Numerical convergence of discrete exterior calculus on arbitrary surface meshes

ABSTRACT Discrete exterior calculus (DEC) is a structure-preserving numerical framework for partial differential equations solution, particularly suitable for simplicial meshes. A longstanding and widespread assumption has been that DEC requires special (Delaunay) triangulations, which complicated the mesh generation process especially for curved surfaces. This paper presents numerical evidence demonstrating that this restriction is unnecessary. Convergence experiments are carried out for various physical problems using both Delaunay and non-Delaunay triangulations. Signed diagonal definition for the key DEC operator (Hodge star) is adopted. The errors converge as expected for all considered meshes and experiments. This relieves the DEC paradigm from unnecessary triangulation limitation.

[1]  Mathieu Desbrun,et al.  HOT: Hodge-optimized triangulations , 2011, ACM Trans. Graph..

[2]  M. Shashkov,et al.  Adjoint operators for the natural discretizations of the divergence gradient and curl on logically rectangular grids , 1997 .

[3]  Xiaoye S. Li,et al.  SuperLU Users'' Guide , 1997 .

[4]  Chandrajit L. Bajaj,et al.  Dual formulations of mixed finite element methods with applications , 2010, Comput. Aided Des..

[5]  Keenan Crane,et al.  Energy-preserving integrators for fluid animation , 2009, ACM Trans. Graph..

[6]  Anil N. Hirani,et al.  Comparison of discrete Hodge star operators for surfaces , 2016, Comput. Aided Des..

[7]  Anil N. Hirani,et al.  Corrigendum to "Delaunay Hodge star" [Comput. Aided Des. 45 (2013) 540-544] , 2018, Comput. Aided Des..

[8]  James Demmel,et al.  A Supernodal Approach to Sparse Partial Pivoting , 1999, SIAM J. Matrix Anal. Appl..

[9]  R. A. Nicolaides,et al.  Covolume techniques for anisotropic media , 1992 .

[10]  M. Shashkov,et al.  Natural discretizations for the divergence, gradient, and curl on logically rectangular grids☆ , 1997 .

[11]  Michael K. Adams,et al.  ( IL ) , 2012 .

[12]  Yiying Tong,et al.  Stable, circulation-preserving, simplicial fluids , 2006, SIGGRAPH Courses.

[13]  J. Cavendish,et al.  The dual variable method for solving fluid flow difference equations on Delaunay triangulations , 1991 .

[14]  T. A. Porsching,et al.  ON A NETWORK METHOD FOR UNSTEADY INCOMPRESSIBLE FLUID FLOW ON TRIANGULAR GRIDS , 1992 .

[15]  B. Perot.  Conservation Properties of Unstructured Staggered Mesh Schemes , 2000 .

[16]  Mark Meyer,et al.  Discrete Differential-Geometry Operators for Triangulated 2-Manifolds , 2002, VisMath.

[17]  Keenan Crane,et al.  Energy-preserving integrators for fluid animation , 2009, SIGGRAPH 2009.

[18]  J. Marsden,et al.  Groups of diffeomorphisms and the motion of an incompressible fluid , 1970 .

[19]  R. A. Nicolaides,et al.  Discretization of incompressible vorticity–velocity equations on triangular meshes , 1990 .

[20]  Anil N. Hirani,et al.  Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes , 2015, J. Comput. Phys..

[21]  L. Kettunen,et al.  Geometric interpretation of discrete approaches to solving magnetostatic problems , 2004, IEEE Transactions on Magnetics.

[22]  J. Cavendish,et al.  A complementary volume approach for modelling three‐dimensional Navier—Stokes equations using dual delaunay/voronoi tessellations , 1994 .

[23]  D. Arnold,et al.  Finite element exterior calculus: From hodge theory to numerical stability , 2009, 0906.4325.

[24]  B. Auchmann,et al.  A geometrically defined discrete hodge operator on simplicial cells , 2006, IEEE Transactions on Magnetics.

[25]  R. Nicolaides.  Direct discretization of planar div-curl problems , 1992 .

[26]  Anil N. Hirani,et al.  Delaunay Hodge star , 2012, Comput. Aided Des..

[27]  Anil N. Hirani,et al.  Discrete exterior calculus , 2005, math/0508341.

[28]  A. Bossavit.  Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements , 1997 .

[29]  Anil N. Hirani,et al.  Numerical Method for Darcy Flow Derived Using Discrete Exterior Calculus , 2008, ArXiv.

[30]  D. Schmidt,et al.  Accuracy and conservation properties of a three-dimensional unstructured staggered mesh scheme for fluid dynamics , 2002 .

[31]  R. A. Nicolaides,et al.  Covolume Discretization of Differential Forms , 2006 .

[32]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[33]  James C. Cavendish,et al.  SOLUTION OF INCOMPRESSIBLE NAVIER‐STOKES EQUATIONS ON UNSTRUCTURED GRIDS USING DUAL TESSELLATIONS , 1992 .

[34]  R. A. Nicolaides,et al.  Flow discretization by complementary volume techniques , 1989 .