thin-slot/thin-layer subcell fdtd algorithms for em...

EMG 23(2) #6315

Electromagnetics, 23:119–133, 2003Copyright © 2003 Taylor & Francis0272-6343/03 $12.00 + .00DOI: 10.1080/02726340390159469

Thin-Slot/Thin-Layer Subcell FDTD Algorithms forEM Penetration through Apertures

MARIOS A. GKATZIANASGERASIMOS I. BALLASCONSTANTINE A. BALANISCRAIG R. BIRTCHER

Department of Electrical EngineeringTelecommunications Research CenterArizona State UniversityTempe, Arizona, USA

THEODOROS D. TSIBOUKIS

Department of Electrical and Computer EngineeringDivision of TelecommunicationsAristotle University of ThessalonikiGreece

Two different methods for implementing a new thin-slot finite-difference time-domain(FDTD) subcell model are presented in this paper. Comparisons are performed withexisting models in the literature, as well as with measurements for Shielding Effective-ness (SE) of cavities. The proposed methods are shown to exhibit superior accuracy,especially at lower frequencies. Furthermore, to model the thickness of the cavitywalls more accurately, a comparative study of three existing one-dimensional thin-layer algorithms is performed, and one of them is selected, generalized, and appliedto three-dimensional SE predictions. The effect of the wall thickness on the field pen-etration is clearly demonstrated.

Introduction

The standard, uniform-cell-size finite-difference time-domain (FDTD) method has beenwidely used in radiation and scattering problems, due to its simplicity, robustness, andability to provide wide-band results with just one simulation. However, since FDTD isan inherently volumetric method, it cannot efficiently model thin features, such as wiresor apertures, without decreasing the cell size. In a uniform-cell-size grid, this may result

Received 20 February 2002; accepted 17 May 2002.This work was sponsored by NASA Grant NAG-1-1781 and Cooperative Agreement NCC-1-

10051. The authors would like to thank Dr. Celeste M. Belcastro and Truong Nguyen of NASALangley Research Center, Hampton, VA for their continued interest and support of the project.

Address correspondence to Marios A. Gkatzianas, Department of Electrical Engineering,Telecommunications Research Center, Arizona State University, Tempe, AZ 85287-7206, USA.E-mail: [email protected]

119

120 M. A. Gkatzianas et al.

in excessive memory and simulation time requirements. The solution to this problem iseither the employment of nonuniform/multigrid algorithms, which use a variable cell sizedepending on the desired resolution or perform local grid refinement, or subcell models,which use a cell size larger than the modeled feature but apply special update equationsto the fields in that region so that the presence of the feature can be taken into account.The first solution is usually more accurate but suffers from dispersion problems, whilethe second solution is easier to implement and results in greater computational savings.In this paper, two subcell models for thin slots and thin conducting layers, respectively,are considered. These subcell models become very useful in EM penetration applications,where the presence of slots may diminish the shielding properties of geometries. Addi-tionally, the thickness of the geometry boundaries (usually they are conducting walls) mayaffect the field penetration inside the structure, especially at higher frequencies where theskin depth becomes smaller.

One of the first thin-slot algorithms was introduced in FDTD by Gilbert and Holland(1981) followed by Taflove et al. (1988). Both algorithms were based on the integralrepresentation of Maxwell’s equations. A time-domain integral equation was used byRiley and Turner (1990) that also took the thickness of the slot into account. The thicknessof the slot was also modeled using a transmission line approach in Warne and Chen (1988,1992). With the exception of the first two aforementioned methods, all other methodswere relatively difficult to implement in FDTD. There is still need for a thin-slot modelthat is conceptually simple and, at the same time, more accurate than the model byGilbert and Holland (1981). One such model is presented in the next section, along withtwo different implementations.

Thin conducting layers have traditionally been modeled using surface impedanceboundary conditions (Maloney & Smith, 1992a). These offer substantial computationalsavings, but they can be used only when the field interior to the conducting surface is notof interest. In all other cases, the wave propagation through the layer must be explicitlymodeled. Three FDTD algorithms have been proposed by Maloney and Smith (1992b),Berghe et al. (1998), and Sarto (1999), respectively. The formulations of these algorithmsare well known, and they will be omitted; the objective is to compare the algorithms andfind possible advantages that one of them may have over the others. To this purpose, aone-dimensional example with an analytical solution will be used as the benchmark forthe three algorithms. Then, one of them will be selected for SE predictions.

Formulation of the Thin-Slot Algorithm

In this section, the formulation for a new thin-slot algorithm is presented. Consider aperfect electric conductor (PEC) plane with a slot that is infinitely long in the x directionand whose height w is less than �y (the FDTD cell size in the y direction), as shownin Figure 1a. It is further assumed that the material surrounding the PEC plane is freespace. From the field locations in the Yee cell, it is obvious that the only componentsthat require special treatment, in order for the slot to be modeled with a height w < �y,are Ey and Hz, since they are the only components that are located on the PEC plane.All other components are displaced with regard to the PEC plane and, thus, are updatedusing the standard FDTD equations. The extension to other orientations can be performedby a simple permutation of the axes, and it will be omitted. The slot is assumed to beinfinitesimally thick in the z direction, although the standard FDTD method simulates theslot with an effective thickness of �z. Assuming that the slot is “thin,” i.e. w � λ, thefield distribution over the slot can be assumed, to a good approximation, to be uniform.

FDTD Algorithms for EM Penetration through Apertures 121

Figure 1. (a) Geometry of an infinitely long, in the x direction, slot mounted on an infinite PECplane. (b) Geometry of the cavity for which the thin-slot models are used.


This approximation is valid in most FDTD applications, since the cell size is usuallyselected in the order of λ/10 for the highest frequency, and w is less than the cellsize. Using Fourier transform techniques, an expression for the per-unit length (hereafterreferred to as pu) slot admittance can be found in Harrington (2001) as

Ypu(ω) = Gpu(ω) + jBpu(ω) = ε0|ω|2

+ 4.232jωε0

2π− jωε0

πln

( |ω|wc0

). (1)

Strictly speaking, this expression is valid only if the slot is infinite in extent and thewave impinges normal to the slot. However, as will become evident from the results, thefirst condition can be relaxed so that even a slot that is finite in the x direction can bemodeled with sufficient accuracy. Nevertheless, the condition of normal incidence mustalways be satisfied.

The slot admittance in (1) can be attributed to a parallel RC circuit, where both Rand C are functions of frequency. Specifically, the pu capacitance of the slot is

Cpu,d(ω) = Bpu(ω)

ω= 4.232ε0

2π− ε0

πln

( |ω|wc0

). (2)

This can be compared to the pu slot capacitance as derived in Smythe (1989) and usedin Gilbert and Holland (1981)

Cpu,s = ε0

K

√1 −

(w

�y

)2

K

(w

�y

) , (3)

where K(·) is the complete elliptic integral of the first kind. This capacitance (hereafterreferred to as the “static” capacitance) is computed by solving the electrostatic problemof the field distribution between two coplanar strips of finite dimensions. The capacitancein (2) will be referred to as the “dynamic capacitance,” since it is based on the solutionof a time-dependent problem. From (3), an effective relative dielectric constant can bedefined as

εreff ,s = �y

�z

Cpu,s

ε0(4)

for the “static” capacitance. The effective relative dielectric constant for the “dynamic”capacitance is, as expected, frequency dependent. For its computation, the lumped elementformulation introduced by Sui et al. (1992) in the FDTD context is employed:

(∇ × H)y(ω) = jωε0Ey(ω) + Jy,L(ω), (5)

Jy,L(ω) = �y

�zYpu(ω)Ey(ω), (5a)


where the y subscript refers to the y component of the curl operator. The effective relativedielectric constant is therefore given by

(∇ × H)y = jωε0εreff ,d(ω)Ey(ω), (6)

εreff ,d(ω) = 1 + �y

�z

Ypu(ω)

jωε0. (6a)

Furthermore, for the reasons mentioned in Gilbert and Holland (1981), the effectiverelative permeability of the slot must be also modified in accordance with

εreff ,d(ω)µreff ,d(ω) = 1, (7)

µreff ,d(ω) = 1

εreff ,d(ω). (7a)

It is apparent that the slot is essentially modeled as a dispersive magnetic material. Itshould be noted, however, that the definition of the effective constitutive parametersis purely mathematical; such materials do not exist in nature, since they violate theKramers–Kronig condition (Jackson, 1998). In fact, the relative permeability can even beless than unity in some cases.

Consequently, the problem of modeling a thin slot in FDTD is reduced to that ofmodeling a dispersive material with parameters given by (6a), (7a). Two alternative meth-ods for implementing this model in FDTD are presented in the section “Implementationof the Thin-Slot Algorithm.”

Thin-Layer Algorithms

The three one-dimensional algorithms proposed by Maloney and Smith (1992b), Bergheet al. (1998), and Sarto (1999) are well known and their derivations are omitted. Onlythe salient points of each algorithm will be briefly presented. The algorithm by Maloneyand Smith (1992b) is based on the integral form of Maxwell’s equations and essentiallymodels the layer with a modified conductivity depending on the layer thickness and con-stitutive parameters. The modified conductivity is then used in the standard FDTD updateequations for the appropriate field components. The main assumption of the algorithmis that the field distribution inside the conductor is uniform. Of course, since the fieldsattenuate exponentially inside the conductor, this assumption is violated, especially forhigh frequencies or conductivities (i.e., when the skin depth becomes small); thereforethe algorithm is expected to perform poorly at high frequencies.

The algorithm by Berghe et al. (1998) assumes that the displacement current inthe layer is negligible with regard to the conductivity current (i.e., the layer is a “goodconductor”). Under this assumption, which is valid for most conducting and/or compositematerials, the field inside the layer can be assumed to propagate normal to the layer.Therefore, the one-dimensional wave equation can be used, and a two-port networkrelationship can be derived between the tangential electric and magnetic fields. Once asufficiently accurate partial fractions expansion has been obtained for the transfer functionof the two-port network, appropriate time-domain first- and second-order differentialequations are obtained. The differential equations are transformed into the correspondingdifference equations in accordance with the temporal separation of the fields, and a newset of update equations is derived for the tangential electric fields.


The algorithm by Sarto (1999) is similar to Berghe et al. (1998) in that a two-portnetwork representation of the layer is derived. Using again a partial fractions expansionfor this expression and transforming into the time domain, a set of convolutions betweenexponentials and the electric fields is performed at each time step. The exponential natureof the convolution arguments results in a recursive evaluation of the convolution integrals,which is very efficient from the computational point of view, and an update equation forthe tangential electric fields is derived.

Implementation of the Thin-Slot Algorithm in FDTD

In this section, the special FDTD update equations for the thin-slot model proposedearlier are derived, using two methods.

Method I

In principle, the inverse Fourier transforms of εreff ,d(ω), µreff ,d(ω) should be computedand two convolutions should be performed in the time domain between εreff ,d(t), Ey(t)

and µreff ,d(t), Hz(t), as explained in Taflove and Hagness (2000). This approach isexpected to be the most accurate but also the most difficult to perform analytically,especially due to the logarithmic dependence of εreff ,d(ω). As an alternative solution, themean value of Cpu,d(ω) can be computed analytically and, from that, a correspondingmean value for the dielectric constant can be determined using (4). Since the resolutionof the FDTD grid imposes an upper bound on the frequency, or equivalently a lowerbound on the wavelength, it is natural to compute the mean value of the aforementionedquantities in the interval [0 !], where ! = 2πfmax, with fmax being the frequencywhere the wavelength is �y/10. Therefore, the mean value of the dynamic capacitanceis found to be

Cpu = 1

!

∫ !

0Cpu,d(ω)dω (8)

and a mean value εr,m for the relative dielectric constant can be computed by using (4) andreplacing Cpu,s with Cpu. The relative permeability µr,m is also changed accordingly sothat the condition in (7) is satisfied. Then, these two mean values are used in the standardupdate equations for Ey and Hz instead of εr , µr , respectively. The update equations arewell known, and they will not be repeated here. All other field components are updatedusing the free-space constitutive parameters ε0, µ0.

Method II

Method I takes the dispersive nature of the effective slot material into account, but inan average sense only. A more accurate approach would be to perform the convolutionbetween εreff ,d(t), Ey(t) or equivalently between Ypu,d(t), Ey(t), either analytically ornumerically, and derive a modified update equation from that. The starting point is thetransformation of (5) into the time domain

(∇ × H)y(t) = ε0∂Ey

∂t+ �y

�zYpu(t) ∗ Ey(t), (9)

Ypu(t) being the inverse Fourier transform (IFT) of Ypu(ω). Due to the logarithmicdependence of Ypu(ω), the IFT of this function does not exist in the classical calculus


context, since the function is not bounded. However, the IFT does exist in the context ofdistribution theory (Richards & Youn, 1990). Specifically, the following Fourier transformrelation is taken from Richards and Youn (1990):

− 1

2|t |F T−−→ ln |ω| + γ, γ = 0.577: Euler’s constant, (10)

so that the IFT can be computed, after some manipulations, as

Ypu(t) = −ε0

2πt2− ε0

2πt |t | + ε0

π

[4.232

2+ γ − ln

(w

c0

)]δ′(t). (11)

Transforming (6a) into the time domain and substituting (11) into it yields

εreff ,d(t) =[

1 + �y

�z

4.232

2π− �y

�z

1

πln

(w

c0

)+ γ

π

]δ(t) + 1

2π

(1

t+ 1

|t |)

. (12)

The dispersive nature of the slot material is evident from the fact that in (12) there arefunctions of infinite duration. If only Dirac delta functions were present, the materialwould be nondispersive. The effect of this on the FDTD simulations is that the fields ateach time step depend on the entire time history and not only the previous time step, asis the case with nondispersive materials.

The update equation for Ey can be derived either from (9) when evaluated at t =(n + 1/2)�t or, equivalently, from

(∇ × H)n+1/2y = ε0

∂

∂t[εreff ,d(t) ∗ Ey(t)]n+1/2. (13)

Using a piecewise-continuous linear interpolation between successive time steps for Ey(t)

of the form

Ey(t) =n∑

k=0

[Ek

y + Ek+1y − Ek

y

�t(t − k�t)

]· [U(t − k�t) − U(t − (k + 1)�t)], (14)

where U(t) is Heaviside’s unit step function, the convolutions in either (9) or (13) canbe performed analytically. Both equations produce the same final update equation, but(9) is slightly easier to start from. It must be noted that there is a singular term in theconvolution at τ = (n+1/2)�t , which requires that the Cauchy principal value be takenin the integration. After some arrangements of terms, the final update equation for Ey

becomes

En+1y = P1

P2En

y + �t

ε0P2(∇ × H)

n+1/2y − 1

πP2

�y

�z(Sn

1 + Sn2 ), (15)

where

P1 = 1 + �y

π�z

[4.232

2− ln

(w

c0

)+ γ − 1 + ln

(�t

2

)],

P2 = P1 + 2

π

�y

�z,

(16)


and

Sn1 =

n∑k=0

Ek

y

n − k + 1

2

− Ek+1y

n − k − 1

2

,

Sn2 =

n∑k=0

(Ek+1y − Ek

y) ln

∣∣∣∣∣∣∣n − k + 1

2

n − k − 1

2

∣∣∣∣∣∣∣ .

(17)

As mentioned before, the relative permeability of the material must also be a function offrequency so that (7) is satisfied. The time-domain form of (7) is

εreff ,d(t) ∗ µreff ,d(t) = 1δ(t), (18)

so that the determination of µreff ,d(t) essentially becomes a deconvolution problem.Unfortunately, this problem is very difficult to solve exactly; as a result, the mean valueof the relative permeability in (7) will be used. Thus, the update equation for Hz becomes

Hn+1/2z = H

n−1/2z − �t

µr,m

(∇ × E)ny. (19)

Of course, this will violate the condition in (18) and some artificial reflections on the slotwill be produced. These reflections are expected to be more severe at higher frequenciesand almost negligible at very low frequencies. This can be explained by the fact that theearly time response is dominated by the high frequencies of the excitation signal, whereasthe late time response corresponds to the low frequencies. Since (7) is satisfied only forthe mean, or almost dc, values, it is expected that the simulation will be accurate for lowfrequencies. However, since (7) is not satisfied for higher frequencies, some errors areintroduced. This will be further verified from the results.

Results

Shielding Effectiveness of a Cavity Using the Thin-Slot Algorithm

In this section, the shielding effectiveness (SE) of the cavity shown in Figure 1b ispredicted using the thin-slot FDTD algorithm described earlier, and the results are com-pared with measurements performed in the ElectroMagnetic Anechoic Chamber (EMAC)of Arizona State University. The SE will be computed for the electric fields only accord-ing to the plane wave definition of SE as detailed in Georgakopoulos et al. (2001). Acell size of �x = �y = �z = 1 cm is chosen, so that the slot’s height is half of thecell size. The standard FDTD method would require a cell size of 0.5 cm at the most,so an eightfold reduction in memory and simulation time is achieved. A plane wave,of Gaussian time dependence, impinges normally to the slot and the Ey component iscomputed at the center of the cavity, both when the cavity is present and when it isabsent. It should be noted that due to the highly resonant nature of the geometry, therate of decay of the time-domain waveforms is slow and, as a result, a large number oftime steps is required. Therefore, a Hanning window is used in a postprocessing stageto force the time-domain fields to decay to zero fast (Georgakopoulos et al., 2001). This


results in only 20,000 time steps being needed for the simulation, whereas the normalnonwindowing procedure of obtaining the SE requires over 100,000 time steps.

A comparison between the static and frequency-dependent values Cpu,s and Cpu,d isdepicted in Figure 2a. It is clear that the static value of the capacitance is approximatelyequal to the mean value of Cpu,d . Specifically, Cpu,s = 11.31 pF, and the mean valueof Cpu,d is Cpu = 12.08 pF. It will become apparent soon that this discrepancy affectsthe results more than would normally be anticipated. Figure 2b shows the imaginary part(susceptance) of the pu slot admittance for Methods I and II (in Method I, the imaginarypart of the slot admittance is just ωCpu) and Figures 3a and 3b compare the performanceof the thin-slot algorithm (labeled as “Static” in Figure 3a) proposed by Gilbert andHolland (1981) with Methods I and II. Method I predicts the resonance at 1420 MHz,which also agrees with measurements; the method by Gilbert and Holland (1981) failsto do so.

Figure 4 presents the relative error with respect to the measurements for the “Static”method, Method I, and Method II, in a frequency range between 250 (the lowest frequencyfor which measurements could be performed in the EMAC) and 650 MHz (just belowthe first resonance). It becomes clear that, in this region, Method II outperforms theother methods in terms of accuracy. It is also surprising that Method I, which uses themean value Cpu, significantly outperforms the “Static” method, which uses the staticvalue Cpu,s , although the difference between the two values of capacitance is very small.Furthermore, as the frequency increases, Method II suffers in accuracy, for the reasonexplained in the section “Formulation of the Thin-Slot Algorithm,” while at the same time,Method I maintains acceptable accuracy. In terms of simulation time, Method I required45 minutes for 20,000 time steps, compared to 160 minutes required by Method II. Thereason for this difference is that, in Method II, the IFT of Ypu,d(ω) cannot be expressedas a sum of exponentials, so the convolution cannot be performed recursively (Sarto,1999) and the entire time-history must be stored. This introduces a severe overhead tothe simulation time. Nevertheless, Method II is still preferable over the standard FDTDmethod, since the latter requires 650 minutes for the same number of time steps.

Shielding Effectiveness of a Plane Slab Using Thin-Layer Algorithms

In this section, a comparison between the three one-dimensional thin-layer algorithmsby Maloney and Smith (1992b), Berghe et al. (1998), and Sarto (1999) is performedfor a simple example. The geometry of interest is an infinite-in-extent plane conductingslab of thickness d = 1 mm and constitutive parameters σ = 104 S/m, εr = 2. Aplane wave impinges normally to the slab and the transmission coefficient is computedusing the above three algorithms and compared to the exact solution (Balanis, 1989).The inverse of the transmission coefficient is defined as the SE. The relative error ofthe three algorithms with regard to the analytical solution is plotted versus frequency inFigure 5a, whereas Figure 5b shows the same error at lower frequencies. The algorithm byBerghe et al. is implemented in two different ways, one using second-order real-valueddifferential equations, and one using first-order complex-valued differential equations,for each pair of complex poles. It is observed that the method by Maloney and Smith(1992) outperforms the other methods at very low frequencies (as shown in Figure 5b).However, at higher frequencies the error rises to unacceptable levels. This is due tothe nonuniformity of the field inside the conductor, as explained earlier. The model byBerghe et al. (1998) takes into account the field variation inside the conductor, so it ismarkedly better than Maloney and Smith (1992b). Furthermore, the implementation with


(a)

(b)

Figure 2. (a) Comparison of the pu slot capacitance for Methods I and II. (b) Comparison of thepu slot susceptance for Methods I and II.


(a)

(b)

Figure 3. (a) SE of the cavity as predicted by the Static method, Method I, and measurements.(b) SE of the cavity as predicted by Method II and measurements.


Figure 4. Relative error of the various methods compared to measurements.

second-order real differential equations is more accurate than the one with first-ordercomplex differential equations. Nonetheless, Sarto (1999) exhibits the best performancein the entire frequency range, except for very low frequencies where Maloney and Smith(1992b) performs at its best. It is also seen that Sarto (1999) is significantly more accuratethan Berghe et al., especially at higher frequencies, although both algorithms are basedon the same two-port representation. A possible reason may be that although Sarto(1999) evaluates the convolution integral of the exponentials with the electric field asthe convolution of the exponentials with a piecewise-continuous approximation of it, theresulting integral is computed analytically; hence, the only error is due to round off.On the other hand, Berghe et al. (1998) is based on a differential equation, which istransformed into finite-difference form, with no further knowledge of the field variationbetween successive time steps. Since no assumption is made regarding the field variation,it is intuitively expected that the method will not perform as well as Sarto (1999).

From the above comparison, it is concluded that the method by Sarto (1999) is, ingeneral, the best of the three examined methods. Therefore, this method is generalizedand is used to perform SE predictions for more complex three-dimensional geometries.Specifically, the method is used to predict the SE of the same cavity geometry as theone shown in Figure 1b but when the slot is absent (i.e., the front face is solid). Thereason that a completely closed structure is selected is because the objective is to study theperformance of the thin-layer algorithms only. The presence of the slot would significantlyaffect the shielding properties of the structure, so that no solid conclusions can be drawn.On the other hand, since the cavity is completely closed, very high levels for SE areexpected. This is verified in Figure 6, where the SE is plotted versus frequency for varyingwalls thickness. It is observed that the resonant frequencies of the cavity remain the same,as they should, since the dimensions of the cavity do not change. Additionally, as the


(a)

(b)

Figure 5. (a) Relative error of the three thin-layer algorithms compared to the analytical solution.(b) Low-frequency zoom on the relative error of the three thin-layer algorithms.


Figure 6. SE of cavity for varying wall thicknesses using the model by Sarto (1999).

wall thickness increases, the average value of the SE also increases. For d = 15 ·10−7 m,the SE levels are similar to the levels for the cavity shown in Figure 1b (when the slot ispresent). This is another verification of the fact that the presence of slots in conductingsurfaces dominates the shielding properties of the geometries.

Conclusions

A new thin-slot model and its implementation in FDTD were presented in this paper andapplied to the problem of SE determination for cavities. It was shown that the new modeloutperforms the model by Gilbert and Holland (1981) in terms of accuracy, especiallyat low frequencies, and although a slight overhead in simulation times was introduced,significant overall savings in computational resources was achieved. The inaccuracy ofthe proposed thin-slot model in higher frequencies was accounted for; alternative, moreaccurate, approaches for performing the deconvolution are currently under investigation.

Furthermore, three existing thin-layer algorithms were compared for a one-dimen-sional example and it was concluded that the algorithm by Sarto (1999) offers the bestperformance. This algorithm was generalized and applied to three-dimensional SE pre-dictions and, although there was no comparative data, the expected trends were observed.

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