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

Bounds on the volume of an inclusion in a body from a complex conductivity measurement

Pages: 863 – 892

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n4.a2

Authors

Andrew E. Thaler (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.; and Schlumberger-Doll Research Center, Cambridge, Massachusetts, U.S.A.)

Graeme W. Milton (Department of Mathematics, University of Utah, Salt Lake City, Ut., U.S.A.)

Abstract

We derive bounds on the volume of an inclusion in a body in two or three dimensions when the conductivities of the inclusion and the surrounding body are complex and assumed to be known. The bounds are derived in terms of average values of the electric field, current, and certain products of the electric field and current. All of these average values are computed from a single electrical impedance tomography measurement of the voltage and current on the boundary of the body. Additionally, the bounds are tight in the sense that at least one of the bounds gives the exact volume of the inclusion for certain geometries and boundary conditions.

Keywords

volume fraction bounds, electrical impedance tomography, size estimation

2010 Mathematics Subject Classification

31A25, 31B20, 35J57, 35Q60

Full Text (PDF format)

Published 12 March 2015