Effective conductivity by stochastic reduced order models (SROMs)

A method is developed for calculating statistics of effective conductivity Σeff for microstructures with conductivity Σ varying randomly in space. The method is based on the representation of Σ by a stochastic reduced order model (SROM) Σ ∼ , that is, a random field with a finite and small number m of samples that, generally, are not equally likely. An optimization algorithm with objective function quantifying the discrepancy between the probability laws of Σ ∼ and Σ is used to construct Σ ∼ . Samples of Σeff corresponding to those of Σ ∼ and their probabilities define a SROM Σ ∼ eff for Σeff. Bounds are develop to quantify the discrepancy between Σ and Σ ∼ and between Σeff and Σ ∼ eff . The method is applied to find statistics of effective conductivity for a two-dimensional specimen with conductivity described by a non-Gaussian homogeneous field. Statistics of effective conductivity obtained from SROMs Σ ∼ eff with m samples are accurate and stable, in contrast to Monte Carlo estimates based on the same number of samples that are unstable and can have large errors.