Abonnement à la biblothèque: Guest
Journal of Machine Learning for Modeling and Computing

Publication de 4  numéros par an

ISSN Imprimer: 2689-3967

ISSN En ligne: 2689-3975

Indexed in

Accéder(open in a new tab)
Suite
Télécharger l'article Télécharger l'archive MARC(open in a new tab) Obtenir des autorisations(open in a new tab)

TENSOR BASIS GAUSSIAN PROCESS MODELS OF HYPERELASTIC MATERIALS

Volume 1, Numéro 1, 2020, pp. 1-17
DOI: 10.1615/JMachLearnModelComput.2020033325
Get accessDownload
Ari L. Frankel
Sandia National Laboratories, Livermore, California 94551, USA
Reese E. Jones
Sandia National Laboratories, Livermore, California, 94551, USA
Laura P. Swiler
Optimization and Uncertainty Quantification Department, Center for Computing Research, Sandia National Laboratories, P.O. Box 5800, Albuquerque, New Mexico 87123-1320, USA

RÉSUMÉ

In this work, we develop Gaussian process regression (GPR) models of isotropic hyperelastic material behavior. First, we consider the direct approach of modeling the components of the Cauchy stress tensor as a function of the components of the Finger stretch tensor in a Gaussian process. We then consider an improvement on this approach that embeds rotational invariance of the stress-stretch constitutive relation in the GPR representation. This approach requires fewer training examples and achieves higher accuracy while maintaining invariance to rotations exactly. Finally, we consider an approach that recovers the strain-energy density function and derives the stress tensor from this potential. Although the error of this model for predicting the stress tensor is higher, the strain-energy density is recovered with high accuracy from limited training data. The approaches presented here are examples of physics-informed machine learning. They go beyond purely data-driven approaches by embedding the physical system constraints directly into the Gaussian process representation of materials models.

MOTS CLÉS: Gaussian process regression, hyperelastic materials, physics-informed machine learning
RÉFÉRENCES

  1. Bachoc, F., Lagnoux, A., and Lopez-Lopera, A., Maximum Likelihood Estimation for Gaussian Processes under Inequality Constraints, Elect. J. Stat., vol. 13, no. 2, pp. 2921-2969, 2019.

  2. Boehler, J., Representations for Isotropic and Anisotropic Non-Polynomial Tensor Functions, in Applications of Tensor Functions in Solid Mechanics, pp. 31-53, Berlin: Springer, 1987.

  3. Bonet, J. and Wood, R., Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge, UK: Cambridge University Press, 1997.

  4. Brunton, S.L., Proctor, J.L., and Kutz, J.N., Discovering Governing Equations from Data by Sparse Identification of Nonlinear Dynamical Systems, Proc. Nat. Acad. Sci., vol. 113, no. 15, pp. 3932-3937, 2016.

  5. Byrd, R., Lu, P., and Nocedal, J., A Limited Memory Algorithm for Bound Constrained Optimization, SIAM J. Sci. Stat. Comput., vol. 16, pp. 1190-1208, 1995.

  6. Da Veiga, S. and Marrel, A., Gaussian ProcessModeling with Inequality Constraints, Annales de la Faculte des Sciences de Toulouse, vol. 21, pp. 529-555, 2012.

  7. Dean, J., Corrado, G., Monga, R., Chen, K., Devin, M., Mao, M., Ranzato, M., Senior, A., Tucker, P., and Yang, K., Large Scale Distributed Deep Networks, Adv. Neural Inf. Process. Syst., pp. 1223-1231, 2012.

  8. Doyle, T. and Ericksen, J., Nonlinear Elasticity, Adv. Appl. Mech., vol. 4, pp. 53-115, 1956.

  9. Frankel, A., Jones, R., Alleman, C., and Templeton, J., Predicting the Mechanical Response of Oligocrystals with Deep Learning, Comput. Mater. Sci., vol. 169, p. 109099, 2019a.

  10. Frankel, A., Tachida, K., and Jones, R., Prediction of the Evolution of the Stress Field of Polycrystals Undergoing Elastic-Plastic Deformation with a Hybrid Neural Network Nodel, 2019b. arXiv: 1910.03172.

  11. Gurtin, M., An Introduction to Continuum Mechanics, vol. 158, Cambridge, MA: Academic Press, 1982.

  12. Hastie, T., Tibshirani, R., and Friedman, J., The Elements of Statistical Learning: Data Mining, Inference and Prediction, 2nd Edition, Berlin: Springer, 2016.

  13. Hill, R., On Constitutive Inequalities for SimpleMaterials-I, J.Mech. Phys. Solids, vol. 16, no. 4, pp. 229-242, 1968.

  14. Jensen, B., Nielsen, J., and Larsen, J., Bounded Gaussian Process Regression, in 2013 IEEE Int. Workshop on Machine Learning for Signal Processing (MLSP), Southampton, UK, 2013.

  15. Jones, R., Templeton, J., Sanders, C., and Ostien, J., Machine Learning Models of Plastic Flow based on Representation Theory, Comput. Model. Eng. Sci., pp. 309-342, 2018.

  16. Karpatne, A., Atluri, G., Faghmous, J., Steinbach, M., Banerjee, A., Ganguly, A., Shekhar, S., Samatova, N., and Kumar, V., Theory-Guided Data Science: A New Paradigm for Scientific Discovery from Data, IEEE Transact. Knowledge Data Eng., vol. 29, no. 10, pp. 2318-2331, 2017.

  17. Lee, K. and Carlberg, K., Model Reduction of Dynamical Systems on Nonlinear Manifolds Using Deep Convolutional Autoencoders, J. Comput. Phys., vol. 404, p. 108973, 2020.

  18. Ling, J., Jones, R., and Templeton, J., Machine Learning Strategies for Systems with Invariance Properties, J. Comput. Phys., vol. 318, pp. 22-35, 2016.

  19. Lopez-Lopera, A., Bachoc, F., Durrande, N., and Roustant, O., Finite-Dimensional Gaussian Approximation with Linear Inequality Constraints, SIAM/ASA J. Uncertain. Quant., vol. 6, no. 3, pp. 1224-1255, 2018.

  20. Lusch, B., Kutz, J.N., and Brunton, S.L., Deep Learning for Universal Linear Embeddings of Nonlinear Dynamics, Nat. Commun., vol. 9, no. 1, p. 4950, 2018.

  21. Magiera, J., Ray, D., Hesthaven, J., and Rohde, C., Constraint-Aware Neural Networks for Riemann Problems, J. Comput. Phys., vol. 409, p. 109345, 2020.

  22. Malvern, L., Introduction to the Mechanics of a Continuous Medium, Engle Cliffs, NJ: Prentice Hall Inc., 1969.

  23. Marckmann, G. and Verron, E., Comparison of Hyperelastic Models for Rubber-Like Materials, Rubber Chem. Technol., vol. 79, no. 5, pp. 835-858, 2006.

  24. Ogden, R., Non-Linear Elastic Deformations, North Chelmsford, MA: Courier Corporation, 1997.

  25. Pan, S. and Duraisamy, K., Data-Driven Discovery of Closure Models, SIAM J. Appl. Dyn. Syst., vol. 17, no. 4, pp. 2381-2413, 2018.

  26. Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., and Duchesnay, E., Scikit-Learn: Machine Learning in Python, J. Mach. Learn. Res., vol. 12, pp. 2825-2830, 2011.

  27. Raissi, M. and Karniadakis, G., Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations, J. Comput. Phys., vol. 357, pp. 125-141, 2018.

  28. Raissi, M., Perdikaris, P., and Karniadakis, G., Machine Learning of Linear Differential Equations Using Gaussian Processes, J. Comput. Phys., vol. 348, pp. 683-693, 2017.

  29. Rasmussen, C. and Williams, C., Gaussian Processes for Machine Learning, Cambridge, MA: MIT Press, 2006.

  30. Riihimaki, J. and Vehtari, A., Gaussian Processes withMonotonicity Information, in Proc. of the Thirteenth Int. Conf. on Artificial Intelligence and Statistics, Sardinia, Italy, 2010.

  31. Solak, E., Murray-Smith, R., Leithead,W., Leith, D., and Rasmussen, C., Derivative Observations in Gaussian Process Models of Dynamic Systems, in Advances in Neural Information Processing Systems, pp. 1057-1064, 2003.

  32. Wu, A., Aoi, M.C., and Pillow, J.W., Exploiting Gradients and Hessians in Bayesian Optimization and Bayesian Quadrature, 2017. arXiv: 1704.00060.

  33. Yang, X., Tartakovsky, G., and Tartakovsky, A., Physics-Informed Kriging: A Physics-Informed Gaussian Process Regression Method for Data-Model Convergence, 2018. arXiv: 1809.03461.

  34. Zheng, Q., Theory of Representations for Tensor Functions-A Unified Invariant Approach to Constitutive Equations, Appl. Mech. Rev., vol. 47, no. 11, pp. 545-587, 1994.

CITÉ PAR

  1. Wang Jikun, Li Tianjiao, Cui Fan, Hui Chung-Yuen, Yeo Jingjie, Zehnder Alan T., Metamodeling of constitutive model using Gaussian process machine learning, Journal of the Mechanics and Physics of Solids, 154, 2021. Crossref

  2. Fuhg Jan N., Marino Michele, Bouklas Nikolaos, Local approximate Gaussian process regression for data-driven constitutive models: development and comparison with neural networks, Computer Methods in Applied Mechanics and Engineering, 388, 2022. Crossref

  3. Fuhg Jan N., Bouklas Nikolaos, On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling, Computer Methods in Applied Mechanics and Engineering, 394, 2022. Crossref

  4. Leng Yue, Tac Vahidullah, Calve Sarah, Tepole Adrian B., Predicting the mechanical properties of biopolymer gels using neural networks trained on discrete fiber network data, Computer Methods in Applied Mechanics and Engineering, 387, 2021. Crossref

  5. Frankel Ari, Hamel Craig M., Bolintineanu Dan, Long Kevin, Kramer Sharlotte, Machine learning constitutive models of elastomeric foams, Computer Methods in Applied Mechanics and Engineering, 391, 2022. Crossref

  6. Vlassis Nikolaos N., Zhao Puhan, Ma Ran, Sewell Tommy, Sun WaiChing, Molecular dynamics inferred transfer learning models for finite‐strain hyperelasticity of monoclinic crystals: Sobolev training and validations against physical constraints, International Journal for Numerical Methods in Engineering, 123, 17, 2022. Crossref

  7. Fuhg J.N., Bouklas N., Jones R.E., Learning hyperelastic anisotropy from data via a tensor basis neural network, Journal of the Mechanics and Physics of Solids, 168, 2022. Crossref

  8. Jones Reese E., Frankel Ari L., Johnson K. L. , A NEURAL ORDINARY DIFFERENTIAL EQUATION FRAMEWORK FOR MODELING INELASTIC STRESS RESPONSE VIA INTERNAL STATE VARIABLES , Journal of Machine Learning for Modeling and Computing, 3, 3, 2022. Crossref

2258 Vues d'articles 2636 Téléchargements d'articles Métrique
2258 VUES 2636 TÉLÉCHARGEMENTS 8 Crossref CITATIONS Google
Scholar CITATIONS

Articles avec un contenu similaire:

MULTISCALE MODEL CALIBRATION BY INVERSE ANALYSIS FOR NONLINEAR SIMULATION OF MASONRY STRUCTURES UNDER EARTHQUAKE LOADING International Journal for Multiscale Computational Engineering, Vol.18, 2020, issue 2
Bassam A. Izzuddin, Corrado Chisari, Lorenzo Macorini
A Multiscale Framework for Analyzing Thermo-Viscoelastic Behavior of Fiber Metal Laminates International Journal for Multiscale Computational Engineering, Vol.7, 2009, issue 4
Anastasia Muliana, Sourabh Sawant
ON THE PRIMAL AND MIXED DUAL FORMATS IN VARIATIONALLY CONSISTENT COMPUTATIONAL HOMOGENIZATION WITH EMPHASIS ON FLUX BOUNDARY CONDITIONS International Journal for Multiscale Computational Engineering, Vol.18, 2020, issue 6
Kristoffer Carlsson, Kenneth Runesson, Fredrik Larsson
MULTISCALE HOMOGENIZATION SCHEMES FOR THE CONSTRUCTION OF SECOND-ORDER GRADE ANISOTROPIC CONTINUUM MEDIA OF ARCHITECTURED MATERIALS International Journal for Multiscale Computational Engineering, Vol.15, 2017, issue 1
Jean-François Ganghoffer, F. Dos Reis, Yosra Rahali
RECONCILED TOP-DOWN AND BOTTOM-UP HIERARCHICAL MULTISCALE CALIBRATION OF BCC FE CRYSTAL PLASTICITY International Journal for Multiscale Computational Engineering, Vol.15, 2017, issue 6
Aaron E. Tallman, David L. McDowell, Laura P. Swiler, Yan Wang

Dernier numéro

DEEP LEARNING MODELING FOR SUBGRID-SCALE FLUXES IN THE LES OF SCALAR TURBULENCE AND TRANSFER LEARNING TO OTHER TRANSPORT REGIMES Ali Akhavan-Safaei, Mohsen Zayernouri DIFFUSION-MODEL-ASSISTED SUPERVISED LEARNING OF GENERATIVE MODELS FOR DENSITY ESTIMATION Yanfang Liu, Minglei Yang, Zezhong Zhang, Feng Bao, Yanzhao Cao, Guannan Zhang PHYSICS-INFORMED NEURAL NETWORKS FOR MODELING OF 3D FLOW THERMAL PROBLEMS WITH SPARSE DOMAIN DATA Saakaar Bhatnagar, Andrew Comerford, Araz Banaeizadeh

Prochains articles

GPT vs Human for Scientific Reviews: A Dual Source Review on Applications of ChatGPT in Science Chenxi Wu, Alan John Varghese, Vivek Oommen, George Em Karniadakis Physics Informed Neural Networks for Modeling of 3D Flow-Thermal Problems with Sparse Domain Data Saakaar Bhatnagar, Andrew Comerford, Araz Banaeizadeh
Portail numérique Bibliothèque numérique eBooks Revues Références et comptes rendus Collections Prix et politiques d'abonnement Begell House Contactez-nous Language English 中文 Русский Português German French Spain