Performance of Neumann Expansion Preconditioners for Iterative Methods with Geotechnical Elastoplastic Applications
Publication: International Journal of Geomechanics
Volume 16, Issue 3
Abstract
Because most geotechnical analyses may involve elastoplastic geomaterials, a robust solution scheme is of critical importance to the efficiency of the entire finite-element (FE) computation. To accelerate the Krylov subspace iterative methods, some preconditioning techniques have been developed based on the factorization of elastic stiffness. However, the idea of constructing a preconditioner based on elastic stiffness is heuristic. In this study, a class of Neumann expansion preconditioners constructed from the constant (elastic) partition and varying (plastic) partition of the elastoplastic stiffness matrix was proposed. Based on two numerical examples, the performances of truncated Neumann expansion preconditioners were examined with associated and nonassociated soil plasticity considered, respectively. It is interesting to note that the convergence behaviors of truncated Neumann expansion preconditioners closely depended on the approximation to the elastic stiffness part as well as the truncated terms. Numerical experiments also revealed that in most large-size elastoplastic FE analyses, zero-order Neumann expansion preconditioners with appropriate approximations to the elastic stiffness appear to be more appealing, particularly for those applications involving a large amount of nonlinear iterations.
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Acknowledgments
This study was supported by the National Natural Science Foundation of China (51179092), the Scientific Research Foundation for the National Basic Research Program of China (2012CB026104), and the National Natural Science Foundation of China (51379103).
References
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.
Axelsson, O., Blaheta, R., and Kohut, R. (1997). “Inexact Newton solvers in plasticity: Theory and experiments.” Numer. Linear Algebr., 4(3), 133–152.
Benzi, M. (2002). “Preconditioning techniques for large linear systems: A survey.” J. Comput. Phys., 182(2), 418–477.
Benzi, M., Kouhia, R., and Tůma, M. (1998). “An assessment of some preconditioning techniques in shell problems.” Commun. Numer. Methods Eng., 14(10), 897–906.
Benzi, M., Cullum, J. K., and Tuma, M. (2000). “Robust approximate inverse preconditioning for the conjugate gradient method.” SIAM J. Sci. Comput., 22(4), 1318–1332.
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.
Brown, P. N., and Saad, Y. (1994). “Convergence theory of nonlinear Newton–Krylov algorithms.” SIAM J. Optim., 4(2), 297–330.
Britto, A. M., and Gunn, M. J. (1987). Critical state soil mechanics via finite elements, Ellis Horwood, Chichester, U.K.
Chan, T. F., and van der Vorst, H. A. (1994). “Approximate and incomplete factorizations.” Technical Rep. 871, Dept. of Mathematics, Univ. of Utrecht, Utrecht, the Netherlands.
Chen, X., and Cheng, Y.-G. (2011). “On accelerated symmetric stiffness techniques for non-associated plasticity with application to soil problems.” Eng. Computation, 28(8), 1044–1063.
Chen, X., and Phoon, K. K. (2012). “Applications of symmetric and nonsymmetric MSSOR preconditioners to large-scale Biot’s consolidation problems with nonassociated plasticity.” J. Appl. Math., 2012, 352081.
Chen, X., Phoon, K.-K., and Toh, K.-C. (2012). “Performance of zero-level fill-in preconditioning techniques for iterative solutions with geotechnical applications.” Int. J. Geomech., 12(5), 596–605.
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.
Chen, X., Wu, Y., Yu, Y., Liu, J., Xu, X. F., and Ren, J. (2014). “A two-grid search scheme for large-scale 3-D finite element analyses of slope stability.” Comput. Geotech., 62, 203–215.
Crisfield, M. A. (1991). Non-linear finite element analysis of solids and structures, Vol. 1, Essentials, John Wiley & Sons, New York.
Dembo, R. S., Eisenstat, S. C., and Steihaug, T. (1982). “Inexact Newton methods.” SIAM J. Numer. Anal., 19(2), 400–408.
Eisenstat, S. C., and Walker, H. F. (1996). “Choosing the forcing terms in an inexact Newton method.” SIAM J. Sci. Comput., 17(1), 16–32.
Ferronato, M. (2012). “Preconditioning for sparse linear systems at the dawn of the 21st century: History, current developments, and future perspectives.” ISRN Appl Math, 2012, 2012, 1–49.
Ferronato, M., Pini, G., and Gambolati, G. (2009). “The role of preconditioning in the solution to FE coupled consolidation equations by Krylov subspace methods.” Int. J. Numer. Anal. Methods Geomech., 33(3), 405–423.
Gambolati, G., Ferronato, M., and Janna, C. (2011). “Preconditioners in computational geomechanics: A survey.” Int. J. Numer. Anal. Methods Geomech., 35(9), 980–996.
Ghanem, R. G., and Spanos, P. D. (2003). Stochastic finite elements: A spectral approach, Courier Dover Publications, Mineola, NY.
Irons, B. M. (1970). “A frontal solution program for finite element analysis.” Int. J. Numer. Methods Eng., 2(1), 5–32.
Knoll, D. A., and Keyes, D. E. (2004). “Jacobian-free Newton–Krylov methods: A survey of approaches and applications.” J. Comput. Physics, 193(2), 357–397.
Krabbenhøft, K. (2002). “Basic computational plasticity.” Lecture notes, Dept. of Civil Engineering, Technical University of Denmark, Kongens Lyngby, Denmark.
Lee, F.-H., Hong, S.-H., Gu, Q., and Zhao, P. (2011). “Application of large three-dimensional finite-element analyses to practical problems.” Int. J. Geomech., 11(6), 529–539.
Mroueh, H., and Shahrour, I. (1999). “Use of sparse iterative methods for the resolution of three-dimensional soil/structure interaction problem.” Int. J. Numer. Anal. Methods Geomech., 23(15), 1961–1975.
Ng, E. G., and Peyton, B. W. (1993). “Block sparse Cholesky algorithms on advanced uniprocessor computers.” SIAM J. Sci. Comput., 14(5), 1034–1056.
Plaxis (2012) “Phi-c reduction and comparison with Bishop’s method.” Validation & verification, Plaxis, Delft, the Netherlands.
Saad, Y. (1994). SPARSKIT–A basic tool-kit for sparse matrix computations (Version 2), University of Minnesota, Minneapolis, MN.
Simo, J. C., and Taylor, R. L. (1985). “Consistent tangent operators for rate-independent elastoplasticity.” Comput. Methods Appl. Mech. Eng., 48(1), 101–118.
Smith, I. M., Griffiths, D. V., and Margetts, L. (2014). Programming the finite element method, 5th Ed., John Wiley & Sons, New York.
Tran, H. H. T., Toh, K. C., and Phoon, K. K. (2013). “Preconditioned IDR(s) iterative solver for non-symmetric linear system associated with FEM analysis of shallow foundation.” Int. J. Numer. Anal. Methods Geomech., 37(17), 2972–2986.
van der Vorst, H. A. (1982). “A vectorizable variant of some ICCG methods.” SIAM J. Sci. Stat. Comput., 3(3), 350–356.
Van Gijzen, M. B., and Sonneveld, P. (2011). “Algorithm 913: An elegant IDR(s) variant that efficiently exploits biorthogonality properties.” ACM Trans. Math. Software, 38(1), 5
White, J. A., and Borja, R. I. (2011). “Block-preconditioned Newton–Krylov solvers for fully coupled flow and geomechanics.” Comput. Geosci., 15(4), 647–659.
Zhao, B., Goh, S. H., and Lee, F. H. (2010). “Implicit domain decomposition scheme for parallel dynamic finite element geotechnical analysis.” Proc., 9th WSEAS Int. Conf. on Software Engineering, Parallel and Distributed Systems, World Scientific and Engineering Academy and Society (WSEAS) Press, Athens, Greece, 127–132.
Zienkiewicz, O. C., Chan, A. H. C., Schrefler, B. A., and Shiomi, T. (1998). Computational geomechanics with special reference to earthquake engineering, John Wiley & Sons, Chichester, U.K.
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Published In
International Journal of Geomechanics
Volume 16 • Issue 3 • June 2016
Copyright
© 2015 American Society of Civil Engineers.
History
Received: Jan 24, 2015
Accepted: Jun 18, 2015
Published online: Dec 3, 2015
Discussion open until: May 3, 2016
Published in print: Jun 1, 2016
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