Novel Buffa-Christiansen Functions for Improving CFIE With Impedance Boundary Condition

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 8, AUGUST 2012 3763

Novel Buffa-Christiansen Functions for ImprovingCFIE With Impedance Boundary ConditionWei-Dong Li, Member, IEEE, Wei Hong, Fellow, IEEE, Hou-Xing Zhou, Member, IEEE, and

Zhe Song, Member, IEEE

Abstract—A novel kind of Buffa-Christiansen (BC) functions isdeveloped for improving the surface combined field integral equa-tion with impedance boundary condition (CFIE-IBC) from three-dimensional imperfectly conducting objects. With difference fromthe original BC function in wide use, the novel BC function is di-mensionless as the RWG function, which makes the and com-ponents of the resulting matrix element from the CFIE-IBC di-mensionally homogeneous. As a result, the eigenvalue conditionnumber of the matrix is very low, which leads to a fast-conver-gence solution whether the surface impedance is low or high. TheCFIE-IBC with the novel BC function is combined with the over-lapped domain decomposition method for reducing the memoryrequirement. In its application to the multilevel fast multipole al-gorithm, the far-field interactions can be simplified for reducingthe computational cost.

Index Terms—Buffa-Christiansen function, electromagneticscattering, impedance boundary condition, multilevel fast multi-pole algorithm, overlapped domain decomposition method.

I. INTRODUCTION

S URFACE integral equations (SIEs) have been widely usedfor formulating electromagnetic (EM) scattering problem

from an imperfectly conducting object or an object with a rela-tively thin coating [1]–[3]. These objects can be modeled in theSIEs by the Leontovitch impedance boundary condition (IBC)[4], higher order IBC [5], generalized IBC [6], and so on. TheLeontovitch IBC defines a simple local relation between tan-gential electric and magnetic fields, and it decouples the elec-tric field integral equation (EFIE) and magnetic field integralequation (MFIE). Consequently, the matrix size in the methodof moments (MoM) solution of the SIEs is reduced by one half.The application of the Rao-Wilton-Glisson (RWG) function

[10] successes in the EFIE-IBC [7], [8]. In the MFIE-IBC andthe combined field integral equation with IBC (CFIE-IBC), anobvious choice for expanding the electric current is the RWGfunction, but the function is not suited for expanding

Manuscript received December 11, 2011; manuscript revised February 23,2012; accepted March 03, 2012. Date of publication May 23, 2012; date ofcurrent version July 31, 2012. This work was supported in part by National973 projects (No. 2010CB327400 and 2009CB320203), in part by NSFC (No.60901013 and 60921063), in part by the project-sponsored by SRF for ROCS,SEM, in part by NNISRP (No. 201110046-2), and in part by NSFJC (No.BK2011019).The authors are with the State Key Laboratory of Millimeter Waves, South-

east University, Nanjing 210096, China (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAP.2012.2201121

themagnetic current because it is not divergence-conformingand then creates fictitious line charges along the border of eachRWG element in the operator [9]. For a closed object withIBC, the EFIE-IBC and MFIE-IBC suffer from internal reso-nances [2], [3]. The CFIE-IBC is free of this problem [9], andis then preferred. For reasons of numerical stability and the or-thogonality required by the IBC, and are expanded withthe RWG and Buffa-Christiansen (BC) [11] functions in theCFIE-IBC, respectively. The fictitious line charges are avoidedin the operator, and the operator is well tested through themixed discretization scheme.The BC function [9], [11]–[17] has attracted much attention

recently because of its good properties: 1) it is approximatelyorthogonal to the RWG function sharing its initial inner edge;2) no normal component flows across the border of its sup-port; 3) total charges vanish within its support. The basis func-tion with similar properties was previously proposed by Chenand Wilton [18], [19]. The BC function is firstly used for con-structing the Calderón preconditoner for the EFIE-based itera-tion solver. It is also available for expanding . The BC func-tion is actually a linear combination of small RWG functionsdefined on the barycentric triangle pairs being completely lo-cated inside its support, and its dimension is the inverse of meterin terms of geometrical size (i.e., ) [9], [12]–[16]. As weknow, the RWG function is dimensionless. The dimensional in-homogeneity of the BC and RWG functions leads to the dimen-sional inhomogeneity of the and components of the re-sulting impedance matrix element after the mixed discretizationof the CFIE-IBC, which is addressed without details in [23].This may be the reason why the CFIE-IBC with the BC func-tion yields an impedance matrix with high eigenvalue conditionnumber and a slow-convergence solution, especially for objectswith high surface impedance. The deficiency of the BC functionmotivates the work in this paper.A novel kind of BC functions is developed for improving

the CFIE-IBC from imperfectly conducting objects. Differentlyfrom the original BC function in wide use [12], the novel BCfunction is dimensionless as the RWG function, which enablesthe and components of the resulting impedance matrix el-ement from the CFIE-IBC to be dimensionally homogeneous.Consequently, the eigenvalue condition number of the matrix isessentially improved, and the matrix equation has a fast-conver-gence rate in an iterative solution.This paper is organized as follows. Section II describes the

novel BC function. In Section III, the approach to the novelBC function is applied to the RWG function. Section IV com-pares the CFIE-IBC with the original and novel BC functions.

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3764 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 8, AUGUST 2012

Fig. 1. BC function is defined on a polygon pair of (right) and (left)consisting of barycentric triangles, and the initial triangles are denoted with dashedges.

In Section V, some numerical examples are presented to vali-date the CFIE-IBC with the novel BC function. Finally, severalconclusions are given in Section VI.

II. BC FUNCTIONS

A BC function is defined on a polygon pair of and(dark gray) formed by connecting the centroids to the midpointsof edges in all initial triangles containing an endpoint of theinner edge of a RWG element in Fig. 1. It is a linear combinationof the RWG functions defined on small barycentric trianglepairs

(1)

where and are the numbers of barycentric triangles in-volved in and , respectively. The weighting coefficients

guarantee the charges neutrality within . Whilewere detailedly addressed in [12], we present an alternative

approach to derive the BC function.On one of polygon pair, for example, in Fig. 2, the chargewithin each barycentric triangle is supposed to be constant,

i.e.,

(2)

where is the area of the corresponding barycentric triangle in, and and are the lengths of inner edges of the supports

of and , respectively. By summing the terms of (2), weobtain the charge within each barycentric triangle

(3)

which is dependent on and .

Fig. 2. Weighting coefficients are determined by enforcing the charge to beconstant within each barycentric triangle of polygon .

A. Original BC Function

When and , the weightingcoefficients can be expressed, recursively by using (2), as

(4)

Similarly, on

(5)

where is the length of inner edge of the support of . TheBC function, defined by (1), (4), and (5), is the original versionwidely used in [9] and [12]–[16]. Since have the dimensionof the inversion of meter in terms of geometrical size (i.e., )and are dimensionless, the original BC function has the di-mension of , which is different from the well-known basisfunctions such as the roof-top function and the RWG function.

B. Novel BC Function

As an alternative choice, , and then theweighting coefficients of small RWG functions on are ex-pressed as

(6)Similarly, on

(7)Since in (6) and (7) are dimensionless, the novel BCfunction obtained is dimensionless, which is obviously dif-ferent from the original BC function and the same as the RWGfunction.The novel BC function has the same properties as the orig-

inal BC function: 1) it is approximately orthogonal to the RWGfunction sharing its initial inner edge; 2) no normal componentflows across the border of the polygon pair; 3) the charges within

LI et al.: NOVEL BUFFA-CHRISTIANSEN FUNCTIONS FOR IMPROVING CFIE 3765

Fig. 3. RWG function is defined on an initial triangle pair of (below) and(above) consisting of twelve barycentric triangles.

barycentric triangles of either or are constant and equalto each other, and the total charges vanish within . Inthe basis function [18]–[20], the first two properties in 1) and2) come into existence. Its different property from the novel BCfunction is that the charge densities within barycentric trianglesof either or are constant and equal to each other.

III. RWG FUNCTIONS

In this section, the idea in Section II will be applied to theRWG function. If there is no danger of confuse, some notationsin Section II are reused here for simplicity.In Fig. 3, an initial triangle pair (dark gray) consists of twelve

small barycentric triangles. A function is defined, on an initialtriangle pair, as a linear combination of twelve RWG functionson the barycentric triangle pairs

(8)

By repeating the procedure in Section II, we obtain the chargewithin each barycentric triangle of in Fig. 4

(9)

If , the weighting coefficients are expressed as

(10)

Easily seen from (10), ; and will become pos-itive if the corresponding current directions in Fig. 4 are re-versed. Similarly, on

(11)

The weighting coefficients in (10) and (11) show that isdimensionless. The right hand terms of (8) are identical to thoseof (42) in [12], which means that is actually a RWG function.Hence, the approach to the novel BC function holds true for theRWG function, which shows its validation and applicability.

Fig. 4. Weighting coefficients are determined by enforcing the charge to beconstant within each barycentric triangle of polygon , where for

, and .

IV. CFIE-IBC FORMULATION

The 3-D EM scattering problem from an imperfectly con-ducting closed object in free space is formulated by the EFIEand MFIE [1], [9] in terms of and

(12)

where “tan” denotes the tangential component on the object sur-face ; and are the incident electric and magneticfields, respectively, and

(13)

(14)

Here is the Green func-tion in free space. and satisfy the Leontovitch impedanceboundary condition (IBC) on

(15)

is the outward unit normal vector on and the surfaceimpedance is assumed to be a scalar constant for simplicity.and are expanded with the RWG functions and the novel

or original BC functions, respectively

(16)

The EFIE and MFIE of (12) are tested with the RWG func-tions and the novel or original BC functions , respectively,and they are converted into the following matrix equations:

(17)


Fig. 5. Test objects. (a) Sphere with radius . (b) Cube with edge . (c) Rotor with six blades with dimension .

TABLE IDIMENSIONS OF , , , AND , RESPECTIVELY, WHEN THEORIGINAL AND NOVEL BC FUNCTIONS ARE USED, RESPECTIVELY

where and are vectors consisting of and , respec-tively. The other notations are similarly defined as in [9].Testing (15) with gives

(18)

where the elements of the matrices are

(19)Since is quasi-curling conforming, is well-conditioned.Thus, (18) can be iteratively solved within a few iterations.Substituting (18) into (17) and combining the first and second

rows yield the matrix equation of the CFIE-IBC

(20)From the dimension of point view, we analyze , ,, , , and in (20), respectively, when the orig-

inal and novel BC functions are used as , respectively. andhave the same dimension of square meter in terms of ge-

ometrical size (i.e., ); has the dimension of whenis the novel BC function, and it is dimensionless when is theoriginal BC function. The dimensions of , , andare listed for both cases in Table I.Conclusions can be drawn from the above description.

and are dimensionally homogeneous when is the novelBC function, and dimensionally inhomogeneous when is theoriginal BC function. The dimensions of and are ho-mogeneous when is the novel BC function, and inhomoge-neous when is the original BC function. The dimensionalinhomogeneity of the and components from the originalBC function may make the impedance matrix in (20) ill-con-ditioned, and then slows down the convergence of an iterationsolver. The dimensional homogeneity from the novel BC func-tion may enable a well-conditioned impedance matrix and thena fast-convergence solution.

TABLE IIECONDS FROM THE CFIE-IBC WITH THE ORIGINAL AND NOVEL BCFUNCTIONS FOR AN IMPEDANCE SPHERE WITH DIFFERENT

TABLE IIIECONDS FROM THE CFIE-IBC WITH THE ORIGINAL AND NOVEL BC

FUNCTIONS FOR AN IMPEDANCE CUBE WITH DIFFERENT

TABLE IVECONDS FROM THE CFIE-IBC WITH THE ORIGINAL AND NOVEL BC

FUNCTIONS FOR AN IMPEDANCE ROTOR WITH DIFFERENT

Next, we compare the eigenvalue condition numbers (econds)of the impedance matrices from the CFIE-IBC with the originaland novel BC functions, respectively, for a sphere, a cube, anda rotor with six blades as shown in Fig. 5. Here the econd is de-fined as the ratio of the maximum and minimum absolute valuesof the eigenvalues of the matrix in (20). The econds are listedin Tables II–IV for the three objects with different , respec-tively. When the novel BC function is used, the econd is lowand varies gradually in small range with increasing. This isbecause that the and components of the matrix element aredimensionally homogeneous. When the original BC function isused, the econd increases drastically with increasing, and ismuch higher than that from the novel BC function, especiallyfor high surface impedance. The reason for this is that the di-mensional inhomogeneity of the and components results inunbalance in the matrix and the unbalance becomes acute withincreasing.

V. NUMERICAL EXAMPLES

In this section, the CFIE-IBC with the novel BC function istested on some impedance objects through comparison with theMie series [21] or the CFIE-IBC with the original BC function.


Fig. 6. Bistatic RCS results from the Mie series and the CFIE-IBC with theoriginal and novel BC functions for an impedance sphere, respectively.

Fig. 7. Convergence of the CFIE-IBC with the original and novel BC functionsfor an impedance sphere, respectively. The curves with (without) circle are forthe case of the novel (original) BC function.

In all cases, the incident wave is a plane wave with -polar-ization and -propagation. The scattering angle is measuredfrom the axis in the plane. The biconjugate gradient stabi-lized method [24] as an iteration solver is terminated when itsrelative residual error reaches the tolerance value of .A sphere of radius as shown in Fig. 5(a) is firstly consid-

ered, where 4764 unknowns are involved. Fig. 6 shows that theRCS results from the CFIE-IBC with the original and novel BCfunctions are in good agreement with those from the Mie series,respectively. Fig. 7 compares the convergence of the CFIE-IBCwith the original and novel BC functions for different . Whenthe original BC function is used, the number of iterations re-quired in the CFIE-IBC increases drastically with increasing,and the stopping criteria cannot be reached within 500 iterationsfor . By using the novel BC function, the numbersof iterations are reduced, significantly. The reason for this is thatthe econds of thematrices from the CFIE-IBCwith the novel BC

Fig. 8. Bistatic RCS results from the CFIE-IBC with the original and novel BCfunctions for an impedance rotor with six blades, respectively.

Fig. 9. Convergence of the CFIE-IBC with the original and novel BC functionsfor an impedance rotor with six blades, respectively.

function are much lower than those from the CFIE-IBC with theoriginal BC function, especially for high surface impedance.A rotor with six blades is shown in Fig. 5(c), where 7164 un-

knowns are involved. Fig. 8 shows that the RCS results fromthe CFIE-IBC with the novel BC function agree well with thosefrom the CFIE-IBC with the original BC function. In Fig. 9,the CFIE-IBC with the original BC function requires a largenumber of iterations to reach the stopping criteria, respectively,for and , and it fails to converge within 500 iter-ations for . The CFIE-IBC with the novel BC func-tion converges within 33 iterations. The different performanceof convergence between the CFIE-IBC with the novel and orig-inal BC functions is in agreement with the difference of econdsin Table IV.The overlapped domain decomposition method (ODDM)

[22] has proven to be an effective and efficient technique basedon the SIEs for solving the EM problem with reasonable storagecost. The ODDM solution of the CFIE-IBC with the novel


TABLE VRELATIVE RESIDUAL ERRORS OF THE OUTER ITERATIONS IN THE CFIE-IBC-ODDM WITH THE

NOVEL BC FUNCTION FOR AN IMPEDANCE ROTOR WITH SIX BLADES WITH DIFFERENT

Fig. 10. Bistatic RCS results from the CFIE-IBC with the novel BC functionand its ODDM version for an impedance rotor with six blades with

, respectively.

Fig. 11. Bistatic RCS results from the CFIE-IBC with the novel BC functionand its ODDM version for an impedance rotor with six blades with ,respectively.

BC function is achieved similarly to the version in [23]. Therotor in Fig. 5(c) is considered as an example, and its surface isdivided into six subdomains by the solid lines.The bistatic RCS results from the CFIE-IBC-ODDMwith the

novel BC function are compared with those from the CFIE-IBCwith the novel BC function in Figs. 10 and 11, respectively,

Fig. 12. Model of airplane with dimension .

Fig. 13. Bistatic RCS results from the CFIE-IBC-MLFMA with the novel BCfunction for an impedance airplane with , , and , respectively.

for and . The CFIE-IBC-ODDMat the third outer iteration gives nearly the same results asthe CFIE-IBC in both cases. Table V summarizes the relativeresidual errors at the first four outer iterations for different .When , the CFIE-IBC is for the case of the perfectlyconducting object. Seen from Table V, the outer iteration inthe CFIE-IBC-ODDM is fast convergent whether is low orhigh. The storage required in the CFIE-IBC-ODDM is reducedto 23.7% of that in the CFIE-IBC.Finally, a large model of airplane is shown in Fig. 12,

where 187986 unknowns are involved. The multilevel fastmultipole algorithm (MLFMA) is used to accelerate the ma-trix-vector products in (20). Two far-field interactions arising in

and can be simplified intothe single ones for further reducing the computational cost (see


Fig. 14. Convergence of the CFIE-IBC-MLFMA with the original and novelBC functions for an impedance airplane with , , , and ,respectively.

the Appendix). Fig. 13 shows that the backscattering RCS fromthe CFIE-IBC-MLFMA with the novel BC function is reducedin the case. Seen from Fig. 14, the CFIE-IBC-MLFMAwith the original BC function is slowly convergent for ,and even not convergent within 500 iterations for ,

, and . The CFIE-IBC with the novel BC functionhas a fast-convergence rate.

VI. CONCLUSION

In this paper, a novel kind of BC functions is investigatedfor improving the CFIE-IBC from imperfectly conducting ob-jects. The electric and magnetic currents are expanded with theRWG and novel BC functions, respectively. Those two kinds ofbasis functions can be constructed through the same approachwithin the initial triangle and polygon pairs, respectively. Withdifference from the original BC function, the novel BC func-tion is dimensionless as the RWG function. The dimension anal-ysis shows that the and components of the matrix elementfrom the CFIE-IBC with the novel BC function are dimension-ally homogeneous and those from the CFIE-IBC with the orig-inal BC function dimensionally inhomogeneous. As a result, theimpedance matrix from the CFIE-IBC with the novel BC func-tion has a low eigenvalue condition number and a fast-conver-gence performance, compared to the CFIE-IBC with the orig-inal BC function. Its combination with the ODDM can yield anaccurate solution within a very few outer iterations with smallstorage cost. The CFIE-IBC-MLFMA is applicable to an electri-cally large and realistic object, where two far-field interactionscan be simplified into the single ones for reducing the compu-tational cost.

APPENDIX

In theMLFMA, the far-field interaction of the electric currenton the RWG function in the bottom cube arising in

is

(A.1)

where denotes the integral over the unit sphere, and thelowest level of the oct-tree in the MLFMA. The incoming waveexpansion is expressed, for in , as

(A.2)

where is the unit dyadic and ; is the RWG ele-ment and is the center of . When

(A.3)

and

(A.4)

where denotes the set of the neighboring cubes of .The incoming wave expansions for in (A.3) originate fromthose for the parent cube and the outgoing wave expan-sions for the unneighboring cube whose parent cubeis neighboring to . is the translation operator [25],where , and and are centers ofand , respectively. The outgoing wave expansion for is

(A.5)

The far-field interaction of the magnetic current on the BCfunction in arising in is

(A.6)

where and are obtained by replacingand with and in (A.2)–(A.5). For simplicity,and in (A.2) and (A.5) are still used to denote the BC

elements due to the same inner edges being contained in theRWG elements, respectively.The far-field interaction of the electric current on the BC

function in arising in is

(A.7)


and the far-field interaction of the magnetic current on the RWGfunction in arising in is

(A.8)

By using (A.1) and (A.7), the far-field interaction arising inis expressed, for and in , as

(A.9)

Using (A.6) and (A.8) yields the far-field interaction arisingin for and in

(A.10)

By combining (A.9) with (A.10), two far-field interactionsarising in (20) can be converted, for and , into the fol-lowing concise form

(A.11)

Obviously, (A.11) serves as a practical approach to the reductionin computational cost, when the MLFMA is used to acceleratethe matrix-vector products in (20).

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Wei-Dong Li (M’09) received the M.S. degree inmathematics and the Ph.D. degree in radio engi-neering from Southeast University, Nanjing, China,in 2003 and 2007, respectively.In April 2007, he joined the State Key Laboratory

of Millimeter Waves, Southeast University, andis currently an Associate Professor of School ofInformation Science and Engineering. From January2008 to January 2009, he was a Visiting Scholar withthe Technische Universität Darmstadt, Germany.His research interests are in computational EM with

focus on integral-equation-based overlapped domain decomposition method,fast and accurate inter/extrapolation techniques, and fast algorithms.


Wei Hong (M’92–SM’07–F’12) received theB.S. degree from the University of InformationEngineering, Zhengzhou, China, in 1982, and theM.S. and Ph.D. degrees from Southeast University,Nanjing, China, in 1985 and 1988, respectively, allin radio engineering.Since 1988, he has been with the State Key Labo-

ratory ofMillimeterWaves and serves for the directorof the lab since 2003, and is currently a Professorand the vice-dean of the School of Information Sci-ence and Engineering, Southeast University. In 1993,

1995, 1996, 1997 and 1998, he was a short-term Visiting Scholar with the Uni-versity of California at Berkeley and at Santa Cruz, respectively. He has beenengaged in numerical methods for electromagnetic problems, millimeter wavetheory and technology, antennas, electromagnetic scattering, RF technology forwireless communications etc. He has authored and co-authored over 200 tech-nical publications, and authored two books of Principle and Application of theMethod of Lines (in Chinese, Southeast University Press, 1993) andDomain De-composition Methods for Electromagnetic Problems (in Chinese, Science Press,2005).Dr. Hong was thrice awarded the first-class Science and Technology Progress

Prizes issued by the Ministry of Education of China and the Jiangsu ProvinceGovernment etc. He also received the Foundations for China DistinguishedYoung Investigators and for “Innovation Group” issued by NSF of China. Heis a Fellow of IEEE, Vice-President of Microwave Society and Antenna So-ciety of CIE, Chairperson of IEEEMTT-S/AP-S/EMC-S Joint Nanjing Chapter,and served for the Associate Editor of IEEE TRANSACTIONS ON MICROWAVETHEORY AND TECHNIQUE during 2007–2010, and an editor board member forIJAP and RFMiCAE etc.

Hou-Xing Zhou (M’09) received the M. S. degreein mathematics from Southwest Normal University,Chongqing, China, in 1995, and the Ph.D. degreefrom Southeast University, Nanjing, China, in 2002.He is currently a Professor of School of Informa-

tion Science and Engineering, Southeast University.He has authored or coauthored 40 journal papers. Hismain research interests are in numerical algorithmsin computational EM, including fast algorithm forspatial domain dyadic Green’s functions of stratifiedmedia, the MLFMA, FFT-based fast algorithm,

the IE-based domain decomposition method, and the FEM-BI-based domaindecomposition method.

Zhe Song (S’08–M’12) received the B.S. andPh.D. degrees from Southeast University, Nanjing,China, in 2005 and 2011, respectively, all in radioengineering.Since 2011, he has beenwith the State Key Labora-

tory of Millimeter Waves, Southeast University. Hiscurrent research interests include full-wave modelingand simulation in multilayered structures by usingmethod of moments, finite element method, and do-main decomposition method.

Novel Buffa-Christiansen Functions for Improving CFIE With Impedance Boundary Condition

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