Evaluation of Singular Potential Integrals with Linear Source Distribution

Progress In Electromagnetics Research Symposium Proceedings, Taipei, March 25–28, 2013

203

Evaluation of Singular Potential Integrals with Linear Source Distribution W. T. Sheng, Z. Y. Zhu, G. C. Wan, and M. S. Tong Department of Electronic Science and Technology, Tongji University 4800 Cao’an Road, Shanghai 201804, China

Abstract— In the method of moments (MoM) with the Rao-Wilton-Glisson (RWG) basis function for solving electromagnetic (EM) surface integral equations (SIEs), one needs to evaluate singular potential integrals with linear source distribution. The weakly singular integrals could be evaluated with the well-known Duffy’s method, but it requires a two-fold numerical integration. In this work, we develop a novel approach to handle the singular potential integrals by using a local polar coordinate system. The approach can automatically cancel the singularity and reduce the integrals to a one-fold numerical integration by deriving a closed-form expression for the integral over the polar coordinate. Numerical example is presented to demonstrate the effectiveness of the approach. 1. INTRODUCTION

Integral equation method is widely used for solving electromagnetic (EM) problems due to its unique features compared to other approaches [1]. The surface integral equations (SIEs) are preferred whenever available because they require less number of unknowns in the domain discretization. One fundamental problem in the integral equation method is the treatment of singularity or evaluation of singular potential integrals related to the Green’s function. In the method of moments (MoM) with Rao-Wilton-Glisson (RWG) basis function [2] for solving the electric field integral equation (EFIE), one needs to handle the 1/R weak singularity, where R is the distance between an observation point and a source point. This is because one can move the gradient operator of the integral kernel, which is the dyadic Green’s function, onto the basis function and testing function and lower the degree of singularity to facilitate the evaluation. The 1/R weak singularity has been widely studied and many efficient evaluation techniques have been developed [3–5]. The well-known Duffy’s method [6] is the earliest work on this subject and it mainly uses a variable change to produce an extra zero in the Jacobian which can be used to cancel the singularity and regularize the integrand. The resultant integrals are regular and can be accurately evaluated with numerical quadrature rules. However, the Duffy’s method requires a two-fold numerical integration after regularizing the integral kernel and may be inconvenient in implementation. In this work, we develop a novel approach to evaluate the singular potential integrals with a linear source distribution by using a special polar coordinate system. The approach can automatically cancel the singularity without using a variable change or coordinate transform and reduce the integral to a one-fold numerical integration by deriving a closed-form expression for the integral over the polar coordinate. The onefold numerical integration is for the angular coordinate and has a very simple integrand which can be easily evaluated. The numerical example for EM scattering by a conducting object is presented to demonstrate the approach and good result can be observed. 2. SURFACE INTEGRAL EQUATIONS

To illustrate the novel approach for evaluating the weakly singular integral, we consider the EM scattering problem for a three-dimensional (3D) conducting object embedded in the free space with a permittivity ²0 and a permeability µ0 . The governing equation for the problem is the electric field integral equation (EFIE) which can be written as [1] Z ¡ ¢ ¡ ¢ ¯ r, r 0 · JS r 0 dS 0 , r ∈ S −n ˆ × Einc (r) = n ˆ × iωµ0 G (1) S

inc

where E (r) is the incident electric field, JS (r 0 ) is the electric current induced on the conducting ¯ surface S, and n ˆ is the unit normal vector of the surface. Also, G(r, r 0 ) is the dyadic Green’s function defined by µ ¶ ¡ ¢ ∇∇ 0 ¯ ¯ G(r, r ) = I + 2 g r, r 0 (2) k0

PIERS Proceedings, Taipei, March 25–28, 2013

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where ¯I is the identity dyad, k0 is the wavenumber in the free space, and g(r, r 0 ) = eik0 R /(4πR) is the 3D scalar Green’s function in which R = |r − r 0 | is the distance between an observation point r and a source point r 0 . If the unknown current JS (r 0 ) is expanded by the RWG basis function, i.e., JS (r 0 ) =

N X

¡ ¢ In Λn r 0

(3)

n=1

where Λn (r 0 ) is the RWG basic function defined over the nth pair of triangular patches, In is the corresponding expansion coefficient, and N is the number of all pairs of triangular patches, then we can obtain the following matrix equation after using the RWG basic function as a testing function to test the EFIE N X ­ ® ­ ¡ ¢ ¡ ¢® inc ¯ r, r 0 , Λn r 0 − Λm (r), E (r) = iωµ0 In Λm (r), G

= iωµ0

n=1 N X

£­ ¡ ¢ ¡ ¢® ­ ¡ ¢ ¡ ¢®¤ In Λm (r), g r, r 0 Λn r 0 + ∇ · Λm (r), g r, r 0 , ∇0 · Λn r 0 (4)

n=1

where we have moved the gradient operator in the dyadic Green’s function onto the basis function and testing function, respectively. The integral kernel in the matrix equation is only in a 1/R weak singularity now and can be handled with the following novel approach. Note that the same singularity appears in the integral equations for dielectric or composite objects, so the developed treatment technique can be applied to those scenarios as well. 3. NOVEL APPROACH FOR EVALUATING THE SINGULAR POTENTIAL INTEGRALS

From Eq. (4), we can see that the singular integral takes two kinds of form resulting from the scalar potential and vector potential, respectively, i.e., Z Z eik0 R eik0 R I1 = dS, I2 = Λn (r) dS (5) R ∆S R ∆S where ∆S is a triangular patch ∆p1 p2 p3 or integral domain as shown in Fig. 1. We establish a local Cartesian coordinate system (u, v, w) and polar coordinate system (ρ, θ) over the triangle plane, in which the projection of the observation point on the plane is chosen as the origin. Fig. 1(a) and Fig. 1(b) illustrate the situations when the origin is outside and inside the triangle, respectively. In such a coordinate system, the observation point is located p at (0, 0, w0 ) while the source point lies on (u, v, 0) or (ρ, θ) within the triangle, leading to R = ρ2 + w02 . Note that we have used the unprimed (u, v, w) or r to indicate a source point and (u0 , v0 , w0 ) to denote an observation point in the above. For the first integral I1 , there is no singularity in fact in the situation of Fig. 1(a) since the origin is outside the triangle, but we can still derive a more friendly formulation to calculate

v

v

p1

p1

l2 O1 ρ2 j

l3 θ3

p2 ρ1 j

θ2

l1

(ρ,θ)

p3

θ1 θ

O

(a)

u

θ2

O

p3

d O1

j j

O2

j

θ0 j

u

θ1 (ρ,θ)

O3

p2

(b)

Figure 1: A local Cartesian coordinate system (u, v, w) and polar coordinate system (ρ, θ) are established over the plane of a triangular patch ∆p1 p2 p3 . The projection of the observation point on the plane is chosen as the origin. (a) The origin is outside the triangle. (b) The origin is inside the triangle.

Progress In Electromagnetics Research Symposium Proceedings, Taipei, March 25–28, 2013

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it, i.e., I1 =

2 Z X j=1

=

=

dθ

dθ

Z

θj+1

R2j (θ)

R1j (θ)

θj

2 Z X j=1

Z

θj+1

√2 2 eik0 ρ +w0 p ρ dρ ρ2 + w02

ρj2 (θ)

ρj1 (θ)

θj

2 Z X j=1

Z

θj+1

dθ

R2j (θ)

R1j (θ)

θj

eik0 R R

q R2

−

w02 d

µq ¶ 2 2 R − w0

eik0 R dR

Z θj+1 h 2 i X j j 1 eik0 R2 (θ) − eik0 R1 (θ) dθ = ik0 θj

(6)

j=1

where R1j (θ)

q q j j 2 = ρ1 (θ) + w0 , R2 (θ) = ρj2 (θ) + w02

(7)

and ρj1 (θ) and ρj2 (θ) are the radial coordinates of the intersection points formed by the polar axis at the angle θ and two sides of the triangle and they can be easily determined by the slopes and intercepts of the sides. If b1 and b2 denote the interceptions and k1 and k2 denote the slopes of the two sides, respectively, then they can be found as b` , ` = 1, 2. sin(θ) − k` cos(θ)

ρj` (θ) =

(8)

From the formulas, we can see that the singularity, if existing, moves to R1j (θ) or R2j (θ) when tan(θ) is equal to one of slopes of the sides. This situation occurs only when one of the sides coincides with the line connecting the point o and the point o1 . Fortunately, this singularity vanishes because the upper bound and lower bound of the integral with respect to θ are the same in this case and this result matches the reality, i.e., no singularity. On the other hand, there exists a singularity in the situation of Fig. 1(b) because the origin is inside the triangle. We divide the triangle patch into three subtriangles by connecting the origin with three vertices of the triangle and each subtriangle is specified by the parameters dj , θ0j , θ1j , and θ2j as shown in the figure. In this case, the integral I1 can be derived as Z dθj (θ) ik0 √ρ2 +w02 3 Z θ2j X e p I1 = dθ ρ dρ j ρ2 + w02 0 j=1 θ1 µq ¶ Z Rj (θ) ik0 R q 3 Z θ2j X e 2 2 2 2 = dθ R − w0 d R − w0 R θ1j R0 j=1

=

3 Z X j=1

θ2j

θ1j

Z

Rj (θ)

dθ

eik0 R dR

R0

3 Z j i 1 X θ2 h ik0 Rj (θ) e − eik0 R0 dθ = ik0 θ1j

(9)

j=1

where R0 = w0 , Rj (θ) =

q [dθj (θ)]2 + w02

(10)

and dθj (θ) = dj / cos(θ − θ0j ). Obviously, there is no singularity in the above final formulation because θ − θ0j cannot be 90◦ or cos(θ − θ0j ) cannot be zero for a triangle and we can use a numerical quadrature rule to conveniently evaluate the one-fold integral with a very simple integrand. The above formulas are valid for all w0 , including w0 = 0 which is the singular case.

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For the second integral I2 , the involved RWG basis function is actually a first-order (linear) polynomial vector and its component could be written as a0 + a1 u + a2 v = a0 + a1 ρ cos θ + a2 ρ sin θ in the local coordinate system, where a0 , a1 , a2 are the known constants. When the basis function combines with the scalar Green’s function, the combination of the first term a0 will result in the integral I1 which we have addressed above. The combination of the second and third term will yield the following integrals ¶ µ ¶ Z ρj2 (θ) ik0 √ρ2 +w02 2 Z θj+1 µ X e cos θ I2a = dθ ρp ρ dρ sin θ I2b 2 + w2 j ρ θ ρ (θ) j 1 0 j=1 ¶ Z Rj (θ) q 2 Z θj+1 µ X 2 cos θ dθ = R2 − w02 eik0 R dR (11) sin θ j θj R (θ) j=1

1

respectively, for the case of Fig. 1(a), and ¶ µ ¶ Z dθj (θ) ik0 √ρ2 +w02 3 Z θ2j µ X e cos θ I2a dθ = ρp ρ dρ sin θ I2b j ρ2 + w2 θ 0 j=1

=

0

1

3 Z X

j 2

θ

µ

j 1

θ

j=1

cos θ sin θ

¶

Z dθ

dθj (θ) q

R2 − w02 eik0 R dR

0

(12)

respectively, for the case of Fig. 1(b). For the nonsingular case or w0 6= 0, we cannot derive a closed-form formula for the inner integrals in the above and they are evaluated with the numerical quadrature rule. For the singular case, namely, w0 → 0 in the situation of Fig. 1(b), we can derive a closed-form formula for the inner integrals in the above, i.e., µ s ¶ X ¶ Z θ2j µ Z dθj (θ) q 3 I2a cos θ = lim dθ R2 − w02 eik0 R dR s I2b sin θ w0 →0 θ j 0 j=1

=

1

3 Z X j=1

j 2

θ

θ1j

µ

cos θ sin θ

¶·

eik0 dj ik0

µ ¶ ¸ 1 1 θ dj (θ) − − 2 dθ. ik0 k0

(13)

4. NUMERICAL EXAMPLE

To demonstrate the effectiveness of the novel approach for evaluating the singular potential integrals, we present a typical numerical example for EM scattering by a perfectly electric conducting (PEC) object, but the technique can also be used in the problems with non-PEC objects. It is assumed that the incident wave is a plane wave with a frequency f = 300 MHz and is propagating along the −z direction in free space. We calculate the bistatic radar cross section (RCS) of the object observed along the principal cut (φ = 0◦ and θ = 0◦ –180◦ ) for the scatterer with both vertical 0

Bistatic Radar Cross Section (dB)

-2

-4

-6

-8

VV, Exact HH, Exact VV, MoM HH, MoM

-10

-12 0

20

40

60

80 100 θ (Degrees)

120

140

160

180

Figure 2: Bistatic RCS solutions for a PEC sphere with a radius a = 0.2758λ.

Progress In Electromagnetics Research Symposium Proceedings, Taipei, March 25–28, 2013

207

polarization (V V ) and horizontal polarization (HH). The scatterer is a PEC sphere with a radius a = 0.2758λ (the origin of the coordinate system is the center of the sphere) and we discretize the spherical surface into 4216 triangular patches. Fig. 2 plots the RCS solutions and they are very close to the corresponding Mie-series solutions. 5. CONCLUSION

Although many techniques have been developed for evaluating the singular potential integrals arising from the SIEs in the MoM with the RWG basis function, we provide an alternative approach to handle the singular integrals in this work. The approach can automatically cancel the singularity without using a variable change or coordinate transform and reduce the integrals to a one-fold numerical integration with a very simple integrand. Compared with the Duffy’s method which requires a two-fold numerical integration, the approach could be more convenient in implementation and more efficient in calculation as illustrated by the numerical example. ACKNOWLEDGMENT

This work was supported by the Program of Shanghai Pujiang Talents, Shanghai, China, with the Project No. 12PJ1408600. REFERENCES

1. Chew, W. C., M. S. Tong, and B. Hu, Integral Equation Methods for Electromagnetic and Elastic Waves, Morgan & Claypool, San Rafael, CA, 2008. 2. Rao, S. M., D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propagat., Vol. 30, No. 3, 409–418, 1982. 3. Wilton, D. R., S. M. Rao, A. W. Glisson, D. H. Schaubert, O. M. Al-Bundak, and C. M. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propagat., Vol. 32, No. 3, 276–281, 1987. 4. Graglia, R. D., “Static and Dynamic potential integrals for linearly varying source distributions in two- and three-dimensional problems,” IEEE Trans. Antennas Propagat., Vol. 35, No. 6, 662–669, 1987. 5. Khayat, M. A. and D. R. Wilton, “Numerical evaluation of singular and near-singular potential integrals,” IEEE Trans. Antennas Propagat., Vol. 53, No. 10, 3180–3190, 2005. 6. Duffy, M. G., “Quadrature over a pyramid or cube of integrands with a singularity at a vertex,” SIAM J. Numer. Anal., Vol. 19, No. 6, 1260–1262, 1982.