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The Finite Element Time Domain Method

The finite-element time-domain (FETD or TDFEM) method combines the advantages of a time-domain technique with the versatile spatial discretization options of the finite element method. A variety of FETD methods have been proposed. These schemes generally fall into two categories. Methods in the first category directly discretize the time-dependent Maxwell"s equations, yielding an explicit, conditionally stable, time-marching algorithm that can be viewed as a generalization of the finite-difference time-domain (FDTD) method for unstructured grids [1-10]. Methods in the second category discretize the second-order vector wave (curl-curl) equation, obtained by eliminating one of the field variables from Maxwell's equations. The solution of a linear system of equations is required at each time step, but this implicit method can be formulated to be unconditionally stable [11-23].

The explicit FETD has less computational complexity, however the maximum time-step must be constrained to insure stability and it can be relatively difficult to achieve convergence. In implicit FETD methods, the time step is not constrained by a stability criterion and these methods can be extended to higher-orders in a relatively straightforward manner. However, implicit schemes have greater computational complexity because they require a global linear system of equations to be solved at each time step. This can make the simulation of large-scale electromagnetic problems relatively inefficient.

Lou [24] presented a dual-field FETD formulation that computes both the electric and magnetic fields in a leapfrog fashion. This formulation has the advantages of implicit FETD schemes while reducing the computational complexity significantly when the computational domain is split into non-overlapping smaller subdomains. He extended the domain decomposition to the element level in [25].

For unbounded problems, the truncated boundaries of the FETD computational domain need to be properly treated. This is usually done by using conventional absorbing boundary operators [26, 27], boundary integral (BI) methods [28], or PML boundaries [29-37]. Absorbing boundary operators are easy to implement but can exhibit large reflection errors [27]. Boundary integral methods are theoretically exact but computationally expensive. PML implementations yield very small reflection errors in other time-domain methods, such as the FDTD method. However, the FETD formulation of the PML has not been thoroughly investigated.

References

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[2] C. Chan, J. Elson, and H. Sangani, "An explicit finite-difference time-domain method using Whitney elements," Proc. IEEE APS Int. Symp. Dig., vol. 3, pp. 1768-1771, Jul. 1994.

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[6] M. Feliziani and F. Maradei, "Point matched finite element-time domain method using vector elements," IEEE Trans. on Magnetics, vol. 30, no. 5, pt. 2, pp. 3184-3187, Sep. 1994.

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[10] W. Artuzi, "Finite element time domain method using piecewise constant basis functions," Proc. IEEE MTT-S Int. Microw. Optoelectr. Conf., vol. 2, pp. 1029-1032, 2003.

[11] O. Zienkiewicz and R. Taylor, The Finite Element Method, 4th ed., New York: McGraw-Hill, 1988.

[12] R. Fernades and G. Fairweather, "An alternating direction Galerkin method for a class of second order hyperbolic equations in two space dimensions," SIAM J. Numer. Anal., vol. 28, no. 5, pp. 1265-1281, 1991.

[13] G. Mur, "The finite-element modeling of three-dimensional time-domain electromagnetic fields in strongly inhomogeneous media," IEEE Trans. on Magnetics, vol. 28, no. 3, pp. 1130-1133, Mar. 1992.

[14] J. F. Lee and Z. Sacks, "Whitney elements time domain (WETD) methods," IEEE Trans. on Magnetics, vol. 31, no. 5, pp. 1325-1329, May 1995.

[15] S. D. Gedney and U. Navsariwala, "An unconditionally stable finite-element time-domain solution of the vector wave equation," IEEE Microw. Guided Wave Lett., vol. 5, no. 5, pp. 332-334, May 1995.

[16] J. Lee, R. Lee, and A. Cangellaris, "Time-domain finite-element methods," IEEE Trans. Antennas Propagat., vol. 45, no. 3, pp. 430-442, Mar. 1997.

[17] W. P. Capers, Jr., L. Pichon, and A. Razek, "A 3-D finite element method for the modeling of bounded and unbounded electromagnetic problems in the time domain," Int. J. Numer. Modeling, vol. 13, pp. 527-540, 2000.

[18] J. M. Jin, The Finite Element Method in Electromagnetics, 2nd ed., New York: Wiley, 2002.

[19] F. Edelvik, G. Ledfelt, P. Lotstedt, and D. Riley, "An unconditionally stable subcell model for arbitrarily oriented thin wires in the FETD method," IEEE Trans. Antennas Propagat., vol. 51, no. 8, pp. 1797-1805, Aug. 2003.

[20] D. Jiao, J. M. Jin, E. Michielssen, and D. Riley, "Time-domain finite-element simulation of three-dimensional scattering and radiation problems using perfectly matched layers," IEEE Trans. Antennas Propagat., vol. 51, no. 2, pp. 296-305, Feb. 2003.

[21] T. Rylander and J. M. Jin, "Perfectly matched layers in three dimensions for the time-domain finite element method," Proc. IEEE APS Int. Symp. Dig., Monterey, CA, Jun. 2004, vol. 4, pp. 3473-3476.

[22] Z. Lou and J. M. Jin, "Modeling and simulation of broadband antennas using the time-domain finite element method," IEEE Trans. Antennas Propagat., vol. 53, no. 12, pp. 4099-4110, Dec. 2005.

[23] A. Peterson and J. M. Jin, "A three-dimensional time-domain finite element formulation for periodic structures," IEEE Trans. Antennas Propagat., vol. 54, no. 1, pp. 12-19, Jan. 2006.

[24] Z. Lou and J. M. Jin, "A novel dual-field time-domain finite-element domain-decomposition method for computational electromagnetics," IEEE Trans. Antennas Propagat., vol. 54, no. 6, pp. 1850-1862, Jun. 2006.

[25] Z. Lou and J. M. Jin, "A new explicit time-domain finite-element method based on element-level decomposition," IEEE Trans. Antennas Propagat., vol. 54, no. 10, pp. 2990-2999, Oct. 2006.

[26] H. Ali and G. Costache, "Finite-element time-domain analysis of axisymmetrical radiators," IEEE Trans. Antennas Propagat., vol. 42, pp. 272-275, Feb. 1994.

[27] S. Caorsi and G. Cevini, "Assessment of the performances of first- and second-order time-domain ABC's for the truncation of finite element grids," Microwave Opt. Technol. Lett., vol. 38, no. 1, pp. 11-16, Jul. 2003.

[28] D. Jiao, M. Lu, E. Michielssen, and J. M. Jin, "A fast time-domain finite element-boundary integral method for electromagnetic analysis," IEEE Trans. Antennas Propagat., vol. 49, no. 10, pp. 1453-1461, Oct. 2001.

[29] M. Kuzuoglu and R. Mittra, "Investigation of nonplanar perfectly matched absorbers for finite-element mesh truncation," IEEE Trans. Antennas Propagat., vol. 45, no. 3, Mar. 1997.

[30] V. Mathis, "An anisotropic perfectly matched layer-absorbing medium in finite element time domain method for Maxwell's equations," Proc. IEEE Antennas and Propagation Soc. Int. Symp., vol. 2, pp. 680-683, Jul. 1997.

[31] D. Jiao and J. M. Jin, "An effective algorithm for implementing perfectly matched layers in time-domain finite-element simulation of open-region EM problems," IEEE Trans. Antennas Propagat., vol. 50, pp. 1615-1623, Nov. 2002.

[32] D. Jiao, J. M. Jin, E. Michielssen, and D. J. Riley, "Time-domain finite element simulation of three-dimensional scattering and radiation problems using perfectly matched layers," IEEE Trans. Antennas Propagat., vol. 51, no. 2, pp. 296-305, Feb. 2003.

[33] H. P. Tsai, Y. Wang, and T. Itoh, "An unconditionally stable extended (USE) finite-element time-domain solution of active nonlinear microwave circuits using perfectly matched layers," IEEE Trans. Microwave Theory Tech., vol. 50, no. 10, pp. 2226-2232, Oct. 2002.

[34] T. Rylander and J.-M. Jin, "Conformal perfectly matched layers for the time domain finite element method," IEEE AP-S Int. Symp. Digest, Jun. 2003, vol. 1, pp. 698-701.

[35] T. Rylander and J.-M. Jin, "Perfectly matched layer for the time domain finite element method applied to Maxwell's equations," J. Comp. Phys., vol. 200, pp. 238-250, May 2004.

[36] T. Rylander and J.-M. Jin, "Perfectly matched layer in three dimensions for the time-domain finite element method applied to radiation problems," IEEE Trans. Antennas Propagat., vol. 53, no. 4, pp. 1489-1499, Apr. 2005.

[37] S. Wang, R. Lee, and F. L. Teixeira, "Anisotropic-medium PML for vector FETD with modified basis functions," IEEE Trans. Antennas Propagat., vol. 54, no. 1, pp. 20-27, Jan. 2006.