芮 錫，胡 俊，柳清伙

（1.中國西南電子技術(shù)研究所，成都610036；2.電子科技大學(xué)電子工程學(xué)院，成都610054；3.杜克大學(xué)電氣和計算機(jī)工程學(xué)院，達(dá)勒姆27708，美國）

混合FEM-FIPWA求解復(fù)雜旋轉(zhuǎn)體散射

芮 錫1，胡 俊2，柳清伙3

（1.中國西南電子技術(shù)研究所，成都610036；2.電子科技大學(xué)電子工程學(xué)院，成都610054；3.杜克大學(xué)電氣和計算機(jī)工程學(xué)院，達(dá)勒姆27708，美國）

提出了一種基于混合有限元與快速非均勻平面波算法求解復(fù)雜旋轉(zhuǎn)體的散射問題。內(nèi)部電場采用基于點(diǎn)元和邊元基函數(shù)的有限元方法計算，同時在旋轉(zhuǎn)體的外表面的場采用基于三角基函數(shù)及脈沖基函數(shù)的快速非均勻平面波算法計算。采用這種方法處理復(fù)雜大尺度的旋轉(zhuǎn)體問題能節(jié)省計算的內(nèi)存和計算時間，算例驗(yàn)證了算法的準(zhǔn)確性和有效性。

復(fù)合材料；旋轉(zhuǎn)體；電磁散射；快速非均勻平面波算法；有限元法

1 Introduction

The electromagnetic problem for bodies of revolution（BOR）of arbitrary shapewith different kinds ofmaterials has been widely discussed for several decades［1－6］.In thiswork，F(xiàn)EM-BI is presented to analyze the scattering of BOR with inhomogeneousmaterials，and extend the fast inhomogeneous plane wave algorithm to accelerate the computation of themethod ofmoment（MoM）.Finite ElementMethod（FEM）isused to analyze the interior electric field.Edge-based and node-based elements are used to represent the interior electric field.While for the exterior region，Boundary Integration（BI）is used as a exact boundary condition.Triangular and pulse basis functions are used to represent the electric and magnetic fields on the boundary.The aggregation and disaggregation factors in Fast Inhomogeneous Plane Wave Algorithm（FIPWA）can be derived analytically.Both thememory requirement and the CPU time are saved for large scale BOR prob-lems.Numerical results are given to demonstrate the validity and the efficiency of the presented method.

2 FEM-FIPWA for Axisymmetric Resonators

2.1 Body of revolution

Because of the symmetry of the geometry，the volume of the BOR is generated by revolving a plane curve about the z-axis as shown in Figure 1.Here（ρ，φ，z）are the variables in cylindrical coordinate system，θincis the angle of incidentwave，＾t and＾φare the unit vectors，S is the surface of the BOR.

Fig.1 Body of revolution and coordinate system圖1 旋轉(zhuǎn)體結(jié)構(gòu)坐標(biāo)系統(tǒng)

The electric and magnetic fields can be expressed in a Fourier series：

where Et，m，Eφ，m，Ht，mand Hφ，mare the electric and magnetic fields in the meridian plane and the azimuthal component of the m-th Fourier mode，respectively.As the fields are decomposed into two parts as shown in E-quation（1－2），only a 2－Dmesh（meridian cross section）is needed for analyzing the3－D axisymmetric problem.These differentmodes can be treated separately because of the orthogonality.In the cylindrical coordinate system，the unit vectors＾t and＾φare defined as whereθis the angle between the z-axis and the unit vector＾t.φis the azimuthal angle as shown in Figure 1.

2.2 FEM for the interior region

As shown in Figure2，the interior region of the BOR is filled with inhomogeneous material with the relative permittivityεrand the relative permeabilityμr.Bothεr andμrare the function of z andρ，but independentofφ.

Fig.2 Themesh for the interior and exterior regions圖2 內(nèi)部區(qū)域與外部區(qū)域的網(wǎng)格劃分

The vector Helmholtz equation for the electric field can be written as

where Seis the source term，and k0is the wave number in free space.If it is source free（Se＝0）in the interior region the weak form of the vector wave equation can be expressed as［7－8］

where Wlis the testing function，andμ0is the permeability of the free space.

The fieldsmust retain the continuity for any values of φon the z-axis（ρ＝0）.Thus，there are three kinds of conditions for different cylindricalmodes：

The basis functions for the electric field are chosen as

where Nnis the number of nodes，Nsis the number of segments（or edges），et，iand eφ，iare the unknown coefficients，and Niand Nirepresent the standard node-based and edge-based element basis functions，respectively. Themagnetic field on the boundary S is expanded as the same as the electric field.

The testing function is chosen as

After substituting the basis and testing functions into Equation（6）and making use of the orthogonality of cylindricalmodes，the wave equation can be rewritten as

can be formed，where m is the index of themode and b is the index of boundary.And thematrix element can be expressed as

2.3 FIPWA for the exterior region

For the exterior region，boundary integration is applied to govern the boundary electricfield E andmagnetic fields M.The fields on the boundary S can bewritten as

where J and M are the equivalent electric and magnetic currents，which will satisfy the electric field integral equation（EFIE）and the magnetic field integral equation（MFIE）

where G is the Green′s function，and the integration in the equation above has remove the contribution of the singular point.The key process of MoM is solving themodal Green′s function gnwhich can be expressed as

For the traditionalMoM，themodalGreen′s function has to be evaluated by numericalmethod，hence it is time consumingwhen the radius of the BOR is large.In this section，fast inhomogeneous plane wave algorithm（FIPWA）is applied to accelerate the computation of the MoM for bodies of revolution.Based on Weyl Identity［9－10］，the Green′s function can be expressed as

Fig.3 The Sommerfeld integration path on the complex u plane圖3 復(fù)平面上的Sommerfeld積分路徑

In order to realize the Fast Inhomogeneous Plane Wave Algorithm（FIPWA），the basis functions are divided into groups.Herewe call rmand rm′are the centers of the groups which contain the source point rjand field point rirespectively.As shown in Figure 4，rji＝rjm＋rmm′＋rm′i.Equation（26）can be rewritten as

Fig.4 The field point and the source point圖4 場點(diǎn)與源點(diǎn)位置分布

The basis functions are divided into M groups along the z direction as shown in Figure 5.In this way，the factor rmm′has z component only.This character will make the integrand decay exponentially away from the real axis in the complex u plane.

Fig.5 The BOR is divided into groups圖5旋轉(zhuǎn)體分組示意圖

Equation（26）can be rewritten as

Here the Bjm（u，v）and Bm′i（u，v）represent the radiation and receiving patterns for the field and source groups，respectively.And f（u）can be considered as the weight function.Both Bjm（u，v）and Bm′i（u，v）are the inhomogeneous planewave as u is complex.With proper numericalmethods for u and v，the integral can be expressed as

The detail of the FIPWA can be found in Reference［5－6］.

2.4 FEM-FIPWA for BOR

where Piis the pulse basis function and Tiis the traditional triangle basis function.Combining the FEM part（interior region）and FIPWA（boundary），thematrix equation can be derived as

The details of thematrix elements can be found in Reference［4－6］.The boundary currents and fields can be derived by solving the equation above，and the far field also can be solved.In the next section，two numerical results will be given to demonstrate the validity and the efficiency of the proposed method.

3 Numerical Results

In this section，two numerical results are presented to show the validity of the proposed FEM-FIPWA method.All problems are solved on the same computer（Intel Core2 DuoCPU P8400＠2.26GHz with 1.92GB RAM）in order to make a fair comparison，with only one core being used.

3.1 An inhomogeneous dielectric sphere with two layer medium

As shown in Figure 6，an inhomogeneous dielectric sphere is computed.The sphere isexcited by the planewave with horizontal polarization（θinc＝00，φinc＝00，λ＝2 m）. The total number of the unknowns is about 40 000.The bistatic RCS is shown in Figure 7，the resultof FEM-BI（or FEM-FIPWA）agreeswellwith analytical result.

Fig.6 An inhomogeneous dielectric spherewith two-layermedium圖6 非均勻雙層介質(zhì)球結(jié)構(gòu)

Fig.7 The bistatic RCSof the dielectric sphere result by FEM-BIcompared with analytic result圖7 介質(zhì)球雙站RCS的計算結(jié)果比較圖

3.2 An inhomogeneous dielectriccylinder

The numerical result proposed above shows the accuracy of FEM-FIPWA for BOR problems.In this section，a more complex example is shown to verify the efficiency of proposedmethod mentioned in this paper.As shown in-Figure 8，there are seven layeredmedium.The thickness of the six inner medium is 0.2m and the height is 2.6m.The thickness of the outermedium is0.3m，and the height is 2m.The cylinder is excited by the plane wavewith horizontal polarization（θinc＝90°，φinc＝0°，λ＝0.5m）.The number of the total unknowns is morethan 100 000 in this2-Dmesh.The field distribution derived by FEM-BI is compared with Wavenologywhich is a famous commercial EM software in USA.As shown in Figure 9，the results agree wellwith each other.

Fig.8 The geometry of the inhomogeneous dielectric cylinder圖8 非均勻介質(zhì)柱結(jié)構(gòu)圖

Fig.9 The field distribution of the dielectric cylinderwith two differentmethods圖9 兩種方法計算介質(zhì)柱的場分布結(jié)果比較

As shown in Table 1，thememory requirements and CPU times for FEM-MoM and FEM-FIPWA are compared.

Table 1 The comparison between FEM-MoM and FEM-FIPWA表1 FEM-MoM與FEM-FIPWA計算比較

4 Conclusion

In this paper，hybrid FEM and FIPWA technique is applied to solve the BOR scattering problem.In this FEM-FIPWA method，the problem is separated into interior and exterior problems.In the interior region，F(xiàn)EM based on hybrid edge-based and node-based elements is used to present the electric field.In the exterior region，boundary integration（BI）is used as the exact boundary condition.Triangular and pulse basis functions are used for representing the electric and magnetic fields on the boundary.FIPWA is added for the BIpart.The proposed method can solve large scale bodies of revolution with inhomogeneous，compositematerials efficiently.

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Biography：

RUIXiwas born in Liyang，Jiangsu Province，in 1983.He received the B.S.degree and the Ph. D.degree from University of Electronic Science and Technology of China in 2005 and 2010，respectively.From 2008 to 2010，he has been workingwith Prof.Q.H.Liu as a visiting scholar in Duke University.He is now an engineer.His research interests include computational electromagnetic and RF stealth.

Email：uestcruixi＠gmail.com

芮錫（1983—），男，江蘇溧陽人，分別于2005年和2010年獲電子科技大學(xué)學(xué)士學(xué)位和博士學(xué)位，2008～2010年在美國北卡州杜克大學(xué)做訪問，現(xiàn)為工程師，主要研究方向?yàn)殡姶庞嬎闩c射頻隱身。

Hybrid FEM-FIPWA for Scattering from Com plex Bodies of Revolution?

RUIXi1，??，HU Jun2，LIU Qing-huo3（1.Southwest China Institute of Electronic Technology，Chengdu 610036，China；
2.School of Electronic Engineering，University of Electronic Science and Technology of China，Chengdu 610054，China；3.Department of Electrical and Computer Engineering，Duke University，Durham，NC 27708，USA）

A method based on hybrid finite elementmethod（FEM）and fast inhomogeneous planewave algorithm（FIPWA）is proposed to solve the electromagnetic scattering problem for bodies of revolution（BOR）with inhomogeneous，compositematerials.The FEM with mixed edge-based and node-based elements is used for representing the interior electric field，while the FIPWA is used as the exact boundary condition，hybrid triangular and pulse basis functions are used for representing the electric field and magnetic field on the boundary.Both thememory and CPU time requirements are reduced for large scale BOR problems.Numerical results are given to demonstrate the validity and the efficiency of the proposedmethod.

compositematerial；body of revolution；electromagnetic scattering；fast inhomogeneous planewave algorithm；finite elementmethod

10.3969/j.issn.1001－893x.2013.01.019

TN

A

1001－893X（2013）01－0093－06

?Received date：2012－11－28；Revised date：2012－12－13

2012－11－28；修回日期：2012－12－13

??Corresponding author：uestcruixi＠gmail.com通訊作者：uestcruixi＠gmail.com