In recent years, there has been a growing interest in understanding complex microstructures and their effect on macroscopic properties. In general, it is difficult to derive an effective constitutive law for such microstructures with reasonable accuracy and meaningful parameters. One numerical approach to bridge the scales is computational homogenization, in which a microscopic problem is solved at every macroscopic point, essentially replacing the effective constitutive model. Such approaches are, however, computationally expensive and typically infeasible in multi-query contexts such as optimization and material design. To render these analyses tractable, surrogate models that can accurately approximate and accelerate the microscopic problem over a large design space of shapes, material and loading parameters are required. In previous works, such models were constructed in a data-driven manner using methods such as Neural Networks (NN) or Gaussian Process Regression (GPR). However, these approaches currently suffer from issues, such as need for large amounts of training data, lack of physics, and considerable extrapolation errors. In this work, we develop a reduced order model based on Proper Orthogonal Decomposition (POD), Empirical Cubature Method (ECM) and a geometrical transformation method with the following key features: (i) large shape variations of the microstructure are captured, (ii) only relatively small amounts of training data are necessary, and (iii) highly non-linear history-dependent behaviors are treated. The proposed framework is tested and examined in two numerical examples, involving two scales and large geometrical variations. In both cases, high speed-ups and accuracies are achieved while observing good extrapolation behavior.

翻译：近年来，人们日益关注复杂微结构及其对宏观性质的影响。通常，为这类微结构推导具有合理精度与有意义参数的有效本构律是困难的。连接尺度的数值方法之一是计算均质化，该方法在每个宏观点上求解微观问题，实质上替代了有效本构模型。然而，这类方法计算成本高昂，在优化与材料设计等多查询场景中通常不可行。为使这些分析易于处理，需要能够在大范围的形状、材料与加载参数设计空间内精确逼近并加速微观问题的代理模型。在以往工作中，此类模型通过神经网络或高斯过程回归等数据驱动方法构建。然而，这些方法目前存在诸多问题，例如需要大量训练数据、缺乏物理机制以及显著的预测误差。本研究开发了一种基于本征正交分解、经验求积方法及几何变换方法的降阶模型，其关键特征包括：(i) 捕获微结构的大尺度形状变化，(ii) 仅需相对少量的训练数据，以及(iii) 处理高度非线性历史依赖行为。所提出的框架在两个涉及双尺度与大几何变化的数值算例中得到测试与验证。在这两种情况下，均实现了高加速比与高精度，同时展现出良好的预测行为。