Unconditional convergence of theoretical solutions to u-p formulation

Abstract

Dynamic soil-water coupling analyses, based on the u-p formulation, are inapplicable to highly permeable soils, causing numerical instability. In this study, it is demonstrated that theoretical solutions to the u-p formulation itself certainly exhibit unconditional convergence regardless of the permeability coefficient. This suggests that the instability is only numerical and can be observed in a temporally discretized system. Firstly, the linearized governing equation for the u-p formulation was proven to be reduced to a damped wave equation under a one-dimensional condition, similar to the Full formulation. Secondly, theoretical solutions for the u-p formulation were derived and their unconditional convergence was confirmed. Then, the essential characteristics of the u-p theoretical solutions, that is, the underestimation of permeability, overestimation of compression wave celerity, and occurrence of negative pore water pressure against positive load application, were described and compared with theoretical solutions for the Full formulation.