Lattice Cleaving: Conforming Tetrahedral Meshes of Multimaterial Domains with Bounded Quality

Abstract

Summary We introduce a new algorithm for generating tetrahedral meshes that conform to physical boundaries in volumetric domains consisting of multiple materials. The proposed method allows for an arbitrary number of materials, produces high-quality tetrahedral meshes with upper and lower bounds on dihedral angles, and guarantees geometric fidelity. Moreover, the method is combinatoric so its implementation enables rapid mesh construction. These meshes are structured in a way that also allows grading, in order to reduce element counts in regions of homogeneity.

An algorithm to mesh interconnected surfaces via the Voronoi interface

“…Combined with a clamping procedure, it was shown that small edges in the initial mesh finish expanding in as few as two to ten iterations, after which the remaining iterations are largely in improving mesh quality globally, where vertices spread apart due to the expansion factor in the rest edge length ' 0 . It may be possible to analyse convergence more rigorously to establish quality bounds, similar to [19,20,21], although such analysis would be subtle, due to the intricate coupling of edge flipping, vertex movement, and clamping. In an extensive application of the meshing algorithm in [1], tens of thousands of meshes were automatically generated and 123 successfully used in finite element methods involving coupled boundary conditions at junctions.…”

“…-Volumetric meshing with guaranteed quality: Another possibility is to mesh the volumes of the individual regions/phases as a tetrahedral mesh, and then extract surface meshes from the boundaries of the volumetric meshes. In the lattice cleaving algorithm [19] (which shares aspects with the single-region isosurface stuffing algorithm of Labelle and Shewchuk [20]), a bodycentred cubic lattice is ''cleaved'', or cut, based on the approximate location of the material boundaries. In particular, vertices at the location of the cuts are ''warped'' by a procedure that guarantees mesh quality, such that minimum dihedral angles are bounded from below.…”

“…2 (right), grid points precisely halfway between C and D, D and E, E and F, F and C, etc., which exist on the background Cartesian grid, are not members of the tessellation. The BCC lattice has been widely used in the literature (see for instance [8,19,20,25]) as the tetrahedra have identical shapes and sizes, are more evenly distributed, and lead to fewer grid-dependent effects, as compared to the 6-tetrahedron split in the standard marching tetrahedra algorithm. These effects, as well as other possible subdivisions of Cartesian grids into tetrahedra, are extensively analysed in [34].…”

Multi-material adaptive volume remesher

“…The isosurface stuffing method [17] uses a similar paradigm to represent single material domains but guarantees theoretical bounds on the tetrahedra' dihedral angles. Inspired by this strategy, Bronson et al [18] offer similar guarantees for labeled volume data.…”

Multiphase surface tracking with explicit contouring

“…Dey et al [2012] applied a recent Delaunay refinement algorithm to generate high quality triangular interface surfaces. Bronson et al [2013] introduced a new algorithm for generating tetrahedral meshes that conform to volumetric domains of multiple materials. Saye and Sethian [2012] extracted the multiphase interfaces with piecewise linear interpolation and further quality-improved [Saye 2013] under the framework of VIIM, which requires the continuous piecewise interpolation of every distance function and mesh abstraction and chopping.…”