Mixed finite element methods for linear viscoelasticity using weak symmetry

Mathematical Models and Methods in Applied Sciences

Volumn 20, Issue 6, 2010, Pages 955-985

  Rognes, Marie E ^a   Winther, Ragnar ^a  

^a Center of Mathematics for Applications   (Norway)

Author keywords

elasticity; Mixed finite elements; viscoelasticity; weak symmetry

Indexed keywords

APRIORI; GENERAL MODEL; KELVIN VOIGT MODEL; LINEAR VISCOELASTICITY; MIXED FINITE ELEMENT METHODS; MIXED FINITE ELEMENTS; NUMERICAL EXPERIMENTS; QUASI-STATIC; STRESS TENSORS; THREE DIMENSIONS; UNIFIED FRAMEWORK; VISCOELASTIC BODIES;

ELASTICITY; VISCOELASTICITY; VISCOSITY;

FINITE ELEMENT METHOD;

EID: 77954339930     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202510004490     Document Type: Article

References (41)

1. S. Adams and B. Cockburn, A mixed nite element method for elasticity in three dimensions, J. Sci. Comput. 25 (2005) 515-521.
2. K. Adolfsson, M. Enelund, S. Larsson and M. Racheva, Discretization of integro-di®erential equations modeling dynamic fractional order viscoelasticity, in Large-Scale Scientic Computing, Lecture Notes in Comput. Sci., Vol.3743 (Springer, 2006), pp. 76-83.
3. M. Amara and J. M. Thomas, Equilibrium nite elements for the linear elastic problem, Numer. Math. 33 (1979) 367-383.
4. D. N. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions, Math. Comp. 77 (2008) 1229-1251.
5. D. N. Arnold, F. Brezzi and J. Douglas, Jr., PEERS: A new mixed nite element for plane elasticity, Jpn. J. Appl. Math. 1 (1984) 347-367.
6. D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006) 1-155.
7. D. N. Arnold, R. S. Falk and R. Winther, Mixed nite element methods for linear elasticity with weakly imposed symmetry, Math. Comp. 76 (2007) 1699-1723 (electronic).
8. D. N. Arnold and R. Winther, Mixed nite elements for elasticity, Numer. Math. 92 (2002) 401-419.
9. R. Bank, W. Coughran, W. Fichtner, D. Grosse, D. Rose and R. Smith, Transient simulation of silicon devices and circuits, IEEE Trans. Computer-Aided Design Integrated Circuits Syst. 4 (1985) 436-451.
10. E. Becache, A. Ezziani and P. Joly, A mixed nite element approach for viscoelastic wave propagation, Comput. Geosci. 8 (2004) 255-299.
11. D. Braess, Finite Elements, 3rd edn. (Cambridge Univ. Press, 2007).
12. K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial-Value Problems in Di®erential-Algebraic Equations, Classics in Applied Mathematics, Vol.14 (SIAM, 1996).
13. F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Fran caise Automat. Informat. Recherche Operationnelle Ser. Rouge 8 (1974) 129-151.
14. F. Brezzi, J. Douglas, Jr. and L. D. Marini, Two families of mixed nite elements for second order elliptic problems, Numer. Math. 47 (1985) 217-235.
15. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, Vol.15 (Springer-Verlag, 1991).
16. S. H. Christiansen and R. Winther, Smoothed projections in nite element exterior calculus, Math. Comp. 77 (2008) 813-829.
17. R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol.1 (Springer-Verlag, 1990).
18. F. de Veubeke, Stress function approach, in Proc. of the World Congress on Finite Element Methods in Structural Mechanics, Vol.1, Bournemouth, Dorset, England, 1975, pp. 1-51.
19. A. D. Drozdov, Mechanics of Viscoelastic Solids (Wiley, 1998).
20. L. C. Evans, Partial Di®erential Equations (Amer. Math. Soc., 2002).
21. M. Fabrizio and A. Morro, Mathematical Problems in Linear Viscoelasticity (SIAM, 1992).
22. R. Falk, Finite element methods for linear elasticity, in Mixed Finite Elements, Compatibility Conditions and Applications (Springer, 2008).
23. M. Farhloul and M. Fortin, Dual hybrid methods for the elasticity and the Stokes problems: A uni ed approach, Numer. Math. 76 (1997) 419-440.
24. FEniCS, FEniCS project. URL: urlhttp//www.fenics.org/.
25. J. D. Ferry, Viscoelastic Properties of Polymers (Wiley, 1970).
26. M. E. Gurtin and E. Sternberg, On the linear theory of viscoelasticity, Arch. Rational Mech. Anal. 11 (1962) 291-356.
27. E. Hairer and G. Wanner, Solving Ordinary Di®erential Equations, II, Springer Series in Computational Mathematics, Vol.14, 2nd edn. (Springer-Verlag, 1996).
28. A. Idesman, R. Niekamp and E. Stein, Finite elements in space and time for generalized viscoelastic Maxwell model, Comput. Mech. 27 (2001) 49-60.
29. A. R. Johnson and A. Tessler, A viscoelastic higher-order beam nite element, in The Mathematics of Finite Elements and Applications, Highlights (Wiley, 1996), pp. 333-345.
30. P. Le Tallec and M. Ravachol, Finite element approximation of viscoelastic progressively incompressibleows, Numer. Math. 56 (1989) 43-66.
31. A. Logg and G. N. Wells, DOLFIN: Automated nite element computing, submitted, 2009.
32. J.-C. Nedelec, A new family of mixed nite elements in R3, Numer. Math. 50 (1986) 57-81.
33. N. Phan-Thien, Understanding Viscoelasticity, Advanced Texts in Physics (Springer-Verlag, 2002).
34. B. Riviere, S. Shaw, M. F. Wheeler and J. R. Whiteman, Discontinuous Galerkin nite element methods for linear elasticity and quasistatic linear viscoelasticity, Numer. Math. 95 (2003) 347-376.
35. B. Riviere, S. Shaw and J. R. Whiteman, Discontinuous Galerkin nite element methods for dynamic linear solid viscoelasticity problems, Numer. Meth. Partial Di®. Eqns. 23 (2007) 1149-1166.
36. S. Shaw, Finite element and discrete time methods for continuum problems with memory and applications to viscoelasticity, Ph.D. thesis, Department of Mathematics and Statistics, Brunel University, 1993.
37. S. Shaw, M. K. Warby, J. R. Whiteman, C. Dawson and M. F. Wheeler, Numerical techniques for the treatment of quasistatic viscoelastic stress problems in linear isotropic solids, Comput. Meth. Appl. Mech. Engrg. 118 (1994) 211-237.
38. S. Shaw and J. R. Whiteman, Numerical solution of linear quasistatic hereditary viscoelasticity problems, SIAM J. Numer. Anal. 38 (2000) 80-97 (electronic).
39. S. Shaw and J. R. Whiteman, A posteriori error estimates for space-time nite element approximation of quasistatic hereditary linear viscoelasticity problems, Comput. Meth. Appl. Mech. Engrg. 193 (2004) 5551-5572.
40. R. Stenberg, A family of mixed nite elements for the elasticity problem, Numer. Math. 53 (1988) 513-538.
41. N. W. Tschoegl, The Phenomenological Theory of Linear Viscoelastic Behavior (Springer-Verlag, 1989).