Stability and dispersion analysis of reproducing kernel collocation method for transient dynamics

doi:10.1007/s10483-011-1457-6

Abstract

Abstract: A reproducing kernel collocation method based on strong formulation is introduced for transient dynamics. To study the stability property of this method, an algorithm based on the von Neumann hypothesis is proposed to predict the critical time step. A numerical test is conducted to validate the algorithm. The numerical critical time step and the predicted critical time step are in good agreement. The results are compared with those obtained based on the radial basis collocation method, and they are in good agreement. Several important conclusions for choosing a proper support size of the reproducing kernel shape function are given to improve the stability condition.

Key words: reproducing kernel collocation method (RKCM), stability analysis, dispersion analysis, transient dynamics

2010 MSC Number:

65M12

LUO Han-Zhong;LIU Wue-Wen;HUANG Xing-Chun. Stability and dispersion analysis of reproducing kernel collocation method for transient dynamics. Applied Mathematics and Mechanics (English Edition), 2011, 32(6): 777-788.

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References

[1] Gingold, R. A. and Monaghan, J. J. Smoothed particle hydrodynamics: theory and application
to nonspherical stars. Royal Astronomical Society, Monthly Notices, 181, 375–389 (1977)

[2] Nayroles, B., Touzot, G., and Villon, P. Generalizing the finite element method: diffuse approximation
and diffuse elements. Computational Mechanics, 10(5), 307–318 (1992)

[3] Belytschko, T., Lu, Y. Y., and Gu, L. Element-free Galerkin methods. International Journal for
Numerical Methods in Engineering, 37(2), 229–256 (1994)

[4] Melenk, J. M. and Babuska, I. The partition of unity finite element method: basic theory and
applications. Computer Methods in Applied Mechanics and Engineering, 139(1-4), 289–314 (1996)

[5] Duarte, C. A. and Oden, J. T. An h-p adaptive method using clouds. Computer Methods in
Applied Mechanics and Engineering, 139(1-4), 237–262 (1996)

[6] Liu, W. K., Jun, S., and Zhang, Y. F. Reproducing kernel particle methods. International Journal
for Numerical Methods in Fluids, 20(8-9), 1081–1106 (1995)

[7] Chen, J. S., Pan, C. H., Wu, C. T., and Liu, W. K. Reproducing kernel particle methods for
large deformation analysis of non-linear structures. Computer Methods in Applied Mechanics and
Engineering, 139(1-4), 195–227 (1996)

[8] Sukumar, N., Moran, B., and Belytschko, T. The natural element method in solid mechanics.
International Journal for Numerical Methods in Engineering, 43(5), 839–887 (1998)

[9] Atluri, S. N. and Zhu, T. L. The meshless local Petrov-Galerkin (MLPG) approach for solving
problems in elasto-statics. Computational Mechanics, 25(2-3), 169–179 (2000)

[10] Kansa, E. J. Multiqudrics — a scattered data approximation scheme with applications to computational
fluid-dynamics — I. surface approximations and partial derivatives. Computers and
Mathematics with Applications, 19(8-9), 127–145 (1990)

[11] Kansa, E. J. Multiqudrics — a scattered data approximation scheme with applications to computational
fluid-dynamics — II. solutions to parabolic, hyperbolic and elliptic partial differential
equations. Computers and Mathematics with Applications, 19(8-9), 147–161 (1990)

[12] Zhang, X., Chen, J. S., and Osher, S. A multiple level set method for modeling grain boundary
evolution of polycrystalline materials. Interaction and Multiscale Mechanics, 1, 178–191 (2008)

[13] Belytschko, T., Kronggaus, Y., Organ, D., and Fleming, M. Meshless methods: an overview and
recent developments. Computer Methods in Applied Mechanics and Engineering, 139(1-4), 3–47
(1996)

[14] Zhang, X., Liu, Y., and Ma, S. Meshfree methods and their applications (in Chinese). Advances
in Mechanics, 39(1), 1–36 (2009)

[15] Liu, G. R. Meshfree Methods: Moving Beyond the Finite Element Method, 2nd ed., CRC Press
(2009)

[16] Zhang, X., Song, K. Z., Lu, M. W., and Liu, X. Meshless methods based on collocation with radial
basis function. Computational Mechanics, 26(4), 333–343 (2000)

[17] Huang, J., Zhang, J., and Chen, G. G. Stability of Schr¨odinger-Poisson type equations. Applied
Mathematics and Mechanics (English Edition), 30(17), 1469–1474 (2009) DOI 10.1007/s10483-
009-1113-y

[18] Ye, Z. A new finite element formulation for planar elastic deformation. International Journal for
Numerical Methods in Engineering, 40(20), 2579–2591 (1997)

[19] Zhu, H. H., Yang, B. H., Cai, Y. C., and Xu, B. Application of meshless natural element method
to elastoplastic analysis (in Chinese). Rock and Soil Mechanics, 25(4), 671–674 (2004)

[20] Xiong, Y. B. and Long, S. Y. An analysis of plates on the winkler foundation with the meshless
local Petrov-Galerkin method (in Chinese). Journal of Hunan University (Natural Science), 31(4),
101–105 (2004)

[21] Li, S. C. and Chen, Y. M. Meshless numerical manifold method based on unit partition (in
Chinese). Acta Mechanica Sinica, 36(4), 496–500 (2004)

[22] Hu, H. Y., Chen, J. S., and Hu, W. Error analysis of collocation method based on reproducing
kernel approximation. Numerical Methods for Partial Differential Equations, 27(3), 554–580
(2009)

[23] Hu, H. Y., Lai, C. K., and Chen, J. S. A study on convergence and complexity of reproducing
kernel particle method. Interaction and Multiscale Mechanics, 2, 295–319 (2009)

[
[24] Lucy, L. A numerical approach to testing the fission hypothesis. The Astronomical Journal, 82(18),
1013–1024 (1977)

[25] Monoghan, J. J. Why particle methods work. SIAM Journal on Scientific and Statistical Computing,
3(4), 422–433 (1982)

[26] Monoghan, J. J. An introduction to SPH. Computer Physics Communications, 48(1), 89–96 (1988)

[27] Randles, P. W. and Libersky, L. D. Smoothed particle hydrodynamics: some recent improvements
and applications. Computer Methods in Applied Mechanics and Engineering, 139(1-4), 375–408
(1996)

[28] Liu, W. K., Jun, S., Li, S., Adee, J., and Belytschko, B. Reproducing kernel particle methods
for structural dynamics. International Journal for Numerical Methods in Engineering, 38(10),
1655–1679 (1995)

[29] Liu, W. K. and Chen, Y. Wavelet and multiple scale reproducing kernel particle methods. International
Journal for Numerical Methods in Fluids, 21(16), 901–931 (1995)

[30] Chen, J. S., Pan, C., Roque, M. O. L., and Wang, H. P. A Lagrangian reproducing kernel particle
method for metal forming analysis. Computational Mechanics, 22(3), 289–307 (1998)

[31] Luo, H. Z., Chen, J. S., Hu, H. Y., and Huang, X. C. Stability of radial basis collocation method
for transient dynamics. Journal of Shanghai Jiaotong University (English Edition), 15(5), 615–621
(2010) DOI 10.1007/s12204-010-1057-4