Sharp bounds on the volume fractions of two materials in a two-dimensional body from electrical boundary measurements: the translation method

TL;DR

This paper develops optimal bounds on the volume fractions of two materials in a 2D body using electrical boundary measurements, employing the translation method and variational principles, with numerical validation.

Contribution

It introduces new sharp bounds for material volume fractions in 2D using boundary data, extending previous results and providing conditions for their attainability.

Findings

Bounds are optimal and attained by configurations with uniform internal fields.

Special boundary conditions reduce bounds to known results by Milton and Capdeboscq-Vogelius.

Numerical experiments confirm the bounds' effectiveness.

Abstract

We deal with the problem of estimating the volume of inclusions using a finite number of boundary measurements in electrical impedance tomography. We derive upper and lower bounds on the volume fractions of inclusions, or more generally two phase mixtures, using two boundary measurements in two dimensions. These bounds are optimal in the sense that they are attained by certain configurations with some boundary data. We derive the bounds using the translation method which uses classical variational principles with a null Lagrangian. We then obtain necessary conditions for the bounds to be attained and prove that these bounds are attained by inclusions inside which the field is uniform. When special boundary conditions are imposed the bounds reduce to those obtained by Milton and these in turn are shown here to reduce to those of Capdeboscq-Vogelius in the limit when the volume fraction tends to zero. The bounds of this paper, and those of Milton, work for inclusions of arbitrary volume fractions. We then perform some numerical experiments to demonstrate how good these bounds are.