A Domain Decomposition Approach for Calculating the Graph Corresponding to a Fibrous Geometry

The effective properties of composite materials/media are in strong demand in engineering, geoscience, and environmental studies to name just a few examples. In [2], we presented an efficient algorithm for computing an approximation of the effective thermal conductivity tensor for high contrast fibrous geometries. The essential idea of the approach is to take into consideration the network-like structure of a given fibrous geometry and to perform all calculations on the induced unstructured grid. More precisely, the intersections of fibers are considered as nodes and the connecting fibers between nodes are considered as edges of an undirected graph. The weight of each edge depends on the diameter and the conductivity of the respective fiber and the distance of the connected nodes. A comparison between the results produced by our algorithm and classical methods, which resolve the fibrous geometry using volumetric elements, yields evidence of its efficiency and reliability for a large class of problems from engineering and science.