A new closed-form formula for calculating a weakly singular static potential integral with linear current source distribution on a triangle

• Autorzy: Grytsko Anna , Słobodzian Piotr

Identyfikatory

DOI

10.24425/bpasts.2022.143645

• Języki publikacji: EN

Abstrakty

EN

The article presents a closed-form formula for solving a weakly singular surface integral with a linear current source distribution associated with the SIE-MoM formulation used for solving electromagnetic (EM) problems. The analytical formula was obtained by transforming the surface integral over a triangular domain into a double integral, and then directly determining formulas for the inner and outer integrals. The solution obtained is marked by high computational efficiency, high accuracy, and very simple implementation. The derived formula, in contrast to the currently available formulas, consists of quantities that have a clear and simple geometric interpretation, related to the geometry of the computational domain.

Słowa kluczowe

EN

static vector potential integral; linear current source distribution; analytic integration; isoparametric transformation; integration over a triangular domain; weakly singular integral

PL

statyczna wektorowa całka potencjalna; liniowy rozkład źródła prądu; integracja analityczna; słabo osobliwe całki; transformacja izoparametryczna; integracja w domenie trójkątnej

• Wydawca: Polska Akademia Nauk, Wydział IV Nauk Technicznych
• Czasopismo: Bulletin of the Polish Academy of Sciences. Technical Sciences
• Rocznik: 2023
• Tom: Vol. 71, nr 1
• Strony: art. no. e143645
• Opis fizyczny: Bibliogr. 26 poz., rys., tab.

Twórcy

autor

Grytsko Anna

• Wroclaw University of Science and Technology, Faculty of Information and Communication Technology, Wybrzeze Wyspiańskiego 27, 50-370 Wrocław, Poland

• https://orcid.org/0000-0002-1107-6575

autor

Słobodzian Piotr

• Wroclaw University of Science and Technology, Faculty of Information and Communication Technology, Wybrzeze Wyspiańskiego 27, 50-370 Wrocław, Poland

• https://orcid.org/0000-0003-3545-9228

Bibliografia

• [1] G. Jaworski, A. Francik, and K. Nowak, “TEm,1 coaxial modes generator for cold-testing of high power components and devices,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 70, no. 2, p. e140467, 2022, doi: 10.24425/bpasts.2022.140467.
• [2] L. Baranowski, “Effect of the mathematical model and integration step on the accuracy of the results of computation of artillery projectile flight parameters,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 61, no. 2, pp. 475–484, 2013, doi: 10.2478/bpasts-2013-0047.
• [3] M.C. Pak, K.I. Kim, H.C. Pak, and K.R. Hong, “Influence of geometric structure, convection, and eddy on sound propagation in acoustic metamaterials with turbulent flow,” Arch. Acoust., vol. 46, no. 4, pp. 637–647, 2021, doi: 10.24425/aoa.2021.139640.
• [4] R.D. Graglia and G. Lombardi, “Machine precision evaluation of singular and nearly singular potential integrals by use of Gauss quadrature formulas for rational functions,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 981–998, Apr. 2008, doi: 10.1109/TAP.2008.919181.
• [5] A. Herschlein, J.V. Hagen, and W. Wiesbeck, “Methods for the Evaluation of Regular, Weakly Singular and Strongly Singular Surface Reaction Integrals Arising in Method of Moments,” Appl. Comput. Electromagn. Soc. Newsl., vol. 17, no. 1, pp. 63–73, Mar. 2002.
• [6] M.A. Khayat and D.R. Wilton, “Numerical evaluation of singular and near-singular potential integrals,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3180–3190, Oct. 2005, doi: 10.1109/TAP.2005.856342.
• [7] A.G. Polimeridis and J.R. Mosig, “Complete semi-analytical treatment of weakly singular integrals on planar triangles via the direct evaluation method,” Int. J. Numer. Methods Eng., vol. 83, no. 12, pp. 1625–1650, Sep. 2010, doi: 10.1002/NME.2877.
• [8] D.J. Taylor, “Accurate and efficient numerical integration of weakly singular integrals in Galerkin EFIE solutions,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1630–1637, Jul. 2003, doi: 10.1109/TAP.2003.813623.
• [9] D.R. Wilton et al., “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag., vol. 32, no. 3, pp. 276–281, Mar. 1984, doi: 10.1109/TAP.1984.1143304.
• [10] R.D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green’s function or its gradient on a plane triangle,” IEEE Trans. Antennas Propag., vol. 41, no. 10, pp. 1448–1455, Oct.1993, doi: 10.1109/8.247786.
• [11] S. Järvenpää, M. Taskinen, and P. Ylä-Oijala, “Singularity subtraction technique for high-order polynomial vector basis functions on planar triangles,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 42–49, Jan. 2006, doi: 10.1109/TAP.2005.861556.
• [12] I. Hänninen, M. Taskinen, and J. Sarvas, “Singularity subtraction integral formulae for surface integral equations with RWG, rooftop and hybrid basis functions,” Prog. Electromagn. Res. PIER, vol. 63, pp. 243–278, 2006. doi: 10.2528/PIER06051901.
• [13] A.G. Polimeridis and T.V. Yioultsis, “On the direct evaluation of weakly singular integrals in Galerkin mixed potential integral equation formulations,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 3011–3019, Sep. 2008, doi: 10.1109/TAP.2008.928782.
• [14] A.G. Polimeridis, F. Vipiana, J.R. Mosig, and D.R. Wilton, “DIRECTFN: Fully numerical algorithms for high precision computation of singular integrals in Galerkin SIE methods,” IEEE Trans. Antennas Propag., vol. 61, no. 6, pp. 3112–3122, Feb. 2013, doi: 10.1109/TAP.2013.2246854.
• [15] S. Caorsi, D. Moreno, and F. Sidoti, “Theoretical and numerical treatment of surface integrals involving the free-space Green’s function,” IEEE Trans. Antennas Propag., vol. 41, no. 9, pp. 1296–1301, Sep. 1993, doi: 10.1109/8.247757.
• [16] M.S. Tong and W.C. Chew, “A novel approach for evaluating hypersingular and strongly singular surface integrals in electromagnetics,” IEEE Trans. Antennas Propag., vol. 58, no. 11, pp. 3593–3601, Nov. 2010, doi: 10.1109/TAP.2010.2071370.
• [17] T.F. Eibert and V. Hansen, “On the calculation of potential integrals for linear source distributions on triangular domains,” IEEE Trans. Antennas Propag., vol. 43, no. 12, pp. 1499–1502, Dec. 1995, doi: 10.1109/8.475946.
• [18] C. Balanis, Advanced Engineering Electromagnetics. New York, Chichester, Brisbane, Toronto, Singapore: Wiley, 1989, pp. 282–283.
• [19] P.M. Słobodzian, Electromagnetic Analysis of Shielde Microwave Structures. The Surface Integral Equation Approach. Wrocław: Oficyna Wydawnicza Politechniki Wrocławskiej, 2007, pp. 135–136.
• [20] A.A. Kucharski, Analiza zagadnień promieniowania i rozpraszania fal elektromagnetycznych za pomocą równań całkowych. Wrocław: Oficyna Wydawnicza Politechniki Wrocławskiej, 2012, pp. 39, 67.
• [21] S.M. Rao, D.R. Wilton, and A.W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 409–418, May 1982, doi: 10.1109/TAP.1982.1142818.
• [22] R.F. Harrington, Field computation by moment methods. New York: Wiley, Sponsor: IEEE Antennas and Propagation Society, 1993, pp. 5–6.
• [23] J.M. Jin, “The reference elements”, in The Finite Element Method in Electromagnetics, 2nd ed., New York: Wiley, 2002, pp. 47–66.
• [24] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington: U.S. Government Printing Office, 1964.
• [25] I.S. Gradshteyn and I.M. Ryzhik, Table of contents for Table of Integrals, Series, and Products. 6th ed., New York: Academic Press, 2000, doi: 10.1016/B978-0-12-294757-5.X5000-4.
• [26] R.P. Brent and P.Zimmermann, “Integer arithmetic,” in Modern Computer Arithmetic. Cambridge University Press, 2011, doi: 10.1017/CBO9780511921698.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).