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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 84
Efficient Preconditioners for Krylov Subspace Methods in the Solution of Coupled Consolidation Problems L. Bergamaschi, M. Ferronato and G. Gambolati
Department of Mathematical Methods and Models for Scientific Applications, University of Padua, Italy L. Bergamaschi, M. Ferronato, G. Gambolati, "Efficient Preconditioners for Krylov Subspace Methods in the Solution of Coupled Consolidation Problems", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 84, 2006. doi:10.4203/ccp.84.84
Keywords: preconditioners, linear system, Krylov subspace method, coupled consolidation model.Summary
The time-dependent distribution of displacements and fluid pore pressure
in porous media was first mathematically described by Biot [1],
who developed the consolidation theory coupling the elastic equilibrium equations with a
continuity or mass balance equation to be solved under appropriate
boundary, initial flow and loading conditions.
A consolidation problem is typically solved in space by a finite element (FE) technique giving rise to a system of first-order differential equations, the solution to which is usually addressed by an appropriate time marching scheme. A major computational effort is the repeated solution in time of the resulting discretized indefinite equations. In particular, with small time integration steps typically required in the early phase of the analysis the final linear system may be severely ill-conditioned [2], so that obtaining an accurate solution may prove difficult by any numerical approach.
Because of the large size of realistic three-dimensional (3-D) consolidation models
(and particularly so in problems related to fluid
withdrawal-injection from-into geological formations) the use of iterative
solvers is generally recommended. Among them, projection (or conjugate
gradient-like) methods based on Krylov subspaces for unsymmetric indefinite
systems, such as Bi-CGSTAB
are attracting a growing interest on the grounds of their robustness and
efficiency.
However, to accelerate convergence a key issue
is the selection of an The aim of the present paper is to investigate the performance and the robustness of a novel block preconditioner developed for the solution to the symmetric indefinite system of equations arising from the FE integration of the coupled consolidation equations. At each time step, the solution of the linear system where
is performed by a Krylov subspace method, preconditioned by
with G a symmetric positive definite
approximation of block K.
The eigenvalues of the preconditioned matrix are proven to be bounded by the eigenvalues of .
Matrix The outcome from two realistic medium and large size 3-D problems is compared to the solution strategy suggested by Gambolati et al. [3], which has proved to be, at present, one of the most effective solution techniques for large size, possibly ill-conditioned, FE coupled consolidation models. In particular, the comparison is performed for both normally conditioned and severely ill-conditioned problems. Preliminary numerical results show that the inexact constraint preconditioners are flexible, robust and reliable tools for the iterative solution to FE consolidation models and possess a promising potential for further possible improvements. References
- 1
- M. A. Biot,
*General theory of three-dimensional consolidation*, J. Appl. Phys., 12 (1941), pp. 155-164. doi:10.1063/1.1712886 - 2
- M. Ferronato, G. Gambolati, and P. Teatini,
*Ill-conditioning of finite element poroelasticity equations.*, Int. J. Solids Struct., 38 (2001), pp. 5995-6014. doi:10.1016/S0020-7683(00)00352-8 - 3
- G. Gambolati, G. Pini, and M. Ferronato,
*Scaling improves stability of preconditioned CG-like solvers for FE consolidation equations.*, Int. J. Numer. Anal. Methods Geomech., 27 (2003), pp. 1043-1056. doi:10.1002/nag.311
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