HIRF penetration and PED coupling analysis for scaled ...rosie/mypapers/paper54.pdfThe FDTD(2,4) method is expected to fail to accurately model infinitesimally thin PEC films. This

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 45, NO. 2, MAY 2003 293

HIRF Penetration and PED Coupling Analysis forScaled Fuselage Models Using a Hybrid Subgrid

FDTD(2,2)/FDTD(2,4) MethodStavros V. Georgakopoulos, Craig R. Birtcher, Constantine A. Balanis, and Rosemary A. Renaut

Abstract—A hybrid method of subgrid FDTD(2,2) withFDTD(2,4) is presented. Both the standard FDTD(2,2) as well asthe hybrid technique are applied to shielding effectiveness analysisof a scaled model of a Boeing 757. Also, analysis of EMI generatedby personal electronic devices is performed on the same scaledfuselage model.

Index Terms—Coupling, FDTD, higher order, penetration, sub-grid.

I. INTRODUCTION

W ITH the clock speed of all electronic equipment in-creasing, classical engineering analysis tools have

become obsolete. As the frequency of operation increases,devices become electrically large. These problems yield largecomputational domains and they require significant amountof computational resources, such as memory and executiontime. Traditional finite methods like the finite-differencetime-domain (FDTD) method and the finite element method(FEM) are second-order accurate, thereby restricting the sizeof the domains that can be handled efficiently.

Numerous attempts have been made in the field of FDTDresearch to minimize phase errors [1], [2]. One of the mostpromising approaches is based on higher order accuracyschemes [3]–[6]. Such schemes theoretically exhibit lowerdispersion errors and can utilize coarser grids as compared tothose needed to achieve comparable levels of accuracy witha second-order scheme. Specifically, the second-order in timeand fourth-order in space FDTD(2,4) method will be usedthroughout this research.

In this paper, a detailed literature review is presented re-garding the challenges of higher order boundary condition(BCs). Furthermore, some of the problems associated with per-fect electric conductor (PEC) BCs in the context of FDTD(2,4)are discussed. Finally, a hybrid technique of FDTD(2,4) withsubgrid FDTD(2,2) is formulated and applied to practicalengineering problems.

Manuscript received December 10, 2001; revised January 10, 2003. Thiswork was sponsored by the National Aeronautic and Space Agency underGrant NAG-1-1781 and Cooperative Agreement NCC-1-01051.

S. V. Georgakopoulos is with the SV Microwave West Palm Beach, FL 33409USA (e-mail: [email protected]).

C. R. Birtcher and C. A. Balanis are with the Department of Electrical Engi-neering, Arizona State University, Tempe, AZ 85287-7206 USA.

R. A. Renaut is with the Department of Mathematics, Arizona State Univer-sity, Tempe, AZ 85287-1804 USA.

Digital Object Identifier 10.1109/TEMC.2003.811308

Specifically, two very important EMI problems are examined.First, the shielding effectiveness of a simplified scaled model ofa Boeing 757 aircraft is calculated. A critical EMI/EMC issuethat is relevant to all aviation, and which has lately attracteda lot of attention, concerns the penetration of high-intensityradiated fields (HIRF) into conducting enclosures via apertures.On numerous occasions, it has been proven that EM sourcesexternal to the aircraft have caused several problems to theequipment of airplanes, such as disrupted communications,disabled navigation equipment, etc. This is a matter of para-mount importance for the safety of commercial airplanes, andits investigation is necessary to ensure the integrity of electricaland electronic equipment under an external EMI/EMC or HIRFthreat. Both the standard FDTD(2,2) and the hybrid of subgridFDTD(2,2) and FDTD(2,4) are used for the predictions whichare validated by comparison with measurements.

Besides the penetration of man-made radio frequency (RF)signals external to the airplane into the fuselage (HIRF), an-other very important EMI issue concerns interference that canoccur on-board. Such EMI can potentially be generated by per-sonal electronic devices (PEDs), such as laptop computers, cellphones, etc. It is a common policy of all commercial airlinesto prohibit the use of PEDs during at least the very sensitivephases of take-off and landing, if not for the entire duration offlights. This policy has been established because it is believedthat radiated emissions from PEDs can interfere with on-boardelectronics and cause problems to their operation; e.g., jammingin communication systems.

One possible mechanism for this PED interference can be es-tablished via the antennas which are mounted on the fuselageand support different types of communication. Electromagneticfields radiated by PEDs can be transmitted to the exterior ofan aircraft through the fuselage apertures (windows). Further-more, these RF fields can be received by communication or nav-igation related antennas mounted on the skin of the fuselage.Again both the standard FDTD(2,2) and the hybrid of subgridFDTD(2,2)/FDTD(2,4) are applied for the predictions which arevalidated by comparison with measurements.

II. BCs FOR HIGHER ORDER SCHEMES

A. Challenges

The modeling of complex structures introduces additionalchallenges in high-order FDTD for the correct formulationsof BCs and discontinuities. Many different approaches for

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294 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 45, NO. 2, MAY 2003

the treatment of BCs have been proposed in the literature.Most of this work has appeared in the field of computationalmathematics and physics. The common method of dealingwith the two issues of BCs and discontinuities is to implementone-sided higher order finite differences. However, suchone-sided stencils cause instabilities which are usually verydifficult to resolve.

When BCs are applied, there are two main numerical issuesthat have to be addressed; accuracy and stability. In [7], it wasshown that in order to retain the formal accuracy of a higherorder scheme, BCs should be implemented with either the sameaccuracy as the one of the interior scheme or at least one orderless. For Cartesian meshes and for most problems, BCs that sat-isfy the overall accuracy of the interior scheme are usually pos-sible to derive. However, the difficulty is to derive high-orderaccurate andstableoperators.

Carpenteret al. [8] examined the stability characteristics ofvarious compact fourth- and sixth-order spatial operators by ap-plying the theory of Gustafsson, Kreiss, and Sundstrom (G-K-S)[9]. It was shown that many higher order schemes that are G-K-Sstable (stable in the classical sense) are not stable in the strictsense. Moreover, it was found later in [10] that most high-orderschemes that were strictly stable for scalar problems, they wereunstable for systems of equations. A new method, for designinghigher order schemes stable both in the classical and the strictsense, was also proposed in [10]. This method uses the sum-mation-by-parts (SAT) procedure to construct derivative oper-ators that satisfy the SAT formula. For higher order explicitformulations, the work by Strand [11] was used, whereas forhigher order compact schemes a new methodology was derived.It was proven that compact schemes satisfying just the SAT for-mula are not necessarily strictly stable unless a specific pro-cedure of imposing BCs is followed. This approach was re-cently generalized to two-dimensional (2-D) problems in [12],[13]. BCs in the context of higher order schemes have alsobeen discussed in [14]–[16]. Additionally, instabilities causedby one-sided high-order BCs were resolved by using an artifi-cial dissipation in [17]–[22] or filtering [23].

A method of dealing with 2-D material discontinuities waspresented in [24]. Special one-sided difference operators werederived and applied to both metal and dielectric interfaces. How-ever, the analysis was limited to one-dimensional (1-D) and 2-Dexamples with metal and/or dielectric interfaces, and a three-di-mensional (3-D) free-space example.

All methods that appeared in the papers discussed above, al-though promising, have not yet been verified for complex 3-Dproblems for which BCs may be required not only on the ex-ternal boundary of the computational space but also in the in-terior of the domain. Their analysis was restricted to 1-D and2-D free-space problems with BCs needed only on the outerboundary of the domain. However, in most practical engineeringproblems, imposition and treatment of BCs is mainly needed in-side the computational domain for both 2-D and 3-D problems.Such problems cannot be solved using methods that have al-ready been proposed in the literature.

B. Description of the Problem

This paper deals with shielding effectiveness analysis of con-ducting rectangular boxes with apertures. In such problems, the

Fig. 1. Field components around a PEC discontinuity.

accuracy of the numerical results depends greatly on the mod-eling accuracy of penetration mechanisms. Most boxes of in-terest are constructed using metals of high conductivity, e.g.,copper, aluminum, etc., and thereby almost all of the wave pen-etration is established through the apertures of the boxes. InFDTD simulations of such boxes, it is common practice to sim-ulate the metal walls as PEC walls, due to again their high con-ductivity. Therefore, modeling of PEC BCs is very importantfor penetration analysis.

Even though the standard FDTD simulates accurately, PECBCs, this does not imply that all higher order FDTD schemesnecessarily do the same. In fact, the higher order FDTD schemeconsidered here, FDTD(2,4), does not implement correctly PECBCs.

The FDTD(2,4) method is expected to fail to accuratelymodel infinitesimally thin PEC films. This can be seen in Fig. 1where a 1-D grid is shown with a PEC plane separating tworegions of free space. The fields on one side of the PEC planewhen updated using the FDTD(2,4) stencil (see Appendix )will couple to the fields on the other side of the plane. This isillustrated graphically in Fig. 1 for one of the components(doubly underlined in the figure) located on the right of the PECplane. This component is updated using the componentswhich are inscribed in a box. It is clearly seen that the stencilreaches across the PEC boundary and therefore coupling occursfrom one side of the PEC boundary to the other. Such couplingis not physical and should not happen for a PEC boundary.

One way to overcome this problem is to implement the PECBCs using one-sided differences. Another way is to use imagetheory to modify the FDTD(2,4) stencil for the fields just be-side the PEC. Both ways do not allow the fields from one sideof the PEC plane to couple to the other. For details on the use ofimage theory implementation see [25] and [26]. These two wayswere found to be effective for most 1-D and 2-D problems butunstable or ineffective for most practical 3-D problems wherePEC planes are not infinite in extent. Specifically, one-sided dif-ferences yield instability problems and this finding is supportedby the line of research discussed above. Moreover, the use ofimage theory is limited to cases where the PEC walls are in-finite in extent. Therefore, it is clear that there is a need for ahigher order method that can simulate correctly PEC BCs. In

GEORGAKOPOULOSet al.: HIRF PENETRATION AND PED COUPLING ANALYSIS FOR SCALED FUSELAGE MODELS 295

Fig. 2. 2-D view of the coarse and fine grids.

Fig. 3. Respective locations in time of the electric and magnetic fields for boththe coarse and the fine grids.

the next section, a method that benefits from the advantages ofboth FDTD(2,2) and FDTD(2,4) is presented.

III. H YBRID OF SUBGRID FDTD(2,2)AND FDTD(2,4)

The proposed approach in this section consists of combininga subgridding technique with a higher order scheme. Subgrid-ding techniques have been used in the past in the context of thestandard FDTD [27], [28]. Also, Chevalieret al. presented in[29] a new subgridding technique that allows the boundary be-tween the coarse and the fine grid to be located in an inhomoge-neous dielectric region. Moreover, odd integer-cell ratios wereused, e.g., 1:3, 1:5, 1:7, etc, so that the fields in the coarse andfine grids stay synchronized in time and collocated in space. Thetime and space arrangement of the fields in the fine and coarsedomains for a 1:3 cell ratio are depicted graphically in Figs. 2and 3, respectively.

In [30], a hybrid formulation of FDTD(2,4) and subgridFDTD(2,2) was presented. The formulation of that hybridmethod assumed that everywhere in the domain FDTD(2,4)is applied except small areas of the domain where subgridFDTD(2,2) is used. These parts where subgrid FDTD(2,2)is used were assumed to be internal to the entire FDTD(2,4)

domain. For example, see in Fig. 4(a), the domain of the twomonopoles that was analyzed in [30], and notice that subgridFDTD(2,2) was used only around the two monopoles. Theareas where subgrid FDTD(2,2) was applied are obviouslyincluded in the FDTD(2,4) domain.

However, there might exist problems where the oppositeconfiguration of meshes (coarse and thin) occurs, i.e., thearea where FDTD(2,4) is used is contained in the subgridFDTD(2,2) domain. One example of such a case comes fromshielding effectiveness analysis of electrically large rectangularenclosures. In these cases, the largest part of the computationaldomain is the interior of the enclosures, and the problem ofBCs arises near the walls of the enclosures. In such cases, itis desired to simulate the propagation inside the box using ahigher order method such as FDTD(2,4). In addition, near thewalls of the boxes a subgrid FDTD(2,2) method should be usedin order to represent accurately the PEC BCs, and successfullysimulate the penetration mechanisms. FDTD(2,2) is used nearthe walls instead of FDTD(2,4) since FDTD(2,4) exhibits aninherent artificial penetration through thin PEC films as shownabove. Also, as discussed in Section II, stable higher order BCsthat would simulate correct PEC discontinuities do not existand they are very challenging to derive. Therefore, subgridFDTD(2,2) is hybridized with FDTD(2,4) to resolve all theseissues. Following such a procedure, yields tremendous savingsin memory and/or time depending on the particular problem.A schematic representation of such a problem is depicted inFig. 4(b). In order to implement this new type of hybrid, a newformulation has to be implemented. This new hybrid is namedsubgrid FDTD(2,2)/FDTD(2,4) which is the reverse of thename of the hybrid method in [30].

A. Method

This new hybrid method resembles the hybrid method pre-sented in [30]. The boundary between the fine and the coarsegrid is collocated with electric field components. The ratio be-tween the coarse and the fine grid cell sizes is chosen to be 1:3.

A 2-D view of the grid is shown in Fig. 5 which representsa cross section of the computational grid. In Fig. 5, several fea-tures should be pointed out as follows.

1) The external thick line represents the walls of the con-ducting enclosure. On and outside this boundary, onlysubgrid FDTD(2,2) is used on a fine grid.

2) The internal thick line represents the boundary betweenthe fine and the coarse grid domains. On and inside thisboundary only FDTD(2,4) is used on a coarse grid. Thedomain that is described by this boundary is labeleddo-main#1 for reference purposes.

3) In the area between the two thick lines, both subgridFDTD(2,2) (on a fine grid) and FDTD(2,4) (on a coarsegrid) are applied. This area is labeled Domain #2 for ref-erence purposes.

4) The distance between the walls of the conducting enclo-sure and the coarse-fine grid boundary is chosen to be 3coarse grid cells, so that the standard FDTD(2,4) usingcentral finite difference spatial scheme can be applied onthe coarse-fine grid boundary.


(a) (b)

Fig. 4. Schematic visualization of the hybrid method of FDTD(2,4) and subgrid FDTD(2,2) presented in the previous section.

Fig. 5. 2-D view of the coarse and fine grids.

To make the method stable, the two types of weighting usedin [29] are applied. First, the electric fields near the coarse–fineboundary are weighted as

(1)

where is the weighted electric field value of the fine grid justadjacent to the coarse-fine grid boundary, and are thefields computed by FDTD(2,2) in the fine grid, and is thefield computed by time and space interpolation of FDTD(2,4)fields on the coarse-fine grid boundary (see Fig. 6). The coef-ficients were empirically found in [29]. This type of weightingcan be considered effectively as a smoothing of the electric fieldvalues when transitioning from the coarse to the fine domain.

Also, another weighting that is used relates to the coarse-gridmagnetic field values adjacent to the coarse-fine domainboundary and inside the fine grid region (see Fig. 5). It is doneusing the following equations:

(2)

(3)

Fig. 6. Electric fielde (near the boundary between the coarse and fine grids)that is linearly weighted.

where the locations of and are illustrated in Fig. 5 byblack dots, and the subscriptscoarseand fine correspond tothe coarse and fine field values computed from FDTD(2,4)and FDTD(2,2), respectively. The coefficients used in (2) and(3) were empirically found in order to provide a stable hybridscheme, and they are similar to the ones found in [29].

The space interpolation of the fine grid values is performedusing a standard 2-D fourth-order accurate spatial interpolation.The time interpolation of the fine grid electric field values on theboundary is performed using the following standard third-orderaccurate interpolation relations:

(4)

(5)

The respective locations in time of the electric and magneticfields for both the coarse and the fine grids are shown in Fig. 3.Also, in order to ensure stability the Courant stability upper limitof the fourth-order FDTD, [4], was reduced by a factor of 2 asfollows:

(6)

This reduction of the Courant stability upper limit was foundnecessary for the stability of the scheme and it was found em-pirically by experimentation.

A brief description of the procedure is given as follows (seealso [29]–[31]).


(a) (b)

Fig. 7. Simplified scaled model of a fuselage: (a) Assembly process. (b) Front view of the entire model.

1) Apply FDTD(2,4) on all the main grid points (includingthe ones inside the shared area by both the coarse and finegrid, i.e., domains #1 and #2) and obtain .

2) Apply FDTD(2,2) on the fine grid to obtain .3) Apply FDTD(2,2) on the fine grid to obtain . Up-

date on the coarse-fine boundary using the space(standard 2-D fourth-order accurate spatial interpolation)and time [based on (4) and (5)] interpolation discussedpreviously. Apply (1) to weight one cell inside thefine grid.

4) Apply FDTD(2,2) on the fine grid to obtain . Use(2) and (3) to weight and collocated onecoarse grid cell into the fine domain. Transfer all finegrid field values to the corresponding collocatedcoarse grid ( ) in the areashared by both the coarse and fine grids (domain #2) ex-cept the fields included in the area described by the blackdots (see Fig. 5).

5) Apply FDTD(2,4) to update at the coarse-fineboundary using the obtained values for fromstep 4. Apply FDTD(2,4) to update inside thecoarse grid domain (domain #1).

6) Repeat step 3 to obtain , step 2 to obtain ,and step 3 again to obtain . Correct the coarse-finegrid boundary values of using a standard 2-D fourth-order accurate spatial interpolation of .

7) Transfer all fine grid field values to the correspondingcollocated coarse grid field values ( ) inthe area that is commonly shared by the fine and coarsegrids (domain #2).

Capital letters and represent coarse grid field values, andsmall letters and represent fine grid field values. It shouldbe pointed out that the space interpolation of the coarse gridvalues on the coarse-fine boundary is performed using a stan-dard second-order accurate interpolation. Also, in all the stepsof the hybrid method where the FDTD(2,4) method is applied ituses the standard FDTD(2,4) scheme [4].

The hybrid method of subgrid FDTD(2,2)/FDTD(2,4) thatwas presented in this section will be validated and applied laterin this paper in the context of shielding effectiveness as well asPED analysis. Deriving a stable hybrid method is a very chal-lenging task. The hybridization procedure, which was outlined

above, is the result of long experimentation with different timeand space interpolation schemes. For the results presented in thismanuscript, the hybrid method was found stable up to severaltens of thousands of time-steps for different applications. Also,it should be mentioned that the outside FDTD(2,2) boundary ofthe hybrid method is terminated using the PML method.

IV. HIRF PENETRATION THROUGH A SCALED FUSELAGE

By examining the shielding effectiveness of various simpleconfigurations of boxes with apertures in [32], the FDTDmethod has been validated and proven accurate through com-parison of the FDTD predictions with measurements. In thissection, a more complex and realistic problem of penetrationis considered.

Direct analysis of HIRF penetration into a full-scale modelof a fuselage is a very complicated problem consisting of aplethora of different materials and very complex geometric fea-tures. In addition, uncertainty for the accuracy of CAD modelsfor full-scale airplanes, as well as uncertainty of the variousmeasurement parameters in full-scale setups, make the mod-eling and numerical simulation of such geometries very chal-lenging. Especially, when numerical results are being validated,a controlled geometric setup as well as measurement environ-ment are desired. This is the reason that stimulated the construc-tion and analysis of a simplified scaled model of a Boeing 757aircraft.

The geometry of the scaled model was constructed and mea-sured in the ASU anechoic chamber thereby providing a com-pletely controlled environment. This model will be used for val-idation of the accuracy of FDTD predictions for HIRF penetra-tion into complex models.

A. Construction and Specifications of the Geometry

The simplified fuselage is much larger than the simple boxesthat were examined in [32]. Its internal dimensions are 155-cmlong by 20-cm wide by 24-cm high. These dimensions are suf-ficient to enclose a 1:20 scale model of a Boeing 757 fuse-lage that is shortened by 25%. There are nine cabin windowsequally spaced along the length of the fuselage on each side.Each cabin window is 5-cm wide by 2-cm high. The top edgesof the cabin windows are 6-cm down from the top of the fuse-lage. The cen-ters of the windows are spaced 16 cm apart, and


(a)

(b)

Fig. 8. Measurement setup. (a) Floorplan view of the anechoic chamberconfiguration. (b) Absorber baffle was used to further reduce the illuminationof the single-plane collimating range (SPCR) reflector by the back-lobes of theauxiliary antenna.

the centers of the first and last windows are 13.5 cm from thenose and tail of the fuselage. The cockpit window is 18-cm wideby 8-cm high. Its top edge is 6 cm down from the inside ceilingof the fuselage and it is centered left and right.

Fig. 9. Geometry of the simplified fuselage model.

The simplified fuselage was assembled using six flat panelsthat were machined from 1.55-mm-thick aluminum stock. Thetop and side panels were clamped together in an upside downorientation, as shown in Fig. 7(a), and epoxy was applied to theinterior corners. Note in the photograph that expanded polysty-rene ribs were used to aid in keeping the sides and top orthog-onal to one another. Next, the bottom panel was fitted to theassembly, and was epoxied into place. Then, the front and backends were attached with aluminum tape, enabling them to beremoved so as to provide access to the inside. A front view ofthe completed model is shown in Fig. 7(b). Ansmaconnectorwas installed in the bottom panel, 50 cm from the front and onthe centerline of the fuselage, as a probe for the measurements.For all the measurements and predictions, the probe length was6 cm.

B. Measurements

The measurements were performed in ASU’s anechoicchamber. The shielding effectiveness measurement configura-tion was a direct illumination of the simplified fuselage by anauxiliary antenna. The fuselage was placed equidistantly fromthe rear and side walls of the chamber. The auxiliary antennawas located on the longitudinal center of the chamber. Thisconfiguration is illustrated in Fig. 8(a). With this setup, thereis no feed spillover. To further reduce the illumination of thereflector, an absorber baffle was placed between it and theauxiliary antenna. A photograph of the auxiliary antenna andabsorber baffle is shown in Fig. 8(b). The absorber reducedthe signal scattered by the reflector by an average (over theband of interest) of about 15 dB, as observed by the monopolereference antenna.

The HP8510 network analyzer can acquire 801 frequencypoints per measurement. This resolution is insufficient to ac-curately sample the rapid variations in the frequency responseof the model. Therefore, 1-GHz bands were measured with 801frequency point resolution, up to 7 GHz.


Fig. 10. Shielding effectiveness of the scaled fuselage for azimuthal incident angle of 0.

C. FDTD(2,2) Predictions Versus Hybrid SubgridFDTD(2,2)/FDTD(2,4) Predictions

In this subsection, predictions of the shielding effectivenessfor the simplified fuselage are presented. All the predictions areinitially performed using the standard second-order accurateboth in time and space FDTD(2,2) method. These FDTD(2,2)predictions are then compared with calculations performedusing the hybrid method of subgrid FDTD(2,2)/FDTD(2,4).All the predictions are compared with measurements. Thecomputations of shielding effectiveness (SE) are based onprocedure #2 described in [32]. Specifically, SE is defined asfollows:

where the frequency response is defined as the magnitude of theelectric-field component of interest (for details see [32]). TheCAD model of the simplified fuselage is illustrated in Fig. 9.

Here, all FDTD(2,2) predictions are performed using a cellsize of 2.5 mm (or at 9 GHz). Therefore, it is expectedthat these FDTD(2,2) calculations will provide accurate resultsat most up to 9 GHz. This FDTD(2,2) mesh yields a verylarge computational domain; 62080 96 cell and requires

MB just for the electric and magnetic fieldcomponents. Therefore, simulating this problem requires avery large amount of computational resources, memory as wellas time. Especially, the memory issue is more restrictive sinceif the required memory for a simulation is not available, thenthe simulation cannot be performed. All the FDTD simulationswere performed for 32 000 time-steps and the time-domain datawere Fourier transformed using the standard FFT algorithm.Predictions and measurements were performed for variousincident angles (see in [31]). However, due to lack of space

only the results for nose incidence (azimuthal angle of 0) arepresented in this manuscript.

The FDTD computations are shown and compared to mea-surements in Figs. 10, and 11(a)–(c). Fig. 11(a)–(c) plots the re-sults in various frequency bands to illustrate clearly the SE vari-ations. It can be observed that the numerical results agree verywell with measurements up to 5.6 GHz. However, their accuracyis already degraded in the 5.6–6.3-GHz band [see Fig. 11(c)].Numerical experiments along with analytical derivations of thephase velocity in the discrete space have shown that a cell sizeof provides satisfactory accuracy for moderate problemswhen using FDTD(2,2). However, this rule of thumb may not berepresentative for electrically large spaces where the accumula-tion of phase errors due to dispersion becomes significant anddeteriorates the accuracy of the FDTD solution. The simplifiedfuselage has electrical dimensions of approximately

at 9 GHz, which make the fuselage an extremely electricallylarge domain. This justifies the inaccurate results of FDTD(2,2)in the frequency band of 5.6–6.3 GHz. Even though the dis-cretization is at least at 9 GHz, the accumulation of phaseerrors through the domain make this discretization insufficient.

Subsequently, the hybrid method of subgridFDTD(2,2)/FDTD(2,4), presented in Section III is applied tocompute the shielding effectiveness of the scaled fuselage.The spatial layout of the methods used by the hybrid in thecomputational domain is depicted in Fig. 4(b). It should bepointed out that the probe exciting the cavity is included inthe fine region where subgrid FDTD(2,2) is applied. This isdone in order to simulate the fields near and on the probe verywell by representing the probe with many cells (fine grid). Allpredictions are again performed for nose incidence.

Here, all FDTD predictions of the hybrid scheme are per-formed using a cell size of 2.5 mm (or at 9 GHz) for thefine grid, and of 7.5 mm (or at 9 GHz) for the coarse grid.


(a) (d)

(b) (e)

(c) (f)

Fig. 11. Shielding effectiveness of the scaled fuselage for azimuthal incident angle of 0. (a)-(c) FDTD(2,2) predictions; (d)-(f) Hybrid subgridFDTD(2,2)/FDTD(2,4) predictions.

Therefore, it is expected that these calculations of the hybridscheme will provide accurate results at most up to 9 GHz.

The new hybrid method of subgrid FDTD(2,2)/FDTD(2,4)provides significant savings in memory compared to the stan-dard FDTD(2,2). The hybrid applies FDTD(2,4) to the entirefuselage but on a coarse grid with a cell size three times larger

than the one required by FDTD(2,2), i.e., a cell size of 7.5 mm(or at 9 GHz). This FDTD(2,4) part of the hybrid code (do-mains #1 and #2 in Fig. 5) requires only 4 MB of memory. Fur-thermore, the subgrid FDTD(2,2) is only applied on a region be-tween the wall of the fuselage and the coarse-fine grid boundary(domain #2 in Fig. 5) and has a thickness of only nine cells.


(a) (b)

Fig. 12. The PED geometry. (a) Cross section of the fuselage at station C. (b) Measurement setup.

TABLE IMEMORY REQUIREMENTS FOR THEDIFFERENT REGIONS OF THE

COMPUTATIONAL DOMAIN FOR A CELL SIZE OF 2.5 MM

This subgrid FDTD(2,2) part of the hybrid code requires 44 MBof memory. Therefore, the total amount of memory for the hy-brid subgrid FDTD(2,2)/FDTD(2,4) method is only

MB which is 2.5 times smaller than the memoryrequired by the standard FDTD(2,2) code (114 MB). This reduc-tion in memory results by not applying FDTD(2,2) along with afine cell size in the part of the computational domain that solelycorresponds to the coarse domain (domain #1 in Fig. 5), whichwould have required MB. Instead, the FDTD(2,4)is applied on the coarse domain using a coarse cell size allo-cating only MB. The comparison of the specificnumerics are illustrated in Table I, where and are thememory requirements for domains #1 and #2, respectively.

The problem examined here exhibits dimensions that are ex-tremely long only in one direction, since the fuselage is a verylong rectangular box. However, in cases where the geometry islarge in two or in all three directions, then the memory savingsare even more significant. Specifically, when the hybrid methodis applied to two different boxes with dimensions: a) 155 cmlong by 155 cm wide by 24-cm high (box #1), and b) 15-cm longby 155-cm wide by 155-cm high (box #2) the memory savingsfactor becomes 4 and 8, respectively. As expected, the memorysavings are tremendous when the domains become large in twoand/or in all three directions.

By comparing Fig. 11(d)-(f) with Fig. 11(a)-(c), it is ob-served that the hybrid method, even with coarser meshing forFDTD(2,4), gives very similar results for shielding effective-ness as the ones of the standard FDTD(2,2) alone (with a cellsize of 2.5 mm). Notice that even though most of the interior

of the fuselage is simulated using a quite coarse cell size of7.5 mm, the accuracy is retained as a higher order schemeis applied [FDTD(2,4)]. Also, the predictions of the hybridmethod agree very well with the measurements. This simulationagain verifies the accuracy and efficacy of the hybrid method.

V. PEDs

In this section, analysis of PED interference mechanisms isperformed using measurements as well as predictions.

A. Geometry and Setup of the Measurements

Two new structures were added to the simplified fuselageexamined in the previous section for the PED measurements:an externally mounted antenna on a small pedestal, and a tallpedestal within the fuselage on which the simulated PED re-sides. In order to be able to mount the antenna at almost anylocation on the fuselage, it was built into a small pedestal. Thepedestal is a 50-mm square aluminum tube with a height of15 mm. This height is as short as possible while still accommo-dating the RF connector within. A brass plate forms the top ofthe pedestal. The antenna extends vertically through the centerof the brass plate. During the measurements, the pedestal istaped to the surface of the simplified fuselage.

To approximate a typical PED, a tall pedestal was constructedso as to raise the interior probe to the level of the cabin windows.This tall pedestal was constructed of 50-mm50-mm alu-minum tubing. The square aluminum tube is 150 mm in height,and it is capped with a plate of aluminum through which thesimulated PED probe is mounted.

A hole was drilled through the bottom of the simplified fuse-lage through which the PED cable passes. To facilitate the de-scription of the PED and external antenna locations, a letterdesignation is associated with each of the cabin windows, asshown in Fig. 9. The current PED location is on the starboardside wall adjacent to window C. A drawing of the fuselage crosssection [Fig. 12(a)] illustrates the relationships of the simulatedPED and the external antenna. In this view, both are at station


(a) (c)

(b) (d)

Fig. 13. Coupling between the PED antenna and the antenna mounted on the exterior of the fuselage. (a) and (b) FDTD(2,2) predictions. (c) and (d) Hybridsubgrid FDTD(2,2)/FDTD(2,4) predictions.

C. Note that both the PED element and the monopole exter-nally mounted to the fuselage are 3-cm tall and have a radiusof 0.60325 mm.

Again, due to the highly overmoded nature of this large cavity,the coupling between the PED and external antenna exhibit ex-tremely rapid variations as a function of frequency. In orderto resolve those variations, a large number of frequency pointswere measured. The network analyzer is capable of measuring801 frequency points per measurement/calibration. The onlyoption available to increase the frequency resolution is to de-crease the bandwidth of the measurement. To cover the fre-quency range of interest, it is separated into many 801-pointbands.

The measurements were performed inside the anechoicchamber, with the instrumentation just outside the door. Thesource and receiver remained outside of the chamber, the testset was relocated to a position immediately under the simplifiedfuselage. This enabled the use of RF cables of four and fivefeet in length, greatly reducing the cable losses. Specifically,from port 2 of the parameter test set, two Adams-Russellgold-tipped cables (flexible instrumentation cables) werejoined together using APC-3.5 connectors (total length 3.5

ft.) followed by a phase-matched APC-3 connected to thesimulated PED within the fuselage (an sma flange connector).A matching female-to-female adapter was used in the ”adapterswap” calibration technique (for the ”thru” measurement) for”noninsertable” devices. From port 1, an sma(f)-to-sma(f)”barrel” connected to a Shurr .141” flexible blue cable (totallength 5 ft.) with sma(m) connectors at each end which thenconnected to the right-angle sma flange connector within theshort pedestal for the external antenna. A photograph of themeasurement setup is shown in Fig. 12(b).

B. FDTD(2,2) Predictions Versus Hybrid SubgridFDTD(2,2)/FDTD(2,4) Predictions

Here, the predictions for the coupling between the PED an-tenna and the antenna mounted on the exterior of the fuselageare presented. A cell size of 5 mm (or at 6 GHz) is used.The radius of the two monopoles is 0.60325 mm, and it is takeninto account in all simulations both along the wire (using a thinwire model) and the excitation (using a source based on the ra-dial electric fields). To speed the simulation times, all sourcesused a internal resistance of 100[33]. The parameters arecomputed using the procedure described in [33].


Both the PED and external antenna are located at station C(see Figs. 9 and 12) as in the measurements. The parameter thatis the focus of a PED analysis is the coupling between the twoantenna elements. Coupling measures the amount of energy thata PED antenna can couple to an antenna that belongs to the com-munication system of an aircraft, and it is represented by theand parameters, which are equal due to reciprocity. To illus-trate clearly the variation of coupling, as well as the agreementbetween the FDTD(2,2) calculations and the measurements, fig-ures that plot coupling in different frequency bands are con-structed [see Fig. 13(a) and (b)]. It is observed that FDTD(2,2)predicts very accurately the coupling between the PED antennaand the antenna mounted on the exterior of the airplane up to3 GHz. The levels of coupling exhibit a highly oscillatory be-havior due to the large number of resonances present inside thefuselage. Also, the maximum level of coupling is approximately

30 dB and occurs at several frequencies. This level of30 dBcan represent a threat to the communication systems of the air-plane. However, the definite interpretation of effects of suchcoupling levels are left to the engineers that deal with and de-sign the communication systems of the aircraft.

Furthermore, the hybrid method of subgridFDTD(2,2)/FDTD(2,4) is applied to the analysis of the PEDproblem. The spatial layout of the methods used by the hybridin the computational domain is depicted in Fig. 4(b). Itshould be pointed out that both antennas and their pedestalsare included in the fine region where subgrid FDTD(2,2) isapplied. This was done in order to simulate the fields near andon the probe very well by representing the probe with manycells (fine grid).

For validation purposes, a cell size of 5 mm (or at 6GHz) for the fine grid and of 15 mm (or at 6 GHz) forthe coarse grid is used. The memory required for storing allthe field components that describe the dimensions of the boxis approximately 14 MB, which is approximately the same asthe one required by FDTD(2,2) alone when a cell size of 5 mmis used. For this cell size no memory savings occur since thecell size of 5 mm does not yield a large computational domain.However, this simulation results are used as validation of theaccuracy of the hybrid method.

By comparing Fig. 13(a) and (b) with Fig. 13(c) and (d), it isobserved that the hybrid method gives almost identical resultsfor PED analysis as the ones of the standard FDTD(2,2) alone(with a cell size of 5 mm). Notice that even though most of theinterior of the fuselage is simulated using a very coarse cell sizeof 15 mm, the accuracy is retained as a higher order scheme isapplied [FDTD(2,4)]. This, once again, validates the accuracyof higher order schemes.

VI. CONCLUSION

In this paper, BCs in the context of higher order schemeswere discussed. BCs represent one of the most important chal-lenges in higher order methods. An extensive literature reviewof this issue was presented, and most of the existing research hasbeen generated in the fields of computational physics and math-ematics. This line of research, which is related to BCs and higherorder schemes, supports the fact that it is very challenging to de-

sign accurate as well as stable higher order BCs. In particular,stability is the most difficult problem.

Furthermore, a method that benefits from the advantagesof both FDTD(2,2) and FDTD(2,4) was presented. The newmethod is a hybrid of FDTD(2,4) with a subgrid FDTD(2,2).After completing the formulation of the hybrid technique, itwas shown that its use yields tremendous memory savingswhen domains become large in all three directions.

Also, HIRF penetration analysis was performed for scaledmodel fuselages using both the standard FDTD(2,2) and thehybrid subgrid FDTD(2,2)/FDTD(2,4) method. All the predic-tions were validated by comparison with measurements. Finally,PED analysis was done using again both FDTD(2,2), the hybridmethod, and measurements.

It should be pointed out that the stability criterion of the hy-brid method was empirically found. Future research concerningthis or similar hybrid techniques should investigate the formula-tion and derivation of their stability criterion. Finally, this paperused a subgrid ratio factor of three in the hybrid formulation.However, other subgrid factors can be used and future researchshould address the prons and cons of smaller versus larger sub-grid ratios.

APPENDIX

Here a second-order in time and fourth-order in spacescheme [FDTD(2,4)] is described. In this scheme the positionsof the electric and magnetic field components remain the samewith the ones of the FDTD(2,2) scheme. The FDTD(2,4) stenciluses central finite differences fourth-order accurate in spaceand second-order accurate in time, respectively

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ACKNOWLEDGMENT

The authors would like to thank Dr. C. M. Belcastro and T.Nguyen of NASA Langley Research Center, Hampton, VA, fortheir continued interest and support of this project.

REFERENCES

[1] R. Holland, L. Simpson, and R. H. St. John, “Code optimization forsolving large 3D EMP problems,”IEEE Trans. Nuclear Sci., vol. NS-26,pp. 4964–4969, Dec. 1979.

[2] D. E. Merewether, R. Fisher, and F. W. Smith, “On implementing a nu-meric Huygen’s source scheme in a finite difference program to illu-minate scattering bodies,”IEEE Trans. Nuclear Sci., vol. NS-27, pp.1829–1833, Dec, 1980.

[3] A. C. Cangellaris and R. Lee, “On the accuracy of numerical wave sim-ulations based on finite methods,”J. Electromagn. Waves Applicat., vol.6, no. 12, pp. 1635–1653, 1992.

[4] J. Fang, “Time domain finite difference computation for Maxwell’sequations,” Ph.D. dissertation, Univ. of California at Berkeley, Berkeley,CA, 1989.


[5] P. G. Petropoulos, “Phase error control for FD-TD methods,” inUltra-Wideband Short-Pulse Electromagnetics, H. L. Bertoni, L. Carin, and L.B. Felsen, Eds. New York: Plenum, 1993, pp. 359–366.

[6] M. F. Hadi and M. Piket-May, “A modified FDTD (2,4) scheme formodeling electrically large structures with high-phase accuracy,”IEEETrans. Antennas Propagat., vol. 45, pp. 254–264, Feb. 1997.

[7] B. Gustafsson, “The convergence rate for difference approximations togeneral mixed initial boundary value problems,”SIAM J. Numer. Anal.,vol. 18, pp. 179–190, 1981.

[8] M. H. Carpenter, D. Gottlieb, and S. Abarbanel, “The stability of nu-merical boundary treatments for compact high-order finite-differenceschemes,”J. Comput. Phys., vol. 108, pp. 272–295, 1993.

[9] B. Gustafsson, H. O. Kreiss, and A. Sundstrom, “Stability theory of dif-ference appoximations for mixed initial boundary value problems II,”Math. Comp., vol. 26, pp. 649–686, 1972.

[10] M. H. Carpenter, D. Gottlieb, and S. Abarbanel, “Time-stable boundaryconditions for finite-difference schemes solving hyperbolic systems:Methodology and application to high-order compact schemes,”J.Comput. Phys., vol. 111, pp. 220–236, 1994.

[11] B. Strand, “Summation by parts for finite difference approximations ford=dx,” J. Comput. Phys., vol. 110, pp. 47–67, 1994.

[12] S. S. Abarbanel and A. E. Chertock, “Strict stability of high-order com-pact implicit finite-difference schemes: The role of boundary conditionsfor hyperbolic pdes, I,”J. Comput. Phys., vol. 160, pp. 42–66, 2000.

[13] S. S. Abarbanel, A. E. Chertock, and A. Yefet, “Strict stability ofhigh-order compact implicit finite-difference schemes: The role ofboundary conditions for hyperbolic pdes, II,”J. Comput. Phys., vol.160, pp. 67–87, 2000.

[14] M. H. Carpenter, J. Nordstrom, and D. Gottlieb, “a stable and conser-vative interface treatment of arbitrary spatial accuracy,” NASA LangleyResearch Center, Hampton, VA, Rep. 98-12, 1998.

[15] D. Darmofal, “Eigenmode analysis of boundary conditions for the one-dimensional preconditioned Euler equations,” NASA Langley ResearchCenter, Hampton, VA, Rep. 98-51, 1998.

[16] J. Nordstrom and M. H. Carpenter, “Boundary and interface conditionsfor high order finite difference methods applied to the Euler and Navier-Stokes equations,” NASA Langley Research Center, Hampton, VA, Rep.98-19, 1998.

[17] D. Gottlieb and E. Turkel, “Dissipative two-four methods for time-de-pendent problems,”Math. Comp., vol. 30, pp. 703–723, 1976.

[18] B. Sjögreen, “High-order centered difference methods for the compress-ible navier-stokes equations,”J. Comput. Phys., vol. 117, pp. 67–78,1995.

[19] J. L. Young, D. Gaitonde, and J. S. Shang, “Toward the constructionof a fourth-order difference scheme for transient EM wave simulation:Staggered grid approach,”IEEE Trans. Antennas Propagat., vol. 45, pp.1573–1580, Nov. 1997.

[20] E. J. Caramana, M. J. Shashkov, and P. P. Whalen, “Formulations ofartificial viscocity for multi-dimensional shock wave computations,”J.Comput. Phys., vol. 144, pp. 70–97, 1998.

[21] R. C. Swanson, R. Radespiel, and E. Turkel, “On some numerical dissi-pation schemes,”J. Comput. Phys., vol. 147, pp. 518–544, 1998.

[22] D. Furihata, “Finite difference schemes for@u=@t = (@u=@x) �g=�uthat inherit energy conservation or dissipation property,”J. Comput.Phys., vol. 156, pp. 181–205, 1999.

[23] J. S. Shang, “High-order compact-difference schemes for time-depen-dent maxwell equations,”J. Comput. Phys., vol. 153, pp. 312–333, 1999.

[24] A. Yefet and P. G. Petropoulos, “A staggered fourth-order accurate ex-plicit finite difference scheme for the time-domain Maxwell’s equa-tions,” J. Comput. Phys., vol. 168, pp. 286–315, 2001.

[25] P. G. Petropoulos, “Phase error analysis control for FD-TD methods ofsecond and fourth order accuracy,”IEEE Trans. Antennas Propagat.,vol. 42, pp. 859–862, June 1994.

[26] G. Haussmann, “A Dispersion Optimized Three-Dimensional Finite-Difference Time-Domain Method for Electromagnetic Analysis,” Ph.D.,University of Colorado at Boulder, Boulder, CO, 1998.

[27] K. S. Kunz and L. Simpson, “A technique for increasing the resolutionof finite-difference solutions of the Maxwell’s equation,”IEEE Trans.Electromagn. Compat., vol. EMC-23, pp. 419–422, Nov. 1981.

[28] S. S. Zivanovic, K. S. Yee, and K. K. Mei, “A subgridding method for thetime-domain finite-difference method to solve Maxwell’s equations,”IEEE Trans. Microwave Theory Tech., vol. 39, pp. 471–479, Mar. 1991.

[29] M. W. Chevalier, R. J. Luebbers, and V. P. Cable, “FDTD local gridwith material traverse,”IEEE Trans. Antennas Propagat., vol. 45, pp.411–421, Mar. 1997.

[30] S. V. Georgakopoulos, R. A. Renaut, C. A. Balanis, and C. R. Birtcher,“A hybrid fourth-order FDTD utilizing a second-order FDTD subgrid,”IEEE Microwave Wireless Comp. Lett., vol. 11, pp. 138–140, Nov. 2001.

[31] S. Georgakopoulos, “Higher-order finite-difference methods,” Ph.D.dissertation, Arizona State Univ., Tempe, 2001.

[32] S. V. Georgakopoulos, C. R. Birtcher, and C. A. Balanis, “HIRF pen-etration through apertures: FDTD versus measurements,”IEEE Trans.Electromagn. Compat., vol. 43, pp. 282–294, Aug. 2001.

[33] S. V. Georgakopoulos, C. A. Balanis, and C. R. Birtcher, “Coupling be-tween transmission line antennas: Analytic solution, FDTD, and mea-surements,”IEEE Trans. Antennas Propagat., vol. 47, pp. 978–985, June1999.

Stavros V. Georgakopoulos(S’93–M’02) was bornin Athens, Greece, in May 1973. He received thediploma in electrical engineering from the Universityof Patras, Patras, Greece, in June 1996, M.S. degreein electrical engineering, and the Ph. D. degree inelectrical engineering (working on computationaland applied electromagnetics in radiation and scat-tering problems), both from Arizona State University(ASU), Tempe, in 1998, and 2001, respectively.

Between 1996 and 2002, he was a Graduate Re-search Assistant in computational electromagnetics

for the Advanced Helicopter Electromagnetics (AHE) Program at ASU’sTelecommunications Research Center (TRC). Since 2002, he has been withSV Microwave, West Palm Beach, FL, where he is a Principal Engineer in theResearch and Development Department. His current research interests relateto the design and analysis of novel passive and active microwave componentssuch as connectors, attenuators, terminations, phase shifters, and mixers.

Craig R. Birtcher received the B.S.E.E. andM.S.E.E. degrees from Arizona State University(ASU), Tempe, in 1983 and 1992, respectively.

He has been at ASU since 1987, where he is nowan Associate Research Professional in charge ofthe ElectroMagnetic Anechoic Chamber (EMAC)facility. His research interests include antenna andradar cross section(RCS) measurement techniquesand near-field to far-field methods.

Constantine A. Balanis(S’62–M’68– SM’74–F’86)received the B.S.E.E. degree from Virginia Tech,Blacksburg, VA, in 1964, the M.E.E. degree fromthe University of Virginia, Charlottesville, VA, in1966, and the Ph.D. degree in electrical engineeringfrom Ohio State University, Columbus, in l969.

From 1964 to 1970, he was with NASA LangleyResearch Center, Hampton VA, and from 1970to 1983 he was with the Department of ElectricalEngineering, West Virginia University, Morgantown,WV. Since 1983, he has been with the Department

of Electrical Engineering, Arizona State University, Tempe, where he is nowRegents’ Professor. His research interests are in low- and high-frequencycomputational methods for antennas and scattering, smart antennas for wirelesscommunication, and high intensity radiated fields. He is the author ofAntennaTheory: Analysis and Design(New York: Wiley, 1997, 1982) andAdvancedEngineering Electromagnetics(New York:Wiley, 1989).

Dr. Balanis received the 2000 IEEE Third Millennium Medal, 1996–1997Arizona State University Outstanding Graduate Mentor Award, 1992 SpecialProfessionalism Award from the IEEE Phoenix Section, the 1989 IEEE Region6 Individual Achievement Award, and the 1987–1988 Graduate TeachingExcellence Award, School of Engineering, Arizona State University. He hasserved as Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND

PROPAGATION(1974–1977) and the IEEE TRANSACTIONS ONGEOSCIENCE AND

REMOTE SENSING (1981–1984), Editor of theNewsletter For The IEEE Geo-science and Remote Sensing Society(1982–1983), as Second Vice-Presidentof the IEEE Geoscience and Remote Sensing Society (1984), Chairman ofthe Distinguished Lecturer Program (1988–1991) and member of the AdCom(1993–95, 1997–1999) of the IEEE Antennas and Propagation Society.


Rosemary A. Renaut received the B.S. (Hons.)degree in mathematics from Durhan University,Durhan, U.K., in 1980, Part III of the AppliedMathematics and Theoretical Physics Tripos ofthe University of Cambridge, Cambridge, U.K., in1981, and the Ph.D. degree in applied mathematics,also from the University of Cambridge, Cambridge,U.K., in 1985.

In 1985, she was a Post-Doctoral Fellow at theTechnical University of Aachen (RWTH), Aachen,Germany, funded by the English Royal Society.

During 1986, she joined the Christian Michelsen Research Institute, Bergen,Norway, as a Research Scientist working on sesimic modeling. Since 1987,she has been with the Department of Mathematics, Arizona State University,Tempe, where she is now a Full Professor, having completed a four year termas Chair of the Department in 2001. She is Director of a new interdisciplinaryProfessional degree program in Computational Biosciences. Her researchinterests are in the design and evaluation of computational methods for thesolution of partial differential equations, with specific emphasis on high-orderand spectral methods, as well as, more recently, the development of novelalgorithms for image reconstruction and restoration in medical imaging. She isthe author of more than 50 research articles covering theoretical analysis andimplementation studies for numerical techniques in these application areas.

Dr. Renaut is a Fellow of the Institute for Mathematics and its Applications,England, through which she is also listed as a Chartered Mathematician, and aMember of the Society for Industrial and Applied Mathematics.

Documents

HIRF penetration and PED coupling analysis for scaled ...rosie/mypapers/paper54.pdfThe FDTD(2,4) method is expected to fail to accurately model infinitesimally thin PEC films. This