路径依赖非均质材料的物理循环神经网络：在数据驱动的代理中嵌入本构模型
Physically recurrent neural networks for path-dependent heterogeneous materials: embedding constitutive models in a data-driven surrogate
摘要：
在加速数值模拟的需求的驱动下，机器学习技术的使用在计算固体力学领域中迅速增长。它们的应用是特别有利的并发多尺度有限元分析（FE $^2$），由于非常高的计算成本往往与它和大量的类似的微观力学分析。为了解决这个问题，使用代理模型来近似微观行为并加速模拟是一种很有前途且越来越受欢迎的策略。然而，与其数据驱动性质相关的一些挑战损害了材料建模中替代模型的可靠性。在这项工作中探索的替代方案是重新引入一些基于物理的知识的经典本构建模到神经网络中，采用实际的材料模型中使用的全阶微观模型引入非线性。因此，路径依赖自然出现，因为层中的每个材料模型都跟踪自己的内部变量。数值算例采用弹性纤维和弹塑性基体材料组成的复合材料代表性体积元作为细观模型。该网络在一系列具有挑战性的场景中进行了测试，并将其性能与最先进的递归神经网络（RNN）进行了比较。新框架的一个显著成果是能够自然地预测卸载/重新加载行为，而不会在训练过程中看到它，这与流行但数据饥渴的模型（如RNN）形成鲜明对比。最后，建议的网络被施加到FE $^2$的例子，以评估其在非线性有限元分析中的应用的鲁棒性。
英文摘要：
Driven by the need to accelerate numerical simulations, the use of machine learning techniques is rapidly growing in the field of computational solid mechanics. Their application is especially advantageous in concurrent multiscale finite element analysis (FE$^2$) due to the exceedingly high computational costs often associated with it and the high number of similar micromechanical analyses involved. To tackle the issue, using surrogate models to approximate the microscopic behavior and accelerate the simulations is a promising and increasingly popular strategy. However, several challenges related to their data-driven nature compromise the reliability of surrogate models in material modeling. The alternative explored in this work is to reintroduce some of the physics-based knowledge of classical constitutive modeling into a neural network by employing the actual material models used in the full-order micromodel to introduce non-linearity. Thus, path-dependency arises naturally since every material model in the layer keeps track of its own internal variables. For the numerical examples, a composite Representative Volume Element with elastic fibers and elasto-plastic matrix material is used as the microscopic model. The network is tested in a series of challenging scenarios and its performance is compared to that of a state-of-the-art Recurrent Neural Network (RNN). A remarkable outcome of the novel framework is the ability to naturally predict unloading/reloading behavior without ever seeing it during training, a stark contrast with popular but data-hungry models such as RNNs. Finally, the proposed network is applied to FE$^2$ examples to assess its robustness for application in nonlinear finite element analysis.
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