The Finite Volume Time Domain Method

The Finite Volume Time Domain (FVTD) method was first applied to electromagnetic problems in the early 1990's [1, 2]. This technique is based on Maxwell's curl equations in their conservative form [3],

(1)

(2)

where δv represents the boundary enclosing V. The FVTD method solves the above form of Maxwell's equations numerically by integration over small elementary volumes. Because there are no limitations for selecting the shape of the elementary volumes, the FVTD is well suited for implementation with unstructured meshes. It has become a powerful alternative to the finite difference time domain (FDTD) method for electromagnetic problems where conformal meshing is advantageous.

Like FDTD, FVTD methods can take advantage of PML absorbing boundaries. Bonnet [4] presented a vertex-centered FVTD model of the PML for scattering problems. Sankaran [5] extended the PML concept to the cell-centered FVTD approach and systematically characterized its performance using both structured and unstructured finite volume meshes. He introduced a uniaxial Maxwellian absorber using PML to solve the waveguide truncation problem [6] and extended the absorber to incorporate radial anisotropy for modeling cylindrical geometries [7]. Fumeaux [8] presented a spherical perfectly matched absorber for finite-volume 3D domain truncation. Pinto [9] incorporated the uniaxial perfectly matched layer in the analysis of light propagation in photonic bandgap devices.

The FVTD method is a promising numerical technique with good potential for the simulation of a variety of complex electromagnetic problems [10-13]. Bonnet [14] presented a method for the resolution of electromagnetic diffraction by complex structures; results obtained for an aircraft were compared with results from a classical FDTD code. Lacour [15] described a multi-domain decomposition method using an FVTD technique for the resolution of an electromagnetic problem on vehicles and evaluated the current on a cable inside the volume of an airplane. Applications of the technique in microwave engineering require both the implementation of electromagnetic sources and the characterization of ports. Baumann [16, 17] introduced new schemes for full-wave field excitation and full-wave S-parameter extraction that make the FVTD method especially well suited for microwave device simulations. To improve the computational efficiency of the FVTD method, Fumeaux [18] introduced a new generalized local time-stepping scheme, which is based on an automatic partition of the computational domain into subdomains where local time steps of the type 2^l-1Δ t ( l = 1, 2, 3, ...) can be applied without violating the stability condition.

References

[1] N. K. Madsen, R.W. Ziolkowski, "A three-dimensional modified finite volume technique for Maxwell's equations," Electromagnetics, vol. 10, pp. 147-161, 1990.

[2] V. Shankar, A.H. Mohammadian, W.F. Hall, "A time-domain, finite-volume treatment for the Maxwell equations," Electromagnetics, vol. 10, pp. 127-145, 1990.

[3] P. Bonnet, X. Ferrières, B. Michielsen, and P. Klotz, Time Domain Electromagnetics, S. M. Rao, Ed., Academic Press, 1997, ch. 9, pp. 307-367.

[4] F. Bonnet and F. Poupaud, "Berenger absorbing boundary condition with time finite-volume scheme for triangular meshes," Appl. Numer. Math., vol. 25, no. 4, pp. 333-354, Dec. 1997.

[5] K. Sankaran, C. Fumeaux, and R. Vahldieck, "Cell-centered finite-volume-based perfectly matched layer for time-domain Maxwell system," IEEE Trans. on Microwave Theory and Tech., vol. 54, no. 3, Mar. 2006.

[6] K. Sankaran, C. Fumeaux, and R. Vahldieck, "Finite-volume Maxwellian absorber on unstructured grid," IEEE MTT-S Int. Microw. Symp. Dig., pp. 169-172, Jun. 2006.

[7] K. Sankaran, C. Fumeaux, and R. Vahldieck, "Uniaxial and radial anisotropy models for finite-volume Maxwellian absorber," IEEE Trans. on Microwave Theory and Tech., vol. 54, no. 12, pp. 4297 - 4304, Dec. 2006.

[8] C. Fumeaux, K. Sankaran, and R. Vahldieck, "Spherical perfectly matched absorber for finite-volume 3-D domain truncation," IEEE Trans. on Microwave Theory and Tech., vol. 55, no.12, Dec. 2007

[9] D. Pinto and S. S. A. Obayya, "Accurate perfectly matched layer finite-volume time-domain method for photonic bandgap devices," IEEE Photonics Technology Letters, vol. 20, no. 5, Mar. 2008.

[10] R. Holland, V.P. Cable, and L.C. Wilson, "Finite-volume time-domain (FVTD) techniques for EM scattering," IEEE Trans. on Electromag. Compat., vol. 33, no. 4, pp. 281-294, Nov. 1991.

[11] P. Bonnet, X. Ferrieres, F. Paladian, J. Grando, J. Alliot, and J. Fontaine, "Electromagnetic wave diffraction using a finite volume method," Electronics Letters, vol. 33, no. 1, pp. 31-32, 1997.

[12] C. Fumeaux, D. Baumann, and R. Vahldieck, "Advanced FVTD simulations of dielectric resonator antennas and feed structures," Journal of the Applied Computational Electromagnetics Society, vol. 19, pp. 155-164. 2004.

[13] D. Baumann, C. Fumeaux, P. Leuchtmann, and R. Vahldieck, "Finite volume time-domain (FVTD) modeling of a broad-band double-ridged horn antenna," Int. J. Numer. Modeling, vol. 17, no. 3, pp. 285-298, 2004.

[14] P. Bonnet, X. Ferrieres, F. Paladian, J. Grando, J.C. Alliot and J. Fontaine, "Electromagnetic wave diffraction using a finite volume method," Electronics Letters, vol. 33, no. 1, pp. 31-32, Jan. 1997.

[15] D. Lacour, X. Ferrieres, P. Bonnet, V. Gobin and J.C. Alliot, "Application of multi-domain decomposition method to solve EMC problem on an aeroplane," Electronics Letters, vol. 33, no. 23, pp.1932-1933, Nov. 1997.

[16] D. Baumann, C. Fumeaux, and R. Vahldieck, "A novel wave-separation scheme for the extraction of S-parameters in non-TEM waveguides for the FVTD method," IEEE MTT-S Int. Microwave Symp. Dig., Long Beach, CA, Jun. 2005.

[17] D. Baumann, C. Fumeaux, and R. Vahldieck, "Field-Based Scattering-Matrix Extraction Scheme for the FVTD Method Exploiting a Flux-Splitting Algorithm," IEEE Trans. on Microwave Theory and Tech., vol. 53, no. 11, Nov. 2005.

[18] C. Fumeaux, D. Baumann, P. Leuchtmann, R. Vahldieck, "A generalized local time-step scheme for efficient FVTD simulations in strongly inhomogeneous meshes," IEEE Trans. on Microwave Theory and Tech., vol. 52, no. 3, pp. 1067-1076, Mar. 2004.