Exotic behaviour of mechanical metamaterials often relies on an internal transformation of the underlying microstructure triggered by its local instabilities, rearrangements, and rotations. Depending on the presence and magnitude of such a transformation, effective properties of a metamaterial may change significantly. To capture this phenomenon accurately and efficiently, homogenization schemes are required that reflect microstructural as well as macro-structural instabilities, large deformations, and non-local effects. To this end, a micromorphic computational homogenization scheme has recently been developed, which employs the particular microstructural transformation as a non-local mechanism, magnitude of which is governed by an additional coupled partial differential equation. Upon discretizing the resulting problem it turns out that the macroscopic stiffness matrix requires integration of macro-element basis functions as well as their derivatives, thus calling for higher-order integration rules. Because evaluation of a constitutive law in multiscale schemes involves an expensive solution of a non-linear boundary value problem, computational efficiency of the micromorphic scheme can be improved by reducing the number of integration points. Therefore, the goal of this paper is to investigate reduced-order schemes in computational homogenization, with emphasis on the stability of the resulting elements. In particular, arguments for lowering the order of integration from expensive mass-matrix to a cheaper stiffness-matrix equivalent are outlined first. An efficient one-point integration quadrilateral element is then introduced and a proper hourglass stabilization is discussed. Performance of the resulting set of elements is finally tested on a benchmark bending example, showing that we achieve accuracy comparable to the full quadrature rules, whereas computational cost decreases proportionally to the reduction in the number of quadrature points used.

中文翻译：

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弹性力学超材料的微形态计算均质化中的简化积分方案

机械超材料的外来行为通常取决于其微观不稳定性，重排和旋转触发的内部微观结构的内部转变。根据此类转换的存在和大小，超材料的有效属性可能会发生重大变化。为了准确有效地捕获此现象，需要使用均质化方案，以反映微观结构以及宏观结构的不稳定性，大变形和非局部效应。为此，最近开发了一种微形态计算均质方案，该方案采用特定的微结构变换作为非局部机制，其大小由附加的耦合偏微分方程控制。离散化结果问题后，事实证明，宏观刚度矩阵需要对宏观元素基础函数及其导数进行积分，因此需要更高阶的积分规则。因为在多尺度方案中评估本构律涉及非线性边值问题的昂贵解决方案，所以可以通过减少积分点的数量来提高微晶方案的计算效率。因此，本文的目的是研究计算均化中的降阶方案，并着重于所得元素的稳定性。特别是，首先概述了将积分的顺序从昂贵的质量矩阵降低到便宜的刚度矩阵等效项的论点。然后介绍了一种有效的单点积分四边形元素，并讨论了适当的沙漏稳定性。最后，在一个基准弯曲示例上测试了所得元素集的性能，表明我们达到了与完整正交规则可比的精度，而计算成本则与所使用的正交点数量的减少成比例地减少。