Advances in Radio Science (2004) 2: 9399 Copernicus GmbH 2004 Advances in

Radio Science

Review of singular potential integrals for method of momentssolutions of surface integral equations

A. Tzoulis and T. F. Eibert

FGAN-Research Institute for High-Frequency Physics and Radar Techniques (FHR), Wachtberg-Werthhoven, Germany

Abstract. Accurate evaluation of singular potential inte-grals is essential for successful method of moments (MoM)solutions of surface integral equations. In mixed potentialformulations for metallic and dielectric scatterers, kernelswith 1/R and 1/R singularities must be considered. Sev-eral techniques for the treatment of these singularities willbe reviewed. The most common approach solves the MoMsource integrals analytically for specific observation points,thus regularizing the integral. However, in the case of 1/Ra logarithmic singularity remains for which numerical eval-uation of the testing integral is still difficult. A recently byYla-Oijala and Taskinen proposed remedy to this issue is dis-cussed and evaluated within a hybrid finite element bound-ary integral technique. Convergence results for the MoMcoupling integrals are presented where also higher-order sin-gularity extraction is considered.

1 Introduction

Exact solutions of electromagnetic antenna and scatteringproblems often rely on integral equations being solved bythe method of moments (MoM) with Galerkins method, us-ing triangular Rao-Wilton-Glisson (RWG) vector basis func-tions (Rao et al., 1982). In mixed potential formulations formetallic and dielectric scatterers, the kernels of surface inte-grals include the Greens function of free space, as well asthe gradient of Greens function. Thus, singularities of order1/R and 1/R must be considered, where R=|rr| is thedistance between observation and source points. Because ofthese terms, the surface integrals become singular if a test-ing point r is near the source element. In order to calculatethe singular surface integrals, special methods must be used,because numerical integration routines lead to inaccurate so-lutions. There are many methods that can be used to evaluatesingular potential integrals, such as the Duffys transforma-

Correspondence to: A. Tzoulis(tzoulis@fgan.de)

tion (Duffy, 1982; Chew et al., 2001) and the singularity ex-traction method (Wilton et al., 1984; Graglia, 1993; Eibertand Hansen, 1995).

In Duffys transformation, the source triangle is dividedinto three subtriangles, with common vertex at the singularpoint. Then, the integrals over each subtriangle are trans-formed into integration over a square. This procedure can-cels the singularity. Duffys transformation has three maindisadvantages. First, it is accurate only for sufficiently reg-ular triangles. Second, new integration points on the sourceintegral have to be generated for each integration point on thetesting integral. This increases the computation time. Third,singularities of order 1/R cannot be easily evaluated, be-cause Duffys transformation is derived for functions havinga point singularity of order 1/R.

In the singularity extraction method, surface integrals areregularized by extracting the singular term from the Greensfunction. The inner source integral of the extracted term iscalculated analytically in primed coordinates for specific ob-servation points. The remaining function is regular and theouter testing integral can be calculated numerically in un-primed coordinates. However, after the extraction of the sin-gular term, the remaining function is not necessarily continu-ously differentiable, which means that a straightforward ap-plication of a numerical integration routine may lead to aninaccurate solution. Further, difficulties in integration mayoccur in the case of 1/R, when the source and test trian-gles have common points and are not in the same plane. Inparticular, after extracting the singularity and calculating theinner source integral analytically, a logarithmic singularityremains on the outer testing integral. Therefore, if higher ac-curacy is required, the testing integral cannot be calculatedby a standard numerical calculation technique.

Yla-Oijala and Taskinen recently proposed a remedy tothis issue (Yla-Oijala and Taskinen, 2003). The surface inte-grals are additionally regularized, by extracting more termsfrom the Greens function and its gradient. The additional ex-tracted terms are integrated analytically over the source trian-gle in primed coordinates. The remaining function is at least

94 A. Tzoulis and T. F. Eibert: Review of singular potential integrals

Figure 1. Division into subtriangles.

Fig. 1. Division into subtriangles.

once continuously differentiable and is integrated over thetesting triangle numerically in unprimed coordinates. Fur-ther, in evaluating the integrals including the gradient of theGreens function, the source integral of the surface compo-nent of the singular term 1/R is transformed into a line in-tegral and the order of integration is being changed. Now, theinner testing integral is evaluated analytically in unprimedcoordinates and the outer source line integral is evaluated bya standard numerical calculation technique. By these modifi-cations logarithmic singularity can be avoided. All singular-ities are being extracted and calculated in closed form andnumerical integration is applied only for regular and con-tinuously differentiable functions. In this work, the abovemethod has been evaluated within a hybrid Finite Element Boundary Integral (FEBI) technique, using Combined FieldIntegral Equation (CFIE). Convergence results of the MoMcoupling integrals are presented for perpendicular source andtesting triangles with common edge. Using this new method,convergence of singular integrals is achieved with signifi-cantly less integration points and, because of this, accuracy,robustness and computation time of the FEBI technique isimproved.

2 CFIE formulation

Consider the problem of electromagnetic scattering or ra-diation including dielectric and arbitrarily shaped three-dimensional bodies. According to Huygens principle, theelectric and magnetic field can be expressed as a function ofthe equivalent current densities on the surface S of the scat-terers and the Greens function of the electromagnetic prob-lem. Then, applying the boundary condition of the surfacecurrent densities for observation points on the surface S, in-tegral equations can be derived with unknown quantities thetangential components of the electric and magnetic field (orequivalently the densities of the Huygens surface currents).

For the equivalent magnetic current density M=En onS, the Electric Field Integral Equation (EFIE) in mixed po-tential formulation is derived:

Figure 2. Triangles for numerical examples.

Fig. 2. Triangles for numerical examples.

1

2(n(r))MS(r)=j

4

S

JS(r)G(r, r)dS

j1

4

S

S JS(r)G(r, r)dS (1)

1

4

S

G(r, r)MS(r)dS+Einc(r),

with r S. Likewise, for the equivalent electric currentdensity J=nH on S the Magnetic Field Integral Equation(MFIE) in mixed potential formulation is derived:

1

2JS(r)=j

4n(r)

S

MS(r)G(r, r)dS

j1

4n(r)

S

S MS(r)G(r, r)dS (2)

1

4n(r)

S

G(r, r)JS(r)dS+n(r)Hinc(r),

with r S. Einc, Hinc is the incident field, G(r, r) =ejkR/R is the Greens function of free space outside thescatterer, n(r) is the normal unit vector of surface S and S denotes the surface divergence with respect to the prime co-ordinates.

The CFIE is the following linear combination of the EFIEand the MFIE:

Z0(1) MFIE+EFIE=0, (3)

where is the combination parameter, with 01, and Z0is the characteristic impedance of free space. Using CFIE,the internal resonance problem, which produces incorrectcomponents in the field solution, can be avoided.

The integral equations are solved by the method of mo-ments. Within the hybrid FEBI technique, the volume of thescatterer is modelled with tetrahedrons, which leads to a sur-face model with triangular elements. The surface currentsare described by RWG functions (Rao et al., 1982) and the

A. Tzoulis and T. F. Eibert: Review of singular potential integrals 95

Figure 3. Convergence of singular terms of EFIE coupling integrals with singularity 1/R

using the new method with single and double singularity extraction, compared with the

corresponding values using the old method.

Fig. 3. Convergence of singular terms of EFIE coupling integralswith singularity 1/R using the new method with single and dou-ble singularity extraction, compared with the corresponding valuesusing the old method.

Galerkins method is used for the testing procedure. Thismeans that the unknown surface currents have the form

JS=N

n=1

inn, MS=N

n=1

unn. (4)

The testing procedure is applied with RWG functions, whichmeans that m=n. After applying this solution to the inte-gral equation, various surface integrals must be solved. In thenext section these integrals are presented and several tech-niques for the treatment of these integrals in singular casesare reviewed.

3 Singular integral treatments

An application of the MoM with the Galerkins method inorder to solve the CFIE using RWG basis functions requirescalculation of the following integrals for the EFIE part:

IEFIE1 =

S

m(r) [n(r)n(r)]dS, (5)

IEFIE2 =

S

m(r)

[ S

n(r)G(r, r)dS

]dS, (6)

IEFIE3 =

S

m(r)

[ S

S n(r)G(r, r)dS

]dS,(7)

IEFIE4 =

S

m(r)

[ S

n(r)G(r, r)dS

]dS. (8)

Similarly, for the MFIE part the following integrals have tobe evaluated:

IMFIE1 =

S

m(