Bounds on the volume fractions of two materials in a three dimensional body from boundary measurements by the translation method

TL;DR

This paper derives bounds on the volume fraction of an inclusion within a 3D conducting body using boundary electrical measurements and the translation method, extending previous bounds under specific boundary conditions.

Contribution

It applies the translation method to obtain volume fraction bounds in 3D, generalizing prior results and connecting to special boundary data cases.

Findings

Derived bounds on inclusion volume fractions using boundary data.

Extended previous bounds to more general boundary conditions.

Connected bounds to special cases like Milton's and Capdeboscq and Vogelius's results.

Abstract

Using the translation method of Tartar, Murat, Lurie, and Cherkaev bounds are derived on the volume occupied by an inclusion in a three-dimensional conducting body. They assume electrical impedance tomography measurements have been made for three sets of pairs of current flux and voltage measurements around the boundary. Additionally the conductivity of the inclusion and surrounding medium are assumed to be known. If the boundary data (Dirichlet or Neumann) is special, i.e. such that the fields inside the body would be uniform were the body homogeneous, then the bounds reduce to those of Milton and thus when the volume fraction is small to those of Capdeboscq and Vogelius.