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Universal meshes for smooth surfaces with no boundary in three dimensions
Author(s) -
Kabaria Hardik,
Lew Adrian J.
Publication year - 2016
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.5350
Subject(s) - polygon mesh , volume mesh , tetrahedron , boundary (topology) , mesh generation , geometry , mathematics , projection (relational algebra) , topology (electrical circuits) , surface (topology) , triangle mesh , face (sociological concept) , laplacian smoothing , t vertices , algorithm , finite element method , mathematical analysis , combinatorics , physics , social science , sociology , thermodynamics
Summary We introduce a method to mesh the boundary Γ of a smooth, open domain inR 3immersed in a mesh of tetrahedra. The mesh follows by mapping a specific collection of triangular faces in the mesh to Γ. Two types of surface meshes follow: (a) a mesh that exactly meshes Γ, and (b) meshes that approximate Γ to any order, by interpolating the map over the selected faces; that is, an isoparametric approximation to Γ. The map we use to deform the faces is the closest point projection to Γ. We formulate conditions for the closest point projection to define a homeomorphism between each face and its image. These are conditions on some of the tetrahedra intersected by the boundary, and they essentially state that each such tetrahedron should (a) have a small enough diameter, and (b) have two of its dihedral angles be acute. We provide explicit upper bounds on the mesh size, and these can be computed on the fly. We showcase the quality of the resulting meshes with several numerical examples. More importantly, all surfaces in these examples were meshed with a single background mesh. This is an important feature for problems in which the geometry evolves or changes, because it could be possible for the background mesh to never change as the geometry does. In this case, the background mesh would be a universal mesh [1][Rangarajan R, 2014] for all these geometries. We expect the method introduced here to be the basis for the construction of universal meshes for domains in three dimensions. Copyright © 2016 John Wiley & Sons, Ltd.