Eigenanalysis of Electromagnetic Structures Based on the Finite Element Method

Affiliation(s)

Department of Electrical and Computer Engineering, Microwaves Lab., Democritus University of Thrace, Xanthi, Greece.

Department of Electrical and Computer Engineering, Microwaves Lab., Democritus University of Thrace, Xanthi, Greece.

ABSTRACT

This article presents a review of our research effort on the eigenanalysis of open radiating waveguides and closed resonating structures. A two dimensional (2-D) hybrid Finite Element method in conjunction with a cylindrical harmonics expansion is established to formulate the open waveguide generalized eigenvalue problem. The key element of this approach refers to the adoption of a vector Dirichlet-to-Neumann map to rigorously enforce the continuity of the two field expansions along a truncation surface. The resulting algorithm was able to evaluate both surface and leaky eigenmodes. The eigenanalysis of three dimensional (3-D) structures involves vast research challenges, especially when they are electrically large and open-radiating. The effort herein is focused on the electrically large case including the losses due to the finite conductivity of metallic walls and objects as well as the loading material losses. The former is introduced through impedance or Leontovich boundary condition, resulting to a non-linear-polynomial generalized eigenvalue problem. A straightforward linearization solution is adopted along with a more efficient alternative technique which mimics analytical approaches. For this one the linear eigenproblem formulated assuming metals as perfect electric conductors is initially solved and their finite conductivity is accounted through impedance boundary conditions enforced locally on the resulting eigenvectors. Finally, some numerical results are presented to verify the performance of these methodologies along with a discussion on their possibilities for extension to open 3D structures as well as to characteristic modes eigenanalysis.

This article presents a review of our research effort on the eigenanalysis of open radiating waveguides and closed resonating structures. A two dimensional (2-D) hybrid Finite Element method in conjunction with a cylindrical harmonics expansion is established to formulate the open waveguide generalized eigenvalue problem. The key element of this approach refers to the adoption of a vector Dirichlet-to-Neumann map to rigorously enforce the continuity of the two field expansions along a truncation surface. The resulting algorithm was able to evaluate both surface and leaky eigenmodes. The eigenanalysis of three dimensional (3-D) structures involves vast research challenges, especially when they are electrically large and open-radiating. The effort herein is focused on the electrically large case including the losses due to the finite conductivity of metallic walls and objects as well as the loading material losses. The former is introduced through impedance or Leontovich boundary condition, resulting to a non-linear-polynomial generalized eigenvalue problem. A straightforward linearization solution is adopted along with a more efficient alternative technique which mimics analytical approaches. For this one the linear eigenproblem formulated assuming metals as perfect electric conductors is initially solved and their finite conductivity is accounted through impedance boundary conditions enforced locally on the resulting eigenvectors. Finally, some numerical results are presented to verify the performance of these methodologies along with a discussion on their possibilities for extension to open 3D structures as well as to characteristic modes eigenanalysis.

KEYWORDS

Eigenanalysis; Finite Element Method; Open Radiating Structures; Electrically Large Cavities

Eigenanalysis; Finite Element Method; Open Radiating Structures; Electrically Large Cavities

Cite this paper

C. Zekios, P. Allilomes and G. Kyriacou, "Eigenanalysis of Electromagnetic Structures Based on the Finite Element Method,"*Applied Mathematics*, Vol. 4 No. 7, 2013, pp. 1009-1022. doi: 10.4236/am.2013.47138.

C. Zekios, P. Allilomes and G. Kyriacou, "Eigenanalysis of Electromagnetic Structures Based on the Finite Element Method,"

References

[1] P. C. Allilomes and G. A. Kyriacou, “A Nonlinear Finite —Element Leaky—Waveguide Solver,” IEEE Transactions on MTT, Vol. 55, 2007, pp. 1496-1510. doi:10.1109/TMTT.2007.900306

[2] C. L. Zekios, P. C. Allilomes and G. A. Kyriacou, “Eigenfunction Expansion for the Analysis of Closed Cavities,” 2010 Loughborough Antennas and Propagation Conference, Loughborough, 14-15 November 2010, pp. 537-540. doi:10.1049/el.2012.1852

[3] C. L. Zekios, P. C. Allilomes and G. A. Kyriacou, “On the Evaluation of Eigenmodes Quality Factor of Large Complex Cavities Based on a PEC Linear Finite Element Formulation,” IET Electronics Letters, Vol. 48, No. 22, 2012, pp. 1399-1401.

[4] D. Givoli, “Numerical Methods for Problems in Infinite Domains,” Elsevier, Amsterdam, 1992.

[5] D. J. B. Keller and D. Givoli, “Exact Non-Reflecting Boundary Conditions,” Journal of Computational Physics, Vol. 82, No. 1, 1989, pp. 172-192. doi:10.1016/0021-9991(89)90041-7

[6] J. D. Jackson, “Classical Electrodynamics,” 3rd Edition, Wiley, New York, 1999, p. 431.

[7] D. M. Pozar, “Microwave Engineering,” 2nd Edition, Wiley, New York, 1998, p. 133.

[8] C. Reddy, M. Deshpande, C. Cockrell and F. Beck, “Finite Elements Method for Eigenvalue Problems in Electromagnetics,” Tech. Report 3485, NASA, Langley Research Center, Hampton, 1994.

[9] R. F. Harrington, “Boundary Integral Formulations for Homogeneous Material Bodies,” Journal of Electromagnetic Waves and Applications, Vol. 3, No. 1, 1989, pp. 115. doi:10.1163/156939389X00016

[10] P. Hager, “Eigenfrequency Analysis: FE-Adaptivity and Nonlinear Eigen-Problem Algorithm,” Ph.D. Dissertation, Chalmers University of Technology, Goteborg, 2001.

[11] R. Lehoucq, K. Maschhoff and D. Sorensen, “ARPACK Homepage.” http://www.caam.rice.edu/software/ARPACK/

[12] E. Darve, “The Fast Multipole Method: Numerical Implementation,” Journal of Computational Physics, Vol. 160, No. 1, 2000, pp. 195-240. doi:10.1006/jcph.2000.6451

[13] Y. Zhu and A. C. Cangellaris, “Multigrid Finite Element Methods for Electromagnetic Field Modelling,” Wiley Interscience, New York, 2006.

[14] R. F. Harrington, “Time Harmonic Electromagnetic Fields,” IEEE Press, John Wiley and Sons, Inc., New York, 2001.

[15] Ch. Bruns, “Three Dimensional Simulation and Experimental Verification of a Reverberation Chamber,” Ph.D. Thesis, University of Fridericiana, Karlsruhe, 2005.

[16] K. Chen, “Matrix Preconditioning Techniques and Applications,” Cambridge University Press, Cambridge, 2005. doi:10.1017/CBO9780511543258

[17] Y. Saad, “Iterative Methods for Sparse Linear Systems,” 2nd Edition, 2000.

[18] L. P. Lampariello, F. Frezza, H. Shigesawa, M. Tsuji and A. Oliner, “A Versatile Leaky-Wave Antenna Based on Stub Loaded Rectangular Waveguide: Part III—Comparison with Measurements,” IEEE Transactions on AP, Vol. 46, No. 7, 1998, pp. 1047-1055. doi:10.1109/8.704806

[19] J. L. Gomez-Tornero, F. D. Quesada-Pereira and A. Alvarez-Melcon, “A Full-Wave Space-Domain Method for the Analysis of Leaky-Wave Modes in Multilayered Planar Open Parallel-Plate Waveguides,” International Journal of RF and Microwave Computer-Aided Engineering, Vol. 15, No. 1, 2005, pp. 128-139.

[20] E. Frezza and P. Lampariello, “On the Modal Spectrum of the Channel Waveguide,” International Journal of Infrared and Millimeter Waves, Vol. 16, No. 3, 1995, pp. 591599. doi:10.1007/BF02066884

[21] L. P. Lampariello, F. Frezza, H. Shigesawa, M. Tsuji and A. Oliner, “A Versatile Leaky-Wave Antenna Based on Stub Loaded Rectangular Waveguide: Part III—Comparison with Measurements,” IEEE Transactions on AP, Vol. 46, No. 7, 1998, pp. 1047-1055. doi:10.1109/8.704806

[22] P. Allilomes, “Electromagnetic Simulation of Open-Radiating Structures Based on the Finite Element Method,” Ph.D. Dissertation, Democritus University of Thrace, Greece, 2007.

[1] P. C. Allilomes and G. A. Kyriacou, “A Nonlinear Finite —Element Leaky—Waveguide Solver,” IEEE Transactions on MTT, Vol. 55, 2007, pp. 1496-1510. doi:10.1109/TMTT.2007.900306

[2] C. L. Zekios, P. C. Allilomes and G. A. Kyriacou, “Eigenfunction Expansion for the Analysis of Closed Cavities,” 2010 Loughborough Antennas and Propagation Conference, Loughborough, 14-15 November 2010, pp. 537-540. doi:10.1049/el.2012.1852

[3] C. L. Zekios, P. C. Allilomes and G. A. Kyriacou, “On the Evaluation of Eigenmodes Quality Factor of Large Complex Cavities Based on a PEC Linear Finite Element Formulation,” IET Electronics Letters, Vol. 48, No. 22, 2012, pp. 1399-1401.

[4] D. Givoli, “Numerical Methods for Problems in Infinite Domains,” Elsevier, Amsterdam, 1992.

[5] D. J. B. Keller and D. Givoli, “Exact Non-Reflecting Boundary Conditions,” Journal of Computational Physics, Vol. 82, No. 1, 1989, pp. 172-192. doi:10.1016/0021-9991(89)90041-7

[6] J. D. Jackson, “Classical Electrodynamics,” 3rd Edition, Wiley, New York, 1999, p. 431.

[7] D. M. Pozar, “Microwave Engineering,” 2nd Edition, Wiley, New York, 1998, p. 133.

[8] C. Reddy, M. Deshpande, C. Cockrell and F. Beck, “Finite Elements Method for Eigenvalue Problems in Electromagnetics,” Tech. Report 3485, NASA, Langley Research Center, Hampton, 1994.

[9] R. F. Harrington, “Boundary Integral Formulations for Homogeneous Material Bodies,” Journal of Electromagnetic Waves and Applications, Vol. 3, No. 1, 1989, pp. 115. doi:10.1163/156939389X00016

[10] P. Hager, “Eigenfrequency Analysis: FE-Adaptivity and Nonlinear Eigen-Problem Algorithm,” Ph.D. Dissertation, Chalmers University of Technology, Goteborg, 2001.

[11] R. Lehoucq, K. Maschhoff and D. Sorensen, “ARPACK Homepage.” http://www.caam.rice.edu/software/ARPACK/

[12] E. Darve, “The Fast Multipole Method: Numerical Implementation,” Journal of Computational Physics, Vol. 160, No. 1, 2000, pp. 195-240. doi:10.1006/jcph.2000.6451

[13] Y. Zhu and A. C. Cangellaris, “Multigrid Finite Element Methods for Electromagnetic Field Modelling,” Wiley Interscience, New York, 2006.

[14] R. F. Harrington, “Time Harmonic Electromagnetic Fields,” IEEE Press, John Wiley and Sons, Inc., New York, 2001.

[15] Ch. Bruns, “Three Dimensional Simulation and Experimental Verification of a Reverberation Chamber,” Ph.D. Thesis, University of Fridericiana, Karlsruhe, 2005.

[16] K. Chen, “Matrix Preconditioning Techniques and Applications,” Cambridge University Press, Cambridge, 2005. doi:10.1017/CBO9780511543258

[17] Y. Saad, “Iterative Methods for Sparse Linear Systems,” 2nd Edition, 2000.

[18] L. P. Lampariello, F. Frezza, H. Shigesawa, M. Tsuji and A. Oliner, “A Versatile Leaky-Wave Antenna Based on Stub Loaded Rectangular Waveguide: Part III—Comparison with Measurements,” IEEE Transactions on AP, Vol. 46, No. 7, 1998, pp. 1047-1055. doi:10.1109/8.704806

[19] J. L. Gomez-Tornero, F. D. Quesada-Pereira and A. Alvarez-Melcon, “A Full-Wave Space-Domain Method for the Analysis of Leaky-Wave Modes in Multilayered Planar Open Parallel-Plate Waveguides,” International Journal of RF and Microwave Computer-Aided Engineering, Vol. 15, No. 1, 2005, pp. 128-139.

[20] E. Frezza and P. Lampariello, “On the Modal Spectrum of the Channel Waveguide,” International Journal of Infrared and Millimeter Waves, Vol. 16, No. 3, 1995, pp. 591599. doi:10.1007/BF02066884

[21] L. P. Lampariello, F. Frezza, H. Shigesawa, M. Tsuji and A. Oliner, “A Versatile Leaky-Wave Antenna Based on Stub Loaded Rectangular Waveguide: Part III—Comparison with Measurements,” IEEE Transactions on AP, Vol. 46, No. 7, 1998, pp. 1047-1055. doi:10.1109/8.704806

[22] P. Allilomes, “Electromagnetic Simulation of Open-Radiating Structures Based on the Finite Element Method,” Ph.D. Dissertation, Democritus University of Thrace, Greece, 2007.