Inexact Block Diagonal Preconditioners to Mitigate the Effects of Relative Differences in Material Stiffnesses
Publication: International Journal of Geomechanics
Volume 13, Issue 3
Abstract
Finite-element analysis of many soil-structure interaction problems involves material zones of widely differing stiffnesses. The large relative differences in material stiffnesses usually result in an ill-conditioned system for which practical preconditioners become ineffective for the iterative solution of such systems. The study suggests that a block diagonal preconditioner can exploit such differences. However, the theoretical preconditioner is expensive for practical use. Less costly block diagonal preconditioners are numerically evaluated for various inexact forms of the blocks in conjunction with conjugate gradient solver. The study includes analysis of two representative soil-structure interaction problems and proposes two simplified block diagonal preconditioners that effectively mitigate such material ill conditionings.
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Acknowledgments
The authors thank Dr. Chen Xi (Beijing Jiaotong University, China) and Dr. Hong Sze Han (GeoSoft Pte. Ltd., Singapore) for help in the implementation of codes into GeoFEA.
References
Addenbrooke, T. I., Potts, D. M., and Puzrin, A. M. (1997). “The influence of pre-failure soil stiffness on the numerical analysis of tunnel construction.” Geotechnique, 47(3), 693–712.
Amarasiri, A., and Kodikara, J. (2011). “Use of material interfaces in DEM to simulate soil fracture propagation in mode I cracking.” Int. J. Geomech., 11(4), 314–322.
Augarde, C. E., Crouch, R. S., Li, T., and Ramage, A. (2008). “The effects of geotechnical material properties on the convergence of iterative solvers.” Proc., 12th Int. Conf. of Int. Association for Computer Methods and Advances in Geomechanics (IACMAG), 587–594.
Augarde, C. E., Ramage, A., and Staudacher, J. (2006). “An element-based displacement preconditioner for linear elasticity problems.” Comp. Struct., 84(31–32), 2306–2315.
Augarde, C. E., Ramage, A., and Staudacher, J. (2007). “Element-based preconditioners for elasto-plastic problems in geotechnical engineering.” Int. J. Numer. Methods Eng., 71(7), 757–779.
Barrett, R., et al. (1994). Templates for the solution of linear systems: Building blocks for iterative methods, SIAM Press, Philadelphia.
Bergamaschi, L., Ferronato, M., and Gambolati, G. (2007). “Novel preconditioners for the iterative solution to FE-discretized coupled consolidation equations.” Comput. Methods Appl. Mech. Eng., 196(25–28), 2647–2656.
Bergamaschi, L., Ferronato, M., and Gambolati, G. (2008). “Mixed constraint preconditioners for the iterative solution of FE coupled consolidation equations.” J. Comput. Phys., 227(23), 9885–9897.
Biot, M. A. (1941). “General theory of three-dimensional consolidation.” J. Appl. Phys., 12(2), 155–164.
Britto, A. M., and Gunn, M. J. (1987). Critical state soil mechanics via finite elements, Ellis Horwood Ltd., Chichester, U.K.
Burd, H. J., Houlsby, G. T., Chow, L., Augarde, C. E., and Liu, G. (1994). “Analysis of settlement damage to masonry structures.” Proc., 3rd European Conf. on Numerical Methods in Geotechnical Engineering, 203–208.
Burkardt, J. (2003). “Reverse Cuthill McKee ordering.” 〈http://people.scs.fsu.edu/~burkardt/f_src/rcm/rcm.html〉 (Aug. 20, 2007).
Chan, S. H., Phoon, K. K., and Lee, F. H. (2001). “A modified Jacobi preconditioner for solving ill-conditioned Biot's consolidation equations using symmetric quasi-minimal residual method.” Int. J. Numer. Anal. Methods Geomech., 25(10), 1001–1025.
Chen, X., Phoon, K. K., and Toh, K. C. (2007). “Partitioned versus global Krylov subspace iterative methods for FE solution of 3-D Biot's problem.” Comput. Methods Appl. Mech. Eng., 196(25–28), 2737–2750.
Chen, X., Toh, K. C., and Phoon, K. K. (2006). “A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations.” Int. J. Numer. Methods Eng., 65(6), 785–807.
Clancy, P., and Randolph, M. F. (1993). “An approximate analysis procedure for piled raft foundations.” Int. J. Numer. Anal. Methods Geomech., 17(12), 849–869.
Dasari, G. R., Rawlings, C. G., and Bolton, M. D. (1996). “Numerical modelling of a NATM tunnel construction in London clay.” Proc., 1st Int. Symp. on the Geotechnical Aspects of Underground Construction in Soft Ground, Balkema, Rotterdam, Netherlands, 491–496.
Eisenstat, S. C. (1981). “Efficient implementation of A class of preconditioned conjugate gradient methods.” SIAM J. Sci. Statist. Comput., 2(1), 1–4.
Ferronato, M., Janna, C., and Gambolati, G. (2008). “Mixed constraint preconditioning in computational contact mechanics.” Comput. Methods Appl. Mech. Eng., 197(45–48), 3922–3931.
Galli, G., Grimaldi, A., and Leonardi, A. (2004). “Three-dimensional modelling of tunnel excavation and lining.” Comput. Geotech., 31(3), 171–183.
GeoFEA [Computer software]. Singapore, GeoSoft Pte. Ltd.
George, A., and Lui, J. (1981). Computer solution of large sparse positive definite systems, Prentice Hall, Englewood Cliffs, NJ.
Golub, G. H., and Van Loan, C. F. (1989). Matrix computations, 2nd Ed., Johns Hopkins University Press, Baltimore.
Hladik, I., Reed, M. B., and Swoboda, G. (1997). “Robust preconditioners for linear elasticity FEM analyses.” Int. J. Numer. Methods Eng., 40(11), 2109–2127.
Katzenbach, R., and Breth, H. (1981). “Nonlinear 3D analysis for NATM in Frankfurt clay.” Proc., 10th Int. Conf. on Soil Mechanics and Foundation Engineering, Stockholm, Sweden, 315–318.
Kayupov, M. A., Bulgakov, V. E., and Kuhn, G. (1998). “Efficient solution of 3-D geomechanical problems by indirect bem using iterative methods.” Int. J. Numer. Anal. Methods Geomech., 22(12), 983–1000.
Kuwabara, F. (1989). “An elastic analysis for piled raft foundations in a homogeneous soil.” Soils Found., 29(1), 82–92.
Lee, F. H., Phoon, K. K., and Lim, K. C. (2006). “Large scale three-dimensional finite element analysis of underground construction.” Numerical modeling of construction processes in geotechnical engineering for urban environment, T. Triantafyllidis, ed., Taylor and Francis Group, 141–153.
Lee, F. H., Phoon, K. K., Lim, K. C., and Chan, S. H. (2002). “Performance of Jacobi preconditioning in Krylov subspace solution of finite element equations.” Int. J. Numer. Anal. Methods Geomech., 26(4), 341–372.
Lee, K. M., and Rowe, R. K. (1990). “Finite element modelling of the three-dimensional ground deformations due to tunnelling in soft cohesive soils: Part I—Method of analysis.” Comput. Geotech., 10(2), 87–109.
Maheshwari, B. K., and Sarkar, R. (2011). “Seismic behaviour of soil-pile-structure interaction in liquefiable soils: Parametric study.” Int. J. Geomech., 11(4), 335–347.
Migliazza, M., Chiorboli, M., and Giani, G. P. (2009). “Comparison of analytical method, 3D finite element model with experimental subsidence measurements resulting from the extension of the Milan underground.” Comput. Geotech., 36(1–2), 113–124.
Möller, S. C. (2006). “Tunnel induced settlements and structural forces in linings.” Ph.D. thesis, Institut für Geotechnik der Universität Stuttgart, Stuttgart, Germany.
Mroueh, H., and Shahrour, I. (1999). “Use of sparse iterative methods for the resolution of three-dimensional soil/structure interaction problems.” Int. J. Numer. Anal. Methods Geomech., 23(15), 1961–1975.
Mroueh, H., and Shahrour, I. (2008). “A simplified 3D model for tunnel construction using tunnel boring machines.” Tunn. Undergr. Space Technol., 23(1), 38–45.
Murphy, M. F., Golub, G. H., and Wathen, A. J. (2000). “A note on preconditioning for indefinite linear systems.” SIAM J. Sci. Comput., 21(6), 1969–1972.
Nazzal, M. D., Abu-Farsakh, M. Y., and Mohammad, L. N. (2010). “Implementation of a critical state two-surface model to evaluate the response of geosynthetic reinforced pavements.” Int. J. Geomech., 10(5), 202–212.
Ng, E. G., and Peyton, B. W. (1993). “Block sparse Cholesky algorithms on advanced uniprocessor computers.” SIAM J. Sci. Comput., 14(5), 1034–1056.
Novak, L. J., Reese, L. C., and Wang, S. T. “Analysis of pile-raft foundations with 3D finite element method.” Proc., 2005 Structures and Congress and 2005 Forensic Engineering Symp., ASCE, Reston, VA.
Ottaviani, M. (1975). “Three-dimensional finite element analysis of vertically loaded pile groups.” Geotechnique, 25(2), 159–174.
Pan, X. D., and Hudson, J. A. (1988). “Plane strain analysis in modelling three-dimensional tunnel excavations.” Int. J. Rock Mech. Min. Sci., 25(5), 331–337.
Papadrakakis, M. (1993). “Solving large-scale linear problems in solid and structural mechanics.” Solving large-scale problems in mechanics: The development and application of computational solution methods, M. Papadrakakis, ed., Wiley, New York, 1–32.
Payer, H. J., and Mang, H. A. (1997). “Iterative strategies for solving systems of linear, algebraic equations arising in 3D BE-FE analyses of tunnel drivings.” Numer. Linear Algebra Appl., 4(3), 239–268.
Phoon, K. K., Chan, S. H., Toh, K. C., and Lee, F. H. (2003). “Fast iterative solution of large undrained soil-structure interaction problems.” Int. J. Numer. Anal. Methods Geomech., 27(3), 159–181.
Phoon, K. K., Lee, F. H., and Chan, S. H. (2006). “Iterative solution of intersecting tunnels using the generalised Jacobi preconditioner.” Numerical modeling of construction processes in geotechnical engineering for urban environment, T. Triantafyllidis, ed., Taylor and Francis Group, 155–163.
Phoon, K. K., Toh, K. C., Chan, S. H., and Lee, F. H. (2002). “An efficient diagonal preconditioner for finite element solution of Biot's consolidation equations.” Int. J. Numer. Methods Eng., 55(4), 377–400.
Potts, D. M., and Zdravković, L. (2001). Finite element analysis in geotechnical engineering: Application, Thomas Telford, London.
Poulos, H. G. (1994). “An approximate numerical analysis of pile-raft interaction.” Int. J. Numer. Anal. Methods Geomech., 18(2), 73–92.
Poulos, H. G. (2001). “Methods of analysis of piled raft foundations.” TC18, International Society on Soil Mechanics and Geotechnical Engineering.
Poulos, H. G., and Davis, E. H. (1980). Pile foundation analysis and design, Wiley, New York.
Randolph, M. F., and Wroth, C. P. (1978). “Analysis of deformation of vertically loaded piles.” J. Geotech. Engrg. Div., 104(12), 1465–1488.
Rowe, R. K., and Lee, K. M. (1992). “An evaluation of simplified techniques for estimating 3-dimensional undrained ground movements due to tunneling in soft soils.” Can. Geotech. J., 29(1), 39–52.
Saad, Y. (1994a). “ILUT: A dual threshold incomplete LU factorization.” Numer. Linear Algebra Appl., 1(4), 387–402.
Saad, Y. (1994b). SPARSKIT: A basic tool kit for sparse matrix computations, 〈http://www-users.cs.umn.edu/~saad/software/SPARSKIT/sparskit.html〉 (Dec. 17, 2006).
Saad, Y. (1996). Iterative methods for sparse linear systems, PWS Publishing, Boston.
Smith, I. M., and Griffiths, D. V. (1997). Programming the finite element method, 3rd Ed., Wiley, New York.
Smith, I. M., and Griffiths, D. V. (2004). Programming the finite element method, 4th Ed., Wiley, New York.
Smith, I. M., and Wang, A. (1998). “Analysis of piled rafts.” Int. J. Numer. Anal. Methods Geomech., 22(10), 777–790.
Toh, K. C., Phoon, K. K., and Chan, S. H. (2004). “Block preconditioners for symmetric indefinite linear systems.” Int. J. Numer. Methods Eng., 60(8), 1361–1381.
Vermeer, P. A., Bonnier, P. G., and Möller, S. C. (2001). “On a smart use of 3D-FEM in tunnelling.” Plaxis Bull., 11(11), 2–7.
Xiao, H. B., Zhang, C. S., Wang, Y. H., and Fan, Z. H. (2011). “Pile-soil interaction in expansive soil foundation: Analytical solution and numerical simulation.” Int. J. Geomech., 11(3), 159–166.
Zienkiewicz, O. C. (2005). The finite element method, 6th Ed., Butterworth Heinemann, Burlington, MA.
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Published In
International Journal of Geomechanics
Volume 13 • Issue 3 • June 2013
Pages: 273 - 291
Copyright
© 2013 American Society of Civil Engineers.
History
Received: Jan 24, 2011
Accepted: Dec 8, 2011
Published online: Dec 10, 2011
Published in print: Jun 1, 2013
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