Fast finite element calculation of effective conductivity of random continuum microstructures: The recursive Poincaré-Steklov operator method

The effective conductivity of a block of composite can be extracted from the Dirichlet-to-Neumann Poincaré-Steklov operator (PSO) for that block. In this paper, a domain decomposition method for computing the PSO over a finite element mesh is discussed. A new numerical strategy is introduced to accelerate the computation of this operator, using the Schur complement to calculate the PSO for the smallest subdomains, then recursively merging subdomain PSOs up to the full domain. At each step of the algorithm, information extraneous to the PSO is discarded. The effective conductivity values computed by this method are identical to those obtained from a basic Finite Element Method, an order of magnitude faster and with much less computer memory consumed. As proof of concept, effective conductivity measurements are presented for a percolating random fractal-like microstructure across a range of phase fractions.