Study of different order of Gaussian Quadrature using Linear Element Interpolation in First Order Polarization Tensor

Abstract

Polarization tensor (PT) has been widely used in engineering application, particularly in electric and magnetic field areas. In this case, suitable method must be employed in the evaluation of PT in order to make sure that the tensor obtained is higher in its accuracy. Our aim in this paper is to provide simple and easy implemented method to compute first order PT which is based on Gaussian quadrature numerical integration involving linear interpolation. This study provides the comparison between two different orders of Gaussian quadrature, which is order one and order three. The numerical technique of higher order Gaussian quadrature as PT being calculated offer more accurate and higher in its convergence. Relative error for PT is calculated by using provided analytical solution and higher order Gaussian quadrature gives high accuracy and convergence compared to lower order Gaussian quadrature. In this study, the validation of the results obtained by using our proposed method is provided by comparing it with the analytical solution derived from previous researcher. We illustrate the behavior of a tensor of a sphere and ellipsoid with graphical representation.