Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Wachspress EL (1975) A rational finite element basis. Academic Press, New York
Sukumar N, Tabarraei A (2004) Conforming polygonal finite elements. Int J Numer Methods Eng 61(12): 2045–2066
Bishop JE (2009) Simulating the pervasive fracture of materials and structures using randomly closed packed Voronoi tessellations. Comput Mech 44(4): 455–471
Talischi C, Paulino GH, Pereira A, Menezes IFM (2010) Polygonal finite elements for topology optimization: a unifying paradigm. Int J Numer Methods Eng 82(6): 671–698
Wicke M, Botsch M, Gross M (2007) A finite element method on convex polyhedra. Comput Graph Forum 26(3): 355–364
Martin S, Kaufmann P, Botsch M, Wicke M, Gross M (2008) Polyhedral finite elements using harmonic basis functions. Comput Graph Forum 27(5): 1521–1529
Milbradt P, Pick T (2008) Polytope finite elements. Int J Numer Methods Eng 73: 1811–1835
Rashid MM, Gullett PM (2000) On a finite element method with variable element topology. Comput Methods Appl Mech Eng 190(11–12): 1509–1527
Rashid MM, Selimotic M (2006) A three-dimensional finite element method with arbitrary polyhedral elements. Int J Numer Methods Eng 67: 226–252
Kaufmann P, Martin S, Botsch M, Gross M (2009) Flexible simulation of deformable models using discontinuous Galerkin FEM. Graph Models 71(4):153–167. Special Issue of ACM SIGGRAPH/Eurographics Symposium on Computer Animation 2008
Voitovich TV, Vandewalle S (2007) Exact integration formulas for the finite volume element method on simplicial meshes. Numer Methods Partial Differ Equ 23(5): 1059–1082
Brezzi F, Lipnikov K, Simoncini V (2005) A family of mimetic finite difference methods on polygonal and polyhedral meshes. Math Models Methods Appl Sci 15(10): 1533–1551
da Veiga LB, Gyrya V, Lipnikov K, Manzini G (2009) Mimetic finite difference method for the Stokes problem on polygonal meshes. J Comput Phys 228: 7215–7232
da Veiga LB, Lipnikov K, Manzini G (2010) Error analysis for a mimetic discretization of the steady Stokes problem on polyhedral meshes. SIAM J Numer Anal 48(4): 1419–1443
Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1): 131–150
Sukumar N, Chopp DL, Moës N, Belytschko T (2001) Modeling holes and inclusions by level sets in the extended finite-element method. Comput Methods Appl Mech Eng 190(46–47): 6183–6200
Lasserre JB (1998) Integration on a convex polytope. Proc Am Math Soc 126(8): 2433–2441
Lasserre JB (1999) Integration and homogeneous functions. Proc Am Math Soc 127(3): 813–818
Baldoni V, Berline N, De Loera JA, Köppe M, Vergne M (2010) How to integrate a polynomial over a simplex. Math Comput. doi:10.1090/S0025-5718-2010-02378-6
Hammer PC, Marlowe OJ, Stroud AH (1956) Numerical integration over simplexes and cones. Math Tables Other Aids Comput 10: 130–137
Liu Y, Vinokur M (1998) Exact integrations of polynomials and symmetric quadrature formulas over arbitrary polyhedral grids. J Comput Phys 140: 122–147
Lasserre JB, Avrachenkov KE (2001) The multi-dimensional version of \({\int_a^b x^p dx}\) . Am Math Mon 108(2): 151–154
Timmer HG, Stern JM (1980) Computation of global geometric properties of solid objects. Comput Aided Des 12(6): 301–304
Cattani C, Paoluzzi A (1990) Boundary integration over linear polyhedra. Comput Aided Des 22(2): 130–135
Bernardini F (1991) Integration of polynomials over n-dimensional polyhedra. Comput Aided Des 23(1): 51–58
Mirtich B (1996) Fast and accurate computation of polyhedral mass properties. J Graph GPU Game Tools 1(2): 31–50
Rathod HT, Govinda Rao HS (1997) Integration of polynomials over n-dimensional linear polyhedra. Comput Struct 65(6): 829–847
Dasgupta G (2003) Integration within polygonal finite elements. J Aerosp Eng 16(1): 9–18
Mousavi SE, Xiao H, Sukumar N (2010) Generalized Gaussian quadrature rules on arbitrary polygons. Int J Numer Methods Eng 82(1): 99–113
Xiao H, Gimbutas Z (2010) A numerical algorithm for the construction of efficient quadratures in two and higher dimensions. Comput Math Appl 59: 663–676
Mousavi SE, Sukumar N (2010) Generalized Duffy transformation for integrating vertex singularities. Comput Mech 45(2–3): 127–140
Mousavi SE, Sukumar N (2010) Generalized Gaussian quadrature rules for discontinuities and crack singularities in the extended finite element method. Comput Methods Appl Mech Eng 199(49–52): 3237–3249
Lyness JN, Jespersen D (1975) Moderate degree symmetric quadrature rules for the triangle. J Inst Math Appl 15: 19–32
Lyness JN, Monegato G (1977) Quadrature rules for regions having regular hexagonal symmetry. SIAM J Numer Anal 14(2): 283–295
Dunavant DA (1985) High degree efficient symmetrical Gaussian quadrature rules for the triangle. Int J Numer Methods Eng 21: 1129–1148
Wandzura S, Xiao H (2003) Symmetric quadrature rules on a triangle. Comput Math Appl 45: 1829–1840
Ventura G (2006) On the elimination of quadrature subcells for discontinuous functions in the eXtended finite-element method. Int J Numer Methods Eng 66: 761–795
Holdych DJ, Noble DR, Secor RB (2008) Quadrature rules for triangular and tetrahedral elements with generalized functions. Int J Numer Methods Eng 73: 1310–1327
Natarajan S, Mahapatra DR, Bordas SPA (2010) Integrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework. Int J Numer Methods Eng 83: 269–294
Cheng KW, Fries TP (2010) Higher-order XFEM for curved strong and weak discontinuities. Int J Numer Methods Eng 82: 564–590
Silvester P (1970) Symmetric quadrature formulae for simplexes. Math Comput 24(109): 95–100
Sunder KS, Cookson RA (1985) Integration points for triangles and tetrahedrons obtained from the Gaussian quadrature points for a line. Comput Struct 21(5): 881–885
Keast P (1986) Moderate-degree tetrahedral quadrature formulas. Comput Methods Appl Mech Eng 55: 339–348
Yip M, Mohle J, Bolander JE (2005) Automated modeling of three-dimensional structural components using irregular lattices. Comput-Aided Civil Infrastruct Eng 20: 393–407
Acknowledgments
The research support of the National Science Foundation through contract grants CMMI-0626481 and DMS-0811025 to the University of California at Davis is gratefully acknowledged. Helpful discussions with Matthias Köppe are also acknowledged.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
National Science Foundation, CMMI-0626481, DMS-0811025.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Mousavi, S.E., Sukumar, N. Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons. Comput Mech 47, 535–554 (2011). https://doi.org/10.1007/s00466-010-0562-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-010-0562-5