O(n) algorithm for identification of domain isolation by fracture surfaces

In this paper, an O(n) algorithm for identification of isolated sub-domains is proposed. A brittle solid specimen is gradually broken into fragments when a dynamic propagation of cracks occurs in it. A sub-domain of the specimen is isolated from other part of it if the sub-domain is completely wrapped up in a set of fracture surfaces. Identification of domain isolation allows us an efficient numerical analysis such as hybridization of DEM (Distinct Element Method) and FEM for dynamic fracture problems. In the case of 2D problems, when a set of line segments forms a closed loop, the interior of the closed loop is isolated from other sub-domains. However, in the 3D case, closed loops can not identify an isolated sub-domain. If a computation and numerical errors are not considered, eigenvalue analysis is a way to identify domain isolation. However, it costs O(n²) computation and causes numerical errors by real number manipulation. So, an algorithm focused on the connectivity of nodal points is proposed. This algorithm has an O(n) computation and operates with only integer processing which does not cause numerical errors. Also it is independent of coordinates of nodal points and the dimension of the analysis domain. Some tests were carried out and it was confirmed that the computation is actually O(n).