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1 This document is downloaded from DR-NTU, Nanyang Technological University Library, ingapore. Title Efficient computation of the impedance matrix of magnetic field integral equation for polyhedral conductors Author(s) Ni, Guyan; hen, Zhongxiang; hi, Jingfeng Citation Ni, G., hen, Z., & hi, J. (2015). Efficient computation of the impedance matrix of magnetic field integral equation for polyhedral conductors. IEEE transactions on antennas and propagation, 63(2), Date 2014 URL Rights 2014 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. The published version is available at: [Article DOI:

2 1 Efficient Computation of the Impedance Matrix of Magnetic Field Integral Equation for Polyhedral Conductors Guyan Ni, Zhongxiang hen, enior Member, IEEE, and Jingfeng hi Abstract The vertical improper integral method is used to formulate a polyhedral magnetic field integral equation (MFIE), which can decrease the number of singular integrals compared with the traditional MFIE. Each element in the impedance matrix resulted from the equation s moment method solution based on Rao-Wilton-Glisson (RWG) basis function is divided into two parts: the induced surface current part and the scattered field part. We obtain the analytical expressions of the induced surface current part through mathematical manipulations, and indicate that some of the integrals in the scattered field part are zero and the remaining non-zero integrals are non-singular. These results can greatly improve the efficiency of the numerical solution. Numerical results show that our new method is more accurate and efficient than the traditional method in computing the impedance matrices. Index Terms Magnetic field integral equation, method of moments, Rao-Wilton-Glisson basis function, scattered field, vertical improper integral. In this paper, we introduce a novel and efficient technique to compute the MOM solution of the polyhedral MFIE with the use of RWG basis functions. We divide each impedance matrix element Z mn of the MFIE into two parts: the induced surface current part Zmn J and the scattered field part Zmn. s We provide the closed-form expression of Zmn J by solving the planar triangular vector integral analytically, and deduce that some of integrals in Zmn s are zero and the remaining non-zero integrals are non-singular integrals by applying the vertical improper integral method [5]. Hence, the polyhedral MFIE can greatly decrease the number of singular integrals compared with the traditional MFIE method, and the analytical elements can also improve the efficiency of the numerical solution. Numerical results show that our new method is more accurate and efficient than the traditional method in computing the impedance matrices. I. INTRODUCTION BOTH electric field integral equation (EFIE) and magnetic field integral equation (MFIE) are widely used for electromagnetic scattering analysis of conducting objects with closed surfaces. In the solution of the MFIE or EFIE by the method of moments (MOM), the Rao-Wilton-Glisson (RWG) basis functions [6] are frequently used because of its accuracy and versatility. However, triangulating the surfaces of conducting bodies and using the RWG basis function to expand the unknown induced currents for the MFIE often give less accurate results compared with those of the EFIE [2], [8]. The key problem is the calculation of singular integrals associated with MFIE. In order to solve this problem, many techniques for treating singular integrals have been developed [4], such as the use of the solid-angle factor [2], singularity-extraction method [3], dual basis function method [7], evaluation the inner integral of the impedance matrix elements by its Taylor series [1]. However, if the conductor is a polyhedron, we find that many singular integrals in the MFIE solution are avoidable. This work was supported by the National Natural cience Foundation of China (No ) and Major tate Basic Research Development Program of China (No ). G. Ni is with the College of cience, National University of Defense Technology, Changsha, Hunan , China Z. hen is with chool of Electrical and Electronic Engineering, Nanyang Technological University, ingapore, J. hi is with Beijing Institute of pecial Electromechanical Technology, Beijing , China II. MFIE FORMULATION For a conducting scatterer with closed surface, the traditional MFIE can be written from the boundary condition ˆn (H i + H s ) = J as ˆn(r) H i (r) = 1 2 J (r) ˆn(r) p.v. g(r, r ) J (r )d, (1) where r, p.v. denotes the Cauchy principal value integral, g(r, r ) is the Green s function defined as g(r, r ) = exp( jk r r ) 4π r r, j = 1. (2) However, if the conductor is a polyhedron and its surface is formed by a number of small planar surfaces {T n : n = 1, 2,, N} with ˆn k being the unit normal pointing outward of T k. Considering an observation point r k T k, for k = 1, 2,, N and based on the vertical improper integral [5], the tangential scattered magnetic field at r k is as follows [5], ˆn k H s (r k ) = J (r k ) 2 + ˆn k ( g(r, r ) J (r )) d.(3) T n n=1,n k ubstituting (3) into the boundary condition, we obtain the

3 2 new MFIE based on the vertical improper integral. ˆn k H i (r k ) = J (r k ) 2 ˆn k ( g(r, r ) J (r )) d.(4) T n n=1,n k olutions of (1) and (4) with the MOM requires the expansion of J (r) into a series of basis functions and projection of the boundary condition onto testing functions. The RWG basis {f m (r)} is defined in [6], as shown in Fig. 1. f m (r) = { ( 1) i l m mi (r r mi ), r mi, i = 1, 2, 0, otherwise, (5) A. Three Basic Integrals III. ANALYTICAL EXPREION We first present the following three basic planar triangular vector integrals (12), (13) and (14), which are used in deriving the analytical expressions of Z J mn and v m. Fig. 2. Two kinds of planar triangles. Fig. 1. The m-th triangle-pair of the RWG basis. Use the RWG functions as the basis, and define J (r) as J (r) = M a n f n (r), (6) n=1 where a n is an unknown coefficient of the n-th basis functions. At the same time, applying the RWG functions as test functions to (1) and (4), one can obtain the following M M matrix equation M (Zmn J Zmn)a s n = v m, m = 1,, M, (7) n=1 where v m = (ˆn m H i (r)) f m (r)d, (8) m1 m2 Zmn J = 1 f n (r) f m (r)d. (9) 2 ( m1 m2 ) ( n1 n2 ) For the traditional MFIE, ( Zmn s = ˆn(r) p.v. g(r, r ) f n (r )d ) f m (r)d. (10) In the new MFIE,we have Z s mn = k=1 l=1,l k (ˆn k T k g(r, r ) T l f n (r )d ) f m (r)d. (11) In the following, we will derive analytical expressions of Z J mn and v m. In addition, we will indicate that some integrals in Z s mn are zero and the remaining integrals are non-singular. (r r 1 )d = A(r c r 1 ) (12) (r r 1 ) 2 d = A 6 [a2 + b 2 + a b] (13) (r r 1 ) (r r 2 )d = A 6 [3a b a2 b 2 ](14) where A denotes the area of, r 1 and r 2 denote two different vertexes of, r c denotes the center point of, (see Fig. 2(a) for (12) and (13), Fig. 2(b) for (14)). Proof. (I) Assume that the triangle of Fig. 2(a) is uniformly partitioned into n triangles i, (i = 1, 2,, n). Then for each triangle i, its area i = A/n. Let point p i i. We have n (r r 1 )d = lim (p i r 1 ) i n i=1 A = lim n n n (p i r 1 ) i=1 = A(r c r 1 ). (15) (II) Assume that the triangle of Fig. 2(a) is partitioned like Fig. 3(a). For each parallelogram ij (1 i n 1, 1 j n i), choose a point p ij ij as in Fig. 3(a). Then p ij r 1 = a(i 1)/n + b(j 1)/n. (16) Each parallelogram ij has an area of ij (r r 1 ) 2 d = lim n n 1 n i n(n 1). Hence, (p ij r 1 ) 2 ij = lim n n 3 (n 1) [(i 1) 2 a 2 n 1 n i +(j 1) 2 b 2 + 2(i 1)(j 1)a b] = A 6 (a2 + b 2 + a b) (17) (III) Assume that the triangle of Fig. 2(b) is also partitioned like Fig. 3(b). For each parallelogram ij (1 i n 1, 1

4 3 C. Computation of Zmn J 1) If m = n, using (9) and (13), we obtain Zmm J = 1 f m (r) f m (r)d 2 m1 + 1 f m (r) f m (r)d 2 m2 ( lm Fig. 3. Two partitioned triangles. = 1 2 m1 m1 ( lm ) 2 (r r m1) 2 d ) 2 (r r m2) 2 d m2 m2 j n i), choose a point p ij ij as in Fig. 3(b). Then p ij r 1 = a(n i)/n bj/n, (18) p ij r 2 = ai/n + b(n j)/n. (19) Each parallelogram ij area ij n(n 1). Hence, (r r 1 ) (r r 2 )d n 1 n i = lim (p ij r 1 ) (p ij r 2 ) ij n = lim n n 3 (n 1) (a(n i) n 1 n i bj) ( ai + b(n j)) = A 6 [3a b a2 b 2 ]. (20) = l 2 m 48A m1 [a 2 m1 + b 2 m1 + a m1 b m1 ] + l2 m 48A m2 [a 2 m2 + b 2 m2 + a m2 b m2 ] (22) where vectors a m1, b m1, a m2 and b m2 are defined in Fig. 1. 2) If m n and ( m1 m2 ) ( n1 n2 ) = ϕ, then Z J mn = 0. (23) 3) If m n and mi = nj, i, j = 1 or 2, then using (9) and (14), we obtain Zmn J = 1 f n (r) f m (r)d 2 mi = 1 ( 1) j+1 l n (r r nj ) ( 1)i+1 l m (r r mi )d 2 mi nj mi = ( 1)i+j l n l m (r r nj ) (r r mi )d 8A nj A mi mi = ( 1)i+j l n l m 48A mi [3a mi b nj a 2 mi b 2 nj] (24) where a mi = r mn r mi, b nj = r mn r nj, as shown in Fig. 4. This completes the proof. B. Computation of v m Using (8) and (12), we have v m (ˆn m1 H i (r c )) f m (r)d m1 +(ˆn m2 H i (r c )) f m (r)d m2 = (ˆn m1 H i (r c m1)) f m (r c m1)a m1 +(ˆn m2 H i (r c m2)) f m (r c m2)a m2 (21) where r c m1and r c m2 denote the center points of m1 and m2 respectively, A m1 and A m2 denote areas of m1 and m2, respectively. Fig. 4. Two triangle-pairs with mi = nj. D. Computation of Zmn. s Using (5), we rewrite Zmn s as 2 Zmn s = Zmi s nj, (25) i,j=1 where Zmi s nj is given in the following. (I) The traditional MFIE case. 1) If mi = nj, then Zmi s nj = (ˆn(r) p.v. g(r, r ) mi nj f n (r )d ) f m (r)d. (26)

5 4 This is a singular integral. 2) Else Zmi s nj = (ˆn(r) mi g(r, r ) nj f n (r )d ) f m (r)d. (27) This is a normal integral. (II) The new MFIE case. Let r mi, r nj, and ˆn mi be the unit normal pointing outward of mi. 1) If mi and nj in the same planar surface T k (k = 1,, N), then ˆn mi ((r r ) J (r )) = 0. It follows that ˆn mi ( g(r, r ) f n (r )) = 0. Hence, 2) Else Z s mi nj = This is again a normal integral. Z s mi nj = 0. (28) (ˆn mi mi g(r, r ) nj f n (r )d ) f m (r)d. (29) IV. NUMERICAL EXAMPLE A. Comparison of Approximate and Analytic olutions In the numerical computation of MFIE, many researchers take the central points r c of triangles to compute of integrals (12), (13) and (14) as their approximate values. Let I 0 = (r c r 1 )d = A(r c r 1 ), (30) I 1 = (r c r 1 ) 2 d = A(r c r 1 ) 2, (31) I 2 = (r c r 1 ) (r c r 2 )d = A(r c r 1 ) (r c r 2 ). (32) produce different errors. Hence, the solution of MFIE is more accurate if one uses the analytical solutions of Zmn J than I 1 and I 2. Table 1. Integral (13) and I 1 for different shape triangles. Int (13) I x Int (13) I 1 1 I a b c Table 2. Integral (14) and I 2 for different shape triangles. Int(14) I x Int (14) I 2 2 I a b c The next example we consider is the electromagnetic scattering by a perfectly electric conducting (PEC) cube and a PEC tetrahedron with a side length 0.1λ, where λ denotes the free-space wavelength. We calculate its bistatic radar crosssection (RC) for both vertical and horizontal polarizations. Numerical results show that RC results with the analytical solution are different from those obtained from I 1 and I 2, as shown in Fig. 6. Fig. 5. Three kinds of integral triangles. It is clear that I 0 is equal to the integral given in (12). But I 1 and I 2, are, in fact, incorrect. Next, we will compare approximate values I 1 and I 2 to analytic solutions (13) and (14) with three different shape triangles, as shown in Fig. 5. We compute analytical solutions of (13) and I 1 for x = 1, 0.1, 0.01, respectively, as shown in Table 1. imilarly, we compute analytical solutions of (14) and I 2 for x = 1, 0.1, 0.01, respectively, and they are given in Table 2. Tables 1 and 2 show that I 1 and I 2 cannot approximate (13) and (14), respectively, even if x is very small. The last columns of Tables 1 and 2 indicate that different shapes Fig. 6. Bistatic RC of a PEC tetrahedron between analytical solution and approximate solution. B. Comparison of Impedance Matrices For the new MFIE, we have that Z s mi nj = 0 if mi and nj in the same planar surface, which implies that Z s mi nj = 0 if mi = nj. However, for the traditional MFIE, Z s mi nj is a singular integral if mi = nj. In the following, we compare

6 5 the difference of two impedance matrices obtained by using two MFIEs, respectively. For example, we consider the electromagnetic scattering by a PEC cube with a side length 0.2λ, where λ denotes the free-space wavelength. We partition the surface of the cube with 300 triangles and have 450 triangle-pairs. We compute the impedance matrices Z 1 based on the traditional MFIE, Z 2 based on the new MFIE, and their error matrix Z 3 = Z 1 Z 2. Two matrices A and B are presented in the following. A is the submatrix of the first 6 columns and 6 rows of Z 1 and B is the submatrix of the first 6 columns and 6 rows of Z 3. It shows that elements of Z 1 and Z 2 are about from 10 4 to 10 7, but elements of Z 3 are about from to Indeed, the results from these two kinds of Z mn are in good agreement, as shown in Fig. 7. Hence, the singular integrals of the traditional MFIE in the numerical computation based on the MOM can be avoided. 0.1λ to 1λ and the partition length is 0.05λ. Numerical result shows that the CPU time used by our new method is much shorter than by traditional method in computing impedance matrices, as shown in Fig. 8. Fig. 8. CPU time required for PEC cubes between the traditional method and our new method. Fig. 7. Numerical results of Zmn s between the traditional method and our new method. C. Comparison of CPU Time For the same example as above, we partition the surface of the cube with 300 triangles and have 450 triangle-pairs. There are integrals to be computed in evaluating Z s mn by the traditional MFIE method, including 2700 singular integrals. However, there are only integrals, 83.3% of the total integrals, to be computed if we use the new MFIE method, and all these integrals are regular integrals. The CPU time used by two different methods in computing the impedance matrix is 5.58s by the traditional method and 1.29s by the new method, respectively. In order to compare the computation time of two methods, we consider PEC cubes with their side length varying from V. CONCLUION In this paper, the vertical improper integral method has been introduced to compute the impedance matrices of polyhedral MFIE. Each impedance matrix element of the polyhedral MFIE is divided into two parts: the induced surface current part Zmn J and the scattered field part Zmn. s Three basic planar triangular vector integral formulations have been presented first and the analytical expressions of v m and Zmn J have been derived. Numerical examples have shown that it is more accurate to compute the RC by these analytical expressions than traditional methods. econd, we have rewritten Zmn s as 2 i,j=1 Zs mi nj, and indicated that Zs mi nj = 0 if mi and nj in the same planar surface T k (k = 1,, N), which implies that each integral of Zmn s is either zero or a regular integral. Comparing with the traditional MFIE, our new method can decrease the number of integrals, especially decrease the number of singular integrals, which in turn reduces the computational time. It may be mentioned that the results may also be used in other general MFIEs. ACKNOWLEDGMENT The authors would like to thank three anonymous reviewers for their valuable comments that greatly improved the manuscript. REFERENCE [1] J.. Asvestas, D. W. Richardson, and O. E. Allen, Calculation of the impedance matrix inner integral of the magnetic-field integral equation to prescribed precision files, IEEE Trans. Antennas Propag., vol. 61, pp , Nov

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