Comparing the Material Modeling Abilities ofFVTD and FIT through the Accuracy Analysis of

the Scattering ParametersChakrapani Bommaraju

Stream of Electronics and Communication EngineeringInternational Institute of Information Technology-Hyderabad

Gachibowli, Hyderabad, Andhra Pradesh - 500032, India

Abstract—In this paper, the scattering parameters computedusing upwind flux finite volume time domain methods of aninhomogeneous coaxial cable with high contrast of materialsare furnished. A convergence study has been performed onthe accuracy of the obtained scattering parameters using finitevolume methods in addition to the finite integration technique.

Keywords-upwind flux; finite volume time domain methods;material modeling; scattering parameters; accuracy of scatteringparameters.

I. INTRODUCTION

The generalized scattering matrix extraction of waveguidedevices using upwind flux finite volume time domain meth-ods [1]–[3] has been presented in “unpublished” [4]. Usingthe same methodology, the scattering parameters of a highlyinhomogeneous coaxial cable are computed [5]. A comparisonis made between the material modeling capabilities of upwindflux second order finite volume time domain methods ontetrahedral meshes and the finite integration technique [6]–[10]in time and frequency domains through the accuracy analysisof obtained scattering parameters.

II. SIMULATION

Five coaxial cables with varying material properties arejoined end to end, centers aligned along the z axis, formingan inhomogeneous coaxial cable as shown in Fig. 1(a). Thecables depicted in red have relative permittivity and relativepermeability of free space whereas cables represented in greenhave the relative permittivity and relative permeability of 2.04and 1, respectively. The relative permittivity and permeabilityof the cable in the center (highlighted in blue) is 10 i.e.,the phase velocity of wave is reduced by a factor of 10 inthe middle cable. The dimensions of all these cables are thesame, and are given in Fig. 1(b) and Fig. 1(c). The cable issurrounded by perfect electric conductor medium except inthe z direction, where ports are located. Fig. 1(a) depicts theobservation port, p2. The port planes are truncated with theSilver-Muller absorbing boundary condition.

1/10 1

p2

(a) Relative velocity profile

p2x

y

320 mil

140 mil

(b) Front view

εr = 1 εr = 2.04 εr = 10 εr = 2.04 εr = 1μr = 1 μr = 1 μr = 10 μr = 1 μr = 1

p2 p1

z

y

300 mil

(c) Side view

Fig. 1. An inhomogeneous coaxial cable with material and parametricdescription. The excitation port, p1, is located at the origin in the xy-plane andthe observation port, p2, is in the +z direction. The ports are truncated withabsorbing boundary conditions whereas the cable is surrounded by perfectelectric conductor medium else where. Note 1 mil is equal to one thousandthof an inch.

978-1-4577-1099-5/11/$26.00 ©2011 IEEE

0 1

(a) Normalized electric field pattern

0 1

(b) Normalized magnetic field pattern

Fig. 2. The fundamental mode at port 1 when the computational domain isdiscretized with tetrahedral mesh at a spatial resolution of 10 CPW at 8 GHz.

The fundamental mode in this structure is a transverseelectromagnetic (TEM) mode. In cylindrical coordinates, thiscan be expressed as follows [11]:

Eρ = −1

ρ(1)

Eϕ = 0 (2)

Ez = 0 (3)

Hρ = 0 (4)

Hϕ =1

ρ(5)

Hz = 0 (6)

In the previous equations ρ represents the radial distance.The signal used to excite both the electric and magnetic fieldcomponents is portrayed in Fig. 3 in time and frequencydomains. The dominant frequency spectrum of the excitationcovers 2–8 GHz. The structure is discretized with a tetrahedralmesh with 10 cells per wavelength (CPW) at 8 GHz and theTEM mode, depicted in Fig. 2, is imposed on the mesh atport 1.

The scattering parameters (S–parameters) are obtainedusing second order upwind flux finite volume time domainmethod with gradient computed from cell center values ontetrahedral mesh (FVTD 22 GCC TET) and second order

0 0.25 0.5 0.75 1−1

−0.5

0

0.5

1

Time (ns)

Am

plitu

de2 3.5 5 6.5 8

0

0.25

0.5

0.75

1

Frequency (GHz)N

orm

aliz

edA

mpl

itude

Fig. 3. Time and frequency domain representations of excitation. Thedominant frequency spectrum is between 2 GHz and 8 GHz.

upwind flux finite volume time domain method with gradientcomputed from face center values on tetrahedral mesh(FVTD 22 GFC TET) [5]. The spatial resolution is changedto 20 CPW and the S–parameters are computed using boththe methods. This process if repeated till the resolution is 60CPW, in steps of 10 CPW.

The domain is discretized with hexahedral mesh andscattering parameters are also obtained with finite integrationtechnique [12]–[15] in time domain (FIT TD) and finiteintegration technique in frequency domain (FIT FD) methods,both employing perfect boundary approximation [16], forvarious spatial resolutions ranging from 10 CPW to 60 CPW,in steps of 10 CPW. All the S–parameters are extracted at themaximum possible time step for all the methods.

In order to make a convergence study, a referencesolution is necessary, which is obtained numerically [5].The computational domain is discretized with a coarsehexahedral mesh and the S–parameters are extracted fromthe CST MWS [16] transient solver (FIT TD). The mesh isrefined and the scattering parameters are computed again.This process is repeated till the minimum absolute differencebetween any two consecutive solutions (S–parameters) fallsbelow 10 per million in the complex plane. The solutionfound with finer mesh among these two is consideredas the reference solution from FIT TD. The reference

solutions are also worked out using the CST MWS frequencydomain solver employing FIT FD (on hexahedral mesh) andfinite element method in frequency domain (FEM FD) ontetrahedral mesh following exactly the same procedure asabove. The reference solutions from FIT TD, FIT FD andFEM FD are compared. Again, if the difference between themlies around 10 per million, then the solution from FIT FDis considered as the reference solution. Here, FIT FD at aspatial resolution of 100 CPW is the chosen reference solution.

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Real

Imag

inar

y

60 CPW

50 CPW

40 CPW

30 CPW20 CPW

10 CPW

Reference Solution

(a) S11 in complex plane.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5

Real

Imag

inar

y

60 CPW

50 CPW

40 CPW

30 CPW

20 CPW

10 CPW Reference Solution

(b) S21 in complex plane.

Fig. 4. The scattering parameters obtained using second order finitevolume time domain method with gradient calculated from cell center valuesfor various spatial resolutions on tetrahedral mesh. The set of scatteringparameters obtained from finite integration technique in frequency domainat a resolution of 100 CPW is considered as the reference solution.

Fig. 4 and Fig. 5 depict the S–parameters computed usingFVTD 22 GCC method for various spatial resolutions along

with the S–parameters obtained using FIT FD at a spatialresolution of 100 CPW. One can observe the poor accuracyof S–parameters obtained from FVTD methods. The strongartificial dissipation present in S21 obtained from FVTDmethods is clearly visible [5].

2 3.5 5 6.5 8−80

−60

−40

−20

0

Frequency (GHz)A

mpl

itude

indB

60 CPW

50 CPW

40 CPW

30 CPW

20 CPW10 CPW

Reference Solution

(a) S11 Magnitude.

2 3.5 5 6.5 8−40

−30

−20

−10

0

Frequency (GHz)

Am

plitu

dein

dB

60 CPW

50 CPW

40 CPW

30 CPW

20 CPW

10 CPW

Reference Solution

(b) S21 Magnitude.

Fig. 5. The magnitude of scattering parameters obtained using second orderFVTD method with gradient calculated from cell center values for variousspatial resolutions on tetrahedral mesh along with reference solution.

The minimum of the absolute difference between the scatter-ing parameters in the complex plane obtained using differentmethods and the reference solution, both sampled at thesame frequency points, is plotted against the cubic root ofthe number of cells in a loglog scale as depicted in Fig. 6and Fig. 7. The slopes of the curves are nothing but theconvergence orders.

101 10210−5

10−4

10−3

10−2

10−1

3√

Number of Cells

Min

imum

Abs

olut

eE

rror

inC

ompl

exPl

ane

��

��

�� ����

����

��

����

���� �� ��

����

����

�� ��

FIT FD HEX PBA, Order 1.7FIT TD HEX PBA, Order 1.7FVTD 22 GCC TET, Order 1.2FVTD 22 GFC TET, Order 2.8

Fig. 6. Convergence of error in S11 of the coaxial cable obtained usingsecond order finite volume time domain methods and finite integrationtechnique in time and frequency domains. Note that FIT methods employPerfect Boundary Approximation.

101 10210−5

10−4

10−3

10−2

10−1

100

3√

Number of Cells

Min

imum

Abs

olut

eE

rror

inC

ompl

exPl

ane

����

����

��

����

��

�� ���� ��

�� ������

�� ���� ��

FIT FD HEX PBA, Order 1.8FIT TD HEX PBA, Order 1.7FVTD 22 GCC TET, Order 1.3FVTD 22 GFC TET, Order 2.1

Fig. 7. Convergence of error in S21 of the coaxial cable obtained usingsecond order finite volume time domain methods and finite integrationtechnique in time and frequency domains. Note that FIT methods employPerfect Boundary Approximation.

III. CONCLUSIONS

The finite integration technique, when employed in bothtime and frequency domains, has less error than finite volumetime domain methods in the mentioned spatial resolutionrange. Note that the convergence order of second order finitevolume time domain method with gradient calculated fromface center values is very high which implies that the errordecreases, eventually to a value lower than all the othermethods. In other words, this finite volume time domainmethod is able to resolve the high contrast of materials moreefficiently than the rest of the methods.

ACKNOWLEDGMENT

The author would like to thank Dr.-Ing. Wolfgang Acker-mann, Dr. rer. nat. Erion Gjonaj and Prof. Dr.-Ing. ThomasWeiland.

REFERENCES

[1] V. Shankar, W. F. Hall, and A. Mohammadian, “A time-domain differ-ential solver for electromagnetic scattering problems,” Proceedings ofthe IEEE, vol. 77, no. 5, pp. 709–721, 1989.

[2] V. Shankar, W. F. Hall, and A. H. Mohammadian, “Development ofcomputational fluid dynamics (CFD) based time-domain algorithms forMaxwell’s equations,” in Antennas and Propagation Society Interna-tional Symposium, 1990. AP-S. ’Merging Technologies for the 90’s’.Digest., Dallas, TX, May 7–11, 1990, pp. 1632–1635.

[3] S. M. Rao, Time Domain Electromagnetics. Academic Press, 1999.[4] C. Bommaraju, “Extraction of scattering parameters using upwind

flux finite volume time domain methods,” in Applied ElectromagneticsConference, 2011. IEEE, Dec. 2011.

[5] ——, “Investigating finite volume time domain methods in compu-tational electromagnetics,” Ph.D. dissertation, Technische UniversitatDarmstadt, Darmstadt, 2009.

[6] T. Weiland, “A discretization method for the solution of Maxwell’sequations for six-component fields,” Electronics and Communication,vol. 31, no. 3, pp. 116–120, 1977.

[7] R. Marklein, “Numerische Verfahren zur Modellierung von akustischen,elektromagnetischen, elastischen und piezoelektrischen Wellenausbre-itungsproblemen im Zeitbereich basierend auf der Finiten Integra-tionstechnik,” Ph.D. dissertation, Universitat Gesamthochschule Kassel,Kassel, 1997.

[8] R. Schuhmann and T. Weiland, “Recent advances in finite integrationtechique for high frequency applications,” in Proceedings of the 4thConference on Scientific Computing in Electrical Engineering, 2002.

[9] T. Weiland and R. Schuhmann, “Discrete electromagnetism by thefinite integration technique,” Journal of the Japan Society of AppliedElectromagnetics and Mechanics, vol. 10, no. 2, pp. 159–169, 2002.

[10] T. Weiland, “Finite integration techniques for time- and frequencydomain,” in Proceedings of the Workshop on Advanced Techniques inElectromagnetic Modeling, May 2004.

[11] K. Zhang and D. Li, Electromagnetic Theory for Microwaves andOptoelectronics. Springer, 2008.

[12] M. Dehler, M. Dohlus, and T. Weiland, “Calculating frequency-domaindata by time-domain methods,” International Journal of NumericalModelling: Electronic Networks, Devices and Fields, vol. 6, no. 1, pp.19–27, Feb. 1993.

[13] B. Geib, M. Dohlus, and T. Weiland, “Calculation of scattering param-eters by orthogonal expansion and finite integration method,” Interna-tional Journal of Numerical Modelling: Electronic Networks, Devicesand Fields, vol. 7, pp. 377–398, 1994.

[14] H. Wolter, M. Dohlus, and T. Weiland, “Broadband calculation of scat-tering parameters in the time domain,” IEEE Transactions on Magnetics,vol. 30, no. 5, pp. 3164–3167, Sept. 1994.

[15] T. Weiland, “The finite integration method and discrete electromag-netism,” in Proceedings of the GAMM Workshop,.

[16] [Online]. Available: http://www.cst.com