A remark on the application of the Newton-Raphson method in non-linear finite element analysis

Computational Mechanics

Volumn 36, Issue 2, 2005, Pages 100-116

  Hartmann, Stefan ^a  

^a UNIVERSITY OF KASSEL   (Germany)

Author keywords

Consistent tangent; Differential algebraic equations; Finite element method; Newton Raphson method; Time integration

Indexed keywords

ALGORITHMS; BOUNDARY VALUE PROBLEMS; COMPUTATIONAL METHODS; DIFFERENTIAL EQUATIONS; ELASTICITY; ELASTOPLASTICITY; INTEGRATION; MATHEMATICAL OPERATORS; NONLINEAR EQUATIONS; OPTIMIZATION; QUADRATIC PROGRAMMING; VISCOELASTICITY;

CONSISTENT TANGENTS; DIFFERENTIAL ALGEBRAIC EQUATIONS; NEWTON-RAPHSON METHOD; TIME INTEGRATION;

FINITE ELEMENT METHOD;

EID: 21144451713     PISSN: 01787675     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00466-004-0630-9     Document Type: Article

References (25)

1. Bathe K-J, Ramm E, Wilson E (1975) Finite element formulations for large deformation dynamic analysis. Int J Numer Methods Eng 9(2):353-386
2. Bicanic N, Johnson K (1979) Who was 'Raphson'? Int J Numer Methods Eng 14:148-152
3. Cash J (1979) Diagonally implicit Runge-Kutta formulae with error estimates. J Institute of Math Appl 24:293-301
4. Christensen P (2000) Computational Nonsmooth Mechanics. Contact, Friction and Plasticity. Doctoral thesis, Department of Mechanical Engineering, Linköping University. Linköping Studies in Science and Technology, No.657
5. Davis T, Duff I (1997) Umfpack version 2.2 unsymmetric-pattern multifrontal package, http://heron.cc.ukans.edu/software/umfpack
6. Eckert S, Baaser H, Gross D, Scherf O (2004). A BDF2 integration method with stepsize control for elastoplasticity. Comput Mech 34: 377-386
7. 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
8. Fritzen P (1997) Numerische Behandlung nichtlinearer Probleme der Elastizitäts- und Plastizitätstheorie. Doctoral thesis, Department of Mathematics, University of Darmstadt
9. Golub G, van Loan CF (1986) Matrix Computations. North Oxford Academic Publ., London
10. Hartmann S (1998a) Nichtlineare Finite-Elemente-Berechnung angewendet auf ein Viskoplastizitätsmodell mit Überspannungen. In: Hartmann S, Tsakmakis C, (eds) Aspekte der Kontinuumsmechanik und Materialtheorie. Gesamthochschul-Bibliothek Verlag, Kassel, pp. 55-80
11. Hartmann S (1998b) Zur Berechnung inelastischer Festkörper mit der Methode der finiten Elemente. In Hartmann S, Haupt P, Ulbricht V, (eds) Modellierung und Simulation. Gesamthochschul-Bibliothek, Kassel pp. 119-130
12. Hartmann S (2002) Computation infinite strain viscoelasticity: finite elements based on the interpretation as differential-algebraic equations. Comput Methods Appl Mech Eng 191(13-14):1439-1470
13. Hartmann S (2003a) Finite-Elemente Berechnung inelastischer Kontinua. Interpretation als Algebro-Differentialgleichungs systeme. Technical Report 1/2003, University of Kassel, Institute of Mechanics, Kassel, Germany
14. Hartmann S (2003b) On displacement control within the DIRK/MLNA approach in non-linear finite element analysis. In: Bathe K-J (ed) Comput. Fluid Solid Mech. 2003, volume 1. Elsevier, Amsterdam pp. 316-319
15. Hartmann S, Lührs G, Haupt P (1997) An efficient stress algorithm with applications in viscoplasticity and plasticity. Int J Num Methods Eng 40:991-1013
16. Hibbitt H, Marcal P, Rice J (1970) A finite element formulation for problems of large strain and large displacement. Int J Solids Struct 6:1069-1086
17. Hoyer W, Schmidt J (1984) Newton-type decomposition methods for equations arising in network analysis. ZAMM Zeitschrift für Angewandte Mathematik und Mechanik 64:397-405
18. Hughes T, Pister K (1979) Consistent linearization in mechanics of solids and structures. Comput Struct 8:391-397
19. Lührs G, Hartmann S, Haupt P (1997) On the numerical treatment of finite deformations in elastoviscoplasticity. Comput Methods Appl Mech Eng 144:1-21
20. Press W, Teukolsky S, Vetterling W, Flannery B (1992) Numerical Recipes in FORTRAN. Cambridge University Press, Cambridge, 2nd edn.
21. Rabbat N, Sangiovanni-Vincentelli A, Hsieh H (1979) A multilevel Newton algorithm with macromodeling and latency for the analysis of large-scale nonlinear circuits in the time domain. IEEE Trans Circuits Sys 26:733-740
22. Scherf O (2000) Numerische Simulation inelastischer Körper. Fortschritt-Berichte VDI, Reihe 20 (Rechnerunterstützte Verfahren) Nr.321. VDI-Verlag, Düsseldorf.
23. Simo J, Taylor R (1985) Consistent tangent operators for rate-independent elastoplasticity. Comput Methods Appl Mech Eng 48:101-118
24. Weber G, Anand L (1990) Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids. Comput Methods Appl Mech Eng 79:173-202
25. Wittekindt J (1991) Die numerische Lösung von Anfangs- Randwertproblemen zur Beschreibung inelastischen Werkstoffverhaltens. Doctoral thesis, Department of Mathematics, University of Darmstadt.