Numerical homogenization for mechanical multiscale modeling by means of the finite element method (FEM) is an elegant way of obtaining structure-property relations, if the behavior of the constituents of the lower scale is well understood. However, the computational costs of this so-called FE$^2$ method are so high that reduction methods are essential. While the construction of a reduced basis for the microscopic nodal displacements using proper orthogonal decomposition (POD) has become a standard technique, the reduction of the computational effort for the projected nodal forces, the so-called hyper reduction, is an additional challenge, for which different strategies have been proposed in the literature. The empirical cubature method (ECM), which has been proven to be very robust, implemented the conservation of the total volume is used as a constraint in the resulting optimization problem, while energy-based criteria have been proposed in other contributions. The present contribution presents a unified integration criteria concept, involving the aforementioned criteria, among others. These criteria are used both with a Gauss point-based as well as with an element-based hyper reduction scheme, the latter retaining full compatibility with the common modular finite element framework. The methods are combined with a previously proposed clustered training strategy and a monolithic solver. Numerical examples empirically demonstrate that the additional criteria improve the accuracy for a given number of modes. Vice verse, less modes and thus lower computational costs are required to reach a given level of accuracy.

翻译：通过有限元法（FEM）进行力学多尺度建模的数值均匀化，若对微观尺度组分行为有充分理解，是获取结构-性能关系的有效途径。然而，这种所谓FE$^2$方法的计算成本极高，因此降阶方法至关重要。虽然利用本征正交分解（POD）为微观节点位移构建降阶基已成为标准技术，但对投影节点力（即所谓超约化）计算量的缩减仍是额外挑战，文献中已提出多种应对策略。经验性立方体法（ECM）已被证明具有很强鲁棒性，其在优化问题中将总体积守恒作为约束条件，而其他研究则提出了基于能量的准则。本文提出了一种统一的积分准则框架，涵盖了上述准则及其他相关准则。这些准则既可与基于高斯点的超约化方案结合，也可与基于单元的超约化方案结合——后者完全兼容通用的模块化有限元框架。所提方法与先前提出的聚类训练策略及整体求解器相结合。数值算例经验性表明：在给定模态数下，附加准则能提升计算精度；反之，要达到给定精度水平，所需模态数更少，计算成本更低。