×

zbMATH — the first resource for mathematics

Computational mechanics enhanced by deep learning. (English) Zbl 1439.74458
Summary: The present paper describes a method to enhance the capability of, or to broaden the scope of computational mechanics by using deep learning, which is one of the machine learning methods and is based on the artificial neural network. The method utilizes deep learning to extract rules inherent in a computational mechanics application, which usually are implicit and sometimes too complicated to grasp from the large amount of available data A new method of numerical quadrature for the FEM stiffness matrices is developed by using the proposed method, where a kind of optimized quadrature rule superior in accuracy to the standard Gauss-Legendre quadrature is obtained on the element-by-element basis. The detailed formulation of the proposed method is given with the sample application above, and an acceleration technique for the proposed method is discussed.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74S99 Numerical and other methods in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
68T05 Learning and adaptive systems in artificial intelligence
Software:
Genocop; PMTK; darch; AdaGrad
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Yagawa, G.; Yoshioka, A.; Yoshimura, S.; Soneda, N., A parallel finite element method with a supercomputer network, Comput. Struct., 47, 3, 407-418 (1993)
[2] Garatani, K.; Nakajima, K.; Okuda, H.; Yagawa, G., Three-dimensional elasto-static analysis of 100 million degrees of freedom, Adv. Eng. Softw., 32, 511-518 (2001)
[4] Bishop, C. M., Pattern Recognition and Machine Learning (2006), Springer
[5] Murphy, K. P., Machine Learning: A Probabilistic Perspective (2012), MIT Press
[7] 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, 484-489 (2016)
[8] Heykin, S., Neural Networks: A Comprehensive Foundation (1999), Prentice Hall
[9] Goldberg, D. E., Genetic algorithms in search, (Optimization & Machine Learning (1989), Addison-Wesley)
[10] Michalewicz, Z., Genetic Algorithms + Data Structures = Evolution Programs (Third, Revised and Extended Edition) (1996), Springer
[11] Koza, J. R., Genetic Programming: On the Programming of Computers By Means of Natural Selection (1992), MIT Press
[12] Koza, J. R., Genetic Programming II: Automatic Discovery of Reusable Programs (1994), MIT Press
[13] Amirjanov, A., Investigation of a changing range genetic algorithm in noisy environments, Internat. J. Numer. Methods Engrg., 73, 26-46 (2008)
[14] Rabinovich, D.; Givoli, D.; Vigdergauz, S., XFEM-based crack detection scheme using a genetic algorithm, Internat. J. Numer. Methods Engrg., 71, 1051-1080 (2007)
[15] Smith, R. E.; Dike, B. A.; Mehra, R. K.; Ravichandran, B.; El-Fallah, A., Classifier systems in combat: two-sided learning of maneuvers for advanced fighter aircraft, Comput. Methods Appl. Mech. Engrg., 186, 421-437 (2000)
[16] Rovira, A.; Valdes, M.; Casanova, J., A new methodology to solve non-linear equation systems using genetic algorithms. Application to combined cycle gas turbine simulation, Internat. J. Numer. Methods Engrg., 63, 1424-1435 (2005)
[17] Furukawa, T.; Yagawa, G., Inelastic constitutive parameter identification using an evolutionary algorithm with continuous individuals, Internat. J. Numer. Methods Engrg., 40, 1071-1090 (1997)
[18] Shim, M. B.; Suh, M. W.; Furukawa, T.; Yagawa, G.; Yoshimura, S., Pareto-based continuous evolutionary algorithms for multiobjective optimization, Eng. Comput., 19, 1, 22-48 (2002)
[19] Ishihara, D.; Jeong, M. J.; Yoshimura, S.; Yagawa, G., Design window search using continuous evolutionary algorithm and clustering - its application to shape design of microelectrostatic actuator, Comput. Struct., 80, 2469-2481 (2002)
[20] Wang, M.; Dutta, D.; Kim, K.; Brigham, J. C., A computationally efficient approach for inverse material characterization combining Gappy POD with direct inversion, Comput. Methods Appl. Mech. Eng., 286, 373-393 (2015)
[21] Peherstorfer, B.; Willcox, K., Dynamic data-driven reduced-order models, Comput. Methods Appl. Mech. Eng., 291, 21-41 (2015)
[22] Wirtz, D.; Karajan, N.; Haasdonk, B., Surrogate modeling of multiscale models using kernel methods, Internat. J. Numer. Methods Engrg., 101, 1-28 (2014)
[23] Peherstorfer, B.; Cui, T.; Marzouk, Y.; Willcox, K., Multifidelity importance sampling, Comput. Methods Appl. Mech. Eng., 300, 490-509 (2016)
[24] Parpinelli, R. S.; Teodoro, F. R.; Lopes, H. S., A comparison of swarm intelligence algorithms for structural engineering optimization, Internat. J. Numer. Methods Engrg., 91, 666-684 (2012)
[25] Vieira, I. N.; Pires de Lima, B. S.L.; Jacob, B. P., Bio-inspired algorithms for the optimization of offshore oil production systems, Internat. J. Numer. Methods Engrg., 91, 1023-1044 (2012)
[26] Zimmermann, M.; von Hoessle, J. E., Computing solution spaces for robust design, Internat. J. Numer. Methods Engrg., 94, 290-307 (2013)
[27] Kohler, D.; Marzouk, Y. M.; Muller, J.; Wever, U., A new network approach to Bayesian inference in partial differential equations, Internat. J. Numer. Methods Engrg., 104, 313-329 (2015)
[28] Franck, I. M.; Koutsourelakis, P. S., Sparse variational Bayesian approximations for nonlinear inverse problems: Applications in nonlinear elastography, Comput. Methods Appl. Mech. Eng., 299, 215-244 (2016)
[29] Tan, L.; Awade, S. R., Response classification of simple polycrystalline microstructures, Comput. Methods Appl. Mech. Eng., 197, 1397-1409 (2008)
[30] Congedo, P. M.; Corre, C.; Martinez, J.-M., Shape optimization of an airfoil in a BZT flow with multiple-source uncertainties, Comput. Methods Appl. Mech. Eng., 200, 216-232 (2011)
[31] Sankaran, S.; Grady, L.; Taylor, C. A., Impact of geometric uncertainty on hemodynamic simulations using machine learning, Comput. Methods Appl. Mech. Eng., 297, 167-190 (2015)
[32] Kirchdoerfer, T.; Ortiz, M., Data-driven computational mechanics, Comput. Methods Appl. Mech. Eng., 304, 81-101 (2016)
[33] Yagawa, G.; Okuda, H., Neural networks in computational mechanics, Arch. Comput. Methods Eng., 3, 4, 435-512 (1996)
[34] Yagawa, G.; Matsuda, A.; Kawate, H., Neural network approach to estimate stable crack growth in welded specimens, Int. J. Press. Vessels Pip., 63, 303-313 (1995)
[35] Yoshimura, S.; Saito, Y.; Yagawa, G., Identification of two dissimilar surface cracks hidden in solid using neural networks and computational mechanics, Comput. Model. Simul. Eng., 1, 477-491 (1996)
[36] Oishi, A.; Yoshimura, S., A new local contact search method using a multi-layer neural network, Comput. Model. Eng. Sci., 21, 2, 93-103 (2007)
[37] Kim, J. H.; Kim, Y. H., A predictor-corrector method for structural nonlinear analysis, Comput. Methods Appl. Mech. Eng., 191, 959-974 (2001)
[38] Lopez, R.; Balsa-Canto, E.; Onate, E., Neural networks for variational problems in engineering, Internat. J. Numer. Methods Engrg., 75, 1341-1360 (2008)
[39] Yoshimura, S.; Matsuda, A.; Yagawa, G., New regularization by transformation for neural network based inverse analyses and its application to structure identification, Internat. J. Numer. Methods Engrg., 39, 3953-3968 (1996)
[40] Furukawa, T.; Yagawa, G., Implicit constitutive modelling for viscoplasticity using neural networks, Internat. J. Numer. Methods Engrg., 43, 195-219 (1998)
[41] Huber, N.; Tsakmakis, Ch., A neural network tool for identifying the material parameters of a finite deformation viscoplasticity model with static recovery, Comput. Methods Appl. Mech. Eng., 191, 353-384 (2001)
[42] Lefik, M.; Schrefler, B. A., Artificial neural network as an incremental non-linear constitutive model for a finite element code, Comput. Methods Appl. Mech. Eng., 192, 3265-3283 (2003)
[43] Jung, S.; Ghaboussi, J., Characterizing rate-dependent material behaviors in self-learning simulation, Comput. Methods Appl. Mech. Eng., 196, 608-619 (2006)
[44] Man, H.; Furukawa, T., Neural network constitutive modelling for non-linear characterization of anisotropic materials, Internat. J. Numer. Methods Engrg., 85, 939-957 (2011)
[45] Lefik, M.; Boso, D. P.; Schrefler, B. A., Artificial neural networks in numerical modelling of composites, Comput. Methods Appl. Mech. Eng., 198, 1785-1804 (2009)
[46] Ghaboussi, J.; Pecknold, D. A.; Zhang, M.; Haj-Ali, R., Autoprogressive training of neural network constitutive models, Internat. J. Numer. Methods Engrg., 42, 105-126 (1998)
[47] Al-Haik, M. S.; Garmestani, H.; Navon, I. M., Truncated-Newton training algorithm for neurocomputational viscoplastic model, Comput. Methods Appl. Mech. Eng., 192, 2249-2267 (2003)
[48] Hashash, Y. M.A.; Jung, S.; Ghaboussi, J., Numerical implementation of a network based material model in finite element analysis, Internat. J. Numer. Methods Engrg., 59, 989-1005 (2004)
[49] Oeser, M.; Freitag, S., Modeling of materials with fading memory using neural networks, Internat. J. Numer. Methods Engrg., 78, 843-862 (2009)
[50] Ootao, Y.; Kawamura, R.; Tanigawa, Y., Optimization of material composition of nonhomogeneous hollow sphere for thermal stress relaxation making use of neural network, Comput. Methods Appl. Mech. Eng., 180, 185-201 (1999)
[51] Gawin, D.; Lefik, M.; Schrefler, B. A., ANN approach to sorption hysteresis within a coupled hygro-thermo-mechanical FE analysis, Internat. J. Numer. Methods Engrg., 50, 299-323 (2001)
[52] Yun, G. J.; Ghaboussi, J.; Elnashai, A. S., Self-learning simulation method for inverse nonlinear modeling of cyclic behavior of connections, Comput. Methods Appl. Mech. Eng., 197, 2836-2857 (2008)
[53] Stavroulakis, G. E.; Antes, H., Neural crack identification in steady state elastodynamics, Comput. Methods Appl. Mech. Eng., 165, 129-146 (1998)
[54] Liu, S. W.; Huang, J. H.; Sung, J. C.; Lee, C. C., Detection of cracks using neural networks and computational mechanics, Comput. Methods Appl. Mech. Eng., 191, 2831-2845 (2002)
[55] Oishi, A.; Yamada, K.; Yoshimura, S.; Yagawa, G., Quantitative nondestructive evaluation with ultrasonic method using neural networks and computational mechanics, Comput. Mech., 15, 6, 521-533 (1995)
[56] Oishi, A.; Yamada, K.; Yoshimura, S.; Yagawa, G.; Nagai, S.; Matsuda, Y., Neural network-based inverse analysis for defect identification with laser ultrasonics, Res. Nondestruct. Eval., 13, 79-95 (2001)
[57] Mera, N. S.; Elliott, L.; Ingham, D. B., The use of neural network approximation models to speed up the optimization process in electrical impedance tomography, Comput. Methods Appl. Mech. Eng., 197, 103-114 (2007)
[58] Zacharias, J.; Hartmann, C.; Delgado, A., Damage detection on crates of beverages by artificial neural networks trained with finite-element data, Comput. Methods Appl. Mech. Eng., 193, 561-574 (2004)
[59] Garijo, N.; Martinez, J.; Garcia-Aznar, J. M.; Perez, M. A., Computational evaluation of different numerical tools for the prediction of proximal femur loads from bone morphology, Comput. Methods Appl. Mech. Eng., 268, 437-450 (2014)
[60] Papadrakakis, M.; Lagaros, N. D.; Tsompanakis, Y., Structural optimization using evolution strategies and neural networks, Comput. Methods Appl. Mech. Eng., 156, 309-333 (1998)
[61] Polini, C.; Giurgevich, A.; Onesti, L.; Pediroda, V., Hybridization of a multi-objective genetic algorithm, a neural network and a classical optimizer for a complex design problem in fluid dynamics, Comput. Methods Appl. Mech. Eng., 186, 403-420 (2000)
[62] Marcelin, J. L., Genetic optimization of stiffened plates and shells, Internat. J. Numer. Methods Engrg., 51, 1079-1088 (2001)
[63] Marcelin, J. L., Genetic optimization of stiffened plates without the FE mesh support, Internat. J. Numer. Methods Engrg., 54, 685-694 (2002)
[64] Papadrakakis, M.; Lagaros, N. D., Reliability-based structural optimization using neural networks and Monte Carlo simulation, Comput. Methods Appl. Mech. Eng., 191, 3491-3507 (2002)
[65] Lagaros, N. D.; Charmpis, D. C.; Papadrakakis, M., An adaptive neural network strategy for improving the computational performance of evolutionary structural optimization, Comput. Methods Appl. Mech. Eng., 194, 3374-3393 (2005)
[66] Giannakoglou, K. C.; Papadimitriou, D. I.; Kampolis, I. C., Aerodynamic shape design using evolutionary algorithms and new gradient-assisted metamodels, Comput. Methods Appl. Mech. Eng., 195, 6312-6329 (2006)
[67] Lagaros, N. D.; Garavelas, A. Th.; Papadrakakis, M., Innovative seismic design optimization with reliability constraints, Comput. Methods Appl. Mech. Eng., 198, 28-41 (2008)
[68] Majorana, C.; Odorizzi, S.; Vitaliani, R., Shortened quadrature rules for finite elements, Adv. Eng. Softw., 4, 52-57 (1982)
[69] Melenk, J. M.; Gerdes, K.; Schwab, C., Fully discrete hp-finite elements: fast quadrature, Comput. Methods Appl. Mech. Eng., 190, 4339-4364 (2001)
[70] Oishi, A.; Yoshimura, S., Finite element analyses of dynamic problems using graphic hardware, Comput. Model. Eng. Sci., 25, 2, 115-131 (2008)
[71] Cecka, C.; Lew, A. J.; Darve, E., Assembly of finite element methods on graphics processors, Internat. J. Numer. Methods Engrg., 85, 640-669 (2011)
[72] Banas, K.; Kruzel, F.; Bielanski, J., Finite element numerical integration for first order approximations on multi- and many-core architectures, Comput. Methods Appl. Mech. Eng., 305, 827-848 (2016)
[73] Mousavi, S. E.; Xiao, H.; Sukumar, N., Generalized Gaussian quadrature rules on arbitrary polygons, Internat. J. Numer. Methods Engrg., 82, 99-113 (2010)
[74] Liu, W. K.; Guo, Y.; Tang, S.; Belytschko, T., A multiple-quadrature eight-node hexahedral finite element for large deformation elastoplastic analysis, Comput. Methods Appl. Mech. Eng., 154, 69-132 (1998)
[75] Hansbo, P., A new approach to quadrature for finite elements incorporating hourglass control as a special case, Comput. Methods Appl. Mech. Eng., 158, 301-309 (1998)
[76] Bittencourt, M. L.; Vanzquez, T. G., Tensor-based Gauss-Jacobi numerical integration for high-order mass and stiffness matrices, Internat. J. Numer. Methods Engrg., 79, 599-638 (2009)
[77] Kikuchi, M., Application of the symbolic mathematics system to the finite element program, Comput. Mech., 5, 41-47 (1989)
[78] Yagawa, G.; Ye, G.-W.; Yoshimura, S., A numerical integration scheme for finite element method based on symbolic manipulation, Internat. J. Numer. Methods Engrg., 29, 1539-1549 (1990)
[79] Rajendran, S., A technique to develop mesh-distortion immune finite elements, Comput. Methods Appl. Mech. Eng., 199, 1044-1063 (2010)
[80] Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric Analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement, Comput. Methods Appl. Mech. Eng., 194, 4135-4195 (2005)
[81] Cottrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric Analysis (2009), Wiley
[82] Sevilla, R.; Fernandez-Mendez, S.; Huerta, A., NURBS-enhanced finite element method (NEFEM), Internat. J. Numer. Methods Engrg., 76, 56-83 (2008)
[83] Sevilla, R.; Fernandez-Mendez, S.; Huerta, A., 3D NURBS-enhanced finite element method (NEFEM), Internat. J. Numer. Methods Engrg., 88, 103-125 (2011)
[84] Sevilla, R.; Fernandez-Mendez, S., Numerical integration over 2D NURBS-shaped domains with applications to NURBS-enhanced FEM, Finite Elem. Anal. Des., 47, 1209-1220 (2011)
[85] Schillinger, D.; Hossain, S. J.; Hughes, T. J.R., Reduced Bezier element quadrature rules for quadratic and cubic splines in isogeometric analysis, Comput. Methods Appl. Mech. Eng., 277, 1-45 (2014)
[86] Antolin, P.; Buffa, A.; Calabro, F.; Martinelli, M.; Sangalli, G., Efficient matrix computation for tensor-product isogeometric analysis: The use of sum factorization, Comput. Methods Appl. Mech. Eng., 285, 817-828 (2015)
[87] Hughes, T. J.R.; Reali, A.; Sangalli, G., Efficient quadrature for NURBS-based isogeometric analysis, Comput. Methods Appl. Mech. Eng., 199, 301-313 (2010)
[88] Auricchio, F.; Calabro, F.; Hughes, T. J.R.; Reali, A.; Sangalli, G., A simple algorithm for obtaining nearly optimal quadrature rules for NURBS-based isogeometric analysis, Comput. Methods Appl. Mech. Eng., 249-252, 15-27 (2012)
[89] Ait-Haddou, R.; Barton, M.; Calo, V. M., Explicit Gaussian quadrature rules for C1 cubic splines with symmetrically stretched knot sequences, J. Comput. Appl. Math., 290, 543-552 (2015)
[90] Johannessen, K. A., Optimal quadrature for univariate and tensor product splines, Comput. Methods Appl. Mech. Eng., 316, 84-99 (2017)
[91] Barton, M.; Calo, V. M., Optimal quadrature rules for odd-degree spline spaces and their application to tensor-product-based isogeometric analysis, Comput. Methods Appl. Mech. Eng., 305, 217-240 (2016)
[92] Nagy, A. P.; Benson, D. J., On the numerical integration of trimmed isogeometric elements, Comput. Methods Appl. Mech. Eng., 284, 165-185 (2015)
[93] Kim, H.-J.; Seo, Y.-D.; Youn, S.-K., Isogeometric analysis for trimmed CAD surfaces, Comput. Methods Appl. Mech. Eng., 198, 2982-2995 (2009)
[94] Kim, H.-J.; Seo, Y.-D.; Youn, S.-K., Isogeometric analysis with trimming technique for problems of arbitrary complex topology, Comput. Methods Appl. Mech. Eng., 199, 2796-2812 (2010)
[95] Liu, G. R., Mesh Free Methods (2003), CRC Press
[96] Belytschko, T.; Lu, Y. Y.; Gu, L., Element-free Galerkin methods, Internat. J. Numer. Methods Engrg., 37, 229-256 (1994)
[97] Yagawa, G.; Yamada, T., Free mesh methods: a new meshless finite element method, Comput. Mech., 18, 383-386 (1996)
[98] Yagawa, G.; Furukawa, T., Recent developments of free mesh method, Internat. J. Numer. Methods Engrg., 47, 1419-1443 (2000)
[99] Funahashi, K., On the approximate realization of continuous mappings by neural networks, Neural Netw., 2, 183-192 (1989)
[100] Hornik, K.; Stinchcombe, M.; White, H., Multilayer feedforward networks are universal approximators, Neural Netw., 2, 359-366 (1989)
[101] Goodfellow, I.; Bengio, Y.; Courville, A., Deep Learning (2016), MIT Press
[102] Rumelhart, D. E.; Hinton, G. E.; Williams, R. J., Learning representations by back-propagating errors, Nature, 323, 533-536 (1986)
[103] Rumelhart, D. E.; McClelland, J. L., The PDP research group, (Parallel Distributed Processing: Explolations in the Microstructure of Cognition, Volume 1: Foundation (1986), MIT Press)
[104] Hinton, G. E.; Osindero, S.; Teh, Y., A fast learning algorithm for deep belief nets, Neural Comput., 18, 1527-1544 (2006)
[106] LeCun, Y.; Bengio, Y.; Hinton, G. E., Deep learning, Nature, 521, 436-444 (2015)
[107] Srivastava, N.; Hinton, G. E.; Krizhevsky, A.; Sutskever, I.; Salakhutdinov, R., Dropout: A simple way to prevent neural networks from overfitting, J. Mach. Learn. Res., 15, 1929-1958 (2014)
[108] Goodfellow, I.; Warde-Farley, D.; Mirza, M.; Courville, A.; Bengio, Y., Maxout network, J. Mach. Learn. Res. W&CP, 28, 3, 1319-1327 (2013)
[109] Duchi, J.; Hazan, E.; Singer, Y., Adaptive subgradient methods for online learning and stochastic optimization, J. Mach. Learn. Res., 12, 2121-2159 (2011)
[115] Gokhale, M. B.; Graham, P. S., Reconfigurable Computing: Accelerating Computation with Field-Programmable Gate Arrays (2005), Springer
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.