Guaranteed-Quality Simplical Mesh Generation with Cell Size and Grading Control

Abstract

Unstructured mesh quality, as measured geometrically, has long been known to influence solution accuracy and efficiency for finite-element and finite-volume simulations. Recent guaranteed-quality unstructured meshing algorithms are therefore welcome tools. However, these algorithms allow no explicit control over mesh resolution or grading. We define a geometric length scale, similar in principle to the local feature size, which allows automatic global control of mesh resolution and grading. We describe how to compute this length scale efficiently and modify Ruppert's two-dimensional and Shewchuk's three-dimensional meshing algorithms to produce meshes matching our length scale. We provide proofs of mesh quality, good grading, and size optimality for both two-and three-dimensions, and present examples, including comparison with existing schemes known to generate good-quality meshes.

3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids

“…We have used the 2D version of GRUMMP in our 2D AMPHI algorithm [20], and will summarize the main ideas of GRUMMP and emphasize features unique to 3D meshing. For further details on GRUMMP, interested readers may consult the work of Ollivier-Gooch et al [39,40] GRUMMP generates a mesh by using Delaunay refinement, and controls the spatial variation of grid size using a scalar field L S , which specifies the intended grid size at each location in the domain. In our study of interfacial dynamics, the grid size should be finest in the interfacial region, and gradually coarsens away from the interface.…”

“…GRUMMP produces tetrahedral elements in 3D based on L S , following the scheme of Shewchuk [41] but with several significant improvements in the areas of cell size and grading control [39]. It begins with enclosing the computational domain Ω inside a large box and implementing an initial tetrahedralization that incorporates all vertices on the domain boundary ∂Ω into the mesh.…”

Guaranteed‐quality triangular mesh generation for domains with curved boundaries

“…Shewchuk also introduced a generalization of Ruppert's algorithm to three dimensions which showed signiÿcantly better quality bounds than a previous 3D algorithm by Mitchell and Vavasis [15]. In previous research [16], we extended Ruppert's and Shewchuk's work to have better control over cell grading and size, in both 2D and 3D. The common downfall of these guaranteed-quality schemes is that they all require the domain to have linear (or planar) boundaries.…”

Coarsening unstructured meshes by edge contraction

“…For example, in their agglomeration algorithm, Venkatakrishnan and Mavriplis [16] construct a maximal independent set of approximately minimal size to reduce the amount of work required on the coarse meshes.…”

“…To ease the transition from anisotropic to isotropic coarse meshes, we set the length scale at all vertices at the end of marching lines to the maximum distance allowed between kept vertices along that line. After all marching lines have been ÿnished, the length scale for vertices near their ends is updated using a grading-based rule (see Reference [16]), and the con ict graph for vertices outside the PSAMF is updated.…”