PIER
 
Progress In Electromagnetics Research
ISSN: 1070-4698, E-ISSN: 1559-8985
Home | Search | Notification | Authors | Submission | PIERS Home | EM Academy
Home > Vol. 117 > pp. 339-355

ON THE ANALYTIC-NUMERIC TREATMENT OF WEAKLY SINGULAR INTEGRALS ON ARBITRARY POLYGONAL DOMAINS

By S. Lopez-Pena, A. G. Polimeridis, and J. R. Mosig

Full Article PDF (525 KB)

Abstract:
An alternative analytical approach to calculate the weakly singular free-space static potential integral associated to uniform sources is presented. Arbitrary oriented flat polygons are considered as integration domains. The technique stands out by its mathematical simplicity and it is based on a novel integral transformation. The presented formula is equivalent to others existing in literature, being also concise and suitable within a singularity subtraction framework. Generalized Cartesian product rules built on the double exponential formula are utilized to integrate numerically the resulting analytical 2D potential integral. As a consequence, drawbacks associated to endpoint singularities in the derivative of the potential are tempered. Numerical examples within a surface integral equation-Method of Moments framework are finally provided.

Citation:
S. Lopez-Pena, A. G. Polimeridis, and J. R. Mosig, "On the Analytic-Numeric Treatment of Weakly Singular Integrals on Arbitrary Polygonal Domains," Progress In Electromagnetics Research, Vol. 117, 339-355, 2011.
doi:10.2528/PIER11050504
http://www.jpier.org/PIER/pier.php?paper=11050504

References:
1. Araujo, M. G., J. M. Taboada, F. Obelleiro, J. M. Bertolo, L. Landesa, J. Rivero, and J. L. Rodriguez, "Supercomputer aware approach for the solution of challenging electromagnetic problems," Progress In Electromagnetics Research, Vol. 101, 241-256, 2010.
doi:10.2528/PIER09121007

2. Taboada, J. M., M. G. Araujo, J. M. Bertolo, L. Landesa, F. Obelleiro, and J. L. Rodriguez, "MLFMA-FFT parallel algorithm for the solution of large-scale problems in electromagnetics,", Vol. 105, 15-30, 2010.
doi:10.2528/PIER10041603

3. ErgÄul, Ä O., T. Malas, and L. GÄurel, "Solutions of large-scale electromagnetics problems using an iterative inner-outer scheme with ordinary and approximate multilevel fast multipole algorithms," Progress In Electromagnetics Research, Vol. 106, 203-223, 2010.
doi:10.2528/PIER10061711

4. Shi, Y., X. Luan, J. Qin, C. J. Lv, and C. H. Liang, "Multilevel Green's function interpolation method solution of volume/surface integral equation for mixed conducting/bi-isotropic objects," Progress In Electromagnetics Research, Vol. 107, 239-252, 2010.
doi:10.2528/PIER10060209

5. Chen, Y., S. Yang, S. He, and Z. Nie, "Fast analysis of microstrip antennas over a frequency band using an accurate MoM matrix interpolation technique," Progress In Electromagnetics Research, Vol. 109, 301-324, 2010.
doi:10.2528/PIER10081107

6. Polimeridis, A. G. and T. V. Yioultsis, "On the direct evaluation of weakly singular integrals in Galerkin mixed potential integral equation formulations," IEEE Trans. Antennas and Propagat., Vol. 56, 3011-3019, 2008.
doi:10.1109/TAP.2008.928782

7. Rossi, L. and P. J. Cullen, "On the fully numerical evaluation of the linear-shape function times the 3-D Green's function on a plane triangle," IEEE Trans. Microw. Theory Tech., Vol. 47, 398-402, 1999.
doi:10.1109/22.754871

8. Khayat, M. A. and D. R. Wilton, "Numerical evaluation of singular and near-singular potential integrals," IEEE Trans. Antennas and Propagat., Vol. 53, 3180-3190, 2005.
doi:10.1109/TAP.2005.856342

9. Graglia, R. D. 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 and Propagat., Vol. 56, 981-998, 2008.
doi:10.1109/TAP.2008.919181

10. Wilton, D. R., S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. AL-Bundak, and C. M. Butler, "Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains," IEEE Trans. Antennas and Propagat., Vol. 32, 276-281, 1984.
doi:10.1109/TAP.1984.1143304

11. triangle, R. D., "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 and Propagat., Vol. 41, 1448-1455, 1993.
doi:10.1109/8.247786

12. Eibert, T. F. and V. Hansen, "On the calculation of potential integrals for linear source distributions on triangular domains," IEEE Trans. Antennas and Propagat., Vol. 43, 1499-1502, 1995.
doi:10.1109/8.475946

13. Arcioni, P., M. Bressan, and L. Perregrini, "On the evaluation of the double surface integrals arising in the application of the boundary integral method to 3-D problems," IEEE Trans. Microw. Theory Tech., Vol. 45, 436-439, 1997.
doi:10.1109/22.563344

14. JÄarvenpÄaÄa, S., M. Taskinen, and P. YlÄa-Oijala, "Singularity subtraction technique for high-order polynomial vector basis functions on planar triangles," IEEE Trans. Antennas and Propagat., Vol. 54, 42-49, 2006.
doi:10.1109/TAP.2005.861556

15. Asvestas, J. S. and H. J. Bilow, "Line-integral approach to computing impedance matrix elements," IEEE Trans. Antennas and Propagat., Vol. 55, 2767-2772.
doi:10.1109/TAP.2007.905815

16. Lopez-Pena, S. and J. R. Mosig, "Analytical evaluation of the quadruple static potential integrals on rectangular domains to solve 3-D electromagnetic problems," IEEE Trans. Magn., Vol. 54, 1320-1323, 2009.
doi:10.1109/TMAG.2009.2012613

17. Harrington, R. F., Field Computation by Moment Methods, Macmillan, New York, FL, Krieger, 1983.

18. Mosig, J. R., "Arbitrarily shaped microstrip structures and their analysis with a mixed potential integral equation," IEEE Trans. Microw. Theory Tech., Vol. 36, 314-323, 1988.
doi:10.1109/22.3520

19. Kolundzija, B. M. and A. R. Djordjevic, Electromagnetic Modeling of Composite Metallic and Dielectric Structures, Artech House, Norwood, MA, USA, 2002.

20. Kolundzija, B. M., M. M. Kostic, B. L. Mrdakovic, and D. S. Sumic, "Comparison of different strategies for conversion of triangular mesh into quadrilateral mesh," EuCAP 2010-The 4th European Conference on Antennas and Propagation, 2010.

21. Adams, T. and J. Singh, "A nonrectangular patch model for scattering from surfaces," IEEE Trans. Antennas and Propagat., Vol. 27, 531-535, 1979.
doi:10.1109/TAP.1979.1142128

22. Tulyathan, P. and E. H. Newman, "A surface patch model for polygonal plates," IEEE Trans. Antennas and Propagat., Vol. 30, 588-593, 1982.
doi:10.1109/TAP.1982.1142841

23. Kolundzija, B. M., "On the locally continuous formulation of surface doublets," IEEE Trans. Antennas and Propagat., Vol. 46, 1879-1883, 1998.
doi:10.1109/8.743838

24. GÄurel, L. and O. Ergul, "Singularity of the magnetic-field integral equation and its extraction," IEEE Antennas Wireless Propag. Lett., Vol. 4, 229-232, 2005.
doi:10.1109/LAWP.2005.851103

25. Gradshteyn, I. S. and I. M. Ryzhik, Tables of Integrals, Series and Products , Academic, New York, 1980.

26. Bailey, D. H., K. Jeyabalan, and X. S. Li, "A comparison of three high-precision quadrature schemes," Experimental Mathematics, Vol. 3, 317-329, 2005.
doi:10.1080/10586458.2005.10128931

27. Polimeridis, A. G. and J. R. Mosig, "Evaluation of weakly singular integrals via generalized Cartesian product rules based on the double exponential formula," IEEE Trans. Antennas and Propagat., Vol. 58, 1980-1988, 2010.
doi:10.1109/TAP.2010.2046866

28. Durand, E., Electrostatique: I. Les Distributions, Masson, Paris, 1964 .

29. Notaros, B. M., "Higher order frequency-domain computational electromagnetics," IEEE Trans. Antennas and Propagat., Vol. 56, 2251-2276, 2008.
doi:10.1109/TAP.2008.926784


© Copyright 2014 EMW Publishing. All Rights Reserved