Well-posedness of integral equations for modeling electromagnetic scattering from cavities

Authors


Abstract

[1] Integral equations for modeling electromagnetic scattering from indented screens or cavity structures are proposed using a symmetric equivalent current model, which is based on image theory with a perfect electrically conducting (PEC) ground plane. The well posedness of the proposed integral equations is theoretically established in this paper. It is known that all resonance modes for cavity problems either due to the presence of a perfect magnetically conducting (PMC) open aperture or a PEC open aperture. In this paper, it is proved that both kinds of resonance modes cannot exist for the proposed integral equations without excitation. To numerically examine this well posedness, the condition numbers of resultant matrices from the generalized network formulation and proposed integral equations are analyzed as functions of frequency and the depth of cavity. The numerical analysis of the condition numbers obtained also shows that the integral equations proposed in this paper are immune to the problem of the interior resonance for three-dimensional cavity problems. Moreover, simulation of scattering from standard cavity structures verifies the validity of the proposed integral equations.

1. Introduction

[2] The integral equation method is one of the most popular techniques for modeling electromagnetic scattering from an indented screen or cavity structure. This important topic of research has attracted much attention over many years. The first and most widely used technique is the generalized network formulation (GNF) [Harrington and Mautz, 1976]. However, it suffers from the interior resonance problem at the frequencies corresponding to the resonant frequencies of the cavity formed by the closed metallic boundary. A variation of this approach, namely, the connection scheme, has been proposed by Wang and Ling to analyze electromagnetic scattering from cavities [Wang and Ling, 1991]. In the latter approach, the cavity is decomposed into subregions, each of which is treated via integral equations on the subregion boundaries. As such, it can be used to analyze a larger cavity problem. However, more singular frequencies will occur using this technique since resonances can occur not only for the whole cavity region but also for all subregions. Asvestas and Kleinman have developed a set of coupled integral equations using purely mathematical vector manipulation in conjunction with Maxwell's equations [Asvestas and Kleinman, 1994]. It has been claimed that the coupled integral equations are expected to be free of interior resonance, but it still remains to be shown. The same integral equations have been set up using a physical model based on image theory with a perfect magnetically conducting (PMC) ground plane and verified by simulating electromagnetic scattering from three-dimensional (3D) rectangular cavities in [Xu et al., 2005]. Wood and Wood [1999] have applied these coupled integral equations to the two-dimensional (2D) cavity filled with a homogeneous material. For the 2D cavity problem, it has been numerically proved that the coupled integral equations are immune to the interior resonance problem. For the three-dimensional (3D) cavity problem, there is no proof to establish that the coupled integral equations are free from the interior resonance problem.

[3] In this paper, we first develop boundary integral equations for electromagnetic cavity problems using the symmetric equivalent current model based on image theory with a perfect electrically conducting (PEC) ground plane developed by Reuster and Thiele [1995]. In this equivalent model, the current on the cavity wall and its image form a continuous current across the rim of the cavity wall. This continuity of the current and its image will relatively simplify the numerical implementation of the proposed integral equations. By combining the integral equations over the aperture, we establish the proposed integral equations for the cavity problems. It is well known that only two kinds of resonance modes exist for cavity problems. One set of modes occurs when the open aperture is replaced by a PEC; these are the modes with zero tangential electrical fields over the open aperture. The other set of modes occur when the open aperture is replaced by a PMC; these are modes with zero tangential magnetic fields over the open aperture. It is proved that neither type of resonant mode can exist for the proposed integral equations if no excitation is present. In the numerical results section, some simulations for the scattering from standard cavity structures are given to verify the correctness of the proposed integral equations. Finally, to numerically verify the well posedness of the proposed integral equations, the condition numbers of matrices obtained from the GNF formulation and the proposed integral equations are analyzed versus frequencies and cavity depth. These condition numbers corroborate our analysis, indicating that the proposed integral equations are immune to the interior resonance problem for 3D cavities.

2. Formulations

[4] Figure 1 shows an open cavity with an infinite ground plane. In Figure 1, S1 and S2 represent the cavity aperture and the cavity walls, respectively. For convenience, we assume that S1 is located in the xoy plane, with region I for the space inside the cavity, and region II for the free space outside the cavity (z > 0).

Figure 1.

Open cavity with an infinite ground plane.

[5] From the equivalence principle [Balanis, 1989; Harrington, 1961], the fields in the two regions can be decoupled by closing the aperture with a perfect conductor and introducing the magnetic current over aperture region, M1 = E × (−equation image). Therefore by using image theory, the total magnetic field in region II can be expressed as

equation image

where Hinc is the incident field, and Href is the field reflected by the ground plane without the aperture. Also, k0 is the free space wavenumber, Y0 is the free space intrinsic admittance, and equation image is the free space dyadic Green's function given by

equation image

where equation image = equation image + equation image + equation image and G(r, r') = equation image.

[6] For the field in region I, the symmetric equivalence model shown in Figure 2 can be obtained by applying image theory. With this equivalence model, the total magnetic field and electric field in region I can be represented as

equation image
equation image

where Z0 = equation image is the free space intrinsic impedance, J2 = equation image × H is electric current on the cavity, equation image is unit normal vector on the interior cavity wall and pointing into the cavity, and J2i is the image of the electric current J2 with respect to the infinite PEC plane located at the aperture.

Figure 2.

The symmetric equivalent model for fields in region I.

[7] Letting rS1 and using the continuity of the tangential magnetic field across the aperture (equation imageHtan(r) = equation imageHtan(r)), the combination of (1) and (2) provides,

equation image

Letting rS2 and using the definition of J2, equation (2) can be rewritten as

equation image

Similarly, letting rS2 and imposing the boundary condition Etan(r) = 0 on S2, equation (3) can be rewritten as

equation image

Equations (4) and (5) or (4) and (6) form the whole integral equation governing the cavity problem. The combination of (4) and (5) is a magnetic field integral equation, while the combination of (4) and (6) involves both magnetic field on S1 and electric field on S2, which may be called a hybrid field integral equation.

3. Well Posedness of the Proposed Integral Equations

[8] To prove the proposed integral equations (4) and (5) or (4) and (6) do not exhibit interior resonances, it is enough to show that no interior resonant mode field satisfies those integral equations without incident fields. In fact, if some interior resonant mode field (E0, H0) satisfies (4) without any incident wave, we have

equation image

Furthermore, there are only two kinds of interior resonance modes associated with the problem under consideration. The first type of resonant mode occurs when the open aperture is closed with a PEC surface, and the second type occurs when the open aperture is closed with a PMC. If (E0, H0) is the first kind of resonant mode field, then E0 × equation image = 0 on S1, and the above equation can be written as

equation image

It is obvious that H0tan = 0 on S1 through comparison of (8) and the representation of the magnetic field on the aperture (2) with rS1. Within region I, any fields can be represented as the radiation from the equivalent currents around S1 and S2, namely,

equation image

Utilizing E0tan = 0 and H0tan = 0 on S1, and letting rS2, we obtain

equation image

Because the resonance problem occurs only when integral equations are applied to closed boundaries [Jin, 2002], equation image × H0 must be zero on S2. Thus, E0, H0 are zero within all of region I.

[9] If (E0, H0) is instead the second kind of resonant mode field, then H0 × equation image = 0 on S1, and (8) still holds in this case. Subtracting (8) from (7) gives rise to

equation image

Therefore E0tan = 0 on S1 since S1 is an open boundary. The remainder of the proof is the same as that provided above for the first type of resonant mode. Therefore it follows that the integral equations (4) and (5) or (4) and (6) are free from the interior resonance problem.

4. Implementation of the Proposed Integral Equations

[10] The symmetric equivalent current model of the cavity problem shown in Figure 2 describes a continuous flow of current J2(r) and its image J2i(r) on a cavity wall across the rim of the cavity. If the RWG vector basis functions [Rao et al., 1982] are employed to expand the currents, it is convenient to use some transitional half RWG basis function to fully characterize the continuity of the electric current across the cavity rim.

[11] A transitional half RWG basis function is only a half RWG basis function. The electric current image of the half RWG basis function forms an entire RWG basis function as shown in Figure 3. However, the equivalent magnetic current M1(r) approaches zero when the common boundary of S1 and S2 is approached, and no transitional half basis function is required to represent M1(r) in this case. Therefore J2(r) and M1(r) can be expanded using the RWG basis functions as follows,

equation image
equation image

where fn, gp are RWG basis functions on S2 and S1, respectively, and hk is a transitional half RWG basis function along the rim of cavity. It should also be noted that J2i(r) is only the image of J2(r), and no extra basis functions are required to represent it. The contribution of J2i(r) is added in the matrix elements associated with J2(r).

Figure 3.

A transitional half RWG basis function and its image.

[12] Substituting (10) and (11) into (4), (5), (6) and applying the standard Galerkin's method to the resultant equations, the following matrix equations can be obtained,

equation image
equation image
equation image
equation image
equation image

It should be noted that (12) and (16) are obtained by using the combination of transitional half basis function and their electric current images, namely, {hk + hki} to match the corresponding equations. All elements in the matrix equations (12)–(16) can be obtained from the standard Galerkin method. The first superscript of matrices U, V and B denotes the observation region and the second one denotes the source region. h denotes the rim of S2 and refers to the contribution of the transitional half basis functions.

[13] The matrix equation obtained by combining (4) and (5) can be written as

equation image

and the matrix equation formed by combining (4) and (6) is

equation image

The factor Z0 in (17) and (18) is used to make sure the unknowns are in the scale of electric current and the right hand sides of (17) and (18) are in the scale of magnetic field. With this scheme, the condition numbers of matrices can be reduced dramatically. In particular, the condition number of (18) can be reduced by about two orders of magnitude.

[14] As discussed in [Jin, 1998], it is convenient to define the scattering pattern using the magnetic fields as follows

equation image

where r = ∣r∣. H1scat is the scattered magnetic field due to the equivalent magnetic current and can be obtained from (1),

equation image

5. Numerical Results and Discussions

[15] Based on the formulations described above, a computer code has been developed to numerically verify the validity and well posedness of the proposed integral equations. We first consider the scattering from 3D cavities and then discuss the well posedness of the integral equations by examining the condition numbers of the associated matrix equations.

5.1. Scattering From Cavities

[16] For the numerical examples considered in this section, it has been observed that the results obtained using the matrix equations (17) and (18) are nearly identical. For this reason, only the results obtained using (18) are shown in the following figures.

[17] The first cavity we consider is the rectangular cavity shown in Figure 4 with a = λ, b = λ and c = 3λ. Figure 5 displays the backscattering patterns obtained using the proposed integral equations in comparison with results obtained using a finite element boundary integral (FE-BI) method [Jin, 1998]. The overall excellent agreement between the results verifies the proposed integral equations and their implementation. To check the contribution from the transitional half RWG basis functions related to the rim of cavity, the backscattering pattern from this rectangular cavity obtained by using the proposed integral equations without transitional half basis functions is also shown in Figure 5 for comparison. Obviously, the difference between results obtained with and without the transitional half basis functions is fairly significant at some angles, especially for θθ polarization.

Figure 4.

Geometry of the 3D rectangular cavity in a ground plane.

Figure 5.

Backscattering patterns of a rectangular cavity with a = λ, b = λ, and c = 3λ(ϕ = 0°).

[18] The second cavity we consider is a circular cavity with radius = λ and depth = 2.1λ. Figure 6 shows its backscattering patterns. It can again be seen that the agreement between the results obtained using the proposed integral equations and the FE-BI method is excellent.

Figure 6.

Backscattering patterns of a cylindrical cavity with radius = λ, and depth = 2.1λ.

5.2. Well Posedness

[19] To further verify that the proposed integral equations are free from the interior resonance problem, we consider the resonance of a rectangular cavity structure with a = 1 m, b = 0.5 m and c = 0.57735 m. It is resonant at 300 MHz, at which frequency a resonant TE101 mode occurs. Figure 7 shows condition numbers of the matrices obtained from GNF and from equations (17) and (18) as functions of frequency. The condition number of the system matrix can be interpreted as an indicator of the location of cavity resonance. It can be seen that when frequency is close to the resonance frequency, the condition number of the GNF matrix equation increases dramatically. (The Galerkin discretization causes the resonance to be at 299.75 MHz rather than 300 MHz.) In contrast, the condition numbers of the matrices obtained from equations (17) and (18) remain almost constant.

Figure 7.

Condition number versus frequency for the matrices obtained using GNF and the proposed integral equations to model a rectangular cavity with a = 1 m, b = 0.5 m, and c = 0.57735 m.

[20] Figure 8 shows the condition numbers of the matrices obtained using GNF and equations (17) and (18) as the function of the depth of rectangular cavity at 299.75 MHz with aperture size a = 1 m, b = 0.5 m. The first spike on the curve for GNF corresponds to the resonant TE101 mode at c = 0.57735 m. The second spike on the GNF curve corresponds to the resonant TE102 mode at c = 1.1547 m. No spike appears on the curves associated with (17) and (18), and the condition numbers of the matrices derived from these equations are almost constant with respect to the depth of the cavity. Thus the numerical results confirm our expectation that the proposed integral equation method is free from the interior resonance problem.

Figure 8.

Condition number versus the depth of rectangular cavity when a = 1 m and b = 0.5 m.

6. Conclusions

[21] The problem of electromagnetic scattering from a 3D cavity, which opens into an infinite ground plane, is formulated into boundary integral equations using a symmetric equivalent current model based on image theory with a PEC ground plane. The newly proposed boundary integral equations involve electric current density on the cavity walls and magnetic current on the aperture of the cavity. The electric current on the cavity walls and its image form a continuous current flow across the rim of cavity. This continuity of the current and its image physically describe the induced current on the cavity wall, which can be easily characterized using transitional half RWG basis functions. Besides all of these advantages, the most important finding in this paper is that the proposed integral equations are free of the interior resonance problem, which has been theoretically and numerically proved. The well posedness of all the integral equations provides a good foundation to develop fast algorithms based on these integral equations, which will result in robust solvers for modeling large cavity structures.

Acknowledgments

[22] The author is grateful to Chao-Fu Wang and R. J. Adams for insightful discussions and useful comments; he also thanks Mr. Fu-Gang Hu for all FE-BI results in this paper.

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