Volume 4, Issue 1, March 2018, Page: 1-7
An Accurate Quadrature for the Numerical Integration of Polynomial Functions
Tahar Latrache, Department of Civil Engineering, University of Tebessa, Tebessa, Algeria
Received: Dec. 15, 2017;       Accepted: Jan. 8, 2018;       Published: Jan. 19, 2018
DOI: 10.11648/j.ijamtp.20180401.11      View  2189      Downloads  211
Abstract
The numerical integration of polynomial functions is one of the most interesting processes for numerical calculus and analyses, and represents thus, a compulsory step especially in finite elements analyses. Via the Gauss quadrature, the users concluded a great inconvenience that is processing at certain points which not required the based in finite element method points for deducting the form polynomials constants. In this paper, the same accuracy and efficiency as the Gauss quadrature extends for the numerical integration of the polynomial functions, but as such at the same points and nods have chosen for the determination of the form polynomials. Not just to profit from the values of the polynomials at those points and nods, but also from their first derivatives, the chosen points positions are arbitrary and the resulted deducted formulas are therefore different, as will be presented bellow and implemented.
Keywords
Integration Quadrature, Numerical Methods, Numerical Integration, Polynomial Functions, Accurate Methods
To cite this article
Tahar Latrache, An Accurate Quadrature for the Numerical Integration of Polynomial Functions, International Journal of Applied Mathematics and Theoretical Physics. Vol. 4, No. 1, 2018, pp. 1-7. doi: 10.11648/j.ijamtp.20180401.11
Copyright
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Reference
[1]
Baldoni, V. N. Berline, De Loera, J. A. K¨oppe, M. and Vergne M. (2010). How to integrate a polynomial over a simplex. Mathematics of Computation, DOI: 10.1090/S0025-5718-2010-02378-6.
[2]
Bernardini F. (1991) Integration of polynomials over n-dimensional polyhedra. Comput Aided Des 23 (1): 51–58.
[3]
Davis, P. J. and Rabinowitz, P. (1984a). Methods of Numerical Integration. Academic Press, San Diego, 2nd edition.
[4]
Davis, P. J. and Rabinowitz, P. (1984b). Methods of Numerical Integration. Academic Press, San Diego, 2nd edition.
[5]
Dunavant D. A. (1985) High degree efficient symmetrical Gaussian quadrature rules for the triangle. Int J Numer Methods Eng 21: 1129–1148.
[6]
Holdych D. J., Noble D. R., Secor R. B. (2008) Quadrature rules for triangular and tetrahedral elements with generalized functions. Int J Numer Methods Eng 73: 1310–1327.
[7]
Keast P. (1986) Moderate-degree tetrahedral quadrature formulas. Comput Methods Appl Mech Eng 55: 339–348.
[8]
Liu Y. and Vinokur M. (1998). Exact integrations of polynomials and symmetric quadrature formulas over arbitrary polyhedral grids. Journal of Computational Physics, 140: 122–147.
[9]
Lubinsky, D. S. and Rabinowitz, P. (1984). Rates of convergence of Gaussian quadrature for singular integrands. Mathematics of Computation, 43 (167): 219–242. Press.
[10]
Lyness J. N., Jespersen D. (1975) Moderate degree symmetric quadrature rules for the triangle. J Inst Math Appl 15: 19–32.
[11]
Lyness J. N., Monegato G. (1977) Quadrature rules for regions having regular hexagonal symmetry. SIAM J Numer Anal 14 (2): 283–295.
[12]
Mousavi S. E., Xiao H., Sukumar N. (2010) Generalized Gaussian quadrature rules on arbitrary polygons. Int J Numer Methods Eng 82 (1): 99–113.
[13]
Mousavi S. E., Sukumar N. (2010) Generalized Duffy transformation for integrating vertex singularities. Comput Mech 45 (2–3): 127–140.
[14]
Mousavi S. E., 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.
[15]
Rathod H. T., Govinda Rao H. S. (1997) Integration of polynomials over n-dimensional linear polyhedra. Comput Struct 65 (6): 829–847
[16]
Silvester P. (1970) Symmetric quadrature formulae for simplexes. Math Comput 24 (109): 95–100.
[17]
Sunder K. S., Cookson R. A. (1985) Integration points for triangles and tetrahedrons obtained from the Gaussian quadrature points for a line. Comput Struct 21 (5): 881–885.
[18]
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.
[19]
Wandzura S., Xiao H. (2003) Symmetric quadrature rules on a triangle. Comput Math Appl 45: 1829–1840.
[20]
Wong, R. (1989). Asymptotic approximation of integrals. Academic Press, San Diego.
[21]
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.
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