This thesis focuses on some mathematical aspects and a few recent applications of the polarization tensor (PT). Here, the main concern of the study is to characterize objects presented in electrical or electromagnetic fields by only using the PT. This is possible since the PT contains significant information about the object such as shape, orientation and material properties. Two main applications are considered in the study and they are electrosensing fish and metal detection. In each application, we present a mathematical formulation of the PT and briefly discuss its properties.The PT in the electrosensing fish is actually based on the first order generalized polarization tensor (GPT) while the GPT itself generalizes the classical PT called as the P\'{o}lyaSzeg\H{o} PT. In order to investigate the role of the PT in electrosensing fish, we propose two numerical methods to compute the first order PT. The first method is directly based on the quadrature method of numerical integration while the second method is an adaptation of some terminologies of the boundary element method (BEM). A code to use the first method is developed in \textit{Matlab} while a script in \textit{Python} is written as an interface for using the new developed code for BEM called as \textit{BEM++}. When comparing the two methods, our numerical results show that the first order PT is more accurate with faster convergence when computed by \textit{BEM++}. During this study, we also give a strategy to determine an ellipsoid from a given first order PT. This is because we would like to propose an experiment to test whether electrosensing fish can discriminate a pair of different objects but with the same first order PT such that the pair could be an ellipsoid and some other object. In addition, the first order PT (or the P\'{o}lyaSzeg\H{o} PT) with complex conductivity (or complex permittivity) which is similar to the PT for Maxwell's equations is also investigated.On the other hand, following recent mathematical foundation of the PT from the eddy current model, we use the new proposed explicit formula to compute the rank 2 PT for a few metallic targets relevance in metal detection. We show that the PT for the targets computed from the explicit formula agree to some degree of accuracy with the PT obtained from metal detectors during experimental works and simulations conducted by the engineers. This suggests to alternatively use the explicit formula which depends only on the geometry and material properties of the target as well as offering lower computational efforts than performing measurements with metal detectors to obtain the PT. By using the explicit formula of the rank 2 PT, we also numerically investigate some properties of the rank 2 PT where, the information obtained could be useful to improve metal detection and also in other potential applications of the eddy current. In this case, if the target is magnetic but nonconducting, the rank 2 PT of the target can also be computed by using the explicit formula of the first order PT.
Date of Award  1 Aug 2017 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  William Lionheart (Supervisor) & Oliver Dorn (Supervisor) 

 conductivity
 metal detector
 integral equation
 matrices
 electrosensing fish
Characterization of Objects by Fitting the Polarization Tensor
Ahmad Khairuddin, T. (Author). 1 Aug 2017
Student thesis: Phd