J. Helsing, ESTIMATING EFFECTIVE PROPERTIES OF COMPOSITES FROM CROSS-SECTIONAL PHOTOGRAPHS, Journal of computational physics, 117(2), 1995, pp. 281-288
An algorithm is presented for the rapid evaluation of certain function
als of three-point correlation functions, measured in a plane. This si
mplifies the estimation of effective physical properties of composite
materials from cross-sectional photographs via bounds. More precisely,
Fourier coefficients of a reduced three-point correlation function ar
e expressed as inner products of polyharmonic fields. The polyharmonic
fields are evaluated with a polyharmonic version of the fast multipol
e method with a CPU requirement proportional to the number p of fields
included and to the number N of points in a discretization of the com
ponent interfaces. The Fourier coefficients are related to structural
parameters which are used in third-order bounds on conductivity and el
astic properties. Inclusion of p polyharmonic fields gives structural
parameters with an error decaying at least as 1/p(3). In a simple appl
ication for disks with p = 10, superalgebraic convergence in 1/N, and
a high-order Gaussian quadrature rule for the inner products, the algo
rithm gives an error of typically 0.05%. A previous algorithm, involvi
ng Monte Carlo integration, gives structural parameters with an error
of typically 2%. (C) 1995 Academic Press, Inc.