High‐order cubature rules for tetrahedra

Jaśkowiec, Jan; Sukumar, N.

doi:10.1002/nme.6313

High‐order cubature rules for tetrahedra

Journal Article · Wed Jan 29 00:00:00 EST 2020 · International Journal for Numerical Methods in Engineering

DOI:https://doi.org/10.1002/nme.6313· OSTI ID:1596161

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Faculty of Civil Engineering Cracow University of Technology Cracow Poland
Department of Civil and Environmental Engineering University of California Davis California

Summary

In this paper, we construct new high‐order numerical integration schemes for tetrahedra, with positive weights and integration points that are in the interior of the domain. The construction of cubature rules is a challenging problem, which requires the solution of strongly nonlinear algebraic (moment) equations with side conditions given by affine inequality constraints. We present a robust algorithm based on a sequence of three modified Newton procedures to solve the constrained minimization problem. In the literature, numerical integration rules for the tetrahedron are available up to order p =15. We obtain integration rules for the tetrahedron from p =2 to p =20, which are computed using multiprecision arithmetic. For p ≤15, our approach provides integration rules that have the same or fewer number of integration points than existing rules; for p =16 to p =20, our rules are new. Numerical tests are presented that verify the polynomial‐precision of the cubature rules. Convergence studies are performed for the integration of exponential, rational, weakly singular and trigonometric test functions over tetrahedra with flat and curved faces. In all tests, improvements in accuracy is realized as p is increased, though in some cases nonmonotonic convergence is observed.

View Accepted Manuscript (Publisher)

Sponsoring Organization:: USDOE

OSTI ID:: 1596161

Journal Information:: International Journal for Numerical Methods in Engineering, Journal Name: International Journal for Numerical Methods in Engineering Journal Issue: 11 Vol. 121; ISSN 0029-5981

Publisher:: Wiley Blackwell (John Wiley & Sons)Copyright Statement

Country of Publication:: United Kingdom

Language:: English

References (32)

Gaussian quadrature formulas for triangles Cowper, G. R. International Journal for Numerical Methods in Engineering, Vol. 7, Issue 3 https://doi.org/10.1002/nme.1620070316	journal	January 1973
Numerical integration on a triangle Hillion, Pierre International Journal for Numerical Methods in Engineering, Vol. 11, Issue 5 https://doi.org/10.1002/nme.1620110504	journal	January 1977
Some criteria for numerically integrated matrices and quadrature formulas for triangles Laursen, M. E.; Gellert, M. International Journal for Numerical Methods in Engineering, Vol. 12, Issue 1 https://doi.org/10.1002/nme.1620120107	journal	January 1978
High degree efficient symmetrical Gaussian quadrature rules for the triangle Dunavant, D. A. International Journal for Numerical Methods in Engineering, Vol. 21, Issue 6 https://doi.org/10.1002/nme.1620210612	journal	June 1985
Generalized Gaussian quadrature rules on arbitrary polygons Mousavi, S. E.; Xiao, H.; Sukumar, N. International Journal for Numerical Methods in Engineering https://doi.org/10.1002/nme.2759	journal	January 2009
Rational Bases and Generalized Barycentrics: Applications to Finite Elements and Graphics Wachspress, Eugene https://doi.org/10.1007/978-3-319-21614-0	book	October 2015
Numerical integration on a tetrahedron Hillion, P. Calcolo, Vol. 18, Issue 2 https://doi.org/10.1007/BF02576492	journal	June 1981
Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra Chin, Eric B.; Lasserre, Jean B.; Sukumar, N. Computational Mechanics, Vol. 56, Issue 6 https://doi.org/10.1007/s00466-015-1213-7	journal	October 2015
Symmetric gaussian quadrature formulae for tetrahedronal regions Yu, Jinyun Computer Methods in Applied Mechanics and Engineering, Vol. 43, Issue 3 https://doi.org/10.1016/0045-7825(84)90072-0	journal	May 1984
Moderate-degree tetrahedral quadrature formulas Keast, Patrick Computer Methods in Applied Mechanics and Engineering, Vol. 55, Issue 3 https://doi.org/10.1016/0045-7825(86)90059-9	journal	May 1986
Monomial cubature rules since “Stroud”: a compilation Cools, Ronald; Rabinowitz, Philip Journal of Computational and Applied Mathematics, Vol. 48, Issue 3 https://doi.org/10.1016/0377-0427(93)90027-9	journal	November 1993
Rotation invariant cubature formulas over the n-dimensional unit cube Cools, Ronald; Kim, K. J. Journal of Computational and Applied Mathematics, Vol. 132, Issue 1 https://doi.org/10.1016/S0377-0427(00)00595-1	journal	July 2001
Monomial cubature rules since “Stroud”: a compilation — part 2 Cools, Ronald Journal of Computational and Applied Mathematics, Vol. 112, Issue 1-2 https://doi.org/10.1016/S0377-0427(99)00229-0	journal	November 1999
Symmetric quadrature rules on a triangle Wandzurat, S.; Xiao, H. Computers & Mathematics with Applications, Vol. 45, Issue 12 https://doi.org/10.1016/S0898-1221(03)90004-6	journal	June 2003
Symmetric Gauss Legendre quadrature formulas for composite numerical integration over a triangular surface Rathod, H. T.; Nagaraja, K. V.; Venkatesudu, B. Applied Mathematics and Computation, Vol. 188, Issue 1 https://doi.org/10.1016/j.amc.2006.10.041	journal	May 2007
Symmetric quadrature rules for tetrahedra based on a cubic close-packed lattice arrangement Shunn, Lee; Ham, Frank Journal of Computational and Applied Mathematics, Vol. 236, Issue 17 https://doi.org/10.1016/j.cam.2012.03.032	journal	November 2012
Symmetric quadrature rules for simplexes based on sphere close packed lattice arrangements Williams, D. M.; Shunn, L.; Jameson, A. Journal of Computational and Applied Mathematics, Vol. 266 https://doi.org/10.1016/j.cam.2014.01.007	journal	August 2014
New fully symmetric and rotationally symmetric cubature rules on the triangle using minimal orthonormal bases Papanicolopulos, Stefanos-Aldo Journal of Computational and Applied Mathematics, Vol. 294 https://doi.org/10.1016/j.cam.2015.08.001	journal	March 2016
A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions Xiao, Hong; Gimbutas, Zydrunas Computers & Mathematics with Applications, Vol. 59, Issue 2 https://doi.org/10.1016/j.camwa.2009.10.027	journal	January 2010
Computation of moderate-degree fully-symmetric cubature rules on the triangle using symmetric polynomials and algebraic solving Papanicolopulos, Stefanos-Aldo Computers & Mathematics with Applications, Vol. 69, Issue 7 https://doi.org/10.1016/j.camwa.2015.02.014	journal	April 2015
On the identification of symmetric quadrature rules for finite element methods Witherden, F. D.; Vincent, P. E. Computers & Mathematics with Applications, Vol. 69, Issue 10 https://doi.org/10.1016/j.camwa.2015.03.017	journal	May 2015
p-FEMs in biomechanics: Bones and arteries Yosibash, Zohar Computer Methods in Applied Mechanics and Engineering, Vol. 249-252 https://doi.org/10.1016/j.cma.2012.09.006	journal	December 2012
New computationally efficient quadrature formulas for triangular prism elements Kubatko, Ethan J.; Yeager, Benjamin A.; Maggi, Ashley L. Computers & Fluids, Vol. 73 https://doi.org/10.1016/j.compfluid.2013.01.002	journal	March 2013
New computationally efficient quadrature formulas for pyramidal elements Kubatko, Ethan J.; Yeager, Benjamin A.; Maggi, Ashley L. Finite Elements in Analysis and Design, Vol. 65 https://doi.org/10.1016/j.finel.2012.10.012	journal	March 2013
On the construction of general cubature formula by flat extensions Abril Bucero, Marta; Bajaj, Chandrajit; Mourrain, Bernard Linear Algebra and its Applications, Vol. 502 https://doi.org/10.1016/j.laa.2015.09.052	journal	August 2016
Symmetric quadrature formulae for simplexes Silvester, P. Mathematics of Computation, Vol. 24, Issue 109 https://doi.org/10.1090/S0025-5718-1970-0258283-6	journal	January 1970
Moderate Degree Symmetric Quadrature Rules for the Triangle Lyness, J. N.; Jespersen, D. IMA Journal of Applied Mathematics, Vol. 15, Issue 1 https://doi.org/10.1093/imamat/15.1.19	journal	January 1975
A compendium of FEM integration formulas for symbolic work Felippa, Carlos A. Engineering Computations, Vol. 21, Issue 8 https://doi.org/10.1108/02644400410554362	journal	December 2004
Invariant Integration Formulas for the n -Simplex by Combinatorial Methods Grundmann, Axel; Möller, H. M. SIAM Journal on Numerical Analysis, Vol. 15, Issue 2 https://doi.org/10.1137/0715019	journal	April 1978
Weight-Adjusted Discontinuous Galerkin Methods: Curvilinear Meshes Chan, Jesse; Hewett, Russell J.; Warburton, T. SIAM Journal on Scientific Computing, Vol. 39, Issue 6 https://doi.org/10.1137/16M1089198	journal	January 2017
A Reflective Newton Method for Minimizing a Quadratic Function Subject to Bounds on Some of the Variables Coleman, Thomas F.; Li, Yuying SIAM Journal on Optimization, Vol. 6, Issue 4 https://doi.org/10.1137/S1052623494240456	journal	November 1996
Decoupling simulation accuracy from mesh quality Schneider, Teseo; Hu, Yixin; Dumas, Jérémie ACM Transactions on Graphics, Vol. 37, Issue 6 https://doi.org/10.1145/3272127.3275067	journal	December 2018

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