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Deep learned finite elements. (English) Zbl 1506.65217

Summary: In this paper, we propose a method that employs deep learning, an artificial intelligence technique, to generate stiffness matrices of finite elements. The proposed method is used to develop 4- and 8-node 2D solid finite elements. The deep learned finite elements practically pass the patch tests and the zero energy mode tests. Through various numerical examples, the performance of the developed elements is investigated and compared with those of existing elements. Computation efficiency is also studied. It was confirmed that the deep learned finite elements can potentially outperform existing finite elements. The proposed method can be applied to generate various types of finite elements in the future.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
68T07 Artificial neural networks and deep learning
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[1] Bathe, K. J., Finite Element Procedures (2006), Prentice Hall: Prentice Hall Upper Saddle River, NJ
[2] Jin, J. M., The Finite Element Method in Electromagnetics (2015), John Wiley & Sons: John Wiley & Sons Hoboken, NJ
[3] Gresho, P. M.; Sani, R. L., Incompressible Flow and the Finite Element Method. Volume 1: Advection-Diffusion and Isothermal Laminar Flow (1998), John Wiley & Sons: John Wiley & Sons Hoboken, NJ · Zbl 0941.76002
[4] Lesaint, P.; Raviart, P. A., On a Finite Element Method for Solving the Neutron Transport Equation (1974), Academic Press: Academic Press Cambridge, MA · Zbl 0341.65076
[5] Donea, J.; Giuliani, S.; Halleux, J. P., An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions, Comput. Methods Appl. Mech. Engrg., 33, 689-723 (1982) · Zbl 0508.73063
[6] Volakis, J. L.; Chatterjee, A.; Kempel, L. C., Finite Element Method Electromagnetics: Antennas, Microwave Circuits, and Scattering Applications (1998), John Wiley & Sons: John Wiley & Sons Hoboken, NJ · Zbl 0966.78504
[7] Reddy, J. N.; Gartling, D. K., The Finite Element Method in Heat Transfer and Fluid Dynamics (2010), CRC press · Zbl 1257.80001
[8] Girault, V.; Raviart, P. A., (Finite Element Approximation of the Navier-Stokes Equations. Finite Element Approximation of the Navier-Stokes Equations, Lecture Notes in Mathematics (1979), Berlin Springer Verlag) · Zbl 0413.65081
[9] McCulloch, W. S.; Pitts, W., A logical calculus of the ideas immanent in nervous activity, Bull. Math. Biophys., 5, 115-133 (1943) · Zbl 0063.03860
[10] Rosenblatt, F., The perceptron: a probabilistic model for information storage and organization in the brain, Psychol. Rev., 65, 386 (1958)
[11] Hinton, G. E.; Osindero, S.; Teh, Y. W., A fast learning algorithm for deep belief nets, Neural Comput., 18, 1527-1554 (2006) · Zbl 1106.68094
[12] Rumelhart, D. E.; Hinton, G. E.; Williams, R. J., Learning representations by back-propagating errors, Nature, 323, 533-536 (1986) · Zbl 1369.68284
[13] Krizhevsky, A.; Sutskever, I.; Hinton, G. E., Imagenet classification with deep convolutional neural networks, (Advances in Neural Information Processing Systems (2012)), 1097-1105
[14] Silver, D.; Huang, A.; Maddison, C. J.; Guez, A.; Sifre, L.; Van Den Driessche, G.; Schrittwieser, J.; Antonoglou, I.; Panneershelvam, V.; Lanctot, M.; Dieleman, S.; Grewe, D.; Nham, J.; Kalchbrenner, N.; Sutskever, I.; Lillicrap, T.; Leach, M.; Kavukcuoglu, K.; Graepel, T.; Hassabis, D., Mastering the game of Go with deep neural networks and tree search, Nature, 529, 484 (2016)
[15] Silver, D.; Schrittwieser, J.; Simonyan, K.; Antonoglou, I.; Huang, A.; Guez, A.; Hubert, T.; Baker, L.; Lai, M.; Bolton, A.; Chen, Y.; Lillicrap, T.; Hui, F.; Sifre, L.; van den Driessche, G.; Graepel, T.; Hassabis, D., Mastering the game of Go without human knowledge, Nature, 550, 354 (2017)
[16] Sirignano, J.; Spiliopoulos, K., DGM: a deep learning algorithm for solving partial differential equations, J. Comput. Phys., 375, 1339-1364 (2018) · Zbl 1416.65394
[17] Raissi, M., Deep hidden physics models: deep learning of nonlinear partial differential equations, J. Mach. Learn. Res., 19, 932-955 (2018) · Zbl 1439.68021
[18] Raissi, M.; Perdikaris, P.; Karniadakis, G. E., Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378, 686-707 (2019) · Zbl 1415.68175
[19] J. Tompson, K. Schlachter, P. Sprechmann, K. Perlin, Accelerating Eulerian fluid simulation with convolutional networks, in: Proceedings of the 34th International Conference on Machine Learning, Vol. 70, 2017, pp. 3424-3433.
[20] X. Guo, W. Li, F. Iorio, Convolutional neural networks for steady flow approximation, in: Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2016, pp. 481-490.
[21] Hennigh, O., Lat-net: compressing lattice Boltzmann flow simulations using deep neural networks (2017), arXiv Preprint arXiv:1705.09036
[22] Ling, J.; Kurzawski, A.; Templeton, J., Reynolds averaged turbulence modelling using deep neural networks with embedded invariance, J. Fluid Mech., 807, 155-166 (2016) · Zbl 1383.76175
[23] Zhang, Z. J.; Duraisamy, K., Machine learning methods for data-driven turbulence modeling, (22nd AIAA Computational Fluid Dynamics Conference (2015)), 2460
[24] Beck, A. D.; Flad, D. G.; Munz, C.-D., Deep neural networks for data-driven turbulence models (2018), arXiv Preprint arXiv:1806.04482
[25] Takeuchi, J.; Kosugi, Y., Neural network representation of finite element method, Neural Netw., 7, 389-395 (1994)
[26] Liang, L.; Liu, M.; Martin, C.; Sun, W., A deep learning approach to estimate stress distribution: a fast and accurate surrogate of finite-element analysis, J. R. Soc. Interface, 15, Article 20170844 pp. (2018)
[27] Chamekh, A.; Salah, H. B.H.; Hambli, R., Inverse technique identification of material parameters using finite element and neural network computation, Int. J. Adv. Manuf. Technol., 44, 173 (2009)
[28] Hashash, Y. M.; Jung, S.; Ghaboussi, J., Numerical implementation of a neural network based material model in finite element analysis, Internat. J. Numer. Methods Engrg., 59, 989-1005 (2004) · Zbl 1065.74609
[29] Chen, D.; Levin, D. I.; Sueda, S.; Matusik, W., Data-driven finite elements for geometry and material design, ACM Trans. Graph., 34, 74 (2015)
[30] Oishi, A.; Yagawa, G., Computational mechanics enhanced by deep learning, Comput. Methods Appl. Mech. Engrg., 327, 327-351 (2017) · Zbl 1439.74458
[31] Zienkiewicz, O. C.; Taylor, R. L., The Finite Element Method: The Basis (2000), Butterworth-Heinemann: Butterworth-Heinemann Oxford, UK · Zbl 0991.74002
[32] Hornik, K.; Stinchcombe, M.; White, H., Multilayer feedforward networks are universal approximators, Neural Netw., 2, 359-366 (1989) · Zbl 1383.92015
[33] Le Roux, N.; Bengio, Y., Deep belief networks are compact universal approximators, Neural Comput., 22, 2192-2207 (2010) · Zbl 1195.68079
[34] Abadi, M.; Barham, P.; Chen, J.; Chen, Z.; Davis, A.; Dean, J.; Devin, M.; Ghemawat, S.; Irving, G.; Isard, M.; Kudlur, M.; Levenberg, J.; Monga, R.; Moore, S.; Murray, D. G.; Steiner, B.; Tucker, P.; Vasudevan, V.; Warden, P.; Wicke, M.; Yu, Y.; Zheng, X., TensorFlow: a system for large-scale machine learning, (12th USENIX Symposium on Operating Systems Design and Implementation (2016)), 265-283
[35] Kingma, D. P.; Ba, J., Adam: a method for stochastic optimization (2014), arXiv Preprint arXiv:1412.6980
[36] X. Glorot, Y. Bengio, Understanding the difficulty of training deep feedforward neural networks, in: Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, 2010, pp. 249-256.
[37] Roylance, D., (Transformation of Stresses and Strains. Transformation of Stresses and Strains, Lecture Notes for Mechanics of Materials (2001))
[38] Zienkiewicz, O. C.; Taylor, R. L., The Finite Element Method: Basic Formulation and Linear Problems (1989), McGraw-Hill: McGraw-Hill New York, NY
[39] Ko, Y.; Lee, P. S.; Bathe, K. J., The MITC4+ shell element and its performance, Comput. Struct., 169, 57-68 (2016)
[40] Lee, C.; Lee, P. S., The strain-smoothed MITC3+ shell element, Comput. Struct., 223, Article 106096 pp. (2019)
[41] Taylor, R. L.; Beresford, P. J.; Wilson, E. L., A non-conforming element for stress analysis, Internat. J. Numer. Methods Engrg., 10, 1211-1219 (1976) · Zbl 0338.73041
[42] Kohnke, P. C., ANSYS Theory Reference: Release 5.5 (1998), ANSYS, Inc.
[43] Bathe, K. J.; Lee, P. S., Measuring the convergence behavior of shell analysis schemes, Comput. Struct., 89, 285-301 (2011)
[44] Lee, P. S.; Bathe, K. J., The quadratic MITC plate and MITC shell elements in plate bending, Adv. Eng. Softw., 41, 712-728 (2010) · Zbl 1195.74184
[45] Ko, Y.; Lee, Y.; Lee, P. S.; Bathe, K. J., Performance of the MITC3+ and MITC4+ shell elements in widely-used benchmark problems, Comput. Struct., 193, 187-206 (2017)
[46] Cook, R. D.; Malkus, D. S.; Plesha, M. E.; Witt, R. J., Concepts and Applications of Finite Element Analysis (2007), John Wiley & Sons: John Wiley & Sons Hoboken, NJ
[47] Walt, S. V.; Colbert, S. C.; Varoquaux, G., The NumPy array: a structure for efficient numerical computation, Comput. Sci. Eng., 13, 22-30 (2011)
[48] Yoon, K.; Lee, P. S., Nonlinear performance of continuum mechanics based beam elements focusing on large twisting behaviors, Comput. Methods Appl. Mech. Engrg., 281, 106-130 (2014) · Zbl 1423.74938
[49] Lee, Y.; Lee, P. S.; Bathe, K. J., The MITC3+ shell finite element and its performance, Comput. Struct., 138, 12-23 (2014)
[50] Ko, Y.; Lee, P. S.; Bathe, K. J., A new 4-node MITC element for analysis of two-dimensional solids and its formulation in a shell element, Comput. Struct., 192, 34-49 (2017)
[51] Kim, S.; Lee, P. S., New enriched 3D solid finite elements: 8-node hexahedral, 6-node prismatic, and 5-node pyramidal elements, Comput. Struct., 216, 40-63 (2019)
[52] Jeon, H. M.; Lee, Y.; Lee, P. S.; Bathe, K. J., The MITC3+ shell element in geometric nonlinear analysis, Comput. Struct., 146, 91-104 (2015)
[53] Kim, D. N.; Bathe, K. J., A triangular six-node shell element, Comput. Struct., 87, 1451-1460 (2009)
[54] Bucalem, M. L.; Bathe, K. J., Higher-order MITC general shell elements, Internat. J. Numer. Methods Engrg., 36, 3729-3754 (1993) · Zbl 0800.73466
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