An improved material point method using moving least square shape functions
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An improved material point method using moving least square shape functions Jae-Uk Song1 • Hyun-Gyu Kim1 Received: 25 May 2020 / Revised: 4 September 2020 / Accepted: 28 September 2020 Ó OWZ 2020
Abstract In this study, moving least square (MLS) shape functions are employed to reduce the cell-crossing error occurred in conventional material point method (MPM) when material particles pass through grid cell boundaries. The level of smoothness of MLS shape functions for mapping information from material particles to a background grid can be controlled by the support size of MLS weight functions. A simple method is proposed to reduce the computational cost for evaluating MLS shape functions at material particles by interpolating pre-computed MLS shape function values at sampling points in a grid cell. Numerical results show that the present method is very effective to reduce the cell-crossing error in MPM computations. Keywords Material point method Cell-crossing error Moving least square Shape functions
1 Introduction Recently, several particle-based methods [1–3] have been proposed to overcome difficulties arising from the distortion of elements in mesh-based methods such as finite element methods (FEMs). Among them, the material point method (MPM) [4, 5] is an effective approach to solve large deformation problems by combining Lagrangian material particles with an Eulerian background grid. In the MPM, variables at material particles are mapped to a background grid, and the deformation field computed on the background grid is mapped back to material particles. Since the background grid is fixed regardless of the motion of material particles, mechanics problems involving large deformations can be effectively simulated with the MPM [6–12]. However, conventional MPM can suffer from the socalled cell-crossing error [13] when material particles pass through the grid cell boundaries. The cell-crossing error is caused by the discontinuity in the derivatives of shape & Hyun-Gyu Kim [email protected] 1
Department of Mechanical and Automotive Engineering, Seoul National University of Science and Technology, 232 Gongneung-ro, Nowon-gu, Seoul 01811, South Korea
functions of grid nodes for mapping information from material particles to a background grid across the grid cell boundaries [14]. Bardenhagen and Kober [15] developed the generalized interpolation material point (GIMP) method to reduce the cell-crossing error. In the GIMP, Petrov–Galerkin discretization scheme is used wherein material particles are defined by particle characteristic functions representing the influence domain of each material particle. The particle characteristic functions have the effect of smoothing the impact of material particles at the grid cell boundaries. However, the GIMP has a difficulty of tracking the supports of particle characteristic functions for finite deformation problems. An instability can be encountered in the GIMP due to a large distortion of particle domains. To eliminate the
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