Continuum Mechanics and Thermodynamics

, Volume 31, Issue 1, pp 239–253 | Cite as

Thermodynamically consistent data-driven computational mechanics

  • David González
  • Francisco Chinesta
  • Elías CuetoEmail author
Original Article


In the paradigm of data-intensive science, automated, unsupervised discovering of governing equations for a given physical phenomenon has attracted a lot of attention in several branches of applied sciences. In this work, we propose a method able to avoid the identification of the constitutive equations of complex systems and rather work in a purely numerical manner by employing experimental data. In sharp contrast to most existing techniques, this method does not rely on the assumption on any particular form for the model (other than some fundamental restrictions placed by classical physics such as the second law of thermodynamics, for instance) nor forces the algorithm to find among a predefined set of operators those whose predictions fit best to the available data. Instead, the method is able to identify both the Hamiltonian (conservative) and dissipative parts of the dynamics while satisfying fundamental laws such as energy conservation or positive production of entropy, for instance. The proposed method is tested against some examples of discrete as well as continuum mechanics, whose accurate results demonstrate the validity of the proposed approach.


Data-driven computational mechanics GENERIC Governing equations 


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This work has been supported by the Spanish Ministry of Economy and Competitiveness through Grants number DPI2017-85139-C2-1-R and DPI2015-72365-EXP and by the Regional Government of Aragon and the European Social Fund, research group T88.


  1. 1.
    Bongard, J., Lipson, H.: Automated reverse engineering of nonlinear dynamical systems. Proc. Nat. Acad. Sci. 104(24), 9943–9948 (2007)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Schmidt, M., Lipson, H.: Distilling free-form natural laws from experimental data. Science 324(5923), 81–85 (2009)ADSCrossRefGoogle Scholar
  3. 3.
    Battista, A., Rosa, L., dell’Erba, R., Greco, L.: Numerical investigation of a particle system compared with first and second gradient continua: deformation and fracture phenomena*. Math. Mech. Solids 22(11), 2120–2134 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Della Corte, A., Battista, A., dellIsola, F.: Referential description of the evolution of a 2d swarm of robots interacting with the closer neighbors: perspectives of continuum modeling via higher gradient continua. Int. J. Non-Linear Mech. 80, 209–220 (2016)ADSCrossRefGoogle Scholar
  5. 5.
    Della Corte, A., Battista, A., dell’Isola, F.: Modeling Deformable Bodies Using Discrete Systems with Centroid-Based Propagating Interaction: Fracture and Crack Evolution, pp. 59–88. Springer, Berlin (2017)zbMATHGoogle Scholar
  6. 6.
    Brunton, S.L., Proctor, J.L., Kutz, J.N.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Nat. Acad. Sci. USA 113(15), 3932–3937 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Peherstorfer, B., Willcox, K.: Data-driven operator inference for nonintrusive projection-based model reduction. Comput. Methods Appl. Mech. Eng. 306, 196–215 (2016)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Daniels, B.C., Nemenman, I.: Automated adaptive inference of phenomenological dynamical models. Nat. Commun. 6, 8133 EP (2015)ADSCrossRefGoogle Scholar
  9. 9.
    Kirchdoerfer, T., Ortiz, M.: Data-driven computational mechanics. Comput. Methods Appl. Mech. Eng. 304, 81–101 (2016)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Ibañez, R., Abisset-Chavanne, E., Aguado, J.V., Gonzalez, D., Cueto, E., Chinesta, F.: A manifold learning approach to data-driven computational elasticity and inelasticity. Arch. Comput. Methods Eng. 25, 1–11 (2016)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Oettinger, H.C.: Beyond Equilibrium Thermodynamics. Wiley, Hoboken (2005)CrossRefGoogle Scholar
  12. 12.
    Grmela, M., Christian Öttinger, H.: Dynamics and thermodynamics of complex fluids. i. development of a general formalism. Phys. Rev. E 56, 6620–6632 (1997)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Öttinger, H.C.: Nonequilibrium thermodynamics: a powerful tool for scientists and engineers. DYNA 79, 122–128 (2012)Google Scholar
  14. 14.
    Romero, I.: Thermodynamically consistent time-stepping algorithms for non-linear thermomechanical systems. Int. J. Numer. Meth. Eng. 79(6), 706–732 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Romero, I.: Algorithms for coupled problems that preserve symmetries and the laws of thermodynamics: Part i: monolithic integrators and their application to finite strain thermoelasticity. Comput. Methods Appl. Mech. Eng. 199(25–28), 1841–1858 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Español, P.: Statistical Mechanics of Coarse-Graining, pp. 69–115. Springer, Berlin (2004)Google Scholar
  17. 17.
    Ibañez, R., Borzacchiello, D., Aguado, J.V., Abisset-Chavanne, E., Cueto, E., Ladeveze, P., Chinesta, F.: Data-driven non-linear elasticity. Constitutive manifold construction and problem discretization. Comput. Mech. 60(5), 813–826 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lopez, E., Gonzalez, D., Aguado, J.V., Abisset-Chavanne, E., Cueto, E., Binetruy, C., Chinesta, F.: A manifold learning approach for integrated computational materials engineering. Arch. Comput. Methods Eng. 25, 1–10 (2016)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Manzoni, A., Lassila, T., Quarteroni, A., Rozza, G.: A reduced-order strategy for solving inverse bayesian shape identification problems in physiological flows. In: Hans Georg B., Xuan Phu H., Rolf R., Johannes P. Schlöder, (eds.) Modeling, Simulation and Optimization of Complex Processes—HPSC 2012: In: Proceedings of the Fifth International Conference on High Performance Scientific Computing, March 5–9, 2012, Hanoi, Vietnam, pp. 145–155. Springer International Publishing, Cham, (2014)Google Scholar
  20. 20.
    Sullivan, T.J.: Introduction to Uncertainty Quantification. Springer, Berlin (2015). [Texts in Applied Mathematics]CrossRefzbMATHGoogle Scholar
  21. 21.
    Soize, C.: The Fokker–Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions, 17th edn. World Scientific, SIngapore (1994)zbMATHGoogle Scholar
  22. 22.
    Owens, R.G., Phillips, T.N.: Computational Rheology. Imperial College Press, London (2002)CrossRefzbMATHGoogle Scholar
  23. 23.
    Walters, K., Webster, M.F.: The distinctive CFD challenges of computational rheology. Int. J. Numer. Meth. Fluids 43(5), 577–596 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Owens, R.G., Phillips, T.N.: Computational Rheology. Imperial College PRess, London (2002)CrossRefzbMATHGoogle Scholar
  25. 25.
    Pasquali, Matteo., Scriven, L.E.: Theoretical modeling of microstructured liquids: a simple thermodynamic approach. Journal of Non-Newtonian Fluid Mechanics, 120(1):101 – 135, (2004). 3rd International workshop on Nonequilibrium Thermodynamics and Complex FluidsGoogle Scholar
  26. 26.
    Vázquez-Quesada, A., Ellero, M., Español, P.: Consistent scaling of thermal fluctuations in smoothed dissipative particle dynamics. J. Chem. Phys. 130(3), 034901 (2009)ADSCrossRefGoogle Scholar
  27. 27.
    Mavrantzas, V.G., Christian Öttinger, H.: Atomistic monte carlo simulations of polymer melt elasticity: their nonequilibrium thermodynamics generic formulation in a generalized canonical ensemble. Macromolecules 35(3), 960–975 (2002)ADSCrossRefGoogle Scholar
  28. 28.
    Kirchdoerfer, T., Ortiz, M.: Data driven computing with noisy material data sets. Comput. Methods Appl. Mech. Eng. 326, 622–641 (2017)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Christian Öttinger, H.: Preservation of thermodynamic structure in model reduction. Phys. Rev. E 91, 032147 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Karhunen, K.: Uber lineare methoden in der wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fennicae, ser. Al. Math. Phys., 37, (1946)Google Scholar
  31. 31.
    Loève, M.M.: Probability theory. The University Series in Higher Mathematics, 3rd edn. Van Nostrand, Princeton, NJ (1963)zbMATHGoogle Scholar
  32. 32.
    Lorenz, E.N.: Empirical Orthogonal Functions and Statistical Weather Prediction. MIT, Departement of Meteorology, Scientific Report Number 1, Statistical Forecasting Project, (1956)Google Scholar
  33. 33.
    Millán, D., Arroyo, M.: Nonlinear manifold learning for model reduction in finite elastodynamics. Comput. Methods Appl. Mech. Eng. 261–262, 118–131 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Laso, M., Öttinger, H.C.: Calculation of viscoelastic flow using molecular models: the connffessit approach. J. Nonnewton. Fluid Mech. 47, 1–20 (1993)CrossRefzbMATHGoogle Scholar
  35. 35.
    Cueto, E., Laso, M., Chinesta, F.: Meshless stochastic simulation of micro macro kinetic theory models. Int. J. Multiscale Comput. Eng. 9(1), 1–16 (2011)CrossRefGoogle Scholar
  36. 36.
    Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500), 2323–2326 (2000)ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Aragon Institute of Engineering ResearchUniversidad de ZaragozaZaragozaSpain
  2. 2.ESI Chair and PIMM LabENSAM ParisTechParisFrance

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