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采用基于计算均匀化的双尺度方法进行研究。由于此类材料具有复杂的微观结构，需要采用如二阶计算均匀化等增强方案来完整捕捉其非线性行为，该行为源于微观结构屈曲或图案化引起的非局部相互作用。在双尺度公式中，微观结构的等效行为通过代表性体积单元（RVE）进行捕捉，并在宏观尺度上考虑均匀化的等效连续体。尽管引入了等效连续体公式，但由于需要在微观尺度上对每个RVE进行大量重复求解，同时求解此类双尺度模型仍然具有较高的计算成本。本研究针对二阶计算均匀化中出现的微观问题，提出了一种降阶模型，该方法采用本征正交分解和一种新型超降阶技术，该技术专为此问题设计并受经验立方体方法的启发。通过两个数值算例，将降阶模型的解与直接数值模拟（完全解析底层微观结构）及完整二阶计算均匀化模型的解进行细致对比，从而系统评估降阶模型的性能。只要训练数据能充分代表所研究问题，降阶模型即可较好地逼近完整计算均匀化的结果。与直接数值模拟相比，任何残余误差均可归因于计算均匀化方案固有的近似误差。在单线程运行时间方面，与直接数值模拟相比，降阶模型实现了约100倍的速度提升。