如果要對異質複合材料進行精確模擬，需要採用並行多尺度方法，像是 FE²。在此類方法中，每個宏觀高斯積分點及每次 Newton--Raphson 迭代，均須求解微觀代表性體積元素（RVE）的邊界值問題。這種巢狀架構對於實際規模的網格而言計算代價極為高昂，因此我們嘗試引入神經網路替代模型，以取代多尺度計算中最為耗費時間的環節。

本論文針對此目標提出三項貢獻。首先，有限元素訊息神經網路（FEINN）以神經網路取代直接剛度矩陣分解，透過訓練使其滿足組裝後的平衡殘差，並以二值遮罩精確施加 Dirichlet 邊界條件，從而免除了標準物理訊息神經網路所需的懲罰權重調整。其次，以資料驅動的深度神經網路（DNN）本構替代模型，基於有限應變 RVE 模擬資料進行訓練，建立從 Green-Lagrange 應變張量至均質化第二 Piola-Kirchhoff應力與一致切線模量的映射關係，使宏觀尺度的 Newton-Raphson 求解器無需任何線上 RVE 求解即可收斂。第三，FE²NN 融合上述兩個建構模組：宏觀位移場以神經網路進行參數化，而應力由預訓練的 DNN 替代模型負責計算，以梯度式優化取代宏觀層次的 Newton--Raphson 求解與線上 RVE 求解，所形成的全可微分管線以 L-BFGS 優化器結合增量載荷步進行求解。

透過與 Ansys 參考解在幾何非線性結構基準算例上的比較驗證，確認 FE²NN 框架能夠對異質非線性材料實現精確的多尺度有限元素分析。

Accurate modeling of heterogeneous composites requires the FE² concurrent multiscale method, in which a microscale representative volume element (RVE) boundary value problem is solved at every macroscopic Gauss point and every Newton–Raphson iteration. This nested scheme is computationally prohibitive for practical mesh sizes, motivating the use of neural network surrogates to replace the most expensive components of the multiscale pipeline.

This thesis presents three contributions toward that goal. First, the Finite Element Informed Neural Network (FEINN) replaces direct stiffness factorization with a network trained to satisfy the assembled equilibrium residual, enforcing Dirichlet boundary conditions exactly through a binary mask rather than soft penalties. Second, a data-driven DNN constitutive surrogate is trained on finite-strain RVE simulation data to map the Green–Lagrange strain tensor to the homogenized second Piola--Kirchhoff stress and consistent material tangent, enabling Newton–Raphson convergence at the macroscale without any online RVE solve. Third, FE²NN unifies the two building blocks by adopting the FEINN parameterization for the macroscale displacement field and routing stress evaluation through the pre-trained DNN surrogate, replacing both repeated Newton–Raphson solves and online RVE solves with gradient-based optimization over incremental load steps.The resulting fully differentiable pipeline is optimized with L-BFGS.

Validation against Ansys reference solutions on geometrically nonlinear structural benchmarks confirms that FE2NN achieves accurate multiscale finite element analysis of heterogeneous nonlinear materials.

[1] Javier Bonet and Richard D. Wood. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, Cambridge, 2nd edition, 2008. ISBN 978-0-521-83870-2.
[2] Chun-Teh Chen and Grace X. Gu. Recurrent neural networks (RNNs) learn the constitutive law of viscoelasticity. Computational Mechanics, 67(3):1009–1019, 2021. doi: 10.1007/s00466-021-01981-y.
[3] Weinan E and Bing Yu. The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Communications in Mathematics and Statistics, 6(1):1–12, 2018. doi: 10.1007/s40304-018-0127-z.
[4] Hamidreza Eivazi, Jendrik-Alexander Tröger, Stefan Wittek, Stefan Hartmann, and Andreas Rausch. FE2 computations with deep neural networks: Algorithmic structure, data generation, and implementation. Mathematical and Computational Applications, 28(4):91, 2023. doi: 10.3390/mca28040091.
[5] Nan Feng, Guodong Zhang, and Kapil Khandelwal. Finite strain FE2 analysis with data-driven homogenization using deep neural networks. Computers & Structures, 263:106742, 2022. doi: 10.1016/j.compstruc.2022.106742.
[6] Frédéric Feyel and Jean-Louis Chaboche. FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Computer Methods in Applied Mechanics and Engineering, 183(3–4):309–330, 2000. doi: 10.1016/S0045-7825(99)00224-8.
[7] Marc G. D. Geers, Varvara G. Kouznetsova, and W. A. M. Brekelmans. Multi-scale computational homogenization: trends and challenges. Journal of Computational and Applied Mathematics, 234(7):2175–2182, 2010. doi: 10.1016/j.cam.2009.08.077.
[8] Christophe Geuzaine and Jean-François Remacle. Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering, 79(11):1309–1331, 2009. doi: 10.1002/nme.2579.
[9] Jamshid Ghaboussi, James H. Garrett, and Xiping Wu. Knowledge-based modeling of material behavior with neural networks. Journal of Engineering Mechanics, 117(1):132–153, 1991. doi: 10.1061/(ASCE)0733-9399(1991)117:1(132).
[10] Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the 13th International Conference on Artificial Intelligence and Statistics (AISTATS), volume 9 of JMLR Workshop and Conference Proceedings, pages 249–256, 2010.
[11] Ehsan Haghighat, Maziar Raissi, Alberto Moure, Hector Gomez, and Ruben Juanes. A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Computer Methods in Applied Mechanics and Engineering, 379:113741, 2021. doi: 10.1016/j.cma.2021.113741.
[12] Rodney Hill. Elastic properties of reinforced solids: some theoretical principles. Journal of the Mechanics and Physics of Solids, 11(5):357–372, 1963. doi: 10.1016/0022-5096(63)90036-X.
[13] Gerhard A. Holzapfel. Nonlinear Solid Mechanics: A Continuum Approach for Engineering. John Wiley & Sons, Chichester, 2000.
[14] George Em Karniadakis, Ioannis G. Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics-informed machine learning. Nature Reviews Physics, 3(6):422–440, 2021. doi: 10.1038/s42254-021-00314-5.
[15] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In 3rd International Conference on Learning Representations (ICLR), 2015. arXiv:1412.6980.
[16] V. Kouznetsova, W. A. M. Brekelmans, and F. P. T. Baaijens. An approach to micro-macro modeling of heterogeneous materials. Computational Mechanics, 27(1):37–48, 2001. doi: 10.1007/s004660000212.
[17] Isaac E. Lagaris, Aristidis Likas, and Dimitrios I. Fotiadis. Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks, 9(5):987–1000, 1998. doi: 10.1109/72.712178.
[18] Ba Anh Le, Julien Yvonnet, and Qi-Chang He. Computational homogenization of nonlinear elastic materials using neural networks. International Journal for Numerical Methods in Engineering, 104(12):1061–1084, 2015. doi: 10.1002/nme.4953.
[19] Dong C. Liu and Jorge Nocedal. On the limited memory BFGS method for large scale optimization. Mathematical Programming, 45(1–3):503–528, 1989. doi: 10.1007/BF01589116.
[20] Zeliang Liu, C. T. Wu, and Masataka Koishi. A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials. Computer Methods in Applied Mechanics and Engineering, 345:1138–1168, 2019. doi: 10.1016/j.cma.2018.09.020.
[21] M. A. Maia, I. B. C. M. Rocha, D. Kovačević, and F. P. van der Meer. Surrogate-based multiscale analysis of experiments on thermoplastic composites under off-axis loading, 2025. arXiv:2501.10193.
[22] Rishith E. Meethal, Anoop Kodakkal, Mohamed Khalil, Aditya Ghantasala, Birgit Obst, Kai-Uwe Bletzinger, and Roland Wüchner. Finite element method-enhanced neural network for forward and inverse problems. Advanced Modeling and Simulation in Engineering Sciences, 10(1):6, 2023. doi: 10.1186/s40323-023-00243-1.
[23] Christian Miehe, Jörg Schröder, and Jörg Schotte. Computational homogenization analysis in finite plasticity Simulation of texture development in polycrystalline materials. Computer Methods in Applied Mechanics and Engineering, 171(3–4):387–418, 1999. doi: 10.1016/S0045-7825(98)00218-7.
[24] Sebastian K. Mitusch, Simon W. Funke, and Miroslav Kuchta. Hybrid FEM-NN models: Combining artificial neural networks with the finite element method. Journal of Computational Physics, 446:110651, 2021. ISSN 0021-9991. doi: 10.1016/j.jcp.2021.110651.
[25] Maziar Raissi, Paris Perdikaris, and George E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686–707, 2019. doi: 10.1016/j.jcp.2018.10.045.
[26] Esteban Samaniego, Cosmin Anitescu, Somdatta Goswami, Vien Minh Nguyen-Thanh, Hongwei Guo, Khader Hamdia, Xiaoying Zhuang, and Timon Rabczuk. 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, 362:112790, 2020. doi: 10.1016/j.cma.2019.112790.
[27] N. Sukumar and Ankit Srivastava. Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks. Computer Methods in Applied Mechanics and Engineering, 389:114333, 2022. doi: 10.1016/j.cma.2021.114333.
[28] Vincent Beng Chye Tan, Karthikayen Raju, and Heow Pueh Lee. Direct FE2 for concurrent multilevel modelling of heterogeneous structures. Computer Methods in Applied Mechanics and Engineering, 360:112694, 2020. ISSN 0045-7825. doi: 10.1016/j.cma.2019.112694.
[29] Jendrik-Alexander Tröger, Hamidreza Eivazi, and Stefan Hartmann. Efficient integration of deep neural networks in sequential multiscale simulations. Proceedings in Applied Mathematics and Mechanics, 23(1):e202300052, 2023. doi: 10.1002/pamm.202300052.
[30] O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu. The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann, Oxford, 6th edition, 2005.