论文标题
空间网络模型的数值均质化
Numerical homogenization of spatial network models
论文作者
论文摘要
我们介绍并分析了一种用于空间网络的数值均质化的方法，例如机械结构的扩散过程和变形。目的是构建网络的准确粗糙模型。通过解决局部子图上的解耦问题，我们构建了具有良好近似特性的解决方案空间的低维子空间。网络的粗糙模型由盖金（Galerkin）公式表示，可用于以低计算成本执行不同源和边界数据的模拟。我们证明，在对网络的同质性，连通性和局部性的粗略假设下，提出的方法在粗糙量表上进行了最佳融合。对于标量值（热传导）和矢量值（结构）模型，可以在数值上确认理论发现。
We present and analyze a methodology for numerical homogenization of spatial networks, modelling e.g. diffusion processes and deformation of mechanical structures. The aim is to construct an accurate coarse model of the network. By solving decoupled problems on local subgraphs we construct a low dimensional subspace of the solution space with good approximation properties. The coarse model of the network is expressed by a Galerkin formulation and can be used to perform simulations with different source and boundary data at a low computational cost. We prove optimal convergence of the proposed method under mild assumptions on the homogeneity, connectivity, and locality of the network on the coarse scale. The theoretical findings are numerically confirmed for both scalar-valued (heat conduction) and vector-valued (structural) models.
