Biot consolidation analysis with automatic time stepping and error control Part 1: theory and implementation

Abstract

SUMMARYA new "nite element algorithm for solving elastic and elastoplastic coupled consolidation problems is described. The procedure treats the governing consolidation relations as a system of "rst-order di!erential equations and is based on the backward Euler and Thomas and Gladwell schemes with automatic subincrementation of a prescribed series of time increments. The prescribed time increments, which are called coarse time steps, serve to start the procedure and are chosen by the user. The automatic consolidation algorithm attempts to select the time subincrements such that, for a given mesh, the time-stepping (or temporal discretisation) error in the displacements lies close to a speci"ed tolerance.Unlike existing solution techniques, the new algorithm computes not only the displacements and pore pressures, but also their derivatives with respect to time. These extra variables permit a family of unconditionally stable integration algorithms to be constructed which automatically provide an estimate of the local truncation error for each time step. This error estimate is inexpensive to compute and may be used to develop a simple and e$cient automatic time stepping mechanism. For the elastic case, the displacements and pore pressures at the end of each subincrement may be solved directly without the need for iteration. For elastoplastic behaviour, however, the governing relationships are non-linear and a system of non-linear equations must be solved to compute the updates.

Aspects of finite element implementation of critical state models

“…Time stepping scheme for mixed displacement±pore pressure elements For coupled analysis of deformation and pore pressure, Sloan and Abbo [30] recently presented a new adaptive time stepping scheme. Using a similar philosophy to the load-stepping scheme described by Abbo and Sloan [29], the adaptive time stepping scheme attempts to choose the time subincrements so that, for a given mesh, the timestepping error in the displacements is close to a speci®ed tolerance.…”

“…In this paper, the adaptive time stepping method of Sloan and Abbo [30] will be used to solve coupled displacement and pore pressure problems in soils whose behavior is simulated by critical state models. Due to the nonlinear behavior in the elastic part, both the elastic and plastic global stiffness matrices have to be reformed at each increment and in each iteration.…”

“…The ®nite element code, SNAC, has been developed at the University of Newcastle over a number of years (see Abbo [28], Abbo and Sloan [29], and Sloan and Abbo [30]). Originally only conventional elastoplastic models, such as the Mohr-Coulomb, Tresca and Von Mises criteria were implemented, but now two critical state (CS) models have also been incorporated.…”

“…Particular attention is focused on the algorithms of Sloan [27] and Abbo [28] for stress integration, Abbo and Sloan [29] for load-displacement integration in uncoupled analysis, and Sloan and Abbo [30] for time integration in coupled analysis. In this paper, these newly developed automatic algorithms have been further re®ned to better handle the nonlinear elasticity in critical state models.…”

Simulation of rock salt dissolution and its impact on land subsidence

“…where j indexes the recursive step size reduction, r max and r min are used to limit multiplication and reduction factors often set equal to 2.0 and 0.1, respectively, and s = 0.9 is a safety factor (Sloan and Abbo, 1999). The temporal truncation error tolerance is set to γ = 0.01.…”