Quadratic Element Integration of Approximated First Order Polarization Tensor for Sphere

Abstract

Polarization tensor is an object-specific property in order to indicate its shape, size and also the material used. This information is beneficial to determine between used and unused material underground for example. 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 was based on the exact solution provided for sphere and ellipsoid geometry by previous researcher. Moreover, numerical computation showed that the quadratic element integration generates more accurate numerical results for the approximated first order polarization tensor and better in its convergence. The numerical results are illustrated in graphical form in order to show the validity of the proposed scheme.