, Volume 53, Issue 6, pp 1333–1355 | Cite as

Space–time model order reduction for nonlinear viscoelastic systems subjected to long-term loading

  • Felix Fritzen
  • Mohammadreza Hassani
Novel Computational Approaches to Old and New Problems in Mechanics


The solution of nonlinear structural problems by means of a space–time model order reduction approach is investigated. The main target is the prediction of the long-term response while reducing both the computation time and the storage requirements considerably. A nonstandard discretization approach is used which treats the internal degrees of freedom as additional unknowns. The resulting nonlinear problem is formulated in a variational setting. The proposed reduced basis represents the behavior of the structure in a complete time interval, e.g. during one load cycle (for cyclic processes). The reduced variables are obtained by a projection of the time-local stationary conditions onto appropriate test functions defined in space–time. This leads to a low-dimensional nonlinear system of equations. Details regarding the theoretical derivation, the discretization and the numerical treatment of the nonlinearity are presented. In the numerical examples the reduced model is compared to FEM reference solutions. Different choices for the test functions are discussed and the postprocessing abilities offered by the reduced model are illustrated.


Model order reduction Space–time compression Nonlinear viscoelasticity Cyclic processes 



This work was effected by the Emmy-Noether-Group EMMA supported through Grant DFG-FR2702/6 within the Emmy-Noether program of the Deutsche Forschungsgemeinschaft (DFG). The authors gratefully acknowledge the financial and personal support of the DFG. The authors acknowledge the input of the anonymous reviewers that helped in the completion of the manuscript.


This study was funded by Deutsche Forschungsgemeinschaft (DFG) (Grant DFG-FR2702/6)

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Ammar A, Mokdad B, Chinesta F, Keunings R (2006) A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J Nonnewton Fluid Mech 139(3):153–176CrossRefzbMATHGoogle Scholar
  2. 2.
    Ammar A, Mokdad B, Chinesta F, Keunings R (2007) A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids: part II: transient simulation using space–time separated representations. J Nonnewton Fluid Mech 144(2–3):98–121CrossRefzbMATHGoogle Scholar
  3. 3.
    Argyris J, Scharpf D (1969) Finite elements in time and space. Nucl Eng Des 10(4):456–464CrossRefGoogle Scholar
  4. 4.
    Benner P, Gugercin S, Willcox K (2015) A survey of projection-based model reduction methods for parametric dynamical systems. SIAM 57:483–531MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boisse P, Bussy P, Ladeveze P (1990) A new approach in non-linear mechanics: the large time increment method. Int J Numer Methods Eng 29(3):647–663CrossRefzbMATHGoogle Scholar
  6. 6.
    Capurso M, Maier G (1970) Incremental elastoplastic analysis and quadratic optimization. Meccanica 5:107–116CrossRefzbMATHGoogle Scholar
  7. 7.
    Chinesta F, Ladevèze P, Cueto E (2011) A short review on model order reduction based on proper generalized decomposition. Arch Comput Methods Eng 18(4):395CrossRefGoogle Scholar
  8. 8.
    Christensen R (1980) A nonlinear theory of viscoelasticity for application to elastomers. J Appl Mech 47(4):762–768MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Comi C, Perego U (1995) A unified approach for variationally consistent finite elements in elastoplasticity. Comput Methods Appl Mech Eng 121(1):323–344ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fried I (1969) Finite-element analysis of time-dependent phenomena. AIAA J 7(6):1170–1173ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Fritzen F, Böhlke T (2010) Three-dimensional finite element implementation of the nonuniform transformation field analysis. Int J Numer Methods Eng 84(7):803–829MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fritzen F, Böhlke T (2013) Reduced basis homogenization of viscoelastic composites. Compos Sci Technol 76:84–91CrossRefGoogle Scholar
  13. 13.
    Fritzen F, Leuschner M (2013) Reduced basis hybrid computational homogenization based on a mixed incremental formulation. Comput Methods Appl Mech Eng 260:143–154ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fritzen F, Hodapp M, Leuschner M (2014) GPU accelerated computational homogenization based on a variational approach in a reduced basis framework. Comput Methods Appl Mech Eng 278:186–217ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Germain P, Nguyen QS, Suquet P (1983) Continuum thermodynamics. J Appl Mech 50(4b):1010–1020CrossRefzbMATHGoogle Scholar
  16. 16.
    Guennouni T (1988) Sur une méthode de calcul de structures soumises à des chargements cycliques: l’homogénéisation en temps. RAIRO Model Math Anal Numer 22(3):417–455MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Haasdonk B (2017) Reduced basis methods for parametrized PDEs—a tutorial introduction for stationary and instationary problems. In: Model reduction and approximation: theory and algorithms, vol 15. p 65Google Scholar
  18. 18.
    Hackl K (1996) Generalized standard media and variational principles in classical and finite strain elastoplasticity. J Mech Phys Solids 45(5):667–688ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Halphen B, Nguyen Q (1975) Sur les Matériaux Standard Généralisés. J Mec 1(14):39–63zbMATHGoogle Scholar
  20. 20.
    Haouala S, Doghri I (2015) Modeling and algorithms for two-scale time homogenization of viscoelastic–viscoplastic solids under large numbers of cycles. Int J Plast 70:98–125CrossRefGoogle Scholar
  21. 21.
    Holzapfe GA, Simo JC (1996) A new viscoelastic constitutive model for continuous media at finite thermomechanical changes. Int J Solids Struct 33(20–22):3019–3034CrossRefzbMATHGoogle Scholar
  22. 22.
    Karhunen K (1947) Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Annales Academiae scientiarum Fennicae. Series A. 1, Mathematica-physica. University of Helsinki, HelsinkiGoogle Scholar
  23. 23.
    Ladevèze P (1989) The large time incmethod for the analysis of structures with non-linear behavior described by internal variables. C R Acad Sci II 309(11):1095–1099MathSciNetzbMATHGoogle Scholar
  24. 24.
    Ladevèze P (1999) Nonlinear computational structural mechanics—new approaches and non-incremental methods of calculation. Springer, New YorkzbMATHGoogle Scholar
  25. 25.
    Ladevèze P, Passieux JC (2010) The LATIN multiscale computational method and the proper generalized decomposition. Comput Methods Appl Mech Eng 199(21–22):1287–1296. doi: 10.1016/j.cma.2009.06.023 (Multiscale models and mathematical aspects in solid and fluid mechanics)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lai J, Findley WN (1968) Stress relaxation of nonlinear viscoelastic material under uniaxial strain. Trans Soc Rheol 12(2):259–280CrossRefGoogle Scholar
  27. 27.
    Lakes RS (1998) Viscoelastic solids, vol 9. CRC press, Boca RatonzbMATHGoogle Scholar
  28. 28.
    Leuschner M, Fritzen F (2017) Reduced order homogenization for viscoplastic composite materials including dissipative imperfect interfaces. Mech Mater 104:121–138.
  29. 29.
    Maier G (1968) Quadratic programming and theory of elastic-perfectly plastic structures. Meccanica 3:265–273MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Michel J, Suquet P (2003) Nonuniform transformation field analysis. Int J Solids Struct 40:6937–6955MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Michel J, Suquet P (2004) Computational analysis of nonlinear composite structures using the nonuniform transformation field analysis. Comput Methods Appl Mech Eng 193:5477–5502ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Miehe C (2002) Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. J Numer Methods Eng 55:1285–1322MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Miehe C (2011) A multi-field incremental variational framework for gradient-extended standard dissipative solids. J Mech Phys Solids 59(4):898–923ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Néron D, Boucard PA, Relun N (2015) Time-space PGD for the rapid solution of 3D nonlinear parametrized problems in the many-query context. Int J Numer Methods Eng 103(4):275–292MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Ortiz M, Stainier L (1999) The variational formulation of viscoplastic constitutive updates. Comput Methods Appl Mech Eng 171:419–444ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Perić D (1993) On a class of constitutive equations in viscoplasticity: formulation and computational issues. Int J Numer Methods Eng 36(8):1365–1393MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Pipkin A, Rogers T (1968) A non-linear integral representation for viscoelastic behaviour. J Mech Phys Solids 16(1):59–72ADSCrossRefzbMATHGoogle Scholar
  38. 38.
    Relun N, Néron D, Boucard PA (2013) A model reduction technique based on the PGD for elastic–viscoplastic computational analysis. Comput Mech 51(1):83–92MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Schapery RA (1969) On the characterization of nonlinear viscoelastic materials. Polym Eng Sci 9(4):295–310CrossRefGoogle Scholar
  40. 40.
    Schüler T, Manke R, Jänicke R, Radenberg M, Steeb H (2013) Multi-scale modelling of elastic/viscoelastic compounds. ZAMM J Appl Math Mech 93(2–3):126–137MathSciNetCrossRefGoogle Scholar
  41. 41.
    Simo J, Honein T (1990) Variational formulation, discrete conservation laws, and path-domain independent integrals for elasto-viscoplasticity. Trans ASME 57:488–497MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Simo J, Hughes TJR (2000) Computational inelasticity, corr. 2. print edn. Springer, BerlinzbMATHGoogle Scholar
  43. 43.
    Touati D, Cederbaum G (1997) Stress relaxation of nonlinear thermoviscoelastic materials predicted from known creep. Mech Time Depend Mater 1(3):321–330ADSCrossRefGoogle Scholar
  44. 44.
    Yu Q, Fish J (2002) Temporal homogenization of viscoelastic and viscoplastic solids subjected to locally periodic loading. Comput Mech 29(3):199–211MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Zerbe P, Schneider B, Moosbrugger E, Kaliske M (2015) Thermoplastics under long-term loading: experiments and viscoelastic-viscoplastic modeling. PAMM 15(1):375–376CrossRefGoogle Scholar
  46. 46.
    Zerbe P, Schneider B, Kaliske M (2016) Viscoelastic–viscoplastic-damage modeling of thermoplastics under long-term cyclic loading. PAMM 16(1):413–414CrossRefGoogle Scholar
  47. 47.
    Zhang R, Wen L, Naboulsi S, Eason T, Vasudevan VK, Qian D (2016) Accelerated multiscale space–time finite element simulation and application to high cycle fatigue life prediction. Comput Mech 58(2):329–349MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Emmy-Noether-Group EMMA, Institute of Applied Mechanics, Faculty 2 (Civil and Environmental Engineering)University of StuttgartStuttgartGermany

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