Response Statistics for Random Heterogeneous Microstructures

A practical method is developed for calculating statistics of material responses at small scale and functionals of these responses. The method is applied to characterize probabilistically potentials in specimens with conductivities that fluctuate randomly in space. Parametric models for microstructures, that is, deterministic functions of space coordinates that depend on random vectors $Z$, and SROMs $\tilde{Z}$ of $Z$, that is, random vectors with finite numbers of samples matching statistics of $Z$, are the essential tools used in analysis. Two surrogate models are constructed for material responses $U(x,Z)$ at small scale providing approximations for the mapping $Z\mapsto U(x,Z)$, where $x$ denotes the space coordinate. Samples of the surrogate models, which are generated by elementary calculations, are employed to characterize $U(x,Z)$. Bounds are developed on the discrepancy between exact solutions and solutions by surrogate models. Numerical examples are presented to demonstrate the implementation o...

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