Coupled Consolidation Equations Research Articles

Comparison of Coupled and Uncoupled Consolidation Equations Using Finite Element Method in Plane-Strain Condition

In the current paper, the consolidation settlement of a strip footing over a finite layer of saturated soil has been studied using the finite element method. In Biot’s coupled consolidation equations, the soil deformation and excess pore pressure are determined simultaneously in every time step which refers to the hydro-mechanical coupling. By considering a constant total stress throughout the time and by assuming that volume strain is a function of isotropic effective stress, uncoupled consolidation equations can be obtained using coupled consolidation equations. In these uncoupled equations, excess pore pressure and deformation are determined separately. In this approach, the excess pore pressure can be identified in the first stage. Using the calculated excess pore pressure, the soil deformation is determined through effective stress-strain analyses. A computer code was developed based on coupled and uncoupled equations that are capable of performing consolidation analyses. To verify the accuracy of these analyses, the obtained results have been compared with the precise solution of Terzaghi’s one-dimensional consolidation theory. The capability of these two approaches in estimation of pore water pressure and settlement and to show Mandel-Crayer’s effect in soil consolidation is discussed. Then, the necessity of utilizing coupled analyses for evaluating soil consolidation analysis was investigated by comparing the coupled and uncoupled analyses results.

RMCP: Relaxed Mixed Constraint Preconditioners for saddle point linear systems arising in geomechanics

A major computational issue in the Finite Element (FE) integration of coupled consolidation equations is the repeated solution in time of the resulting discretized indefinite system. Because of ill-conditioning, the iterative solution, which is recommended in large size 3D settings, requires the computation of a suitable preconditioner to guarantee convergence. In this paper the coupled system is solved by a Krylov subspace method preconditioned by a Relaxed Mixed Constraint Preconditioner (RMCP) which is a generalization based on a parameter ω of the Mixed Constraint Preconditioner (MCP) developed in [7]. Choice of optimal ω is driven by the spectral distribution of suitable symmetric positive definite (SPD) matrices. Numerical tests performed on realistic 3D problems reveal that RMCP accelerates Krylov subspace solvers by a factor up to three with respect to MCP.

FSAI-based parallel Mixed Constraint Preconditioners for saddle point problems arising in geomechanics

In this paper we propose and describe a parallel implementation of a block preconditioner for the solution of saddle point linear systems arising from Finite Element (FE) discretization of 3D coupled consolidation problems. The Mixed Constraint Preconditioner developed in [L. Bergamaschi, M. Ferronato, G. Gambolati, Mixed constraint preconditioners for the solution to FE coupled consolidation equations, J. Comput. Phys., 227(23) (2008), 9885–9897] is combined with the parallel FSAI preconditioner which is used here to approximate the inverses of both the structural (1, 1) block and an appropriate Schur complement matrix. The resulting preconditioner proves effective in the acceleration of the BiCGSTAB iterative solver. Numerical results on a number of test cases of size up to 2×10 6 unknowns and 1.2×10 8 nonzeros show the perfect scalability of the overall code up to 256 processors.

Mixed Constraint Preconditioners for the iterative solution of FE coupled consolidation equations

The Finite Element (FE) integration of the coupled consolidation equations requires the solution of linear symmetric systems with an indefinite saddle point coefficient matrix. Because of ill-conditioning, the repeated solution in time of the FE equations may be a major computational issue requiring ad hoc preconditioning strategies to guarantee the efficient convergence of Krylov subspace methods. In the present paper a Mixed Constraint Preconditioner (MCP) is developed combining implicit and explicit approximations of the inverse of the structural sub-matrix, with the performance investigated in some representative examples. An upper bound of the eigenvalue distance from unity is theoretically provided in order to give practical indications on how to improve the preconditioner. The MCP is efficiently implemented into a Krylov subspace method with the performance obtained in 2D and 3D examples compared to that of Inexact Constraint Preconditioners and Least Square Logarithm scaled ILUT preconditioners. Two variants of MCP (T-MCP and D-MCP), developed with the aim at reducing the cost of the preconditioner application, are also tested. The results show that the MCP variants constitute a reliable and robust approach for the efficient solution of realistic coupled consolidation FE models, and especially so in severely ill-conditioned problems.

The role of preconditioning in the solution to FE coupled consolidation equations by Krylov subspace methods

AbstractThe repeated solution in time of the linear system arising from the finite element integration of coupled consolidation equations is a major computational effort. This system can be written in either a symmetric or an unsymmetric form, thus calling for the implementation of different preconditioners and Krylov subspace solvers. The present paper aims at investigating when either a symmetric or an unsymmetric approach should be better used. The results from a number of representative numerical experiments indicate that a major role in selecting either form is played by the preconditioner rather than by the Krylov subspace method itself. Two other important issues addressed are the size of the time integration step and the possible lumping of the flow capacity matrix. It appears that ad hoc block constrained preconditioners provide the most robust algorithm independently of the time step size, lumping, and symmetry. Copyright © 2008 John Wiley & Sons, Ltd.

Novel preconditioners for the iterative solution to FE-discretized coupled consolidation equations

A major computational issue in the finite element (FE) integration of coupled consolidation equations is the repeated solution in time of the resulting discretized indefinite system. Because of ill-conditioning, the iterative solution, which is recommended in large size 3D settings, requires the computation of a suitable preconditioner to guarantee convergence. In this paper the coupled system is solved by a Krylov subspace method preconditioned by an inexact constraint preconditioner (ICP) preserving the same block structure as the native FE matrix. The conditioning number of the preconditioned coupled problem depends on the quality of the approximation of the block corresponding to the structural stiffness matrix. An efficient algorithm to implement ICP into a Krylov subspace method is developed. Numerical tests performed on realistic 3D problems reveal that ICP typically outperforms standard ILUT preconditioners and proves much more robust in severely ill-conditioned problems.

Scaling improves stability of preconditioned CG‐like solvers for FE consolidation equations

AbstractPreconditioned projection (or conjugate gradient like) methods are increasingly used for the accurate and efficient solution to finite element (FE) coupled consolidation equations. Theory indicates that preliminary row/column scaling does not affect the eigenspectrum of the iteration matrix controlling convergence as long as the preconditioner relies on the incomplete factorization of the FE coefficient matrix. However, computational experience with mid‐large size problems shows that the above inexpensive operation can significantly accelerate the solver convergence, and to a minor extent also improve the final accuracy, as a result of a better solver stability to the accumulation and propagation of floating point round‐off errors. This is demonstrated with the aid of the least square logarithm (LSL) scaling algorithm on FE consolidation problems of increasing size up to more than 100 000. It is shown that a major source of numerical instability rests with the sub‐matrix which couples the structural to the fluid part of the underlying mathematical model. It is concluded that for mid‐large size, possibly difficult, FE consolidation problems left/right LSL scaling is to be always recommended when the incomplete factorization is used as a preconditioning technique. Copyright © 2003 John Wiley & Sons, Ltd.

Direct, partitioned and projected solution to finite element consolidation models

AbstractDirect, partitioned, and projected (conjugate gradient‐like) solution approaches are compared on unsymmetric indefinite systems arising from the finite element integration of coupled consolidation equations. The direct method is used in its most recent and computationally efficient implementations of the Harwell Software Library. The partitioned approach designed for coupled problems is especially attractive as it addresses two separate positive definite problems of a smaller size that can be solved by symmetric conjugate gradients. However, it may stagnate and when converging it does not prove competitive with a global projection method such as Bi‐CGSTAB, which may take full advantage of its flexibility in working on scaled and reordered equations, and thus may greatly improve its computational performance in terms of both robustness and convergence rate. The Bi‐CGSTAB superiority to the other approaches is discussed and demonstrated with a few representative examples in two‐dimensional (2‐D) and three‐dimensional (3‐D) coupled consolidation problems. Copyright © 2002 John Wiley & Sons, Ltd.