Abstract: We present a deep energy method (DEM) to solve functionally graded porous beams. We use the Euler-Bernoulli assumptions with varying mechanical properties across the thickness. DEM is subsequently developed, and its performance is demonstrated by comparing the analytical solution, which was adopted from our previous work. The proposed method completely eliminates the need of a discretization technique, such as the finite element method, and optimizes the potential energy of the beam to train the neural network. Once the neural network has been trained, the solution is obtained in a very short amount of time.

一种功能梯度多孔梁的深度能量方法

[1]Alshorbagy AE, Eltaher MA, Mahmoud FF, 2011. Free vibration characteristics of a functionally graded beam by finite element method. Applied Mathematical Modelling, 35(1):412-425.

[2]Altenbach H, Ochsner A, 2010. Cellular and Porous Materials in Structures and Processes. Springer Verlag, Wien, Austria.

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

[4]Chakraverty S, Pradhan KK, 2016. Vibration of Functionally Graded Beams and Plates. Academic Press, London, UK, p.33-66.

[5]Chen D, Yang J, Kitipornchai S, 2016. Free and forced vibrations of shear deformable functionally graded porous beams. International Journal of Mechanical Sciences, 108-109:14-22.

[6]Galeban MR, Mojahedin A, Taghavi Y, et al., 2016. Free vibration of functionally graded thin beams made of saturated porous materials. Steel and Composite Structures, 21(5):999-1016.

[7]Ghannadpour SAM, Mohammadi B, Fazilati J, 2013. Bending, buckling and vibration problems of nonlocal Euler beams using Ritz method. Composite Structures, 96:584-589.

[8]Goodfellow I, Bengio Y, Courville A, 2016. Deep Learning. MIT Press and Cambridge, Cambridge, USA.

[9]Khatir S, Boutchicha D, Le Thanh C, et al., 2020. Improved ANN technique combined with Jaya algorithm for crack identification in plates using XIGA and experimental analysis. Theoretical and Applied Fracture Mechanics, 107:102554.

[10]Kingma DP, Jimmy B, 2015. Adam: a Method for Stochastic Optimization. Proceedings of the 3rd International Conference on Learning Representations, p.1-41.

[11]Lagaris IE, Likas A, Fotiadis DI, 1998. Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks, 9(5):987-1000.

[12]Li SR, Zhou YH, Zheng XJ, 2002. Thermal post-buckling of a heated elastic rod with pinned-fixed ends. Journal of Thermal Stresses, 25(1):45-56.

[13]Liu ZY, Yang YT, Cai QD, 2019. Solving differential equation with constrained multilayer feedforward network. arXiv:1904.06619.

[14]Mojahedin A, Jabbari M, Rabczuk T, 2018. Thermoelastic analysis of functionally graded porous beam. Journal of Thermal Stresses, 41(8):937-950.

[15]Nabian MA, Meidani H, 2018. A deep neural network surrogate for high-dimensional random partial differential equations. arXiv:1806.02957.

[16]Nguyen DK, 2014. Large displacement behaviour of tapered cantilever Euler-Bernoulli beams made of functionally graded material. Applied Mathematics and Computation, 237:340-355.

[17]Nguyen HX, Nguyen TN, Abdel-Wahab M, et al., 2017. A refined quasi-3D isogeometric analysis for functionally graded microplates based on the modified couple stress theory. Computer Methods in Applied Mechanics and Engineering, 313:904-940.

[18]Nguyen-Le DH, Tao QB, Nguyen VH, et al., 2020. A data-driven approach based on long short-term memory and hidden Markov model for crack propagation prediction. Engineering Fracture Mechanics, 235:107085.

[19]Phung-Van P, Tran LV, Ferreira AJM, et al., 2017. Nonlinear transient isogeometric analysis of smart piezoelectric functionally graded material plates based on generalized shear deformation theory under thermo-electro-mechanical loads. Nonlinear Dynamics, 87(2):879-894.

[20]Phung-Van P, Thai CH, Nguyen-Xuan H, et al., 2019. Porosity-dependent nonlinear transient responses of functionally graded nanoplates using isogeometric analysis. Composites Part B: Engineering, 164(1):215-225.

[21]Reddy JN, 2017. Energy Principles and Variational Methods in Applied Mechanics, 3rd Edition. John Wiley and Sons, Hoboken, USA.

[22]Sankar BV, 2001. An elasticity solution for functionally graded beams. Composites Science and Technology, 64(5):689-696.

[23]Shirvany Y, Hayati M, Moradian R, 2009. Multilayer perceptron neural networks with novel unsupervised training method for numerical solution of the partial differential equations. Applied Soft Computing, 9(1):20-29.

[24]Sirignano J, Spiliopoulos K, 2018. DGM: a deep learning algorithm for solving partial differential equations. Journal of Computational Physics, 357:1339-1364.

[25]Thanh CL, Phung-Van P, Thai CH, et al., 2018. Isogeometric analysis of functionally graded carbon nanotube reinforced composite nanoplates using modified couple stress theory. Composite Structures, 184:633-649.

[26]Thanh CL, Tran LV, Bui TQ, et al., 2019. Isogeometric analysis for size-dependent nonlinear thermal stability of porous FG microplates. Composite Structures, 221:110838.

[27]Tran-Ngoc H, Khatir S, de Roeck G, et al., 2019. An efficient artificial neural network for damage detection in bridges and beam-like structures by improving training parameters using cuckoo search algorithm. Engineering Structures, 199:109637.

[28]Weinan E, Yu B, 2018. The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics, 6(1):1-12.