Estimating effective properties of composites from cross-sectional photographs (Q1891062)

The authors present a new algorithm to evaluate structural parameters of statistically isotropic composites from cross-sectional photographs. In particular, they discuss in detail the evaluation of structural parameters \(\zeta_ 2\) and \(\eta_ 2\) which enter into third-order bounds on the conductivity and elastic moduli of two-component composites, but the algorithm also applies to the evaluation of structural parameters for multicomponent composites. The suggested algorithm works (roughly) as follows. It discretizes the component interfaces, as seen on the photograph. Polyharmonic fields \(\phi_ n\) are evaluated with a polyharmonic version of the fast multipole method. Fourier coefficients \(a_ n\) are computed via inner products of \(\phi_ n\) over one of the components. Finally, the parameters \(\zeta_ 2\) and \(\eta_ 2\) are expressed as a sum over the \(a_ n\). Two numerical examples are discussed. First the random aggregate of permeable spheres and for second an analysis of a photograph that looks like a square array of disks.