×

Element-based preconditioners for elasto-plastic problems in geotechnical engineering. (English) Zbl 1194.74360

Summary: Iterative solvers are widely regarded as the most efficient way to solve the very large linear systems arising from finite element models. Their memory requirements are small compared to those for direct solvers. Consequently, there is a major interest in iterative methods and particularly the preconditioning necessary to achieve rapid convergence. In this paper we present new element-based preconditioners specifically designed for linear elasticity and elasto-plastic problems. The study presented here is restricted to simple associated plasticity but should find wide application in other plasticity models used in geotechnics.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
74L05 Geophysical solid mechanics
65F10 Iterative numerical methods for linear systems

Software:

Matlab
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Iterative Solution Methods. Cambridge University Press: Cambridge, 1994. · doi:10.1017/CBO9780511624100
[2] Iterative Methods for Solving Linear Systems. SIAM: Philadelphia, PA, 1997. · doi:10.1137/1.9781611970937
[3] Iterative Methods for Sparse Linear Systems. PWS: MA, 1996.
[4] Saad, Journal of Computational and Applied Mathematics 123 pp 1– (2000)
[5] , . A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. In Sparse Matrix Computations, (eds). Academic Press: New York, 1976.
[6] Hestenes, Journal of Research of the National Bureau of Standards 49 pp 409– (1952) · Zbl 0048.09901 · doi:10.6028/jres.049.044
[7] . Finite Element Solution of Boundary Value Problems. SIAM: Philadelphia, PA, 2001. · Zbl 0981.65130 · doi:10.1137/1.9780898719253
[8] Dickinson, International Journal for Numerical Methods in Engineering 37 pp 2211– (1994)
[9] Graham, International Journal for Numerical Methods in Engineering 44 pp 77– (1999)
[10] Hladik, International Journal for Numerical Methods in Engineering 40 pp 2109– (1997)
[11] Saint-Georges, International Journal for Numerical Methods in Engineering 39 pp 1313– (1996)
[12] Cai, SIAM Journal on Numerical Analysis 35 pp 320– (1997)
[13] Augarde, Computers and Structures (2006)
[14] Chan, International Journal for Numerical and Analytical Methods in Geomechanics 25 pp 1001– (2001)
[15] Smith, Engineering Computations 17 pp 75– (2000)
[16] Mroueh, International Journal for Numerical and Analytical Methods in Geomechanics 23 pp 1961– (1999)
[17] Paige, SIAM Journal on Numerical Analysis 12 pp 617– (1975)
[18] Saad, SIAM Journal on Scientific and Statistical Computing 7 pp 856– (1986)
[19] van der Vorst, SIAM Journal on Scientific and Statistical Computing 13 pp 631– (1992)
[20] Borja, Computer Methods in Applied Mechanics and Engineering 86 pp 27– (1991)
[21] Ferronato, International Journal of Solids and Structures 38 pp 5995– (2001)
[22] Gambolati, International Journal for Numerical and Analytical Methods in Geomechanics 25 pp 1429– (2001)
[23] Phoon, International Journal for Numerical Methods in Engineering 55 pp 377– (2002)
[24] Chen, International Journal for Numerical Methods in Engineering 65 pp 785– (2006)
[25] . Finite Element Analysis in Geotechnical Engineering: Theory. Thomas Telford: London, 1999. · doi:10.1680/feaiget.27534
[26] Phoon, International Journal for Numerical and Analytical Methods in Geomechanics 27 pp 159– (2003)
[27] Beauwens, Journal of Computational and Applied Mathematics 26 pp 257– (1989)
[28] Axelsson, Computer Methods in Applied Mechanics and Engineering 15 pp 241– (1978)
[29] . Using Korn’s inequality for an efficient iterative solution of structural analysis problems. In Iterative Methods in Linear Algebra, (eds). North-Holland: Amsterdam, 1992. · Zbl 0785.65039
[30] Gustafsson, Numerical Linear Algebra with Applications 5 pp 123– (1998)
[31] . Programming the Finite Element Method. Wiley: New York, 2004.
[32] Hughes, Computer Methods in Applied Mechanics and Engineering 36 pp 241– (1983)
[33] Gustafsson, Computer Methods in Applied Mechanics and Engineering 55 pp 201– (1986)
[34] Kaasschieter, BIT 29 pp 824– (1989)
[35] The Symmetric Eigenvalue Problem. SIAM: Philadelphia, PA, 1998. · doi:10.1137/1.9781611971163
[36] Ramage, SIAM Journal on Matrix Analysis and Applications 15 pp 909– (1994)
[37] Wathen, Computer Methods in Applied Mechanics and Engineering 74 pp 271– (1989)
[38] Singular element preconditioning for the finite element method. In Iterative Methods in Linear Algebra, (eds). North-Holland: Amsterdam, 1992. · Zbl 0785.65106
[39] MATLAB, Mathworks Inc., http://www.mathworks.com/
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.