Multipoint matrix Padé approximant bounds on effective anisotropic transport coefficients of two-phase media

S Tokarzewski

doi:10.1007/s00033-012-0214-z

Abstract

It is well known that a tensor Stieltjes function f represents an effective transport coefficient q of an inhomogeneous medium consisting of two isotropic components. In this paper, we investigate multipoint matrix Padé approximants to matrix expansions of f. We prove that matrix Padé ones to f estimate f from the top and below. Consequently the Padé approximants to q form upper and lower bounds on q. The inequalities for matrix Padé bounds on f and q are established. They reduce to the inequalities for scalar Padé ones Tokarzewski (ZAMP 61:773–780, 2010). As an illustrative example, matrix Padé estimates of an effective conductivity of a specially laminated two-phase medium are computed.

Highlights

One of the most popular method of a calculation of effective physical properties q of inhomogeneous media is a method of bounds
Bergman [3, 4] introduced a method for obtaining bounds on qjj, j = 1, 2, 3 which does not rely on variational principles
The aim of this paper is to establish multipoint matrix Pade bounds [1,2] on an effective, anisotropic transport coefficient q of a two-phase medium for the case, when matrix power expansions of q at a number of real points are given

Summary

Introduction

One of the most popular method of a calculation of effective physical properties q of inhomogeneous media is a method of bounds. Bergman [3, 4] introduced a method for obtaining bounds on qjj, j = 1, 2, 3 which does not rely on variational principles. Instead, he exploited the analyticity of qjj, j = 1, 2, 3 as functions of the moduli of phases. Matrix continued fraction methods for computing matrix bounds on an effective, anisotropic transport coefficient q are reported in [6] and [17, Section 28.4], see [15,16]. Milton [17, Section 28.4] deals mostly with threedimensional composites He computes the bounds on q from a power expansion of q(z) at z = 0. The aim of this paper is to establish multipoint matrix Pade bounds [1,2] on an effective, anisotropic transport coefficient q of a two-phase medium for the case, when matrix power expansions of q at a number of real points are given

Anisotropic transport coefficients as a matrix Stieltjes functions

Conclusion

Matrix continued fractions to a matrix Stieltjes function

Inequalities for matrix bounds on a matrix Stieltjes function

Illustrative example

Conclusions and final remarks