Stabilized explicit $$u-p_w$$ solution in soil dynamic problems near the undrained-incompressible limit

Traditionally, Biot’s formulation is employed to model the behavior of saturated soils. The $$u-p_w$$ u - p w (solid displacement–pore water pressure) formulation can be considered as the standard one, since involves a good computational performance together with excellent accuracy for slow and moderate speed phenomena. Dynamic processes can be studied even if the acceleration of the water is neglected, what occurs in the undrained limit. It is well-known that $$u-p_w$$ u - p w formulation might display instabilities in the undrained-incompressible limit. Several techniques have been proposed to overcome this issue, principally within an implicit time integration scheme for small strains. In this paper, a robust implementation of the divergence of the momentum equation technique is presented for an explicit $$u-p_w$$ u - p w approach within the framework of optimal transportation meshfree scheme at finite strain. Several examples are provided in order to assess the good performance of the proposed methodology.