Shape optimization of the total scattering cross section for cylindrical scatterers

Authors


Abstract

[1] We propose and test a gradient-based shape optimization algorithm for the total scattering cross section of infinitely long cylinders, by means of changing the shape of the cylinder's cross section. On the basis of the optical theorem, we derive sensitivity expressions for both dielectric and metal cylinders given an incident plane wave, where the wave vector is perpendicular to the cylinder axis. Both the transverse electric (TE) case and the transverse magnetic case are considered. The sensitivity expressions are based on the continuum form of Maxwell's equations, and they provide the sensitivity with respect to an arbitrary number of shape parameters in terms of the field solution of the original scattering problem and an adjoint scattering problem. These results are used to construct a gradient-based optimization algorithm that we exploit for the reduction of the total scattering cross section in the TE case for metal cylinders, e.g., struts used in reflector antennas. We present optimized cross sections that are oblong in the direction of the incident wave vector, and some of these designs feature corrugations that are parallel to the cylinder axis. We show designs with asymmetric cross sections that yield a low monostatic scattering cross section for certain directions in combination with a low total scattering cross section, which can be used to reduce the noise temperature contributions from the upper strut in an inverted Y tripod reflector antenna.

1. Introduction

[2] Reflector antennas normally involve struts that hold the microwave head. Such struts may cause significant degradation of the antenna's performance [Kildal et al., 1988] in terms of losses, sidelobes and cross polarization. In turn, such effects may also increase the noise temperature of the reflector antenna [Moreira et al., 1996].

[3] The design of the strut's cross section can be crucial to fulfill the overall requirements on the performance of reflector antennas that are used in modern communication systems and radio telescopes. In particular, it is advantageous to find designs that yield a low total scattering cross section [Kildal et al., 1988]. For the transverse electric (TE) case, the magnetic field is parallel to the cylinder axis of the strut and satisfies the homogeneous Neumann boundary condition at the surface of a perfect electric conductor (PEC). For this case, it is feasible to achieve a relatively small total scattering cross section by making the strut's cross section oblong in the direction of propagation of the incoming wave [Rusch et al., 1976; Kildal et al., 1988; Moreira et al., 1996]. For the transverse magnetic (TM) case, the electric field is parallel to the cylinder axis and it satisfies the homogeneous Dirichlet boundary condition at the surface of a PEC. This type of boundary condition at the surface of a strut yields substantial scattering and changing the shape of the metal strut's cross section is not a competitive way to reduce the total scattering cross section. However, it is feasible to construct surfaces that yield, at least in a narrow frequency band, an artificial magnetic conductor [Kildal, 1990] and this effectively gives a homogeneous Neumann boundary condition for the electric field in the TM case. Equipped with such an artificial magnetic conductor surface for the strut in the TM case, it is feasible to reuse the oblong shapes that yield a low total scattering cross section for the TE case. For example, the total scattering cross section can be reduced by coating the metal strut with a dielectric material [Lin and Yaqoob, 1992; Kildal et al., 1996], which effectively yields an artificial magnetic conductor for a narrow frequency band [Kildal, 1990]. To reduce the total scattering cross section for the TE and TM case simultaneously, metal strips that circumvent the strut can be added to a metal cylinder coated with a dielectric. This creates an anisotropic boundary condition that allows for reduction of the total scattering cross section for both the TE and TM case simultaneously [Kildal et al., 1996; Kishk, 2003, 2004; Wang, 1985].

[4] Shape optimization is an important tool for design in engineering. In particular, efficient methods can be formulated if the sensitivities are expressed in terms of the field solution to the original field problem combined with the solution to an adjoint field problem [Pironneau, 1984]. Thus, only two field computations yield the sensitivity with respect to an arbitrary number of design variables. For Maxwell's equations, shape optimization has been applied to several classes of problems: (1) magnetostatics [Park et al., 1991], (2) steady state eddy current problems [Park et al., 1994], (3) passive microwave circuits [Lee and Itoh, 1997; Nair and Webb, 2003], (4) monostatic radar cross section [Bondeson et al., 2004], and (5) optical resonators [Kao and Santosa, 2008]. In other areas, such as structural design, shape and topology optimization [Neves et al., 2002] are widely used tools and a popular way of describing the geometry is to use a level-set method [Allaire et al., 2004a, 2002; Osher and Fedkiw, 2001]. Optimization using gradient-based methods that exploit an adjoint problem [Dorn et al., 2000; Burger, 2001] or heuristic methods is also an important tool in shape reconstruction in inverse problems. The list given here is not complete and the interested reader is referred to books and review articles on this topic [e.g., Haslinger and Mäkinen, 2003; Allaire and Henrot, 2001; Sokolowski and Zolesio, 1992].

[5] In this article, we propose and test a gradient-based optimization algorithm for the total scattering cross section of infinitely long cylinders. It changes the shape that describes the cross section of the cylinder iteratively and, for the tests presented in this article, a significant reduction of the goal function is attained in approximately 10–20 iterations. We present a continuum sensitivity analysis for the total scattering cross section with respect to the shape of the cross section for both dielectric and PEC cylinders, which forms the basis of our optimization algorithm. With the optical theorem as a basis, the detailed derivations are presented for the TM case and the final results for both polarizations (see the work by Bondeson et al. [2004], where the related but different problem for the monostatic radar cross section is studied). The continuum formulation of the sensitivity expressions decouples the optimization procedure from the field solver and, here, we solve Maxwell's equations with a stable hybridization [Rylander and Bondeson, 2002] of the finite element method (FEM) [Jin, 2002] and the finite difference time domain (FDTD) scheme [Taflove, 2005]. It is formulated in the time domain and one computation solves the scattering problem for more than four octaves in frequency. We test the optimization algorithm on metal struts for the TE case and find different types of solutions: (1) conventional oblong shapes; and (2) oblong shapes with parts covered by corrugations that are parallel to the cylinder axis.

[6] This article features a couple of main contributions: (1) the sensitivity expressions for the total scattering cross section with respect to changes in the shape of the cylinder for both penetrable and impenetrable scatterers, where the sensitivities are given for the TE and TM case separately; (2) optimized shapes that feature conventional oblong cross sections; and (3) struts that have corrugations parallel to the cylinder axis which lower the total scattering cross section for higher frequencies.

2. Gradients for the Total Scattering Cross Section

[7] We align the z axis with the cylinder axis of the strut, which is assumed to be infinitely long. In antenna applications, this assumption is motivated by the fact that the distance between the antenna and the cylinder is large in terms of wavelengths, yielding a local scattering problem. In the following, we consider an incident plane wave with the wave vector ki = k0 (equation image cos ϕi + equation image sinϕi), i.e., it is perpendicular to the z axis and the plane wave is incident on the strut at an angle ϕi with respect to the x axis. For the TM case, the incident plane wave is described by Ei = equation imageE0 exp(−jki · r), where we assume and suppress the time dependence exp(jωt). Similarly, we have the incident plane wave Hi = equation imageH0 exp(−jki · r) for the TE case.

[8] For this situation, we derive expressions for the total scattering cross section σt and its variation with respect to changes in the shape of the scatterer's cross section for the TM case. The derivation for the TE case is analogous. The total scattering cross section is

equation image

when formulated in terms of the bistatic scattering cross section σ = 2πG2/∣E02. Here, the scattering amplitude [Taflove, 2005] is

equation image

where ks = k0 (equation image cos ϕs + equation image sin ϕs) and the angle ϕs is the direction of the scattered field with respect to the x axis. To simplify the notation, the electric field's z component is denoted E. The contour integral above is evaluated for a curve LNTF that encloses the scatterer. In the far field, the scattered electric field is given by Ess) = r−1/2e−jkrGs).

[9] For optimization purposes, we exploit the optical theorem [Ishimaru, 1991] to relate σt to the scattered field in the forward direction ϕs = ϕi and the relation is given by

equation image

We emphasize that the optical theorem allows for the computation of the total scattering cross section without the need to calculate the scattered field for many scattering angles. In the context of shape optimization, this approach reduces the computational cost dramatically since it suffices to solve only one adjoint problem. In contrast, a formulation that directly exploits the bistatic scattering cross section requires the solution of a large number of adjoint problems, since the bistatic scattering cross section must be evaluated at a relatively high resolution with respect to its azimuthal variation.

[10] Another way to characterize the scattering from the strut is by means of the equivalent blockage width [Kildal et al., 1996] denoted by Weq, which is related to the total scattering cross section by Re[Weq] = σt/2. The equivalent blockage width is a scaled version of the forward scattered field and its real part tends to the physical width of the scatterer in the high-frequency limit, where physical optics and similar approximations are typically used.

2.1. Original Scattering Problem

[11] The electromagnetic field problem can be expressed in terms of the scalar Helmholtz equation for E on a domain Ω. The outer boundary ∂Ω of the domain Ω is located at a sufficiently large distance from the scatterer and we apply a radiation boundary condition equation image · (∇E + jk0equation imageE) = 0 at ∂Ω. The weak formulation [Jin, 2002] that corresponds to this scattering problem is given by

equation image

where

equation image
equation image

for appropriately chosen test functions wi. Here, we impose the incident field on a Huygens surface S [Taflove, 2005] that encloses the strut and is entirely located inside Ω. In the following, we assume that the interior of the strut is homogeneous and characterized by the permittivity ɛ1 and the permeability μ1. The region outside the strut is free space with the permittivity ɛ0 and the permeability μ0. The cross section of the strut is represented by the boundary Γ.

2.2. Adjoint Scattering Problem

[12] In order to calculate the first-order variation of the total scattering cross section (3), we introduce the adjoint problem

equation image

where the operator equation image0 is identical to the corresponding operator in the original problem and it is given by equation (5). The source term for the adjoint problem we consider is given by

equation image

Physically, the adjoint problem is a scattering problem with the same geometry as the original scattering problem. However, the incident plane wave is Ea = equation image, equation image which propagates in the direction opposite to the incident plane wave in the original problem. It should be noted that this adjoint problem (7) differs from the adjoint problem associated with the monostatic scattering cross section [Bondeson et al., 2004], where the adjoint problem is identical to the original field problem as a result of reciprocity.

2.3. Sensitivities for Dielectric Cylinder

[13] We are now in position to calculate the first-order variation of σt expressed by equation (3) with respect to a normal displacement of the boundary Γ of the strut. We express the first-order variation of the total scattering cross section as

equation image

where δG is the variation of the scattering amplitude in the forward direction given by

equation image

[14] By means of the adjoint solution Ea to the adjoint problem (7), we write the variation of the scattering amplitude as

equation image

We proceed according to the work of Bondeson et al. [2004] and express the bilinear form in equation (11) as

equation image

where Γ is the boundary of the scatterer and δξ is a small normal displacement of Γ as shown in Figure 1. Here, we use M = − equation image × E, J = equation image × H, ϱ = μequation image · H. Thus, the first-order variation of the total scattering cross section for the TM case is given by

equation image
Figure 1.

The boundary Γ between the strut (characterized by ɛ1 and μ1) and the surrounding free space is displaced a small distance δξ, where δξ is the normal displacement of the boundary.

[15] The total scattering cross section and its variation in the TE case is derived in the same way as for the TM case and the variation of the total scattering cross section with respect to a normal displacement is given by

equation image

where ρ = ɛequation image ·E.

2.4. Sensitivities for Perfect Electric Conductor

[16] In the limit ɛ1 → ∞ and μ1 → 0, we get the expression for the variation of the total scattering cross section with respect to a normal displacement of the surface of a PEC strut [Bondeson et al., 2004]. For the TM case, we get from equation (13) the expression

equation image

where η0 is the wave impedance in vacuum. The corresponding expression for the TE case given a normal displacement of a PEC surface is deduced from equation (14) and it is given by

equation image

2.5. Parameterization of the Strut

[17] It is beneficial to use a parameterization ξ = ξ(r, a0, …, aN) of the surface in terms of a set of design variables a0, …, aN. In general, it is not appropriate to use the node coordinates directly as design variables in the optimization algorithm. The variation with respect to the shape design parameters is then given by

equation image

where ξm is the normal displacement of the mth node on the boundary Γ, which has in total M nodes in the discretized problem.

[18] For testing purposes, we express the boundary Γ of the PEC strut in polar coordinates and use the parameterization

equation image

where β influences the convergence properties of the optimization procedure. We constrain the area

equation image

to a given value A0 in order to achieve a certain mechanical rigidity of the strut. This constraint is enforced by setting the mean radius of the parameterization (18) to

equation image

Thus, a0 is not a design variable but is expressed in terms of A0 and the free design variables a1, a2, …, aN.

[19] It is often desirable to have a hollow interior of the strut, which, e.g., can host cables or waveguides that are used to connect to the feed of a reflector antenna. For such purposes, we also require that r(ϕ) > rmin for all angles ϕ. This constraint also avoids designs that are too thin and, therefore, may have poor mechanical rigidity. The requirement on the minimum radius is enforced as a set of nonlinear constraints in the optimization algorithm, since a0 is given by the nonlinear relation in equation (20).

3. Numerical Tests

[20] We test the shape sensitivities for PEC struts and limit the investigation to the TE case, where shape optimization can yield significant reductions in the total scattering cross section. We optimize the root-mean-square (RMS) value of the total scattering cross section for a frequency interval [fmin, fmax] of length Δf = fmaxfmin. Thus, the goal function is given by

equation image

where σt is the total scattering cross section. The variation of g with respect to the design parameters an can be computed by means of equation (16) in combination with the chain rule (17) and the parameterization (18). In the following, we use β = 1 in equation (18) and let the incident wave propagate in the direction of the x axis, i.e., ϕi = 0.

[21] The field problem is solved by a stable FEM-FDTD hybrid method [Rylander and Bondeson, 2002] formulated in the time domain. It combines the efficiency of the FDTD scheme in large homogeneous regions outside the scatterer with the body-conforming ability of the FEM in the vicinity of the curved boundary of the scatterer. In the numerical simulations, we use a resolution of 30 cells per wavelength at the highest frequency and truncate the computational domain by a perfectly matched layer [Taflove, 2005]. The use of a time domain method allows us to obtain the frequency domain solution for a wide frequency range by the Fourier transform of the time domain solution. For the optimization, we use the routine SNOPT [Gill et al., 2005], which is part of the larger package of optimization algorithms provided by TOMLAB. SNOPT implements the sequential quadratic programming (SQP) [Bazaraa et al., 2006; Murray, 1997] algorithm and it works well for nonlinear problems with many constraints where the goal function is expensive to compute. The SQP algorithm is a local optimization algorithm, which implies that it is only guaranteed to find a local optimum. As a consequence, different initial shapes may result in different optimized shapes that represent local optima to the optimization problem.

[22] We note that Sohl et al. [2007] present results on the integrated extinction for broadband scattering of acoustic waves. In contrast to equation (21) they integrate the weighted total scattering cross section σt (k)/k2 from k = 0 to infinity. This allows for simplifications of the broadband scattering description and their results depend only on the material properties and geometry in the static or low-frequency limit.

[23] In the numerical tests that follow, the total area of the strut's cross section is fixed to A0 = ξAmin, where Amin = πrmin2 is the area of the smallest circle that satisfies the constraint on the radius. The optimization is performed for the frequency interval 1 GHz < f < 20 GHz and we use N = 50 terms in the Fourier series expansion (18) of the cross section boundary for the strut. On the basis of the values A0 and rmin used as constraints for an optimized shape, we use a reference strut for comparisons and its cross section shown in Figure 2 is symmetric with respect to the x and y axis. The reference strut is a well established shape [Kildal et al., 1996] and features sharp leading and trailing edges (with zero radius of curvature) and a midsection that conforms to a circular cylinder of radius rmin. The length of the reference strut is chosen such that its area is equal to the area constraint A0 used for the optimized shape.

Figure 2.

Cross section of reference strut that features sharp corners connected by straight lines to the midsection that conform to a circle (indicated by the dashed line) of radius rmin.

3.1. Investigation of Convergence and Initial Design

[24] In order to investigate the convergence and influence of the initial shape of the optimization procedure, the case with rmin = 30 mm and ξ = 2 is tested with 83 different initial shapes. The initial shapes are constructed using five base geometries shown in Figure 3: a circle, two rhomb-like shapes oblong in the y direction with different radius of curvature at the corners, and two similar rhomb-like shapes oblong in the x direction. Zero-mean Gaussian noise with a standard deviation of the same magnitude as the Fourier coefficients for the shapes in Figure 3 are added to the first ten coefficients to generate 83 different asymmetric initial designs. The initial designs occupy the annular region bounded by the inner and outer dashed curves shown in Figure 3.

Figure 3.

The five base shapes used to generate the 83 initial shapes by perturbing the Fourier coefficients shown by solid lines. For each azimuthal angle, the smallest and largest values of the radius for the 83 shapes are indicated by dashed lines.

[25] The major part of the optimized shapes can be divided into four groups of typical shapes. Figure 4 shows the goal function value for each optimized shape together with the representative shapes from the four groups of typical optimized shapes denoted equation image2, …, equation image5. The optimized shape that has the lowest goal function value in this case is also shown. It is denoted equation image1 and it differs somewhat from the four representative shapes. The five shapes equation image1…, equation image5 are shown together in Figure 5 to facilitate comparisons. The mean value of the goal function for the initial shapes is 87 mm and the 83 optimized shapes have a goal function value in the interval 34.5–39 mm. This can be compared to reference strut, shown in Figure 2, which has a goal function value of 36.4 mm. Although different initial shapes for the optimization algorithm result in different optimized shapes, the goal function values are comparable for most of the optimized shapes and they show similar features. We notice that the constraint r(ϕ) > rmin is active at the top and bottom parts of the boundary Γ for all the optimized designs. The real part of the equivalent blockage width equation image[Weq] of the five shapes equation image1, …, equation image5 and the reference strut are shown as a function of frequency in Figure 6. We notice that the frequency dependence of the equivalent blockage width is rather different for the optimized designs despite the similar values of the goal function.

Figure 4.

The goal function values for the optimized shapes for the 83 initial shapes. Five shapes denoted equation image1, equation image2, equation image3, equation image4, and equation image5 are shown.

Figure 5.

Five shapes obtained by optimizing with the initial 83 designs. Thick solid line, equation image1; thick dashed line, equation image2; solid line, equation image3; dashed line, equation image4; dash-dotted line, equation image5.

Figure 6.

The equivalent blockage width as a function of frequency for the five shapes shown in Figure 5 and the reference strut shown in Figure 2. Thick solid line, equation image1; thick dashed line, equation image2; thick dash-dotted line, equation image3; solid line, equation image4; dashed line, equation image5; dash-dotted line, reference strut.

[26] The evolution of the goal function as a function of the number of goal function evaluations is shown in Figure 7 for the five shapes in Figure 5. It should be noted that Figure 7 also includes values of the goal function that are associated with the line search implemented by SNOPT, where regions with larger values of the goal function may also be explored. It can be noted that goal function decreases rapidly during the beginning of the optimization and it is reduced significantly after 10–20 goal function evaluations.

Figure 7.

The evolution of the goal function for the five shapes shown in Figure 5. Thick solid line, equation image1; thick dashed line, equation image2; solid line, equation image3; dashed line, equation image4; dash-dotted line, equation image5.

3.2. Rhombic Cross Section

[27] We continue to investigate different constraints used in the optimization of the strut. First, we let the minimum radius of the strut be rmin = 15 mm and optimize the RMS value of the total scattering cross section for the frequency range 1 GHz < f < 20 GHz. This combination yields a minimum width wmin = 2rmin of the strut and, given the frequency range, it is in the interval 0.1 < wmin/λ < 2 when expressed in terms of the wavelength λ. Here, we use initial designs with an = 0 for odd n in the parameterization (18). This yields initial designs that are symmetric with respect to the plane x = 0.

[28] Figure 8 shows the optimized shapes for different area constraints: ξ = 2 (solid line); ξ = 4 (dashed line); and ξ = 6 (dash-dotted line). It is clear that the cross section of the strut becomes longer when the area is increased. Basically, the optimizer attempts to approach a thin plate that is aligned with the plane y = 0, which yields a zero total scattering cross section in the limit of zero thickness.

Figure 8.

Optimized shapes for the minimum width wmin = 2rmin = 30 mm with different area constraints A0 = ξAmin with Amin = πrmin2. Solid line, ξ = 2; dashed line, ξ = 4; dash-dotted line, ξ = 6.

[29] For the smallest area constraint, we achieve a rhombic shape and its sharp edge that points toward the incident wave is known to yield low monostatic radar cross section [Bondeson et al., 2004]. The main features of the bistatic scattering cross section of the rhombic shape are (1) the forward scattering that corresponds to the shadow of the strut and (2) the scattering from the two oblique planar faces of the strut that are directly illuminated by the incident plane wave. The optimized shapes for the two larger area constraints basically have the same characteristic features, although somewhat less well pronounced.

[30] The real part of the equivalent blockage width Re[Weq] is shown in Figure 9 by thick curves for the optimized struts shown in Figure 8 given the different area constraints: ξ = 2 (solid line); ξ = 4 (dashed line); and ξ = 6 (dash-dotted line). The thin curves in Figure 9 show Re[Weq] for the corresponding reference struts, which are constructed given the constraints on the area and the minimum radius used for the optimized shapes.

Figure 9.

Real part of the equivalent blockage widths for the optimized shapes with wmin = 30 mm shown in Figure 8 are represented by thick lines. Solid line, ξ = 2; dashed line, ξ = 4; dash-dotted line, ξ = 6. The thin lines represent the corresponding reference struts with the same constraints as the optimized shapes.

[31] Figure 9 shows that the reference struts yield a total scattering cross section that is very similar to the results achieved by the optimized struts. Clearly, the representation (18) of the boundary Γ cannot yield edges with zero radius of curvature. The optimized shapes get sharper leading and trailing edges for ξ = 4 and ξ = 6 if the number of design variables in equation (18) is increased beyond N = 50.

3.3. Truncated Rhombic Cross Section

[32] It may be desirable to find strut designs that fulfill two conflicting requirements: (1) a large cloaked area (controlled by rmin) in order to be able to create a hollow cylinder that can fit cables and waveguides; and (2) an area A0 of the cross section that is sufficiently small to fulfill cost and weight requirements.

[33] Therefore, we keep the area constraint A0 = 2Amin fixed and increase the minimum radius rmin. First, we consider the value wmin = 2rmin = 45 mm that gives 0.15 < wmin/λ < 3 for the frequency interval 1 GHz < f < 20 GHz used in the goal function (21). Here, we use initial designs with an = 0 for odd n in the parameterization (18). This yields initial designs that are symmetric with respect to the plane x = 0. Figure 10 shows two different optimized shapes for this case: optimized shape with a circle as the initial design (solid line); and optimized shape with the reference strut as the initial design (dashed line). We refer to the two shapes shown in Figure 10 as equation image45,2 (shown by the solid line) and equation image45,2 (shown by the dashed line) in the following. The notation equation image refers to the optimized shapes resembling a rhomb with a minimum width of wmin = 2rmin and area constraint A0 = ξAmin. In the same way, the notation equation image refers to the optimized shape that resembles a truncated rhomb.

Figure 10.

Optimized shapes given two different initial designs for wmin = 45 mm. Solid line, equation image45,2; dashed line, equation image45,2.

[34] Figure 11 shows the real part of the equivalent blockage width of the two optimized struts in comparison with the corresponding reference strut: optimized strut equation image45,2 (solid line); optimized strut equation image45,2 (dashed line); and reference strut (dash-dotted line). It is interesting to notice that equation image45,2 yields a lower equivalent blockage width for the higher-frequency range 13 GHz < f < 20 GHz, as compared to both the reference strut and the optimized shape equation image45,2. All three designs yield rather similar values for the goal function, i.e., the RMS value of the total scattering cross section for the frequency interval 1 GHz < f < 20 GHz.

Figure 11.

Real part of the equivalent blockage width for the two designs in Figure 10 together with the corresponding reference strut. Solid line, equation image45,2; dashed line, equation image45,2; dash-dotted line, reference rhomb.

[35] Before we discuss the different characteristics of the truncated rhomb equation image45,2, we consider also a case with even thicker struts enforced by wmin = 60 mm, which yields 0.2 < wmin/λ < 4. Again, we use the area constraint A0 = 2Amin. The optimized shapes are shown in Figure 12 optimized shape equation image60,2 achieved with a circle as the initial shape (solid line); and optimized shape equation image60,2 achieved with a reference strut as the initial design (dashed line). Figure 13 shows the real part of the equivalent blockage widths for the case with wmin = 60 mm: optimized strut equation image60,2 (solid line); optimized strut equation image60,2 (dashed line); and reference strut (dash-dotted line). As for wmin = 45 mm, equation image60,2 is better than equation image60,2 for the higher-frequency range 11 GHz < f < 20 GHz. All three designs yield rather similar values for the goal function, i.e., the RMS value of the total scattering cross section for the frequency interval 1 GHz < f < 20 GHz.

Figure 12.

Optimized shapes given two different initial designs for wmin = 60 mm. Solid line, equation image60,2; dashed line, equation image60,2.

Figure 13.

Real part of the equivalent blockage width for the two designs in Figure 12 together with the corresponding reference strut. Solid line, equation image60,2; dashed line, equation image60,2; dash-dotted line, reference rhomb.

[36] As a result of the simultaneous constraints on the radius and the area, equation image45,2 and equation image60,2 connect the upper part of the boundary with the lower part by means of a corrugated surface. For the frequency f = 15 GHz, the corrugations in Figures 10 and 12 have an electrical depth of d/λequation image 1/4, where λ is the wavelength. For oblique angles of incidence, the quarter wavelength deep corrugations yield an approximate magnetic conductor [Kildal, 1990] that prevents electric currents to flow along the circumference of the strut. Field computations confirm that the shapes equation image45,2 and equation image60,2 yield very small currents on the corrugated surface that is located on the shadow side of the strut. Furthermore, the upper and lower parts that connect the two sides with corrugations are rather well aligned with the wave vector of the incident field. These features yield a considerable reduction of the scattering in the forward direction and the nearby scattering lobes, when compared to the reference struts equation image45,2 and equation image60,2. It should be noted that, in comparison with equation image45,2 and equation image60,2, this is achieved at the expense of a considerably higher back scattering due to the corrugation on the side of the strut that is directly illuminated by the incident wave. However, the shapes equation image45,2 and equation image60,2 yield an overall reduction in the total scattering cross section for a frequency range where the corrugations can effectively approximate a magnetic conductor. We note that the shapes equation image2,…,equation image5 in Figure 5 feature corrugations similar to equation image60,2 and that equation image1 features one large corrugation. Figure 6 shows that the equivalent blockage width for equation image1 is significantly lower for the frequency f = 6 GHz corresponding to a corrugation depth of approximately λ/4.

3.4. Asymmetrical Cross Section

[37] The terms with an odd index in the parameterization (18) of the boundary Γ allow for a cross section that is not symmetric with respect to the plane x = 0. For the designs considered in sections 3.2 and 3.3, we have used initial designs with an = 0 for all odd n and this resulted in optimized shapes that are symmetric with respect to the plane x = 0. It deserves to be emphasized that the reciprocity of Maxwell's equations implies that the total scattering cross section is identical for the two different incident waves with the wave vectors ±ki, regardless of the strut's shape.

[38] Here, we construct an initial shape from (1) the left part of equation image60,2 shown in Figure 12 and (2) the right part of equation image60,2 in Figure 12. Figure 14 shows the optimized asymmetric shape denoted equation image60,2 by the thick solid line, where we also include the shapes equation image60,2 (shown by the dashed line) and equation image60,2 (shown by the dash-dotted line) from Figure 12 in order to facilitate comparisons. As a result of the optimization procedure, some material is moved from the side with the corrugations in order to make the tip on the other side a bit longer. Figure 15 shows the corresponding real part of the equivalent blockage widths: equation image60,2 (solid line); equation image60,2 (dashed line); and equation image60,2 (dash-dotted line). The shapes denoted equation image, equation image, and equation image are local optima to the same optimization problem and their goal function values are comparable. The frequency dependence of the blockage width, however, differs. The shapes denoted equation image and equation image have a lower total scattering cross section at higher frequencies and the maximum of the total scattering cross section over the frequencies considered is lower. These characteristics could be useful for certain applications.

Figure 14.

Optimized shapes with asymmetric initial design for wmin = 60 mm. Solid line, equation image60,2; dashed line, equation image60,2 (also shown in Figure 12); dash-dotted line, equation image60,2 (also shown in Figure 12).

Figure 15.

Real part of the equivalent blockage width for the three designs in Figure 14. Solid line, equation image60,2; dashed line, equation image60,2 (also shown in Figure 13); dash-dotted line, equation image60,2 (also shown in Figure 13).

[39] It is interesting to notice that the equivalent blockage width for the asymmetric shape is located in between the corresponding results for the two symmetric shapes, i.e., the optimized shapes with and without corrugations. The symmetric optimized shape with corrugations (i.e., equation image60,2) yields a rather large equivalent blockage width around the frequency f = 5 GHz and, in comparison, the asymmetric shape provides a significant reduction in the total scattering cross section for the frequency interval 3 GHz < f < 10 GHz. The RMS value of the equivalent blockage width (for the frequency interval 1 GHz < f < 20 GHz) is g = 35 mm for equation image60,2, which is somewhat lower than the value g = 36 mm achieved for both the symmetric shapes equation image60,2 and equation image60,2.

[40] The asymmetric cross section features a low back-scattering cross section when the illuminating plane wave is incident on the sharp edge of the strut, which corresponds to illumination from the left in Figure 14. However, the reverse direction of the incident wave yields a substantially higher back-scattering cross section. This feature is important in some cases and, in particular, we mention the upper strut in the inverted Y tripod that is used in many reflector antennas. In such a case, the low back-scattering cross section can reduce the noise temperature of the antenna significantly [Moreira et al., 1996].

4. Conclusion

[41] We present a method for the reduction of the total scattering cross section for struts by means of shape optimization in two dimensions. We use the optical theorem to formulate the total scattering cross section in terms of the forward scattering amplitude. The gradient of the total scattering cross section with respect to an arbitrary number of design parameters is calculated from the field solution of the original scattering problem and an appropriate adjoint field problem. For both polarizations, the sensitivities are derived for dielectric cylinders and, as a special case, we also present the corresponding formulas for perfect electric conductor (PEC) cylinders.

[42] The sensitivities are used for shape optimization of PEC cylinders in the transverse electric (TE) case, where the wave vector of the incident plane wave is perpendicular to the cylinder axis. The goal function is the root-mean-square (RMS) value of the total scattering cross section for a frequency interval 1 GHz < f < 20 GHz. The shape of the strut's cross section is described in polar coordinates, where the radial coordinate is expressed in a Fourier series with respect to the polar angle and the coefficients are used as design variables. The shape is constrained to have a given constant area and its radial coordinate is restricted by a lower bound.

[43] We show optimized designs that fall into two main categories based on the cross section of the strut. The first category equation image , is a rather conventional shape that is oblong in the direction of propagation of the incident wave and this type of shape is often equipped with rather sharp tips, which makes it possible to approximate its shape by a rhomb. The other category equation image is also oblong but features corrugations parallel to the cylinder axis on part of the circumference. At the frequency where the corrugation depth is a quarter of the wavelength, the corrugated surface yields an artificial magnetic conductor for surface waves that circumvent the strut and, consequently, prevents electric currents to flow along that part of the strut's boundary. Many of the optimized shapes are symmetric with respect to the plane that is perpendicular to the incident wave vector and coincides with the strut's cylinder axis. However, optima with asymmetric shapes also exist which can yield slightly lower values for the RMS value of the total scattering cross sections. These designs also have the benefit of a very low back-scattering cross section when illuminated by a plane wave that is incident on the sharp edge of the strut.

[44] The shapes presented in this article can also be used for the transverse magnetic (TM) case, should it be feasible to realize an artificial magnetic conductor that is efficient for a sufficiently large frequency interval. Also, the designs can be exploited in acoustics, for cylinders constructed by materials that can be modeled by a hard-surface boundary condition.

Acknowledgments

[45] This work was supported in part by the Strategic Research Center CHARMANT, financed by the Swedish Foundation for Strategic Research.

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