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Numerical performance of projection methods in finite element consolidation models. (English) Zbl 1112.74513

Summary: Projection, or conjugate gradient like, methods are becoming increasingly popular for the efficient solution of large sparse sets of unsymmetric indefinite equations arising from the numerical integration of (initial) boundary value problems. One such problem is soil consolidation coupling a flow and a structural model, typically solved by finite elements (FE) in space and a marching scheme in time (e.g. the Crank-Nicolson scheme). The attraction of a projection method stems from a number of factors, including the ease of implementation, the requirement of limited core memory and the low computational cost if a cheap and effective matrix preconditioner is available. In the present paper, biconjugate gradient stabilized (Bi-CGSTAB) is used to solve FE consolidation equations in 2-D and 3-D settings with variable time integration steps. Three different nodal orderings are selected along with the preconditioner ILUT based on incomplete triangular factorization and variable fill-in. The overall cost of the solver is made up of the preconditioning cost plus the cost to converge which is in turn related to the number of iterations and the elementary operations required by each iteration. The results show that nodal ordering affects the perfor mance of Bi-CGSTAB. For normally conditioned consolidation problems Bi-CGSTAB with the best ILUT preconditioner may converge in a number of iterations up to two order of magnitude smaller than the size of the FE model and proves an accurate, cost-effective and robust alternative to direct methods.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74L10 Soil and rock mechanics
76S05 Flows in porous media; filtration; seepage

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[1] Biot, Journal of Applied Physics 12 pp 155– (1941) · JFM 67.0837.01 · doi:10.1063/1.1712886
[2] Sandhu, Journal of Engineering Mechanics Division ASCE 95 pp 641– (1969)
[3] Christian, Journal of Soil Mechanics and Foundations Division ASCE 96 pp 1435– (1970)
[4] Hwang, Canadian Geotechnology Journal 8 pp 109– (1971) · doi:10.1139/t71-009
[5] Analysis of consolidation by numerical methods. Proceedings of the Symposium on Recent Developments in the Analysis of Soil Behavior and Application to Geotechnical Structures, University of New South Wales, Sidney, 1975.
[6] Finite element analysis of soil consolidation. Geotechnology Engineering Report to NSF 6, The Ohio State University, 1976.
[7] Smith, Geotechnique 26 pp 149– (1976) · doi:10.1680/geot.1976.26.1.149
[8] Generation and dissipation of pore water pressure. In Finite Elements in Geomechanics, (ed.). John Wiley: London, 1977; 293-317.
[9] The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, 2nd edn. John Wiley: Chichester, UK, 1998. · Zbl 0935.74004
[10] Ghaboussi, International Journal of Numerical Methods in Engineering 5 pp 419– (1973) · Zbl 0248.76037 · doi:10.1002/nme.1620050311
[11] Vermeer, International Journal for Numerical and Analytical Methods in Geomechanics 5 pp 1– (1981) · Zbl 0456.73060 · doi:10.1002/nag.1610050103
[12] Reed, International Journal for Numerical and Analytical Methods in Geomechanics 8 pp 243– (1984) · Zbl 0536.73089 · doi:10.1002/nag.1610080304
[13] Sloan, International Journal for Numerical and Analytical Methods in Geomechanics 23 pp 467– (1999) · Zbl 0955.74067 · doi:10.1002/(SICI)1096-9853(199905)23:6<467::AID-NAG949>3.0.CO;2-R
[14] Ferronato, International Journal of Solids and Structures 38 pp 5995– (2001) · Zbl 1075.74643 · doi:10.1016/S0020-7683(00)00352-8
[15] A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type. Proceedings of the Cambridge Philosophical Society, vol. 40, 1947; 50-67. · Zbl 0029.05901
[16] Booker, International Journal of Solids and Structures 11 pp 907– (1975) · Zbl 0311.73047 · doi:10.1016/0020-7683(75)90013-X
[17] Direct Methods for Sparse Matrices. Clarendom Press: Oxford, 1986. · Zbl 0604.65011
[18] Iterative Methods for Sparse Linear Systems. PWS Publishing Company: Boston, MA, 1996.
[19] Projection methods for the finite element solution of the dual-porosity model in variably saturated porous media. In Advances in Groundwater Pollution Control and Remediation, Vol. 9 of NATO ASI Series 2: Environment, (ed.) Kluwer Academic: Dordrecht, Holland, 1996; 97-125.
[20] Gambolati, Water Resources Research 36 pp 2443– (2000) · doi:10.1029/2000WR900127
[21] Local rock expansion in non-homogeneous productive gas/oil fields. In et al., editors, Proceedings of 10th Conference International Association of Computer Methods and Advances in Geomechanics, vol. 2, A. A. Balkema Publ., Rotterdam, Holland, 2001; 1295-1299.
[22] Iterative Solution Methods. Cambridge University Press: New York, NY, 1994. · doi:10.1017/CBO9780511624100
[23] van der Vorst, SIAM Journal of Scientific and Statistical Computing 13 pp 631– (1992) · Zbl 0761.65023 · doi:10.1137/0913035
[24] van der Knaap, Petroleum Transactions of the AIME 216 pp 179– (1959)
[25] Problems of rock mechanics in petroleum production engineering. Proceedings of the First Congress International Society of Rock Mechanics, Lisbon, 1966; 585-594.
[26] The Finite Element Method in Engineering Geoscience. McGraw Hill: London, 4th edition, 1991.
[27] Difference Methods for Initial-value Problems. Interscience: New York, 1957.
[28] Krylov subspace methods: Theory, algorithms, and applications. In Computing Methods in Applied Sciences and Engineering, (eds). SIAM: Philadelphia, 1990; 24-41. · Zbl 0741.65038
[29] Conjugate gradient-like methods for the numerical solution of the two site model in sorbing porous media. In Computational Mechanics ’95. Theory and Applications, (eds). Springer-Verlag: Berlin, New York, 1995; 748-753.
[30] Saad, SIAM Journal of Scientific and Statistical Computing 7 pp 856– (1986) · Zbl 0599.65018 · doi:10.1137/0907058
[31] Freund, SIAM Journal of Scientific Computing 14 pp 470– (1993) · Zbl 0781.65022 · doi:10.1137/0914029
[32] Iterative solution of linear systems. Technical Report 91.21, RIACS, NASA Ames Research Center, Mottet Field, 1991.
[33] Duff, BIT 29 pp 635– (1989) · Zbl 0687.65037 · doi:10.1007/BF01932738
[34] Brand, Numerical Mathematics 61 pp 433– (1992) · Zbl 0772.65018 · doi:10.1007/BF01385519
[35] Intoduction to Parallel Computing. Benjamin/Cunnings Publishing Company: CA, 1994.
[36] Karypis, Journal of Parallel and Distributed Computing 48 pp 71– (1998) · doi:10.1006/jpdc.1997.1403
[37] Lanczos, Journal of Research of the National Bureau of Standards 49 pp 33– (1952) · doi:10.6028/jres.049.006
[38] Sonneveld, SIAM Journal of Scientific and Statistical Computing 10 pp 36– (1989) · Zbl 0666.65029 · doi:10.1137/0910004
[39] Kershaw, Journal of Computational Physics 26 pp 43– (1978) · Zbl 0367.65018 · doi:10.1016/0021-9991(78)90098-0
[40] Saad, Numerical Linear Algebra Applications 1 pp 387– (1994) · Zbl 0838.65026 · doi:10.1002/nla.1680010405
[41] Johnson, SIAM Journal of Numerical Analysis 20 pp 362– (1983) · Zbl 0563.65020 · doi:10.1137/0720025
[42] Grote, SIAM Journal of Scientific Computing 18 pp 838– (1997) · Zbl 0872.65031 · doi:10.1137/S1064827594276552
[43] Benzi, SIAM Journal of Scientific Computing 19 pp 968– (1998) · Zbl 0930.65027 · doi:10.1137/S1064827595294691
[44] Chou, SIAM Journal of Scientific Computing 19 pp 995– (1998) · Zbl 0922.65034 · doi:10.1137/S1064827594270415
[45] Gambolati, International Journal for Numerical Methods in Fluids 29 pp 343– (1999) · Zbl 0946.76044 · doi:10.1002/(SICI)1097-0363(19990215)29:3<343::AID-FLD789>3.0.CO;2-W
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