First order polarization tensor approximation using multivariate polynomial interpolation method via least square minimization technique

Abstract

This paper proposes a new numerical approach useful in dealing with nearly singular integrals, specifically, the integral of the first order polarization tensor (PT). Polarization tensor represents the integral equations in an asymptotic series, and it can also define the boundary value problem of a partial differential equation (PDE). Since PT has been widely used and implemented in many engineering areas, particularly electric and magnetic field areas, it is crucial to estimate the first order PT solutions accurately. In this regard, the computation of PT for different geometry types is basically from the quadratic interpolation and the multivariate polynomial fitting using the least square method. The numerical calculation of the integral of the singular integral operator, ∗ which is one of the primary integral processes before we obtained the solution of PT uses the multivariate polynomial fitting. This paper aims to provide an accurate numerical solution for first order PT for different geometry types, particularly sphere and ellipsoid geometry. The numerical results of the proposed method are shown together with the comparison of its analytical solutions. From the results obtained, the numerical solution of first order PT shows higher accuracy and higher convergence as the number of surface elements increases. The numerical and the analytical solution of first order PT for a sphere is discussed and represented in graphical form. The utilization of two different software types throughout this study is Netgen Mesh Generator and MATLAB to aid the numerical computation process. The simulation and the numerical examples verify the effectiveness and efficiency of the proposed method.