SPECTRAL STOCHASTIC HOMOGENIZATION

Abstract eng:
An original method of stochastic homogenization is developed and applied to a class of PDEs that is representative of a large number of problems in science and engineering. The starting point is a governing differential equation whose coefficients are modeled as stochastic processes with multi-scale oscillatory components. Even the deterministic versions of these equations present great chal lenges to numerical analysis methodologies, justifying various approaches to homogenization. The difficulty in the present situation is compounded by the explicit stochastic modeling of the variability in the heterogeneous coefficients. The present situation, therefore, corresponds to a situation involving highly heterogeneous materials with stochastic variability. A new system of stochastic differential equations is then obtained with random variables as effective coefficients. These coefficients are obtained through an asymptotic analysis that effectively identifies a macro-scale system as the limit of a micro-scale representative volume (REV) as this volume goes to zero. This resulting system of equations can then be resolved by relying on standard numerical integration schemes. The above methodology is integrated with the polynomial chaos representation of stochastic processes resulting in computationally tractable algorithms for evaluating the homogenized coefficients. These algorithms involve simple quadratures over the REV. An application to a multi-scale stochastic diffusion equation is used to highlight the theory, implementation and features of the proposed methodology.

• Export as: BibTeX | MARC | MARCXML | DC | EndNote | NLM | RefWorks
• Add to your basket