Bridging finite element and deep learning: High-resolution stress distribution prediction in structural components

Abstract

Finite-element analysis (FEA) for structures has been broadly used to conduct stress analysis of various civil and mechanical engineering structures. Conventional methods, such as FEA, provide high fidelity results but require the solution of large linear systems that can be computationally intensive. Instead, Deep Learning (DL) techniques can generate results significantly faster than conventional run-time analysis. This can prove extremely valuable in real-time structural assessment applications. Our proposed method uses deep neural networks in the form of convolutional neural networks (CNN) to bypass the FEA and predict high-resolution stress distributions on loaded steel plates with variable loading and boundary conditions. The CNN was designed and trained to use the geometry, boundary conditions, and load as input to predict the stress contours. The proposed technique’s performance was compared to finite-element simulations using a partial differential equation (PDE) solver. The trained DL model can predict the stress distributions with a mean absolute error of 0.9% and an absolute peak error of 0.46% for the von Mises stress distribution. This study shows the feasibility and potential of using DL techniques to bypass FEA for stress analysis applications.

Graphical abstract

Keywords

Deep Learning / finite element analysis / stress contours / structural components

References

[1] LeCun Y, Bengio Y, Hinton G E. Deep learning. Nature, 2015, 521(7553): 436–444

[2] Schmidhuber J. Deep learning in neural networks: An overview. Neural Networks, 2015, 61: 85–117

[3] Umetani N. Exploring generative 3D shapes using autoencoder networks. Pages, 2017, 24: 1–4

[4] YuYHurTJungJ. Deep learning for topology optimization design. 2018, arXiv:1801.05463

[5] Zhang W, Jiang H, Yang Z, Yamakawa S, Shimada K, Kara L B. Data-driven upsampling of point clouds. Computer-Aided Design, 2019, 112: 1–13

[6] UluEZhangRYumerM EKaraL B. A data-driven investigation and estimation of optimal topologies under variable loading configurations. In: Computational Modeling of Objects Presented in Images. Fundamentals, Methods, and Applications. Pittsburgh: Springer International Publishing, 2014

[7] Roy A G, Conjeti S, Karri S P K, Sheet D, Katouzian A, Wachinger C, Navab N. Relaynet: retinal layer and fluid segmentation of macular optical coherence tomography using fully convolutional networks. Biomedical Optics Express, 2017, 8(8): 3627–3642

[8] ArvindT. Mohan and Datta V. Gaitonde. A Deep Learning-based approach to reduced order modeling for turbulent flow control using LSTM neural networks. 2018, arXiv:1804.09269

[9] FarimaniA BGomesJPandeV S. Deep learning the physics of transport phenomena. 2017, arXiv:1709.02432

[10] KimBAzevedoV CThuereyNKimTGrossMSolenthalerB. Deep fluids: A generative network for parameterized fluid simulations. Computer Graphics Forum. 2019, 38(2): 59–70

[11] Goh G B, Hodas N O, Vishnu A. Deep learning for computational chemistry. Journal of Computational Chemistry, 2017, 38(16): 1291–1307

[12] Mardt A, Pasquali L, Wu H, Noé F. VAMPnets for deep learning of molecular kinetics. Nature Communications, 2018, 9(1): 1–11

[13] Montavon G, Rupp M, Gobre V, Vazquez-Mayagoitia A, Hansen K, Tkatchenko A, Müller K R, Anatole von Lilienfeld O. Machine learning of molecular electronic properties in chemical compound space. New Journal of Physics, 2013, 15(9): 095003

[14] Tribello G A, Ceriotti M, Parrinello M. A self-learning algorithm for biased molecular dynamics. Proceedings of the National Academy of Sciences of the United States of America, 2010, 107(41): 17509–17514

[15] Mohammadi Bayazidi A, Wang G G, Bolandi H, Alavi A H, Gandomi A H. Multigene genetic programming for estimation of elastic modulus of concrete. Mathematical Problems in Engineering, 2014, 474289

[16] Sarveghadi M, Gandomi A H, Bolandi H, Alavi A H. Development of prediction models for shear strength of SFRCB using a machine learning approach. Neural Computing & Applications, 2019, 31(7): 2085–2094

[17] Mousavi S M, Aminian P, Gandomi A H, Alavi A H, Bolandi H. A new predictive model for compressive strength of HPC using gene expression programming. Advances in Engineering Software, 2012, 45(1): 105–114

[18] Bolandi H, Banzhaf W, Lajnef N, Barri K, Alavi A H. An Intelligent model for the prediction of bond strength of FRP bars in concrete: A soft computing approach. Technologies, 2019, 7(2): 42

[19] Atalla M J, Inman D J. On model updating using neural networks. Mechanical Systems and Signal Processing, 1998, 12(1): 135–161

[20] Levin R I, Lieven N A J. Dynamic finite element model updating using neural networks. Journal of Sound and Vibration, 1998, 210(5): 593–607

[21] FanZWuYLuJLiW. Automatic pavement crack detection based on structured prediction with the convolutional neural network. 2018, arXiv:1802.02208

[22] Dung C V, Anh L D. Autonomous concrete crack detection using deep fully convolutional neural network. Automation in Construction, 2019, 99: 52–58

[23] Gulgec N S, Takáč M, Pakzad S N. Convolutional neural network approach for robust structural damage detection and localization. Journal of Computing in Civil Engineering, 2019, 33(3): 04019005

[24] Cha Y J, Choi W, Büyüköztürk O. Deep learning-based crack damage detection using convolutional neural networks. Computer-Aided Civil and Infrastructure Engineering, 2017, 32(5): 361–378

[25] Javadi A A, Tan T P, Zhang M. Neural network for constitutive modelling in finite element analysis. Computer Assisted Mechanics and Engineering Sciences, 2003, 10(4): 523–530

[26] Oishi A, Yagawa G. Computational mechanics enhanced by deep learning. Computer Methods in Applied Mechanics and Engineering, 2017, 327: 327–351

[27] Madani A, Bakhaty A, Kim J, Mubarak Y, Mofrad M R. Bridging finite element and machine learning modeling: stress prediction of arterial walls in atherosclerosis. Journal of Biomechanical Engineering, 2019, 141(8): 084502

[28] Liang L, Liu M, Martin C. A deep learning approach to estimate stress distribution: A fast and accurate surrogate of finite-element analysis. Journal of The Royal Society Interface, 2018, 15(138): 20170844

[29] Samuel A L. Some studies in machine learning using the game of checkers. IBM Journal of Research and Development, 1959, 3(3): 210–229

[30] AlpaydinE. Introduction to machine learning. Cambridge, MA: MIT Press, 2014

[31] KimY. Convolutional neural networks for sentence classification. 2014, arXiv:1408.5882

[32] KrizhevskyASutskeverIHintonG E. Imagenet classification with deep convolutional neural networks. In: Advances in Neural Information Processing Systems 25. Lake Tahoe: Curran Associates, 2012

[33] LeCun Y, Bottou L, Bengio Y, Haffner P. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 1998, 86(11): 2278–2324

[34] SimonyanKZissermanA. Very deep convolutional networks for large-scale image recognition. 2014, arXiv:1409.1556

[35] ZeilerM DFergusR. Visualizing and understanding convolutional networks. In: European Conference on Computer Vision. New York: Springer, 2014

[36] SzegedyCLiuWJiaYSermanetPReedSAnguelovDErhanDVanhouckeVRabinovichA. Going deeper with convolutions. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. New York: IEEE, 2015

[37] HeKZhangXRenSSunJ. Deep residual learning for image recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. Boston: IEEE, 2015

[38] JenkinsW M. Neural network-based approximations for structural analysis. In: Developments in Neural Networks and Evolutionary Computing for Civil and Structural Engineering. Cambridge: Civil-Comp Press Edinburgh, 1995

[39] WaszczyszynZZiemiańskiL. Neural networks in mechanics of structures and materials—New results and prospects of applications. Computers & Structures, 2001, 79(22−25): 2261−2276

[40] Goh A T C, Wong K S, Broms B B. Multivariate modelling of FEM data using neural networks. Computers & Structures, 2001, 79(22−25): 2261–2276

[41] Huber N, Tsakmakis Ch. Determination of constitutive properties from spherical indentation data using neural networks. Part II: Plasticity with nonlinear isotropic and kinematic hardening. Journal of the Mechanics and Physics of Solids, 1999, 47(7): 1589–1607

[42] Settgast C, Abendroth M, Kuna M. Constitutive modeling of plastic deformation behavior of open-cell foam structures using neural networks. Mechanics of Materials, 2019, 131: 1–10

[43] AbendrothMKunaM. Determination of deformation and failure properties of ductile materials by means of the small punch test and neural networks. Computational Materials Science, 2003, 28(3–4): 633–644

[44] Wu X, Ghaboussi J, Garrett J H Jr. Use of neural networks in detection of structural damage. Computers & Structures, 1992, 42(4): 649–659

[45] Modarres C, Astorga N, Droguett E L, Meruane V. Convolutional neural networks for automated damage recognition and damage type identification. Structural Control and Health Monitoring, 2018, 25(10): e2230

[46] Zang C, Imregun M. Structural damage detection using artificial neural networks and measured for data reduced via principal component projection. Journal of Sound and Vibration, 2001, 242(5): 813–827

[47] KhadilkarAWangJRaiR. Deep learning-based stress prediction for bottom-up sla 3d printing process. International Journal of Advanced Manufacturing Technology, 2019, 102(5−8): 2555−2569

[48] Nie Z, Jiang H, Kara L B. Stress field prediction in cantilevered structures using convolutional neural networks. Journal of Computing and Information Science in Engineering, 2020, 20(1): 011002

[49] GuoHZhuangXRabczukT. A deep collocation method for the bending analysis of Kirchhoff plate. 2021, arXiv:2102.02617

[50] Anitescu C, Atroshchenko E, Alajlan N, Rabczuk T. Artificial neural network methods for the solution of second-order boundary value problems. Computers, Materials & Continua, 2019, 59(1): 345–359

[51] Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh V M, Guo H, Hamdia K, Zhuang X, Rabczuk T. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 2020, 362: 112790

[52] Zhuang X, Guo H, Alajlan N, Zhu H, Rabczuk T. Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning. European Journal of Mechanics. A, Solids, 2021, 87: 104225

[53] GuoHZhuangXRabczukT. Stochastic analysis of heterogeneous porous material with modified neural architecture search (NAS) based physics-informed neural networks using transfer learning. 2020, arXiv:2010.12344

[54] Guo H, Zhuang X, Chen P, Alajlan N, Rabczuk T. Analysis of three-dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis. Engineering with Computers, 2022, 1–22

[55] Zahraei S M, Heidarzadeh M. Destructive effects of the 2003 bam earthquake on structures. Asian Journal of Civil Engineering, 2007, 8(3): 329–342

[56] Zahrai S M, Bolandi H. Towards lateral performance of CBF with unwanted eccentric connection: A finite element modeling approach. KSCE Journal of Civil Engineering, 2014, 18(5): 1421–1428

[57] Zahrai S M, Bolandi H. Numerical study on the impact of out-of-plane eccentricity on lateral behavior of concentrically braced frames. International Journal of Steel Structures, 2019, 19(2): 341–350

[58] Bolandi H, Zahrai S M. Influence of in-plane eccentricity in connection of bracing members to columns and beams on the performance of steel frames. Journal of Civil Engineering, 2013, 24(1): 91–102

[59] MasciJMeierUCireşanDSchmidhuberJ. Stacked convolutional auto-encoders for hierarchical feature extraction. In: International Conference on Artificial Neural Networks. Heidelberg: Springer, 2011

[60] HuJShenLSunG. Squeeze-and-excitation networks. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. Salt Lake City: IEEE, 2018

[61] SandlerMHowardAZhuMZhmoginovAChenL C. Mobilenetv2: Inverted residuals and linear bottlenecks. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. Salt Lake City: IEEE, 2018

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The Author(s). This article is published with open access at link.springer.com and journal.hep.com.cn