Extended B-spline-based mixed material point method stabilized by the variational multiscale method for compressible and nearly incompressible hyperelastic materials

To further improve the stability of the extended B-spline (EBS) based material point method (EBSMPM), a displacement-pressure ($ \mathit{\boldsymbol{u}} $-$ p $)-mixed formulation incorporating the variational multiscale (VMS) method is formulated within the finite strain framework. In this formulation, we define a constraint on the pressure field that can be applied to both compressible and nearly incompressible materials, taking into account the extensibility to material models that exhibit plastic incompressibility. In order to satisfy the LBB conditions for ensuring stability in the mixed formulation with equal-order interpolation, the VMS method is formulated within finite strain framework in computational solid mechanics. The adoption of the VMS method makes it possible to use higher-order EBS basis functions of the same order for interpolation of both displacement and pressure, and this successfully prevents the occurrence of numerical instability. Three representative numerical examples are presented to demonstrate the performance of the proposed method for solid mechanics problems using compressible and nearly incompressible materials, comparing the formulation whose independent variable is only the displacement field. Additionally, through these numerical examples, the ability to suppress volumetric locking and pressure oscillation is also verified.