Reproducing kernel particle methods for structural dynamics. (English) Zbl 0840.73078

Reproducing kernel particle method (RKPM) is investigated, and the theory is presented with numerical experiments which confirm the derived equations. The method can be applied to a class of dynamic problems. In RKPM, the classical mesh of generated elements is changed by a mesh free system which only requires a set of nodes or particles in space. Flexible window functions are implemented using Gaussian or cubic spline functions in order to provide the refinement in the solution process. The method is able to analyse a specific frequency range in dynamic problems by reducing the computer time required. The stability of the window function as well as the critical time step formula are investigated. Many numerical experiments are performed to confirm the theoretical statements referring to reconstitution of given functions and solving elastic and elastic-plastic one-dimensional bar problems for both small and large deformation as well as two-dimensional large deformation nonlinear elastic problems.
Numerical and theoretical results show that the proposed reproducing kernel interpolation functions satisfy the consistency conditions and provide the critical time step prediction; moreover, the RKPM exhibits better stability properties than smooth particles hydrodynamics methods.


74S30 Other numerical methods in solid mechanics (MSC2010)
74H45 Vibrations in dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
Full Text: DOI


[1] Nayroles, Comput. Mech. 10 pp 307– (1992)
[2] Belytschko, Int. j. numer. methods eng. 37 pp 229– (1994)
[3] Belytschko, Comput. Methods Appl. Mech. Eng. 113 pp 397– (1994)
[4] Belytschko, Modelling Simul. Mater. Sci. Eng. 2 pp 519– (1994)
[5] Lancaster, Math. Comput. 37 pp 141– (1981)
[6] Gingold, Mon. Not. Roy. Astron. Soc. 181 pp 375– (1977) · Zbl 0421.76032
[7] Monaghan, SIAM J. Sci. Stat. Comput. 3 pp 422– (1982)
[8] Monaghan, Comp. Phys. Comm. 48 pp 89– (1988)
[9] and , ’Reproducing kernel and wavelet particle methods’, in and (eds.), Aerospace Structures: Nonlinear Dynamic and System Response, AD 33, ASME, 1993, pp. 39-56.
[10] and , ’Reproducing kernel particle methods’, Int. j. numer. methods fluids, accepted for publication.
[11] and , ’Reproducing kernel particle methods for elastic and plastic problems’, in and (eds.), Advanced Computational Methods for Material Modeling, AMD 180 and PVP 268, ASME, 1993 pp. 175-190.
[12] and , ’Smooth particle hydrodynamics with strength of materials’, in Advances in the Free-Lagrange Method, Lecture Notes in Physics, Vol. 395, 1990, pp. 248-257.
[13] , and , ’Coupling of smooth particle hydrodynamics with pronto’, preprint, 1993.
[14] Libersky, J. Comput. Phys. 109 pp 67– (1993)
[15] private communication, Berlin, 1993.
[16] An Introduction to Wavelets, Academic Press, New York, 1992. · Zbl 0925.42016
[17] Liu, Int. j. numer. methods eng. 32 pp 969– (1991)
[18] Ten Lectures on Wavelets, CBMS/NSF Series in Applied Mathematics, No. 61, SIAM, Philadelphia, PA, 1992. · Zbl 0776.42018
[19] and , ’Wavelet and multiple scale reproducing kernel methods’, Int. j. numer. methods fluids, accepted for publication. · Zbl 0885.76078
[20] Hughes, J. Appl. Mech. ASME 45 pp 371– (1978) · Zbl 0392.73076
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.