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

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.