Von Neumann stability analysis of the u–p reproducing kernel formulation for saturated porous media

Abstract

This paper introduces the von Neumann method to investigate the temporal stability of the displacement---pressure $$(u{-}p)$$(u-p) reproducing kernel formulation for saturated porous media. Both dynamic and quasi-static formulations are considered and the critical time steps are derived. The effect of lumped and consistent matrices on temporal stability is analyzed under explicit temporal discretization. It is shown that lumped matrices have better temporal stability than consistent matrices. The study also shows that nodal support size greatly affects the critical time step size of the formulations. For consistent matrices, larger support size results in smaller critical time step size; however, opposite relation occurs if lumped scheme is used. The numerical study shows that stabilization parameter of the stabilized nodal integration methods reduces the critical time step size. Transient analyses are performed to verify the results from von Neumann analysis.