Iterative solvers within sequences of large linear systems in non-linear structural mechanics

ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

Volumn 89, Issue 9, 2009, Pages 711-728

  Hartmann, Stefan ^a   Duintjer Tebbens, Jurjen ^b   Quint, Karsten J ^c   Meister, Andreas ^c  

^a CLAUSTHAL UNIVERSITY OF TECHNOLOGY   (Germany)

^b INSTITUTE OF COMPUTER SCIENCE   (Czech Republic)

^c UNIVERSITY OF KASSEL   (Germany)

Author keywords

Finite elements; Finite strains; Iterative solver; Non symmetric matrices; Sequences of linear systems

Indexed keywords

EID: 68849103142     PISSN: 00442267     EISSN: 15214001     Source Type: Journal    
DOI: 10.1002/zamm.200800211     Document Type: Article

References (66)

1. L. Armijo, Minimization of functions having Lipschitz-continuous first partial derivatives, Pacific J. Math. 16, 1-3 (1966).
2. C.E. Augarde, A. Ramage, and J. Staudacher, An element-based displacement preconditioner for linear elasticity problems, Comput. Struct. 84, 2306-2315 (2006).
3. C.E. Augarde, A. Ramage, and J. Staudacher, Element-based preconditioners for elasto-plastic problems in geotechnical engineering, Int. J. Numer. Meth. Eng. 71, 757-779 (2007).
4. K.J. Bathe, Finite Element Procedures in Engineering Analysis (Prentice Hall, Englewood Cliffs, 1982).
5. T. Belytschko, W.K. Liu, and B. Moran, Nonlinear Finite Elements for Continua and Structures (John Wiley and Sons, Chichester, 2000).
6. M. Benzi and D. Bertaccini, Approximate inverse preconditioning for shifted linear systems, BIT 43(2), 231-244 (2003).
7. M. Benzi and M. Tůma, A sparse approximate inverse preconditioner for nonsymmetric linear systems, SIAM J. Sci. Comput. 19(3), 968-994 (electronic) (1998).
8. D. Bertaccini, Efficient preconditioning for sequences of parametric complex symmetric linear systems, Electron. Trans. Numer. Math. 18, 49-64 (2004).
9. P. Birken, J. Duintjer Tebbens, A. Meister, and M. Tůma, Preconditioner updates applied to CFD model problems, Appl. Numer. Math. 58(11), 1628-1641 (2008).
10. J. Büttner and B. Simeon, Runge-Kutta methods in elastoplasticity, Appl. Numer. Math. 41, 443-458 (2002).
11. F. Cameron, Low-order Runge-Kutta Methods for Differential-Algebraic Equations, PhD thesis, Tampere University of Technology, Department of Mathematics, Tampere, Finland, 1999.
12. M. Clemens, M. Wilke, and T. Weiland, Efficient extrapolation methods for electro- and magneto-quasistatic field simulations, Adv. Radio Sci. 1, 81-86 (2003).
13. M.A. Crisfield, Finite Elements and Solution Procedures for Structural Analysis (Pineridge Press, Swansea, 1986).
14. T.A. Davis and I.S. Duff, A combined/multifrontal method for unsymmetric sparse matrices, Technical Report TR-97-016 (REP-1997-252), Computer and Information Science and Engineering Department, University of Florida, Gainesville, Florida, 1997.
15. R.S. Dembo, S.C. Eisenstat, and T. Steihaug, Inexact Newton methods, SIAM J. Numer. Anal. 19(2), 400-408 (1982).
16. J.W. Demmel, S.C. Eisenstat, J.R. Gilbert, X.S. Li, and J.W.H. Liu, A supernodal approach to sparse partial pivoting, SIAM J. Matrix Anal. Appl. 20(3), 720-755 (1999).
17. J.E. Dennis and R.B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations, Classics in Applied Mathematics, Vol. 16 (SIAM Society for Industrial and Applied Mathematics, Philadelphia, 1996).
18. J. Duintjer Tebbens and M. Tůma, Efficient preconditioning of sequences of nonsymmetric linear systems, SIAM J. Sci. Comput. 25(5), 1918-1941 (2007).
19. S.C. Eisenstat and H.F. Walker, Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput. 17(1), 16-32 (1996), Special issue on iterative methods in numerical linear algebra (Breckenridge, CO, 1994).
20. P. Ellsiepen, Zeit- und ortsadaptive Verfahren angewandt auf Mehrphasenprobleme poröser Medien, Doctoral thesis, Institute of Mechanics II, University of Stuttgart, 1999, Report No. II-3.
21. P. Ellsiepen and S. Hartmann, Remarks on the interpretation of current non-linear finite-element-analyses as differential-algebraic equations, Int. J. Numer. Meth. Eng. 51, 679-707 (2001).
22. S.K. Esche, G.L. Kinzel, and T. Altan, Issues in convergence improvements for non-linear finite element programs, Int. J. Numer. Meth. Eng. 40, 4577-4594 (1997).
23. R.M. Ferencz and T.J.R. Hughes, Iterative finite element solutions in nonlinear solid mechanics, in: Numerical Methods for Solids (Part 3), edited by P.G. Ciarlet and J.L. Lions, Handbook of Numerical Analysis Vol.VI (Elsevier Science, 1998), pp. 3-179.
24. P. Fritzen, Numerische Behandlung nichtlinearer Probleme der Elastizitäts- und Plastizitätstheorie, Doctoral thesis, Department of Mathematics, University of Darmstadt, 1997.
25. G.H. Golub and C.F. van Loan, Matrix Computation, 3rd edition (The Johns Hopkins University Press, Baltimore, 1996).
26. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, 2nd revised edition (Springer, Berlin, 1996).
27. S. Hartmann, Zur Berechnung inelastischer Festkörper mit der Methode der finiten Elemente, in: Modellierung und Identifikation, edited by S. Hartmann, P. Haupt, and V. Ulbricht (Gesamthochschul-Bibliothek, Kassel, 1998), pp. 119-130.
28. S. Hartmann, Computation in finite strain viscoelasticity: finite elements based on the interpretation as differential-algebraic equations, Comput. Meth. Appl. Mech. Eng. 191(13-14), 1439-1470 (2002).
29. S. Hartmann, A remark on the application of the Newton-Raphson method in non-linear finite element analysis, Comput. Mech. 36(2), 100-116 (2005).
30. S. Hartmann, A thermomechanically consistent constitutive model for polyoxymethylene: experiments, material modeling and computation, Arch. Appl. Mech. 76, 349-366 (2006).
31. S. Hartmann and W. Bier, High-order time integration applied to metal powder plasticity, Int. J. Plasticity 24(1), 17-54 (2008).
32. S. Hartmann, K.J. Quint, and M. Arnold, On plastic incompressibility within time-adaptive finite elements combined with projection techniques, Comput. Meth. Appl. Mech. Eng. 198(2), 178-193 (2008).
33. M.R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Res. Natl. Bur. Stand., Section B 49, 409-436 (1952).
34. C.T. Kelley, Iterative methods for linear and nonlinear equations (SIAM Society for Industrial and Applied Mathematics, Philadelphia, 1995).
35. C.T. Kelley and D.E. Keyes, Convergence analysis of pseudo-transient continuation, SIAM J. Numer. Anal. 35, 508-523 (1998).
36. D.V. Kulkarni, D. Tortorelli, and M. Wallin, A Newton-Schur alternative to the consistent tangent approach in computational plasticity, Comput Meth. Appl. Mech. Eng. 196, 1169-1177 (2007).
37. A. Lion, Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models, Int. J. Plast. 16, 469-494 (2000).
38. G. Lührs, S. Hartmann, and P. Haupt, On the numerical treatment of finite deformations in elastoviscoplasticity, Comput. Meth. Appl. Mech. Eng. 144, 1-21 (1997).
39. R. Mahnken, A Newton-multigrid algorithm for elasto-plastic/viscoplastic problems, Comput. Mech. 15, 408-425 (1995).
40. A. Meister, Numerik linearer Gleichungssysteme, 3rd edition (Vieweg Verlag, Wiesbaden, 2008).
41. G. Meurant, On the incomplete Cholesky decomposition of a class of perturbed matrices, SIAM J. Sci. Comput. 23(2), 419-429 (electronic) (2001), Copper Mountain Conference (2000).
42. K. Miller, Nonlinear krylov and moving nodes in the method of lines, JCAM 183, 275-287 (2005).
43. K. Mocellin, L. Fourment, T. Coupez, and J.L. Chenot, Toward large scale F.E. computation of hot forging process using iterative solvers, parallel computation and multigrid algorithms, Int. J. Numer. Meth. Eng. 52, 473-488 (2001).
44. P. Neff and C. Wieners, Comparison of models for finite plasticity, Comput. Vis. Sci. 6, 23-35 (2003).
45. M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, and S. Maiti, Recycling Krylov subspaces for sequences of linear systems, SIAM J. Sci. Comput. 28, 1651-1674 (2006).
46. N.B.G. Rabbat, A.L. Sangiovanni-Vincentelli, and H.Y. Hsieh, A multilevel Newton algorithm with macromodeling and latency for the analysis of large-scale nonlinear circuits in the time domain, IEEE Trans. Circ. Syst. 26, 733-740 (1979).
47. Y. Saad, Iterative methods for sparse linear systems, 2nd edition (SIAM Society for Industrial and Applied Mathematics, Philadelphia, 2003).
48. Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 7, 856-869 (1986).
49. Y. Saad, ILUT: a dual threshold incomplete LU factorization, Numer. Linear Algebra Appl. 1(4), 387-402 (1994).
50. P. Saint-Georges, G. Warzee, R. Beauwens, and Y. Notay, High-performance PCG solvers for FEM structural analysis, Int. J. Numer. Meth. Eng. 39, 1313-1340 (1996).
51. O. Schenk, K. Gärtner, and W. Fichtner, Efficient sparse LU factorization with left-right looking strategy on shared memory multiprocessors, BIT 40(1), 158-176 (2000).
52. O. Scherf, Numerische Simulation inelastischer Körper, Fortschritt-Berichte VDI, Reihe 20 (Rechnerunterstützte Verfahren) Nr.321 (VDI-Verlag, Düsseldorf, 2000).
53. J. Schöberl, Robust multigrid methods for parameter dependent problems, PhD thesis, Institut für Analysis und Numerik, Universität Linz, Linz, Austria, 1999.
54. J.C. Simo and T.J.R. Hughes, Computational Inelasticity (Springer Verlag, New York, 1998).
55. J.C. Simo and R.L. Taylor, Consistent tangent operators for rate-independent elastoplasticity, Comput. Meth. Appl. Mech. Eng. 48, 101-118 (1985).
56. J.A. Stricklin, W.E. Haisler, and W.A. von Riesemann, Evaluation of solution procedures for material and for geometrically nonlinear structural analysis, AIAA 11, 292-299 (1973).
57. C. Tsakmakis and A. Willuweit, A comparative study of kinematic hardening rules at finite deformations, Int. J. Non-Lin. Mech. 39, 539-554 (2004).
58. A.H. van den Boogard, T. Meinders, and J. Huétink, Efficient implicit finite element analysis of sheet forming processes, Int. J. Numer. Meth. Eng. 56, 1083-1107 (2003).
59. H.A. van der Vorst, BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput. 13, 631-644 (1992).
60. J.G. Verwer, Convergence and order reduction of diagonally implicit runge-kutta schemes in the method of lines, Tech. Rep. NM-R8506, Department of Numerical Mathematics, Centrum for Mathematics and Computer Science, 1986.
61. L.T. Watson, S.C. Billups, and A. P. Morgan, Algortihm 652: HOMPACK: A suite of codes for globally convergent homotopy algorithms, ACM Trans. Math. Software 13, 281-310 (1987).
62. C. Wieners, M. Ammann, S. Diebels, and W. Ehlers, Parallel 3-d simulations for porous media models in solid mechanics, Comput. Mech. 29, 75-87 (2002).
63. C. Wieners, M. Ammann, T. Graf, and W. Ehlers, Parallel Krylov methods and the application to 3-d simulations of a triphasic porous media model in soil mechanics, Comput. Mech. 36, 409-420 (2005).
64. J. Wittekindt, Die numerische Lösung von Anfangs-Randwertproblemen zur Beschreibung inelastischen Werkstoffverhaltens, Doctoral thesis, Department of Mathematics, University of Darmstadt, 1991.
65. P. Wriggers, Nichtlineare Finite-Elemente Methoden (Springer Verlag, Berlin, 2001).
66. M. Yannakakis, Computing the minimum fill-in is NP-complete, SIAM J. Algebraic Discrete Methods 2(1), 77-79 (1981).