Finite element analysis of viscoelastic structures using Rosenbrock-type methods

Computational Mechanics

Volumn 40, Issue 2, 2007, Pages 383-398

  Hartmann, Stefan ^a   Wensch, Jörg ^b  

^a UNIVERSITY OF KASSEL   (Germany)

^b UNIVERSITY OF POTSDAM   (Germany)

Author keywords

Differential algebraic equations; Finite elements; Integration schemes; Rosenbrock; Viscoelasticity

Indexed keywords

DIFFERENTIAL EQUATIONS; EVOLUTIONARY ALGORITHMS; FINITE ELEMENT METHOD; INTEGRATION; NONLINEAR SYSTEMS; PROBLEM SOLVING;

DIFFERENTIAL ALGEBRAIC EQUATIONS; DISCRETISATION; INTEGRATION SCHEMES; ROSENBROCK-TYPE METHODS;

VISCOELASTICITY;

EID: 34247468411     PISSN: 01787675     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00466-006-0117-y     Document Type: Article

References (39)

1. Drozdov A (1998) Viscoelastic structures. Mechanics of growth and aging. Academic, San Diego
2. Durran DR (1999) Numerical methods for wave equations in geophysical fluid dynamics. Springer, Berlin Heidelberg New york
3. Eckert S, Baaser H, Gross D, Scherf O (2004) A BDF2 integration method with stepsize control for elastoplasticity. Comput Mech 34(5):377-386
4. Ellsiepen P, Hartmann S (2001) Remarks on the interpretation of current non-linear finite-element-analyses as differential-algebraic equations. Int J Numer Methods Eng 51:679-707
5. Fritzen P (1997) Numerische Behandlung nichtlinearer Probleme der Elastizitäts- und Plastizitätstheorie. Doctoral Thesis, Department of Mathematics, University of Darmstadt
6. Gustafsson K, Lundh M, Söderlind G (1988) A pi stepsize control for the numerical solution of ordinary differential equations. BIT 28:270-287
7. Hairer E, Wanner G (1991) Solving ordinary differential equations IT. Springer, Berlin Heidelberg New york
8. Hairer E, Wanner G (1996) Solving ordinary differential equations II, 2nd eds. Springer, Berlin Heidelberg New york
9. Hartmann S (2000) A time adaptive finite-element procedure applied to creep and relaxation processes. ZAMM Z Angew Math Mech 80(Suppl. 2):S515-S516
10. Hartmann S (2002) Computation in finite strain viscoelasticity: finite elements based on the interpretation as differential-algebraic equations. Comput Methods Appl Mech Eng 191(13-14):1439-1470
11. Hartmann S (2002) On displacement control within the DIRK/MLNA approach in non-linear finite element analysis. In: Bathe K-J (ed) Computational Fluid and Solid Mechanics 2003, vol 1. Elsevier, Amsterdam, pp 316-319
12. Hartmann S (2005) A remark on the application of the Newton-Raphson method in non-linear finite element analysis. Comput Mech 36(2):100-116
13. Haupt P (2000) Continuum Mechanics and theory of materials. Springer, Berlin Heidelberg New york
14. Haupt P, Lion A (1995) Experimental identification and mathematical modelling of viscoplastic material behavior. J Continuum Mech Thermodyn 7:73-96
15. Higham DJ (1993) Error control for initial value problems with discontinuities and delays. Appl Numer Math 12(4): 315-330
16. Hundsdorfer W, Verwer J (2003) Numerical solution of time-dependent advection-diffusion-reaction equations. Springer series in computational mathematics, vol 33 In: Springer, Berlin Heidelberg New york
17. Kaps P, Ostermann A (1989) Rosenbrock methods using few LU-decompositions. IMA J Numer Anal 9:15-27
18. Kaps P, Rentrop P (1979) Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations. Numer Math 38:55-68
19. Kaps P, Wanner G (1981) A study of Rosenbrock-type methods of high order. Numer Math 38:279-298
20. Kavetski D, Binning P, Sloan SW (2004) Truncation error and stability analysis of iterative and non-iterative Thomas-Gladwell methods for first-order nonlinear differential equations. Int J Numer Methods Eng 60:2031-2043
21. Kirchner E, Simeon B (1999) A higher-order time integration method for viscoplasticity. Comput Methods Appl Mech Eng 175:1-18
22. Lang J (2001) Adaptive multilevel solution of nonlinear parabolic PDE systems: theory, algorithms and applications. Lecture notes in computational science and engineering, vol 16 In: Springer, Berlin Heidelberg New york
23. Lang J, Verwer J (2001) Ros3p-an accurate third-order rosenbrock solver designed for parabolic problems. BIT 41:731-738
24. Nrsett SP, Wolfbrandt A (1979) Order conditiones for Rosen-brock-type methods. Numer Math 32:1-15
25. Rabbat NBG, Sangiovanni-Vincentelli AL, Hsieh HY (1979) 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
26. Reich S (2006) Linearly implicit time stepping methods for numerical weather prediction. BIT (to appear)
27. Roche M (1988) Rosenbrock methods for differential-algebraic systems. Numer Math 52:45-63
28. Scherf O (2000) Numerische Simulation inelastischer Körper. Fortschritt-Berichte VDI, Reihe 20 (Rechnerunterstützte Verfahren) Nr. 321. VDI-Verlag, Düsseldorf
29. Simo JC, Taylor RL (1985) Consistent tangent operators for rate-independent elastoplasticity. Comput Methods Appl Mech Eng 48:101-118
30. Steihaug T, Wolfbrandt A (1979) An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff ordinary differential equations. Math Comput 33: 521-534
31. Strehmel K, Weiner R (1995) Numerik gewöhnlicher Differentialgleichungen. Teubner Verlag, Stuttgart
32. Strehmel K, Weiner R, Dannehl I (1988) A study of B- convergence of linearly implicit Runge-Kutta methods. Computing 40:241-253
33. Strehmel K, Weiner R, Dannehl I (1990) On error behaviour of partitioned linearly implicit Runge-Kutta methods for stiff and differential algebraic systems. BIT 30:358-375
34. Verwer JG, Hundsdorfer WH, Blom JG (2001) Numerical time integration for air pollution models. Surv Math Ind 10(2):107-147
35. Wensch J (1998) An eight stage fourth order partitioned rosenbrock method for multibody systems in index-3 formulation. Appl Numer Math 27(2):171-183
36. Wensch J (2004) Beiträge zur geometrischen integration und anwendungen in der numerischen simulation. Habilitation Thesis, Mensch und Buch Verlag, Berlin
37. Wensch J (2005) Krylov-ROW methods for DAEs of index 1 with applications to viscoelasticity. Appl Numer Math 53(2-4):527-541
38. Wensch J, Podhaisky H, Hartmann S (2003) Time integration of index 1 DAEs with Rosenbrock methods using Krylov subspace techniques. PAMM Proc Appl Math Mech 3:573-574
39. Wittekindt J (1991) Die numerische Lösung von Anfangs- Randwertproblemen zur Beschreibung inelastischen Werkstoffverhaltens. Doctoral Thesis, Department of Mathematics, University of Darmstadt