagrams for Polygonal Finite Element Computations

“…To this end, a number of very successful mesh improvement approaches have been proposed, being based on either Delaunay refinement [28,29,9] or some variational optimization [1,38,22,39,10,41]. So-called pliant methods, which combine local topological changes (e.g., Delaunay refinement) and vertex relocation (e.g., Laplacian smoothing or Lloyd relaxation) have been found to be superior over methods involving one of the techniques only [5,22,38].…”

Quality encoding for tetrahedral mesh optimization

“…Minimizing the objective function through vertex repositioning leads to improvement in those mesh properties. Combining the two strategies can often result in higher quality meshes [19,22].…”

“…Furthermore, previous methods [12,13,17] often result in bad boundary tetrahedra since boundary vertices are fixed for boundary conformation and hence boundary tetrahedra benefit less from the optimization than interior ones. Some methods [19,22] relax the problem to approximately preserve the boundary shape. They employ constrained smoothing of boundary vertices, in which a boundary vertex can be moved within a common plane or edge of its neighbor vertices to avoid boundary shape distortion [19].…”

“…They employ constrained smoothing of boundary vertices, in which a boundary vertex can be moved within a common plane or edge of its neighbor vertices to avoid boundary shape distortion [19]. However, curved boundaries can not benefit from such an approach as few neighboring vertices share a common plane or edge [22].…”

“…Some methods [13,17,16,23,28] choose to fix all boundary vertices during optimization of tetrahedral element quality, which constrains the processing of boundary tetrahedra. Other methods [19,22] optimize the boundary tetrahedra by repositioning boundary vertices or by changing boundary mesh connectivity under necessary constraints. However, these methods can still produce degenerate boundary tetrahedra.…”