Computational multiscale methods for analyzing and deriving constitutive responses have been used as a tool in engineering problems because of their ability to combine information at different length scales. However, their application in a nonlinear framework can be limited by high computational costs, numerical difficulties, and/or inaccuracies. In this paper, a hybrid methodology is presented which combines classical constitutive laws (model-based), a data-driven correction component, and computational multiscale approaches. A model-based material representation is locally improved with data from lower scales obtained by means of a nonlinear numerical homogenization procedure, leading to a model-data-driven approach. Therefore, macroscale simulations explicitly incorporate the true microscale response, maintaining the same level of accuracy that would be obtained with online micro-macro simulations but with a computational cost comparable to classical model-driven approaches. In the proposed approach, both model and data play a fundamental role allowing for the synergistic integration between a physics-based response and a machine learning black-box. Numerical applications are implemented in two dimensions for different tests investigating both material and structural responses in large deformations. Overall, the presented model-data-driven methodology proves to be more versatile and accurate than methods based on classical model-driven, as well as pure data-driven techniques. In particular, a lower number of training samples is required and robustness is higher than for simulations which solely rely on data.

用于分析和推导本构响应的计算多尺度方法已被用作工程问题的工具，因为它们能够组合不同长度尺度的信息。然而，它们在非线性框架中的应用可能会受到高计算成本、数值困难和/或不准确性的限制。在本文中，提出了一种混合方法，它结合了经典本构律（基于模型）、数据驱动的校正组件和计算多尺度方法。基于模型的材料表示通过非线性数值均匀化程序获得的较低尺度的数据进行局部改进，从而形成模型数据驱动的方法。因此，宏观模拟明确地结合了真实的微观响应，保持与在线微观宏观模拟相同的精度水平，但计算成本与经典模型驱动方法相当。在所提出的方法中，模型和数据都发挥着基本作用，允许基于物理的响应和机器学习黑盒之间的协同集成。数值应用在二维中实现，用于研究大变形中的材料和结构响应的不同测试。总体而言，所提出的模型数据驱动方法被证明比基于经典模型驱动和纯数据驱动技术的方法更加通用和准确。特别是，与仅依赖数据的模拟相比，需要较少数量的训练样本并且鲁棒性更高。