One-step and linear multistep methods for nonlinear consolidation

Abstract

The behavior of numerical ODE methods for the solution of nonlinear consolidation is investigated. One-step algorithms based on the generalized trapezoidal method and linear multistep methods based on the backward differentiation formulas (BDF) are considered. Mechanical deformation is coupled with fluid flow via a formulation based on Biot's three-dimensional consolidation theory. The resulting nonlinear FE matrix equations are cast in such a form that the one-step and the BDF methods can be coded in a similar fashion via the use of a displacement difference operator. Accuracy and stability analyses are presented for both the one-step and BDF methods. In particular, the effect of variable step size on the stability of the BDF methods is discussed in the context of a zero-stability criterion so that bounds on a uniformly growing step size can be established. The generalized trapezoidal methods and the backward differentiation formulas of order up to three are shown to converge with the expected order, save for round-off errors, for nonlinear consolidation problems where the soil is modeled as an elasto-plastic material of the Cam-Clay type.