Overview and applications of the reproducing Kernel Particle methods

  • W. K. Liu
  • Y. Chen
  • S. Jun
  • J. S. Chen
  • T. Belytschko
  • C. Pan
  • R. A. Uras
  • C. T. Chang


Multiple-scale Kernel Particle methods are proposed as an alternative and/or enhancement to commonly used numerical methods such as finite element methods. The elimination of a mesh, combined with the properties of window functions, makes a particle method suitable for problems with large deformations, high gradients, and high modal density. The Reproducing Kernel Particle Method (RKPM) utilizes the fundamental notions of the convolution theorem, multiresolution analysis and window functions. The construction of a correction function to scaling functions, wavelets and Smooth Particle Hydrodynamics (SPH) is proposed. Completeness conditions, reproducing conditions and interpolant estimates are also derived. The current application areas of RKPM include structural acoustics, elastic-plastic deformation, computational fluid dynamics and hyperelasticity. The effectiveness of RKPM is extended through a new particle integration method. The Kronecker delta properties of finite element shape functions are incorporated into RKPM to develop a C m kernel particle finite element method. Multiresolution and hp-like adaptivity are illustrated via examples.


Shape Function Timoshenko Beam Window Function Correction Function Smooth Particle Hydrodynamic 
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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  1. Amsden, A.A. and Harlow, F.H. (1970), “The SMAC Method, A Numerical Technique for Calculating Incompressible Fluid Flow”, Tech. Report LA-4370, Los Alamos Scientific Laboratory.Google Scholar
  2. Babuska, I. and Melenk, J.M. (1995), “The Partition of Unity Finite Element Method”, Univ. of Maryland, Technical Note BN-1185.Google Scholar
  3. Babuška, I. (1973), “The Finite Element Method with Lagrangian Multiplies,”Num. Math.,20, pp. 179–192.CrossRefGoogle Scholar
  4. Beklkin, G.R., Coifman, I., Daubechies, S., Mallat, Y., Meyer, L., Raphael, and Ruskai, B. (eds) (1992), “Wavelets and Their Applications”, Cambridge, MA.Google Scholar
  5. Belytschko, T. (1994), “Are Finite Elements Passé?”,USACM Bulletin,7, 3, pp. 4–7.Google Scholar
  6. Belytschko, T., Gu, L. and Lu, Y.Y. (1994a), “Fracture and Crack Growth by EFG Methods,”Modelling Simul. Mater. Sci. Eng.,2, pp. 519–534.CrossRefGoogle Scholar
  7. Belytschko, T., Lu, Y.Y. and Gu, L. (1994b), “Element Free Galerkin Methods”,International Journal for Numerical Methods in Engineering,37, pp. 229–256MATHCrossRefMathSciNetGoogle Scholar
  8. Belytschko, T., Lu, Y.Y. and Gu, L. (1994c), “A New Implementation of the Element Free Galerkin Method”,Computer Methods in Applied Mechanics and Engineering 113, pp. 397–414.MATHCrossRefMathSciNetGoogle Scholar
  9. Belytschko, T. (1983), “An Overview of Semidiscretization and Time Integration Procedures”, eds. Belytschko, T. and Hughes, T.J.R.,Computational Methods for Transient Analysis, North-Holland, Amsterdam, pp. 1–63.Google Scholar
  10. Belytschko, T. and Kennedy, J.M. (1978), “Computer Methods for Subassembly Simulation,”Nuclear Engrg. Des.,49, pp. 17–38.CrossRefGoogle Scholar
  11. Brezzi, F. (1974), “On the Existance, Uniqueness and Approximation of Saddle-Point Problems Arising from Lagrange Multipliers”,R. A. I. R. O.,8-R2, pp. 129–151.MathSciNetGoogle Scholar
  12. Chen, J.S. and Pan, C. (1995), “A Pressure Projection Method for Nearly Incompressible Rubber Hyperelasticity, Part I: Theory”, accepted byJournal of Applied Mechanics.Google Scholar
  13. Chen, J.S., Pan, C. and Chang, T.Y.P. (1995b), “On The Control of Pressure Oscillation in Bilinear-Displacement Constant-Pressure Element”,Comput. Meth. Appl. Mech. Engng., in press.Google Scholar
  14. Chen, J.S., Wu, C.T. and Pan, C. (1995c), “A Pressure Projection Method for Nearly Incompressible Rubber Hyperlasticity, Part II: Applications”, accepted by Journal of Applied Mechanics.Google Scholar
  15. Chen, J.S., Han, H., Wu, C.T. and Duan, W. (1995d), “On the Perturbed Lagrangian Formulation for Nearly Compressible and Incompressible Hyperelasticity”, submitted toComputer Methods in Applied Mechanics and Engineering.Google Scholar
  16. Chen, J.S., Satyamurthy, K.S. and Hirschfelt, L.R. (1994), “Consistent Finite Element Procedures for Nonlinear Rubber Elasticity with a Higher Order Strain Energy Function”,Comput. & Struct.,50, pp.715–727.MATHCrossRefGoogle Scholar
  17. Chui, C.K. (1992), “An Introduction to Wavelets”, Academic Press.Google Scholar
  18. Daubechies (1992), “Ten Lectures on Wavelets”,CBMS/NSF Series in Applied Mathematics,61, SIAM Publication.Google Scholar
  19. Duarte, C.A. and Oden, J.T. (1995), “Hp Clouds— A Meshless Method to Solve Boundary-Value Problems”,TICAM Report 95-05.Google Scholar
  20. Dym, C.L. and Shames, I.H. (1973), “Solid Mechanics: A Variational Approach”, McGraw-Hill, Inc.Google Scholar
  21. Gent, A.N. and Lindley, P.B. (1959), “The Compression of Bonded Rubber Blocks”,Proc. Inst. Mech. Engrs.,173, pp. 11–122.Google Scholar
  22. Glowinski, R., Lawton, W.M., Ravachol, M. and Tenenbaum, E. (1990), “Wavelets solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension”. Glowinski, R. and Lichnewsky, A. eds.,Computing Methods in Applied Sciences and Engineering, SIAM, Philadelphia, pp. 55–120.Google Scholar
  23. Gingold, R.A. and Monaghan, J.J. (1982), “Kernel Estimates as a Basis for General Particle Methods in Hydrodynamics”,J. Comp. Phys.,46, pp. 429–453.MATHCrossRefMathSciNetGoogle Scholar
  24. Gingold, R.A. and Monaghan, J.J. (1977), “Smoothed Particle Bydrodynamics: Theory and Application to Non-Spherical Stars”,Mon. Not. Roy. Astron. Soc.,181, pp. 375–389.MATHGoogle Scholar
  25. Grossmann, A. and Morlet, J. (1984), “Decomposition of Hardy Functions into Square Integrable Wavelets of Constant Shape”,SIAM J. Math.,15, pp. 723–736.MATHCrossRefMathSciNetGoogle Scholar
  26. Haar, A. (1910), “Zur Theorie dr orthogonalen Funktionsysteme”,Math. Ann.,69, pp. 331–371.CrossRefMathSciNetGoogle Scholar
  27. Harlow, F.H., Amsden, A.A. and Nix, J.R. (1976), “Relativistic Fluid Dynamics Calculation with Particle-in-Cell Technique”,J. Comput. Phys.,20.Google Scholar
  28. Harlow, F.H., and Welch, J.E. (1965), “Numberical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface”,The Physics of Fluids,8, pp. 2182–2189.CrossRefGoogle Scholar
  29. Hirt, C.W. (1983), “Flow Analysis for Non-Experts”,Engineering Foundation Conference Proceedings, Modeling and Control of Casting and Welding Processes II.Google Scholar
  30. Hirt, C.W. (1975), “SOLA-A Numerical Solution Algorithm for Transient Fluid Flows”, Los Alamos Scientific Laboratory Report LA-5852.Google Scholar
  31. Huerta, A. and Liu, W.K. (1988), “Viscous Flow with Large Free Surface Motion”,Computer Methods in Applied Mechanics and Engineering,69, pp. 277–324.MATHCrossRefGoogle Scholar
  32. Hughes, T.J.R., Liu, W.K. and Zimmerman, T.K. (1981), “Lagrangian-Eulerian Finite Element Formulations for Incompressible Viscous Flows”,Computer Methods in Applied Mechanics and Engineering,29, pp. 329–349.MATHCrossRefMathSciNetGoogle Scholar
  33. Hughes, T.J.R. (1987), “The Finite Element Method”, Prentice-Hall.Google Scholar
  34. Hughes, T.J.R. (1980), “Generalization of Selective Integration Procedures to Anisotropic and Nonlinear Medium”,Int. J. Numer. Mech. Engng.,15, pp. 141–1418.Google Scholar
  35. Johnson, G.R., Peterson, E.H. and Stryrk, R.A. (1993), “Incorporation of an SPH Option into the EPIC code for a Wide Range of High Velocity Impact Computations”, preprint.Google Scholar
  36. Lancaster, P. and Salkauskas, K. (1981), “Surfaces Generated by Moving Least Squares Methods.”Mathematics of Computation,37, pp. 141–158.MATHCrossRefMathSciNetGoogle Scholar
  37. Libersky, L. and Petchek, A.G. (1990), “Smooth Particle Hydrodynamics with Strength of Materials”,Proceedings of The next Free-Lagrange Conference, Jackson Lake Lodge, Moran, Wyoming, June 3–7.Google Scholar
  38. Liu, W.K. (1995), “An Introduction to Wavelet Reproducing Kernel Particle Methods”,USACM Bulletin,8, 1, pp. 3–16.Google Scholar
  39. Liu, W.K. and Chen, Y. (1995), “Wavelet and Multiple Scale Reproducing Kernel Methods”,International Journal for Numerical Methods in Fluids, Vol.21, pp. 901–932.MATHCrossRefMathSciNetGoogle Scholar
  40. Liu, W.K., Jun, S., Li, S., Adee, J. and Belytschko, T. (1995a), “Reproducing Kernel Particle Methods for Structural Dynamics”,International Journal of Numerical Methods for Engineering,38, pp. 1655–1679.MATHCrossRefMathSciNetGoogle Scholar
  41. Liu, W.K., Jun, S. and Zhang, Y.F. (1995b), “Reproducing Kernel Particle Methods”,International Journal of Numerical Methods in Fluids,20, pp. 1081–1106.MATHCrossRefMathSciNetGoogle Scholar
  42. Liu, W.K., Li, Shaofan and Belytschko, T. (1995c), “Moving Least Square Kernel Galerkin Method (I) Methodology and Convergence”, submitted toComputer Methods in Applied Mechanics and Engieering.Google Scholar
  43. Liu, W.K., Chen, Y. and Uras, R.A. (1995d), “Enrichment of the Finite Element Method with Reproducing Kernel Particle Method”,Current Topics in Computational Mechanics, eds. Cory, J.F. Jr. and Gordon, J.L., ASME PVP,305, pp. 253–258.Google Scholar
  44. Liu, W.K. and Hu, Y.K. (1994), “Finite Element Hydrodynamic Friction Model for Metal Forming,”International Journal of Numerical Methods for Engineering,37, pp. 4015–4037.MATHCrossRefMathSciNetGoogle Scholar
  45. Liu, W.K., Hu, Y.K. and Belytschko, T. (1994), “Multiple Quadrature Underintegrated Finite Elements”,Int. J. Numer. Mech. Engng. 37, pp. 3262–3289.MathSciNetGoogle Scholar
  46. Liu, W.K. and Hu, Y.K. (1993), “An ALE Hydrodynamic Lubricated Finite Element Method with Application to Strip Rolling”,International Journal of Numerical Methods for Engineering,36, pp. 855–880.MATHCrossRefGoogle Scholar
  47. Liu, W.K. and Oberste-Brandenburg, C. (1993), “Reproducing Kernel and Wavelet Particle Methods”,Aerospace Structures: Nonlinear Dynamics and System Response, eds. Cusumano, J.P., Pierre, C., and Wu, S.T., AD 33, ASME, pp. 39–56.Google Scholar
  48. Liu, W.K., Adee, J. and Jun, S. (1993), “Reproducing Kernel Particle Methods for Elastic and Plastic Problems”,Advanced Computational Methods for Material Modeling, eds. Beson, D.J. and Asaro, R.A., AMD 180 and PVP 268, ASME, pp. 175–190.Google Scholar
  49. Liu, W.K. (1992), “Arbitrary Lagrangian-Eulerian Finite Elements for Fluid-Shell Interaction Problems”,J. of the Braz. Soc. of Mech. Sc.,XIV-4, pp. 347–368.Google Scholar
  50. Liu, W.K. and Haeussermann, U. (1992), “Multiple Temporal and Spatial Scale Methods,”New Methods in Transient Analysis, P. Smolinski, W. K. Liu, G. Hulbert and K. Tamma, eds., PVP 246/AMD 143, ASME, pp. 51–64.Google Scholar
  51. Liu, W.K. and Hu, Y.K. (1992), “ALE Finite Element Formulation for Ring Rolling Analysis,”International Journal of Numerical Methods for Engineering,33, pp. 1217–1236.MATHCrossRefGoogle Scholar
  52. Liu, W.K., Hu, Y.K. and Belytschko, T. (1992a), “ALE Finite Elements with Hydrodynamic Lubricated for Metal Forming”,Nuclear Engineering and Design,138, pp. 1–10.CrossRefGoogle Scholar
  53. Liu, W.K., Zhang, Y.F. and Ramirez, M.R. (1991a), “Multiple Scale Finite Element Methods”,International Journal for Numerical Methods in Engineering,32, pp. 969–990.MATHCrossRefGoogle Scholar
  54. Liu, W.K., Zhang, Y.F., Belytschko, T. and Chen, J.S. (1991b), “Adaptive ALE Finite Elements with Particular Reference to External Work Rate on Frictional Interface”,Computer Methods in Applied Mechanics and Engineering,93, pp. 189–216.MATHCrossRefGoogle Scholar
  55. Liu, W.K., Belytschko, T. and Chen, J.S. (1988a), “Nonlinear Versions of Flexurally Superconvergent Elements”,Comput. Meth. Appl. Mech. Engng.,71, pp. 24–256.CrossRefGoogle Scholar
  56. Liu, W.K., Chang, H. and Belytschko, T. (1988b), “Arbitrary Lagrangian and Eulerian Petrov-Galerkin Finite Elements for Nonlinear Continua”,Computer Methods in Applied Mechanics and Engineering,68, pp. 259–310.MATHCrossRefGoogle Scholar
  57. Liu, W.K., Belytschko, T. and Chang, H. (1986), “An Arbitrary Lagrangian Eulerian Finite Element Method for Path-Dependent Materials”,Computer Methods in Applied Mechanics and Engineering,58, pp. 227–246.MATHCrossRefGoogle Scholar
  58. Liu, W.K., Belytschko, T., Ong, J.S.J. and Law, E. (1985a), “Use of Stabilization Matrices in Nonlinear Finite Element Analysis”,Engineering Computations,2, pp. 47–55.Google Scholar
  59. Liu, W.K., Ong, J.S.J. and Uras, R.A. (1985b), “Finite Element Stabilization Matrices—A Unification Approach,”Comput. Meth. Appl. Mech. Engng.,53, pp. 13–46.MATHCrossRefMathSciNetGoogle Scholar
  60. Liu, W.K. and Belytschko, T. (1984), “Efficient Linear and Nonlinear Heat Conduction with a Quadrilateral Element”,International Journal for Numerical Methods in Engineering,20, pp. 931–948.MATHCrossRefMathSciNetGoogle Scholar
  61. Liu, W.K. (1981), “Finite Element Procedures for Fluid-Structure Interactions and Applications to Liquid Storage Tanks”,Nuclear Engineering and Design,64, 2, pp. 221–238.CrossRefGoogle Scholar
  62. Lucy, L. (1977), “A Numerical Approach to Testing the Fission Hypothesis”,A.J.,82, pp. 1013–1024.CrossRefGoogle Scholar
  63. Lu, Y.Y., Belytschko, T. and Gu, L. (1994), “A New Implementation of the Element Free Galerkin Method”,Comput. Meth. Appl. Mech. Engng.,113, pp. 397–414.MATHCrossRefMathSciNetGoogle Scholar
  64. Mallat, S. (1989), “Multi-resolution Approximations and Wavelet Orthogonal Bases of L2(R)”,Trans. Amer. Math. Soc.,315, pp. 69–87.MATHCrossRefMathSciNetGoogle Scholar
  65. Monaghan, J.J. (1982), “Why Particle Methods Work”,SIAM J. Sci. Stat. Comput.,3, pp. 422–433.MATHCrossRefMathSciNetGoogle Scholar
  66. Monaghan, J.J. and Gingold, R.A. (1983), “Shock Simulation by the Particle Method SPH”,J. Comp. Phys.,52, pp. 374–389.MATHCrossRefGoogle Scholar
  67. Monaghan, J.J. (1988), “An Introduction to SPH”,Comp. Phys. Comm.,48, pp. 89–96.MATHCrossRefGoogle Scholar
  68. Mooney, M. (1940), “A Theory of Large Elastic Deformation”,J. Appl. Phys.,11, pp. 582–592.CrossRefGoogle Scholar
  69. Nayroles, B., Touzot, G. and Villon, P. (1992), “Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Elements”,Computational Mechanics,10, pp.307–318.MATHCrossRefGoogle Scholar
  70. Nichols, B.D. and Hirt, C.W. (1971), “Improved Free Surface Boundary Conditions for Numerical Incompressible Flow Calculations”,J. Comp. Phys.,8.Google Scholar
  71. Oñate, E., Idelsohn, S. and Zienkiewicz, O.C. (1995), “Finite Point Methods in Computational Mechanics”,International Center for Numerical Methods in Engineering, July.Google Scholar
  72. Penn, R. W. (1970), “Volume Changes Accompanying the Extension of Rubber”,Trans. Soc. Rheo.,14, 4, pp. 50–517.Google Scholar
  73. Poularikas, A.D. and Seely, S., “Signals and Systems”, PWS-KENT Publishing Company, 2nd edition.Google Scholar
  74. Rivlin, R.S. (1956), “Rheology Theory and Applications”, ed. Eirich, F.R. 1, Chap. 10, pp. 351–385, Academic Press, New York.Google Scholar
  75. Shodja, H.M., Mura, T. and Liu, W.K. (1995), “Multiresolution analysis of a Micromechanical Model”,Computational Methods in Micromechanics, ASME AMD 212/MD 62, eds. S. Ghosh and M. Ostoja-Starzewski, pp. 33–54.Google Scholar
  76. Stellingwerf, R.F. and Wingate, C.A. (1993), “Impact Modeling with Smooth Particle Hydrodynamics”,Int. J. Impact Engng.,14, pp. 707–718.CrossRefGoogle Scholar
  77. Strang, G. (1989), “Wavelets and Dilation Equations: a Brief Introduction”,SIAM Rev.,31, 4, pp. 614–627.MATHCrossRefMathSciNetGoogle Scholar
  78. Subbiah, S.,et al.(1989), “Non-isothermal Flow of Polymers into Two-dimensional, Thin Cavity Molds: A Numerical Grid Generation Approach”,Int. J. Heat Mass Transfer,32, 3, pp. 415–434.CrossRefGoogle Scholar
  79. Sulsky, D., Chen, Z. and Schreyer, H.L. (1992), “The Application of a Material-Spatial Numerical Method to Penetration”,New Methods in Transient Analysis, eds. Smolinski, P., Liu, W.K., Hulbert, G. and Tamma, K., ASME, PVP, Vol. 246/AMD143, pp 91–102.Google Scholar
  80. Sussman, T.S. and Bathe, K.J. (1987), “A Finite Element Formulation for Incompressible Elastic and Inelastic Analysis”,Comput. & Struct.,26, pp. 357–409.MATHCrossRefGoogle Scholar
  81. Williams, J.R. and Amaratunga, K. (1994), “Introduction to Wavelets in Engineering”,International Journal for Numerical Methods in Engineering,37, pp. 2365–2388.MATHCrossRefMathSciNetGoogle Scholar
  82. Yagawa, G., Yamada, T. and Kawai (1995), “Some Remarks on Free Mesh Method: A kind of Meshless Finite Element Methods”, To be presented at ICES-95.Google Scholar
  83. Yeoh, O.H., “Characterization of Elastic Properties of Carbon Black Filled Rubber Vulcanizates”,Rubb. Chem. Technol.,63, pp. 79–805.Google Scholar
  84. Yeoh, O.H. (1993), “Some Forms of the Strain Energy Function for Rubber”,Rubb. Chem. Technol.,66, pp. 754–771.Google Scholar

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© CIMNE 1996

Authors and Affiliations

  • W. K. Liu
    • 1
  • Y. Chen
    • 1
  • S. Jun
    • 1
  • J. S. Chen
    • 1
  • T. Belytschko
    • 1
  • C. Pan
    • 1
  • R. A. Uras
    • 1
  • C. T. Chang
    • 1
  1. 1.Department of Mechanical EngineeringNorthwestern UniversityEvanston

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