Universal meshes for smooth surfaces with no boundary in three dimensions

• Published in: International journal for numerical methods in engineering Vol. 110; no. 2; pp. 133 - 162
• Main Authors: Kabaria, Hardik, Lew, Adrian J.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 13-04-2017; Wiley Subscription Services, Inc
• Subjects: Approximation; arbitrary Lagrangian–Eulerian; Boundaries; Collection; immersed boundary; Mathematical analysis; Mathematical models; mesh motion; moving boundary; Projection; Showcases; Tetrahedrons
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.5350

Summary
We introduce a method to mesh the boundary Γ of a smooth, open domain in
R3 immersed 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 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.