Variable‐order variable‐step algorithms for second‐order systems. Part 1: The methods

Num Anal Meth Geomechanics

Gens

et al. 2003

172

106

SUMMARYThis paper presents a complete finite-element treatment for unsaturated soil problems. A new formulation of general constitutive equations for unsaturated soils is first presented. In the incremental stress-strain equations, the suction or the pore water pressure is treated as a strain variable instead of a stress variable. The global governing equations are derived in terms of displacement and pore water pressure. The discretized governing equations are then solved using an adaptive time-stepping scheme which automatically adjusts the time-step size so that the integration error in the displacements and pore pressures lies close to a specified tolerance. The non-linearity caused by suction-dependent plastic yielding, suction-dependent degree of saturation , and saturation-dependent permeability is treated in a similar way to the elastoplasticity. An explicit stress integration scheme is used to solve the constitutive stress-strain equations at the Gauss point level. The elastoplastic stiffness matrix in the Euler solution is evaluated using the suction as well as the stresses and hardening parameters at the start of the subincrement, while the elastoplastic matrix in the modified Euler solution is evaluated using the suction at the end of the subincrement. In addition, when applying subincrementation, the same rate is applied to all strain components including the suction.

Section: Coupled Equationsmentioning

confidence: 99%

Finite element formulation and algorithms for unsaturated soils. Part I: Theory

Sheng

Num Anal Meth Geomechanics

Gens

et al. 2003

172

106

“…Thomas and Gladwell [1] presented a family of multistage time-stepping schemes for the solution of second-order ODE systems …”

Section: Introductionmentioning

confidence: 99%

Truncation error and stability analysis of iterative and non‐iterative Thomas–Gladwell methods for first‐order non‐linear differential equations

Kavetski

Binning

Numerical Meth Engineering

2004

SUMMARYThe consistency and stability of a Thomas-Gladwell family of multistage time-stepping schemes for the solution of first-order non-linear differential equations are examined. It is shown that the consistency and stability conditions are less stringent than those derived for second-order governing equations. Second-order accuracy is achieved by approximating the solution and its derivative at the same location within the time step. Useful flexibility is available in the evaluation of the non-linear coefficients and is exploited to develop a new non-iterative modification of the Thomas-Gladwell method that is secondorder accurate and unconditionally stable. A case study from applied hydrogeology using the non-linear Richards equation confirms the analytic convergence assessment and demonstrates the efficiency of the non-iterative formulation.

“…where C is given by (19). Although much slower to converge, this approach does not require a fresh factorisation for each iteration and always has a well conditioned Jacobian matrix.…”

Section: Theory For Elastoplastic Algorithmmentioning

confidence: 99%

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

Int. J. Numer. Anal. Meth. Geomech.

Abbo

1999

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.