v-p material point method for weakly compressible problems

•A vp-MPM is proposed for weakly compressible problems.•A slope limiter is employed to suppress spurious pressure oscillations.•The vp formulation is incorporated into the ICFEMP to suppress pressure oscillation.•The vp-MPM has much less extra variables than a Hu-Washizu principle based method. The weakly compressible material point method (WCMPM) suffers from volumetric-locking and numerical oscillation in modeling fluid flow and fluid-structure interaction problems. In this paper, a v-pformulation of the material point method (vp-MPM) is proposed for weakly compressible problems based on a two-field variational principle. As only the velocity v and the pressure p are the independent variables, the v-p formulation has much less extra variables than those based on the Hu-Washizu multi-field variational principle which takes the velocity, strain and stress as independent variables. The pressure is assumed independently in the control volume of each gird node. Spurious pressure oscillation reduces but still occurs at the interface of discontinuity due to large pressure gradient difference across the interface. Therefore, a slope limiter is employed to suppress the oscillation and the general interpolation functions are used to eliminate the cell-crossing error. In order to extend the method to the fluid-structure interaction problems, the v-p formulation is incorporated into the improved coupled finite element material point method. Several numerical examples are presented to validate the vp-MPM.