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Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings

Caplan, Philip Claude, Robert Haimes, David L. Caplan, Marshall C. Galbraith

Proceedings, 26th International Meshing Roundtable, Elsevier, Science Direct, September 18-21 2017

INTERNATIONAL
MESHING
ROUNTABLE

26th International Meshing Roundtable
Barcelona, Spain
September 18-21, 2017

Philip Claude Caplan, Massachusetts Institute of Technology, US, pcaplan@mit.edu
Robert Haimes, Massachusetts Institute of Technology, US, haimes@mit.edu
David L. Caplan, Massachusetts Institute of Technology, US, darmofal@mit.edu
Marshall C. Galbraith, Massachusetts Institute of Technology, US, galbramc@mit.edu

Abstract
We develop a dimension-independent, Delaunay-based anisotropic mesh generation algorithm suitable for integration with adaptive numerical solvers. As such, the mesh produced by our algorithm conforms to an anisotropic metric prescribed by the solver as well as the domain geometry, given as a piecewise simplicial complex. Motivated by the work of L\'evy and Dassi \cite{Levy_2013_Vorpaline,Dassi_2014_Curvature_adapted_remeshing_of_CAD_surfaces,Dassi_2015_Anisotropic_Finite_Element_Mesh_Adaptation_via_Higher_Dimensional_Embedding,Dassi_2016_An_anisotropic_surface_remeshing_strategy_combining_higher_dimensional_embedding_with_radial_basis_functions}, we use a discrete manifold embedding algorithm to transform the anisotropic problem to a uniform one. This work differs from previous approaches in several ways. First, the embedding algorithm is driven by a Riemannian metric field instead of the Gauss map, lending itself to general anisotropic mesh generation problems. Second we describe our method for computing restricted Voronoi diagrams in a dimension-independent manner which is used to compute constrained centroidal Voronoi tessellations. In particular, we compute restricted Voronoi simplices using exact arithmetic and support our data structures using convex polytope theory. Finally, since adaptive solvers require geometry-conforming meshes, we offer a Steiner vertex insertion algorithm for ensuring the extracted dual Delaunay triangulation is homeomorphic to the input geometries.\newline The two major contributions of this paper are a method for isometrically embedding arbitrary mesh-metric pairs in higher dimensional Euclidean spaces along with a dimension-independent vertex insertion algorithm for producing geometry-conforming Delaunay meshes. The former is demonstrated on a two-dimensional anisotropic problem whereas the latter is demonstrated on both $3d$ and $4d$ problems.\newline

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