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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Bongard, J., Lipson, H.: Automated reverse engineering of nonlinear dynamical systems. Proc. Nat. Acad. Sci. 104(24), 9943–9948 (2007)
Schmidt, M., Lipson, H.: Distilling free-form natural laws from experimental data. Science 324(5923), 81–85 (2009)
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)
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)
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)
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)
Peherstorfer, B., Willcox, K.: Data-driven operator inference for nonintrusive projection-based model reduction. Comput. Methods Appl. Mech. Eng. 306, 196–215 (2016)
Daniels, B.C., Nemenman, I.: Automated adaptive inference of phenomenological dynamical models. Nat. Commun. 6, 8133 EP (2015)
Kirchdoerfer, T., Ortiz, M.: Data-driven computational mechanics. Comput. Methods Appl. Mech. Eng. 304, 81–101 (2016)
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)
Oettinger, H.C.: Beyond Equilibrium Thermodynamics. Wiley, Hoboken (2005)
Grmela, M., Christian Öttinger, H.: Dynamics and thermodynamics of complex fluids. i. development of a general formalism. Phys. Rev. E 56, 6620–6632 (1997)
Öttinger, H.C.: Nonequilibrium thermodynamics: a powerful tool for scientists and engineers. DYNA 79, 122–128 (2012)
Romero, I.: Thermodynamically consistent time-stepping algorithms for non-linear thermomechanical systems. Int. J. Numer. Meth. Eng. 79(6), 706–732 (2009)
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)
Español, P.: Statistical Mechanics of Coarse-Graining, pp. 69–115. Springer, Berlin (2004)
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)
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)
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)
Sullivan, T.J.: Introduction to Uncertainty Quantification. Springer, Berlin (2015). [Texts in Applied Mathematics]
Soize, C.: The Fokker–Planck Equation for Stochastic Dynamical Systems and its Explicit Steady State Solutions, 17th edn. World Scientific, SIngapore (1994)
Owens, R.G., Phillips, T.N.: Computational Rheology. Imperial College Press, London (2002)
Walters, K., Webster, M.F.: The distinctive CFD challenges of computational rheology. Int. J. Numer. Meth. Fluids 43(5), 577–596 (2003)
Owens, R.G., Phillips, T.N.: Computational Rheology. Imperial College PRess, London (2002)
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 Fluids
Vázquez-Quesada, A., Ellero, M., Español, P.: Consistent scaling of thermal fluctuations in smoothed dissipative particle dynamics. J. Chem. Phys. 130(3), 034901 (2009)
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)
Kirchdoerfer, T., Ortiz, M.: Data driven computing with noisy material data sets. Comput. Methods Appl. Mech. Eng. 326, 622–641 (2017)
Christian Öttinger, H.: Preservation of thermodynamic structure in model reduction. Phys. Rev. E 91, 032147 (2015)
Karhunen, K.: Uber lineare methoden in der wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fennicae, ser. Al. Math. Phys., 37, (1946)
Loève, M.M.: Probability theory. The University Series in Higher Mathematics, 3rd edn. Van Nostrand, Princeton, NJ (1963)
Lorenz, E.N.: Empirical Orthogonal Functions and Statistical Weather Prediction. MIT, Departement of Meteorology, Scientific Report Number 1, Statistical Forecasting Project, (1956)
Millán, D., Arroyo, M.: Nonlinear manifold learning for model reduction in finite elastodynamics. Comput. Methods Appl. Mech. Eng. 261–262, 118–131 (2013)
Laso, M., Öttinger, H.C.: Calculation of viscoelastic flow using molecular models: the connffessit approach. J. Nonnewton. Fluid Mech. 47, 1–20 (1993)
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)
Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500), 2323–2326 (2000)
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.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by Francesco dell’Isola.
About this article
Cite this article
González, D., Chinesta, F. & Cueto, E. Thermodynamically consistent data-driven computational mechanics. Continuum Mech. Thermodyn. 31, 239–253 (2019). https://doi.org/10.1007/s00161-018-0677-z
- Data-driven computational mechanics
- Governing equations