Product Gauss quadrature rules vs. cubature rules in the meshless local Petrov-Galerkin method

Journal of Complexity

Volumn 26, Issue 1, 2010, Pages 82-101

  Mazzia, Annamaria ^a   Pini, Giorgio ^a  

^a UNIVERSITY OF PADOVA   (Italy)

Author keywords

Cubature formulas on the disk; Meshless Local Petrov Galerkin method

Indexed keywords

CODES (SYMBOLS); GAUSSIAN DISTRIBUTION; NUMERICAL METHODS; PARTIAL DIFFERENTIAL EQUATIONS; PLASTIC DEFORMATION;

CUBATURE FORMULA; DOMAIN INTEGRALS; GAUSS QUADRATURE RULES; MESH-LESS METHODS; MESHLESS LOCAL PETROV-GALERKIN METHOD; MESHLESS LOCAL PETROV-GALERKIN METHODS; NUMERICAL QUADRATURE; QUADRATURE FORMULA;

GALERKIN METHODS;

EID: 71649107095     PISSN: 0885064X     EISSN: 10902708     Source Type: Journal    
DOI: 10.1016/j.jco.2009.07.002     Document Type: Article

References (30)

1. Gingold R.A., and Monaghan J.J. Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. Roy. Astron. Soc. 181 (1977) 375-389
2. Nayroles B., Touzot G., and Villon P. Generalizing the finite element method: diffuse approximation and diffuse elements. Comput. Mech. 10 (1992) 307-318
3. Lu Y.Y., Belytschko T., and Gu L. A new implementation of the element free Galerkin method. Comput. Methods Appl. Mech. Eng. 113 (1994) 397-414
4. Liu W.K., Chen Y., Uras R.A., and Chang C.T. Generalized multiple scale reproducing kernel particle methods. Comput. Methods Appl. Mech. Eng. 139 (1996) 91-157
5. Babuska I., and Melenk J.M. The partition of unity method. Int. J. Numer. Methods Eng. 40 (1997) 727-758
6. Duarte C., and Oden J.T. Hp-cloud - a meshless method to solve boundary-value problems. Comput. Methods Appl. Mech. Eng. 139 (1996) 237-262
7. Sukumar N., Moran B., and Belytschko T. The natural element method in solid mechanics. Int. J. Numer. Methods Eng. 43 (1998) 839-887
8. Wendland H. Meshless galerkin methods using radial basis function. Math. Comput. 68 228 (1999) 1521-1531
9. Atluri S.N., and Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput. Mech. 22 (1998) 117-127
10. Atluri S.N., and Shen S. The Meshless Local Petrov Galerkin (MLPG) Method (2002), Tech Science Press, Forsyth, GA, USA
11. Atluri S.N., and Shen S. The Meshless Method (MLPG) for Domain and BIE Discretizations (2004), Tech Science Press, Forsyth, GA, USA
12. Ferronato M., Mazzia A., Pini G., and Gambolati G. A meshless method for axi-symmetric poroelastic simulations: numerical study. Int. J. Numer. Methods Eng. 70 11 (2007) 1346-1365
13. Lancaster P., and Šalkauskas K. Surfaces generated by moving least squares methods. Math. Comput. 37 155 (1981) 141-158
14. De S., and Bathe K.J. The method of finite spheres. Comput. Mech. 25 (2000) 329-345
15. De S., and Bathe K.J. The method of finite spheres with improved numerical integration. Comput. Struct. 79 (2001) 2183-2196
16. Mazzia A., Ferronato M., Pini G., and Gambolati G. A comparison of numerical integration rules for the Meshless Local Petrov-Galerkin method. Numer. Algorithms 45 1-4 (2007) 61-74
17. Pecher R. Efficient cubature formulae for MLPG and related methods. Int. J. Numer. Methods Eng. 65 (2006) 566-593
18. Stroud A.H. Approximate Calculation of Multiple Integrals (1971), Prentice-Hall, Inc, Englewood Cliffs, New Jersey
19. Cools R. Monomial cubature rules since 'Stroud'-part 2. J. Comput. Appl. Math. 112 (1999) 21-27
20. Cools R. An encyclopaedia of cubature formulas. J. Complexity 19 3 (2003) 445-453
21. Cools R., and Rabinowitz P. Monomial cubature rules since 'Stroud': A compilation. J. Comput. Appl. Math. 48 (1993) 309-326
22. Atluri S.N., Han Z.D., and Rajendran A.M. A new implementation of the meshless finite volume method, through the MLPG Mixed approach. Comput. Modeling Eng. Sci. 6 6 (2004) 491-513
23. De S., and Bathe K.J. Towards an efficient meshless computational technique: the method of finite spheres. Eng. Comput. 18 12 (2001) 170-192
24. Peirce W.H. Numerical integration over the planar annulus. J. Soc. Indust. Appl. Math. 5 2 (1957) 66-73
25. Haegemans A. Circularly symmetrical integration formulas for two-dimensional circularly symmetrical regions. BIT 16 (1976) 52-59
26. Cools R., and Kim K. A survey of known and new cubature formulas for the unit disk. Korean J. Comput. Appl. Math. 7 3 (2000) 477-485
27. Kim K., and Song M. Symmetric quadrature formulas over a unit disk. Korean J. Comput. Appl. Math. 4 1 (1997) 179-192
28. Belytschko T., Krongauz Y., Organ D., Fleming M., and Krysl P. Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng. 139 (1996) 3-47
29. Rabinowitz P., and Richter N. Perfectly symmetric two-dimensional integration formulas with minimal number of points. Math. Comput. 23 108 (1969) 765-779
30. Cools R., and Haegemans A. Automatic computation of knots and weights of cubature formulae for circular symmetric planar regions. J. Comput. Appl. Math. 20 (1987) 153-158