A Bayesian multiscale CNN framework to predict local stress fields in structures with microscale features. (English) Zbl 07492692

Summary: Multiscale computational modelling is challenging due to the high computational cost of direct numerical simulation by finite elements. To address this issue, concurrent multiscale methods use the solution of cheaper macroscale surrogates as boundary conditions to microscale sliding windows. The microscale problems remain a numerically challenging operation both in terms of implementation and cost. In this work we propose to replace the local microscale solution by an Encoder-Decoder Convolutional Neural Network that will generate fine-scale stress corrections to coarse predictions around unresolved microscale features, without prior parametrisation of local microscale problems. We deploy a Bayesian approach providing credible intervals to evaluate the uncertainty of the predictions, which is then used to investigate the merits of a selective learning framework. We will demonstrate the capability of the approach to predict equivalent stress fields in porous structures using linearised and finite strain elasticity theories.


74-XX Mechanics of deformable solids
Full Text: DOI arXiv


[1] Barrault, M.; Maday, Y.; Nguyen, NC; Patera, AT, An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations, Comptes Rendus Mathematique, 339, 9, 667-672 (2004) · Zbl 1061.65118
[2] Bessa, M.; Bostanabad, R.; Liu, Z.; Hu, A.; Apley, DW; Brinson, C.; Chen, W.; Liu, W., A framework for data-driven analysis of materials under uncertainty: countering the curse of dimensionality, Comput Methods Appl Mech Eng, 320, 633-667 (2017) · Zbl 1439.74014
[3] Bishop, CM, Neural networks for pattern recognition (1995), Oxford: Oxford University Press Inc, Oxford · Zbl 0868.68096
[4] Blundell C, Cornebise J, Kavukcuoglu K, Wierstra D (2015) Weight uncertainty in neural networks. arXiv preprint arXiv:1505.05424
[5] Cheng X, Li X, Yang J, Tai Y (2018) SESR: single image super resolution with recursive squeeze and excitation networks. In: 2018 24th International conference on pattern recognition (ICPR), pp 147-152
[6] Constantine P, Dow E (2013) Erratum: active subspace methods in theory and practice: applications to kriging surfaces. SIAM J Sci Comput 36 · Zbl 1464.62049
[7] Gal Y, Ghahramani Z (2016) Dropout as a Bayesian approximation: representing model uncertainty in deep learning. arXiv preprint arXiv:1506.02142
[8] Gal Y, Islam R, Ghahramani Z (2017) Deep Bayesian active learning with image data. arXiv preprint arXiv:1703.02910
[9] Garbin, C.; Zhu, X.; Marques, O., Dropout vs. batch normalization: an empirical study of their impact to deep learning, Multimed Tools Appl, 79, 1-39 (2020)
[10] Ghanem RG, Spanos PD (1991) Stochastic finite elements: a spectral approach · Zbl 0722.73080
[11] Goury, O.; Amsallem, D.; Bordas, SPA; Liu, WK; Kerfriden, P., Automatised selection of load paths to construct reduced-order models in computational damage micromechanics: from dissipation-driven random selection to Bayesian optimization, Comput Mech, 58, 2, 213-234 (2016) · Zbl 1398.74053
[12] Graves A (2011) Practical variational inference for neural networks. In: Shawe-Taylor J, Zemel RS, Bartlett PL, Pereira F, Weinberger KQ (eds) Advances in neural information processing systems, vol 24. Curran Associates Inc, pp 2348-2356
[13] He K, Zhang X, Ren S, Sun J (2015) Deep residual learning for image recognition. arXiv preprint arXiv:1512.03385
[14] Hennigh O, Narasimhan S, Nabian MA, Subramaniam A, Tangsali K, Rietmann M, del Aguila Ferrandis J, Byeon W, Fang Z, Choudhry S (2020) NVIDIA \(SimNet^{TM}\): an AI-accelerated multi-physics simulation framework. arXiv preprint arXiv:2012.07938
[15] Hesthaven, J.; Zhang, S.; Zhu, X., Reduced basis multiscale finite element methods for elliptic problems, SIAM J Multiscale Model Simul, 13, 316-337 (2015) · Zbl 1317.65238
[16] Hinton GE, van Camp D (1993) Keeping neural networks simple by minimizing the description length of the weights. In: Proceedings of the 16th annual conference on learning theory (COLT)
[17] Hochreiter S, Bengio Y, Frasconi P (2001) Gradient flow in recurrent nets: the difficulty of learning long-term dependencies. In: Kolen J, Kremer S (eds) Field guide to dynamical recurrent networks. IEEE Press
[18] Hu J, Shen L, Albanie S, Sun G, Wu E (2019) Squeeze-and-excitation networks. arXiv preprint arXiv:1709.01507
[19] Ioffe S, Szegedy C (2015) Batch normalization: accelerating deep network training by reducing internal covariate shift. arXiv preprint arXiv:1502.03167
[20] Islam R (2016) Active learning for high dimensional inputs using Bayesian convolutional neural networks. PhD. dissertation, Dept. Eng., Univ. Cambridge, Cambridge, UK
[21] Jiang H, Nie Z, Yeo R, Farimani AB, Kara LB (2020) StressGAN: a generative deep learning model for 2d stress distribution prediction. arXiv preprint arXiv:2006.11376
[22] Jones, D.; Schonlau, M.; Welch, W., Efficient global optimization of expensive black-box functions, J Glob Optim, 13, 455-492 (1998) · Zbl 0917.90270
[23] Kabel M, Fliegener S, Schneider M (2016) Mixed boundary conditions for FFT-based homogenization at finite strains. Comput Mech 57 · Zbl 1359.74356
[24] Kerfriden, P.; Allix, O.; Gosselet, P., A three-scale domain decomposition method for the 3d analysis of debonding in laminates, Comput Mech, 44, 343-362 (2009) · Zbl 1166.74039
[25] Kharitonov V, Molchanov D, Vetrov D (2018) Variational dropout via empirical Bayes. arXiv preprint arXiv:1811.00596
[26] Kim J, Lee JK, Lee KM (2016) Deeply-recursive convolutional network for image super-resolution. arXiv preprint arXiv:1511.04491
[27] Kingma DP, Welling M (2014) Auto-encoding variational Bayes. arXiv preprint arXiv:1312.6114
[28] Li X, Chen S, Hu X, Yang J (2018a) Understanding the disharmony between dropout and batch normalization by variance shift. arXiv preprint arXiv:1801.05134
[29] Li X, Wu J, Lin Z, Liu H, Zha H (2018b) Recurrent squeeze-and-excitation context aggregation net for single image deraining. In: Proceedings of the European conference on computer vision (ECCV)
[30] Li H, Kafka O, Gao J, Yu C, Nie Y, Zhang L, Tajdari M, Tang S, Li G, Tang S, Cheng G, Liu W (2019) Clustering discretization methods for generation of material performance databases in machine learning and design optimization. Comput Mech 64 · Zbl 1470.74073
[31] Liang, L.; Minliang, L.; Caitlin, M.; Wei, S., A deep learning approach to estimate stress distribution: a fast and accurate surrogate of finite-element analysis, J Roy Soc Interface, 15, 138 (2018)
[32] Lim B, Son S, Kim H, Nah S, Lee KM (2017) Enhanced deep residual networks for single image super-resolution. arXiv preprint arXiv:1707.02921
[33] Liu, Z.; Bessa, M.; Liu, WK, Self-consistent clustering analysis: an efficient multi-scale scheme for inelastic heterogeneous materials, Comput Methods Appl Mech Eng, 306, 319-341 (2016) · Zbl 1436.74070
[34] Liu Z, Fleming M, Liu W (2018) Microstructural material database for self-consistent clustering analysis of elastoplastic strain softening materials. Comput Methods Appl Mech Eng 330:547-577. Funding Information: Z.L. and W.K.L. warmly thank the support from AFOSR Grant No. FA9550-14-1-0032 and National Institute of Standards and Technology and Center for Hierarchical Materials Design (CHiMaD) under Grant Nos. 70NANB13Hl94 and 70NANB14H012 . Publisher Copyright: 2017 Elsevier B.V
[35] Machiels, L.; Maday, Y.; Oliveira, IB; Patera, A.; Rovas, D., Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems, Comptes Rendus De L Academie Des Sciences Serie I-mathematique, 331, 153-158 (2000) · Zbl 0960.65063
[36] Meister F, Passerini T, Mihalef V, Tuysuzoglu A, Maier A, Mansi T (2018) Towards fast biomechanical modeling of soft tissue using neural networks. arXiv preprint arXiv:1812.06186 · Zbl 1441.74263
[37] Mendizabal, A.; Márquez-Neila, P.; Cotin, S., Simulation of hyperelastic materials in real-time using deep learning, Med Image Anal, 59 (2020)
[38] Nie Z, Jiang H, Kara LB (2019) Stress field prediction in cantilevered structures using convolutional neural networks. J Comput Inf Sci Eng 20(1)
[39] Oden, J.; Prudhomme, S.; Romkes, A.; Bauman, P., Multiscale modeling of physical phenomena: adaptive control of models, SIAM J Sci Comput, 28, 6, 2359-2389 (2006) · Zbl 1126.74006
[40] Paladim D, Almeida J, Bordas S, Kerfriden P (2016) Guaranteed error bounds in homogenisation: an optimum stochastic approach to preserve the numerical separation of scales. Int J Numer Methods Eng 110 · Zbl 1365.80011
[41] Pilkey W, Pilkey D (2008) Peterson’s stress concentration factors, 3rd edn. Peterson’s Stress Concentration Factors, pp 1-522
[42] Raghavan, P.; Ghosh, S., Concurrent multi-scale analysis of elastic composites by a multi-level computational model, Comput Methods Appl Mech Eng, 193, 6, 497-538 (2004) · Zbl 1060.74590
[43] Rocha, I.; Kerfriden, P.; van der Meer, F., On-the-fly construction of surrogate constitutive models for concurrent multiscale mechanical analysis through probabilistic machine learning, J Comput Phys X, 9 (2021)
[44] Roewer-Despres F, Khan N, Stavness I (2018) Towards finite-element simulation using deep learning. In: 15th International symposium on computer methods in biomechanics and biomedical engineering, Lisbon, Portugal
[45] Ryckelynck, D., Hyper-reduction of mechanical models involving internal variables, Int J Numer Methods Eng, 77, 75-89 (2009) · Zbl 1195.74299
[46] Saha S, Gan Z, Cheng L, Gao J, Kafka O, Xie X, Li H, Tajdari M, Kim H, Liu W (2020) Hierarchical deep learning neural network (hidenn): an artificial intelligence (AI) framework for computational science and engineering. Comput Methods Appl Mech Eng 373 · Zbl 07337747
[47] Sanchez-Palencia, E., Homogenization in mechanics, a survey of solved and open problems, Rendiconti del Seminario Matematico, 44, 1, 1-45 (1986) · Zbl 0615.73009
[48] Sanchez-Palencia É (1987) General introduction to asymptotic methods, vol 272
[49] Santurkar S, Tsipras D, Ilyas A, Madry A (2019) How does batch normalization help optimization? arXiv preprint arXiv:1805.11604
[50] Sun Y, Hanhan I, Sangid MD, Lin G (2020) Predicting mechanical properties from microstructure images in fiber-reinforced polymers using convolutional neural networks. arXiv preprint arXiv:2010.03675
[51] Sussillo D, Abbott LF (2015) Random walk initialization for training very deep feedforward networks. arXiv preprint arXiv:1412.6558
[52] Tang, S.; Li, Y.; Qiu, H.; Yang, H.; Saha, S.; Mojumder, S.; Liu, W., MAP123-EP: a mechanistic-based data-driven approach for numerical elastoplastic analysis, Comput Methods Appl Mech Eng, 364 (2020) · Zbl 1442.74044
[53] Tang, S.; Yang, H.; Qiu, H.; Fleming, M.; Liu, WK; Guo, X., MAP123-EPF: a mechanistic-based data-driven approach for numerical elastoplastic modeling at finite strain, Comput Methods Appl Mech Eng, 373 (2021) · Zbl 07337767
[54] Tsymbalov E, Panov M, Shapeev A (2018) Dropout-based active learning for regression. In: Analysis of images, social networks and texts, pp 247-258
[55] Wang Y, Oyen D, Guo WG, Mehta A, Scott CB, Panda N, Fernández-Godino MG, Srinivasan G, Yue X (2021) StressNet-Deep learning to predict stress with fracture propagation in brittle materials. npj Mater Degrad 5
[56] Xiao, M.; Breitkopf, P.; Coelho, R.; Knopf-Lenoir, C.; Sidorkiewicz, M.; Villon, P., Model reduction by CPOD and kriging, IntJStruc Multidisc Optim, 41, 555-574 (2009) · Zbl 1274.90365
[57] Yan, J.; Mu, L.; Wang, L.; Ranjan, R.; Zomaya, A., Temporal convolutional networks for the advance prediction of ENSO, Sci Rep, 10, 8055 (2020)
[58] Zagoruyko S, Komodakis N (2017) Wide residual networks. arXiv preprint arXiv:1605.07146
[59] Zohdi T, Wriggers P (2005) An introduction to computational micromechanics, vol 20 · Zbl 1085.74001
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