Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

We construct efficient quadratures for the integration of polynomials over irregular convex polygons and polyhedrons based on moment fitting equations. The quadrature construction scheme involves the integration of monomial basis functions, which is performed using homogeneous quadratures with minimal number of integration points, and the solution of a small linear system of equations. The construction of homogeneous quadratures is based on Lasserre’s method for the integration of homogeneous functions over convex polytopes. We also construct quadratures for the integration of discontinuous functions without the need to partition the domain into triangles or tetrahedrons. Several examples in two and three dimensions are presented that demonstrate the accuracy and versatility of the proposed method.

[1]  Joseph E. Bishop,et al.  Simulating the pervasive fracture of materials and structures using randomly close packed Voronoi tessellations , 2009 .

[2]  F. Brezzi,et al.  A FAMILY OF MIMETIC FINITE DIFFERENCE METHODS ON POLYGONAL AND POLYHEDRAL MESHES , 2005 .

[3]  Markus H. Gross,et al.  Polyhedral Finite Elements Using Harmonic Basis Functions , 2008, Comput. Graph. Forum.

[4]  Jean B. Lasserre,et al.  Integration on a convex polytope , 1998 .

[5]  P. M. Gullett,et al.  On a finite element method with variable element topology , 2000 .

[6]  Gianmarco Manzini,et al.  Mimetic finite difference method for the Stokes problem on polygonal meshes , 2009, J. Comput. Phys..

[7]  Konstantin Avrachenkov,et al.  The Multi-Dimensional Version of ∫ba xp dx , 2001, Am. Math. Mon..

[8]  Zydrunas Gimbutas,et al.  A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions , 2010, Comput. Math. Appl..

[9]  Sundararajan Natarajan,et al.  Integrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework , 2010, 1107.4732.

[10]  Gautam Dasgupta,et al.  Integration within Polygonal Finite Elements , 2003 .

[11]  H. Timmer,et al.  Computation of global geometric properties of solid objects , 1980 .

[12]  N. Sukumar,et al.  Generalized Gaussian Quadrature Rules for Discontinuities and Crack Singularities in the Extended Finite Element Method , 2010 .

[13]  N. Sukumar,et al.  Generalized Gaussian quadrature rules on arbitrary polygons , 2010 .

[14]  N. Sukumar,et al.  Conforming polygonal finite elements , 2004 .

[15]  S. Wandzurat,et al.  Symmetric quadrature rules on a triangle , 2003 .

[16]  G. Ventura On the elimination of quadrature subcells for discontinuous functions in the eXtended Finite‐Element Method , 2006 .

[17]  Brian Mirtich,et al.  Fast and Accurate Computation of Polyhedral Mass Properties , 1996, J. Graphics, GPU, & Game Tools.

[18]  T. Belytschko,et al.  MODELING HOLES AND INCLUSIONS BY LEVEL SETS IN THE EXTENDED FINITE-ELEMENT METHOD , 2001 .

[19]  Fausto Bernardini,et al.  Integration of polynomials over n-dimensional polyhedra , 1991, Comput. Aided Des..

[20]  K. Sunder,et al.  Integration points for triangles and tetrahedrons obtained from the gaussian quadrature points for a line , 1985 .

[21]  Marcel Vinokur,et al.  Exact Integrations of Polynomials and Symmetric Quadrature Formulas over Arbitrary Polyhedral Grids , 1998 .

[22]  Ted Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[23]  Markus H. Gross,et al.  Eurographics/ Acm Siggraph Symposium on Computer Animation (2008) Flexible Simulation of Deformable Models Using Discontinuous Galerkin Fem , 2022 .

[24]  P. Milbradt,et al.  Polytope finite elements , 2008 .

[25]  Glaucio H. Paulino,et al.  Polygonal finite elements for topology optimization: A unifying paradigm , 2010 .

[26]  D. A. Dunavant High degree efficient symmetrical Gaussian quadrature rules for the triangle , 1985 .

[27]  Jean B. Lasserre,et al.  Integration and homogeneous functions , 1999 .

[28]  James N. Lyness,et al.  Moderate degree symmetric quadrature rules for the triangle j inst maths , 1975 .

[29]  L. J. Comrie,et al.  Mathematical Tables and Other Aids to Computation. , 1946 .

[30]  H. Rathod,et al.  Integration of polynomials over n-dimensional linear polyhedra , 1997 .

[31]  Carlo Cattani,et al.  Boundary integration over linear polyhedra , 1990, Comput. Aided Des..

[32]  Markus H. Gross,et al.  A Finite Element Method on Convex Polyhedra , 2007, Comput. Graph. Forum.

[33]  John E. Bolander,et al.  Automated Modeling of Three‐Dimensional Structural Components Using Irregular Lattices , 2005 .

[34]  P. Silvester,et al.  Symmetric Quadrature Formulae for Simplexes , 1970 .

[35]  Jesús A. De Loera,et al.  How to integrate a polynomial over a simplex , 2008, Math. Comput..

[36]  E. Wachspress,et al.  A Rational Finite Element Basis , 1975 .

[37]  David R. Noble,et al.  Quadrature rules for triangular and tetrahedral elements with generalized functions , 2008 .

[38]  O. J. Marlowe,et al.  Numerical integration over simplexes and cones , 1956 .

[39]  M. M. Rashid,et al.  A three‐dimensional finite element method with arbitrary polyhedral elements , 2006 .

[40]  Giovanni Monegato,et al.  Quadrature Rules for Regions Having Regular Hexagonal Symmetry , 1977 .

[41]  Stefan Vandewalle,et al.  Exact Integration Formulas for the Finite Volume Element Method on Simplicial Meshes , 2007 .

[42]  P. Keast Moderate-degree tetrahedral quadrature formulas , 1986 .

[43]  Thomas-Peter Fries,et al.  Higher‐order XFEM for curved strong and weak discontinuities , 2009 .

[44]  Gianmarco Manzini,et al.  Error Analysis for a Mimetic Discretization of the Steady Stokes Problem on Polyhedral Meshes , 2010, SIAM J. Numer. Anal..

[45]  N. Sukumar,et al.  Generalized Duffy transformation for integrating vertex singularities , 2009 .