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 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 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 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 of the macroscopic problem.

最近引入的一系列Wang切片表示法（对传统的基于周期单位细胞的方法的概括）可以用作具有随机异质微观结构的材料的简化几何模型，从而可以有效地合成微观结构实现。为了方便使用Wang瓷砖生成的完全解析的微观结构进行宏观分析，我们利用瓷砖的预先计算的特征开发了降阶建模方案。在离线阶段，受计算均一化的启发，我们从压缩的微结构表示中提取连续的波动场，作为对由一阶和二阶宏观梯度表示的广义载荷的响应。在在线阶段，使用广义有限元方法的ansatz，我们将这些场与粗糙的有限元离散化相结合，以创建特定于给定宏观问题的，微观结构相关的简化模式。考虑二维标量椭圆问题，我们证明了我们的方案在两个相对误差均小于3％的情况下与完全解决的问题相比，\（L_2 \）和能量范数只有0.01％的未知数。通过局部细化宏观离散化和/或采用更多的预先计算的波动场，可以进一步提高精度。最后，与标准的基于快照的降序方法不同，我们的方案无需更改脱机阶段即可处理宏观几何或载荷的重大变化，因为在没有任何宏观问题的先验知识的情况下提取了波动场。