A particle digitization-based computational method for continuum percolation

Abstract

A new theoretical model and its computational implementation in two dimensions (2D) for the study of continuum percolation phenomena is presented. The aim was the development of a model which has inherent similarity with lattice percolation. The physical medium is simulated as an (infinite) grid comprising of representative surface elements (RSEs). Assuming medium’s homogeneity the RSEs average propagation probability can be interpreted and generalized as the occupation probability for the infinite medium. The RSE’s resulting from a Monte Carlo iterative process involving the creation of the relative small samples and their propagation ability checked individually from their top to bottom. The propagation in the actual physical medium takes place when the calculated probability (p) is higher than the critical propagation probability (pc≈0.5927). The proposed method treats the low dimensional material system as a 2D infinite homogenized medium which can be further reduced leading to a mapping on a square lattice with site occupation. The proposed numerical algorithm considers the particles in the RSE as digitized using sites-pixels without contacts. Following the digitization procedure, traditional computational methods like Depth First Search are involved for the detection of possible propagation paths in the randomly selected square samples. For the confirmation of the theoretical model as well as the algorithm, problems known from the literature were used and it was found that regardless of microstructure at the critical concentration Φc the percolation probability on the RSE converges to the anticipated pc≈0.5927 value. In addition, the results obtained from the proposed methodology compare very well with available predictions in the literature. New results are reported covering a wide range of particle geometrical types (circular, elliptical, rectangular) and surface fractions in matrix-filler or matrix-fillers systems proving the robustness and applicability of the proposed methodology.