Reproducing kernel particle methods for structural dynamics.

*(English)*Zbl 0840.73078Reproducing 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.

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.

Reviewer: G.V.Vasiliev (Bucureşti)

##### MSC:

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74H45 | Vibrations in dynamical problems in solid mechanics |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

##### Keywords:

elastic-plastic large deformations; mesh free system; window functions; cubic spline functions; stability; critical time step; consistency conditions
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\textit{W. K. Liu} et al., Int. J. Numer. Methods Eng. 38, No. 10, 1655--1679 (1995; Zbl 0840.73078)

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