GENERALIZED POLARIZATION TENSORS 3
element or finite difference methods. This is because such features require refined
meshes in their neighborhoods, with their attendant problems [93].
The analysis of the GPT's associated with anisotropic conductivities is parallel
to that for the isotropic conductivity problems, apart from some technical difficul-
ties due to the fact that we are dealing with a system, not a single equation, and
the fundamentals solutions inside and outside the inclusion are different.
Let us straight away explain what makes the GPT's interesting in electrical
impedance imaging. In all of the reconstruction algorithms based on dipole-type
approximations (first-order boundary perturbations), the locations of the inclusions
are found with an error of the common order of magnitude of their diameters, and
little about their shapes can be reconstructed. Making use of the GPT's (higher-
order boundary perturbations), we are able to reconstruct the small inclusions with
higher resolution [33]. Indeed, this allows us to identify quite general conductivity
inclusions without restrictions on their sizes.
The concepts of higher-order polarization tensors generalize those of classical
P6lya-Szego polarization tensors which have been extensively studied in the liter-
ature by many authors for various purposes [52, 29, 55, 66, 111, 110, 73, 98,
101, 120, 63]. The notion of P6lya-Szego polarization tensor appeared in prob-
lems of potential theory related to certain problems arising in hydrodynamics and
in electrostatics. If the conductivity is zero, namely, if the inclusion is insulated,
the polarization tensor of P6lya-Szego is called the virtual mass.
We provide a survey of important symmetric properties and positivity of the
GPT's and present certain inequalities satisfied by the tensor elements of the GPT's.
These relations can be used to find bounds on the weighted volume.
The concept of polarization tensors also occurs in several other interesting
contexts, in particular in asymptotic models of dilute composites [116, 29, 65].
The determination of the effective or macroscopic property of a two-phase medium
consisting of inclusions of one material of known shape embedded homogeneously
into a continuous matrix of another having physical properties different from its
own has been one of the classical problems in physics. When the inclusions are
well-separated d-dimensional spheres and their volume fraction is small, the effec-
tive electrical conductivity of the composite medium is given by the well-known
Maxwell-Garnett formula.
Despite the importance of calculating the effective or macroscopic properties
of composites there has been very little work addressing the influence of inclu-
sion shape. Most theoretical treatments focus on generalizing the Maxwell-Garnett
formula to finite concentrations. The methods include bounds on the effective prop-
erties of the mixtures and many effective medium type models have been proposed
[115]. Indeed, there are effective medium calculations that attempt to extend the
Maxwell-Garnett formula to higher powers of the volume fraction, but only for the
case of d-dimensional spherical inclusions [81, 126].
Until recently, ellipsoids are the only family of inclusions that could be rigor-
ously and accurately estimated [136]. Douglas and Garboczi [66, 74, 111] made
an important advance in treating more complicated shape inclusions by formally
finding that the leading order term in the expansion of the effective conductivity
(and other effective properties) in terms of the volume fraction could be expressed
by means of the polarization tensors of the inclusion shape. See also, in connection
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