# The Effect of Different Scale on Object to the Approximation of the First Order Polarization Tensor of Sphere, Ellipsoid, and Cube

## Authors

• Suzarina Ahmed Sukri Department of Mathematical Science, Faculty of Science, Universiti Teknologi Malaysia (UTM), 81310 Skudai, Johor Bahru
• Taufiq Khairi Ahmad Khairuddin UTM Centre for Industrial and Applied Mathematics (UTM-CIAM), Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia
• Mukhiddin Muminov Mathematical Faculty, Samarkand State University, Samarkand, 140104, Uzbekistan
• Yeak Su Hoe Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia,81310 Johor Bahru, Johor, Malaysia
• Syafina Ahmad Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia,81310 Johor Bahru, Johor, Malaysia

## Keywords:

Integral Equations, Matrices, Conductivity

## 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.

taufiq@utm.my

mukhiddin@utm.my

s.h.yeak@utm.my

## References

Pólya, G. "A minimum problem about the motion of a solid through a fluid." Proceedings of the National Academy of Sciences of the United States of America 33, no. 7 (1947): 218-221. https://doi.org/10.1073/pnas.33.7.218

Ammari, Habib, and Hyeonbae Kang. Polarization and moment tensors: with applications to inverse problems and effective medium theory. Vol. 162. Springer Science & Business Media, 2007.

Ammari, Habib, Hyeonbae Kang, Hyundae Lee, and Mikyoung Lim. "Enhancement of near cloaking using generalized polarization tensors vanishing structures. Part I: The conductivity problem." Communications in Mathematical Physics 317, no. 1 (2013): 253-266. https://doi.org/10.1007/s00220-012-1615-8

Ammari, Habib, Michael S. Vogelius, and Darko Volkov. "Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter II. The full Maxwell equations." Journal de Mathématiques Pures et Appliquées 80, no. 8 (2001): 769-814. https://doi.org/10.1016/S0021-7824(01)01217-X

Ammari, Habib, Junqing Chen, Zhiming Chen, Darko Volkov, and Han Wang. "Detection and classification from electromagnetic induction data." Journal of Computational Physics 301 (2015): 201-217. https://doi.org/10.1016/j.jcp.2015.08.027

Dekdouk, Bachir, Liam A. Marsh, David W. Armitage, and Anthony J. Peyton. "Estimating magnetic polarizability tensor of buried metallic targets for land mine clearance." In UltraWideband, Short-Pulse Electromagnetics 10, pp. 425-432. Springer, New York, NY, 2014. https://doi.org/10.1007/978-1-4614-9500-0_38

Ledger, Paul D., and William RB Lionheart. "Characterizing the shape and material properties of hidden targets from magnetic induction data." The IMA Journal of Applied Mathematics 80, no. 6 (2015): 1776-1798. https://doi.org/10.1093/imamat/hxv015

Ammari, Habib, Junqing Chen, Zhiming Chen, Josselin Garnier, and Darko Volkov. "Target detection and characterization from electromagnetic induction data." Journal de Mathématiques Pures et Appliquées 101, no. 1 (2014): 54-75. https://doi.org/10.1016/j.matpur.2013.05.002

Ammari, Habib, Junqing Chen, Zhiming Chen, Darko Volkov, and Han Wang. "Detection and classification from electromagnetic induction data." Journal of Computational Physics 301 (2015): 201-217. https://doi.org/10.1016/j.jcp.2015.08.027

Yunos, Nurhazirah Mohamad, Taufiq Khairi Ahmad Khairuddin, and William RB Lionheart. "Identification of a Spheroid based on the First Order Polarization Tensor." Journal of Science and Technology 9, no. 3 (2017): 154-159.

Khairuddin, Taufiq K. Ahmad, and William RB Lionheart. "Characterization of objects by electrosensing fish based on the first order polarization tensor." Bioinspiration & Biomimetics 11, no. 5 (2016): 055004. https://doi.org/10.1088/1748-3190/11/5/055004

Khairuddin, Taufiq Khairi Ahmad, and William RB Lionheart. "Computing the first order polarization tensor: welcome BEM++!." Menemui Matematik (Discovering Mathematics) 35, no. 2 (2013): 15-20.

Ledger, Paul D., and WR Bill Lionheart. "Understanding the magnetic polarizability tensor." IEEE Transactions on Magnetics 52, no. 5 (2015): 1-16. https://doi.org/10.1109/TMAG.2015.2507169

Khairuddin, Taufiq Khairi Ahmad, and William Lionheart. "Some properties of the first order polarization tensor for 3-D domains." MATEMATIKA: Malaysian Journal of Industrial and Applied Mathematics 29, no. 1 (2013): 1-18.

Khairuddin, Taufiq K. A., and William R. B. Lionheart. "Numerical comparisons for the approximated first order polarization tensor for ellipsoids." Applied Mathematics and Computational Intelligence 4, no. 1 (2015): 341-354.

Khairuddin, Taufiq Khairi Ahmad, Nurhazirah Mohamad Yunos, and Suzarina Ahmed Sukri. "Characterization of Objects based on the Polarization Tensor: Nature versus Artificial Intelligence." In IOP Conference Series: Materials Science and Engineering, vol. 1051, no. 1, p. 012033. IOP Publishing, 2021. https://doi.org/10.1088/1757-899X/1051/1/012033

Sukri, Suzarina Ahmed, Taufiq Khairi Ahmad Khairuddin, and Yeak Su Hoe. "The Effect of Translation on the Approximated First Order Polarization Tensor of Sphere and Cube." Open Journal of Science and Technology 3, no. 3 (2020): 274-282.

Bahuriddin, Nur Safirah, Mukhiddin Muminov, Taufiq Khairi Ahmad Khairuddin, Syafina Ahmad, and Wan Rohaizad Wan Ibrahim. "An Extended Method for Fitting the First Order Polarization Tensor to a Spheroid." Journal of Advanced Research in Applied Sciences and Engineering Technology 23, no. 1 (2021): 8-17. https://doi.org/10.37934/araset.23.1.817

Kang, Hyeonbae. "Layer potential approaches to interface problems." Inverse problems and imaging, Panoramas et Syntheses, Societe Mathematique de France, to appear (2015).

Schöberl, Joachim. "NETGEN An advancing front 2D/3D-mesh generator based on abstract rules." Computing and Visualization in Science 1, no. 1 (1997): 41-52. https://doi.org/10.1007/s007910050004

Sukri, Suzarina Ahmed, Yeak Su Hoe, and Taufiq Khairi Ahmad Khairuddin. "Quadratic Element Integration of Approximated First Order Polarization Tensor for Sphere." Malaysian Journal of Fundamental and Applied Sciences 16, no. 5 (2020): 560-565.

Huang, Cheng-Hung. "Hypersingular boundary integral equation for axisymmetric elasticity." International Journal for Numerical Methods in Engineering 52, no. 11 (2001): 1337-1354. https://doi.org/10.1002/nme.259

2021-09-10

## How to Cite

Ahmed Sukri, S., Ahmad Khairuddin, T. K. ., Muminov, M., Yeak Su Hoe, & Ahmad, S. (2021). The Effect of Different Scale on Object to the Approximation of the First Order Polarization Tensor of Sphere, Ellipsoid, and Cube. Journal of Advanced Research in Fluid Mechanics and Thermal Sciences, 87(1), 108–117. https://doi.org/10.37934/arfmts.87.1.108117

Articles