Book contents
- Frontmatter
- Dedication
- Contents
- Preface to the second edition
- Preface to the first edition
- Acknowledgements
- To the reader
- List of notation
- 1 An overview of computational electromagnetics for RF and microwave applications
- 2 The finite difference time domain method: a one-dimensional introduction
- 3 The finite difference time domain method in two and three dimensions
- 4 A one-dimensional introduction to the method of moments: modelling thin wires and infinite cylinders
- 5 The application of the FEKO and NEC-2 codes to thin-wire antenna modelling
- 6 The method of moments for surface modelling
- 7 The method of moments and stratified media: theory
- 8 The method of moments and stratified media: practical applications of a commercial code
- 9 A one-dimensional introduction to the finite element method
- 10 The finite element method in two dimensions: scalar and vector elements
- 11 The finite element method in three dimensions
- 12 A selection of more advanced topics in full-wave computational electromagnetics
- Appendix A The Whitney element
- Appendix B The Newmark-β time-stepping algorithm References
- Appendix C On the convergence of the MoM Reference
- Appendix D Useful formulas for simplex coordinates
- Appendix E Web resources
- Appendix F MATLAB files supporting this text
- Index
- References
11 - The finite element method in three dimensions
Published online by Cambridge University Press: 05 July 2014
Book contents
- Frontmatter
- Dedication
- Contents
- Preface to the second edition
- Preface to the first edition
- Acknowledgements
- To the reader
- List of notation
- 1 An overview of computational electromagnetics for RF and microwave applications
- 2 The finite difference time domain method: a one-dimensional introduction
- 3 The finite difference time domain method in two and three dimensions
- 4 A one-dimensional introduction to the method of moments: modelling thin wires and infinite cylinders
- 5 The application of the FEKO and NEC-2 codes to thin-wire antenna modelling
- 6 The method of moments for surface modelling
- 7 The method of moments and stratified media: theory
- 8 The method of moments and stratified media: practical applications of a commercial code
- 9 A one-dimensional introduction to the finite element method
- 10 The finite element method in two dimensions: scalar and vector elements
- 11 The finite element method in three dimensions
- 12 A selection of more advanced topics in full-wave computational electromagnetics
- Appendix A The Whitney element
- Appendix B The Newmark-β time-stepping algorithm References
- Appendix C On the convergence of the MoM Reference
- Appendix D Useful formulas for simplex coordinates
- Appendix E Web resources
- Appendix F MATLAB files supporting this text
- Index
- References
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.
- Type
- Chapter
- Information
- Computational Electromagnetics for RF and Microwave Engineering , pp. 407 - 450Publisher: Cambridge University PressPrint publication year: 2010
References
[1] and , “Higher-order vector finite elements for tetrahedral cells,” IEEE Trans. Microwave Theory Tech., 44, 874–879, June 1996.Google Scholar
[2] and , “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] and , Finite Elements for Electrical Engineers. Cambridge: Cambridge University Press, 2nd edn., 1990.Google Scholar
[4] , and , Finite Element Method for electromagnetics: Antennas, Microwave Cicuits and Scattering Applications. Oxford & New York: Oxford University Press and IEEE Press, 1998.Google Scholar
[5] , and , “Finite element method programming made easy???,” IEEE Antennas Propag. Soc. Mag., 45, 73–79, August 2003.Google Scholar
[6] , and , Fields and Waves in Communication Electronics. Chichester: Wiley, 3rd edn., 1994.Google Scholar
[8] and , Finite Elements for Electrical Engineers. Cambridge: Cambridge University Press, 3rd edn., 1996.Google Scholar
[9] , “Edge elements and what they can do for you,” IEEE Trans. Magn., 29, 1460–1465, March 1993.Google Scholar
[10] , “Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements,” IEEE Antennas Propagat., 47, 1244–1253, August 1999.Google Scholar
[11] , “An evaluation of mixed-order versus full-order vector finite elements,” IEEE Trans. Antennas Propagat., 51, 2430–2441, September 2003.Google Scholar
[12] , and , “Higher order interpolatory vector bases for computational electromagnetics,” IEEE Trans. Antennas Propagat., 45, 329–342, March 1997.Google Scholar
[13] , “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] and , “Hierarchal scalar and vector tetrahedra,” IEEE Trans. Magn., 29, 1495–1498, March 1993.Google Scholar
[15] and , “Hierarchical tangential vector finite elements for tetrahe-dra,” IEEE Microwave Guided Wave Lett., 8, 127–129, March 1998.Google Scholar
[16] , and , “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] , “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] , “Fully hierarchical divergence-conforming basis functions on tetrahedral cells, with applications,” Int. J. Numer. Meth. Eng., 71, 127–148, 2007.Google Scholar
[19] and , “Scale factors and matrix conditioning associated with triangular-cell hierarchical vector basis functions,” IEEE Antennas Wireless Propag. Lett., 9, 40–43, 2010.Google Scholar
[21] , “Moderate-degree tetrahedral quadrature formulas,” Computer Meth. Appl. Mechan. Eng., 55(3), 339-348, 1986.Google Scholar
[22] , “Matching a given field using hierarchal vector basis functions,” Electromagnetics, 24, 113–122, January-March 2004.Google Scholar
[23] and , “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] , 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] , Waveguide Handbook. London: Peter Peregrinus, on behalf of IEE, 1986. Originally published 1951.Google Scholar
[26] , The Finite Element Method in Electromagnetics. New York: Wiley, 2nd edn., 2002.Google Scholar
[27] , “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] , “P-adaptive methods for electromagnetic wave problems using hierarchal tetra-hedral edge elements,” Electromagnetics, 22, 443–151, July 2002.Google Scholar
[29] and , “Radiation boundary conditions for wave-like equations,” Commun. Pure Appl. Mathem., 33, 707–725, 1980.Google Scholar
[30] , “Absorbing boundary conditions for the vector wave equation,” Microwave Optical Technol. Lett., 1, 62–64, April 1988.Google Scholar
[31] and , “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] and , “Implementation and derivation of conformal absorbing boundary conditions for the vector wave equation,” J. Electromagn. Waves Appl., 12, 1653–1677, 1998.Google Scholar
[33] and , “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] and , “PML-FDTD in cylindrical and spherical grids,” IEEE Microwave Guided Wave Lett., 7(9), 285-287, 1997.Google Scholar
[35] , “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] , “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] and , “Evaluation of a spherical PML for vector FEM applications,” IEEE Trans. Antennas Propagat., 55, 494–498, Feb 2007.Google Scholar
[38] , , , and , Iterative and Self-Adaptive Finite-Elements in Electromagnetic Modelling. Boston, MA: Artech House, 1998.Google Scholar
[39] 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] , “Canonical construction of finite elements,” Mathem. Comput., 68, 1325–1346, May 1999.Google Scholar
[41] , and , “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] , “Finite element methods for junctions of microwave and optical waveguides,” IEEE Trans. Magn., 26, 1754–1758, September 1990.Google Scholar
[43] and , “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] , “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] , “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] , , and , “Application of extended Huygens' principle to scattering discontinuities in waveguide,” in IEEE AFRICON-02 Proceedings, pp. 555-558, October 2002.
[47] , , and , “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] and , “Curl-conforming mixed-order edge elements for discretizing the 2D and 3D vector Helmholtz equation,” in Finite Element Software for Microwave Engineering (, and , eds.), Chapter 5. New York: Wiley, 1996.Google Scholar