The finite element method in three dimensions

David B. Davidson

doi:10.1017/CBO9780511778117.015

11 - The finite element method in three dimensions

Published online by Cambridge University Press: 05 July 2014

David B. Davidson

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David B. Davidson: Affiliation:
University of Stellenbosch, South Africa

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Summary

In this penultimate chapter, the preceding discussion of the finite element method is extended to three dimensions. We start by extending the eigenanalysis of the preceding chapter to three dimensions, using Whitney elements. Although a 3D FEM code is non-trivial to implement, simple problems can nonetheless be addressed, and the eigenanalysis of a PEC cavity is used to illustrate the development of a 3D FEM code. Higher-order vector elements have already been introduced in the preceding chapter for two-dimensional analysis; in this chapter, a more detailed theoretical treatment will be presented for the more general three-dimensional case, and an LT/QN element is applied to the cavity eigenanalysis problem. Following this, the stationary functional formulation for deterministic (driven) problems will be outlined. In this and the preceding chapter, eigenvalue problems have been used to illustrate the FEM; in this chapter, a deterministic three-dimensional problem will also be discussed, namely the analysis of waveguide obstacles. Finite element analysis is ideal for this problem, and good results have been obtained by a number of workers. Results for two waveguide problems computed using FEM codes incorporating higher-order elements will be shown. The chapter concludes with a discussion on the use of absorbing boundary conditions for open-region problems, and a first-order ABC is presented. Treatment of a more sophisticated scheme is deferred to the final chapter.

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Type: Chapter
Information: Computational Electromagnetics for RF and Microwave Engineering , pp. 407 - 450

DOI: https://doi.org/10.1017/CBO9780511778117.015 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2010

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References

[1] J. S., Savage and A. F., Peterson, “Higher-order vector finite elements for tetrahedral cells,” IEEE Trans. Microwave Theory Tech., 44, 874–879, June 1996.Google Scholar

[2] J.-F., Lee and R., Mittra, “A note on the application of edge-elements for modeling three-dimensional inhomogeneously-filled cavities,” IEEE Trans. Microwave Theory Tech., 40, 1767–1773, September 1992.Google Scholar

[3] P. P., Silvester and R. L., Ferrari, Finite Elements for Electrical Engineers. Cambridge: Cambridge University Press, 2nd edn., 1990.Google Scholar

[4] J., Volakis, A., Chatterjee and L., Kempel, Finite Element Method for electromagnetics: Antennas, Microwave Cicuits and Scattering Applications. Oxford & New York: Oxford University Press and IEEE Press, 1998.Google Scholar

[5] A., Awadhiya, P., Barba and L., Kempel, “Finite element method programming made easy???,” IEEE Antennas Propag. Soc. Mag., 45, 73–79, August 2003.Google Scholar

[6] S., Ramo, J. R., Whinnery and T., van Duzer, Fields and Waves in Communication Electronics. Chichester: Wiley, 3rd edn., 1994.Google Scholar

[7] D. M., Pozar, Microwave Engineering. New York: Wiley, 2nd edn., 1998.Google Scholar

[8] P. P., Silvester and R. L., Ferrari, Finite Elements for Electrical Engineers. Cambridge: Cambridge University Press, 3rd edn., 1996.Google Scholar

[9] J. P., Webb, “Edge elements and what they can do for you,” IEEE Trans. Magn., 29, 1460–1465, March 1993.Google Scholar

[10] J. P., Webb, “Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements,” IEEE Antennas Propagat., 47, 1244–1253, August 1999.Google Scholar

[11] D. B., Davidson, “An evaluation of mixed-order versus full-order vector finite elements,” IEEE Trans. Antennas Propagat., 51, 2430–2441, September 2003.Google Scholar

[12] R. D., Graglia, D. R., Wilton and A. F., Peterson, “Higher order interpolatory vector bases for computational electromagnetics,” IEEE Trans. Antennas Propagat., 45, 329–342, March 1997.Google Scholar

[13] J. S., Savage, “Comparing high order vector basis functions,” in Proceedings of the 14th Annual Review of Progress in Applied Computational Electromagnetics, Monterey, CA, pp. 742-749, March 1998.Google Scholar

[14] J. P., Webb and B., Forghani, “Hierarchal scalar and vector tetrahedra,” IEEE Trans. Magn., 29, 1495–1498, March 1993.Google Scholar

[15] L. S., Andersen and J. L., Volakis, “Hierarchical tangential vector finite elements for tetrahe-dra,” IEEE Microwave Guided Wave Lett., 8, 127–129, March 1998.Google Scholar

[16] D., Sun, J., Lee and Z., Cendes, “Construction of nearly orthogonal Nedelec bases for rapid convergence with multilevel preconditioned solvers,” Siam. J. Sci. Comput., 23(4), 1053–1076, 2001.Google Scholar

[17] P., Ingelström, “A new set of H(curl)-conforming hierarchical basis functions for tetrahedral meshes,” IEEE Trans. Microwave Theory Tech., 54, 106–114, January 2006.Google Scholar

[18] M. M., Botha, “Fully hierarchical divergence-conforming basis functions on tetrahedral cells, with applications,” Int. J. Numer. Meth. Eng., 71, 127–148, 2007.Google Scholar

[19] A. F., Peterson and R. D., Graglia, “Scale factors and matrix conditioning associated with triangular-cell hierarchical vector basis functions,” IEEE Antennas Wireless Propag. Lett., 9, 40–43, 2010.Google Scholar

[20] J. C., Nedelec, “Mixed finite elements in 9t3,” Numerische Mathematik, 35, 315–341, 1980.Google Scholar

[21] P., Keast, “Moderate-degree tetrahedral quadrature formulas,” Computer Meth. Appl. Mechan. Eng., 55(3), 339-348, 1986.Google Scholar

[22] J. P., Webb, “Matching a given field using hierarchal vector basis functions,” Electromagnetics, 24, 113–122, January-March 2004.Google Scholar

[23] L. E. R., Petersson and J., Jin, “An efficient procedure for the projection of a given field onto hierarchical vector basis functions of arbitrary order,” Electromagnetics, 25, 81–91, 2005.Google Scholar

[24] M. M., Botha, Efficient finite element electromagnetic analysis of antennas and microwave devices: the FE-BI-FMM formulation and a posteriori error estimation for p adaptive analysis. PhD thesis, Dept. Electrical & Electronic Engineering, University of Stellenbosch, 2002.Google Scholar

[25] N., Marcuvitz, Waveguide Handbook. London: Peter Peregrinus, on behalf of IEE, 1986. Originally published 1951.Google Scholar

[26] J.-M., Jin, The Finite Element Method in Electromagnetics. New York: Wiley, 2nd edn., 2002.Google Scholar

[27] D. B., Davidson, “Higher-order (LT/QN) vector finite elements for waveguide analysis,” Special Issue on Approaches to Better Accuracy/Resolution in Computational Electromagnetics, Appl. Comput. Electromagn. Soc. J., 17, 1–10, March 2002.Google Scholar

[28] J. P., Webb, “P-adaptive methods for electromagnetic wave problems using hierarchal tetra-hedral edge elements,” Electromagnetics, 22, 443–151, July 2002.Google Scholar

[29] A., Bayliss and E., Turkel, “Radiation boundary conditions for wave-like equations,” Commun. Pure Appl. Mathem., 33, 707–725, 1980.Google Scholar

[30] A. F., Peterson, “Absorbing boundary conditions for the vector wave equation,” Microwave Optical Technol. Lett., 1, 62–64, April 1988.Google Scholar

[31] J. P., Webb and V. N., Kanellopoulos, “Absorbing boundary conditions for the finite element solution of the vector wave equation,” Microwave Optical Technol. Lett., 2, 370–372, October 1989.Google Scholar

[32] B., Stupfel and M., Mognot, “Implementation and derivation of conformal absorbing boundary conditions for the vector wave equation,” J. Electromagn. Waves Appl., 12, 1653–1677, 1998.Google Scholar

[33] M. M., Botha and D. B., Davidson, “Rigorous, auxiliary variable-based implementation of a second-order ABC for the vector FEM,” IEEE Trans. Antennas Propagat., 54, 3499–3504, November 2006.Google Scholar

[34] F. L., Teixeira and W. C., Chew, “PML-FDTD in cylindrical and spherical grids,” IEEE Microwave Guided Wave Lett., 7(9), 285-287, 1997.Google Scholar

[35] C. W., Teixeira, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett., 7(11), 371-373, 1997.Google Scholar

[36] P. G., Petropoulos, “Reflectionless sponge layers as absorbing boundary conditions for the numerical solution of Maxwell equations in rectangular, cylindrical, and spherical coordinates,” SIAMJ. Appl. Mathem., 60(3), 1037-1058, 2000.Google Scholar

[37] D. B., Davidson and M. M., Botha, “Evaluation of a spherical PML for vector FEM applications,” IEEE Trans. Antennas Propagat., 55, 494–498, Feb 2007.Google Scholar

[38] M., Salazar-Palma, T. K., Sarkar, L. E., García-Castillo, T., Roy and A., Djordjević, Iterative and Self-Adaptive Finite-Elements in Electromagnetic Modelling. Boston, MA: Artech House, 1998.Google Scholar

[39] T. V., Yioultsis and T. D. Tsiboukis, “Development and implementation of second and third order vector finite elements in various 3-D electromagnetic field problems,” IEEE Trans. Magn., 33, 1812–1815, March 1997.Google Scholar

[40] R., Hiptmair, “Canonical construction of finite elements,” Mathem. Comput., 68, 1325–1346, May 1999.Google Scholar

[41] K., Ise, K., Inoue and M., Koshiba, “Three-dimensional finite-element method with edge elements for electromagnetic waveguide discontinuities,” IEEE Trans. Microwave Theory Tech., 39, 1289–1295, August 1991.Google Scholar

[42] J. P., Webb, “Finite element methods for junctions of microwave and optical waveguides,” IEEE Trans. Magn., 26, 1754–1758, September 1990.Google Scholar

[43] Ü., Pekel and R., Lee, “An a posteriori error reduction scheme for the three-dimensional finite-element solution of Maxwell's equations,” IEEE Trans. Microwave Theory Tech., 43, 421–427, February 1995.Google Scholar

[44] W. R., Scott, “Accurate modelling of axisymmetric two-port junctions in coaxial lines using the finite element method,” IEEE Trans. Microwave Theory Tech., 40, 1712–1716, August 1992.Google Scholar

[45] R. L., Ferrari, “An extended Huygens' principle for modeling scattering from general discontinuities within hollow waveguides,” Int. J. Numer. Modelling: Electronic Networks, Devices Fields, 14(5), 411-422, 2002.Google Scholar

[46] R. H., Geschke, R. L., Ferrari, D. B., Davidson and P., Meyer, “Application of extended Huygens' principle to scattering discontinuities in waveguide,” in IEEE AFRICON-02 Proceedings, pp. 555-558, October 2002.

[47] R. H., Geschke, R. L., Ferrari, D., Davidson and P., Meyer, “The solution of waveguide scattering problems by application of an extended Huygens formulation,” IEEE Trans. Microwave Theory Tech., 54(10), 3698-3705, 2006.Google Scholar

[48] A. F., Peterson and D. R., Wilton, “Curl-conforming mixed-order edge elements for discretizing the 2D and 3D vector Helmholtz equation,” in Finite Element Software for Microwave Engineering (T., Itoh, G., Pelosi and P. P., Silvester, eds.), Chapter 5. New York: Wiley, 1996.Google Scholar

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