An investigation of the stability of numerical solutions of Biot's equations of consolidation

Booker, J. R.; Small, J. C.

doi:10.1016/0020-7683(75)90013-x

Cited by 182 publications

(48 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The earliest of these studies is due to Sandhu and Wilson (1969) with other noteworthy studies by Ghaboussi and Wilson (1973), Booker and Small (1975), Verruijt (1977), Huyakorn and Pinder (1983), Valliappan et al (1974) and Simon et al (1986). Complete discussions of the computational aspects of poroelasticity are also given by Desai and Christian (1977) and Lewis and Schrefler (1998).…”

Section: Finite Element Formulationsmentioning

confidence: 92%

Stationary damage modelling of poroelastic contact

Selvadurai

2004

International Journal of Solids and Structures

View full text Add to dashboard Cite

Section: Finite Element Formulationsmentioning

confidence: 92%

Stationary damage modelling of poroelastic contact

Selvadurai

2004

International Journal of Solids and Structures

View full text Add to dashboard Cite

“…Generally speaking, for small t values, the matrix A can be much more illconditioned than for large t. In coupled consolidation problems, the time step size is usually increased as the simulation proceeds [54]; hence, conditioning of system 1 generally tends to improve during the solution of a complete transient problem. To investigate a representative situation, two t values are selected for each test problem so as to obtain both an ill-and a relatively wellconditioned case.…”

Section: Numerical Resultsmentioning

confidence: 99%

Parallel inexact constraint preconditioning for ill-conditioned consolidation problems

2012

View full text Add to dashboard Cite

Constraint preconditioners have proved very efficient for the solution of ill-conditioned finite element (FE) coupled consolidation problems in a sequential computing environment. Their implementation on parallel computers, however, is not straightforward because of their inherent sequentiality. The present paper describes a novel parallel inexact constraint preconditioner (ParICP) for the efficient solution of linear algebraic systems arising from the FE discretization of the coupled poro-elasticity equations. The ParICP implementation is based on the use of the block factorized sparse approximate inverse incomplete Cholesky preconditioner, which is a very recent and effective development for the parallel preconditioning of symmetric positive definite matrices. The ParICP performance is experimented with in real 3D coupled consolidation problems, proving a scalable and efficient implementation of the constraint preconditioning for highperformance computing. ParICP appears to be a very robust algorithm for solving ill-conditioned large-size coupled models in a parallel computing environment

show abstract

“…0:5 for unconditional stability (Booker and Small, 1975). The backward Euler scheme, with h ¼ 1, has excellent damping characteristics (Wood, 1990) and leads to…”

Section: Explicit Methods (Exbe)mentioning

confidence: 99%

“…These explicit methods differ from one another in the way the unknown X is approximated. Although they are only conditionally stable for consolidation analysis (see Booker and Small, 1975), explicit time stepping schemes are still used in some commercial software. In this paper, the performance of the explicit backward Euler scheme (EXBE) is investigated.…”

Section: Explicit Methods (Exbe)mentioning

confidence: 99%

Time stepping schemes for coupled displacement and pore pressur? analysis

Sheng

Sloan

2003

Computational Mechanics

View full text Add to dashboard Cite

The paper investigates the performance of various time stepping schemes for coupled displacement and pore pressure analysis. A number of alternative forms of the automatic time stepping method proposed by Sloan and Abbo (1999a) are also presented. These alternative schemes use different updates for the displacements and pore pressures and also adopt different starting conditions for the iterations. The automatic schemes are compared with an implicit h-method, as well as an explicit method, through analysis of a variety of problems involving undrained loading, drained loading, and consolidation for Mohr-Coulomb and critical state models. As expected, the numerical results confirm that the explicit scheme is neither accurate nor robust. Although the implicit h-method is accurate and fast, it fails to give a solution in a number of cases where the time step is large. The automatic schemes are shown to be accurate, fast and generally robust. Two of the automatic schemes proposed never fail to furnish a solution for the cases considered. In addition, all the automatic schemes are able to constrain the time-stepping (temporal integration) error in the displacements and pore pressures to lie near a prescribed tolerance, provided the iteration error tolerance is properly chosen. For complex soil models, it is important that the latter is set sufficiently small in order for the schemes to be able to constrain the time-stepping error to lie within a prescribed tolerance. IntroductionIn geotechnical engineering, deformation is often coupled with the flow of pore fluid. Finite element analysis of geotechnical problems therefore often uses a mixed formulation where the nodal variables are displacements and pore pressures. Such a formulation eventually leads to a set of stiff, non-linear ordinary differential equations in terms of these variables. One critical issue for solving such a system accurately is the choice of time discretisation. In the literature, as well as in commercial software, the discretisation in the time domain is usually carried out using the generalised trapezoidal method (also called the h-method).With the h-method, all time dependent variables are estimated at some intermediate point within the interval, depending on the actual value of h. The non-iterative explicit method, which estimates time dependent variables at the start of the interval, is usually avoided due to its poor stability properties. Therefore, the h-method is usually used in an implicit form with the need for iterations in each time step. The key disadvantage of the conventional h-method is the need for a trial-and-error process to find an accurate and efficient sequence of time steps. This is not a trivial task in practice, as coupled consolidation problems often span long durations with vary large gradients in pore pressures and displacements at certain times.In an effort to address the problems associated with coupled consolidation analysis, Sloan and Abbo (1999a) proposed an automatic time stepping scheme. This attempts to choose the...

show abstract

An investigation of the stability of numerical solutions of Biot's equations of consolidation

Cited by 182 publications

References 11 publications

Stationary damage modelling of poroelastic contact

Stationary damage modelling of poroelastic contact

Parallel inexact constraint preconditioning for ill-conditioned consolidation problems

Time stepping schemes for coupled displacement and pore pressur? analysis

Contact Info

Product

Resources

About