Implementation of the asymptotic corrugation boundary condition in the finite difference time domain method

Authors


Abstract

[1] Geometry description in the finite difference time domain method is a tedious task if the geometry contains fine details, such as the case of corrugated objects. Such fine details constrain the cell size. The corrugated object can be modeled using the asymptotic corrugation boundary condition (ACBC) with a correction due to the width-over-period ratio. The ACBC forces certain field distributions inside the corrugation and allows for the removal of the corrugation teeth to have a homogeneous region with enforced field behavior that represents the actual corrugations. The ACBC approach is found to be accurate when the number of corrugations per wavelength is large (typically around 10 corrugations per wavelength). Computed results using ACBC are in good agreement with detailed simulations, which demonstrates the validity of the asymptotic approximations. Last, a major improvement in the computation time is achieved when using the ACBC to model structures that have a large number of corrugations per wavelength.

1. Introduction

[2] Analysis of periodic corrugated structures can be performed using several methods such as using the anisotropic surface impedance or by analyzing the corrugated structure with all its details. The surface impedance method requires knowledge of the surface impedance, and can only be used for plane wave incidence with known angle of arrival, which is very rarely present in general field problems. It does not provide any information about the field within the corrugation region. Its accuracy depends on how accurately we predict the surface impedance. Another approach could be using the periodic boundary conditions, but this will force the periodicity to the corrugations and will not allow the study of nonperiodic structures in the presence of the periodic structures. In addition, if the corrugated region is finite, a detailed numerical analysis must be used. In some numerical methods such as the method of moments, such a detailed structure can be very large and require excessive computational resources in terms of memory and run time. If the finite difference time domain (FDTD) method is to be used, the details of the corrugations might require very fine cell sizes to accurately capture all the details of the structure and consequently great amounts of computational resources are required.

[3] Recently, an asymptotic corrugation boundary condition (ACBC) has been used to analyze corrugated structures even if the corrugation depth is varying or the corrugation region is finite. Typically, there should be more than 10 corrugations per wavelength for the asymptotic method to be accurate [Kishk et al., 1998], but it will often work satisfactorily even for 2 or 4 corrugations per wavelength if the corrugation width is smaller than λ/2 in the dielectric material inside the corrugations. The ACBC is able to overcome the limitations in geometry and complexity of the other methods. The ACBC was formulated on the basis of the study of structures that exhibit anisotropic characteristics similar to soft and hard surfaces [Kildal, 1990]. It is an anisotropic averaging type of boundary condition, but in the FDTD, certain field behavior is enforced in the corrugated region and on the surface. The ACBC may be mixed with other conventional boundary conditions when dealing with composite materials or structures [Kishk et al., 1998; Kishk and Kildal, 1995]. Kishk et al. [1998] used an analytical series expansion technique with the ACBC to model corrugations used as hard surfaces and was able to get excellent agreement with a solution based on the exact Floquet mode expansion and the mode matching technique. The comparison also made apparent that the ACBC could be applied in certain cases when there are fewer than 10 corrugations per wavelength. Kishk [2004] applied the ACBC in the method of moments for cylinders with arbitrarily cross section.

[4] We applied the ACBC in the FDTD. The updating field expressions are given through the discretized field equations. In addition, the ratio between the corrugation widths to the period width (w/p factor) is added as a weighting factor that was shown by Kishk et al. [1998] to improve the accuracy of the ACBC. It was added intuitively by Kishk et al. [1998] by comparing the surface impedance approach that used the factor when analyzing corrugated surfaces. We demonstrate our analysis by an example of a strut with corrugations placed on it to create an artificial hard surface for purposes of reducing the forward scattering [Kildal et al., 1996]. Results using the FDTD simulation for the ACBC cases are compared with their corresponding detailed corrugated cases; simulations that describe the detailed corrugations using dielectric and conducting materials.

2. Formulation

[5] In the process of implementing the ACBC, the corrugations (see Figure 1a) are removed as seen in Figure 1b, and the field behavior inside the corrugated region is enforced. Inside the corrugated region, the electric field is orthogonal (equation image) to the corrugation grooves. Furthermore, this field has no derivatives in the orthogonal direction. Thus the electric field within the grooves is said to be transverse electric to the direction equation image along the corrugations. The electric field that is tangent to the walls within the corrugations is zero. As seen in Figure 1c, the corrugated region can be looked at as a homogeneous anisotropic region with field distribution similar to the field distribution in the corrugations. To account for the removal of the corrugation teeth, the electric field is weighted by the factor w/p, which is the ratio between the corrugation widths to the corrugation period.

Figure 1.

Corrugated object and the steps of transformations to ACBC model.

[6] The following are the boundary conditions at the interface of the corrugations (corr) with the fields outside (air):

equation image

where the corrugation side is denoted by “corr” and the outside is denoted by “air.” Within the corrugation region the electric field is forced to have only one component in the direction of equation image.

[7] Kishk et al. [1998] showed that introducing the w/p factor improves the accuracy of results for corrugated surfaces using the mode-matching technique. Introducing this factor inside the corrugations averages the actual field in the corrugated region after removing the corrugated grooves as seen in Figure 1b. The w/p factor is only applied to the orthogonal electric field component, as the other electric field components within the region are zero.

[8] To formulate the FDTD updating equations, we start with Maxwell's equations to derive the modified electric and magnetic updating equations for the ACBC by introducing the w/p factor to the orthogonal electric field to denote a reduced field due to the presence of conducting materials:

equation image
equation image

where equation imagec = Enequation image + Eequation image + equation imageEoequation image with equation image, being the normal unit vector to the surface, and equation image and equation image are unit vectors tangential to the surface, which are along and orthogonal to the corrugations, respectively, ɛr is the relative permittivity, μr is the relative permeability, and (Ji, Mi) are the impressed electric and magnetic sources.

[9] By applying the standard procedure of the FDTD that is based on the Yee cell approximation [Yee, 1966] and using the central difference approximation of the derivatives, (2) and (3) can be written in the following forms (for simplicity and readability, assume n, ℓ, and o as the x, y, and z components, respectively, in the rectangular coordinate system):

equation image
equation image
equation image
equation image
equation image
equation image

where equation image, equation image, equation image, and equation image.

[10] These equations are the scattered fields if the impressed current sources are suppressed and the total fields are obtained by adding the incident fields, which is computed analytically. It should be noted that if the local impressed sources (Ji and Mi) are removed from this formulation, the equations reduce to model a plane wave excitation for use in scattering problems. In contrast, if the incident fields are removed, the equations reduce to model a local source excitation for antennas or other microwave circuit problems. The time derivatives of the incident fields were determined analytically instead of using the central difference.

[11] When updating the magnetic fields near the ACBC interface, extra care must be given to the position of the fields with respect to the interface as modified curl equations with the w/p ratio must be used on the appropriate electric field. For example, equation (6) applies the ratio (w/p − 1) to both the Ez(i, j, k) and Ez(i, j + 1, k) component in the discretized curl equation. At the interface, however, this ratio should be applied to only one Ez component, with the other being outside the ACBC region. A similar situation occurs for the updating equations of the Hy component at the interface. An alternate modification that may be made to the implementation of the updating equations was introduced to simplify the tedious task of classifying the respective field components as separate objects when building the geometry. However, the cost is more computation time needed to go through a single time step. By replacing the Ez component as follows after it is updated with equation (5), the task of classifying a whole other magnetic mesh for the corrugated region is avoided and the magnetic fields may be updated assuming free space without the hassle of applying the w/p ratio to the appropriate fields in the discretized curl equation.

equation image

3. Scattering Example

[12] In applying the ACBC in the FDTD method, we use the example of the strut given by Kildal et al. [1996], which is a two-dimensional (2-D) object, but here the strut has a finite length and has been analyzed as a 3-D object. Figure 2 shows the geometry and the parameters of the strut, which reduces forward scattering for both the TMz and TEz polarizations. It should be mentioned that in all the results we used segmentation for the structure to meet the criteria of at least 10 cells per the anticipated shortest wavelength. The time step used satisfies Courant stability criteria, Δt ≤ 1/equation image where c is the speed of light. In addition, the actual corrugated period is about 3 mm, which provides 11 corrugations per the longest wavelength and 7 corrugations per the shortest wavelength. The time domain Gaussian pulse is chosen to cover the required frequency band and used with the detailed structure, and the structure uses the ACBC model.

Figure 2.

Strut of parallel-plate waveguide with the outer surface made of dielectric filled corrugations.

[13] The term “equivalent width” is given below, which is defined by Kildal et al. [1996] to be the width of an equivalent ideal shadow that produces the same forward scattering as that produced from the actual structure. The equation for Weq is given by Kildal et al. [1996] and is related to the actual width, w, as follows:

equation image

where equation image* is the complex conjugate of the polarization vector equation image, which is in the direction of z for the TMz and in the direction of x for the TEz case. Eequation image is the forward scattered field from the object and Eequation image forward scattered field from a conducting strip of width w. There is a relation between the equivalent width and the radar cross section (RCS) in the forward direction as given by Kildal et al. [1996] as RCS = kWeq2 where k is the wave number.

[14] The equivalent blockage width (Weq) is computed for the case of the detailed strut (actual structure with all the details without any approximations) and the strut based on the ACBC. Figure 3 shows the magnitude of the equivalent blockage width for the struts with L = 42 mm, W = 20 mm, Lz = 200 mm and ɛr = 2.54. Figure 3 shows that the ACBC results and the detailed case results are identical for both TMz and TEz polarizations. The ACBC model uses w/p = 1 to compare to the detailed model since the strut was modeled using thin corrugated plates. The period of the corrugations in the detailed case is 1 mm, lending to the strut having 20 periods per wavelength at 15 GHz. Figure 4 shows a detailed strut having a period of 3 mm with corrugation tooth of 1 mm. Figure 4 shows the comparison of the equivalent blockage widths of both the ACBC and detailed models with w/p = 0.667 and p = 3 mm with normal incidence (θ = 90°, ϕ = 270°). The results using the detailed strut and the ACBC show excellent agreement for the TEz polarization while the comparison for the TMz polarization is fairly good. The results vary slightly from the w/p = 1 case. Figure 5 shows the equivalent width for the same strut with oblique plane wave incidence (θ = 60°, ϕ = 270°). The level of the blockage width increases because the strut was designed for normal incident. The above results are obtained from the scattered field in the forward direction. The TE case showed that as the ratio w/p changes from 1 to 0.677, no graphical difference between the results is noticed. The bistatic scattering cross sections are computed for the strut at different frequencies as shown in Figure 6. The ACBC solution is compared with the solution from the detailed structure. Within the whole angular rage one can see excellent agreement between both solutions.

Figure 3.

Computed magnitude of the equivalent blockage width using the ACBC model with w/p = 1 compared to the equivalent struts (L = 42 mm, W = 20 mm, Lz = 200 mm, and ɛr = 2.54) due to normal incidence (θ = 90°, ϕ = 270°).

Figure 4.

Computed magnitude of the equivalent blockage width using the ACBC model with w/p = 0.667 compared to the detailed strut model (p = 3 mm, L = 42 mm, W = 20 mm, Lz = 200 mm, and ɛr = 2.54) due to normal incidence (θ = 90°, ϕ = 270°).

Figure 5.

Computed magnitude of the equivalent blockage width using the ACBC model with w/p = 0.667 (actual ratio) and w/p = 1 compared to the detailed strut model (p = 3 mm, L = 42 mm, W = 20 mm, Lz = 200 mm, and ɛr = 2.54) due to oblique incidence (θ = 60°, ϕ = 270°): (a) TMz and (b) TEz.

Figure 6.

Computed bistatic radar cross section for the ACBC model with w/p = 0.67 compared to a strut with detailed corrugations (p = 3 mm, L = 42 mm, W = 20 mm, Lz = 200 mm, and ɛr = 2.54) excited with an oblique incident plane wave (θ = 60°, ϕ = 270°): (a) TMz (10 GHz), (b) TEz (10 GHz), (c) TMz (12 GHz), (d) TEz (12 GHz), (e) TMz (14 GHz), and (f) TEz (14 GHz).

[15] Besides accurately modeling periodic structures of corrugations, one of the greatest benefits of using the ACBC is the ability to use larger spatial increments due to the assumed homogeneity of the corrugation region. This reduces the memory requirements and computational time as shown in Table 1. The model parameters used are L = 30 mm, W = 72 mm, Lz = 210 mm and ɛr = 2.54. This strut is not optimized for low blockage width for TEz polarization but the ACBC model is still expected to give results that agree with the detailed strut model. Figure 7 shows the magnitude of the equivalent width of the ACBC model using w/p = 0.667 with different spatial increments for the TMz and TEz polarizations. It should be pointed out that the criteria of using a minimum number of segments per wavelength were reserved in all computations. This number is typically in the order of 10 cells per shortest wavelength. In the simulation, this number of segments is 23.5 when the spatial increment is set at 1 mm. At 2 mm spatial increments, 12 segments per wavelength were able to give good results when compared to the 1 mm increment case. The case using 3 mm spatial increments had too few segments per wavelength to accurately simulate results in the ACBC with the TMz polarization, but were able to in the TEz polarization. The cases that benefit the most are those having a large number of corrugated segments per wavelength being predetermined by fine details when building the geometry with thin grooves. In those cases, using the ACBC would greatly relieve computation costs.

Figure 7.

Computed equivalent blockage width of the ACBC model with w/p = 0.667 using different spatial increments (L = 30 mm, W = 72 mm, Lz = 210 mm, and ɛr = 2.54) due to normal incidence (θ = 90°, ϕ = 270°): (a) TMz and (b) TEz.

Table 1. Summary of Computation Time and Memory Requirementsa
Δx, mm (= Δy = Δz)Δt, psTime StepsComputational TimeMemory, kb
  • a

    The time increment is calculated from the spatial increments using the Courant stability criterion and multiplied by a factor of 0.9. The processor used for this simulation is a Pentium IV 1.0 GHz on a Windows NT 2000 system. Memory used is related to the number of frequency points needed for postprocessing, in this case 71 points.

11.733600016 hours, 19 min186,452
23.46630002 hours, 3 min65,292
35.200200040 min34,660

4. Conclusion

[16] The asymptotic corrugation boundary condition (ACBC) was successfully formulated and implemented in the FDTD method. Computed results were encouraging as compared to detailed models, which demonstrates the validity of the asymptotic approximations when compared to detailed models where individual corrugated plates were modeled. The approximations made with the ACBC simply require a large number of periods per wavelength before the asymptotic boundary conditions are ensured to be accurate in its modeling. The benefits of using the ACBC are namely the ease of object descriptions in the FDTD formulation and the added flexibility in selecting the spatial increments. By removing the constraints in describing the object under investigation, huge savings in computational costs were exploited by selecting larger spatial increments that were possible through the replacement of the fine details with a homogenous region by the ACBC. This latter fact is what makes these unique boundary conditions very useful and practical in the FDTD method. It is reasonable to say that a design process using the ACBC can be accomplished faster than a design with detailed models, with a comparable accuracy and fewer computer resources.

Acknowledgments

[17] This work was partially supported by the National Science Foundation under grant ECS-0220218.

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