The structural properties of mechanical metamaterials are typically studied with two-scale methods based on computational homogenization. Because such materials have a complex microstructure, enriched schemes such as second-order computational homogenization are required to fully capture their non-linear behavior, which arises from non-local interactions due to the buckling or patterning of the microstructure. In the two-scale formulation, the effective behavior of the microstructure is captured with a representative volume element (RVE), and a homogenized effective continuum is considered on the macroscale. Although an effective continuum formulation is introduced, solving such two-scale models concurrently is still computationally demanding due to the many repeated solutions for each RVE at the microscale level. In this work, we propose a reduced-order model for the microscopic problem arising in second-order computational homogenization, using proper orthogonal decomposition and a novel hyperreduction method that is specifically tailored for this problem and inspired by the empirical cubature method. Two numerical examples are considered, in which the performance of the reduced-order model is carefully assessed by comparing its solutions with direct numerical simulations (entirely resolving the underlying microstructure) and the full second-order computational homogenization model. The reduced-order model is able to approximate the result of the full computational homogenization well, provided that the training data is representative for the problem at hand. Any remaining errors, when compared with the direct numerical simulation, can be attributed to the inherent approximation errors in the computational homogenization scheme. Regarding run times for one thread, speed-ups on the order of 100 are achieved with the reduced-order model as compared to direct numerical simulations.

翻译：机械超材料的结构特性通常通过基于计算均质化的双尺度方法进行研究。由于此类材料具有复杂的微结构，需要采用诸如二阶计算均质化等增强型方案来充分捕捉其非线性行为——这类行为源于微结构屈曲或构型产生的非局部相互作用。在双尺度框架中，微结构的有效行为通过代表性体积单元表征，而在宏观尺度上则考虑均质化的等效连续介质。尽管引入了等效连续介质理论，同时求解此类双尺度模型仍具有较高计算成本，这是因为需要在微观层面反复求解每个代表性体积单元。本文针对二阶计算均质化中的微观问题提出了一种降阶模型，采用本征正交分解方法，并基于经验积分法则思想创新性地设计了一种针对该问题的超降阶方法。通过两个数值算例，将降阶模型的解与直接数值模拟（完整解析底层微结构）及全阶二阶计算均质化模型的结果进行详细比较，系统评估了其性能。研究表明：当训练数据充分表征目标问题时，降阶模型能够较好地逼近全阶计算均质化的结果；与直接数值模拟相比，残余误差可归因于计算均质化方案固有的近似误差。在单线程计算时间方面，降阶模型相比直接数值模拟实现了约100倍的加速比。