Meshing 3D Domains Bounded by Piecewise Smooth Surfaces*

Abstract

This paper proposes an algorithm to mesh 3D domains bounded by piecewise smooth surfaces. The algorithm handle multivolume domains defined by surfaces that may be non connected or non manifold. The boundary and subdivision surfaces are assumed to be described by a complex formed by surface patches stitched together along curve segments. The meshing algorithm is a Delaunay refinement and it uses the notion of restricted Delaunay triangulation to approximate the input curve segments and surface patches. The algorithm yields a mesh with good quality tetrahedra and offers a user control on the size of the tetrahedra. The vertices in the final mesh have a restricted Delaunay triangulation to any input feature which is a homeomorphic and accurate approximation of this feature. The algorithm also provides guarantee on the size and shape of the facets approximating the input surface patches. In its current state the algorithm suffers from a severe angular restriction on input constraints. It basically assumes that two linear subspaces that are tangent to non incident and non disjoint input features on a common point form an angle measuring at least 90 degrees.

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“…One way to achieve this goal is to keep the boundary elements while meshing the input domain after meshing or remeshing the input surface boundary. An alternative way is to operate the surface and volume mesh instantaneously by getting new boundary mesh through restricted Delaunay triangulation [53], which is a preferred solution for computational mesh generation.…”

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“…Due to its flexibility, we use the Delaunay refinement package [10] of CGAL 4 . In particular, it was possible to provide our own type for D and our own set of criteria to classify triangles as bad.…”

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“…Delaunay-based approaches [8,24] for meshing this domain have been able to provide topological correctness guarantees using either weighted Delaunay triangulations [8] or bounding the angle deviations between smooth patches [24]. A missing piece to the implementations of these algorithms is the ability to adapt to a sizing field, primarily because there is no consensus on what is the correct sizing field for non-smooth shapes and how best to compute it.…”