PIER B
 
Progress In Electromagnetics Research B
ISSN: 1937-6472
Home | Search | Notification | Authors | Submission | PIERS Home | EM Academy
Home > Vol. 24 > pp. 351-367

SOLVING HELMHOLTZ EQUATION BY MESHLESS RADIAL BASIS FUNCTIONS METHOD

By S.-J. Lai, B.-Z. Wang, and Y. Duan

Full Article PDF (537 KB)

Abstract:
In this paper, we propose a brief and general process to compute the eigenvalue of arbitrary waveguides using meshless method based on radial basis functions (MLM-RBF) interpolation. The main idea is that RBF basis functions are used in a point matching method to solve the Helmholtz equation only in Cartesian system. Two kinds of boundary conditions of waveguide problems are also anlyzed. To verify the e┬▒ciency and accuracy of the present method, three typical waveguide problems are analyzed. It is found that the results of various waveguides can be accurately determined by MLM-RBF.

Citation:
S.-J. Lai, B.-Z. Wang, and Y. Duan, "Solving Helmholtz Equation by Meshless Radial Basis Functions Method," Progress In Electromagnetics Research B, Vol. 24, 351-367, 2010.
doi:10.2528/PIERB10062303

References:
1. Arlett, P. L., A. K. Bahrani, and O. C. Zienkiewicz, "Application of finite element to the solution of Helmholtzs equation," Proc. IEE., Vol. 115, No. 12, 1762-1766, 1968.

2. Chang, H. W., Y. H. Wu, S. M. Lu, W. C. Cheng, and M. H. Sheng, "Field analysis of dielectric waveguide devices based on coupled transverse-mode integral equation --- Numerical investigation," Progress In Electromagnetics Research, Vol. 97, 159-176, 2009.
doi:10.2528/PIER09091402

3. Khalaj-Amirhosseini, M., "Analysis of longitudinally inhomogeneous waveguides using the method of moments," Progress In Electromagnetics Research, Vol. 74, 57-67, 2007.
doi:10.2528/PIER07042101

4. Lavranos, C. S. and G. A. Kyriacou, "Eigenvalue analysis of curved open waveguides using a finite difference frequency domain method employing orthogonal curvilinear coordinates," PIERS Online, Vol. 1, No. 3, 271-275, 2005.
doi:10.2529/PIERS041210160317

5. Seydou, F., T. Seppanen, and O. M. Ramahi, "Computation of the Helmholtz eigenvalues in a class of chaotic cavities using the multipole expansion technique," IEEE Transactions on Antennas and Propagation, Vol. 57, No. 4, 1169-1177, 2009.
doi:10.1109/TAP.2009.2015801

6. Reutskiy, S. Y., "The methods of external excitation for analysis of arbitrarily-shaped hollow conducting waveguides," Progress In Electromagnetics Research,, Vol. 82, 203-226, 2008.
doi:10.2528/PIER08022701

7. Miao, Y., Y. Wang, and Y. H. Wang, "A meshless hybrid boundary-node method for Helmholtz problems," Eng. Anal. Boundary Elem., Vol. 33, No. 2, 120-127, 2009.
doi:10.1016/j.enganabound.2008.05.009

8. Hon, Y. C. and W. Chen, "Boudary knot method for 2D and 3D helmholtz and convection-diffusion problems under complicated geometry," Int. J. Numer. Methods Eng., Vol. 56, No. 13, 1931-1948, 2003.
doi:10.1002/nme.642

9. Christina, W. and O. Von Estorff, "Dispersion analysis of the meshfree radial point interpolation method for the Helmholtz equation," Int. J. Numer. Methods Eng., Vol. 77, No. 12, 1670-1689, 2009.
doi:10.1002/nme.2463

10. Shu, C., W. X. Wu, and C. M. Wang, "Analysis of metallic waveguides by using least square-based finite difference method," Comput. Mater. Continua., Vol. 2, No. 3, 189-200, 2005.

11. Ooi, B. L. and G. Zhao, "Element-free method for the analysis of partially-filled dielectric waveguides," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 2, 189-198, 2007.
doi:10.1163/156939307779378772

12. Liu, X. F., B. Z. Wang, and S. J. Lai, "Element-free Galerkin method for transient electromagnetic field simulation," Microw. Opt. Techn. Let., Vol. 50, No. 1, 134-138, 2008.
doi:10.1002/mop.23017

13. Kansa, E. J., "Multiqudrics --- A scattered data approximation scheme with applications to computational fluid-dynamics --- I: Surface approximations and partial derivatives," Computer Math. Applic., Vol. 9, No. 8--9, 127-145, 1992.

14. Zhao, G., B. L. Ooi, Y. J. Fan, Y. Q. Zhang, I. Ang, and Y. Gao, "Application of conformal meshless RBF coupled with coordinate transformation for arbitrary waveguide analysis," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 1, 3-14, 2007.
doi:10.1163/156939307779391731

15. Jiang, P. L., S. Q. Li, and C. H. Chuan, "Analysis of elliptical waveguides by a meshless collocation method with the Wendland radial basis funcitons," Microwave Opt. Technol. Lett., Vol. 32, No. 2, 162-165, 2002.
doi:10.1002/mop.10119

16. Yu, J. H. and H. Q. Zhang, "Solving waveguide eigenvalue problem by using radial basis function method," World Autom. Congr. WAC, Vol. 1, 1-5, 2008.

17. Lai, S. J., B. Z. Wang, and Y. Duan, "Application of the RBF-based meshless method to solve 2-D time domain maxwells equations," Int. Conf. Microw. Millimeter Wave Technol. Proc. ICMMT, Vol. 2, 749-751, 2008.

18. Lai, S. J., B. Z. Wang, and Y. Duan, "Meshless radial basis function method for transient electromagnetic computations," IEEE Trans. Magn., Vol. 44, No. 10, 2288-2295, 2008.
doi:10.1109/TMAG.2008.2001796

19. Fornberg, B., T. A. Driscoll, G. Wright, and R. Charles, "Observations on the behavior of radial basis function approximation near boundaries ," Comput. Math. Appl., Vol. 43, No. 3, 473-490, 2002.
doi:10.1016/S0898-1221(01)00299-1

20. Wu, Z. M., "Compactly supported positive definite radial functions," Adv. Comput. Math., Vol. 4, 283-292, 1995.
doi:10.1007/BF03177517

21. Zhang, K. Q. and D. J. Li, Electromagnetic Theory for Microwaves and Optoelectronics, No. 4, Springer Press, New York, 2008.

22. Zhang, S. J. and Y. C. Shen, "Eigenmode sequence for an elliptical waveguide with arbitrary ellipticity," IEEE Trans. MTT., Vol. 43, No. 1, 221-224, 1995.

23. Lin, W. G., Microwave Theory and Techniques, 158-162, Science Press, Beijing, 1979 (in Chinese).


© Copyright 2010 EMW Publishing. All Rights Reserved