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Microstructure-informed reduced modes synthesized with Wang tiles and the Generalized Finite Element Method
arXiv - CS - Computational Engineering, Finance, and Science Pub Date : 2020-09-28 , DOI: arxiv-2010.02690
M. Do\v{s}k\'a\v{r} (1), J. Zeman (1), P. Krysl (2), and J. Nov\'ak (1 and 3) ((1) Czech Technical University in Prague, (2) University of California, San Diego, (3) University of Luxembourg)

A recently introduced representation by a set of Wang tiles -- a generalization of the traditional Periodic Unit Cell based approach -- serves as a reduced geometrical model for materials with stochastic heterogeneous microstructure, enabling an efficient synthesis of microstructural realizations. To facilitate macroscopic analyses with a fully resolved microstructure generated with Wang tiles, we develop a reduced order modelling scheme utilizing pre-computed characteristic features of the tiles. In the offline phase, inspired by the computational homogenization, we extract continuous fluctuation fields from the compressed microstructural representation as responses to generalized loading represented by the first- and second-order macroscopic gradients. In the online phase, using the ansatz of the Generalized Finite Element Method, we combine these fields with a coarse finite element discretization to create microstructure-informed reduced modes specific for a given macroscopic problem. Considering a two-dimensional scalar elliptic problem, we demonstrate that our scheme delivers less than a 3% error in both the relative $L_2$ and energy norms with only 0.01% of the unknowns when compared to the fully resolved problem. Accuracy can be further improved by locally refining the macroscopic discretization and/or employing more pre-computed fluctuation fields. Finally, unlike the standard snapshot-based reduced-order approaches, our scheme handles significant changes in the macroscopic geometry or loading without the need for recalculating the offline phase, because the fluctuation fields are extracted without any prior knowledge on the macroscopic problem.

中文翻译:

用王瓦片和广义有限元方法合成的微观结构信息简化模式

最近引入的一组 Wang 瓷砖表示 - 基于传统周期单位单元的方法的概括 - 作为具有随机异质微观结构的材料的简化几何模型,从而能够有效地合成微观结构实现。为了使用 Wang 瓷砖生成的完全解析的微观结构促进宏观分析,我们开发了一种利用瓷砖的预先计算的特征特征的降阶建模方案。在离线阶段,受计算均匀化的启发,我们从压缩的微观结构表示中提取连续波动场,作为对由一阶和二阶宏观梯度表示的广义载荷的响应。在线阶段,使用广义有限元法的 ansatz,我们将这些场与粗略的有限元离散化相结合,以创建特定于给定宏观问题的微观结构信息简化模式。考虑一个二维标量椭圆问题,我们证明了我们的方案在相对 $L_2$ 和能量范数方面的误差都小于 3%,与完全解决的问题相比,只有 0.01% 的未知数。通过局部细化宏观离散和/或采用更多预先计算的波动场,可以进一步提高精度。最后,与标准的基于快照的降阶方法不同,我们的方案无需重新计算离线阶段即可处理宏观几何或负载的重大变化,因为提取波动场时无需对宏观问题有任何先验知识。我们证明,与完全解决的问题相比,我们的方案在相对 $L_2$ 和能量范数方面的误差均小于 3%,而未知数仅为 0.01%。通过局部细化宏观离散和/或采用更多预先计算的波动场,可以进一步提高精度。最后,与标准的基于快照的降阶方法不同,我们的方案无需重新计算离线阶段即可处理宏观几何或负载的重大变化,因为提取波动场时无需对宏观问题有任何先验知识。我们证明,与完全解决的问题相比,我们的方案在相对 $L_2$ 和能量范数方面的误差都小于 3%,只有 0.01% 的未知数。通过局部细化宏观离散和/或采用更多预先计算的波动场,可以进一步提高精度。最后,与标准的基于快照的降阶方法不同,我们的方案无需重新计算离线阶段即可处理宏观几何或负载的重大变化,因为提取波动场时无需对宏观问题有任何先验知识。通过局部细化宏观离散和/或采用更多预先计算的波动场,可以进一步提高精度。最后,与标准的基于快照的降阶方法不同,我们的方案无需重新计算离线阶段即可处理宏观几何或负载的重大变化,因为提取波动场时无需对宏观问题有任何先验知识。通过局部细化宏观离散和/或采用更多预先计算的波动场,可以进一步提高精度。最后,与标准的基于快照的降阶方法不同,我们的方案无需重新计算离线阶段即可处理宏观几何或负载的重大变化,因为提取波动场时无需对宏观问题有任何先验知识。
更新日期:2020-10-07
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