Structure-preserving time integration of non-isothermal finite viscoelastic continua related to variational formulations of continuum dynamics

Original Paper
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Abstract

A structure-preserving time integration scheme is a modified standard time stepping scheme, which fulfills the balance laws of a dynamical problem also in a discrete setting. Hence, such a scheme represents a space-time discretization of a continuum with all of its central properties. This paper deals with a special second-order accurate structure-preserving time integration of thermo-mechanical couplings in a moving continuum, satisfying all discrete thermodynamical balance laws in contrast to the underlying second-order accurate midpoint rule. More accurately, we consider isotropic heat conduction and the Gough–Joule effect in moving continua. The considered time evolution is described by five differential equations with respect to five independent variables, including the entropy field of the continuum. The bodies in the numerical examples are subjected to Dirichlet- or Neumann boundary conditions in the displacement as well as the temperature field. Although this structure-preserving time integrator is based on a time discrete spatially weak finite element formulation, it fulfills the same balance laws as the underlying five differential equations. Namely, in addition to the balances of entropy, linear momentum and angular momentum, also the balances of total energy and Lyapunov’s function are fulfilled in the discrete case.

Keywords

Entropy Thermo-mechanical coupling Structure-preserving time integration 

Notes

Acknowledgements

This research project is provided by DFG grant GR 3297/2-1. This support is gratefully acknowledged. We also thank Melanie Krüger for fruitful discussions about energy-entropy-consistent time integration.

References

  1. 1.
    Stuart AM, Humphries AR (1991) Dynamical systems and numerical analysis. Cambridge University Press, CambridgeMATHGoogle Scholar
  2. 2.
    Hairer E, Lubich C, Wanner G (2006) Geometric numerical integration. Springer, BerlinMATHGoogle Scholar
  3. 3.
    Gonzalez O (2000) Exact energy and momentum conserving algorithms for general models in nonlinear elasticity. Comput Methods Appl Mech Eng 190:1763–1783MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Armero F, Romero I (2001) On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part I. Comput Methods Appl Mech Eng 190:2603–2649CrossRefMATHGoogle Scholar
  5. 5.
    Marsden JE, West M (2001) Discrete mechanics and variational integrators. Acta Numer 10:357–514Google Scholar
  6. 6.
    Romero I (2010) Algorithms for coupled problems that preserve symmetries and the laws of thermodynamics. Part I. Comput Methods Appl Mech Eng 199:1841–1858CrossRefMATHGoogle Scholar
  7. 7.
    Betsch P, Steinmann P (2002) Conservation properties of a time FE method. Part III. Int J Numer Methods Eng 53:2271–2304CrossRefMATHGoogle Scholar
  8. 8.
    Groß M, Betsch P (2011) Galerkin-based energy–momentum consistent time-stepping algorithms for classical nonlinear thermo-dynamics. Math Comput Simul 82(4):718–770CrossRefMATHGoogle Scholar
  9. 9.
    Bauchau OA, Bottasso CL (1999) On the design of energy preserving and decaying schemes for flexible nonlinear multi-body systems. Comput Methods Appl Mech Eng 169:61–79MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ober-Blöbaum S, Saake N (2013) Construction and analysis of higher order Galerkin variational integrators. arXiv: 1304.1398 [math.NA]
  11. 11.
    Krüger M, Groß M, Betsch P (2016) An energy-entropy-consistent time stepping scheme for nonlinear thermo-viscoelastic continua. ZAMM 96(2):141–178MathSciNetCrossRefGoogle Scholar
  12. 12.
    Öttinger HC, Grmela M (1997) Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Phys Rev E 56(6):6633–6655MathSciNetCrossRefGoogle Scholar
  13. 13.
    Babuska I (1973) The finite element method with Lagrangian multipliers. Numer Math 20:179–192MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Holzapfel GA (2000) Nonlinear solid mechanics. Wiley, ChichesterMATHGoogle Scholar
  15. 15.
    Reese S, Govindjee S (1998) A theory of finite viscoelasticity and numerical aspects. Int J Solids Struct 35:3455–3482CrossRefMATHGoogle Scholar
  16. 16.
    Hahn W (1967) Stability of motion. Springer, BerlinCrossRefMATHGoogle Scholar
  17. 17.
    Gurtin ME (1975) Thermodynamics and stability. Arch Rational Mech Anal 59:63–96MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Simo JC (1998) Numerical analysis and simulation of plasticity. In: Ciarlet PG, Lions JL (eds) Handbook of numerical analysis, vol VI. Elsevier, AmsterdamGoogle Scholar
  19. 19.
    Giaquinta M, Hildebrandt S (2004) Calculus of variations I. Springer, BerlinCrossRefMATHGoogle Scholar
  20. 20.
    Marsden JE, Ratiu TS (1999) Introduction to mechanics and symmetry. Springer, New YorkCrossRefMATHGoogle Scholar
  21. 21.
    Reese S (2001) Thermomechanische Modellierung gummiartiger Polymerstrukturen. Habilitationsschrift, F 01/4, Institut für Baumechanik und Numerische Mechanik, Universität HannoverGoogle Scholar
  22. 22.
    Truesdell C (2012) Rational thermodynamics. Springer, BerlinMATHGoogle Scholar
  23. 23.
    Kuhl E (2004) Theory and numerics of open system thermodynamics. Lehrstuhl für Technische Mechanik, Technische Universität Kaiserslautern, HabilitationsschriftGoogle Scholar
  24. 24.
    Haupt P (2002) Continuum mechanics and theory of materials. Springer, BerlinCrossRefMATHGoogle Scholar
  25. 25.
    Wriggers P (2001) Nichtlineare finite-elemente-methoden. Springer, BerlinCrossRefMATHGoogle Scholar
  26. 26.
    Simo JC, Taylor RL, Pister KS (1985) Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comput Methods Appl Mech Eng 51:177–208MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Armero F (2008) Assumed strain finite element methods for conserving temporal integrations in non-linear solid dynamics. Int J Numer Methods Eng 74:1795–1847MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Groß M (2009) Higher-order accurate and energy–momentum consistent discretisation of dynamic finite deformation thermo-viscoelasticity. Habilitation thesis, Series of the chair for computational mechanics, vol 2, Department of Mechanical Engineering, University of Siegen, urn:nbn:de:hbz:467-3890Google Scholar
  29. 29.
    Hughes TJR (2000) The finite element method. Dover, MineolaMATHGoogle Scholar
  30. 30.
    Krüger M, Groß M, Betsch P (2011) A comparison of structure-preserving integrators for discrete thermoelastic systems. Comput Mech 47(6):701–722MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Betsch P, Steinmann P (2000) Inherently energy conserving time finite elements for classical mechanics. J Comput Phys 160:88–116MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Gonzalez O (1996) Design and analysis of conserving integrators for nonlinear Hamiltonian systems with symmetry. Ph.D. dissertation, SUDAM report no. 96-x, Department of Mechanical Engineering, Stanford University, Stanford CaliforniaGoogle Scholar
  33. 33.
    Groß M, Dietzsch J (2017) Variational-based energy–momentum schemes of higher-order for elastic fiber-reinforced continua. Comput Methods Appl Mech Eng 320:509–542MathSciNetCrossRefGoogle Scholar
  34. 34.
    Maugin GA, Kalpakides VK (2002) A Hamiltonian formulation for elasticity and thermoelasticity. J Phys A Math Gen 35:10775–10788MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Greenspan D (1973) Discrete Models. Addison-Wesley, BostonMATHGoogle Scholar
  36. 36.
    Rank E, Katz C, Werner H (1983) On the importance of the discrete maximum principle in transient analysis using finite elements. Int J Numer Methods Eng 19(12):1771–1782MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Goldstein H (1980) Classical mechanics. Addison-Wesley, BostonMATHGoogle Scholar
  38. 38.
    Schröder B, Kuhl D (2015) Small strain plasticity: classical versus multifield formulation. Arch Appl Mech 85(8):1127–1145CrossRefMATHGoogle Scholar
  39. 39.
    Maugin GA (2011) Config Forces. CRC Press, Boca RatonGoogle Scholar
  40. 40.
    Schlögl T, Leyendecker S (2016) Electrostatic–viscoelastic finite element model of dielectric actuators. Comput Methods Appl Mech Eng 299:421–439MathSciNetCrossRefGoogle Scholar
  41. 41.
    Yavari A (2008) On geometric discretization of elasticity. J Math Phys 49:022901MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Applied Mechanics and DynamicsTechnische Universität ChemnitzChemnitzGermany
  2. 2.Institute of MechanicsKarlsruhe Institute of TechnologyKarlsruheGermany

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