Development of an implicit cell-based material point method

  • Song, Jae-Uk (Seoul National University of Sci. and Tech.)
  • Kim, Hyun-Gyu (Seoul National University of Sci. and Tech.)

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As an extension of particle-in-cell methods, Sulsky et al. [1] developed the material point method (MPM) based on the combination of Lagrangian and Eulerian material descriptions to solve large deformation problems. In the conventional MPM, the volume integrals in the weak form are evaluated at material particles with the corresponding material volumes. The numerical integration at material particles causes an inaccurate evaluation of the weak form when material particles are unequally placed over a background grid. In addition, the cell-crossing error occurs when material particles cross the grid cell boundaries due to the lack of smoothness of the piecewise linear shape functions defined on a background grid. In order to resolve these issues, a grid cell-based integration scheme is proposed in this study. Because the shape functions have continuous differential values inside the cell, cell-crossing errors do not occur when the integration points are located inside the grid cells. A further advantage of the cell-based integration method is that quadrature errors are reduced when material particles move largely through the background grid. In this paper, we propose a novel implicit MPM using a cell-based integration scheme for solving large deformation static problems. When material particles are located at the boundary of a background grid, the convergence problem arises in the conventional MPM due to an inaccurate evaluation of the weak form near the boundary. However, the convergence problem does not occur in the novel MPM using the cell-based integration scheme. A methodology for modelling the boundaries of the analysis domains is also proposed in this paper. The domain boundaries are defined using the level set values of material particles, and the integration points are refined in the grid cells associated with the domain boundaries. Since the active grid cells in conventional MPMs are taken as the analysis domains, the domain boundaries cannot be precisely defined. However, the present method can be efficiently and effectively used to apply boundary tractions and contact interactions on the domain boundaries.