1999 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

IEEE Transactions on Antennas and Propagation
Volume 47 Number 8, August 1999

Table of Contents for this issue

Complete paper in PDF format

Hierarchal Vector Basis Functions of Arbitrary Order for Triangular and Tetrahedral Finite Elements

Jon P. Webb, Member, IEEE

Page 1244.

Abstract:

New vector finite elements are proposed for electromagnetics. The new elements are triangular or tetrahedral edge elements (tangential vector elements) of arbitrary polynomial order. They are hierarchal, so that different orders can be used together in the same mesh and p-adaption is possible. They provide separate representation of the gradient and rotational parts of the vector field. Explicit formulas are presented for generating the basis functions to arbitrary order. The basis functions can be used directly or after a further stage of partial orthogonalization to improve the matrix conditioning. Matrix assembly for the frequency-domain curl-curl equation is conveniently carried out by means of universal matrices. Application of the new elements to the solution of a parallel-plate waveguide problem demonstrates the expected convergence rate of the phase of the reflection coefficient, for tetrahedral elements to order 4. In particular, the full-order elements have only the same asymptotic convergence rate as elements with a reduced gradient space (such as the Whitney element). However, further tests reveal that the optimum balance of the gradient and rotational components is problem-dependent.

References

  1. C. W. Crowley, P. P. Silvester, and H. Hurwitz, "Covariant projection elements for 3D vector field problems," IEEE Trans. Magn., vol. 24, no. 1, pp. 397-400, Jan. 1988.
  2. P. T. S. Liu and J. P. Webb, "Hierarchal vector finite elements for 3D electromagnetics," Proc. Inst. Elec. Eng., Microwaves, Antennas Propagat., vol. 142, no. 5, pp. 373-378, Oct. 1995.
  3. J. P. Webb, "Edge elements and what they can do for you," IEEE Trans. Magn., vol. 29, pp. 1460-1465, Mar. 1993.
  4. A. Bossavit, "A rationale for `edge-elements' in 3-D fields computations," IEEE Trans. Magn., vol. 24, pp. 74-79, Jan. 1988.
  5. G. Mur, "A finite-element method for computing three-dimensional electromagnetic fields in inhomogeneous media," IEEE Trans. Magn., vol. MAG-21, no. 6, p. 2188, 1985.
  6. J. F. Lee, "Analysis of passive microwave devices by using three-dimensional tangential vector finite elements," Int. J. Numerical Modeling: Electronic Networks (Devices and Fields), vol. 3, no. 4, pp. 235-246, Dec. 1990.
  7. J. F. Lee, D. K. Sun, and Z. J. Cendes, "Full-wave analysis of dielectric waveguides using tangential vector finite elements," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 1262-1271, Aug. 1991.
  8. J. P. Webb and B. Forghani, "Hierarchal scalar and vector tetrahedra," IEEE Trans. Magn., vol. 29, pp. 1495-1498, Mar. 1993.
  9. A. Ahagon and T. Kashimoto, "Three-dimensional electromagnetic wave analysis using high order edge elements," IEEE Trans. Magn., vol. 21, pp. 1753-1756, May 1995.
  10. A. Kameari, "Symmetric second order edge elements in triangles and tetrahedrons," presented at the Eighth Bienniel IEEE Conference on Electromagnetic Field Computation, Tucson, AZ, June 1-3, 1998.
  11. I. Babuska, B. A. Szabo, and I. N. Katz, "The p-version of the finite element method," SIAM J. Numer. Anal., vol. 18, pp. 515-545, 1981.
  12. S. McFee and J. P. Webb, "Adaptive finite element analysis of microwave and optical devices using hierarchal triangles," IEEE Trans. Magn., vol. 28, pp. 1708-1711, Mar. 1992.
  13. O. C. Zienkiewicz, J. Z. Zhu, and N. G. Gong, "Effective and practical h-p version adaptive analysis procedures for the finite element method," Int. J. Numer. Methods in Eng., vol. 28, pp. 879-891, 1989.
  14. J. C. Nedelec, "Mixed finite elements in R3," Numerische Mathematik, vol. 35, pp. 315-341, 1980.
  15. --, "A new family of mixed finite elements in R3," Numerische Mathematik, vol. 50, pp. 57-81, 1986.
  16. J. P. Webb and B. Forghani, "A T-Omega method using hierarchal edge elements," Proc. Inst. Elec. Eng.,, Sci. Meas. Technol., vol. 142, no. 2, pp. 133-141, Mar. 1995.
  17. G. Peng, R. Dyczij-Edlinger, and J.-F. Lee, "Hierarchical methods for solving matrix equations from TVFEM's for microwave components," IEEE Trans. Magn., vol. 35, pp. 1474-1477, May 1999.
  18. R. D. Graglia, D. R. Wilton, and A. F. Peterson, "Higher order interpolatory vector bases for computational electromagnetics," IEEE Trans. Antennas Propagat., vol. 45, pp. 329-342, Mar. 1997.
  19. 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., vol. 33, pp. 1812-1815, Mar. 1997.
  20. J. S. Wang and N. Ida, "Curvilinear and higher order `edge' finite elements in electromagnetic field computation," IEEE Trans. Magn., vol. 29, pp. 1491-1494, Mar. 1993.
  21. D. Sun, J. Manges, X. Yuan, and Z. Cendes, "Spurious modes in finite-element methods," IEEE Antennas Propagat. Mag., vol. 37, pp. 12-24, Oct. 1995.
  22. P. Silvester, "High-order polynomial triangular elements for potential problems," Int. J. Eng. Sci., vol. 7, pp. 849-861, 1969.
  23. L. S. Andersen and J. L. Volakis, "Hierarchical tangential vector finite elements for tetrahedra," IEEE Microwave Guided Wave Lett., vol. 8, pp. 127-129, Mar. 1998.
  24. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers, 3rd ed.Cambridge, U.K.: Cambridge Univ. Press, 1996.
  25. R. Albanese and C. Rubinacci, "Solution of three dimensional eddy current problems by integral and differential methods," IEEE Trans. Magn., vol. 24, pp. 98-101, Jan. 1988.
  26. J. P. Webb and R. Abouchacra, "Hierarchal triangular elements using orthogonal polynomials," Int. J. Numer. Methods in Eng., vol. 38, no. 2, pp. 245-257, Jan. 1995.
  27. P. P. Silvester, "Universal finite element matrices for tetrahedra," Int. J. Num. Met. Eng., vol. 18, pp. 1055-1061, 1982.
  28. C. Geuzaine, B. Meys, P. Dular, and W. Legros, "Convergence of high order curl-conforming finite elements," IEEE Trans. Magn., vol. 35, pp. 1442-1445, May 1999.
  29. B. M. Irons, "A frontal solution program for finite element analysis," Int. J. Num. Met. Eng., vol. 2, pp. 5-32, 1970.