Enhancement of the material point method using B‐spline basis functions

Abstract

The material point method (MPM) enhanced with B-spline basis functions, referred to as B-spline MPM (BSMPM), is developed and demonstrated using representative quasi-static and dynamic example problems. Smooth B-spline basis functions could significantly reduce the cell-crossing error as known for the original MPM. A Gauss quadrature scheme is designed and shown to be able to diminish the quadrature error in the BSMPM analysis of largedeformation problems for the improved accuracy and convergence, especially with the quadratic B-splines. Moreover, the increase in the order of the B-spline basis function is also found to be an effective way to reduce the quadrature error and to improve accuracy and convergence. For plate impact examples, it is demonstrated that the BSMPM outperforms the generalized interpolation material point (GIMP) and convected particle domain interpolation (CPDI) methods in term of the accuracy of representing stress waves. Thus, the BSMPM could become a promising alternative to the MPM, GIMP, and CPDI in solving certain types of transient problems. KEYWORDS B-spline basis functions, Gauss quadrature, large deformation, material point method, mesh refinement, transient problems

Comparison and unification of material-point and optimal transportation meshfree methods

“…The choice of BSMPM is motivated by previous studies. First of all, a number of studies have indicated that BSMPM is a viable alternative not only for MPM, but also for its more advanced versions such as dual domain material-point (DDMP) [60] and convected particledomain interpolation (CPDI) [40] methods [20,45,53,57]. Secondly, Cyron et al [13] pointed out the strong similarities between B-spline and maxent basis functions.…”

Temporal and null‐space filter for the material point method

“…However, in the simulations made with the original MPM, there are numerical noises when material points crossing the cell boundaries. These numerical inaccuracies led to the development of other MPM variants, including the generalized interpolation material point method (GIMP), the dual domain material point method (DDMP), and the B‐spline MPM . Those implementations improved the errors directly related to cell‐crossing, yet some high frequency noises are still present in the solutions.…”

“…Tran et al have shown that, if there are more material points than nodes, null‐space errors exist despite the choice of the shape function. Therefore, as the null‐space existence is due to the larger number of the material points than the grid nodes, the null‐space error is present in all the variants of the MPM, including GIMP, DDMP, and B‐spline MPM . To remove the null‐space errors, Gritton et al proposed a null‐space filter using the single‐value‐decomposition (SVD) operator in the 1D formulation of the original MPM.…”

“…Therefore, as the null-space existence is due to the larger number of the material points than the grid nodes, the null-space error is present in all the variants of the MPM, including GIMP, 8 DDMP, 9 and B-spline MPM. [10][11][12][13] To remove the null-space errors, Gritton et al 23,25 proposed a null-space filter using the single-value-decomposition (SVD) operator in the 1D formulation of the original MPM. This paper shows how to apply a null-space filter to higher-order interpolation function, for example, in the GIMP or DDMP formulations of the MPM.…”

Extension of B-spline Material Point Method for unstructured triangular grids using Powell–Sabin splines

“…Moreover, spline basis functions are known to provide higher accuracy per degree of freedom as compared to C 0 -finite elements [14]. On structured rectangular grids, adopting B-spline basis functions within MPM not only eliminates grid-crossing errors but also yields higher-order spatial convergence [27,28,35,10,3]. Previous research also demonstrates that BSMPM is a viable alternative to the GIMP, CPDI and DDMP methods [29,17,10,36].…”