Identification of a Spheroid based on the First Order Polarization Tensor

Polarization tensor (PT) has a lot of useful and important practical applications. In this case, it must be firstly determined by some appropriate method. Besides, understanding some properties of the PT might also be very useful in order to apply it. In this study, we investigate the first order PT for ellipsoid and use it to describe the first order PT for spheroid as well as identify the spheroid. Numerical examples are also given to further justify our results

The depolarization factors for ellipsoids and some of their properties

The terminology depolarization factors was firstly highlighted in the study of problems involving magnetic, where, it was initially used to describe magnetic properties of material. Recently, this terminology was investigated to describe composites, improve imaging techniques, and other field of researches related to potential theory in mathematics and physics. Due to our interest in electrical imaging using polarization tensor (PT) and since PT is actually related to the depolarization factors, in this paper, some properties of the depolarization factors are investigated for future applications. The values of these depolarization factors are firstly proven to be non-negative. Based on the previous studies which consider the incomplete elliptic integrals of the first and second kind with some suitable identities, the summation of the depolarization factors are shown to be equal to one. By using these two properties, the value for each depolarization factor for ellipsoid is then explained to be between zero and one. It is also shown in this paper that the depolarization factors can be characterized based on the values of the semi principal axes of the ellipsoid. Reversely, the semi principal axes of the ellipsoid can be classified based on the values of the depolarization factors. All properties presented in this paper could be useful and important in the future especially to use the depolarization factors in any related applications

The effect of different scale on object to the approximation of the first order polarization tensor of sphere, ellipsoid, and cube

Polarization tensor (PT) is a classical terminology in fluid mechanics and theory of electricity that can describe geometry in a specific boundary domain with different conductivity contrasts. In this regard, the geometry may appear in a different size, and for easy characterizing, the usage of PT to identify particular objects is crucial. Hence, in this paper, the first order polarization tensor for different types of objects with a diverse range of sizes are presented. Here, we used three different geometries: sphere, ellipsoid, and cube, with fixed conductivity for each object. The software Matlab and Netgen Mesh Generator are the essential mathematical tools to aid the computation of the polarization tensor. From the analytical results obtained, the first order PT for sphere and ellipsoid depends on the size of both geometries. On the other hand, the numerical investigation is conducted for the first order PT for cube, since there is no analytical solution for the first order PT related to this geometry, to further verify the scaling property of the first order PT due to the scaling on the size of the original related object. Our results agree with the previous theoretical result that the first order polarization tensor of any geometry will be scaled at a fixed scaling factor according to the scaling on the size of the original geometry

An extended method for fitting the first order polarization tensor to a spheroid

Polarization Tensor (PT) has been widely used in some of the applications of electric and electromagnetic such as electrical imaging, metal detection and electrosensing fish. Furthermore, in these applications, polarization tensors can capture significant information such as material, shape and orientation about the related objects (medical images in electrical imaging or metallic target in metal detection). Some physical information about the unknown objects can then be characterized from the given first order polarization tensor that representing the object. Therefore, it is beneficial to determine an ellipsoid based on the given first order polarization tensor due to the possible similarities between the ellipsoid and the unknown object. The main objective of this study is to present a method in order to determine the semi axes of the spheroid, which is an ellipsoid with two identical axes. The method is an extension of the previous method which is only applicable to two types of spheroid. Using the rotation of the first order polarization tensor, we will show that this extended method can be used to determine all semi axes for any types of spheroid, depending on the given first order PT. After that, some numerical examples are provided specifically to compare between the first order PT for the spheroid with the given first order PT for verifying the results obtained. It is expected that the computed first order PT for the spheroid will be almost similar to the given first order PT

Forced convection boundary layer flow in a thin nanofluid film on a stretching sheet under the effects of suction and injection

The forced convection thin-film hybrid nanofluid flow over a stretching sheet with heat transfer is investigated in the present study. The effect of the suction and injection is considered. The concerned hybrid nanoparticles are copper and alumina which are dissolved in blood base fluid. Suitable similarity variables are applied to convert the nonlinear governing partial differential equations subject to appropriate boundary conditions into a set of ordinary differential equations. The MATLAB solver bvp4c is utilized to solve the similarity transformed governing equations numerically. There exists a great agreement when the present computed findings are compared with the published results for a limiting condition. Dual solutions are obtained for the velocity and temperature profiles. Conflict behavior is observed for the effect of the unsteadiness parameter and mass transfer parameter on both solutions of the velocity and temperature distributions. The increment of the mass transfer parameter has enhanced the velocity profile in the injection case, while an opposite trend is detected in the suction situation

Characterization of Objects by Fitting the Polarization Tensor

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 �olya-Szeg �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 Matlab while a script in Python is written as an interface for using the new developed code for BEM called as BEM++. When comparing the two methods, our numerical results show that the first order PT is more accurate with faster convergence when computed by 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 �olya-Szeg �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 non-conducting, the rank 2 PT of the target can also be computed by using the explicit formula of the first order PT

Polarization tensor: between biology and engineering

There has been a lot of interest over recent years in the study of mathematical aspects and applications of the polarization tensor. This promising terminology appears widely in electric and electromagnetic inverse problems. Our main purpose in this paper then is to review the polarization tensor biologically in electro-sensing by a weakly electric fish and in the engineering problems which are based on the Eddy current principle. Here, the mathematical formulations of the polarization tensor for both cases are firstly presented. At the same time, a few related applications will also be briefly explained

Quadratic element integration of approximated first order polarization tensor for sphere

Polarization tensor is an object-specific property in order to indicate its shape, size and also the material used. In this paper, we describe an accurate and easy-implemented method based on numerical integration in order to compute the first order polarization tensor. We proposed an alternative method to deal with boundary integral equation of first order polarization tensor which is quadratic element numerical integration. This method uses standard three points Gaussian quadrature in order to generate the singular integral operator matrix of polarization tensor. Different values of object's conductivity are used in order to study the behavior of the polarization tensor. The validation of the results is based on the exact solution provided for sphere and ellipsoid geometry by previous researcher. Moreover, numerical computation showed that the quadratic element integration generates high accuracy numerical results for the approximated first order polarization tensor. The numerical results are illustrated in graphical form in order to show the validity of the proposed scheme