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


  • Suzarina Ahmed Sukri
  • Yeak Su Hoe
  • Taufiq Khairi Ahmad Khairuddin



Polarization Tensor, Boundary Integral Equation, Gaussian Quadrature, Numerical Integration


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.


Abdel-rehim OA, Davidson JL, Marsh LA, Toole MDO, Peyton AJ. Magnetic Polarizability Tensor Spectroscopy for Low Metal Anti-Personnel Mine Surrogates. IEEE Sens J 2016;16:3775–83.

Polya G, Szego G. Isoperimetric inequalities in mathematical physics. B Am MathSoc 1953;59:588–602.

Ammari H, Kang H. Polarization and Moment Tensors. vol. 162. New York, NY: Springer New York; 2007.

Ammari H, Kang H. Properties of the generalized polarization tensors. Multiscale Model Simul 2003;1:335–48.

Ammari H, Kang H, Lee H, Lim M. Enhancement of Near Cloaking Using Generalized Polarization Tensors Vanishing Structures . Part I : The Conductivity Problem 2013;266:253–66.

Khairuddin TKA, Lionheart WRB. Characterization of objects by electrosensing fish based on the first order polarization tensor. Bioinspir Biomim 2016;11:55004.

Marsh LA, Ktistis C, Järvi A, Armitage DW, Peyton AJ. Determination of the magnetic polarizability tensor and three dimensional object location for multiple objects using a walk-through metal detector. Meas Sci Technol 2014;25.

Makkonen J, Marsh LA, Vihonen J, Järvi A, Armitage DW, Visa A, et al. KNN classification of metallic targets using the magnetic polarizability tensor. Meas Sci Technol 2014;25:055105.

Marsh LA, Ktistis C, Järvi A, Armitage DW, Peyton AJ. Three-dimensional object location and inversion of the magnetic polarizability tensor at a single frequency using a walk-through metal detector. Meas Sci Technol 2013;24:045102.

Dekdouk B, Marsh LA, Armitage DW, Peyton AJ. Estimating Magnetic Polarizability Tensor of Buried Metallic Targets for Land Mine Clearance. Springer 2014:425–32.

Dekdouk B, Ktistis C, Marsh LA, Armitage DW, Peyton AJ. Towards metal detection and identification for humanitarian demining using magnetic polarizability tensor spectroscopy. Meas Sci Technol 2015;26:115501.

Khairuddin TKA, Lionheart WRB. Characterization of objects by electrosensing fish based on the first order polarization tensor. Bioinspiration and Biomimetics 2016;11:1–8.

Von Der Emde G, Fetz S. Distance, shape and more: Recognition of object features during active electrolocation in a weakly electric fish. J Exp Biol 2007;210:3082–95.

Khairuddin TKA, Lionheart WRB. Does electro-sensing fish use the first order polarization tensor for object characterization? Object discrimination test. Sains Malaysiana 2014;43:1775–9.

Ammari H, Boulier T, Garnier J, Wang H. Mathematical modelling of the electric sense of fish: The role of multi-frequency measurements and movement. Bioinspiration and Biomimetics 2017;12:1–16.

Khairuddin TKA, Lionheart WRB. Polarization Tensor : Between Biology and Engineering. Malaysian J Math Sci 2016;10:179–91.

Ammari H, Boulier T, Garnier J, Wang H. Shape recognition and classification in electro-sensing. Proc Natl Acad Sci 2014;111:11652–11657.

Lu M, Zhao Q, Hu P, Yin W, Peyton AJ. Prediction of the asymptotical magnetic polarization tensors for cylindrical samples using the boundary element method. SAS 2015 - 2015 IEEE Sensors Appl Symp Proc 2015:1–4.

Khairuddin TKA, Lionheart WRB. Computing the First Order Polarization Tensor : Welcome BEM ++ ! Menemui Mat 2013;35:15–20.

Capdeboscq Y, Karrman AB, Nedelec JC. Numerical computation of approximate generalized polarization tensors. Appl Anal 2012;91:1189–203.

Sukri SA, Hoe YS, Khairuddin TKA. Quadratic Element Integration of Approximated First Order Polarization Tensor for Sphere . 2020;16:560–5.

Mohamad Yunos N, Ahmad Khairuddin TK, Aziz ZA, Ahmad T, Lionheart WRB. Identification of a Spheroid based on the First Order Polarization Tensor. J Phys Conf Ser 2017;890:154–9.

Khairuddin TKA, Yunos NM, Lionheart WRB. Classification of Material and Type of Ellipsoid based on the First Order Polarization Tensor. J Phys Conf Ser 2018;1123:0–8.

Ammari H, Kang H, Kim K. Polarization tensors and effective properties of anisotropic composite materials 2005;215:401–28.

Adler A, Gaburro R, Lionheart W. Handbook of mathematical methods in imaging: Volume 1, second edition. 2015.

Holder D. Electrical Impedance Tomography: Methods, History and Applications. Med Phys 2005;32:2731–2731.

Schoberl J. NETGEN-4.X. RWTH Aachen Univ Ger 2009:39.