Abstract
The study of thermoelasticity problems holds significant importance in the field of engineering. When analyzing non-Fourier thermoelastic problems, it was found that as the thermal relaxation time increases, the finite element solution will face convergence difficulties. Therefore, it is necessary to use alternative methods to solve. This paper proposes a physics-informed neural network (PINN) based on the DeepXDE deep learning library to analyze thermoelastic problems, including classical thermoelastic problems, thermoelastic coupling problems, and generalized thermoelastic problems. The loss function is constructed based on equations, initial conditions, and boundary conditions. Unlike traditional data-driven methods, this approach does not rely on known solutions. By comparing with analytical and finite element solutions, the applicability and accuracy of the deep learning method have been validated, providing new insights for the study of thermoelastic problems.
Funding source: United Innovation Center Project
Award Identifier / Grant number: AR963
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 52072265
Acknowledgments
Bailin Zheng is grateful for the support from the United Innovation Center Project under grant number AR963. Ke Song is grateful for the support from the National Natural Science Foundation of China under grant number 52072265.
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Research ethics: Not applicable.
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Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Competing interests: The authors state no conflict of interest.
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Research funding: None declared.
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Data availability: The raw data can be obtained on request from the corresponding author.
The details of PINN will be briefly explained in this section. The temperature and displacement partial differential equations involved in this study are set as follows (taking one of the cases as an example):
The boundary conditions and initial conditions are set as follows, the displacement boundary condition is controlled to be zero on the left boundary, the radial stress on the right boundary is zero, the temperature boundary condition is zero flux on the left boundary, the constant heat flux on the right boundary, the initial displacement is zero, and the initial temperature is T00.
The network structure is set as follows, which is a deep neural network with five hidden layers, with 80 neurons per layer. The specific parameters can be seen in Table 1.
References
[1] H. Wang, Introduction to Thermoelasticity, Beijing, China, Tsinghua University Press, 1989.Search in Google Scholar
[2] O. A. Ganilova, M. P. Cartmell, and A. Kiley, “Application of a dynamic thermoelastic coupled model for an aerospace aluminium composite panel,” Compos. Struct., vol. 288, p. 115423, 2022, https://doi.org/10.1016/j.compstruct.2022.115423.Search in Google Scholar
[3] G. X. Jian, Y. Q. Wang, P. Zhang, Y. K. Li, and H. Luo, “Thermal elastohydrodynamic lubrication of modified gear system considering vibration,” J. Cent. S. Univ., vol. 27, no. 11, pp. 3350–3363, 2020. https://doi.org/10.1007/s11771-020-4551-3.Search in Google Scholar
[4] Y. Xu, B. Zheng, K. Song, K. Zhang, and R. Fang, “Non-Fourier heat conduction and thermal-stress analysis of a spherical ice particle subjected to thermal shock in PEM fuel cell at quick cold start-up,” J. Energy Eng., vol. 147, no. 5, p. 04021028, 2021. https://doi.org/10.1061/(asce)ey.1943-7897.0000773.Search in Google Scholar
[5] X. Lu, M. Chiumenti, M. Cervera, G. Zhang, and X. Lin, “Mitigation of residual stresses and microstructure homogenization in directed energy deposition processes,” Eng. Comput., vol. 38, no. 6, pp. 4771–4790, 2022. https://doi.org/10.1007/s00366-021-01563-9.Search in Google Scholar
[6] T. Mukherjee, W. Zhang, and T. DebRoy, “An improved prediction of residual stresses and distortion in additive manufacturing,” Comput. Mater. Sci., vol. 126, pp. 360–372, 2017, https://doi.org/10.1016/j.commatsci.2016.10.003.Search in Google Scholar
[7] S. Prussin, “Generation and distribution of dislocations by solute diffusion,” J. Appl. Phys., vol. 32, no. 10, pp. 1876–1881, 1961. https://doi.org/10.1063/1.1728256.Search in Google Scholar
[8] K. Zhang and B. Zheng, “Effect of irreversible electrochemical reaction on diffusion and diffusion-induced stresses in spherical composition–gradient electrodes,” Z. Naturforsch. A, vol. 75, no. 1, pp. 55–63, 2019. https://doi.org/10.1515/zna-2019-0215.Search in Google Scholar
[9] N. Bazarra, J. R. Fernández, and R. Quintanilla, “Analysis of a thermoelastic problem of type III,” Eur. Phys. J. Plus, vol. 135, pp. 1–21, 2020, https://doi.org/10.1140/epjp/s13360-020-00475-9.Search in Google Scholar
[10] N. Bazarra, J. R. Fernández, and R. Quintanilla, “A type III thermoelastic problem with mixtures,” J. Comput. Appl. Math., vol. 389, p. 113357, 2021, https://doi.org/10.1016/j.cam.2020.113357.Search in Google Scholar
[11] N. Bazarra, J. R. Fernández, A. Magaña, and R. Quintanilla, “A poro-thermoelastic problem with dissipative heat conduction,” J. Therm. Stresses, vol. 43, no. 11, pp. 1415–1436, 2020. https://doi.org/10.1080/01495739.2020.1780176.Search in Google Scholar
[12] Y. Li, T. He, and X. Tian, “A generalized thermoelastic diffusion theory of porous materials considering strain, thermal and diffusion relaxation, and its application in the ultrashort pulse laser heating,” Acta Mech., vol. 234, no. 3, pp. 1083–1103, 2023. https://doi.org/10.1007/s00707-022-03433-x.Search in Google Scholar
[13] J. P. Carter and J. R. Booker, “Finite element analysis of coupled thermoelasticity,” Comput. Struct., vol. 31, no. 1, pp. 73–80, 1989. https://doi.org/10.1016/0045-7949(89)90169-7.Search in Google Scholar
[14] P. Hosseini-Tehrani and M. Eslami, “BEM analysis of thermal and mechanical shock in a two-dimensional finite domain considering coupled thermoelasticity,” Eng. Anal. Bound. Elem., vol. 24, no. 3, pp. 249–257, 2000. https://doi.org/10.1016/s0955-7997(99)00063-6.Search in Google Scholar
[15] B. Mavrič and B. Šarler, “Application of the RBF collocation method to transient coupled thermoelasticity,” Int. J. Numer. Methods Heat Fluid Flow, vol. 27, no. 5, pp. 1064–1077, 2017. https://doi.org/10.1108/hff-03-2016-0110.Search in Google Scholar
[16] Y. Liu, G. Zhang, H. Lu, and Z. Zong, “A cell-based smoothed point interpolation method (CS-PIM) for 2D thermoelastic problems,” Int. J. Numer. Methods Heat Fluid Flow, vol. 27, no. 6, pp. 1249–1265, 2017. https://doi.org/10.1108/hff-02-2016-0042.Search in Google Scholar
[17] K. Hasanpour and D. Mirzaei, “A fast meshfree technique for the coupled thermoelasticity problem,” Acta Mech., vol. 229, pp. 2657–2673, 2018, https://doi.org/10.1007/s00707-018-2122-6.Search in Google Scholar
[18] J. Lei, X. Wei, Q. Wang, Y. Gu, and C. M. Fan, “A novel space–time generalized FDM for dynamic coupled thermoelasticity problems in heterogeneous plates,” Arch. Appl. Mech., vol. 92, pp. 287–307, 2022, https://doi.org/10.1007/s00419-021-02056-3.Search in Google Scholar
[19] S. Panghal and M. Kumar, “Optimization free neural network approach for solving ordinary and partial differential equations,” Eng. Comput., vol. 37, pp. 2989–3002, 2021, https://doi.org/10.1007/s00366-020-00985-1.Search in Google Scholar
[20] Y. Liang, R. Niu, J. Yue, and M. Lei, “A physics-informed recurrent neural network for solving time-dependent partial differential equations,” Int. J. Comput. Methods, vol. 20, p. 2341003, 2023, https://doi.org/10.1142/s0219876223410037.Search in Google Scholar
[21] J. Choi, N. Kim, and Y. Hong, “Unsupervised Legendre–Galerkin neural network for solving partial differential equations,” IEEE Access, vol. 11, pp. 23433–23446, 2023, https://doi.org/10.1109/access.2023.3244681.Search in Google Scholar
[22] M. Raissi, P. Perdikaris, and G. M. Karniadakis, “Physics informed deep learning (part I): data-driven solutions of nonlinear partial differential equations,” arXiv − CS − Mach. Learn., pp. 1–22, 2017, https://doi.org/10.48550/arXiv.1711.10561.Search in Google Scholar
[23] M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics informed deep learning (part II): data-driven discovery of nonlinear partial differential equations,” arXiv − CS − Artif. Intell., pp. 1–19, 2017, https://doi.org/10.48550/arXiv.1711.10566.Search in Google Scholar
[24] M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” J. Comput. Phys., vol. 378, pp. 686–707, 2019, https://doi.org/10.1016/j.jcp.2018.10.045.Search in Google Scholar
[25] S. Cai, Z. Wang, S. Wang, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks for heat transfer problems,” J. Heat Transfer, vol. 143, no. 6, p. 060801, 2021. https://doi.org/10.1115/1.4050542.Search in Google Scholar
[26] E. Haghighat, M. Raissi, A. Moure, H. Gomez, and R. Juanes, “A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics,” Comput. Methods Appl. Mech. Eng., vol. 379, p. 113741, 2021, https://doi.org/10.1016/j.cma.2021.113741.Search in Google Scholar
[27] Y. Xue, Y. Li, K. Zhang, and F. Yang, “A physics-inspired neural network to solve partial differential equations–application in diffusion-induced stress,” Phys. Chem. Chem. Phys., vol. 24, no. 13, pp. 7937–7949, 2022. https://doi.org/10.1039/d1cp04893g.Search in Google Scholar PubMed
[28] H. Huang, Y. Li, Y. Xue, K. Zhang, and F. Yang, “A deep learning approach for solving diffusion-induced stress in large-deformed thin film electrodes,” J. Energy Storage, vol. 63, p. 107037, 2023, https://doi.org/10.1016/j.est.2023.107037.Search in Google Scholar
[29] Y. Diao, J. Yang, Y. Zhang, D. Zhang, and Y. Du, “Solving multi-material problems in solid mechanics using physics-informed neural networks based on domain decomposition technology,” Comput. Methods Appl. Mech. Eng., vol. 413, p. 116120, 2023, https://doi.org/10.1016/j.cma.2023.116120.Search in Google Scholar
[30] S. Shi, D. Liu, and Z. Huo, “Simulation of thermoelastic coupling in silicon single crystal growth based on alternate two-stage physics-informed neural network,” Eng. Appl. Artif. Intell., vol. 123, p. 106468, 2023, https://doi.org/10.1016/j.engappai.2023.106468.Search in Google Scholar
[31] K. Eshkofti and S. M. Hosseini, “A gradient-enhanced physics-informed neural network (gPINN) scheme for the coupled non-fickian/non-fourierian diffusion-thermoelasticity analysis: a novel gPINN structure,” Eng. Appl. Artif. Intell., vol. 126, p. 106908, 2023, https://doi.org/10.1016/j.engappai.2023.106908.Search in Google Scholar
[32] S. Rezaei, A. Harandi, A. Moeineddin, B. X. Xu, and S. Reese, “A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method,” Comput. Methods Appl. Mech. Eng., vol. 401, p. 115616, 2022, https://doi.org/10.1016/j.cma.2022.115616.Search in Google Scholar
[33] E. Samaniego, et al.., “An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications,” Comput. Methods Appl. Mech. Eng., vol. 362, p. 112790, 2020, https://doi.org/10.1016/j.cma.2019.112790.Search in Google Scholar
[34] V. M. Nguyen-Thanh, C. Anitescu, N. Alajlan, T. Rabczuk, and X. Zhuang, “Parametric deep energy approach for elasticity accounting for strain gradient effects,” Comput. Methods Appl. Mech. Eng., vol. 386, p. 114096, 2021, https://doi.org/10.1016/j.cma.2021.114096.Search in Google Scholar
[35] N. Kriegeskorte and T. Golan, “Neural network models and deep learning,” Curr. Biol., vol. 29, no. 7, pp. R231–R236, 2019. https://doi.org/10.1016/j.cub.2019.02.034.Search in Google Scholar PubMed
[36] F. Aldakheel, R. Satari, and P. Wriggers, “Feed-forward neural networks for failure mechanics problems,” Appl. Sci., vol. 11, no. 14, p. 6483, 2021. https://doi.org/10.3390/app11146483.Search in Google Scholar
[37] D. Marijanović, E. K. Nyarko, and D. Filko, “Wound detection by simple feedforward neural network,” Electronics, vol. 11, no. 3, p. 329, 2022. https://doi.org/10.3390/electronics11030329.Search in Google Scholar
[38] D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” Nature, vol. 323, no. 6088, pp. 533–536, 1986. https://doi.org/10.1038/323533a0.Search in Google Scholar
[39] Y. Bengio, “Learning deep architectures for AI,” Found. Trends® Mach. Learn., vol. 2, no. 1, pp. 1–127, 2009. https://doi.org/10.1561/9781601982957.Search in Google Scholar
[40] R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput., vol. 16, no. 5, pp. 1190–1208, 1995. https://doi.org/10.1137/0916069.Search in Google Scholar
[41] L. Lu, X. Meng, Z. Mao, and G. E. Karniadakis, “DeepXDE: a deep learning library for solving differential equations,” SIAM Rev., vol. 63, no. 1, pp. 208–228, 2021. https://doi.org/10.1137/19m1274067.Search in Google Scholar
[42] H. Guo, X. Zhuang, P. Chen, N. Alajlan, and T. Rabczuk, “Stochastic deep collocation method based on neural architecture search and transfer learning for heterogeneous porous media,” Eng. Comput., vol. 38, pp. 5173–5198, 2022, https://doi.org/10.1007/s00366-021-01586-2.Search in Google Scholar
[43] M. Abadi, et al.., “{TensorFlow}: a system for {Large-Scale} machine learning,” in 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16), 2016, pp. 265–283.Search in Google Scholar
[44] Y. Cheng and M. W. Verbrugge, “Evolution of stress within a spherical insertion electrode particle under potentiostatic and galvanostatic operation,” J. Power Sources, vol. 190, pp. 453–460, 2009, https://doi.org/10.1016/j.jpowsour.2009.01.021.Search in Google Scholar
[45] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. Oxford, Clarendon Press, 1959.Search in Google Scholar
[46] N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” J. Mach. Learn. Res., vol. 15, no. 1, pp. 1929–1958, 2014.Search in Google Scholar
[47] L. Tisza, “Sur la Supraconductibilit e thermique de l’helium II liquide et la statistique de Bose-Einstein,” CR Acad. Sci., vol. 207, no. 22, p. 1035, 1938.Search in Google Scholar
[48] L. Landau, “Theory of the superfluidity of helium II,” Phys. Rev., vol. 60, no. 4, pp. 356–358, 1941. https://doi.org/10.1103/physrev.60.356.Search in Google Scholar
[49] V. Peshkov, “Second sound in helium II,” J. Phys. USSR, vol. 8, p. 381, 1944.Search in Google Scholar
[50] C. Cattaneo, “A form of heat-conduction equations which eliminates the paradox of instantaneous propagation,” Comptes Rendus, vol. 247, p. 431, 1958.Search in Google Scholar
[51] P. Vernotte, “Les paradoxes de la theorie continue de l’equation de la chaleur,” Comptes Rendus, vol. 246, p. 3154, 1958.Search in Google Scholar
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