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

Abstract

An enhancement of the extended B-spline-based implicit material point method (EBS-MPM) is developed to avoid pressure oscillation and volumetric locking. The EBS-MPM is a stable implicit MPM that enables the imposition of arbitrary boundary conditions thanks to the higher-order EBS basis functions and the help of Nitsche's method. In particular, by means of the higher-order EBS basis functions, the EBS-MPM can suppress the cell-crossing errors caused by material points crossing the background grid boundaries and can avoid both the stress oscillations arising from inaccurate numerical integration and the ill-conditioning of the resulting tangent matrices. Although the higher-order EBS basis functions are known to avoid volumetric locking, the problem of pressure oscillation has not yet been resolved. Therefore, to suppress pressure oscillation due to quasi-incompressibility, we propose the incorporation of the F-bar projection method into the EBS-MPM, which is compatible with the higher-order EBS basis functions. Three representative numerical examples are presented to demonstrate the capability of the proposed method in suppressing both the pressure oscillation and volumetric locking. The results of the proposed method are compared to those of the finite element method with F-bar elements and those of isogeometric analysis with quadratic NURBS elements.

An implicit locking‐free B‐spline Material Point Method for large strain geotechnical modelling

“…The F-bar method was originally developed by de Souza Neto et al [32] to counteract volumetric locking in FEM. Coombs et al [33] firstly applied the Fbar approach to stabilise MPM, which became a popular method to overcome this issue in MPM [34][35][36][37][38]. However, Xie et al [25] discovered that the F-bar method cannot fully eliminate the stress oscillation in MPM under very large deformation.…”

“…This approach relies on the triangular mesh, however, the rectangular mesh is a more popular choice in the MPM literature. To the best of our knowledge, there are currently only three available pieces of research 28,30,31 related to the mitigation of volumetric locking for a higherorder MPM. Two of them 30,31 are based on the F-bar projection method proposed by Elguedj et al, 32 which was originally developed for B-spline FEM.…”

“…To the best of our knowledge, there are currently only three available pieces of research 28,30,31 related to the mitigation of volumetric locking for a higher‐order MPM. Two of them 30,31 are based on the F‐bar projection method proposed by Elguedj et al., 32 which was originally developed for B‐spline FEM. However, the application of this method is not straightforward due to the introduction of an additional background grid with low‐order shape functions 28 .…”

B-spline-based material point method with dynamic load balancing technique for large-scale simulation

“…Here, "the cell-crossing error" refers to the numerical error that occurs when a material point crosses a background grid cell due to the continuous nature of C 0 in the linear shape function employed in standard MPMs, while GIMP achieves C 1 near the element boundary. Another strategy to address this is to use more versatile B-spline basis functions, which can be extended to the extended B-spline basis function [41][42][43]. These bases are effective in preventing a loss of integration accuracy in regions with a small number of material points.…”