Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings

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 smooth complex. Motivated by the work of Levy and Dassi [10-12,20], 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 use data structures based on 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. The two major contributions of this paper are: a method for isometrically embedding arbitrary mesh-metric pairs in higher dimensional Euclidean spaces and 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 3 d and 4 d problems.

[1]  Mael Rouxel-Labbé.  Anisotropic mesh generation , 2016 .

[2]  Philip Claude Delhaye Caplan,et al.  An adaptive framework for high-order, mixed-element numerical simulations , 2014 .

[3]  Joshua A. Levine,et al.  A Practical Delaunay Meshing Algorithm for aLarge Class of Domains* , 2007, IMR.

[4]  Jean-Daniel Boissonnat,et al.  An Obstruction to Delaunay Triangulations in Riemannian Manifolds , 2018, Discret. Comput. Geom..

[5]  N. Kuiper,et al.  On C1-isometric imbeddings. II , 1955 .

[6]  Tamal K. Dey,et al.  Delaunay Refinement for Piecewise Smooth Complexes , 2007, SODA '07.

[7]  Todd R. Michal,et al.  Anisotropic Mesh Adaptation Through Edge Primitive Operations , 2012 .

[8]  Jonathan Richard Shewchuk,et al.  Incrementally constructing and updating constrained Delaunay tetrahedralizations with finite-precision coordinates , 2013, Engineering with Computers.

[9]  Nikos Chrisochoides,et al.  4D space–time Delaunay meshing for medical images , 2015, Engineering with Computers.

[10]  Chenglei Yang,et al.  On centroidal voronoi tessellation—energy smoothness and fast computation , 2009, TOGS.

[11]  Alexander M. Bronstein,et al.  Numerical Geometry of Non-Rigid Shapes , 2009, Monographs in Computer Science.

[12]  Franco Dassi,et al.  Curvature-adapted Remeshing of CAD Surfaces☆ , 2014 .

[13]  Frédéric Alauzet,et al.  A decade of progress on anisotropic mesh adaptation for computational fluid dynamics , 2016, Comput. Aided Des..

[14]  B. Lévy,et al.  L p Centroidal Voronoi Tessellation and its applications , 2010, SIGGRAPH 2010.

[15]  Darren Engwirda,et al.  Conforming restricted Delaunay mesh generation for piecewise smooth complexes , 2016, ArXiv.

[16]  Tamal K. Dey,et al.  Localized Delaunay Refinement for Volumes , 2011, Comput. Graph. Forum.

[17]  Adrien Loseille,et al.  Cavity-Based Operators for Mesh Adaptation , 2013 .

[18]  Brielin C. Brown,et al.  Fast segment insertion and incremental construction of constrained Delaunay triangulations , 2015, Comput. Geom..

[19]  Eric O. Postma,et al.  Dimensionality Reduction: A Comparative Review , 2008 .

[20]  B. Lévy,et al.  Lp Centroidal Voronoi Tessellation and its applications , 2010, ACM Trans. Graph..

[21]  Franco Dassi,et al.  An Anisoptropic Surface Remeshing Strategy Combining Higher Dimensional Embedding with Radial Basis Functions , 2016 .

[22]  H. Borouchaki,et al.  Adaptive triangular–quadrilateral mesh generation , 1998 .

[23]  Tamal K. Dey,et al.  Delaunay Mesh Generation , 2012, Chapman and Hall / CRC computer and information science series.

[24]  Bruno Lévy,et al.  Variational Anisotropic Surface Meshing with Voronoi Parallel Linear Enumeration , 2012, IMR.

[25]  Franco Dassi,et al.  Curvature-adapted remeshing of CAD surfaces , 2018, Engineering with Computers.

[26]  Bruno Lévy,et al.  Robustness and efficiency of geometric programs: The Predicate Construction Kit (PCK) , 2016, Comput. Aided Des..

[27]  J. Nash.  C 1 Isometric Imbeddings , 1954 .

[28]  Simona Perotto,et al.  Anisotropic finite element mesh adaptation via higher dimensional embedding , 2015 .

[29]  Jonathan Richard Shewchuk,et al.  Constrained Delaunay Tetrahedralizations and Provably Good Boundary Recovery , 2002, IMR.

[30]  Frédéric Alauzet,et al.  Size gradation control of anisotropic meshes , 2010 .

[31]  Marina Meila,et al.  Nearly Isometric Embedding by Relaxation , 2016, NIPS.

[32]  Jonathan Richard Shewchuk,et al.  Updating and constructing constrained delaunay and constrained regular triangulations by flips , 2003, SCG '03.

[33]  Adrien Loseille,et al.  Metric-orthogonal Anisotropic Mesh Generation☆ , 2014 .

[34]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[35]  Jonathan Richard Shewchuk,et al.  A condition guaranteeing the existence of higher-dimensional constrained Delaunay triangulations , 1998, SCG '98.

[36]  U. Diego.  Towards an Algorithmic Realization of Nash ’ s Embedding Theorem , 2011 .

[37]  D. Engwirda.  Voronoi-Based Point-Placement for Three-Dimensional Delaunay-Refinement , 2015 .

[38]  G. Ziegler,et al.  Basic properties of convex polytopes , 1997 .

[39]  J. Shewchuk,et al.  Incrementally constructing and updating constrained Delaunay tetrahedralizations with finite-precision coordinates , 2013, Engineering with Computers.