Quality tetrahedral mesh smoothing via boundary-optimized Delaunay triangulation

Abstract

Despite its great success in improving the quality of a tetrahedral mesh, the original optimal Delaunay triangulation (ODT) is designed to move only inner vertices and thus cannot handle input meshes containing “bad” triangles on boundaries. In the current work, we present an integrated approach called boundary-optimized Delaunay triangulation (B-ODT) to smooth (improve) a tetrahedral mesh. In our method, both inner and boundary vertices are repositioned by analytically minimizing the error between a paraboloid function and its piecewise linear interpolation over the neighborhood of each vertex. In addition to the guaranteed volume-preserving property, the proposed algorithm can be readily adapted to preserve sharp features in the original mesh. A number of experiments are included to demonstrate the performance of our method.

Propagating Updates of Residential Areas in Multi-Representation Databases Using Constrained Delaunay Triangulations

“…There are three operations to meet cartographical constraints during the construction of small-scale objects: (1) Exclusion of unsuitable triangles: unsuitable triangles (e.g., small long narrow triangles connected to the main polygon) are possibly introduced when building or trimming constrained Delaunay triangulation [34]. If the triangle has a very small interior angle and if it does not share a common side with other triangles or it only shares one very short side with other triangles, the corresponding triangle is considered as an unsuitable triangle and is excluded during the construction of the small-scale objects; (2) Rectangularity: given a near-rectangular interior angle, rectangularity is to convert the interior angle to a rectangle, which is done by drawing a vertical line from the corresponding vertex that connects to the shorter side to the longer side and regenerating the shape of the polygon; (3) Elimination of small objects: if the object in a topographic map has an area smaller than the minimum representable area, the corresponding object is removed because we do not consider the operation of exaggeration in the current study.…”

“…There are three operations to meet cartographical constraints during the construction of smallscale objects: (1) Exclusion of unsuitable triangles: unsuitable triangles (e.g., small long narrow triangles connected to the main polygon) are possibly introduced when building or trimming constrained Delaunay triangulation [34]. If the triangle has a very small interior angle and if it does not share a common side with other triangles or it only shares one very short side with other triangles, the corresponding triangle is considered as an unsuitable triangle and is excluded during the construction of the small-scale objects.…”

Curved optimal delaunay triangulation

“…Dirichlet conditions using a fixed sampling of the boundary of Ω produces nicely-shaped tetrahedra throughout the domain, but generates many degenerate elements near ∂Ω (in particular, "slivers" in 3D, i.e., almost flat tets with nearly equal edge lengths) as boundary vertices are not optimized, preventing the formation of equilateral elements. Reprojecting boundary vertices [Alliez et al 2005] or letting them slide along the boundary by only taking the tangential part of the ODT gradient [Chen and Holst 2011;Gao et al 2012] improves the resulting meshes, but leaves degenerate elements as vertices that reach the boundary tend to stay on the boundary. The work of [Tournois et al 2009] exploits a degree of freedom in the boundary terms of the update of a vertex to enforce a "circumsphere property" on ∂Ω.…”

“…Depending on the application for which a mesh is constructed, vertices may have to be precisely on the boundary, or the mesh boundary just needs to approximate ∂Ω. While the former can be achieved using methods as in [Alliez et al 2005;Chen et al 2014;Gao et al 2012], we derive a simple approach to the latter by exploiting the stability of our non-shrinking minimization.…”

“…Its implementation in 3D along with details on the sizing field computations and boundary handling was studied by Alliez et al [2005], and a hybrid approach mixing Delaunay refinement and ODT optimization was later introduced [Tournois et al 2009] to accelerate convergence and improve results. Recent improvements include an alternative boundary treatment [Gao et al 2012] and various numerical accelerations [Chen and Holst 2011;Chen et al 2014]. Today, the isotropic meshes obtained via ODT optimization consistently outperform other approaches (based on advancing front, bubble packing, etc) in terms of element quality (dihedral angles, aspect ratio, etc) for a given vertex budget.…”