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Stability of DPD and SPH

  • Philip W. Randles
  • Albert G. Petschek
  • Larry D. Libersky
  • Carl T. Dyka
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 26)

Abstract

It is shown that DPD (Dual Particle Dynamics) and SPH (Smoothed Particle Hydrodynamics) are conditionally stable for Eulerian kernels and linear fields. This result is important because it is highly desirable to move and change neighbors where the material deformation is large. For higher dimensions (than 1D), stability for general neighborhoods is shown to require a two-step update, such a predictor-corrector. Co-locational methods (all field variables calculated on every particle) benefit from the completeness property also. We show that SPH with corrected derivatives is conditionally stable. Linear completeness of interpolations is shown to assert itself as a powerful ally with respect to stability as well as accuracy.

Keywords

Smooth Particle Hydrodynamic Background Field Move Little Square Stress Point Motion Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Lucy L. (1977) A Numerical Approach to the Fission Hypothesis. Astron. J. 82: 1013CrossRefGoogle Scholar
  2. 2.
    Gingold R.A., Monaghan J.J. (1977) Smoothed Particle Hydrodynamics: Theory and Application to N-n-Spherical Stars. Mon. Not. Roy. Astron. Soc. 181:375zbMATHGoogle Scholar
  3. 3.
    Liszka T., Orkisz J., Modified finite difference Methods at arbitrary Irregular Meshes and its Application i Applied Mechanics, Proc. Of the 18th Polish Conf. On Mechanics of Solids, Wisla, Poland (1976)Google Scholar
  4. 4.
    Harlow F. H. (1964) The Particle-in-Cell Method for Fluid Dynamics. Methods in Computational Physics, 3, B. Adler, S. Fernbach, and M. Rotenberg editors, Academic Press, New YorkGoogle Scholar
  5. 5.
    Libersky L.D., Petschek A.G. (1990) Smoothed Particle Hydrodynamics With Strength of Materials. In: Trease, Fritts, and Crowley, (Eds.), Proceedings of the Next Free-Lagrange Conference, Jackson Hole, WY, USA, June 3-7, 1990. Advances in the Free-Lagrange Method Lecture Notes in Physics, 395, Springer Verlag, 248–257Google Scholar
  6. 6.
    Randies P.W., Libersky L.D., Carney T.C. (1995) SPH Calculations of Fragmentation in the MK82 bomb. In: APS Topical Conf. On Shock Compression of Condensed Matter, Seattle, WA,Google Scholar
  7. 7.
    Libersky L.D., Randies P.W., Carney T.C. (1995) SPH Calculations of Fragmentation, Proc. Of 3rd U. S. Congress on Computational Mechanics (USACM), Dallas, TX, June 1995Google Scholar
  8. 8.
    Belytschko T., Lu Y.Y., Gu L. (1994) Element Free Galerkin Methods, International Journal for Numerical Methods in Engineering, 37:229MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Liu W.K., Li S, Belytschko T. (1997) Moving Least-Square Reproducing Kernel Methods, Part I: Methodology and Convergence. Computer Methods in Applied Mechanics and Engineering 143:113MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Duarte C.A., Oden J.T. (1996) An h-p Adaptive Method Using Clouds. Computer Methods in Applied Mechanics and Engineering 139:237MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Nayroles B., Touzot G., Villon P. (1992) Generalizing the FEM: Diffuse Approximation and Diffuse Elements. Computational Mechanics 10:307zbMATHCrossRefGoogle Scholar
  12. 12.
    Onate E., Idelsohn S., Zienkiewicz O.C., Taylor R.L. (1996) A Finite Point Method in Computational Mechanics. Applications to Convective Transport and Fluid Flow, International Journal for Numerical Methods in Engineering. 39:3839MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Atmri S.N., Cho J.Y., Kim H. (1999) Analysis of Thin Beams, using the Mesh-less Local Petrov-Galerkin Method, with Generalized Moving Least Squares Interpolations. Computational Mechanics 24:334CrossRefGoogle Scholar
  14. 14.
    De S., Bathe K.J. (2000) The Method of Finite Spheres. Computational Mechanics 25:329MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Babuška I., Melenk J.M. (1997) The Partition of Unity Method, International Journal for Numerical Methods in Engineering. 40:727MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Swegle J.W., Hicks D.L. Attaway S.W. (1995) Smoothed Particle Hydrodynamics Stability Analysis, J. Comp. Phys. 116:123MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Johnson G.R., Bissell S.R. (1996) Normalized Smoothing Functions for Impact Calculations. Int. J. Num. Meth. Engng. 39:2725zbMATHCrossRefGoogle Scholar
  18. 18.
    Randies P.W., Libersky L.D. (1996) Smoothed Particle Hydrodynamics: Some Recent Improvements and Applications. Computer Methods in Applied Mechanics and Engineering, 139:375MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bonet J., Kulasegaram S. (2000) Correction and Stabilization of Smooth Particle Hydrodynamics Methods with Applications in Metal Forming Simulations. International Journal for Numerical Methods in Engineering, 47:1189zbMATHCrossRefGoogle Scholar
  20. 20.
    Dilts G.A. (1997) Moving Least Squares-Particle Hydrodynamics I, Consistency and Stability, Los Alamos Report LA-UR-97-4168Google Scholar
  21. 21.
    Dyka C.T. and Ingel R.P. (1995) An Approach for Tension Instability in Smooth Particle Hydrodynamics (SPH), Computers and Structures, Volume 57:573zbMATHCrossRefGoogle Scholar
  22. 22.
    Dyka C.T., Randies P.W., Ingel R.P. (1997) Stress Points for Tension Instability in SPH, International Journal for Numerical Methods in Engineering, Volume 40:2325zbMATHCrossRefGoogle Scholar
  23. 23.
    Belytschko T., Guo Y., Liu W.K., Xiao S.P (2000) A unified stability analysis of meshless particle methods, International Journal for Numerical Methods in Engineering, Volume 48:1359MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Bonet J., Kulasegaram S. (2001) Remarks on Tension Instability of Eulerian and Lagrangian Corrected Smoothed Particle (CSPH) Methods, International Journal for Numerical Methods in Engineering 52, Issue: 11:1203zbMATHCrossRefGoogle Scholar
  25. 25.
    Richtmeyer R.D., Morton K.W. (1957) Difference Methods for Initial-Value Problems, Interscience, New YorkGoogle Scholar
  26. 26.
    Balsara D.S. (1995) von-Neumann Stability Analysis of Smoothed Particle Hydrodynamics — Suggestions for Optimal Algorithms. J. Comp. Phys. 121:357zbMATHCrossRefGoogle Scholar
  27. 27.
    Morris J.P. (1996) Stability properties of SPH. Publ. Astron. Soc. Aust. 13:97Google Scholar
  28. 28.
    Randies P.W. and Libersky L.D. (2000) Normalized SPH with Stress Points, International Journal of Numerical Methods in Engineering, Volume 48:1445CrossRefGoogle Scholar
  29. 29.
    Caramana E.J. (2001) Private communications, Los Alamos National LaboratoryGoogle Scholar
  30. 30.
    Wolfram S. (1991) Mathematica, A System for Doing Mathematics by Computer, Second Addition, Addison-WesleyGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Philip W. Randles
    • 1
  • Albert G. Petschek
    • 2
  • Larry D. Libersky
    • 2
  • Carl T. Dyka
    • 3
  1. 1.Defense Threat Reduction AgencyKirtland AFBUSA
  2. 2.Los Alamos National LaboratoryLos AlamosUSA
  3. 3.Naval Surface Warfare CenterIndian HeadUSA

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