Modified Method of Physical Optics for Calculating Electromagnetic Wave Scattering on Non-Convex Objects

Abstract

In order to solve the problem of electromagnetic wave scattering on non-convex objects, a combined method based on the method of physical optics with corrections for multiple reflections was discussed. The calculation of these corrections was carried out by the ray method. The article considered the effect of taking into account the curvature of the surface on the calculation error. The results of the calculations of the test example were compared with the calculations by the rigorous method of moments.

1. Introduction

Despite significant progress in the development of computer technology, approximate methods for calculating the radar cross section (RCS) can still compete with rigorous calculation methods due to considerably shorter computation time. The only requirement for the applicability of approximate methods is to meet the necessary accuracy standards.

Therefore, developing methods to reduce the calculation errors for approximate methods is becoming increasingly relevant. This article presents one of the possible methods of error reduction for the approximate method of physical optics (PO). An obvious way to reduce the error is to introduce corrections to the basic method. As follows from the above, we chose the PO method as the basic method. The added corrections should compensate for the shortcomings and limitations of the basic method. The following limitations of the PO method should be noted. Firstly, it is only applicable to convex bodies and, thus, fails to account for multiple reflections between surface areas of the object. Secondly, the principle of mirror reflection of the electric field, which is used when calculating the physical optics current on the surface, is only valid for flat surfaces and, hence, introduces errors when applied to those curved. Although other limitations of the PO method exist, they are beyond the scope of this discussion.

Given the above considerations, the formula for the combined method with improved calculation accuracy is as follows:

$${\overrightarrow{E}}^p={\overrightarrow{E}}_{PO}^p+{\overrightarrow{E}}_{ms}^p+{\overrightarrow{E}}_{2D}^p+{\overrightarrow{E}}_{3D}^p,$$

(1)

${\overrightarrow{E}}^p$—field scattered by an object;

${\overrightarrow{E}}_{PO}^p$—zeroth-order approximation, calculated using the PO method;

${\overrightarrow{E}}_{ms}^p$—corrections for multiple reflections;

${\overrightarrow{E}}_{2D}^p$—corrections for 2D-discontinuity;

${\overrightarrow{E}}_{3D}^p$—corrections for 3D-discontinuity.

In addition to the physical optics current, non-uniform currents are induced on areas of the object’s surface where the radius of curvature of the surface is less than several wavelengths (in particular, sharp edges). This division of the total current into physical optics and non-uniform currents was proposed by P. Y. Ufimtsev [1] and is convenient for calculating corrections to the PO method. Since these corrections are related to violations of surface smoothness, we called such surface areas 2D- and 3D-discontinuity. The difference between 2D- and 3D-discontinuities is determined by whether one or both principal radii of curvature fail to satisfy the necessary conditions, respectively. A description of the method for calculating the correction for 2D-discontinuities can be found elsewhere [2].

If the conditions of convexity of the object surface are violated, it is necessary to take into account the correction for multiple reflections. When selecting a method for calculating this correction, the main requirement was the ability to carry out scattering calculations on objects of complex shapes with a parameter kL reaching 10⁴–10⁵ (where k—wave number, L—characteristic size of the object). The effect of the current on one surface area on the current of another surface area, known as multiple reflections, can be accounted for by applying the radiated field integral in the near field, as carried out in previous works [3,4,5]. The iterative application of this method, starting from the physical optics current as the zero approximation, offers high accuracy. However, our studies showed that this method does not provide the required counting rate for the required values of the parameter kL. Therefore, in this work, the ray method was applied to calculate the corrections of the multiple reflections.

This article primarily focuses on the method for calculating the correction ${\overrightarrow{E}}_{ms}^p$ of multiple reflections, while also discussing specific features of the implementation of the PO method. The definition of corrections for multiple reflections is provided, along with expressions for their calculation. The article includes examples of calculations and a comparison of results with those obtained by rigorous methods. In addition, some features of the calculation algorithms that affect the accuracy are addressed. However, the methods for calculating the corrections for 2D- and 3D-discontinuities are beyond the scope of this work and are discussed in other articles.

All further conclusions relate to monochromatic electromagnetic fields with a time dependence $exp\left(i\omega t\right)$.

2. Materials and Methods

Since, in a combined method, the PO method comprises the primary procedure, let us consider several features of the calculation algorithm on its basis.

From Maxwell’s equations, the Helmholtz equation for monochromatic fields in isotropic media is derived, the solution of which comprises the source representation of the field in the form of an integral over the current distribution area. However, given the equivalence theorem, a volume integral can be reduced to a closed-surface integral S, covering this volume [6,7,8,9]. For the electric component of the field, the integral is presented as follows.

$$\overrightarrow{E}(\overrightarrow{r})=-ikW{\displaystyle {\int }_S\left(1+\frac{1}{k^2}\overrightarrow{\nabla }\overrightarrow{\nabla }\right)}G\left(\overrightarrow{r},{\overrightarrow{r}}_q\right)\cdot \overrightarrow{J}\left({\overrightarrow{r}}_q\right)d{s}_q+{\displaystyle {\int }_S{\overrightarrow{J}}^m\left({\overrightarrow{r}}_q\right)\times \overrightarrow{\nabla }}G\left(\overrightarrow{r},{\overrightarrow{r}}_q\right)d{s}_q,$$

(2)

where

$G=\frac{\exp (-ikR)}{4\pi R}$—free-space Green’s function;

$R=\left|{\overrightarrow{r}}_q-\overrightarrow{r}\right|$—distance between the observation point and source point;

$k$—wave number;

$W=\sqrt{\frac{{\mu }_a}{{\epsilon }_a}}$—wave impedance of the medium;

${\epsilon }_a,{\mu }_a$—dielectric and magnetic absolute permittivity of the medium;

$\overrightarrow{J}=[\widehat{n}\times \overrightarrow{H}],{\overrightarrow{J}}^m=-[\widehat{n}\times \overrightarrow{E}]$—electric and magnetic equivalent currents;

$\widehat{n}$–surface normal.

Expression (2) comprises the basis for various rigorous and approximate methods, including the PO method. In order to obtain the integral of the PO method in the far field, a Cartesian coordinate system for the observation point is established, in which the observation point lies on the ${\widehat{z}}_p$ axis. Assuming the condition of the far field, vector Formula (2) is transformed into two scalar expressions

$$E_x=\frac{-ik}{4\pi R}{{\displaystyle \int }}_S\left\{W{J}_x\left({\overrightarrow{r}}_q\right)+J_y^m\left({\overrightarrow{r}}_q\right)\right\}\exp \left(ik{z}_q\right)d{s}_q,$$

(3)

$$E_y=\frac{-ik}{4\pi R}{{\displaystyle \int }}_S\left\{W{J}_y\left({\overrightarrow{r}}_q\right)-J_x^m\left({\overrightarrow{r}}_q\right)\right\}\exp \left(ik{z}_q\right)d{s}_q,$$

(4)

where

$J_x,J_y$—components of the electric current on the surface S;

$J_x^m,J_y^m$—components of the magnetic current on the surface S;

${\overrightarrow{r}}_q=\left(x_q,y_q,z_q\right)$—variable of integration on the surface S.

Expressions (3) and (4) correspond to the horizontal and vertical components of the scattered electric field in the coordinate system of the observation point, respectively. In the PO method, the integration surface coincides with the surface of the object. In order to formulate the PO method, the approximate values of the total field on the surface of the object are determined. It is assumed that in the shaded area, the field and the corresponding current are equal to zero, while on the illuminated surface, the field is divided into the incident and mirror-reflected fields.

$$\begin{array}{c}\overrightarrow{E}={\overrightarrow{E}}_i+{\overrightarrow{E}}_s\\ \overrightarrow{H}={\overrightarrow{H}}_i+{\overrightarrow{H}}_s\end{array},$$

(5)

where

${\overrightarrow{E}}_i, {\overrightarrow{H}}_i$—incident field;

${\overrightarrow{E}}_s, {\overrightarrow{H}}_s$—mirror-reflected field.

The surface of the object is approximated by flat triangular panels, each having a local coordinate system as shown in Figure 1. The figure shows the axes ${\widehat{y}}_1,\widehat{n}$ of this local coordinate system.

In order to calculate the reflected field, the systems of the incident ${\widehat{k}}_i,{\widehat{y}}_1,{\widehat{e}}_i$ and reflected ${\widehat{k}}_s,{\widehat{y}}_1,{\widehat{e}}_s$ waves are defined. Given the introduced notations, following a simple transformation, the mirror-reflected field can be written as

$$\begin{array}{l}{\overrightarrow{E}}_s=R_{\perp }{E}_{\perp }^i{\widehat{y}}_1+R_{\parallel }{E}_{\parallel }^i{\widehat{e}}_s\\ {\overrightarrow{H}}_s=\frac{1}{W}\left(R_{\perp }{E}_{\perp }^i{\widehat{e}}_s-R_{\parallel }{E}_{\parallel }^i{\widehat{y}}_1\right)\end{array}$$

(6)

where R_⊥(θ) and R_‖(θ)—complex reflection coefficients, being a function of the angle of incidence θ at the current point on the surface.

It should be noted that, in order to account for coating materials on surfaces, the complex reflection coefficients were used, since such an approach is more universal than a simple impedance condition on the surface. In a computer program, coatings are represented as a set of dielectric layers having specified complex dielectric and magnetic permittivities. In addition, impedance films characterized by a given square resistance can be placed between the layers. Formulas for calculating the functions R_⊥(θ) and R_‖(θ) for multilayer coatings can be found in the literature [10]. This algorithm has no limitations on the methods for obtaining complex reflection coefficients, including experimental ones.

Thus, by substituting Equation (6) into Equation (5), the total field and corresponding equivalent currents on the surface of the object can be determined. Further, the obtained currents should be recalculated in the coordinate system of the observation point. The scattered field can be found by using Formulas (3) or (4), depending on the desired polarization.

In order to simplify the further description, Expressions (3) and (4) for the scattered field can be written in a generalized form

$$E^p={{\displaystyle \int }}_{S}f\left(\overrightarrow{E},\overrightarrow{H},\widehat{n},{\overrightarrow{r}}_q\right)d{s}_q,$$

(7)

In Equations (2)–(4), the integration surface S represents the entire surface of the object. However, in the PO method, the current in the shadowed areas of the surface is assumed to be zero. Therefore, in Formula (7) for the PO method, the integration surface S only represents the illuminated part.

Since the surface of the object is represented by flat triangular panels, Expression (7) for the PO method can be transformed to the sum of integrals over each panel

$$E_{PO}^p=\sum _{j=1}^{N_p}{{\displaystyle \int }}_{S_j}f\left(\overrightarrow{E},\overrightarrow{H},{\widehat{n}}_j,{\overrightarrow{r}}_q\right)d{s}_q,$$

(8)

where $E_{PO}^p$—scattered field calculated by PO method, ${\widehat{n}}_j$—panel normal, $S_j$—illuminated part of the panel surface, $N_p$—number of panels on the entire surface. Here, the integrals over triangular panels are calculated analytically. The formulas can be found, for example, in the work [9].

For convex bodies, it is relatively simple to divide panels into illuminated and shaded, since shaded panels satisfy the following condition:

$$\left({\widehat{k}}_i\cdot {\widehat{n}}_j\right)>0,$$

(9)

However, for non-convex objects, Condition (9) cannot guarantee the correct accounting of shading. Therefore, it is necessary to calculate the shaded areas for each panel individually, since, in general, the panel can be shaded only partially. To divide the panel into illuminated and shaded areas, we used the Weiler–Atherton algorithm for cutting off areas with non-convex contours [11]. Then, the illuminated part of each panel is divided into triangles, and the formulas for calculating the fully illuminated triangle are applied.

Figure 2 shows the calculation of shaded areas on the example of an object consisting of two cylinders, where, in Figure 2a, the viewing angle coincides with the direction of the incident wave, while in Figure 2b, the viewing angle is shifted in order to show the separated shaded parts of the panels.

3. Corrections for Multiple Reflections

When electromagnetic waves are scattered on non-convex objects, multiple reflections of waves between different parts of the surface may occur, which remain unaddressed by the PO method. In order to increase the accuracy of calculations, it is necessary to include multiple reflections, as, for example, was carried out in the work [12]. Although it is assumed that the wave reflected from one panel illuminates the other panel entirely, it is most often only its part that is illuminated. In contrast to the method given in [12], to calculate the corrections for multiple reflections, we applied the ray method, in which the cross section of the rays is much smaller than the size of the panels, which avoids such a problem.

Let us consider the calculation of scattering by the geometric optics (GO) method, which comprises a special case of ray methods. In the GO method, the incident plane wave is divided into individual rays, followed by an independent tracing of each ray per geometric optics laws. When a ray hits the surface of an object, a reflection spot $S_{ij}^{sec}$ is formed (i—ray number, and j—reflection number of that ray). The induced field ${\overrightarrow{E}}_{ij},{\overrightarrow{H}}_{ij}$ is calculated according to Formulas (5) as the sum of the incident and reflected waves. The scattered field induced by this field in the reflection spot is calculated using Integral (7), where the integration area is equal to $S_{ij}^{sec}$. In this case, the total scattered field for the entire object, calculated by the GO method, is determined as the sum of all reflections of one ray, followed by the summation of all rays.

$$E_{GO}^p=\sum _{i=1}^{N_r}\left\{\sum _{j=1}^{N_i}{{\displaystyle \int }}_{S_{ij}^{\sec }}f\left({\overrightarrow{E}}_{ij},{\overrightarrow{H}}_{ij},\widehat{n},{\overrightarrow{r}}_q\right)d{s}_q\right\},$$

(10)

where $E_{GO}^p$—scattered field calculated by GO method, N_r—number of rays, N_i—number of reflections for the ray i.

Let us extract from the sum (10) the contribution of the first reflection of all participating rays:

$$E_{GO}^p=\sum _{i=1}^{N_r}{\left\{{{\displaystyle \int }}_{s_{i1}^{\sec }}f\left({\overrightarrow{E}}_{ij},{\overrightarrow{H}}_{ij},\widehat{n},{\overrightarrow{r}}_q\right)d{s}_q\right\}}_{j=1}^{}+\sum _{i=1}^{N_i}\left\{\sum _{j=2}^{N_{ij}}{{\displaystyle \int }}_{S_{ij}^{\sec }}f\left({\overrightarrow{E}}_{ij},{\overrightarrow{H}}_{ij},\widehat{n},{\overrightarrow{r}}_q\right)d{s}_q\right\},$$

(11)

Note that the total integration surface in the first term of expression (11) corresponds to the part of the surface illuminated by all rays at the first touch of the ray. This illuminated surface area corresponds to the integration surface in Expressions (7) and (8) for the PO method, i.e.,

$$S=\sum _{i=1}^{N_r}{S}_{i1}^{sec}=\sum _{j=1}^{N_p}{S}_j,$$

where S—illuminated surface of the object.

In addition, the total fields in the integrals are calculated using the same Formula (5). Thus, the first sum in Expression (11) is equal to physical optics Expression (8). If the first sum in Expression (11) for the GO method is replaced with Expression (8) for the PO method, an expression for calculating the scattered field by the combined method can be obtained:

$$E_{COMB}^p=E_{PO}^p+\sum _{i=1}^{N_r}\left\{\sum _{j=2}^{N_i}{{\displaystyle \int }}_{S_{ij}^{\sec }}f\left({\overrightarrow{E}}_{ij},{\overrightarrow{H}}_{ij},\widehat{n},{\overrightarrow{r}}_q\right)d{s}_q\right\},$$

(12)

where $E_{COMB}^p$—scattered field calculated by the combined method. As we can see, the combined method is equivalent to the GO method, but has a different calculation algorithm. The second term of the Expression (12) corresponds to the desired correction for multiple reflections:

$${\overrightarrow{E}}_{ms}^p=\sum _{i=1}^{N_r}\left\{\sum _{j=2}^{N_i}{{\displaystyle \int }}_{S_{ij}^{sec}}f\left({\overrightarrow{E}}_{ij},{\overrightarrow{H}}_{ij},\widehat{n},{\overrightarrow{r}}_q\right)d{s}_q\right\},$$

(13)

When tracing rays, reflections from the surface are calculated according to the mirror law:

$${\widehat{k}}_s={\widehat{k}}_i-2\left({\widehat{k}}_i\cdot \widehat{n}\right)\cdot \widehat{n}$$

The analysis of the expression shows that the direction of the reflected ray ${\widehat{k}}_s$ depends on the direction of the normal to the surface at the reflection point. This leads to a significant change in the trajectory of the ray at a small difference between the normal to the real surface and the normal to a flat panel. To reduce the error associated with this problem, we used a linear approximation to calculate the normal at the reflection point of the ray. For this, real or average normals to the surface at the vertices of the panels are used.

$$\widehat{n}=\gamma {\widehat{n}}_1+\alpha {\widehat{n}}_2+\beta {\widehat{n}}_3,$$

where $\widehat{n}$—normal in the reflection point, ${\widehat{n}}_1,{\widehat{n}}_2,{\widehat{n}}_3$—normals in vertices of panel, $\gamma ,\alpha ,\beta$—simplex coordinate.

The approximation of normals performed in this way corresponds to an approximate allowance for the curvature of the surface. An example of the calculations for the trajectories of reflected rays from a sphere consisting of 190 triangular panels is shown in Figure 3.

Figure 3a shows the trajectories of reflected rays when a plane wave is incident on the sphere without normals approximation, while Figure 3b shows trajectories with normals approximation. To simplify the picture, the incident rays, which represent a plane wave, are not shown. As can be seen, the difference in the trajectories of the rays is significant. Later in the text, an example is given to demonstrate how the inclusion or exclusion of the approximation affects the results of calculations for the scattered field.

4. Example: Two Cylinders

By using an example of two perfectly conducting cylinders, the calculation of the scattering of an electromagnetic wave taking into account multiple reflections was carried out (Figure 2). Cylinder dimensions: diameter 100 mm, length 300 mm. The angle between the axes of the cylinders was 90°, the distance between the cylinders was 10 mm. The axes of the cylinders were rotated 45° to the horizontal plane. A plane electromagnetic wave having a wavelength of 30 mm was incident upon the cylinders. The back RCS diagrams were calculated in the horizontal plane, where an incident angle of 0° corresponded to the direction along the normal to both cylinder axes.

First, the calculations of the scattered field by the PO method were carried out. On the left-hand and right-hand sides in Figure 4, the RCS diagrams for horizontal and vertical polarization, respectively, are presented.

As is known, for perfectly conducting objects, the PO method provides the same results for both horizontal and vertical polarizations (Curves 1 on the left and right graphs). However, as can be seen for RCS diagrams used as a reference (Curves 2), the results obtained by the MoM depended on the orientation of polarization. The comparison of the diagrams in Figure 4 shows that the PO method for this case gave an error of up to 15 dB.

To increase the accuracy, we calculated corrections for multiple reflections and 2D-discontinuities. The method for taking into account 2D-discontinuities was described in [2]. Here, the 2D-discontinuities were represented by the sharp edges of the cylinders.

The calculation of corrections for multiple reflections was carried out using Formula (13). The linear ray density in the calculations amounted to 30 rays per wavelength. Figure 5 shows, on the left, the corrections to the PO for horizontal polarization and, on the right, the corrections for vertical polarization.

Curves 1 correspond to corrections for multiple reflections, while Curves 2 correspond to corrections for 2D-discontinuities. For clarity, the correction diagrams shown in the figure were scaled to RCS. However, when adding these corrections to the PO solution, scattered fields were used as per Expression (1), since it is necessary to account for the phase. Digital noise on the resulting diagrams can be reduced by increasing the ray emission density during the correction calculation.

The results of calculating the RCS for the two cylinders by the combined method, i.e., the total of the physical optics solution and the corrections, are shown in Figure 6.

As in the above figures, on the left-hand and right-hand sides, the RCS diagrams for horizontal and vertical polarization, respectively, are presented. Curve 1 corresponds to calculations using the combined method, while Curve 2 represents the solution obtained by the rigorous method of moments (MoM), which is known for its accuracy. The difference between the diagrams reaches 2–3 dB if interference gaps are excluded. This suggests that the approximate methods provided acceptable accuracy compared to that of the more accurate method of moments, especially given the much shorter calculating time.

When calculating the correction for multiple reflections, the linear approximation of the normal on the panel surface was taken into account, which is equivalent to considering the surface curvature. If the normal approximation is disabled during ray tracing, the calculation result shown in Figure 7 is obtained.

As in the above figures, the diagrams for horizontal polarization are shown on the left-hand side, while on the right-hand side, the diagrams for vertical polarization are presented. Curves 1 represent calculation results obtained using the combined method without the approximation of normals, while Curves 2 correspond to the results obtained using the rigorous MoM. The significant difference between the diagrams of up to 5–10 dB emphasizes the importance of accounting for the surface curvature, i.e., the approximation of normal in the ray-tracing methods.

5. Example: Cylinder with a Hemisphere

The RCS diagram calculation error can be divided into two components: analytical and numerical. The analytical component encompasses the error arising from admissible simplifications made in the approximate method. The numerical component comprises all errors related to the numerical algorithms used in the computer calculations, such as the approximation of curved surfaces by flat panels, limited ray emission density, insufficient steps in calculating the RCS diagram, etc.

The correction for multiple reflections proposed in this study was based on the GO method. Therefore, it can be noted that the analytical error of the PO method with corrections for multiple reflections corresponded to the analytical error of the GO method, as discussed above. However, there were significant differences in the computational implementations of these methods, resulting in different properties of the computational programs themselves. To clarify the advantages of the combined PO method with the correction of multiple reflections over the GO method, an example can be considered.

The maximum advantage of combining PO with multiple reflections over the GO method was observed in the case of large bodies with a limited area of multiple reflection influence. For instance, the case of a cylinder with a hemisphere in the middle can be examined (Figure 8).

The cylinder had a length of 10 m and a diameter of 1 m, and the hemisphere in the middle had a diameter of 0.5 m. The mathematical model of the surface was composed of 6240 triangular panels, and calculations were carried out at a wavelength of 30 mm. Only the correction for multiple reflections was applied in the calculation.

Figure 9 depicts the RCS diagrams for vertical wave polarization. The RCS diagram on the left was calculated using the combined PO method with correction for multiple reflections, while the diagram on the right was obtained using the GO method.

The selection of appropriate calculation parameters for both methods resulted in a good match of the RCS diagrams, indicating their related origin. However, there was a significant difference in the calculating time between the programs.

The GO method took 1 h and 12 min to calculate the RCS diagrams, while the PO method with multiple reflections took only 5 min and 30 s, which was 13 times faster. The calculations were performed on a computer equipped with an Intel i7-4930K processor running at 3.4 GHz with 6 cores.

Let us now examine the gain in the counting rate in more detail. The calculating time of the program by the GO method can be influenced by two parameters: the step size for calculating the RCS diagram and the ray emission density used in the calculation. It is important to note that selecting appropriate parameters is crucial to avoid compromising the numerical accuracy of the calculations.

The step for calculating the RCS diagram should be at least three times smaller than the width of the narrowest diffraction-pattern lobes. The minimum possible width of diffraction-pattern lobes in radians was estimated as λ/L (λ—illumination wavelength, L—maximum size of the object). Therefore, the required step for the diagram calculation should be less than 0.06°.

The linear ray emission density was set to 30 rays per wavelength, since a lower density leads to noticeable numerical errors.

In the program based on the PO method, the calculation step for the diagrams was set to 0.01°, and the step for recalculation of panel shading was set to 1°. Under these conditions, the calculating time for the PO method equaled 24 s.

When calculating the correction for multiple reflections, the linear ray density was also set to 30 rays per wavelength.

In the GO method, the area of ray emission is determined by the dimensions of the entire object, while when calculating the correction for multiple reflections, the area of consideration can be limited to the vicinity of the hemisphere. This results in a significant reduction in the number of rays taken into account compared to those in the GO method. Figure 10 depicts a semi-transparent cylinder representing the area for considering multiple reflections.

Moreover, limiting the area for calculating multiple reflections allows the step size for computing corrections to be increased to 0.2°, which further reduces the calculating time. When using these parameters, the computing time for the corrections of multiple reflections was 5 min and 7 s.

Figure 11 displays the RCS diagram calculated by the PO method on the left, while on the right is the correction for multiple reflections for the vertical polarization of the wave.

Here, the main part of the diagram depicting the corrections for multiple reflections is smooth enough to allow for a larger step size in the calculation. Thus, when the correction was calculated with a step of 1°, the calculating time decreased to 53 s. Therefore, the total calculating time for the combined method, including the time spent on the PO method, was 1 min and 17 s, while the RCS diagram (left in Figure 9) remained almost unchanged.

6. Conclusions

In order to improve the accuracy of the calculation of electromagnetic wave scattering, a combined method based on the PO method with corrections was proposed. This approach addresses the challenge of calculating scattering on non-convex objects and introduces equations for computing the necessary corrections based on ray methods.

The PO method with correction for multiple reflections was expected to provide calculation accuracy comparable to that of the GO method. However, due to significant differences in the numerical calculation algorithms used in computer programs based on these methods, a noticeable reduction in calculating time was achieved. For the example of a cylinder with a hemisphere, the proposed combined method was over 10 times faster than the GO method.

In order to evaluate the accuracy of the calculation of the RCS by the combined method, it was assumed that rigorous methods offer a guaranteed lower error than approximate methods. As an example, the RCS diagram for two crossed cylinders was calculated by the combined method. Verification showed that the difference from the RCS diagram calculated by the rigorous method of MoM, in general, was less than 2–3 dB

In the above example, the cylinders had a curved surface, highlighting the importance of accounting for curvature in ray-tracing calculations. When calculating the RCS diagram without considering curvature, the error increased to 5–10 dB.

On the basis of the experience of using the combined method, it was found that the disadvantage of the ray tracing algorithm was the increase in the calculation error up to 5 dB and more when the number of ray reflections exceeded 6 (average number per one ray). This limits the application of the method to the calculation of cavity RCS.