Model order reduction of nonlinear homogenization problems using a Hashin–Shtrikman type finite element method
Model order reduction of nonlinear homogenization problems using a Hashin–Shtrikman type finite element method
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DOI：
10.1016/j.cma.2017.10.019
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发表时间：
2018-03
影响因子：
7.2
通讯作者：
S. Wulfinghoff;F. Cavaliere;S. Reese
中科院分区：
文献类型：
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作者：
S. Wulfinghoff;F. Cavaliere;S. Reese
本文提出了一种计算非线性均匀化方法，其出发点是基于数据聚类的模型降阶方法。为此，分析了数值实验（快照）的微观力学数据，以识别特征的微观结构变形模式。这些描述了宏观应变通常如何在微观结构中定位。该过程的结果是将微观结构细分为一组材料点簇。然后，在每个簇内，应变近似为常数。力学问题在引入线弹性参考介质的基础上，用三场Hashin-Shtrikman型变分公式来表述。离散化后，大多数全局未知数可以通过静态凝聚消除，留下分段恒定的簇应变作为主要未知数。所得到的均匀化方案包括，作为特殊情况，有限元法以及Hashin-Shtrikman和Talbot-Willis型均匀化方法与相位恒定的试验场（以及相关的边界）。极限情况下的“有限元法”允许从有限元技术转移知识，从而为参考材料的刚度选择提供新的策略。将该方法应用于几种具有不同夹杂物体积分数和不同程度各向异性的非线性微观结构。结果与现场有限元模拟结果吻合较好。此外，该方法用于计算Talbot-Willis型的精细化上界（与相向常数试验场相比），该上界收敛于越来越精细的离散化有限元解。
This work presents a computational nonlinear homogenization approach, the starting point of which is a model order reduction method based on data-clustering. To this end, the micromechanical data from numerical experiments (snapshots) is analyzed in order to identify characteristic microstructural deformation patterns. These describe how the macroscopic strain typically localizes within the microstructure. The outcome of the procedure is a subdivision of the microstructure into a set of clusters of material points. Within each cluster the strain is then approximated as being constant.The mechanical problem is formulated in terms of a three-field Hashin–Shtrikman type variational formulation which is based on the introduction of a linear-elastic reference medium. After discretization, most of the global unknowns can be eliminated via static condensation leaving the piecewise constant cluster strains as the primary unknowns. The resulting homogenization scheme includes, as special cases, the finite element method as well as Hashin–Shtrikman and Talbot–Willis type homogenization approaches with phase-wise constant trial fields (as well as related bounds). The limit case ’finite element method’ allows to transfer knowledge from finite element technology and thus provides new strategies for the choice of the stiffness of the reference material. The method is applied to several nonlinear microstructures with different inclusion volume fractions and varying degree of anisotropy. The results are shown to be in good agreement with full-field FE-simulations. Furthermore, the method is used to compute a refined upper bound of the Talbot–Willis type (compared to phase-wise constant trial fields), which converges to the finite element solution with increasingly refined discretization.