×

Stability analysis of particle methods with corrected derivatives. (English) Zbl 1073.76619

Summary: The stability of discretizations by particle methods with corrected derivatives is analyzed. It is shown that the standard particle method (which is equivalent to the element-free Galerkin method with an Eulerian kernel and nodal quadrature) has two sources of instability: (1) rank deficiency of the discrete equations; and (2) distortion of the material instability. The latter leads to the so-called tensile instability. It is shown that a Lagrangian kernel with the addition of stress points eliminates both instabilities. Examples that verify the stability of the new formulation are given.

MSC:

76M28 Particle methods and lattice-gas methods
74S30 Other numerical methods in solid mechanics (MSC2010)
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Belytschko, T.; Lu, Y.-Y.; Gu, L., Element-free Galerkin methods, Int. J. Numer. Meth. Engng., 37, 229-256 (1994) · Zbl 0796.73077
[2] Duarte, C. A.; Oden, J. T., HP clouds—A meshless method to solve boundary-value problems, Technical Report 95-05 (1995), University of Texas: University of Texas Austin
[3] Duarte, C. A.; Oden, J. T., An H-P adaptive method using clouds, Comput. Meths. Appl. Mech. Engrg., 139, 237-262 (1996) · Zbl 0918.73328
[4] Lucy, L. B., A numerical approach to the testing of the fission hypothesis, J. Astron., 82, 1013-1024 (1977)
[5] Monaghan, J. J., An introduction to SPH, Comput. Phys. Commun., 48, 89-96 (1988) · Zbl 0673.76089
[6] Belytschko, T.; Krongauz, K.; Organ, D.; Fleming, M.; Krysl, P., Meshless methods: An overview and recent developments, Comput. Meths. Appl. Mech. Engrg., 139, 3-47 (1996) · Zbl 0891.73075
[7] Liu, W. K.; Li, S. F.; Belytschko, T., Moving least square reproducing kernel method. (I) Methodology and convergence, Comput. Meths. Appl. Mech. Engrg., 143, 113-154 (1997) · Zbl 0883.65088
[8] Johnson, G. R.; Beissel, S. R., Normalized smoothing functions for SPH impact calculations, Int. J. Numer. Meth. Engng., 39, 2725-2741 (1996) · Zbl 0880.73076
[9] Randles, P.; Libersky, L., Smoothed particle hydrodynamics: Some recent improvements and applications, Comput. Meths. Appl. Mech. Engrg., 139, 375-408 (1996) · Zbl 0896.73075
[10] Krongauz, Y.; Belytschko, T., Consistent pseudo-derivatives in meshless methods, Comput. Meth. Appl. Mech. Engng., 146, 371-386 (1968) · Zbl 0894.73156
[11] Belytschko, T.; Krongauz, K.; Dolbow, J.; Gerlach, C., On the completeness of meshfree particle methods, Int. J. Numer. Meth. Engng., 43 (1998) · Zbl 0939.74076
[12] Black, T.; Belytschko, T., Convergence of corrected derivative methods for second-order linear partial differential equations, Int. J. Numer. Meth. Engng., 44, 177-203 (1999) · Zbl 0938.65134
[13] Dyka, C. T.; Ingel, R. P., An approach for tension instability in smoothed particle hydrodynamics (SPH), Comput. Struct., 57, 573-580 (1995) · Zbl 0900.73945
[14] Dyka, C. T.; Randles, P. W.; Ingel, R. P., Stress points for tension instability in SPH, Int. J. Numer. Meth. Engng., 40, 2325-2341 (1997) · Zbl 0890.73077
[15] Belytschko, T.; Liu, W. K.; Moran, B., Finite Element Methods for Nonlinear Continua and Structures (2000), Wiley: Wiley New York
[16] Shepard, D., A two dimensional function for irregularly spaced data, ACM National Conf. (1968)
[17] Belytschko, T.; Krongauz, Y.; Dolbow, J.; Gerlach, C., On the completeness of meshfree particle methods, Int. J. Numer. Mech. Engng., 43, 785-819 (1998) · Zbl 0939.74076
[18] Beissel, S.; Belytschko, T., Nodal integration of the element-free Galerkin method, Comput. Meths. Appl. Mech. Engrg., 139, 49-74 (1996) · Zbl 0918.73329
[19] Seydel, R., From Equilibrium to Chaos, Practical Bifurcation and Stability Analysis (1988), Elsevier: Elsevier New York · Zbl 0652.34059
[20] Thompson, J. M.T.; Hunt, G. W., Elastic Instability Phenomena (1984), Chichester, Wiley: Chichester, Wiley New York · Zbl 0636.73034
[21] Ogden, R. W., Non-Linear Elastic Deformations (1984), Ellis Horwood · Zbl 0541.73044
[22] Swegle, J. W.; Hicks, D. L.; Attaway, S. W., Smoothed particle hydrodynamics stability analysis, J. Comput. Phys., 116, 123-134 (1995) · Zbl 0818.76071
[23] Kachanov, L. M., Time of the rupture process under creep conditions, Izv. Akad. Nauk SSR Otd. Tech., 8, 26-31 (1958), Nauk
[24] Kachanov, L. M., Foundations of the Theory of Plasticity (1971), North Holland: North Holland London · Zbl 0231.73015
[25] Grady, D. E.; Benson, D. A., Fragmentation of metal rings by electromagnetic loading, Exp. Mech., 12, 393-400 (1983)
[26] Mott, N. F., Fragmentation of shell cases, Proc. Roy. Soc. London, 300, 300-308 (1947)
[27] Hao, S.; Liu, W. K.; Chang, C. T., Computer implementation of damage models by finite element and meshfree methods, Comput. Meths. Appl. Mech. Engrg., 187, 401-440 (2000) · Zbl 0980.74063
[28] Fried, I.; Johnson, A. R., A note on elastic energy density function for largely deformed compressible rubber solids, Comput. Meths. Appl. Mech. Engrg., 69, 53-64 (1988) · Zbl 0653.73028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.