The complex variable reproducing kernel particle method for two-dimensional elastodynamics

doi:10.1088/1674-1056/19/9/090204

Abstract On the basis of the reproducing kernel particle method (RKPM), a new meshless method, which is called the complex variable reproducing kernel particle method (CVRKPM), for two-dimensional elastodynamics is presented in this paper. The advantages of the CVRKPM are that the correction function of a two-dimensional problem is formed with one-dimensional basis function when the shape function is obtained. The Galerkin weak form is employed to obtain the discretised system equations, and implicit time integration method, which is the Newmark method, is used for time history analysis. And the penalty method is employed to apply the essential boundary conditions. Then the corresponding formulae of the CVRKPM for two-dimensional elastodynamics are obtained. Three numerical examples of two-dimensional elastodynamics are presented, and the CVRKPM results are compared with the ones of the RKPM and analytical solutions. It is evident that the numerical results of the CVRKPM are in excellent agreement with the analytical solution, and that the CVRKPM has greater precision than the RKPM.

Keywords: meshless method reproducing kernel particle method complex variable reproducing kernel particle method elastodynamics

Received: 28 January 2010
Revised: 24 February 2010
Accepted manuscript online:

PACS:	0260
	0270
	4630C

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 10871124), and the Innovation Program of Shanghai Municipal Education Commission, China (Grant No. 09ZZ99).

Cite this article:

Chen Li(陈丽) and Cheng Yu-Min(程玉民) The complex variable reproducing kernel particle method for two-dimensional elastodynamics 2010 Chin. Phys. B 19 090204

[1]	Yang Z J and Deeks A J 2007 Comput. Mech. 40 725
[2]	Belytschko T, Krongauz Y, Organ D, Fleming M and Krysl P 1996 Comput. Meth. Appl. Mech. Eng. 139 3
[3]	Li S F and Liu W K 2002 Appl. Mech. Rev. 55 1
[4]	Gingold R A and Monaghan J J 1977 Mon. Not. R. Astron. Soc. 18 375
[5]	Nayroles B, Touzot G and Villon P G 1992 Comput. Mech. 10 307
[6]	Belytschko T, Lu Y Y and Gu L 1994 Int. J. Numer. Meth. Eng. 37 229
[7]	Liu W K, Jun S and Zhang Y F 1995 Int. J. Numer. Meth. Eng. 20 1081
[8]	Onate E 1996 Int. J. Numer. Meth. Eng. 39 3839
[9]	Atluri S N and Zhu T L 1998 Comput. Mech. 22 117
[10]	Aluru N R 2000 Int. J. Numer. Meth. Eng. 47 1083
[11]	Liu W K and Chen Y J 1995 Int. J. Numer. Meth. Fluids 21 901
[12]	Wendland H 1999 Math. Comput. 68 1521
[13]	Idelsohn S R, Onate E and Calvo N 2003 Int. J. Numer. Meth. Eng. 58 893
[14]	Cheng Y M and Li J H 2005 Acta Phys. Sin. 54 4463 (in Chinese)
[15]	Cheng Y M and Li J H 2006 Sci. China Ser. G Phys. Mech. Astron. 49 46
[16]	Liew K M, Feng C, Cheng Y M and Kitipornchai S 2007 Int. J. Numer. Meth. Eng. 70 46
[17]	Chati M K, Mukherjee S and Mukherjee Y X 1999 Int. J. Numer. Meth. Eng. 46 1163
[18]	Zhu T, Zhang J and Atluri S N 1999 Eng. Anal. Bound. Elem. 23 375
[19]	Gu Y T and Liu G R 2003 Struct. Eng. Mech. 15 535
[20]	Liew K M, Cheng Y M and Kitipornchai S 2006 Int. J. Numer. Meth. Eng. 65 1310
[21]	Kitipornchai S, Liew K M and Cheng Y M 2005 Comput. Mech. 36 13
[22]	Cheng Y M and Peng M J 2005 Sci. China Ser. G Phys. Mech. Astron. 48 641
[23]	Liew K M, Cheng Y M and Kitipornchai S 2005 Int. J. Numer. Meth. Eng. 64 1610
[24]	Liew K M, Cheng Y M and Kitipornchai S 2007 Int. J. Solids Struct. 44 4220
[25]	Peng M J and Cheng Y M 2009 Eng. Anal. Bound. Elem. 33 77
[26]	Cheng Y M, Liew K M and Kitipornchai S 2009 Int. J. Numer. Meth. Eng. 78 1258
[27]	Ren H P and Zhang W 2009 Chin. Phys. B 18 4065
[28]	Lu Y Y, Belytschko T and Tabbara M 1995 Comput. Meth. Appl. Mech. Eng. 126 131
[29]	Liu W K, Jun S, Li S, Adee J and Belytschko T 1995 Int. J. Numer. Meth. Eng. 38 1655
[30]	Han Z D and Atluri S N 2004 Comput. Mater. Continua 1 129
[31]	Liu K Y, Long S Y and Li G Y 2006 Eng. Anal. Bound. Elem. 30 72
[32]	Gu Y T and Liu G R 2001 Comput. Mech. 27 188
[33]	Bueche D, Sukumar N and Moran B 2000 Comput. Mech. 25 207
[34]	Li S C, Li S C and Zhu W S 2006 Chinese J. R. Mech. Eng. 25 141 (in Chinese)
[35]	Li S C, Cheng Y M and Li S C 2006 Acta Phys. Sin. 55 4760 (in Chinese)
[36]	Sladek J, Sladek V and Keer R V 2003 Int. J. Numer. Meth. Eng. 57 235
[37]	Sellountos E J and Polyzos D 2005 Comput. Mech. 35 265
[38]	Li H, Wang Q X and Lam K Y 2004 Comput. Meth. Appl. Mech. Eng. 193 2599
[39]	Liew K M and Cheng Y M 2009 Comput. Meth. Appl. Mech. Eng. 198 3925
[40]	Monaghan J J 1988 Comput. Phys. Commun. 48 89
[41]	Liu W K, Chen Y, Jun S, Chen J S and Belytschko T 1996 Arch. Comput. Meth. Eng. 3 3
[42]	Zhou J X, Zhang H Y and Zhang L 2005 J. Sound Vib. 279 389
[43]	Zhang J P, Gong S G and Huang Y Q 2008 Struct. Multidiscip. Optim. 36 307
[44]	Kiani K, Nikkhoo A and Mehri B 2009 J. Sound Vib. 320 632
[45]	Gan N F, Li G Y and Long S Y 2009 Eng. Anal. Bound. Elem. 33 1211
[46]	Chen J S, Chen C, Wu C T and Liu W K 1996 Comput. Meth. Appl. Mech. Eng. 139 195
[47]	Liu W K and Jun S 1998 Int. J. Numer. Meth. Eng. 41 1339
[48]	Liu W K, Jun S, Thomas S D, Chen Y and Hao W 1997 Int. J. Numer. Meth. Fluids 24 1391
[49]	Chen L and Cheng Y M 2008 Acta Phys. Sin. 57 1 (in Chinese)
[50]	Chen L and Cheng Y M 2008 Acta Phys. Sin. 57 6047 (in Chinese)
[51]	Danielson K T, Hao S, Liu W K, Uras R A and Li S F 2000 Int. J. Numer. Meth. Eng. 47 1323
[52]	Ching H K 2002 Solution of Linear Elastostatic and Elastodynamic Plane Problems by the Meshless Local Petrov--Galerkin Method (Ph.D. Thesis) (Blacksburg: Virginia Polytechnic Institute and State University) endfootnotesize

[1]	Improved reproducing kernel particle method for piezoelectric materials Ji-Chao Ma(马吉超), Gao-Feng Wei(魏高峰), Dan-Dan Liu(刘丹丹). Chin. Phys. B, 2018, 27(1): 010201.
[2]	Topology optimization using the improved element-free Galerkin method for elasticity Yi Wu(吴意), Yong-Qi Ma(马永其), Wei Feng(冯伟), Yu-Min Cheng(程玉民). Chin. Phys. B, 2017, 26(8): 080203.
[3]	Meshless analysis of an improved element-free Galerkin method for linear and nonlinear elliptic problems Yao-Zong Tang(唐耀宗), Xiao-Lin Li(李小林). Chin. Phys. B, 2017, 26(3): 030203.
[4]	Two-dimensional fracture analysis of piezoelectric material based on the scaled boundary node method Shen-Shen Chen(陈莘莘), Juan Wang(王娟), Qing-Hua Li(李庆华). Chin. Phys. B, 2016, 25(4): 040203.
[5]	Solving unsteady Schrödinger equation using the improved element-free Galerkin method Rong-Jun Cheng(程荣军) and Yu-Min Cheng(程玉民). Chin. Phys. B, 2016, 25(2): 020203.
[6]	Hybrid natural element method for large deformation elastoplasticity problems Ma Yong-Qi (马永其), Zhou Yan-Kai (周延凯). Chin. Phys. B, 2015, 24(3): 030204.
[7]	Analysis of elastoplasticity problems using an improved complex variable element-free Galerkin method Cheng Yu-Min (程玉民), Liu Chao (刘超), Bai Fu-Nong (白福浓), Peng Miao-Juan (彭妙娟). Chin. Phys. B, 2015, 24(10): 100202.
[8]	Hybrid natural element method for viscoelasticity problems Zhou Yan-Kai (周延凯), Ma Yong-Qi (马永其), Dong Yi (董轶), Feng Wei (冯伟). Chin. Phys. B, 2015, 24(1): 010204.
[9]	A meshless algorithm with moving least square approximations for elliptic Signorini problems Wang Yan-Chong (王延冲), Li Xiao-Lin (李小林). Chin. Phys. B, 2014, 23(9): 090202.
[10]	An interpolating reproducing kernel particle method for two-dimensional scatter points Qin Yi-Xiao (秦义校), Liu Ying-Ying (刘营营), Li Zhong-Hua (李中华), Yang Ming (杨明). Chin. Phys. B, 2014, 23(7): 070207.
[11]	A meshless method based on moving Kriging interpolation for a two-dimensional time-fractional diffusion equation Ge Hong-Xia (葛红霞), Cheng Rong-Jun (程荣军). Chin. Phys. B, 2014, 23(4): 040203.
[12]	Analysis of variable coefficient advection–diffusion problems via complex variable reproducing kernel particle method Weng Yun-Jie (翁云杰), Cheng Yu-Min (程玉民). Chin. Phys. B, 2013, 22(9): 090204.
[13]	A complex variable meshless local Petrov-Galerkin method for transient heat conduction problems Wang Qi-Fang (王启防), Dai Bao-Dong (戴保东), Li Zhen-Feng (栗振锋). Chin. Phys. B, 2013, 22(8): 080203.
[14]	A meshless Galerkin method with moving least square approximations for infinite elastic solids Li Xiao-Lin (李小林), Li Shu-Ling (李淑玲). Chin. Phys. B, 2013, 22(8): 080204.
[15]	Analysis of the generalized Camassa and Holm equation with the improved element-free Galerkin method Cheng Rong-Jun, Wei Qi. Chin. Phys. B, 2013, 22(6): 060209.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

The complex variable reproducing kernel particle method for two-dimensional elastodynamics

Cite this article:

Online attention