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In this paper, combining the moving Kriging interpolation method and meshless local Petrov-Galerkin method, an improved meshless local Petrov-Galerkin method is presented, in which the Heaviside step function is used as test function over the local weak form. The present method is applied to two-dimensional potential problems and the corresponding discrete equations are derived. Because the shape functions so-obtained possess the Kronecker delta property, the essential boundary conditions can be enforced as the FEM; furthermore, the Heaviside step function is used as the test function, there is no domain integral, and only a regular boundary integral is involved. In this paper, the choice of the important parameters is studied. Numerical examples show that the present method has simpler numerical procedures and lower computation cost, in addition, the essential boundary conditions can be implemented directly.
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Keywords:
- meshless local Petrov-Galerkin method /
- moving Kriging interpolation method /
- Meshless method /
- potential problems
[1] Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P 1996 Comput. Methods Appl.Mech. Engrg. 139 3
[2] Zhang X, Liu Y, Ma S 2009 Advance in Mechanics 39 1(in Chinese) [张 雄、刘 岩、马 上 2009 力学进展 39 1]
[3] Lu Y Y, Belytschko T, Tabbara M 1995 Comput. Methods Appl.Mech. Engrg 126 131
[4] Li S, Hao W, Liu W K 2000 Comput. Mech. 25 102
[5] Cheng R J, Cheng Y M 2008 Acta Phys. Sin. 57 6037 (in Chinese) [程荣军、程玉民 2008 57 6037]
[6] Cheng R J, ChengY M 2007 Acta Phys. Sin. 56 5569 (in Chinese) [程荣军、程玉民 2007 56 5569]
[7] Ren H P, Cheng Y M, Zhang W 2009 Chin. Phys. B 18 4065
[8] Peng M J, Cheng Y M 2009 Eng. Anal. Bound. Elem. 33 77
[9] Cheng Y M, Peng M J 2005 Sci.Chin. G Phys.Mech. and Astron. 48 641
[10] Belytscko T, Lu Y Y, Gu L 1994 Int.J.Numer.Meth.Engng. 37 229
[11] Cheng R J, Ge H X 2009 Chin. Phys. B 18 4059
[12] Liu W K, Jun S, Zhang Y F 1995 Int.J.Numer.Meth.Fluids. 20 1081
[13] Chen L, Cheng Y M 2008 Acta Phys. Sin. 57 1(in Chinese) [陈 丽、程玉民 2008 57 1]
[14] Onate E, Idelsohn S, Zienkiewicz O Z, Taylor R L 1996 Inter. J. for Num. Meth. in Engin. 39 3839
[15] Babuska I, Melenk J M 1997 Int.J.Numer.Meth.Engng. 40 727
[16] Liu G R, Gu Y T. 2001 Int.J.Numer.Meth.Engng. 50 937
[17] Zhang X, Liu X H and Song K Z 2001 Int.J.Numer.Meth.Engng. 51 1089
[18] Cheng Y M, Li J H 2005 Acta Phys. Sin. 54 4463 (in Chinese) [程玉民、李九红 2005 54 4463]
[19] Atluri S N, Zhu T L 1998 Comput. Mech. 22 117
[20] Mukherjee Y X, Mukherjee S 1997 Int.J.Numer.Meth.Engng. 40 797
[21] Zhu T, Zhang J, Atluri S N 1999 Eng. Anal. Bound. Elem. 23 375
[22] Qin Y X, Cheng Y M 2006 Acta Phys. Sin. 55 3215 (in Chinese) [秦义校、程玉民 2006 55 3215]
[23] Zhang Z, Zhao P, Liew K M 2009 Eng. Anal. Bound. Elem. 33 547
[24] Atluri S N 2004 The Meshless Method for Domain & BIE Discretizations (Irvine: University of California)
[25] Lancaster P L, Salkauskas K 1981 Math. Comput. 37 141
[26] Krige D G 1951 Journal of the Chemical Metallurgical and Mining Society of South Africa 52 119
[27] Gu L 2003 Int.J.Numer.Meth.Engng. 56 1
[28] Bui Q T, Nguyen N T, Nguyen-Dang H 2009 Int.J.Numer.Meth.Engng.77 1371
[29] Dai K Y, Liu G R, Lim K M, Gu Y T 2003 Comput. Mech. 32 60
[30] Sacks J, Susannah S B, Welch W J 1989 Technometrics 31 41
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[1] Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P 1996 Comput. Methods Appl.Mech. Engrg. 139 3
[2] Zhang X, Liu Y, Ma S 2009 Advance in Mechanics 39 1(in Chinese) [张 雄、刘 岩、马 上 2009 力学进展 39 1]
[3] Lu Y Y, Belytschko T, Tabbara M 1995 Comput. Methods Appl.Mech. Engrg 126 131
[4] Li S, Hao W, Liu W K 2000 Comput. Mech. 25 102
[5] Cheng R J, Cheng Y M 2008 Acta Phys. Sin. 57 6037 (in Chinese) [程荣军、程玉民 2008 57 6037]
[6] Cheng R J, ChengY M 2007 Acta Phys. Sin. 56 5569 (in Chinese) [程荣军、程玉民 2007 56 5569]
[7] Ren H P, Cheng Y M, Zhang W 2009 Chin. Phys. B 18 4065
[8] Peng M J, Cheng Y M 2009 Eng. Anal. Bound. Elem. 33 77
[9] Cheng Y M, Peng M J 2005 Sci.Chin. G Phys.Mech. and Astron. 48 641
[10] Belytscko T, Lu Y Y, Gu L 1994 Int.J.Numer.Meth.Engng. 37 229
[11] Cheng R J, Ge H X 2009 Chin. Phys. B 18 4059
[12] Liu W K, Jun S, Zhang Y F 1995 Int.J.Numer.Meth.Fluids. 20 1081
[13] Chen L, Cheng Y M 2008 Acta Phys. Sin. 57 1(in Chinese) [陈 丽、程玉民 2008 57 1]
[14] Onate E, Idelsohn S, Zienkiewicz O Z, Taylor R L 1996 Inter. J. for Num. Meth. in Engin. 39 3839
[15] Babuska I, Melenk J M 1997 Int.J.Numer.Meth.Engng. 40 727
[16] Liu G R, Gu Y T. 2001 Int.J.Numer.Meth.Engng. 50 937
[17] Zhang X, Liu X H and Song K Z 2001 Int.J.Numer.Meth.Engng. 51 1089
[18] Cheng Y M, Li J H 2005 Acta Phys. Sin. 54 4463 (in Chinese) [程玉民、李九红 2005 54 4463]
[19] Atluri S N, Zhu T L 1998 Comput. Mech. 22 117
[20] Mukherjee Y X, Mukherjee S 1997 Int.J.Numer.Meth.Engng. 40 797
[21] Zhu T, Zhang J, Atluri S N 1999 Eng. Anal. Bound. Elem. 23 375
[22] Qin Y X, Cheng Y M 2006 Acta Phys. Sin. 55 3215 (in Chinese) [秦义校、程玉民 2006 55 3215]
[23] Zhang Z, Zhao P, Liew K M 2009 Eng. Anal. Bound. Elem. 33 547
[24] Atluri S N 2004 The Meshless Method for Domain & BIE Discretizations (Irvine: University of California)
[25] Lancaster P L, Salkauskas K 1981 Math. Comput. 37 141
[26] Krige D G 1951 Journal of the Chemical Metallurgical and Mining Society of South Africa 52 119
[27] Gu L 2003 Int.J.Numer.Meth.Engng. 56 1
[28] Bui Q T, Nguyen N T, Nguyen-Dang H 2009 Int.J.Numer.Meth.Engng.77 1371
[29] Dai K Y, Liu G R, Lim K M, Gu Y T 2003 Comput. Mech. 32 60
[30] Sacks J, Susannah S B, Welch W J 1989 Technometrics 31 41
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