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The Effect of Different Scale on Object to the Approximation of the First Order Polarization Tensor of Sphere, Ellipsoid, and Cube
Abstract
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 object 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.
... Conversely, some researchers focus on implementing different approaches to numerically compute the PT since it cannot be computed analytically [12,13]. Meanwhile, some researchers focus on exploring the properties of the PT theoretically [14,15] and validating the theoretical findings by presenting some numerical examples [16]. ...
The polarization tensor (PT) has been utilized in numerous applications involving electric and electromagnetic fields, such as metal detection, landmine detection, and electrical imaging. In these applications, the PT is implemented in the process of identifying object, where the object can be represented in the form of the first order PT. Thus, it is crucial to have an application that provides easy access to calculate the first order PT for the object. However, the existing application in the literature has some limitations, focusing solely on computing the first order PT for a prolate spheroid with semi axes and an oblate spheroid with semi axes . Therefore, the purpose of this study is to invent a graphical user interface (GUI) for the Spheroidal First Order Polarization Tensor (SFOPT) Toolkit which facilitates efficient computations and visualizations related to spheroids. The SFOPT Toolkit will be developed by using App Designer in MATLAB. The SFOPT interface integrates four essential functions: computation of the first order PT for a spheroid, classification of spheroidal types, three-dimensional visualization of spheroids, and determination of semi axes from the computed first order PT. Through illustrative examples, we demonstrate the effectiveness and versatility of the SFOPT framework, offering insights into its practical utility and potential applications in diverse fields. The reliability of the toolkit is also presented, and the findings shows the error of computations are small. The toolkit is a user-friendly application since the users can easily access it by downloading the application instead of running the coding themselves. Moreover, this toolkit can be a reference for researchers to compute the first order PT for a spheroid and determine the semi axes (size) of the spheroid from the given first order PT.
... Many problems in applied science and engineering involve geometric objects. Most of the time, the problems cannot be solved analytically and when numerical methods are implemented to solve the problems, mesh objects are used to approximated the related objects involved in the problem (see the previous studies by [1][2][3][4][5]). For example, in a study of heat transfer by [4], the domain involved, as a three-dimensional object, is firstly partitioned into small pieces of polyhedron before simulation is made using the partitioned object. ...
In many applications of science and engineering, the real physical related objects are usually modelled in terms of mesh objects, where, the mesh will be used during computation or simulation. For most cases, a good mesh is required in order to increase the accuracy of the results. As the mesh represents the original object, the mesh should have similar physical properties of the original object such as the surface area or the volume. However, finding the value of these parameters could require extra computational cost and increase overall time taken to solve the problem. Thus, it might be useful if these parameters can be determined separately to ensure good mesh is used before any simulation is performed. In this study, a Matlab GUI is developed to enable user for easily computing the volume and the surface area of three-dimensional mesh objects. In order to use the GUI, user will firstly need to import the Mesh that is generated by the software Netgen Mesh generator. Once any mesh is imported as an input to the GUI, the program will use the concepts in vectors, which includes dot and cross products, to compute both parameters. Apart from presenting briefly the steps in creating the GUI with Matlab, this paper will also provide some examples to demonstrate the use of the GUI.
... Similarly, the numerical examples of the first order PT for translated and rotated objects was also conducted by Khairuddin et al., [25]. Besides, by using both linear and quadratic elements in numerical method for computing the PT, Sukri et al., [26] have investigated how the first order PT associated to a few objects including sphere, ellipsoid and cube are effected after each object is scaled. ...
In order to enhance identification of objects in electrical imaging or metal detection, the polarization tensor is used to characterize the perturbation in electric or electromagnetic field due to the presence of the conducting objects. This is similar as describing the uniform fluid flow that is disturbed after a solid is immersed in the fluid during the study of fluid mechanics. Moreover, in some applications, it is beneficial to determine a spheroid based on the first order polarization tensor in order to understand what is actually represented by the tensor. The spheroid could share similar physical properties with the actual object represented by the polarization tensor. The purpose of this paper is to present how scaling on the matrix for the first order polarization tensor will affect the original spheroid represented by that first order polarization tensor. In the beginning, we revise the mathematical property regarding how scaling the semi axes of a conducting spheroid has an effect to its first order polarization tensor. After that, we give theoretical results with proofs on how scaling the matrix for the first order polarization tensor affects the volume and semi axes of the spheroid. Following that, some numerical examples are provided to further justify the theory. Here, different scalar factors will be used on the given first order polarization tensor before the new volume and semi axes of the spheroid are computed. In addition, we also investigate how the size of the scale on the first order polarization tensor influence the accuracy of computing the related volume and semi axes. In this case, it is found that a large error could occur to the volume and the semi axes when finding them by solving the first order PT with that has being scaled by a very large scaling factor or a too small scaling factor. A suggestion is then given on how to reduce the errors.
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