A mixed Material Point Method for the Stokes equations: a stabilized velocity-pressure formulation

The Material Point Method (MPM) is a particle-based numerical technique, usually chosen for its ability to efficiently handle large deformation problems; for this reason, it has historically been used in a variety of applications within the field of solid mechanics and geomechanics. However, it has also been employed successfully in the field of fluid dynamics, with numerous recent developments available in the literature. This thesis proposes a novel MPM formulation for fluid dynamics problems. As the governing equations are solved using a Lagrangian description, this translates into numerically solving the Stokes problem using a Galerkin formulation. In particular, a mixed velocity-pressure formulation has been developed, as it’s the most convenient approach to solve the Stokes equations. Mixed formulations in an MPM setting have been employed in several other works, both in solid mechanics and in fluid dynamics; it should be noted, however, that mixed velocity-pressure formulations like the one presented here have yet to be seen in the MPM literature. Well-known instability issues arise in mixed formulations of the Stokes problem due to the violation of the discrete inf-sup condition. To address this, various stabilization techniques have been developed. Among them a Variational-Multiscale (VMS) approach has been adopted here. The developed numerical scheme has been implemented within the KRATOS Multiphysics framework, and validated using the Method of Manufactured Solutions (MMS).