Simulating the mechanical response of advanced materials can be done more accurately using concurrent multiscale models than with single-scale simulations. However, the computational costs stand in the way of the practical application of this approach. The costs originate from microscale Finite Element (FE) models that must be solved at every macroscopic integration point. A plethora of surrogate modeling strategies attempt to alleviate this cost by learning to predict macroscopic stresses from macroscopic strains, completely replacing the microscale models. In this work, we introduce an alternative surrogate modeling strategy that allows for keeping the multiscale nature of the problem, allowing it to be used interchangeably with an FE solver for any time step. Our surrogate provides all microscopic quantities, which are then homogenized to obtain macroscopic quantities of interest. We achieve this for an elasto-plastic material by predicting full-field microscopic strains using a graph neural network (GNN) while retaining the microscopic constitutive material model to obtain the stresses. This hybrid data-physics graph-based approach avoids the high dimensionality originating from predicting full-field responses while allowing non-locality to arise. By training the GNN on a variety of meshes, it learns to generalize to unseen meshes, allowing a single model to be used for a range of microstructures. The embedded microscopic constitutive model in the GNN implicitly tracks history-dependent variables and leads to improved accuracy. We demonstrate for several challenging scenarios that the surrogate can predict complex macroscopic stress-strain paths. As the computation time of our method scales favorably with the number of elements in the microstructure compared to the FE method, our method can significantly accelerate FE2 simulations.

翻译：使用并发多尺度模型模拟先进材料的力学响应比单尺度模拟更为精确，但计算成本阻碍了该方法的实际应用。这些成本源于必须在每个宏观积分点求解的微观有限元模型。现有大量替代建模策略试图通过学习从宏观应变预测宏观应力来完全替代微观模型，从而降低计算成本。本研究提出一种替代建模策略，该策略保留了问题的多尺度特性，可随时与有限元求解器互换使用。我们的替代模型提供所有微观量，再通过均匀化得到关注的宏观量。针对弹塑性材料，我们采用图神经网络预测全场微观应变，同时保留微观本构模型以获取应力。这种基于图的数据-物理混合方法避免了全场响应预测带来的高维度问题，同时允许非局部效应的产生。通过在多种网格上训练图神经网络，该网络学会泛化到未见过的网格，使得单一模型可用于多种微结构。嵌入在神经网络中的微观本构模型隐式追踪历史相关变量，并提升了预测精度。针对多个具有挑战性的场景，我们证明了该替代模型能预测复杂的宏观应力-应变路径。由于相比有限元方法，本方法的计算时间随微结构单元数量增长的扩展性更优，因此可显著加速FE²模拟。