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A complex variable meshless method for fracture problems. (English) Zbl 1147.74410

Summary: Based on the moving least-square (MLS) approximation, the complex variable moving least-square approximation (CVMLS) is discussed in this paper. The complex variable moving least-square approximation cannot form ill-conditioned equations, and has greater precision and computational efficiency. Using the analytical solution near the tip of a crack, the trial functions in the complex variable moving least-square approximation are extended, and the corresponding approximation function is obtained. And from the minimum potential energy principle, a complex variable meshless method for fracture problems is presented, and the formulae of the complex variable meshless method are obtained. The complex variable meshless method in this paper has greater precision and computational efficiency than the conventional meshless method. Some examples are given.

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
74Rxx Fracture and damage
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[1] Belytschko, T., Krongauz, Y., Organ, D. et al., Meshless methods: An overview and recent developments, Computer Methods in Applied Mechanics and Engineering, 1996, 139:3–47. · Zbl 0891.73075 · doi:10.1016/S0045-7825(96)01078-X
[2] Li Shaofan, Liu, W. K., Meshfree and particle methods and their applications, Applied Mechanics Review, 2002, 55:1–34. · doi:10.1115/1.1431547
[3] Nayroles, B., Touzot, G., Villon, P. G., Generalizing the finite element method: diffuse approximation and diffuse elements, Computational Mechanics, 1992, 10:307–318. · Zbl 0764.65068 · doi:10.1007/BF00364252
[4] Belytschko, T., Lu, Y. Y., Gu, L., Element-free Galerkin methods, International Journal for Numerical Methods in Engineering, 1994, 37:229–256. · Zbl 0796.73077 · doi:10.1002/nme.1620370205
[5] Duarte, C. A., Oden, J. T., Hp clouds-a meshless method to solve boundary-value problems, Technical Report 95-05, Texas Institute for Computational and Applied Mathematics, University of Texas at Austin, 1995.
[6] Onarte, E., A finite point method in computational mechanics, International Journal for Numerical Methods in Engineering, 1996, 39:3839–3866. · Zbl 0884.76068 · doi:10.1002/(SICI)1097-0207(19961130)39:22<3839::AID-NME27>3.0.CO;2-R
[7] Atluri, S. N., Zhu, T. L., A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Computational Mechanics, 1998, 22:117–127. · Zbl 0932.76067 · doi:10.1007/s004660050346
[8] Liu, W. K., Chen, Y., Uras, R. A. et al., Generalized multiple scale reproducing kernel particle, Computer Methods in Applied Mechanics and Engineering, 1996, 139:91–157. · Zbl 0896.76069 · doi:10.1016/S0045-7825(96)01081-X
[9] Liu, W. K., Chen, Y. J., Wavelet and multiple scale reproducing kernel methods, International Journal for Numerical Methods in Fluids, 1995, 21:901–931. · Zbl 0885.76078 · doi:10.1002/fld.1650211010
[10] Chen, W., New RBF Collocation methods and kernel RBF with applications, in Meshfree Methods for Partial Differential Equations (Griebel M and Schweitzer M A Eds.), Vol.1, New York: Springer-Verlag, 2000
[11] Idelsohn, S. R., Onate, E., Calvo, N. et al., The meshless finite element method, International Journal for Numerical Methods in Engineering, 2003, 58:893–912. · Zbl 1035.65129 · doi:10.1002/nme.798
[12] Hao, S., Park, H. S., Liu, W. K., Moving particle finite element method, International Journal for Numerical Methods in Engineering, 2002, 53(8):1937–1958. · Zbl 1169.74606 · doi:10.1002/nme.368
[13] Zhu, T., Zhang, J. D., Atluri, S. N., A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach, Computational Mechanics, 1998, 21:223–235. · Zbl 0920.76054 · doi:10.1007/s004660050297
[14] Kothnur, V. S., Mukherjee, S., Mukherjee, Y. X., Two-dimensional linear elasticity by the boundary node method, International Journal of Solids and Structures, 1999, 36:1129–1147. · Zbl 0937.74074 · doi:10.1016/S0020-7683(97)00363-6
[15] Zhang Xiong, Liu Xiao-Hu, Song Kang-Zu et al., Least-squares collocation meshless method, International Journal for Numerical Methods in Engineering, 2001, 51:1089–1100. · Zbl 1056.74064 · doi:10.1002/nme.200
[16] Zhang Xiong, Hu Wei, Pan Xiaofei et al., Meshless weighted least-square method, Acta Mechanica Sinica, 2003, 35(4):425–431.
[17] Cai Yongcang, Wang Jianhua, The meshless Local-Petrov Galerkin method based on the Voronoi cells, Acta Mechanica Sinica (in Chinese), 2003, 35(2):187–193.
[18] Zhang Jianming, Yao Zhenhan, Li Hong, A hybrid boundary node method, International Journal for Numerical Methods in Engineering, 2002, 53:751–763. · doi:10.1002/nme.313
[19] Long Shuyao, Xu Jingxiao, A Local boundary integral equation method for the elasticity problem, Acta Mechanica Sinica, 2000, 32(5):566–578.
[20] Cheng Yumin, Chen Meijuan, A boundary element-free method for elasticity, Acta Mechanica Sinica (in Chinese), 2003, 35(2):181–186.
[21] Li Shuchen, Cheng Yumin, Meshless numerical manifold method based on unit partition, Acta Mechanica Sinica (in Chinese), 2004, 36(4):496–500.
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