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Model-data-driven constitutive responses: application to a multiscale computational framework. (English) Zbl 07411518

Summary: Computational multiscale methods for analyzing and deriving constitutive responses have been used as a tool in engineering problems because of their ability to combine information at different length scales. However, their application in a nonlinear framework can be limited by high computational costs, numerical difficulties, and/or inaccuracies. In this paper, a hybrid methodology is presented which combines classical constitutive laws (model-based), a data-driven correction component, and computational multiscale approaches. A model-based material representation is locally improved with data from lower scales obtained by means of a nonlinear numerical homogenization procedure, leading to a model-data-driven approach. Therefore, macroscale simulations explicitly incorporate the true microscale response, maintaining the same level of accuracy that would be obtained with online micro-macro simulations but with a computational cost comparable to classical model-driven approaches. In the proposed approach, both model and data play a fundamental role allowing for the synergistic integration between a physics-based response and a machine learning black-box. Numerical applications are implemented in two dimensions for different tests investigating both material and structural responses in large deformations. Overall, the presented model-data-driven methodology proves to be more versatile and accurate than methods based on classical model-driven, as well as pure data-driven techniques. In particular, a lower number of training samples is required and robustness is higher than for simulations which solely rely on data.

MSC:

74-XX Mechanics of deformable solids
86-XX Geophysics

Software:

DACE; AceFEM; GitHub; AceGen
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ababou, R.; Bagtzoglou, A. C.; Wood, E. F., On the condition number of covariance matrices in kriging, estimation, and simulation of random fields, Mathematical geology, 26, 1, 99-133 (1994) · Zbl 0970.86543
[2] Arruda, E. M.; Boyce, M. C., A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials, Journal of the mechanics and physics of solids, 41, 2, 389-412 (1993) · Zbl 1355.74020
[3] Ayensa-Jiménez, J.; Doweidar, M. H.; Sanz-Herrera, J. A.; Doblaré, M., A new reliability-based data-driven approach for noisy experimental data with physical constraints, Computer methods in applied mechanics and engineering, 328, 752-774 (2018) · Zbl 1439.65217
[4] Bernardo, J.; Berger, J.; Dawid, A.; Smith, A., Some bayesian numerical analysis, Bayesian statistics, 4, 345-363 (1992)
[5] Bouhlel, M. A.; Martins, J. R., Gradient-enhanced kriging for high-dimensional problems, Engineering with computers, 35, 1, 157-173 (2019)
[6] Boyce, M. C.; Arruda, E. M., Constitutive models of rubber elasticity: A review, Rubber Chemistry and Technology, 73, 3, 504-523 (2000)
[7] Chen, L.; Qiu, H.; Gao, L.; Jiang, C.; Yang, Z., A screening-based gradient-enhanced kriging modeling method for high-dimensional problems, Applied mathematical modelling, 69, 15-31 (2019) · Zbl 1470.74069
[8] Chinesta, F.; Cueto, E.; Abisset-Chavanne, E.; Duval, J. L.; El Khaldi, F., Virtual, digital and hybrid twins: A new paradigm in data-based engineering and engineered data, Archives of Computational Methods in Engineering, 1-30 (2018)
[9] Conti, S.; Müller, S.; Ortiz, M., Data-driven finite elasticity, Archive for rational mechanics and analysis, 237, 1, 1-33 (2020) · Zbl 1437.35654
[10] Drucker, H.; Burges, C. J.; Kaufman, L.; Smola, A. J.; Vapnik, V., Support vector regression machines, Advances in neural information processing systems 9, 155-161 (1997), MIT Press
[11] Eggersmann, R.; Kirchdoerfer, T.; Reese, S.; Stainier, L.; Ortiz, M., Model-free data-driven inelasticity, Computer methods in applied mechanics and engineering, 350, 81-99 (2019) · Zbl 1441.74048
[12] Fernández, M.; Jamshidian, M.; Böhlke, T.; Kersting, K.; Weeger, O., Anisotropic hyperelastic constitutive models for finite deformations combining material theory and data-driven approaches with application to cubic lattice metamaterials, Computational mechanics, 67, 2, 653-677 (2021) · Zbl 07360523
[13] Flory, P. J., Statistical thermodynamics of random networks, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 351, 1666, 351-380 (1976)
[14] Fuhg, J. N.; Fau, A.; Nackenhorst, U., State-of-the-art and comparative review of adaptive sampling methods for kriging, Archives of Computational Methods in Engineering, 1-59 (2020)
[15] Fuhg, J. N.; Marino, M.; Bouklas, N., Local approximate gaussian process regression for data-driven constitutive laws: Development and comparison with neural networks (2021), https://arxiv.org/abs/2105.04554
[16] Geers, M. G.D.; Kouznetsova, V. G.; Matouš, K.; Yvonnet, J., Homogenization methods and multiscale modeling: Nonlinear problems, 1-34 (2017), Encyclopedia of Computational Mechanics, Second Edition · doi:10.1002/9781119176817.ecm2107
[17] González, D.; Chinesta, F.; Cueto, E., Learning corrections for hyperelastic models from data, Frontiers in Materials, 6, 14 (2019)
[18] González, D.; Chinesta, F.; Cueto, E., Thermodynamically consistent data-driven computational mechanics, Continuum Mechanics and Thermodynamics, 31, 1, 239-253 (2019)
[19] Haykin, S., Neural networks: A comprehensive foundation (1994), Prentice Hall PTR · Zbl 0828.68103
[20] Huang, D.; Fuhg, J. N.; Weißenfels, C.; Wriggers, P., A machine learning based plasticity model using proper orthogonal decomposition, Computer methods in applied mechanics and engineering, 365, 113008 (2020) · Zbl 1442.74042
[21] Ibáñez, R.; Abisset-Chavanne, E.; González, D.; Duval, J.-L.; Cueto, E.; Chinesta, F., Hybrid constitutive modeling: Data-driven learning of corrections to plasticity models, International Journal of Material Forming, 12, 4, 717-725 (2019)
[22] Karniadakis, G. E.; Kevrekidis, I. G.; Lu, L.; Perdikaris, P.; Wang, S.; Yang, L., Physics-informed machine learning, Nature Reviews Physics, inpress (2021)
[23] Karpatne, A.; Atluri, G.; Faghmous, J. H.; Steinbach, M.; Banerjee, A., Theory-guided data science: A new paradigm for scientific discovery from data, IEEE transactions on knowledge and data engineering, 29, 10, 2318-2331 (2017)
[24] Kirchdoerfer, T.; Ortiz, M., Data-driven computational mechanics, Computer methods in applied mechanics and engineering, 304, 81-101 (2016) · Zbl 1425.74503
[25] Kirchdoerfer, T.; Ortiz, M., Data driven computing with noisy material data sets, Computer methods in applied mechanics and engineering, 326, 622-641 (2017) · Zbl 1464.62282
[26] Kirchdoerfer, T.; Ortiz, M., Data-driven computing in dynamics, International journal for numerical methods in engineering, 113, 11, 1697-1710 (2018)
[27] Kleijnen, J. P., Kriging metamodeling in simulation: A review, European journal of operational research, 192, 3, 707-716 (2009) · Zbl 1157.90544
[28] Kleijnen, J. P.; Mehdad, E., Multivariate versus univariate kriging metamodels for multi-response simulation models, European journal of operational research, 236, 2, 573-582 (2014) · Zbl 1317.62048
[29] Korelc, J. (2020). AceFEM and AceGen user manuals. http://symech. fgg.uni-lj.si/, version 7.0.
[30] Korelc, J.; Wriggers, P., Automation of finite element methods (2016), Springer, Cham · Zbl 1367.74001
[31] Le, B.; Yvonnet, J.; He, Q.-C., Computational homogenization of nonlinear elastic materials using neural networks, International journal for numerical methods in engineering, 104, 12, 1061-1084 (2015) · Zbl 1352.74266
[32] Lee, T.; Bilionis, I.; Tepole, A. B., Propagation of uncertainty in the mechanical and biological response of growing tissues using multi-fidelity gaussian process regression, Computer methods in applied mechanics and engineering, 359, 112724 (2020) · Zbl 1441.74128
[33] Liu, Z.; Wu, C., Exploring the 3D architectures of deep material network in data-driven multiscale mechanics, Journal of the mechanics and physics of solids, 127, 20-46 (2019) · Zbl 1477.74006
[34] Liu, Z.; Wu, C.; Koishi, M., A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials, Computer methods in applied mechanics and engineering, 345, 1138-1168 (2019) · Zbl 1440.74340
[35] Logarzo, H. J.; Capuano, G.; Rimoli, J. J., Smart constitutive laws: Inelastic homogenization through machine learning, Computer methods in applied mechanics and engineering, 373, 113482 (2021) · Zbl 1506.74336
[36] Lophaven, S. N.; Nielsen, H. B.; Søndergaard, J.; Dace, A., A matlab kriging toolbox, Article IMM-TR-2002-12 (2002), Technical University of Denmark Report
[37] Lu, X.; Giovanis, D. G.; Yvonnet, J.; Papadopoulos, V.; Detrez, F.; Bai, J., A data-driven computational homogenization method based on neural networks for the nonlinear anisotropic electrical response of graphene/polymer nanocomposites, Computational mechanics, 64, 2, 307-321 (2019) · Zbl 1464.74404
[38] Lupera, G.; Shokry, A.; Medina-González, S.; Vyhmeister, E.; Espuña, A., Ordinary kriging: A machine learning tool applied to mixed-integer multiparametric approach, Computer Aided Chemical Engineering, 43, 531-536 (2018), Elsevier · doi:10.1016/B978-0-444-64235-6.50094-2
[39] Marino, M., Constitutive modeling of soft tissues, (Narayan, R., Encyclopedia of biomedical engineering (2019), Elsevier: Elsevier Oxford), 81-110
[40] Matérn, B., Spatial variation, Lecture Notes in Statistics, 36 (1986) · Zbl 0608.62122
[41] Nachar, S.; Boucard, P.-A.; Néron, D.; Bordeu, F., Coupling multi-fidelity kriging and model-order reduction for the construction of virtual charts, Computational mechanics, 64, 6, 1685-1697 (2019) · Zbl 1468.74088
[42] Nguyen, L. T.K.; Keip, M.-A., A data-driven approach to nonlinear elasticity, Computers & Structures, 194, 97-115 (2018)
[43] Ogden, R. W.; Hill, R., Large deformation isotropic elasticity: On the correlation of theory and experiment for incompressible rubberlike solids, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 326, 1567, 565-584 (1972) · Zbl 0257.73034
[44] Peng, G. C.; Alber, M.; Tepole, A. B.; Cannon, W. R.; De, S., Multiscale modeling meets machine learning: What can we learn?, Archives of Computational Methods in Engineering, 28, 1017-1037 (2021)
[45] Platzer, A.; Leygue, A.; Stainier, L.; Ortiz, M., Finite element solver for data-driven finite strain elasticity, Computer methods in applied mechanics and engineering, 379, 113756 (2021) · Zbl 1506.74430
[46] PyKrige Developers (2021). Kriging toolkit for python. https://github.com/GeoStat-Framework/PyKrige.
[47] Rai, R.; Sahu, C. K., Driven by data or derived through physics? a review of hybrid physics guided machine learning techniques with cyber-physical system (cps) focus, IEEE Access, 8, 71050-71073 (2020)
[48] Ramakrishnan, R.; Dral, P. O.; Rupp, M.; von Lilienfeld, O. A., Big data meets quantum chemistry approximations: The \(\operatorname{\Delta} \)-machine learning approach, Journal of chemical theory and computation, 11, 5, 2087-2096 (2015)
[49] Rasmussen, C. E.; Williams, C. K.I., Gaussian processes for machine learning (adaptive computation and machine learning) (2005), The MIT Press
[50] Rocha, I. B.C. M.; Kerfriden, P.; van der Meer, F. P., On-the-fly construction of surrogate constitutive models for concurrent multiscale mechanical analysis through probabilistic machine learning, Journal of Computational Physics: X, 9, 100083 (2021)
[51] Santner, T. J.; Williams, B. J.; Notz, W.; Williams, B. J., The design and analysis of computer experiments. The design and analysis of computer experiments, Springer Series in Statistics (2003), Springer: Springer New York, NY · Zbl 1041.62068 · doi:10.1007/978-1-4757-3799-8
[52] Schröder, J.; Wick, T.; Reese, S.; Wriggers, P.; Müller, R., A selection of benchmark problems in solid mechanics and applied mathematics, Archives of Computational Methods in Engineering, 28, 2, 713-751 (2021)
[53] Seryo, N.; Sato, T.; Molina, J. J.; Taniguchi, T., Learning the constitutive relation of polymeric flows with memory, Phys. Rev. Research, 2, 033107 (2020)
[54] Solak, E.; Murray-Smith, R.; Leithead, W.; Leith, D.; Rasmussen, C., Derivative observations in gaussian process models of dynamic systems, Advances in neural information processing systems, 15, 1057-1064 (2002)
[55] Šolinc, U.; Korelc, J., A simple way to improved formulation of FE \({}^2\) analysis, Computational mechanics, 56, 5, 905-915 (2015) · Zbl 1329.65279
[56] Stein, M., Large sample properties of simulations using latin hypercube sampling, Technometrics, 29, 2, 143-151 (1987) · Zbl 0627.62010
[57] Svenson, J. D.; Santner, T. J., Multiobjective optimization of expensive black-box functions via expected maximin improvement, The Ohio State University, Columbus, Ohio, 32 (2010)
[58] Yvonnet, J.; Gonzalez, D.; He, Q.-C., Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials, Computer methods in applied mechanics and engineering, 198, 33-36, 2723-2737 (2009) · Zbl 1228.74067
[59] Toal, D.; Bressloff, N.; Keane, A.; Holden, C., The development of a hybridized particle swarm for kriging hyperparameter tuning, Engineering Optimization, 43, 6, 675-699 (2011)
[60] Yang, Z.; Yabansu, Y. C.; Al-Bahrani, R.; Liao, W.-k.; Choudhary, A. N.; Kalidindi, S. R.; Agrawal, A., Deep learning approaches for mining structure-property linkages in high contrast composites from simulation datasets, Computational Materials Science, 151, 278-287 (2018)
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