volume | PIER Journals

Abstract

This paper presents a convex meshfree framework for solving the scalar Helmholtz equation in the waveguide analysis of electromagnetic problems. The generalized meshfree approximation (GMF) method using inverse tangent basis functions and cubic spline weight functions is employed to construct the first-order convex approximation which exhibits a weak Kronecker-delta property at the waveguide boundary and allows a direct enforcement of homogenous Dirichlet boundary conditions for the transverse magnetic (TM) mode analyses. Three arbitrary waveguide examples are analyzed to demonstrate the accuracy of the presented formulation, and comparison is made by the analytical, finite element and meshfree solutions.

1. Ihlenburg, F. and I. Babuska, "Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of FEM," Comp. Math. Appl., Vol. 30, No. 9, 9-37, 1995.
doi:10.1016/0898-1221(95)00144-N

2. Harari, I. and T. J. R. Hughes, "Finite element method for the Helmholtz equation in an exterior domain: Model problems," Comp. Meth. Appl. Mech. Eng., Vol. 87, 59-96, 1991.
doi:10.1016/0045-7825(91)90146-W

3. Harari, I. and T. J. R. Hughes, "Galerkin/least squares finite element method for the reduced wave equation with non-reflecting boundary conditions," Comp. Meth. Appl. Mech. Eng., Vol. 92, 441-454, 1992.

4. Babuska, I., F. Ihlenburg, E. Paik, and S. Sauter, "A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution," Comp. Meth. Appl. Mech. Eng., Vol. 128, 325-359, 1995.
doi:10.1016/0045-7825(95)00890-X

5. Farhat, C., I. Harari, and L. P. Franca, "The discontinuous enrichment method," Comp. Meth. Appl. Mech. Eng., Vol. 190, 6455-6479, 2001.
doi:10.1016/S0045-7825(01)00232-8

6. Zienkiewicz, O. C., "Achievements and some unsolved problems of the finite element method," Int. J. Numer. Meth. Engrg., Vol. 47, 9-28, 2000.
doi:10.1002/(SICI)1097-0207(20000110/30)47:1/3<9::AID-NME793>3.0.CO;2-P

7. 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

8. Miao, Y., Y. Wang, and H. Wang, "A meshless hybrid boundary-node method for Helmholtz problems," Eng. Anal. Boundary Elem, Vol. 33, 120-127, 2009.
doi:10.1016/j.enganabound.2008.05.009

9. Correa, B. C., E. J. Silva, A. R. Fonseca, D. B. Oliveria, and R. C. Mesquita, "Meshless local Petrov-Galerkin in solving microwave guide problems," IEEE Trans. Magnetics, Vol. 47, 1526-1529, 2011.
doi:10.1109/TMAG.2010.2091496

10. Belytschko, T., Y. Y. Lu, and L. Gu, "Element-free Galerkin methods," Int. J. Numer. Meth. Engrg., Vol. 37, 229-256, 1994.
doi:10.1002/nme.1620370205

11. Liu, W. K., S. Jun, and Y. F. Zhan, "Reproducing kernel particle methods," Int. J. Numer. Meth. Fluids, Vol. 20, 1081-1106, 1995.
doi:10.1002/fld.1650200824

12. Kansa, E. J., "Multiqudrics --- A scattered data approximation scheme with applications to computational fluid-dynamics --- I: Surface approximations and partial derivatives," Comp. Math. Appl., Vol. 9, 127-145, 1992.

13. Jiang, P. L., S. Q. Li, and C. H. Chan, "Analysis of elliptic waveguides by a meshless collocation method with the Wendland adial basis functions," Microwave and Optical Technology Letters, Vol. 32, No. 2, 162-165, 2002.
doi:10.1002/mop.10119

14. Lai, S. J., 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

15. Kaufmann, T., Y. Yu, C. Engstrom, Z. Chen, and C. Fumeaux, "Recent developments of meshless radial point interpolation method for time-domain electromagnetics," Int. J. Numer. Model: Elect. Networks, Devices and Fields, Vol. 25, 468-489, 2012.
doi:10.1002/jnm.1830

16. Wu, C. T., C. K. Park, and J. S. Chen, "A generalized meshfree approximation for the meshfree analysis of solids," Int. J. Numer. Meth. Engrg., Vol. 85, 693-722, 2011.
doi:10.1002/nme.2991

17. Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in Fortran 90, 2nd Ed., Cambridge University Press, New York, 1996.

18. Harrington, R. F., Time-harmonic Electromagnetic Fields, Wiley-IEEE Press, New York, 2001.
doi:10.1109/9780470546710

19. Marcuvitz, N., Waveguide Handbook, Peter Peregrinus Ltd., London, 1993.

20. Shaw, A. and D. Roy, "NURBS-based parametric mesh-free methods," Comput. Methods Appl. Mech. Engrg., Vol. 197, 1541-1567, 2008.
doi:10.1016/j.cma.2007.11.024

Dr. Li-Fang Wang