×

An efficient diagonal preconditioner for finite element solution of Biot’s consolidation equations. (English) Zbl 1076.74558

Summary: Finite element simulations of very large-scale soil-structure interaction problems (e.g. excavations, tunnelling, pile-rafts, etc.) typically involve the solution of a very large, ill-conditioned, and indefinite Biot system of equations. The traditional preconditioned conjugate gradient solver coupled with the standard Jacobi (SJ) preconditioner can be very inefficient for this class of problems. This paper presents a robust generalized Jacobi (GJ) preconditioner that is extremely effective for solving very large-scale Biot’s finite element equations using the symmetric quasi-minimal residual method. The GJ preconditioner can be formed, inverted, and implemented within an ‘element-by-element’ framework as readily as the SJ preconditioner. It was derived as a diagonal approximation to a theoretical form, which can be proven mathematically to possess an attractive eigenvalue clustering property. The effectiveness of the GJ preconditioner over a wide range of soil stiffness and permeability was demonstrated numerically using a simple three-dimensional footing problem. This paper casts a new perspective on the potentialities of the simple diagonal preconditioner, which has been commonly perceived as being useful only in situations where it can serve as an approximate inverse to a diagonally dominant coefficient matrix.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76S05 Flows in porous media; filtration; seepage
74L10 Soil and rock mechanics

Software:

KELLEY
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Biot, Journal of Applied Physics 12 pp 155– (1941)
[2] The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media (2nd edn). Wiley: New York, 1998. · Zbl 0935.74004
[3] Lewis, International Journal for Numerical Methods in Engineering 27 pp 195– (1989)
[4] Lewis, International Journal for Numerical and Analytical Methods in Geomechanics 13 pp 1– (1989)
[5] Three-dimensional analysis of deep excavation in soft clay. Proceedings of the 11th African Regional Conference on Soil Mechanics and Foundation Engineering, Cairo, Egypt, 1995; 519-533.
[6] Three dimensional finite element analysis of pile groups and piled-rafts. Ph.D. Thesis, University of Manchester, 1996.
[7] Three-dimensional analysis of twin tunnels. In Proceedings of the Eighth KKNN Seminar on Civil Engineering, National University of Singapore, Kent Ridge, Singapore, (eds), 30th November and 1st December 1998; 452-457.
[8] Three dimensional analysis of building settlement caused by shaft construction. In Proceedings of the International Symposium on Geotechnical Aspects of Underground Construction in Soft Ground-IS-Tokyo’99, Tokyo, Japan, (eds), 19-21 July 1999. A.A. Balkema: Rotterdam, Brookfield, 2000; 607-612.
[9] Mroueh, International Journal for Numerical and Analytical Methods in Geomechanics 23 pp 1961– (1999)
[10] Linear system solvers: sparse iterative methods. In Parallel Numerical Algorithms, (eds), ICASE/LaRC Interdisciplinary Series in Science and Engineering, vol. 4. Kluwer Academic Publishers: Dordrecht, 1997; 91-118. · Zbl 0865.65014
[11] Closer to the solution: iterative linear solvers. In The State of the Art in Numerical Analysis, (eds). Clarendon Press: Oxford, 1997; 63-92. · Zbl 0881.65025
[12] Iterative Solution Methods. Cambridge University Press: Cambridge, 1994.
[13] Freund, Acta Numerica 1 pp 57– (1992)
[14] Fox, American Institute of Aeronautics and Astronautics Journal 6 pp 1036– (1968) · Zbl 0157.55905
[15] Fried, American Institute of Aeronautics and Astronautics Journal 7 pp 565– (1969) · Zbl 0185.52602
[16] Hughes, Journal of Engineering Mechanics 109 pp 576– (1983)
[17] Hughes, Computer Methods in Applied Mechanics and Engineering 36 pp 241– (1983)
[18] Winget, Computer Methods in Applied Mechanics and Engineering 52 pp 711– (1985)
[19] Computer solution of large linear systems. In Studies in Mathematics and Its Applications, vol. 28, (eds). Elsevier: Amsterdam, 1999.
[20] Template for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM Press: Philadelphia, PA, 1994.
[21] Iterative Methods for Sparse Linear Systems. PWS Publishing Company: Boston, 1996.
[22] Iterative Methods for Solving Linear Systems. SIAM Press: Philadelphia, PA, 1997.
[23] Iterative Methods for Linear and Nonlinear Equations. SIAM Press: Philadelphia, PA, 1995.
[24] Lee, International Journal for Numerical and Analytical Methods in Geomechanics 26 pp 341– (2002)
[25] Chan, International Journal for Numerical and Analytical Methods in Geomechanics 25 pp 1001– (2001)
[26] Hughes, Computer Methods in Applied Mechanics and Engineering 61 pp 215– (1987)
[27] Nour-Omid, SIAM Journal on Scientific and Statistical Computing 6 pp 761– (1985)
[28] Ortiz, Computer Methods in Applied Mechanics and Engineering 36 pp 223– (1983)
[29] Dayde, SIAM Journal on Scientific Computing 18 pp 1767– (1997)
[30] A new Krylov-subspace method for symmetric indefinite linear systems. Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics, Atlanta, U.S.A., Ames WF (ed.), 11-15 July 1994; 1253-1256.
[31] Critical State Soil Mechanics via Finite Elements. Ellis Horwood Ltd: Chichester, UK, 1952.
[32] Programming the Finite Element Method (3rd edn) Wiley: Chichester, UK, 1997.
[33] Murphy, SIAM Journal on Scientific Computing 21 pp 1969– (2000)
[34] Fischer, BIT 38 pp 527– (1998)
[35] Geotechnical Engineering: Soil Mechanics. Wiley: New York, 1995.
[36] Veiledning Ved Losing av Fandamenteringsoppgaver. Norwegian Geotechnical Institute Publication No. 16, Oslo, 1956.
[37] Paige, SIAM Journal on Numerical Analysis 12 pp 617– (1975)
[38] Cryer, Quarterly Journal of Mechanics and Applied Mathematics 16 pp 401– (1963)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.