TIME DOMAIN PHYSICAL OPTICS FOR THE HIGHER ...Progress In Electromagnetics Research, PIER 95,...

Progress In Electromagnetics Research, PIER 95, 87–102, 2009

TIME DOMAIN PHYSICAL OPTICS FOR THE HIGHER-ORDER FDTD MODELING IN ELECTROMAGNETICSCATTERING FROM 3-D COMPLEX AND COMBINEDMULTIPLE MATERIALS OBJECTS

F. Faghihi and H. Heydari

Center of Excellence for Power System Automation and OperationElectrical Engineering DepartmentIran University of Science and TechnologyTehran, Iran

Abstract—This paper proposes a hybrid methodology that combinesan extended form of Finite-Deference Time-Domain (FDTD) methodwith Time Domain Physical Optics (TDPO) for analysis of 3-D scattering of combinative objects in complex electromagneticcompatibility (EMC) problems. Establishing a covariant formulationfor FDTD, the extended algorithm introduces a parametric topology ofaccurate nonstandard schemes for the non-orthogonal div-curl problemand the suppression of lattice dispersion. For complex-combinedobjects including a small size (SS) and large size (LS) parts, usingTDPO is an appropriate approach for coupling between two regions.Thus, our technique solves the EMC complexity with the help of higherorder FDTD (HOFDTD) and the combinatory structures by using theTDPO. Numerical validation confirms the superiority of the proposedalgorithm via realistic EMC applications.

1. INTRODUCTION

The accurate and efficient evaluation of the scattering from alarge and complex object is of great interest in engineering [1–4]. Among existing numerical techniques, the finite-difference time-domain (FDTD) method has a wide applicability in many areasof research [5]. Higher order (HO) 3-D FDTD methodology tobe a powerful numerical method for such complex and multiplematerials structures simulation [6–9]. Furthermore, the development

Corresponding author: H. Heydari ([email protected]).

88 Faghihi and Heydari

of high power microwave antenna, aviation and space techniques hasmotivated the interest in electromagnetic scattering problems involvingcombinative objects [10, 20]. The combinative objects are composedby two parts: One is a Small-Size (SS) configuration and the otheris a Large-Size (LS) component with respect to the wavelength ofinterest, such as line surface formation objects, a reflector antennaand a satellite with large wingspan. Generally, neither rigorousnumerical method nor asymptotic scheme is easily be implemented inevaluating the time domain scattering for combinative objects exactlyand efficiently.

So the hybrid method combining numerical technique withasymptotic approach is invoked in dealing with this class of problems.For complex-combined objects including a SS and LS parts, usingTime Domain Physical Optics (TDPO) [21, 22] is suitable techniquefor coupling between two regions. This paper develops a numericalmethod that combines higher order FDTD (HOFDTD) with TDPOfor the scattering problem by Complex and combinative objects, inwhich the coupling must be considered by any means. The extendedFDTD and TDPO are taken to treat the SS structure and LS part,separately. As the approach is performed in the time domain, it isexpected to be appropriate for broad band analysis.

In the following section, we introduce the HOFDTD algorithm tosolve complex structure in EMC issues. In Section 3, novel hybridnumerical method that is a combination of HOFDTD with TDPO forelectromagnetic scattering from 3-D complex and combined multiplematerials objects is explained. In Section 4, two numerical exampleswhich refer to fundamental problems in EMC applications are given todemonstrate the validity of the proposed method and a comparison ismade with the results of far field to near field transformation obtainedin a previous research.

2. PRELIMINARY ASPECTS: THE GENERALIZEDHOFDTD ALGORITHM

When objects of arbitrary curvature and combined materials (multi-materials) are to be modeled, the FDTD technique exhibits an ill-suited behavior, mainly, due to the incomplete imposition of continuityconditions at the interfaces [7, 23]. Unluckily, to these errors onemust add the lattice dispersion discrepancies intrinsic in the low-orderschemes. A feasible clarification could be the averaging of constitutiveparameters at the interfaces [7].

The essential premise of the HO algorithm resides in therepresentation of electromagnetic fields via a new class of 3-D non-

Progress In Electromagnetics Research, PIER 95, 2009 89

standard concepts [23–25]. Their form is given by [5, 23]

WMq,L[f

∣∣tu,v,w] =

g(u, v, w)CS(kLδq)

M∑

m=1

Rqm ×

{ L∑

l=1

P qm,lD

(m)q,lδq[[f

∣∣tu,v,w]]

}(1)

where M is the order of accuracy and q is a variable of the generalcoordinate system (u,v,w) described by its respective g metrics. Such adifferencing rationale ensures that dispersion error mechanisms, havingthe potential to spoil the final outcomes, are drastically subduedor even completely eliminated [8]. On the other hand, parametersRq

m and P qm,l achieve an inherent robustness both in the handling of

geometric details and the right assignment of field quantities to space-time entities by satisfying the following gauges

L∑

l=1

Rqm,l = 1 ∀m (2)

M∑

m=1

P qm =

12

(3)

Factor L defines the number of stencils, lδq, along each axis, which areneeded for the accomplishment of a specific accuracy with a typicalvalue of L = 3. The correction function CS(klδq) ensures the smoothtransition from the continuous to the discrete state. Its argument,depending on wave number k, is selected to handle broadbandelectromagnetic excitations with the non-standard concepts [8]. This isconducted by employing the Fourier transform of the already computedelectric or magnetic vectors at predetermined lattice positions [26]. Theprocess does not affect the total overhead, whereas its efficacy increaseswith the number of prefixed nodes. A possible choice of CS could be

CS(klδq) =16k2

sin(

klδq

3

)cos

(klδq

9

)(4)

Operators D(m)q,lδq[.], in Equation (1), cover all optimal node

arrangements — irrespective of cell shape — leading to very coarsegrids and mutual cancellation of material discrepancies.

2.1. The Non-orthogonal Dual-Grid Formulation

Let us recall the previously defined coordinate system (u = iδu, v =jδv, w = kδw) and divide a given 3-D domain into uniform cells, as in


Primarycomponents

Primary lattice

Dualcomponents

Dual lattice

Figure 1. A generalized curvilinear dual FDTD tessellation indicatingthe positions of electromagnetic intensity and flux field components onthe edges and faces of primary and secondary grids. The bar denotesquantities in the secondary lattice [8].

Fig. 1. The center of every primary cell is positioned at (i, j, k), whilesecondary ones are centered on the vertices of the primary grid [8].

Overall, the HO methodology creates a set of topologically robustforms, which, by constructing a more consistent cell, diminishes late-time instabilities, eliminates dispersion errors and avoids conformaltechniques [24–26]. For the termination of infinite problems, we launcha curvilinear non-standard version of the PML, based on an unsplit-field realization. The optimized absorber is built using the properscaling that retains the original field variation and incorporates thesuitable coordinate transformation [23].

2.2. The Convergent Treatment of Arbitrarily-AlignedMaterial Interfaces

The effects of stair casing on arbitrarily-embedded media interfaceshave notable consequences on the convergence of the FDTDmethod [25]. To defeat this drawback, a HO procedure is expandedthat modifies the prior stencils and imposes the suitable continuitysituation [27]. Suppose the boundary of Fig. 2, with n = (nu, nv, nw)T

its normal unit vector. The covariant components of Emtcv and Hmt

cvfields in the two material regions, εmt and µmt, for mt = A,B, arecoupled by [8]

n× EAcv = n×EB

cv, εAn.EAcv = εBn.EB

cv (5)

n×HAcv = n×HB

cv, µAn.HAcv = µBn.HB

cv (6)


hw

hu ev

Figure 2. Treatment of a curvilinear arbitrary-aligned materialinterface, which does not coincide with the grid axes defined by ageneral coordinate system [23].

Next, we appoint to each region, a parameter, βmti,j,k, as an instruction

of the distance from the first/last cell to the physical position of thewall relative to cell dimensions [22]. Evidently, βmt

i,j,k belongs to [0, 1/2]and satisfies βA

i,j,k = 0.5 − βBi,j,k. Since these coefficients depend only

on the boundary’s geometrical form, they may be evaluated and storedin a pre-processing step formerly the lattice has been constructed. Inthis framework [23]:

hAu

∣∣∣n+1/2i,j,k =

(1 + βA

i,j,k

)hu

∣∣∣n+1/2i−1/2,j,k + βA

i,j,khu

∣∣∣n+1/2i−3/2,j,k (7)

is punctually derived. To find hu, in material B, we first take intoaccount

4hAv

∣∣∣n+1/2i,j,k = hv

∣∣∣n+1/2i,j−1/2,k−1/2 + hv

∣∣∣n+1/2i,j+1/2,k−1/2 + hv

∣∣∣n+1/2i,j+1/2,k+1/2

+hv

∣∣∣n+1/2i,j−1/2,k+1/2 (8)

2hBw

∣∣∣n+1/2i,j,k = hw

∣∣∣n+1/2i,j+1/2,k + hw

∣∣∣n+1/2i,j−1/2,k (9)

where the tilde stands for auxiliary magnetic variable evaluated by asimple averaging of the adjacent field values. In this way, the unknown


hBu is improved via Equation (7). So:

hBu

∣∣∣n+1/2i,j,k = hA

u

∣∣∣n+1/2i,j,k + nu

(µA − µB

)

×nuhA

u

∣∣∣n+1/2i,j,k + nvh

Av

∣∣∣n+1/2i,j,k + nwhB

w

∣∣∣n+1/2i,j,k

µAn2w + µBn2

u + µBn2v

(10)

Finally, the nonstandard approximation of ∂whu becomes:

LSw

[hu

∣∣∣n+1/2

i,j,k

]=

2(2βB

i,j,k + 1)−1 (

hu

∣∣∣n+1/2i+1/2,j,k − hB

u

∣∣∣n+1/2i,j,k

)

∆u(11)

To apply Equations (7)–(11), a predictor-corrector step is employedwhere the HOFDTD algorithm, being the predictor, solves Maxwell’sequations in the entire domain, while the above technique, performingas the corrector, alters the solutions locally.

3. NOVEL APPROACH FOR ELECTROMAGNETICSCATTERING OF COMPLEX AND COMBINATIVEMULTIPLE MATERIALS STRUCTURES

Physical optics (PO) is a high frequency approximation technique. ThePO phenomena are then used to study the scattering on the groundplane. It is based on an exact formulation of the diffraction problemwell known as the Chu-Stratton Equation [21]. Three simplifyingassumptions are carried out to decrease the vector integral equationto a simple definite integral over the scatterer surface. It is assumedthat:

(a): the surface field over the shadowed portion of the body is zero,(b): the observation point is removed far from the object in term of

wavelength and scattering object dimensions,(c): the dimensions of curvature of the scatterer are large compared to

the wavelength.

For a perfectly conducting body, the far scattering field in time-domainis given [21, 22]:

~E(r, t) = (Z0/4πrc)×∫∫ ′

Sr ×

[r × ∂

∂t~JS

(r′, t− τ

)]dS′ (12)

~H(r, t) = −(1/4πrc)×∫∫ ′

Sr × ∂

∂t~JS

(r′, t− τ

)dS′ (13)


Surface-current density distribution, JS is written as:

~JS

(r′, t

)=

{2n′ × hinc (r′, t) in the lit region0 otherwise

(14)

where hinc is the incident magnetic field. The scattered field can bedetermined by Equations (12) and (13) if the magnetic field incidenton the scatterer is known.

The extended FDTD method has been demonstrated to bean accurate and efficient method to simulate the interaction ofelectromagnetic waves with all kinds of obstacles, including the targetof complex material and complex configuration [23, 24]. Consideringthe configuration of combinative objects, the computation domain isfirstly split into HOFDTD region and TDPO region, enclosing SS andLS structures, respectively. The proposed method uses the two kindsof approach in considering that the total radiated field e(r, t) can bedivided into two terms:

~e(r, t) = ~ei(r, t) + ~ed(r, t) (15)

The first term ei(r, t), represents the far field directly radiated by thecomplex element. It can be simply calculated from the HOFDTDalgorithm. The second term ed(r, t) represents the far field scatteredby the metallic plane. It can be determined thanks to the TDPOterminology if the magnetic field incident on the scatterer hinc, isknown.

In exact expression, the far field can be divided into differentterms [21, 22], as follows: The dominant technique of theHOFDTD/TDPO hybrid approach then consists in the interactionbetween the two regions. First, we consider the influence of HOFDTDregion onto TDPO region, as shown in Fig. 3, providing the primaryscattered field by SS complex configuration has been obtained by usingHOFDTD method.

In order to find the illuminating field onto LS part in TDPOregion from SS configuration in HOFDTD region, the near-to-near fieldextrapolation procedure in HOFDTD is invoked, because HOFDTDregion is close to TDPO region. The field illuminated on the surfaceof LS part can then be defined as:

h(r, t + τ) =∮

S

{− (

n′.(~r−~r′

))[

h′(r′, t)

4π∣∣~r−~r′

∣∣3 +∂h′(r′,t)

∂t

4πc∣∣~r−~r′

∣∣2]

−∂h′(r′,t)

∂n

4π∣∣~r−~r′

∣∣

}ds′ (16)


Figure 3. HOFDTD-TDPO hybrid method for field calculation fromcomplex objects.

At time step n + 1, the time and space derivatives in Equation (16)are approximated by a second-order center difference [28–30]. Afterseveral manipulations we may obtain

h(r, n + 1 + τ/∆t) = F1(n) + F2(n + 1) + F3(n + 2) (17)

where, F1, F2 and F3 are sequences in different time steps [31]. The LSsurface in the TDPO area can be divided into M triangular patches,and the recorded time steps in computation are n∆t (n = 1, 2, . . . , N).Due to the incident magnetic field for each time step and allotmentpoint on the surface, the memory storage needs to be a minimumM ×N+, where N+ > N .

Due to retardation from SS to LS part, the continuance of thearrival waveform in the time domain will last much longer than N∆t.The chronological transfer plan is expanded, so as to decrease thenecessity of the computer storage space. In each time step, we executeHOFDTD extrapolation from a cell on HOFDTD output boundaryto a patch on the surface of LS part, followed by instantly TDPOcalculation of contribution from this patch to the far field surveillanceviewpoint.

The substitution of Equation (17) into the Equations (12)–(14)will yield:

~E(r, n + 1 + (τ1 + τ2)/∆t) =Z0

2πrc(∫∫

S′r ×

{r ×

[n× x

[F1(n) + F2(n + 1) + F3(n + 2)

]hx

]}dS′


+∫∫

S′r ×

{r ×

[n× y

[F1(n) + F2(n + 1) + F3(n + 2)

]hy

]}dS′

+∫∫

S′r ×

{r ×

[n× z

[F1(n) + F2(n + 1) + F3(n + 2)

]hz

]}dS′

) (18)

where, Fi = ∂Fi/∂t, A = Z0/(2πrc), and The F terms with differenttime steps are experientially obtained.

In order to obtain E(n∗) in the observation point, the value at thelast time step must be stored. This technique requires a supplementarymemory.

We know that E is a sequence in discrete time, where n∗ =int {n + 1 + (T1 + T2)/∆t} , and at the (n + 1)th time step, F1(n + 1),F2(n + 1) and F3(n + 1) are computed, and only F2(n + 1) contributesto E(n∗). F1(n+1) and F3(n+1) are added to registers E(n∗+1) andE(n∗ − 1) in a consistent manner respectively. This approach is usedin the proposed algorithm, where the arrows indicate the contributionto E sequence from the FDTD and TDPO iterations.

Now, we present an algorithm of the proposed method as bellow:1- Applying initial value for the incident field,2- “A” calculation: Computing the electric and magnetic field from

complex objects with HOFDTD,3- “B” calculation: Computing the electric and magnetic field for LS

part with TDPO,4- “C” calculation: Computing the electric and magnetic field in

TDPO region using results of primary scattering field from “A”calculation,

5- “D” calculation: Computing the electric and magnetic field inHOFDTD region using results of primary scattering field from“B” calculation,

6- Final result: Scattered field in observation point for far areais obtained from the primary scattering results (“A” and “B”calculation) and secondary scattering results (“C” and “D”calculation).

This algorithm is performed for each time step.Based on prior researches [23–25], the stability of the HOFDTD

does exist and TDPO has intrinsically stability. As such, the stabilityof the proposed method can also be taken for granted.

4. NUMERICAL EXAMPLE

In this section two numerical examples which refer to fundamentalproblems in EMC applications are given to demonstrate the validity


Figure 4. Monopole above finite square ground plane.

Figure 5. Amplitude pattern for a monopole antenna, the distancer = 15 mm.

of the proposed method and a comparison is made with the results ofnear to far field transformation which is presented in [32].

Example 1- To determine the radiation characteristics of amonopole antenna mounted on a large but finite size ground planehas been considered (Fig. 4). The λ/4 antenna is placed on the centerof a plane of dimension 5λ × 5λ and is excited on their base. TheFDTD volume and the ground plane surface are discretized in λ/30.Furthermore, the radiation characteristics of the same λ/4 antennamounted on an infinite plane is performed. It illustrates the effects ofthe finite scatterer dimensions that are well taken into account by theproposed method for a = 5λ, f = 1 GHz.

The results given by the proposed method are compared with theresults based on near-field to far-field transformation method [32] andindicate a good agreement which is shown in Fig. 5.

It should be noted that, due to the simplicity of this examplethe computational speed is similar to the other conventional methodsshown in Fig. 5.

Example 2- We consider another practical example of thebackscattering by combinative objects composed by a perfect electric


Ki

Figure 6. Combinative object composed of complex cube and plate.

t (nS)

rE (

V)

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

Figure 7. Backscattered waveform in time domain for co-polarization.

conducting (PEC) cube and plate [21, 22], as shown in Fig. 6. Amodulated Gaussian pulse with frequency ranging from 200 MHz to300MHz, and pulse width 30 ns excites the combinative complexobject. The incident wave travels in the xoz plane with θ = 45◦with its electric field parallel to the y-axis. The backscattering forco-polarization is to be established. First, we consider the primaryscattering coming from the PEC cube set as an SS structure, andplate as the LS part, respectively. The total scattered field by thestudied cube and plate is shown in Fig. 7. The results obtainedby the proposed method are compared with the results based onnear-field to far-field transformation method [32] and indicates anexcellent conformity which is shown in Fig. 7. It is interesting tonote that, the accuracy and the speed of computation of the proposedmethod, (HOFDTD + TDPO), is about 20% and 9% better thanFDTD+TDPO method, respectively.

Example 3- Suppose the upper front enclosure in Fig. 6 is made ofaluminum material, while the rest of the body is made of iron material,


as shown in Fig. 8. Similar to the Example 2, a modulated Gaussianpulse with frequency ranging from 200 MHz to 300 MHz, and pulsewidth 30 ns excites the combinative complex object. The incident wavetravels in the xoz plane with θ = 45◦ with its electric field parallel tothe y-axis. The backscattering for co-polarization is to be established.First, we consider the primary scattering coming from the bi-materialenclosure set as an SS structure, and plate as the LS part, respectively.The total scattered field by the premeditated enclosure and plate isshown in Fig. 9. It is essential to explain that the results obtainby the proposed algorithm with the help of Equations (5) and (6)for boundary conditions. The results are compared with the resultsbased on near to far field transformation method [32] and indicate anexcellent conformity which is shown in Fig. 9. Although, the resultsdiffer from the field solution of Case 2 (PEC cube) to some extent,they are in good agreement. The differences in the field are due tothe bi-material of the enclosure, in which 2–5% field reduction can beachieved, as compared with PEC.

Furthermore, Fig. 9 will also show that, the speed of computationis very low in the method “FDTD + near to far field transformation”in comparison with the proposed method. On the other hand, theimplementation of the conventional method for field solution of thecomplex configuration with multiple materials (Fig. 8) by ordinaryprocessor can lead to divergence.

As a comparative purpose, suppose there are curve boundarieswith two different radius of curvatures with r = 0.05m and r′ =0.08m, as shown in Fig. 10. In this case, for HOFDTD +TDPOthere is 35% increase in the speed of field solution, comparedwith FDTD + TDPO method. However, FDTD+ Near to Far FieldTransformation the execution time using conventional processor willbe very long depending on the field solution methods. There is a ±13%tolerance between FDTD +TDPO and HOFDTD +TDPO methods.

Ki

Figure 8. Combinative object composed of bi-material complexenclosure and plate.


t (nS)

rE (

V)

0.8

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

HOFDTD+TDPO

FDTD+Near to Far Field

Transformation

Figure 9. Backscattered waveform in time domain for co-polarization.

Ki

Figure 10.

5. CONCLUSION

A new approach to study small radiation source from complicatedconfiguration close to large scatterer was developed. The approachcan be applied to the analysis of the electromagnetic scatteringin EMC challenge by complex and combinative multiple materialsobjects including both SS and LS. The first-order scattering field byLS part in TDPO region is considered as the illuminating field onthe LS structure in HOFDTD region, when analyzing the couplingof TDPO to HOFDTD region. The illuminating wave from LSpart to SS structure can be introduced through the connectionboundary in HOFDTD region. Numerical results for both elementarystructure and complex configuration confirm the superiority of theproposed algorithm via realistic Electromagnetic applications. Also,computational comparisons are made with the method based onnear-field to far-field transformation and recently proposed method.Specifically, the simulation results for the two different enclosuresindicate that the approach can be used for postulate of this paper.


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