Element-based preconditioners for elasto-plastic problems in geotechnical engineering

International Journal for Numerical Methods in Engineering

Volumn 71, Issue 7, 2007, Pages 757-779

  Augarde, C E ^a   Ramage, A ^b   Staudacher, J ^a  

^a DURHAM UNIVERSITY   (United Kingdom)

^b UNIVERSITY OF STRATHCLYDE   (United Kingdom)

Author keywords

Elasto plasticity; Finite elements; Iterative solvers; Preconditioning

Indexed keywords

FINITE ELEMENT METHOD; GEOTECHNICAL ENGINEERING; ITERATIVE METHODS; LINEAR SYSTEMS; MATHEMATICAL MODELS;

ITERATIVE SOLVERS; PRECONDITIONING;

ELASTOPLASTICITY;

ELASTOPLASTICITY; FINITE ELEMENT METHOD; GEOTECHNICAL ENGINEERING; ITERATIVE METHODS; LINEAR SYSTEMS; MATHEMATICAL MODELS;

EID: 34547899146     PISSN: 00295981     EISSN: 10970207     Source Type: Journal    
DOI: 10.1002/nme.1947     Document Type: Article

References (39)

1. Axelsson O. Iterative Solution Methods. Cambridge University Press: Cambridge, 1994.
2. Greenbaum A. Iterative Methods for Solving Linear Systems. SIAM: Philadelphia, PA, 1997.
3. Saad Y. Iterative Methods for Sparse Linear Systems. PWS: MA, 1996.
4. Saad Y, van der Vorst HA. Iterative solution of linear systems in the 20th century. Journal of Computational and Applied Mathematics 2000; 123:1-33.
5. Concus P, Golub GH, O'Leary DP. A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations. In Sparse Matrix Computations, Bunch JR, Rose DJ (eds). Academic Press: New York, 1976.
6. Hestenes MR, Stiefel E. Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards 1952; 49:409-436.
7. Axelsson O, Barker VA. Finite Element Solution of Boundary Value Problems. SIAM: Philadelphia, PA, 2001.
8. Dickinson JK, Forsyth PA. Preconditioned conjugate gradient methods for three-dimensional linear elasticity. International Journal for Numerical Methods in Engineering 1994; 37(13):2211-2234.
9. Graham E, Forsyth PA. Preconditioning methods for very ill-conditioned three-dimensional linear elasticity problems. International Journal for Numerical Methods in Engineering 1999; 44:77-99.
10. Hladik I, Reed MB, Swoboda G. Robust preconditioners for linear elasticity FEM analyses. International Journal for Numerical Methods in Engineering 1997; 40:2109-2127.
11. Saint-Georges P. Warzee G, Beauwens R, Notay Y. High-performance PCG solvers for FEM structural analysis. International Journal for Numerical Methods in Engineering 1996; 39:1313-1340.
12. Cai Z, Manteuffel TA, McCormick SF, Parter SV. First-order system least squares (FOSLS) for planar linear elasticity: pure traction problem. SIAM Journal on Numerical Analysis 1997; 35:320-335.
13. Augarde CE, Ramage A, Staudacher J. An element-based displacement preconditioner for linear elasticity problems. Computers and Structures, in press. Available online 27 October 2006 on http:// www.sciencedirect.com/science/article/B6V28-4M6SB97-5/2/ 68cd187bb650ec2cd404dfc1b43b4b29
14. Chan SH, Phoon KK, Lee FH. A modified Jacobi preconditioner for solving ill-conditioned Biot's consolidation equations using symmetric quasi-minimal residual method. International Journal for Numerical and Analytical Methods in Geomechanics 2001; 25:1001-1025.
15. Smith IM. A general purpose system for finite element analyses in parallel. Engineering Computations 2000; 17(1):75-91.
16. Mroueh H, Shahrour I. Use of sparse iterative methods for the resolution of three-dimensional soil/structure interaction problems. International Journal for Numerical and Analytical Methods in Geomechanics 1999; 23:1961-1975.
17. Paige CC, Saunders MA. Solution of sparse indefinite systems of linear equations. SIAM Journal on Numerical Analysis 1975; 12:617-629.
18. Saad Y, Schultz MH. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing 1986; 7:856-869.
19. van der Vorst HA. Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing 1992; 13:631-644.
20. Borja RI. Composite Newton-PCG and quasi-Newton iterations for nonlinear consolidation. Computer Methods in Applied Mechanics and Engineering 1991; 86:27-60.
21. Ferronato M, Gambolati G, Teatini P. III-conditioning of finite element poroelasticity equations. International Journal of Solids and Structures 2001; 38:5995-6014.
22. Gambolati G, Pini G, Ferronato M. Numerical performance of projection methods in finite element consolidation models. International Journal for Numerical and Analytical Methods in Geomechanics 2001; 25:1429-1447.
23. Phoon KK, Toh KC, Chan SH, Lee FH. An efficient diagonal preconditioner for finite element solution of Biot's consolidation equations. International Journal for Numerical Methods in Engineering 2002; 55:377-400.
24. Chen X, Toh KC, Phoon KK. A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations. International Journal for Numerical Methods in Engineering 2006; 65:785-807.
25. Potts DM, Zdravković L. Finite Element Analysis in Geotechnical Engineering: Theory. Thomas Telford: London, 1999.
26. Phoon KK, Chan SH, Toh KC, Lee FH. Fast iterative solution of large undrained soil-structure interaction problems. International Journal for Numerical and Analytical Methods in Geomechanics 2003; 27(3):159-181.
27. Beauwens R, Wilmet R. Conditioning analysis of positive definite matrices by approximate factorizations. Journal of Computational and Applied Mathematics 1989; 26:257-269.
28. Axelsson O, Gustafsson I. Iterative methods for the solution of the Navier equations of elasticity. Computer Methods in Applied Mechanics and Engineering 1978; 15:241-258.
29. Shlafman S, Efrat I. Using Korn's inequality for an efficient iterative solution of structural analysis problems. In Iterative Methods in Linear Algebra, Beauwens R, de Groen P (eds). North-Holland: Amsterdam, 1992.
30. Gustafsson I, Lindskog G. On parallel solution of linear elasticity problems. Part I: Theory. Numerical Linear Algebra with Applications 1998; 5:123-139.
31. Smith IM, Griffiths DV. Programming the Finite Element Method. Wiley: New York, 2004.
32. Hughes TJR, Levit I, Winget J. An element-by-element solution algorithm for problems of structural and solid mechanics. Computer Methods in Applied Mechanics and Engineering 1983; 36:241-254.
33. Gustafsson I, Lindskog G. A preconditioning technique based on element matrix factorisations. Computer Methods in Applied Mechanics and Engineering 1986; 55:201-220.
34. Kaasschieter EF. A general finite element preconditioning for the conjugate gradient method. BIT 1989; 29: 824-849.
35. Parlett BN. The Symmetric Eigenvalue Problem. SIAM: Philadelphia, PA, 1998.
36. Ramage A, Wathen AJ. On preconditioning for finite element equations on irregular grids. SIAM Journal on Matrix Analysis and Applications 1994; 15(3):909-921.
37. Wathen AJ. An analysis of some element-by-element techniques. Computer Methods in Applied Mechanics and Engineering 1989; 74:271-287.
38. Wathen AJ. Singular element preconditioning for the finite element method. In Iterative Methods in Linear Algebra, Beauwens R, de Groen P (eds). North-Holland: Amsterdam, 1992.
39. MATLAB, Mathworks Inc., http://www.mathworks.com/