Generalized Gaussian quadrature rules on arbitrary polygons

International Journal for Numerical Methods in Engineering

Volumn 82, Issue 1, 2010, Pages 99-113

  Mousavi, S E ^a   Xiao, H ^b   Sukumar, N ^a  

^a University of California   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Convex and concave polygons; Gaussian quadrature; Gram Schmidt orthogonalization; Newton's method; Numerical integration; Polygonal finite elements

Indexed keywords

CONCAVE POLYGONS; GAUSSIAN QUADRATURES; GRAM-SCHMIDT ORTHOGONALIZATIONS; NEWTON'S METHODS; NUMERICAL INTEGRATIONS; POLYGONAL FINITE ELEMENT;

ALGEBRA; ALGORITHMS; FLUID STRUCTURE INTERACTION; FUNCTION EVALUATION; INTEGRATION; NEWTON-RAPHSON METHOD; POLYNOMIAL APPROXIMATION; RATIONAL FUNCTIONS; STIFFNESS MATRIX;

COMPUTATIONAL EFFICIENCY;

EID: 77149178430     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.2759     Document Type: Article

References (27)

1. Stroud AH, Secrest D. Gaussian Quadrature Formulas. Prentice-Hall, Inc.: Englewood Cliffs, NJ, 1966.
2. Ma J, Rokhlin V, Wandzura S. Generalized Gaussian quadrature of systems of arbitrary functions. SIAM Journal on Numerical Analysis 1996; 33(3):971-996.
3. Yarvin N, Rokhlin V. Generalized Gaussian quadratures and singular value decompositions of integral operators. SIAM Journal on Scientific Computing 1998; 20(2):669-718.
4. Wandzura S, Xiao H. Symmetric quadrature rules on a triangle. Computers and Mathematics with Applications 2003; 45:1829-1840.
5. Silvester P. Symmetric quadrature formulae for simplexes. Mathematics of Computation 1970; 24(109):95-100.
6. Cowper GR. Gaussian quadrature formulas for triangles. International Journal for Numerical Methods in Engineering 1973; 7(3):405-408.
7. Lyness JN, Jespersen D. Moderate degree symmetric quadrature rules for the triangle. Journal of the Institute of Mathematics and its Applications 1975; 15:19-32.
8. Hillion P. Numerical integration on a triangle. International Journal for Numerical Methods in Engineering 1977; 11:797-815.
9. Laursen ME, Gellert M. Some criteria for numerically integrated matrices and quadrature formulas for triangles. International Journal for Numerical Methods in Engineering 1978; 12:67-76.
10. Dunavant DA. High degree efficient symmetrical Gaussian quadrature rules for the triangle. International Journal for Numerical Methods in Engineering 1985; 21:1129-1148.
11. Berntsen J, Espelid TO. Degree 13 symmetric quadrature rules for the triangle. Technical Report 44, Department of Informatics, University of Bergen, Norway, 1990.
12. Rathod HT, Nagaraja KV, Venkatesudu B. Symmetric Gauss Legendre quadrature formulas for composite numerical integration over a triangular surface. Applied Mathematics and Computation 2007; 188:865-876.
13. Dasgupta G. Integration within polygonal finite elements. Journal of Aerospace Engineering 2003; 16(1):9-18.
14. Sukumar N, Tabarraei A. Conforming polygonal finite elements. International Journal for Numerical Methods in Engineering 2004; 61(12):2045-2066.
15. Tabarraei A, Sukumar N. Extended finite element method on polygonal and quadtree meshes. Computer Methods in Applied Mechanics and Engineering 2008; 197(5):425-438.
16. Talischi C, Paulino GH, Le C. Honeycomb Wachspress finite elements for structural topology optimization. Structural and Multidisciplinary Optimization 2009; 37:569-583.
17. Lyness JN, Monegato G. Quadrature rules for regions having regular hexagonal symmetry. SIAM Journal on Numerical Analysis 1977; 14(2):283-295.
18. Nooijen M, Velde GT, Baerends EJ. Symmetric numerical integration formulas for regular polygons. SIAM Journal on Numerical Analysis 1990; 27(1):198-218.
19. Xiao H, Gimbutas Z. A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions. Computers and Mathematics with Applications 2008; in review.
20. Koornwinder T. Two-variable analogues of the classical orthogonal polynomials. Theory and Applications of Special Functions. Academic Press, Inc.: New York, 1975; 435-495.
21. Penrose R. A generalized inverse for matrices. Proceedings of the Cambridge Philosophical Society 1955; 51:406-413.
22. Martinsson PG, Rokhlin V, Tygert M. On interpolation and integration in finite-dimensional spaces of bounded functions. Technical Report 1317, Department of Computer Science, Yale University, March 2005.
23. Stroud AH. Approximate Calculation of Multiple Integrals. Prentice-Hall: Englewood Cliffs, NJ, 1971.
24. Cools R, Haegemans A. Why do so many cubature formulae have so many positive weights? BIT Numerical Mathematics 1988; 28:792-802.
25. Sommariva A, Vianello M. Product Gauss cubature over polygons based on Green's integration formula. BIT Numerical Mathematics 2007; 47:441-453.
26. Talischi C, Paulino GH, Pereira A, Menezes IFM. Polygonal finite elements for topology optimization: a unifying paradigm. International Journal for Numerical Methods in Engineering 2009; in press.
27. Holdych DJ, Noble DR, Secor RB. Quadrature rules for triangular and tetrahedral elements with generalized functions. International Journal for Numerical Methods in Engineering 2008; 73:1310-1327.