Reproducing kernel enhanced material point method with improved artificial viscosity formulations for shock wave and vortical compressible flows

The modeling of inviscid compressible flows under a Lagrangian description with shock waves and/or vortical structures is challenging when using conventional mesh-based methods owing to severe mesh distortion and the instability caused by moving discontinuities. By contrast, meshfree methods such as the material point method (MPM) reduce mesh sensitivity but may suffer from low accuracy and instability due to cell-crossing instability and the inadequacy of the existing artificial viscosity (AV) formulations for stabilization. In this study, we present a reproducing kernel (RK)-stabilized MPM that incorporates an enhanced tensorial AV model to address these challenges. A mixed formulation is developed to solve the momentum and energy conservation equations, while the smooth RK approximation is used to mitigate cell-crossing instability. To address the instability induced by moving discontinuities, we improve the classical AV by incorporating tensorial forms of the gradient/divergence operator and following a vorticity-based blending approach. This allows us to appropriately regulate numerical dissipation in terms of dilatational and deviatoric strain rates. The proposed formulation is validated by applying it to classical benchmark problems involving shock and rarefaction waves, interfacial flows, and vortical structures.