Fast computation of scattering from 3D complex structures by MLFMA*

Journal of Systems Engineering and Electronics Vol. 19, No. 5, 2008, pp.872–877

Fast computation of scattering from 3D complex structures by MLFMA∗ Hu Jun, Nie Zaiping, Que Xiaofeng & Meng Min Coll. of Electronic Engineering, Univ. of Electronic Science and Technology of China, Chengdu 610054, P. R. China (Received June 18, 2007)

Abstract: This paper introduces the research work on the extension of multilevel fast multipole algorithm (MLFMA) to 3D complex structures including coating object, thin dielectric sheet, composite dielectric and conductor, cavity. The impedance boundary condition is used for scattering from the object coated by thin lossy material. Instead of volume integral equation, surface integral equation is applied in case of thin dielectric sheet through resistive sheet boundary condition. To realize the fast computation of scattering from composite homogeneous dielectric and conductor, the surface integral equation based on equivalence principle is used. Compared with the traditional volume integral equation, the surface integral equation reduces greatly the number of unknowns. To compute conducting cavity with electrically large aperture, an electric field integral equation is applied. Some numerical results are given to demonstrate the validity and accuracy of the present methods.

Keywords: impedance boundary condition, equivalence theory, surface integral equation, dielectric, cavity.

1. Introduction Since 1990’s, multilevel fast multipole algorithm (MLFMA) [1] has been applied to solve electromagnetic scattering and radiation problems successfully. The primary researches concentrate on the following two aspects. One is to improve the efficiency of MLFMA and reduce the memory requirement, as higher-order MLFMA[2] , fast far field approximation (FAFFA) and ray propagation technique[3] , local MLFMA (LMLFMA)[4] , MLFMA combined with preconditioner technique[5] , parallel MLFMA[6] and so on. The other is to implement the MLFMA into the scattering solution from 3D complex structures as coating object[7−8] , thin dielectric sheet, composite dielectric and conductor[9−10] , cavity[11] and so on. This study on solving 3D complex structures by MLFMA is summarized in this article.

2. A brief introduction of MLFMA The mathematical basis of MLFMA is the addition

theorem. By the addition theorem, the dyadic Green function for non-nearby regions can be represented in a formula in which the observation point and source point are separate. Based on the formula, MLFMA can be applied to realize fast computation of matrix-vector multiplication in iterative methods, as conjugate gradient (CG) method. In detail, the matrix-vector multiplication is computed by two parts. The first part represents the interactions from nearby regions; the second part represents the interactions from non-nearby regions, computed by aggregating the contributions from sources in each non-empty box into their centers, translating the outgoing wave from source groups to the incoming wave of observation groups, and disaggregating the contributions of the center of observation groups into the observation points. In MLFMA, the matrix-vector multiplication is implemented in a multilevel multistage fashion. For N unknowns, the computational complexity and storage requirement are O(N lgN ). More details are shown

This project was supported partly by the National Natural Science Foundation of China (60431010, 60601008), New Century Excellent Talent Support Plan of China (NCET-05-0805), the International Joint Research Project(607048), in part by the “973” Programs(61360, 2008CB317110), Research Founding (9110A03010708DZ0235) and Young Doctor Discipline Platform of UESTC.

Fast computation of scattering from 3D complex structures by MLFMA

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in Refs.[1,12−13].

3. MLFMA for 3D complex structures 3.1

Scattering from thin lossy coating object

The research on scattering from 3D thin lossy coating conducting body is very important for practical problems, such as radar cross section (RCS) reduction. The impedance boundary condition (IBC) can be used to approximate the coating material region[14] . Under IBC, no material region is discretized, only the surface electric current and magnetic current need to solve. For thin lossy coating, it can be modeled as an IBC with[15]  √ ηs = −i μr /εr tan ( εr μr k0 d)

(1)

where ηs is the relative surface impedance, εr and μr are the relative permittivity and permeability of the coating material, respectively, k0 is the wavenumber in free space, and d is the coating thickness. For the IBC problem, the electric field on the surface of scatterer is related by surface impedance n · E) n ˆ = η0 ηs (ˆ n × H) Etan = E − (ˆ

(2)

Fig. 1

3.2

Monostatic RCS of coating plate

Thin dielectric sheet (TDS)

For scattering from very thin dielectric, the efficiency of volume integral equation (VIE) method is very low. For this problem, resistive sheet (RS) boundary condition can be applied. Similar to the IBC, only surface discretization is required, and MLFMA can be used to expedite the efficiency of surface IE. For the TDS, the electric field integral equation (EFIE) is as follows

ˆ is the unit external normal vector where η0 = 377Ω, n on surface S.

 tol inc Etan + L (J) tan = Etan

The electric and magnetic surface currents (density) are

the RS boundary condition (RS-BC) is yielded easily

J (r) = n ˆ ×H (r) ,

M (r) = −ˆ n×E (r) ,

r ∈ S (3)

So the relation between the electric and magnetic current can be shown from Eqs. (2) and (3) M (r) = −ηs η0 (ˆ n × J (r)) ,

r∈S

(4)

Based on the above equations, the IBC integral equation (IE) can be yielded. MLFMA can also be implemented to expedite the matrix-vector multiplication. More details are shown in Refs.[7−8]. Figure 1 shows the monostatic RCS of plate (300 mm×100 mm×1 mm), only one face is coated. εr = 29.78 − i2.31,μr = 1.87 − i1.96, and the coating thickness is 0.5 mm, horizontal polarized case. The frequency is 10 GHz. It is shown that the present result agrees well with the measure result.

tol = Etan

(5)

Js Jv = = Rs Js −iω(ε − ε0 ) −iω(ε − ε0 )t

(6)

1 −iω(ε − ε0 )t

(7)

Rs =

In the above equations, Rs represents the surface impedance of TDS, ω is the angular frequency and t is the coating thickness. Figure 2 shows the bistatic RCS of 3D dielectric sphere shell in horizontal polarization. The outer radius and inner radius are 2.005, 1.995 wavelength in free space, respectively,εr = 2.5. The results computed by FEKO software are also given for comparison. It is shown that the results by MLFMA based RS-BC agree very well with the one by FEKO.

Hu Jun, Nie Zaiping, Que Xiaofeng & Meng Min

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Fig. 2

3.3

Bistatic RCS of dielectric sphere shell, HH-pol

Composite dielectric and conductor

Except coating structures and TDS, composite dielectric and conductor is also of ten seen in the field of engineering. In general, there are two kinds of IE available for this problem. One method is hybrid volume IE and surface IE (VSIE), suitable to solve arbitrarily inhomogeneous dielectric. The disadvantage of the VSIE is that too many unknowns are required in dielectric region. Another method is SIE-based equivalence theory. Although the SIE reduces the unknowns compared with VSIE, homogeneous dielectric in each region is required. For the limit of contents, only SIEbased equivalence theory is introduced.

homogeneous medium has a permittivity ε1 and a permeability μ1 . Applying the equivalence principle, the total field in a certain region can be expressed in terms of the incident field in this region and the equivalent electric and magnetic current on the inner boundary of this region using the Green’s function for this region. Now the equivalent electric and magnetic current Jd and Md are introduced on the surface of the dielectric part, Jc and Jdc on the conductor surface and the interface. Using the PMCHW approach, the integral equations on Sd are obtained as Z1 Ls1 (Jc ) − Z2 Ls2 (Jdc ) + Z1 Ls1 (Jd ) + Z2 Ls2 (Jd )− K1s (Md ) − K2s (Md ) = −E inc

(8)

K1s (Jc ) − K2s (Jdc ) + K1s (Jd ) + K2s (Jd )+ 1/Z1 · Ls1 (Md ) + 1/Z2 · Ls2 (Md ) = −H inc

(9)

 where Zi = μi /εi . The two operators L and K are defined as    Lsi (X) = jki X(r )+1/ki2 ·∇∇ ·X(r ) Gi (r, r )dS  s

 Kis (X) =

(10a) ∇Gi (r, r ) × X(r )dS 

(10b)

s

where, Gi is the Green’s function in region i. Then EFIE is required on Sc and Sdc , respectively.   0 = Z1 Ls1 (Jc ) + Z1 Ls1 (Jd ) − K1s (Md ) + E inc × nc (11) 0 = [Z2 Ls2 (Jdc ) + Z2 Ls2 (−Jd ) − K2s (−Md )] × ndc (12)

Fig. 3

Equivalent problem of composite object

A 3D composite object is considered, which consists of conducting and homogeneous dielectric bodies, in Fig. 3(a). The dielectric part is characterized by a permittivity ε2 and a permeability μ2 and the background

The TE-PMCHW equations which consist of (8), (9), (11) and (12) can be discretized in the MOM procedure. The surfaces Sd , Sc and Sdc are separated into planar triangular patches and RWG bases are selected to represent the electric and magnetic currents on the surface. Using Galerkin’s method, it is convenient to write the whole set of TE-PMCHW equations in matrix forms Z1 P1T E Jc + Z1 P1T E Jd − Z0 QT1 E Md = bT E

(13a)

−Z2 P2T E Jdc + Z2 P2T E Jd − Z0 QT2 E Md = 0

(13b)

Fast computation of scattering from 3D complex structures by MLFMA   3.4 Scattering from conducting cavity Z1 P1T E Jc − Z2 P2T E Jdc + Z1 P1T E + Z2 P2T E Jd −   Z0 QT1 E + QT2 E Md = bT E

(13c)

  Z0 QT1 E Jc − Z0 QT2 E Jdc + Z0 QT1 E + QT2 E Jd +   2 Z0 /Z1 · P1T E + Z02 /Z2 · P2T E Md = bT H

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The problem of electromagnetic wave scattering from a conducting object with a cavity is considered illustrated in Fig.5.

(13d)

where, gn and gm are the basis and testing functions, respectively. The matrix and vector elements are defined as  TE Pimn =

 E QTimn

s

= s

gm · Lsi (gn )ds

(14a)

gm · Kis (gn )ds

(14b) Fig. 5



bTmE = −

gm · E i ds

(14c)

s

 bTmH

= −Z0

gm · H i ds

(14d)

s

Figure 4 shows the result of composite dielectric and conducting sphere calculated by MLFMA. The diameter of the PEC sphere is R = 4λ0 . The relative permittivity of the coating is εr = 2 and the thickness is d = 0.1λ0 . The incident plane wave propagates along the z-axis and is characterized by a unit amplitude electric field (E0 = 1), parallel to the x-axis. The number of unknowns is 54936. It is shown that the results by MLFMA agree very well with the analytical solution.

A conducting object with a cavity

There are three main processes to calculate the equivalent magnetic current on the aperture surface. First, the object is separated by a hypothetical interface on the aperture of the cavity. According to the surface equivalence principle, the effects of the field in the cavity can be replaced by the effects of the equivalent source on the interface with null field in the cavity. The field produced by the equivalent source is identical to that of the original problem. When the object is electrically large, the field of the surface far from the aperture has little effect to the magnetic current. So, only the effect of the innersurface and outer-surface around the aperture is calculated, namely partial coupling modeling. The error brought by partial coupling approximation will decrease by increasing the working frequency and will not make the RCS unaccepted. Of course, the more the partial surface considered, the higher accuracy is attained. The considered partial surface (real line shown in Fig.5) is open, so EFIE instead of CFIE is used to get the induced current,

 inc  ˆ ˆ t · E (r) + t · ds iωμ Js (r )ψ(r, r )+ s

 1    ∇ · Js (r )∇ψ(r, r ) == k2  ikη ¯ r )·J(r )dS  = 0 G(r, tˆ·E inc (r)+ tˆ· 4π S Fig. 4

Bistatic RCS of composite dielectric and conducting sphere, HH-pol

(15)

where tˆ denotes the weight function, k is the free-space wavenumber, ψ(r, r ) is the scalar Green function.

Hu Jun, Nie Zaiping, Que Xiaofeng & Meng Min

876 Solving the inner problem by surface integral equation makes the electric size of the cavity aperture not limited by the inversion of traditional aperture matrix. The only limitation is the total unknowns in solving the outer problem. Secondly, the total electric field on the aperture surface is Et = Ei + Es

(16)

Finally, the magnetic current is obtained by applying the equivalent principle as follows Ms = −ˆ n×E

t

(17)

where n ˆ denotes the outward unit normal to the aperture surface. To calculate the far field of the whole cavity, GCFIE[11] is used after the magnetic current was solved. Figure 6(b) shows the monostatic RCS results of a conducting cylinder with a coaxial large cavity. The diameter and height of cavity are 8λ, 2λ. The diameter and height of cylinder are 10λ, 3λ, as shown in Fig.6(a).

and conductor, cavity. For these cases, the computational complexity and storage requirement of MLFMA are also O(N log N ), where N is the number of unknowns. So the MLFMA is very efficient for the above problems, especially with electrically large sizes.

References [1] Song J M, Chew W C. Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering. Microwave Opt. Tec. Lett., 1995, 10(1): 14–19. [2] Kang G, Song J M, Chew W C, et al. A novel grid-robust higher order vector basis function for the method of moments. IEEE Trans. on Antennas & Propagation, 2001, 49(6): 908–915. [3] Cui T J, Chew W C, Chen Guang, et al. Efficient MLFMA, RPFMA, and FAFFA algorithms for EM scattering by very large structures. IEEE Trans. on Antennas and Propagation, 2004, 52(3): 759–770. [4] Hu J, Nie Z P, Lei L, et al. Three- dimensional local multilevel fast multipole algorithm. Systems Engineering and Electronics, 2006, 28(3): 329–335. [5] Lee J, Zhang J, Lu C C. Sparse inverse pre- conditioning of multilevel fast multipole algorithm for hybrid integral equations in electro- magnetics. IEEE Trans. on Antennas and Propagation, 2004, 52(9): 2277–2287. [6] Velamparambil S V, Chew W C, Song J M. 10 million unknowns: is it that big? IEEE Antennasand Propagation Magazine, 2003, 45(2): 43–58. [7] Hu Jun, Nie Zaiping, Wang Jun, et al. Solution of scattering from 3-D coating conductor with arbitrary shape with FMM and IBC. Chinese Journal of Electronics, 2000, 9(3): 337–340. [8] Hu Jun, Nie Zaiping, Lei Lin, et al. Fast solution of scattering from 3D coating object by MLFMA with diagonal element approximation method. IEEE Antennas and Propagation Symposium, 2004: 4196–4199.

Fig. 6

Monostatic RCS of a conducting cylinder with a coaxial ylinder cavity in the E-plane

4. Conclusions The MLFMA is extended to accelerate the calculation of scattering from complex structures including coating object, thin dielectric sheet, composite dielectric

[9] Yao H Y, Sheng X Q, Yung E K, et al. A rigorous analysis of dielectric tapered rod antennas. Journal of Electromagnetic Waves and Applications, 2001, 15(7): 989–999. [10] Que Xiaofeng, Nie Zaiping, Hu Jun. Analysis of scattering and radiation of mixed conducting/dielectric objects using MLFMA. the 7th International Symposium on Antennas, Propagation, and EM Theory, Guilin, China, 2006: 988–

Fast computation of scattering from 3D complex structures by MLFMA

Nie Zaiping received the B.S. degree in radar engi-

991. [11] Wang Haogang, Nie Zaiping, Wang Jun.

877

New tech-

nique of solving the integral equation for electromagnetic scattering from the targets with combined sources. Acta Electronica Sinica, 2004, 32(6): 907–910. [12] Hu J, Nie Z P. Multilevel fast multipole algorithm for solving electromagnetic scattering from 3-D electrically large object. Chinese Journal of Radio Science, 2004, 19(5): 509–514.

neering and M.S. degree in electromagnetic field and microwave technique from Chengdu Radio Engineering Institute, now University of Electronic Science and Technology of China (UESTC), in 1968 and 1981, respectively. From 1987 to 1989, he was a visiting scholar in the Electromagnetic Laboratory, University of Illinois, Urban Champagne. He is currently a full professor in the School of Electronic Engineering,

[13] Chew W C, Jin J M, Michielssen Eric, et al. Fast and effi-

UESTC. He is the author or coauthor of more than

cient algorithms in computational electromagnetics. Nor-

300 technical papers. His current research interests include antenna theory and technique, field and wave in

wood: Artech House, 2001. [14] Medgyesi-Mitschang L N, Putnam J M. Integral equation formulations for imperfectly conducting scatterers. IEEE

inhomogeneous media, computational electromagnetics, electromagnetic scattering, and inverse scattering.

Trans. on Antennas and Propagation, 1985, 33(2): 206– 214. [15] Song J M, Lu C C, Chew W C, et al. Fast illinois solver code (FISC). IEEE Antennas and Propagation Magazine, 1998, 40(3): 27–34.

Hu Jun received the B.S., M.S., and Ph.D. degrees in electromagnetic field and microwave technique from the Department of Microwave Engineering, University of Electronic Science and Technology of China (UESTC), in 1995, 1998, and 2000, respectively. During 2001, he was a research assistance in the Center of Wireless Communication, City University of Hong Kong. He is currently a full professor in the School of

Que Xiaofeng received the Ph.D. degree in electromagnetic field and microwave technique from the Department of Microwave Engineering, University of Electronic Science and Technology of China (UESTC), in June, 2008. Now he is a lecturer of School of Electronic Engineering, UESTC. His current research interests include computational electromagnetics, electromagnetic scattering, antenna theory and technique. Meng Min received the M.S. degree in electromagnetic field and microwave technique from the Department of Microwave Engineering, University of Elec-

Electronic Engineering, UESTC. He is the author or

tronic Science and Technology of China (UESTC), in June, 2005. Now she is working toward her Ph.D.

coauthor of more than 100 technical papers. His current research interests include computational electro-

degree in electromagnetic field and microwave technique, in School of Electronic Engineering, UESTC.

magnetics, electromagnetic scattering and radiation, and electromagnetic compatibility.

Her current research interests include computational electromagnetics, electromagnetic scattering.