A unified stability analysis of meshless particle methods

International Journal for Numerical Methods in Engineering

Volumn 48, Issue 9, 2000, Pages 1359-1400

  Belytschko, Ted ^a   Guo, Yong ^a   Liu, Wing Kam ^a   Xiao, Shao Ping ^a  

^a Northwestern University   (United States)

Author keywords

Kernel; Particle methods; Stability

Indexed keywords

KERNELS; MESHLESS PARTICLE METHODS; SMOOTHED PARTICLE METHOD;

BOUNDARY CONDITIONS; CONVERGENCE OF NUMERICAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; FOURIER TRANSFORMS; GALERKIN METHODS; INTEGRAL EQUATIONS; LAGRANGE MULTIPLIERS; PARTIAL DIFFERENTIAL EQUATIONS; PERTURBATION TECHNIQUES; STIFFNESS MATRIX; TENSILE STRESS; TENSORS;

LEAST SQUARES APPROXIMATIONS;

EID: 0034230204     PISSN: 00295981     EISSN: None     Source Type: Journal    
DOI: 10.1002/1097-0207(20000730)48:9<1359::AID-NME829>3.0.CO;2-U     Document Type: Article

References (44)

1. Lucy LB. A numerical approach to the testing of the fission hypothesis. Journal of Astronomy 1977; 82:1013-1024.
2. Gingold RA, Monaghan JJ. Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly Notice of the Royal Astronomical Society 1977; 181:375-389.
3. Mas-Gallic S, Raviart P. A particle method for first-order symmetric systems. Numerische Mathematik 1987; 51: 323-352.
4. Belytschko T, Krongauz K, Dolbow J, Gerlach C. On the completeness of meshfree particle methods. International Journal for Numerical Methods in Engineering 1998; 43:785-819.
5. Liu WK, Jun S, Zhang Y-F. Reproducing kernel particle methods. International Journal for Numerical Methods in Fluids, 1995; 21:901-931.
6. Krongauz K, Belytschko T. Consistent pseudo-derivatives in meshless methods. Computer Methods in Applied Mechanics and Engineering 1997; 146:371-386.
7. Randles P, Libersky L. Smoothed particle hydrodynamics: some recent improvements and applications. Computer Methods in Applied Mechanics and Engineering 1996: 139:375-408.
8. Belytschko T, Lu Y-Y, Gu L. Element-free Galerkin methods. International Journal for Numerical Methods in Engineering 1994; 37:229-256.
9. Belytschko T, Krongauz K, Organ D, Fleming M, Krysl P. Meshless methods: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering 1996; 139:3-47.
10. Liu WK, Chen Y, Chang C-T, Belytschko T. Advances in multiple scale kernel particle methods. Computational Mechanics 1996; 18(2):31-111.
11. Hockney R, Eastwood J. Computer Simulation Using Particles. McGraw-Hill: New York, 1981.
12. Harlow FH. PIC and its progeny. Computer Physics Communications 1988; 48:1.
13. Duarte CA, Oden JT. Hp clouds - a meshless method to solve boundary-value problems. Technical Report 95-05, University of Texas at Austin, 1995.
14. Duarte CA, Oden JT. An H-P adaptive method using clouds. Computer Methods in Applied Mechanics and Engineering 1996; 139:237-262.
15. Melenk JM, Babuška I. The partition of unity finite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering 1996; 139:289-314.
16. Babuška I, Melenk JM. The partition of unity finite element method. International Journal for Numerical Methods in Engineering 1997; 40:727-758.
17. Swegle JW, Hicks DL, Attaway SW. Smoothed particle hydrodynamics stability analysis. Journal of Computational Physics 1995; 116:123-134.
18. Dyka CT, Ingel RP. An approach for tension instability in smoothed particle hydrodynamics (SPH). Computers and Structures, 1995, 57:573-580.
19. Dyka CT, Randles PW, Ingel RP. Stress points for tension instability in SPH. International Journal for Numerical Methods in Engineering 1997; 40:2325-2341.
20. Dilts GA. Moving-least-squares-particle hydrodynamics: I. Consistency and stability. International Journal for Numerical Methods in Engineering 1999; 44:1115-1155.
21. Swegle JW, Attaway SW, Heinsein MW, Mello FJ, Hicks DL. An analysis of smoothed particle hydrodynamics. Report No. SAND93-2513-UC-705, Sandia National Laboratories, Albuquerque, NM, 1994.
22. Wen Y, Hicks DL, Swegle JW. Stabilizing SPH with conservative smoothing. Report No. SAND94-1932-UC-705, Sandia National Laboratories, Albuquerque, NM, 1994.
23. Morris JP. A study of the stability properties of SPH, Applied Mathematics Reports and Preprints, Monash University, 1994.
24. Morris JP. A study of the stability properties of smoothed particle hydrodynamics. Publications of the Astronomical Society of Australia 1996; 13.
25. Beissel S, Belytschko T. Nodal integration of the element-free Galerkin method. Computer Methods in Applied Mechanics and Engineering 1996; 139:49-74.
26. Hill R. Journal of the Mechanics and Physics of Solids 1962; 10:1.
27. Rice JR. In Theoretical and Applied Mechanics, Koher WT (ed.). North-Holland: Amsterdam, 1977; 207.
28. Rudnicki JW, Rice JR. Journal of the Mechanics and Physics of Solids, 1975; 23:371.
29. Belytschko T, Liu WK, Moran B. Finite Element Methods for Nonlinear Continua and Structures. Wiley: New York, 2000, to be published.
30. Krongauz K, Belytschko T. Enforcement of essential boundary conditions in meshless approximations using finite elements, Computer Methods in Applied Mechanics and Engineering 1996; 131:133-145.
31. Belytschko T, Krongauz K, Fleming M, Organ D, Liu WK. Smoothing and accelerated computations in the element-free Galerkin method. Journal of Computational and Applied Mathematics 1996; 74:111-126.
32. Monaghan JJ. An introduction to SPH. Computer Physics Communications 1988; 48:89-96.
33. Bonet J, Lok T-SL. Variational and momentum preserving aspects of smoothed particle hydrodynamics (SPH) formulations. Computer Methods in Applied Mechanics and Engineering 1999; 180(1-2):97-116.
34. Seydel R. From Equilibrium to Chaos, Practical Bifurcation and Stability Analysis. Elsevier: New York, 1988.
35. Thompson JMT, Hunt GW. Elastic Instability Phenomena. Wiley: New York, 1984.
36. Belytschko T, Ong J-S, Liu WK, Kennedy JM. Hourglass control in linear and nonlinear problems. Computer Methods in Applied Mechanics and Engineering 1984; 43:251-276.
37. Guo Y. Composite finite elements and stabilization of meshfree methods. Ph. D. Dissertation, Northwestern University, Evanston, IL, 1999.
38. Hallquist JO. DYNA2D and DYNA3D user's manuals. Technical Report, Lawrence Livermore National Laboratories, Livermore, CA, 1980.
39. Souter K, Krachman H. Two-dimensional water-impact tests of flexible cylinders. EPRI Report NP-1612, November, 1980.
40. Monaghan JJ. Simulating free surface flows with SPH. Journal of Computational Physics 1994; 110:399.
41. Belytschko T, Neal MO. The vectorized pinball contact-impact routine. In Transactions of the 10th International Conference on Structural Mechanics in Reactor Technology, Los Angeles, CA, Hadjian AH (ed.), Vol. B. AASMiRT: Anaheim, CA, 1989; 161-166.
42. Belytschko T, Neal MO. Contact-impact by the pinball algorithm with penalty and Lagrangian methods. International Journal for Numerical Methods in Engineering 1991; 31:547-572.
43. Kachanov LM. Time of the rupture process under creep conditions. Izvestiya Akademii Nauk SSR Otdelenie Technicheskikh Nauk 1958; 8:26-31.
44. Lemaitre J. Evaluation of dissipation and damage in metal submitted to dynamic loading. Proceedings ICM 1, Kyoto, Japan, 1971.