An efficient method to calculate surface currents on a PEC cylinder with flat end caps

Authors


Abstract

[1] In this paper, an efficient method is developed to model the scattering of electromagnetic waves by large three-dimensional perfectly conducting (PEC) bodies of revolution with sharp edges. The magnetic field integral equation (MFIE) is implemented and applied to a finite solid cylinder, including the upper and lower end caps. Point matching is used to construct the interaction matrix. The numerical instabilities that generally occur to the current behavior near sharp edges are completely eliminated by the proper choice of special base functions near the edges. Not taking the singular behavior of the currents into account leads to a bad convergence or to incorrect results.

1. Introduction

[2] In the past, several integral equation methods have been used to calculate the induced surface current on a PEC object when illuminated by an arbitrary electromagnetic wave. The basic equations are the Magnetic Field Integral Equation (MFIE) and the Electrical Field Integral Equation (EFIE) or a combined formulation of them (Combined Field Integral Equation or CFIE) [Peterson et al., 1998]. When the body has sharp edges, applying the classical method of moments leads to erroneous currents and anomalies due to the singular behavior of the currents near the edges. Meixner [1972] and Van Bladel [1991] have studied the behavior of the fields and currents near sharp edges in detail. Wilton and Govind [1977] introduced the use of special base functions near edges to calculate the scattering of a TM-wave by a straight, plane, infinite PEC strip. More recently, Ingber and Ott [1991] studied a boundary element method (BEM) for the solution of electromagnetic scattering problems using the MFIE. The surface of the body was discretized by constructing a mesh of quadrilaterals and triangles. The use of superparametric elements was introduced and a special treatment of edges was discussed. In order to treat the current behavior near edges semicontinuous boundary elements were employed. The method is rather laborious and the curved surface is approximated by a polynomial expansion. Brown and Wilton [1999] developed a method to take sharp edges into account when the body is discretized by triangular curvilinear patches. Special base functions for the currents near the edges are derived. The EFIE was used to calculate the current distribution.

[3] The procedure we use to calculate the scattering by a finite 3D PEC cylinder with flat end caps builds further on the method introduced by Wilton and Govind [1977]. The method is expanded to the scattering by a three-dimensional body of revolution with curved surfaces and sharp edges. We use the MFIE, since this equation provides better numerical stability than the EFIE.

[4] In section 2, the method together with the choice of the regular and the special base functions is explained. The procedure to apply the method to a finite cylinder is demonstrated. In section 3, the results are shown and the method is validated on solutions published in the literature.

2. Method

2.1. MFIE Equation

[5] Consider the configuration of Figure 1, where a perfectly conducting body is illuminated by an arbitrary incident electromagnetic wave. The following equation is applicable (the time-harmonic dependency ejωt is suppressed) [Ishimaru, 1991]:

equation image

with

equation image

Js(r) is the surface current at location r, Hinc(r) is the incident magnetic field at location r, un(r) is the unit vector pointing outward and normal to the surface at r and Sδ is an infinitesimal surface excluding the singularity at r = r′ for the surface integration. The contribution of this singularity is the term equation imageJs(r) (the so-called self-patch). Equation (1) is the MFIE for exterior problems (such as scattering by a solid PEC body). For typical interior problems (such as fields in a waveguide), the factor 1/2 in the first term of equation (1) must be replaced by −1/2. If the structure is not resonant at the considered frequency, the boundary conditions for the electric and magnetic field at the perfectly conducting structure are satisfied since one of the major steps in obtaining equation (1) is the enforcement of these boundary conditions [Yaghjian, 1981]. We have decided to use this type of integral equation above the electrical field integral equation (EFIE) for several reasons [Jones, 1994]. The first reason is that the EFIE is a singular integral equation of the first kind, whereas the MFIE is one of the second kind, for which the theory has more sound foundations. A second reason is related to numerical considerations. The system matrix of the EFIE tends to become ill-conditioned when the number of mesh points is increased. The first term of equation (1) contributes in the same way to the diagonal elements of the interaction matrix, irrespective of the mesh size. As a consequence, the condition number of the system matrix does not degrade when the mesh size decreases when using the MFIE. A disadvantage of the MFIE is that it can only be used for closed surfaces, whereas the EFIE can also be applied for open bodies. As one or several dimensions of the body become small (relative to the wavelength), it is more difficult to obtain convergence with the MFIE, as stated by Poggio and Miller [1973]. Our results confirm these observations.

Figure 1.

Scattering of an electromagnetic wave by a PEC conductor.

2.2. Analytical Behavior of Currents Near Sharp Edges

[6] In general, the method of moments involves rooftop or pulsed base functions for the current and the point-matching or the Galerkin method as test procedure [Harrington, 1993]. Though, it is well known that currents parallel to a sharp edge can show a singular behavior [Meixner, 1972; Van Bladel, 1991]. If one tries to use the classical rooftop or pulsed base functions near the edge, obtaining correct results is not assured if the grid is refined and the resulting current often shows anomalies [Wilton and Govind, 1977]. If an analytical treatment of the current near the edge is carried out, the simple procedure of point matching gives rapidly converging results. This means that one locally applies special base functions, which include the analytical behavior of currents near an edge. The procedure was first carried out by Wilton and Govind [1977] on a straight infinite strip. Let us consider the wedge configuration of Figure 2 [Van Bladel, 1991]. At distances much smaller than the main radius of curvature of the edge, the latter may be considered as “straight” and the wedge becomes two-dimensional. A local cylindrical coordinate system is defined with the z-axis perpendicular to Figure 2 and the origin at the edge point. Van Bladel [1991] has shown that the current near the edge shows the following dependency on r (an exception occurs when the excitation shows specific symmetries with regard to the edge, which does not occur in the general case. More information is given by Van Bladel [1991]):

equation image
Figure 2.

Sharp edge configuration.

[7] The z-component of the magnetic field does not vanish at the edge [Bowman et al., 1969] so the complete behaviour of the current is:

equation image

with C representing the constant field contribution.

2.3. Application to a Finite Circular Cylinder

[8] When the MFIE is applied to the cylindrical configuration of Figure 3, the following coupled set of integral equations is obtained at r(r, ϕ, z) located on the cylindrical surface (κ(r, r′) = κr(r, r′)ur(r′) + κϕ′(r, r′)uϕ(r′) + κz(r, r′)uz):

equation image
equation image

For the observation point r located on the cap surfaces, equations (7) and (8) are applicable. The lower or upper sign is used when r is on the bottom respectively top cap.

equation image
equation image

On Figure 4, the chosen base functions and the exact positioning of the point-matching point are indicated. The standard rooftop base functions are used. An exception is made for the base functions near the edges. Figure 5 zooms into these base functions. Mathematically (based on equation (4)), Jz on the body of the cylinder and Jr on the caps behave as:

equation image
equation image

The Jϕ shows the following singular behavior at the edges (also based on equation (4)):

equation image
equation image

The singularities in expressions (11) and (12) are weak, making integration over the patch feasible. For the construction of the interaction matrix, the appropriate number of point-matching points is placed on the surface of the cylinder. If one chooses Nz, Nr and Nϕ match points in respectively the z-, r- and ϕ-direction, the size of the square interaction matrix is 2NzNϕ + 4NrNϕ. The current perpendicular to the edge introduces one extra unknown, namely the constant part of its analytical behavior. This requires an additional equation obtained by selecting an extra match point. This additional match point is indicated as a cross on Figures 4 and 5. For reasons of symmetry, this point is situated at Δr/8 (when located on a cap) or Δz/8 (when located on the cylindrical surface) away from the edge. The additional edge match point leads to an extra vector equation. One can project this equation on the z and r-axis (respectively for the cylindrical surface and cap surface) or on the ϕ-axis. The projection on the z or r-axis is preferred since projection on the ϕ-axis leads to an ill-conditioned matrix. The system matrix has a dimension of 2NzNϕ + 4NrNϕ + 4Nϕ.

Figure 3.

Geometry of the finite cylinder under study.

Figure 4.

Base functions shown on (a),(b) a cut by a plane containing the z-axis through point-matching points, (c),(d) on a cut by a plane for constant z and through point-matching points on the cylindrical surface, and (e),(f) on a curved line on a cap for constant r and through point-matching points.

Figure 4.

(continued)

Figure 4.

(continued)

Figure 4.

(continued)

Figure 4.

(continued)

Figure 4.

(continued)

Figure 5.

Detailed view on the special base functions near an edge.

Figure 5.

(continued)

3. Results

[9] In the literature, very little is published on the scattering by finite cylinders with sharp edges, shown in Figure 3. The cylindrical configuration studied is two-dimensional or does not have sharp edges. Nevertheless, we will use these results as a verification of the correctness of our approach and make interesting conclusions on approximations for certain configurations of the PEC cylinder.

3.1. Plane Wave Scattering by a Cylinder With and Without Sharp Edges

[10] A first example compares the scattering of a plane wave by a cylinder with and without sharp edges. In Figures 6a and 6b, a cylindrical rod with rounded ends as studied by Andreasen [1965] and a cylinder with sharp edges are respectively shown. A normalized perimeter s is defined, which describes the longitudinal position on the cylindrical surface. The length of the cylinder and the rod are chosen such that the paths s are evenly long for both configurations with identical diameters. We can compose the current out of two components, namely Js and Js, being the surface currents parallel respectively perpendicular to the path s. It is clear that for the cylinder with the sharp edges, Js is Jz for the body surface and ±Jr for the caps. Js and Js are plotted along the paths where the currents are maximal in Figure 7. The magnitude of the incident magnetic field is chosen 1 A/m. We observe that the standing wave pattern in Js on the cylindrical surface is similar. The magnitude of the oscillation is larger for the cylinder due to the sharp edges. The magnitude of the currents on the top cap is smaller compared with the currents on the rod.

Figure 6.

(a) Cylindrical rod with rounded edges, and (b) cylinder with sharp edges.

Figure 7.

(a) Js and (b) Js for the cylindrical rod and sharp edged cylinder.

Figure 7.

(continued)

3.2. Long Cylinder Under TE Illumination

[11] The second example illustrates the effect of the finiteness of a long cylinder. When a plane TE-wave illuminates an infinite cylinder, an exact solution can be easily found [Van Bladel, 1985]:

equation image

[12] The only existing current component is Jϕ. The finite cylinder will however contain non-zero Jz-currents, due to the effect of the currents on the caps. The longer the cylinder, the better the finite cylinder approaches the infinitely long cylinder. This is illustrated in Figure 8, where the Jϕ-component as a function of ϕ is plotted for the infinitely long cylinder with a radius of λ/4 and in the middle of the finite cylinder with the same radius but with different lengths (h = λ/2, λ, 2λ and 3λ). One can observe that the phase of Jϕ corresponds well (even for smaller cylinders) with the phase of the Jϕ of an infinitely long cylinder. The modulus of Jϕ converges more slowly towards the magnitude of Jϕ for an infinitely long cylinder. For a length of 3λ, the current is nearly identical to the one of the infinitely long cylinder. The distribution of ∣Jz∣ for h = 3λ is plotted in Figure 8c. One can clearly observe the damped standing wave pattern in Jz.

Figure 8.

(a) (b) Dependency of the central Jϕ as a function of the length of the cylinder; distribution of ∣Jz∣ for a (c) 3λ long cylinder.

Figure 8.

(continued)

Figure 8.

(continued)

3.3. Scattering of a Plane Wave on a Thin Straight Wire

[13] For a thin-wire configuration, valid approximations can be made without losing accuracy [Harrington, 1993]. As a first approximation, it is assumed that the current only flows in the direction of the wire axis. The current and charge densities are approximated by filaments of current and charge on the wire axis. Secondly, the boundary condition for the electrical field is applied only to the axial component at the wire surface. This leads to a set of equations that can be solved by the method of moments. We will apply our method to a thin straight wire, without making any approximation. The results of our method can be compared with the thin-wire approximation by integrating Jz(z, ϕ) over the circumference of the cylinder. This is the total current flowing in the axial direction at a specific location z. The position of the cylinder with regard to the co-ordinate system is shown in Figure 3. The following conventions are used: the wave vector k is located in the yz-plane (with positive k.uy and negative k.uz). The incident magnetic field has only an x component with magnitude 1 A/m and the angle of incidence is defined as the angle between uy and k. The specific dimensions of the cylinder we consider are the same as those treated by Harrington [1993], namely, h = 2λ and a = λ/74.2. In Figure 9, one can see the good correspondence between our method and the results of the thin-wire approximation for an angle of incidence of 60 degrees. Good correspondence was also found for other angles.

Figure 9.

Axial current distribution at an angle of incidence of 60 degrees.

[14] It is interesting to investigate in how far the Jϕ-component can be neglected. We have, totally arbitrary, focused on one angle of incidence, namely 15 degrees. In Figure 10a, we have plotted the distribution of Jz over the cylindrical surface. One clearly can observe that the current densities are maximal at ϕ = 270° since this position is in the middle of the illuminated side of the cylinder. At the opposite side, the currents are still relatively high. This effect is caused by the high coupling of the currents flowing on a strongly curved surface (relative to the wavelength). When larger diameters are considered, the difference between the illuminated and shady side is more pronounced, as can be seen on Figure 11a, where the radius equals λ/3 and h remains 2λ. On Figure 10b, the logarithm of the magnitude of Jϕ is plotted for different values of ϕ as a function of z. It is indeed justified in this case to neglect Jϕ relative to Jz away from the sharp edges. The ratio of magnitude is of the order of 10−5. In Figure 11b, the typical magnitude of Jϕ is depicted for a cylinder with a larger diameter (2λ/3). In this case, Jϕ has no negligible contribution to the total current distribution.

Figure 10.

(a) Distribution of ∣Jz∣ and (b) ∣Jϕ∣ over the cylindrical surface for an angle of incidence of 15 degrees and a = λ/74.2.

Figure 10.

(continued)

Figure 11.

(a) Distribution of ∣Jz∣ and (b) ∣Jϕ∣ over the cylindrical surface for an angle of incidence of 15 degrees and a = λ/3.

Figure 11.

(continued)

[15] We have further studied the relative importance of Jz and Jϕ as the radius increases. The length of the cylinder under study is λ/2 and a plane wave under an angle of incidence of 0 degrees is scattered. We have taken the ratio ρ of the maximal values of Jz and Jϕ at z = λ/8 (z = 0 corresponds to the center of the cylinder).

equation image

[16] On Figure 12, ρ is plotted as a function of a/λ. A first observation is that for very small radii, ρ converges asymptotically to zero. For the radius smaller than λ/20, one can see that Jz is typically more than 100 times larger than Jϕ. For the radius equal to λ/10, which is sometimes used as a rule of thumb to apply the thin-wire approximation, Jϕ is 5% of the value of Jz.

Figure 12.

Ratio of maximal values of ∣Jϕ∣ and ∣Jz∣ for r = a and z = λ/8 as a function of a/λ.

4. Conclusion

[17] In this paper, it is shown that by making use of the knowledge of the analytical behavior of currents near a sharp edge, correct results for general scattering problems by three-dimensional bodies of revolution with sharp edges are easily obtained. The method we explained is applied to a finite cylinder with sharp edges. The simple method of point matching leads to correct results for different kinds of configurations. The influence of the length of the cylinder on the currents has been studied. The scattering of a plane wave by a thin wire also provided interesting insights in the approximations that are commonly used for this kind of configuration. Further investigations must indicate whether the method can be applied to general bodies (not only BOR) with sharp edges. Limitations of the method are purely of numerical nature. When the scattering object becomes very large (order 10λ or beyond), the size of the interaction matrix will demand high memory storage capacity.

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