Communications in Mathematical Sciences

Volume 13 (2015)

Number 4

Special Issue in Honor of George Papanicolaou’s 70th Birthday

Guest Editors: Liliana Borcea, Jean-Pierre Fouque, Shi Jin, Lenya Ryzhik, and Jack Xin

Spectral measure computations for composite materials

Pages: 825 – 862

DOI: https://dx.doi.org/10.4310/CMS.2015.v13.n4.a1

Authors

N. Benjamin Murphy (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)

Elena Cherkaev (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)

Christel Hohenegger (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)

Kenneth M. Golden (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)

Abstract

The analytic continuation method of homogenization theory provides Stieltjes integral representations for the effective parameters of composite media. These representations involve the spectral measures of self-adjoint random operators which depend only on the composite geometry. On finite bond lattices, these random operators are represented by random matrices and the spectral measures are given explicitly in terms of their eigenvalues and eigenvectors. Here we provide the mathematical foundation for rigorous computation of spectral measures for such composite media, and develop a numerically efficient projection method to enable such computations. This is accomplished by providing a unified formulation of the analytic continuation method which is equivalent to the original formulation and holds for finite and infinite lattices, as well as in continuum settings. We also introduce a family of bond lattices and directly compute the associated spectral measures and effective parameters. The computed spectral measures are in excellent agreement with known theoretical results. The behavior of the associated effective parameters is consistent with the symmetries and theoretical predictions of models, and the computed values fall within rigorous bounds. Some previous calculations of spectral measures have relied on finding the boundary values of the imaginary part of the effective parameter in the complex plane. Our method instead relies on direct computation of the eigenvalues and eigenvectors which enables, for example, statistical analysis of the spectral data.

Keywords

composite materials, random resistor network, percolation, homogenization, spectral measure, random matrix

2010 Mathematics Subject Classification

00B15, 30B40, 47B15, 60K35, 65C60, 78A48, 80M40

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Published 12 March 2015