  The material point method (MPM) has been increasingly used for the simulation
of large deformation processes in fluid-infiltrated porous materials. For
undrained poromechanical problems, however, standard MPMs are numerically
unstable because they use low-order interpolation functions that violate the
inf-sup stability condition. In this work, we develop stabilized MPM
formulations for dynamic and quasi-static poromechanics that permit the use of
standard low-order interpolation functions notwithstanding the drainage
condition. For the stabilization of both dynamic and quasi-static formulations,
we utilize the polynomial pressure projection method whereby a stabilization
term is augmented to the balance of mass. The stabilization term can be
implemented with both the original and generalized interpolation material point
(GIMP) methods, and it is compatible with existing time-integration methods.
Here we use fully-implicit methods for both dynamic and quasi-static
poromechanical problems, aided by a block-preconditioned Newton-Krylov solver.
The stabilized MPMs are verified and investigated through several numerical
examples under dynamic and quasi-static conditions. Results show that the
proposed MPM formulations allow standard low-order interpolation functions to
be used for both the solid displacement and pore pressure fields of
poromechanical formulations, from undrained to drained conditions, and from
dynamic to quasi-static conditions.
