A local grid refinement scheme for B‐spline material point method

Abstract

A local grid refinement scheme for the material point method with B-spline basis functions (BSMPM) is developed based on the concept of bridging domain approach. The fine grid is defined in the local large-deformation regions to accurately capture the complex material responses, whereas other spatial domains are discritized by coarse grids. In the overlapping domain between the fine and coarse grids, the constraint of particle displacements obtained with different grids is enforced using the Lagrange multiplier method to eliminate the spurious stress reflection at the fine/coarse grid interface. Representative numerical examples have shown that the BSMPM simulations with the proposed local grid refinement scheme could provide the solutions in good agreement with those obtained with the uniformly fine grid, and that no significant spurious stress reflection is induced at the fine/coarse grid interface, even for the bridging domain size as small as the cell size of the fine grid. It is also found that the proposed local grid refinement method can significantly reduce the BSMPM computational time compared with the cases for uniformly fine grids. A multitime-step algorithm is presented and shown to considerably enhance the efficiency of the present local grid refinement scheme with no compromise in accuracy.

Mapped material point method for large deformation problems with sharp gradients and its application to soil‐structure interactions

“…To locally refine an MPM discretization, one must derive and implement quite a complicated scheme devoted to a specific type of basis function used in the background grid, as done in the literature for linear basis functions, 26,33 the generalized interpolation material point (GIMP) method's basis functions, 13,34,35 and B-splines basis functions. 36,37 As such, a local refinement scheme formulated for one type of basis function cannot be applied to other types of basis functions. Furthermore, a locally refined background grid is not compatible with some MPM schemes that are originally built on basis functions with a uniform background grid.…”

“…Local refinement, however, is a less attractive way for the MPM than that for mesh‐based methods like the FEM. To locally refine an MPM discretization, one must derive and implement quite a complicated scheme devoted to a specific type of basis function used in the background grid, as done in the literature for linear basis functions, 26,33 the generalized interpolation material point (GIMP) method's basis functions, 13,34,35 and B‐splines basis functions 36,37 . As such, a local refinement scheme formulated for one type of basis function cannot be applied to other types of basis functions.…”

Extended B‐spline‐based implicit material point method enhanced by F‐bar projection method to suppress pressure oscillation

“…As an alternative, Steffen et al 19 employed B‐spline basis functions in the MPM to suppress the cell‐crossing instability. Subsequently, this approach seems to have become mainstream; see References 20‐23.…”

Multiple‐GPU parallelization of three‐dimensional material point method based on single‐root complex

“…Structured elements, used as often as the unstructured elements, 36,37 bring extra computational loads with identical mesh size from the concerning domain to the far field. Elements in singularity zone around structures need to be further refined for soil‐structure interaction problems 38,39 . Although an initial assignment of four particles in per element is often sufficient to obtain a smooth stress/strain field in many cases of MPM simulations, 40 the configuration of 16 particles in each element sometimes is necessary for high‐speed impacting problems 14,41 .…”