forward backward domain decomposition method for finite element solution of electromagnetic boundary...

respectively. The performance of the frequency synthesizer isgiven in Table 2 and compared with recently published WLANfrequency synthesizer. The power consumption of this frequencysynthesizer includes on-chip band gap and output buffer. Thephase noise performance can be improved if stable quartz oscilla-tor and voltage regulator are used for measurement. Signal gener-ator is used as signal source in this measurement setup. Withreferred to the frequency synthesizer designed with the sameprocess as well as the architecture but operating at 5 GHz [10], thiswork shows great reduction in the power consumption. Note thatthe performance of this frequency synthesizer is comparable withother WLAN frequency synthesizer. The optimization from thesystem level, which halves the operating frequency, is the mainreason for this improvement.

5. CONCLUSION

We have demonstrated a design of frequency synthesizer forwireless network application, such as WLAN 802.11a. By reduc-ing the LO operating frequency and its frequency range to half,high-performance and low-power frequency synthesizer is imple-mented. The frequency synthesizer is fully on-chip, including boththe VCO and LPF. The performance is, however, more robust andless susceptible to process variation. It is achieved through digitalprogrammable charge pump and VCO. This frequency synthesizercan be further optimized to cover WLAN 802.11b/g.

REFERENCES

1. B. Goldberg, RF synthesizers: PLL switching speed and speed-uptechniques, a short review, In: 2001 IEEE MTT-S International Mi-crowave Symposium, 2001, p. 693–696.

2. M. Goldfarb, E. Balboni, and J. Cavey, Even harmonic double-bal-anced active mixer for use in direct conversion receivers. IEEE JSolid-State Circ 38 (2003), 1762–1766.

3. S. Ye, et al., A 200–2250 MHz/400–4500 MHz regenerative fre-quency doubler in a 0.35 �m SiGe process, In: 4th InternationalConference on Microwave and Millimeter Wave Technology, 2004.

4. B. Razavi, RF Microelectronics, Prentice-Hall, Upper Saddle River,NJ, 1998.

5. T. Aytur and B. Razavi, A 2-GHz, 6-mW BiCMOS frequency syn-thesizer. IEEE J Solid-State Circ 30 (1995), 1457–1462.

6. H.R. Rategh, H. Samavati, and T.H. Lee, A CMOS frequency synthe-sizer with an injection-locked frequency divider for a 5-GHz wirelessLAN receiver. IEEE J Solid-State Circ 35 (2000), 780–787.

7. T.H. Teo, et al., Characterization of symmetrical spiral inductor in 0.35�m CMOS technology for RF application. Int J Solid-State Electron48 (2004), 1643–1650.

8. D. Coolbaugh, et al., Advanced passive devices for enhanced inte-grated RF circuit performance, In: 2002 IEEE Radio Frequency Inte-grated Circuits Symposium, 2002.

9. T. H. Teo, et al., LC oscillator design at 10-GHz using substrate

capacitance with scalable varactor parameters extraction technique, In:35th European Microwave Conference, Paris, France 2005.

10. H. Ainspan and M. Soyuer. A fully-integrated 5-GHz frequency syn-thesizer in SiGe BiCMOS, In: Bipolar/BiCMOS Circuits and Tech-nology Meeting, 1999.

11. T. Schwanenberger, et al., A multi standard single-chip transceivercovering 5.15–5.85 GHz, In: IEEE International Solid-State CircuitsConference, 2003.

12. P. Zhang, et al., A direct conversion CMOS transceiver for IEEE802.11a WLANs, In: IEEE International Solid-State Circuits Confer-ence, 2003.

© 2007 Wiley Periodicals, Inc.

FORWARD–BACKWARD DOMAINDECOMPOSITION METHOD FOR FINITEELEMENT SOLUTION OFELECTROMAGNETIC BOUNDARYVALUE PROBLEMS

Ozlem Ozgun and Mustafa KuzuogluDepartment of Electrical and Electronics Engineering, Middle EastTechnical University, 06531 Ankara, Turkey; Corresponding author:[email protected]

Received 6 March 2007

ABSTRACT: We introduce the forward–backward domain decomposi-tion method (FB-DDM), which is basically an improved version of theclassical alternating Schwarz method with overlapping subdomains, forelectromagnetic boundary value problems. The proposed method is non-iterative in some cases involving smooth geometries, or it usually con-verges in a few iterations in other cases involving challenging geome-tries, via the utilization of the locally-conformal PML method. We reportsome numerical results for two-dimensional electromagnetic scatteringproblems. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett49: 2582–2590, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22757

Key words: domain decomposition method (DDM); electromagneticscattering; finite element method (FEM); perfectly matched layer(PML); Schwarz method

1. INTRODUCTION

Obtaining efficient and accurate simulations of electromagneticboundary value problems (BVPs) employing electrically-large andgeometrically complicated objects is still a challenging problem.The finite element method (FEM) is a powerful numerical methodto solve such problems. However, to apply the FEM to large-scale

TABLE 2 Frequency Synthesizer Performance Summary

This work Ainspan et al. [10] Schwanenberger et al. [11] Zhang et al. [12]

Frequency ranges (GHz) 2.59–2.66, 2.8725–2.9025 4.850–5.325 5.150–5.825 5.150–5.350Supply voltage (V) 3.0 3.3 3.3 1.8Phase noise at 1 MHz (dBc/Hz) �115 �101 �96 �113Reference frequency (MHz) 2.5 12.5 40.0 13.3Reference spur (dBc) �56 �47 � �66Settling time (�s) 32 10 – –Power dissipation (mW) 42 255 207 56Active area (mm2) 2.34 (1.8 � 1.3) 2 – –LPF On-chip On-chip Off-chip Off-chipVCO LC-tank On-chip On-chip Off-chip On-chipTechnology 0.35-�m SiGe BiCMOS 0.50-�m SiGe BiCMOS 0.18-�m CMOS

2582 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 10, October 2007 DOI 10.1002/mop

electromagnetic problems, it is necessary to develop some strate-gies to increase the efficiency of the FEM by decreasing thememory storage requirements and computation time. A powerfulapproach is to employ a domain decomposition method (DDM),which is generally based on the partitioning of the original (whole)problem into several coupled subproblems which can be solvedindependently and easily. The subproblems are solved iteratively(or noniteratively in some cases) by suitably communicating withthe others. Then, the solution of the original problem is assembledusing the solutions of all subproblems.

Although DDMs have been widely used in computational me-chanics, they have been utilized in electromagnetic BVPs duringthe last few years. There are several domain decomposition strat-egies devised for problems governed by partial differential equa-tions. The oldest class of the DDMs is a group of approachesknown as Schwarz methods [1], developed for solving BVPs bydecomposing the original spatial domain into smaller subdomains.The classical alternating Schwarz method solves several overlap-ping subproblems iteratively with Dirichlet boundary conditions(BCs) in each subdomain. The algorithm starts with an arbitraryinitial guess for BCs (usually zero BCs), and solves the subdo-mains iteratively by modifying the BCs in each iteration until acertain equilibrium condition is satisfied in the solution. Overlap-ping Schwarz methods have also some variants, such as additiveand multiplicative Schwarz methods. The number of studies basedon the overlapping Schwarz methods is limited in the literature ofelectromagnetics. Up to now, these methods have been usuallyimplemented in microwave interconnect structures in the staticregime [2, 3]. The original Schwarz method has been furtherextended to an iterative algorithm employing nonoverlapping sub-domains (overlap only at an interface) with transmission condi-tions (TCs) across the subdomain interfaces. The nonoverlappingSchwarz method has been implemented in computational electro-magnetics in the modeling of certain problems [4, 5]. However, thesuitable choice of TCs in the nonoverlapping Schwarz method iscrucial and challenging to achieve stability and convergence in the

solution. Apart from the Schwarz methods, various DDMs havebeen developed to solve large-scale FEM problems in electromag-netic simulations [6, 7].

In this article, we propose a novel DDM technique, which wecall as “forward–backward domain decomposition method (FB-DDM),” for the finite element solution of electromagnetic scatter-ing problems using the locally-conformal PML approach [8]. TheFB-DDM is basically established on the classical alternatingSchwarz method with overlapping subdomains. However, the FB-DDM implements the Schwarz method by taking into accountcertain physical aspects of the original (whole) problem. That is,the Schwarz method is improved in terms of efficiency and rate ofconvergence by considering the geometry of the problem and theexpected field behavior inside the original domain. This isachieved by the implementation of the locally-conformal PMLmethod along the boundaries of the subdomains. In some problemsinvolving smooth geometries, the FB-DDM converges in just asingle forward iteration (i.e., the method is noniterative), where theproblem in each subdomain is solved only once by appropriatelydefined subdomains and additional PML regions attached to eachsubdomain. In other challenging geometries, the initial forwarditeration of the FB-DDM provides an initial guess ‘close’ to theexact values of the BCs of each subdomain, unlike the ‘arbitrary’initial guess in the Schwarz method. After the initial forwarditeration, the problems defined on the subdomains are solvediteratively, similar to the Schwarz method, in a forward–backwardfashion until convergence is achieved. It is obvious that the ‘better’initial guess for the BCs provides an increased convergence rate bydecreasing the number of the forward–backward iterations duringthe field refinement process. In addition, the proposed method

Figure 1 FB-DDM partitioning with two subdomains: (a) Original prob-lem, (b) Partitioned problem, (c) First subdomain having additional PMLregion, (d) First subdomain with emphasis on the new PML-free spaceinterface

Figure 2 FB-DDM partitioning with three subdomains: (a) Partitionedproblem, (b) First subdomain having additional PML region, (c) Secondsubdomain having additional PML region

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 10, October 2007 2583

provides a considerable reduction in the memory requirements andcomputation time.

This article is structured as follows: In Section 2, we introducethe implementation procedure of the FB-DDM algorithm in theproblem of two-dimensional (2D) TMz electromagnetic scatteringfrom an infinitely-long cylindrical perfect electric conductor (PEC)object with arbitrary cross-section. Although the algorithm can be

extended to cases involving objects with arbitrarily-defined con-stitutive parameters in a straightforward manner, we restrict ouranalysis to PEC objects for the purpose of brevity. In Section 3, wediscuss some important issues related to the successful implemen-tation of the method (such as partitioning scheme, order of sub-domains, etc.) with regard to the physics of the problem. Then, inSection 4, we present some numerical applications demonstratingthe effectiveness of the method in scattering problems. Finally, wedraw some conclusions in Section 5.

2. FORWARD–BACKWARD DDM ALGORITHM

The algorithm starts with the partitioning of the original (whole)problem into a suitable number of overlapping subdomains. Forthe purpose of illustrating a partitioning scheme with two subdo-mains, we show the original problem and the partitioned problemfor a square cylinder in Figure 1. In this figure, the subdomains �i

include both the free-space and the PML regions which alreadyexist in the original domain. In this partitioning scheme, a smallpart of the scatterer boundary remains inside the PML region(��Si,PML). Along this boundary, the applied incident field decayssmoothly because of the implementation of the locally-conformalPML method.

To understand the crucial role of the PML concept in theFB-DDM algorithm, it is useful to overview briefly the locally-conformal PML method [8]. A PML is an artificial layer whichabsorbs outgoing plane waves irrespective of their frequency and

Figure 3 Flowchart of the FB-DDM algorithm

Figure 4 Scattering from circular PEC cylinder: (a) Partitioned problem,(b) First subdomain having additional PML region, (c) Reflected rays andthe rays of shadow forming field, (d) Diffracted rays emanating from P1

Figure 5 Bistatic RCS profile of circular cylinder with N subdomains inthe first forward iteration

TABLE 1 Err1 Values for N Number of Subdomains inCircular Cylinder (First Forward Iteration)

Err1 (%)

N � 2 0.0045N � 3 0.0393N � 4 0.0178N � 5 0.1088N � 6 0.1043N � 7 0.1573N � 8 0.1776N � 9 0.1884N � 10 0.2504


angle of incidence without any reflection. The locally-conformalPML method is based on the locally-defined complex coordinatetransformation, where the PML action is achieved by the analyticcontinuation of the frequency-domain Maxwell’s equations tocomplex space. That is, the method is implemented by just replac-ing the real coordinates (r� � �2) inside the PML region with theircomplex counterparts calculated by the following complex coor-dinate transformation

r� � r� ��m

jkmdPMLm�1a� (1)

where r� � C2, � is the distance between r� and r�0 (r�0 is the point onthe PML-free space interface which is closest to r�), a� is the unitvector along the direction of decay (i.e. in the direction r� � r�0),k is the wavenumber, � is a positive parameter, m is a positiveinteger (typically 2 or 3), and dPML is the local PML thickness forthe corresponding PML point. In the FB-DDM algorithm, in orderto implement a decay in the incident field along the scattererboundary which is inside the PML region, the incident field alongthis boundary is simply evaluated using the complex coordinates,which is equivalent to the multiplication of the field with a decay-ing real-valued function along the boundary.

To better grasp the implementation procedure of the algorithmwithout loosing any generality, we assume that the original prob-lem is decomposed into three subdomains as demonstrated inFigure 2 for a circular cylinder. In the first forward iteration of theFB-DDM algorithm, we first solve the problem in the first domain�1 using the additional PML region (��PML

a )) attached to theboundary �1 [see Fig. 2(b)]. More specifically, for the 2D TMz

case (i.e., E� � azE), we solve the following BVP governed by theHelmholtz equation with a Dirichlet type BC

Inside the free-space region:

�2E1�r�� k2E1�r�� 0 in �FS1 (2.a)

with BC: E1�r�� Einc�r�� on ��S1 (2.b)

Inside the PML region:

�2E1c�r�� k2E1

c�r�� 0 in �PML1 � �PMLa (2.c)

with BC: E1c�r�� Einc�r�� on ��S1,PML (2.d)

where Einc�r�� and Einc�r�� are the incident plane wave functionsevaluated in real and complex coordinates, respectively, E1�r�� andE1

c�r�� are scattered fields in real and complex coordinates, respec-tively, � is the nabla operator in complex space, and the super-script ‘c’ represents the analytical continuation to complex space.It is clear from (2.d) that the field decays smoothly along thescatterer boundary which is inside the PML region (i.e., ��S1,PML)due to the locally-conformal PML implementation. In addition, theregion �PML

a provides a smooth field decay along the direction ofthe arrows pointing to the �PML

a symbols in Figures 1 and 2. Ifarbitrary ‘incorrect’ initial values were used along �1 (as in theclassical Schwarz method) instead of using �PML

a , there wouldoccur some artificial reflections from the boundary �1. Thus, �PML

a

gets rid of these reflections, and provides a more accurate solutioninside the first subdomain.

Since the solution of the problem on �1 induces field valuesalong the boundary �2a, we then solve the problem on the secondsubdomain �2 using the field values along �2a, and employing�PML

a attached to the boundary �2b. More precisely, we solve thefollowing BVP


�2E2�r�� k2E2�r�� 0 in �FS2 (3.a)

with BC: E2�r�� Einc�r�� on ��S2

E1�r�� on �2a(3.b)


Figure 6 Scattering by circular cylinder with different illuminations

TABLE 2 Err1 Values for Circular Cylinder with DifferentIlluminations (First Forward Iteration)

Err1 (%)

�inc�180° 0.0045�inc�135° 4.5686�inc�90° 4.0355�inc�45° 3.0451�inc�0° 2.3146

Figure 7 Bistatic RCS profile of rectangular cylinder in the first forwarditeration


�2E2c�r�� k2E2

c�r�� 0 in �PML2 � �PMLa (3.c)

with BC: E2c�r�� Einc�r�� on ��S2,PML on��S2,PML (3.d)

Since the solution of the problem on �2 yields field valuesalong the boundary �3, we then solve the problem on the thirdsubdomain �3 using the values along �3 (without using �PML

a ).More precisely, we solve the following BVP


�2E3�r�� k2E3�r�� 0 in �FS3 (4.a)

with BC: E3�r�� Einc�r�� on ��S3

E2�r�� on �3(4.b)


�2E3c�r�� k2E3

c�r�� 0 in �PML3 (4.c)

In this way, the first forward iteration of the FB-DDM algorithm iscompleted. In problems involving ‘smooth’ geometries, the firstforward iteration is sufficient to get accurate results by properlydefining the subdomains and the additional PML regions. We leavethe detailed discussion of these issues to Section 3, and we hereconsider only the basic steps of the algorithm. We should also notethat we calculate the mean value of the field values in the over-lapping regions. For instance, in the overlapping region where �1

and �2 intersect, we assume that the value of the field is the meanof E1 and E2.

Next, let us assume that the first forward iteration fails toprovide an accurate approximate solution. In this case, similar tothe classical Schwarz method, we carry out the backward iterationby first solving the problem on �2 using the new boundary valuesalong �2a and �2b, and then solving the problem on �1 using thenew boundary values along �1. Then, we implement the forwarditeration by solving the problem on �2 using the new boundaryvalues along �2a and �2b, and then solving the problem on �3

using the new boundary values along �3. Similarly, we continuewith the subsequent forward–backward iterations until conver-gence is achieved. In these iterations, we do not employ anyadditional PML region. To be more precise, for the ith subdomain

at the nth step (i.e., the step denotes each stage where a singlesubproblem is solved), we solve the following BVP.


�2Ei�r�� k2Ei�r�� 0 in �FS,i (5.a)

with BC: Ei�r�� Einc�r�� on ��S,i

En�1�r�� on �i(5.b)


�2Eic�r�� k2Ei

c�r�� 0 in �PML,i (5.c)

where En�1�r�� denotes the most recent field values inside thewhole domain at the (n�1)th step. Since the first forward iterationprovides an ‘almost correct’ initial guess for the solution along theinterfaces as well as inside the original domain, the algorithmconverges faster after the first forward iteration, compared to theclassical Schwarz method with arbitrary initial guess. The FB-Figure 8 Convergence profile of T-shaped cylinder

Figure 9 Bistatic RCS profile of T-shaped cylinder: (a) Selected itera-tions, (b) Converged iteration


DDM algorithm can be generalized to cases containing an arbitrarynumber of subdomains in a straightforward manner. The basicsteps of the FB-DDM algorithm for N subdomains are summarizedas a flowchart in Figure 3.

3. IMPLEMENTATION ISSUES

In the implementation of the FB-DDM algorithm in a scatteringproblem, some key issues should be taken into consideration toincrease the efficiency and the convergence rate of the algo-rithm. It should be obvious that the success of the FB-DDMalgorithm directly depends on the first forward iteration, wherethe BVPs over the subdomains are solved by introducing addi-tional PML regions (except in the last subdomain), because thefirst forward iteration provides an initial guess for the forth-coming forward– backward iterations analogous to those of theclassical Schwarz method. Therefore, if the first forward itera-tion provides a guess ‘close’ to the global solution, the follow-ing forward– backward iterations are expected to convergefaster. To perform a successful first forward iteration; thepartitioning of the original domain, the selection of the order ofthe subdomains, and the design of the additional attached PMLregions should be done in accordance with the expected fieldbehavior inside the original domain. If we consider the problemof scattering from ‘smooth’ objects, the first subdomain should

be coincident with the most illuminated part of the originaldomain. In this manner, the additional PML region attached tothe first subdomain will be able to absorb properly the scatteredfield, if this PML region is designed in such a way that thedirection of the decay is in conformity with the original fieldbehavior in this PML region. If these conditions are met, wemay obtain an almost accurate field solution inside the firstsubdomain. This is also applicable for the remaining subdo-mains, but the design of the additional PML region is obviouslymore critical for the regions where the scattered field has aconsiderable magnitude. To be more precise, let us consider the2D scattering from an infinitely-long cylinder with a circularcross-section. Assuming that the cylinder cross-section is ‘elec-trically-large, ’ the principles of ray optics can be employed.We assume that the plane wave is incident from the left side ofthe cylinder, as shown in Figure 4. The scattered field consistsof four terms: the reflected field, the shadow forming field, thediffracted field originating at P1, and the diffracted field origi-nating at P2 [9]. The plot in Figure 4(c) shows the rays reflectedby the circular cylinder and the rays of shadow forming field,and the plot in Figure 4(d) shows the diffracted rays emanatingfrom P1. Thus, if we choose the subdomains as shown in Figure4(a) in conformity with the field behavior, we can get an almostaccurate result in the first forward iteration. This is due to the

Figure 10 Scattered field contours of T-shaped cylinder (magnitude): (a) First forward iteration of FB-DDM, (b) 7th iteration of FB-DDM, (c) 15thiteration of FB-DDM, (d) Wholedomain FEM. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com]


fact that the direction of decay in the additional PML region[implied by the direction of arrows in Fig. 4(b)] matches thedirection of scattered field in this region.

The main reason why the first forward iteration is sufficient toget accurate results in the aforementioned manner is that none ofthe subdomains experiences any considerable field returns fromother subdomains (excluding surface diffracted waves because oftheir rapid decaying behavior, especially in the shadow region).Therefore, the subdomains are almost independent from eachother, particularly at high frequencies where the laws of ray opticsdominate. If, for instance, there happens to be a considerable fieldreturn to the first subdomain from other subdomains in the form ofa reflected field, the first subdomain would not be informed of thisreflected field, and we would solve the boundary value problem inthe first subdomain as if it is isolated from other subdomains. Inthis case, the first forward iteration is not sufficient for obtaining anaccurate solution and we need additional forward–backward iter-ations. To clarify this discussion, we consider two specific scat-tering applications: (i) scattering from a T-shaped cylinder asshown in the figure which is the inset of Figure 8, and (ii)scattering from an H-shaped cylinder as shown in the figure whichis the inset of Figure 11. We suppose that the plane wave isincident from the right side of the scatterer in both cases. If thereis a large amount of field activity (field interactions such asreflections, diffractions) between different subdomains in the illu-minated part of the original domain, the first subdomain shouldalways be chosen to be coincident to the region where the fieldactivity is dominant, as in the case of T-shaped cylinder. If there isno reflected field return to the first subdomain from other subdo-mains in the illuminated part (even if there occurs some fieldactivity in the shadow region in the form of diffractions), the firstsubdomain should be chosen as the most illuminated part of theoriginal domain as in the case of H-shaped cylinder. In both cases,the algorithm requires some forward–backward sweeps for con-vergence due to the field interactions between the subdomains.However, we expect that the problem for the H-shaped cylindershould converge faster because the first subdomain can be consid-ered as ‘more isolated’ as compared to the problem for T-shapedcylinder because of the geometry of the problem and the directionof the plane wave illumination. Therefore, we can conclude thatthe performance of the first subdomain is crucial to improve theconvergence rate of the algorithm. The implications of these ob-servations related to these examples will be demonstrated numer-ically in Section 4.

4. NUMERICAL EXPERIMENTS

In this section, we report the results of some numerical experiments totest the accuracy of the FB-DDM procedure in 2D TMz electromag-netic scattering problems involving infinitely-long cylindrical PECscatterers with arbitrary cross-section. All simulations are performedusing our FEM software employing isoparametric triangular ele-ments. In all examples, the locally-conformal PML method is imple-mented using the parameters � � 7k and m � 3, and the approximatePML thickness is chosen as /4. The wavenumber k is set to 20 (i.e.,the wavelength is 0.1 m). The element size is approximately set to/60. In addition, the incident plane wave is assumed to be E� inc

� az exp jk�x cos�inc � y sin�inc)] where �inc is the angle of inci-dence with respect to the x-axis.

The performance of the algorithm is tested in terms of the radarcross section (RCS) calculations, and is compared with the resultsof the whole-domain FEM, analytical Mie series solution, as wellas those of a standard method of moments (MoM) code employinga fine boundary mesh to minimize numerical errors. In addition,the accuracy of the algorithm is measured by means of twoFigure 11 Convergence profile of H-shaped cylinder

Figure 12 Bistatic RCS profile of H-shaped cylinder: (a) Selected iter-ations, (b) Converged iteration


different kinds of mean-square error criteria. The first error crite-rion is defined as

Err1 ��c

�EDDM � Ewhole�2

��c�Ewhole�2

(6)

where EDDM and Ewhole are the scattered field values in the originalcomputational domain �c (�c � �FS � �PML) calculated by theFB-DDM algorithm and the original whole-domain FEM, respec-tively. The second error criterion is employed only in the case ofmultiple forward–backward iterations, and is defined as

Err2 ��c

�En � En�1�2

��c�En�1�2

(7)

where En and En�1 are the scattered field values in the originalcomputational domain �c calculated by the FB-DDM algorithm atthe nth and (n�1)th iterations, respectively. The second errorcriterion also provides a proper way of finding out when toterminate the algorithm. In other words, when Err2 is less than acertain value (i.e., Err2 10�2 based on our numerical observa-tions), we assume that convergence is achieved since the scatteredfield values inside the whole domain do not change significantly

for smaller values of this constant. However, if we need to con-sider only the RCS values, we have observed that Err2 10�1 issufficient for the convergence of the algorithm. This is because thefar-field calculation which is performed in the RCS calculation hassome smoothing effect, and may reflect a reduction in the magni-tude of errors present in the near field terms.

The first example is a benchmark scattering problem where aplane wave (�inc � 180�) is incident to an infinitely-long PECcylinder of circular cross-section whose diameter is 30. Thedecomposition of the original problem into N number of subdo-mains is illustrated in the figure which is the inset of Figure 5. Weimplement the FB-DDM algorithm in just the first forward itera-tion, and we tabulate the error values Err1 for different number ofsubdomains in Table 1. We plot the bistatic RCS profiles of allcases in Figure 5 by comparing them with the whole-domain FEMsolution and the analytical Mie series solution. In this plot, theresults of the whole-domain FEM and the results of all FB-DDMcases coincide almost exactly, and they all have some tiny dis-crepancies when compared to the results of the Mie series solution.These examples demonstrate that a single forward iteration issufficient to get reliable results with a suitable partitioning scheme,irrespective of the number of subdomains. However, it is obviousthat as the number of subdomains increases, the error also in-creases because of the effect of the cumulative error as we sweep

Figure 13 Scattered field contours of H-shaped cylinder (magnitude): (a) First forward iteration of FB-DDM, (b) 4th iteration of FB-DDM, (c) 8th iterationof FB-DDM, (d) Wholedomain FEM. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com]


through the subdomains. In spite of this increase, the error valuesremain at an acceptable level.

Considering again the problem of scattering from a circularcylinder, we show the importance of the order and the positions ofthe subdomains by varying the direction of incidence of the planewave while keeping the positions and numberings of the subdo-mains fixed in all cases (Fig. 6). We decompose the whole domaininto two subdomains, and then we run the FB-DDM algorithm injust the first forward iteration. We tabulate the error values Err1 fordifferent cases in Table 2. As expected, the case where �inc � 180°shows the optimal choice for this partitioning scheme and providesthe best result for the analysis. This example demonstrates theimportance of the problem physics on the performance of thealgorithm, and proves our claim that the first subdomain should beplaced in the most illuminated part of the original domain in orderto get an acceptably accurate result in the first forward iteration.Thus, all cases except the optimal case need further forward–backward iterations to refine the solution.

The second example is a scattering problem where a planewave (�inc � 0°) is incident to an infinitely-long 16 � 4rectangular PEC cylinder. The original problem is decomposedinto four subdomains (see the inset in Fig. 7). In the first forwarditeration of the FB-DDM algorithm, we calculate Err1 as 3.1 �10�4%, and we plot the bistatic RCS profiles in Figure 7. Becauseof the absence of reflections, a single forward sweep is sufficientto yield an accurate result in this case.

The third example is a scattering problem where a plane wave (�inc � 0°) is incident to a T-shaped cylinder. The original problem isdecomposed into four subdomains (see the inset in Fig. 8). We plotthe error values Err1 and Err2 versus the number of iterations (i.e.,convergence profile) in Figure 8. In this plot, a single iteration denotesa single forward–backward iteration. In terms of the scattered fieldvalues inside the whole domain, the algorithm converges at the 15thiteration. In terms of the RCS values, the algorithm converges at the7th iteration. We also plot the bistatic RCS profiles in Figure 9, andthe scattered field contours corresponding to the first forward iteration,7th and 15th iterations of the FB-DDM algorithm, as well as thewhole-domain FEM in Figure 10. These contours serve as an indica-tion of a large amount of field activity in the illuminated part of thewhole domain.

The last example is a scattering problem where a plane wave(�inc � 0° ) is incident to a H-shaped cylinder. The original problemis decomposed into four subdomains (see the inset in Fig. 11). We plotthe convergence profile in Figure 11. In terms of the scattered fieldvalues inside the whole domain, the algorithm converges at the 8th

iteration. In terms of the RCS values, the algorithm converges in thefirst forward iteration. We also plot the bistatic RCS profiles in Figure12, and the contours of the scattered field in Figure 13. We canconclude that the algorithm in this example converges faster than theone in the previous example, as expected.

5. CONCLUSIONS

In this article, we have introduced the FB-DDM algorithm for thefinite element solution of electromagnetic BVPs using the locally-conformal PML technique. The proposed method may converge inthe first forward iteration in some problems containing smoothgeometries, or convergence may be achieved in a few iterations inother problems having challenging geometries, with the aid of asuitable partitioning scheme. We have investigated the accuracy ofthe method by means of some numerical comparisons, and wehave demonstrated that the results calculated by the FB-DDMalgorithm are sufficiently close to the reference results, and arevery reliable even in handling challenging geometries.

REFERENCES

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7. S.-C. Lee, M.N. Vouvakis, and J.-F. Lee, A non-overlapping domaindecomposition method with non-matching grids for modeling largefinite antenna arrays, J Comput Phys 203 (2005), 1-21.

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© 2007 Wiley Periodicals, Inc.

TAPERED SLOTLINE ANTENNAMODIFICATION FOR RADIATIONPATTERN IMPROVING

E. Garcıa, E. De Lera, and E. RajoDepartamento de Teorıa de la Senal y Comunicaciones, UniversidadCarlos III de Madrid, Avda. de la Universidad 30, 28911, Leganes,Madrid, Spain; Corresponding author: [email protected]

Received 6 March 2007

ABSTRACT: A novel structure of tapered slotline antenna is presented.Using “palm tree” shaped metallization cross-polarization levels areimproved. Also their influence on radiation pattern is considered.Matching performance is at least as good as a simple exponential ta-pered slotline (Vivaldi) antenna. Frequency coverage for return loss �10 dB exceeded 5.5:1 (2.9–15 GHz) and measured gain reached val-ues higher than 6 dB from 5 to 15 GHz. Radiation pattern is stablefrom 6 GHz to 12 GHz and cross-polarization is below �20 dB for the3–11 GHz band in the main direction. © 2007 Wiley Periodicals, Inc.Microwave Opt Technol Lett 49: 2590–2595, 2007; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22756

Key words: antenna; radiation pattern; tapered slotline; ultra-wideband (UWB)

1. INTRODUCTION

VIVALDI antennas are a special kind of tapered slot antennas (TSA),which are end-fire traveling wave antennas. The TSA features lowprofile, light weight, easy fabrication, conformal installation, andcompatibility with microwave integrated circuits (MIC). Prasad andMahapatra [1] introduced the linearly tapered slot antenna (LTSA) at1983. In the same year, Gibson [2] reported a TSA with exponentially


forward backward domain decomposition method for finite element solution of electromagnetic boundary...

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