COMSOL 6.4 - RCS of a Metallic Sphere Using the Boundary Element Method

RCS of a Metallic Sphere Using the Boundary Element Method

This model illustrates the process of evaluating the radar cross section (RCS) of a metallic sphere through the utilization of the boundary element method (BEM). By taking advantage of a vertical symmetry plane that is parallel to the polarization of an incident background field, the model reduces computational expenses. The computed RCS values are compared with analytical values within the Mie RCS region. See Table 1 below for a discussion of the characteristic sphere size a for the three RCS scattering regions.

Table 1: Radar terminology scattering regions.
Region	Sphere size
Rayleigh	r0 << λ0
Mie	Between Rayleigh and optical region
Optical	r0 >> λ0, conventionally 2πr0/λ0 > 10

Figure 1: Only the half of the metallic sphere is modeled using the symmetric plane.

This classic benchmark problem in computational electromagnetics is about computing the RCS of a perfectly conducting sphere in free space, illuminated by a linearly polarized plane wave. The simulation result can be directly compared to the analytically solution presented in many electromagnetics textbooks to verify the simulation accuracy.

The analytical solution to the RCS of a perfectly conducting sphere is given by

where S = πr02 is the cross sectional area of the sphere, ρ = 2πr0f/c is the normalized frequency, r0 is the sphere radius, f is the frequency, and c is the speed of light.

In addition, the coefficients an and bn are given by

and

where jn is the spherical Bessel function of the first kind and hn(2) is the spherical Hankel function of the second kind.

Model Definition

The linearly z-polarized plane wave in the background field defines the incident field on the metallic spherical object. The conductive sphere is set up using the default perfect electric conductor (PEC) boundary condition. For the simulation, the sphere is divided in half, and only this half size is utilized. The boundary for far-field calculation is applied to the sphere’s exterior boundaries but excludes the cut plane due to the symmetry plane. As the symmetry plane aligns parallel to the polarization of the background field, a zero tangential magnetic field (PMC) type of symmetry plane is employed. Unlike the conventional finite element method (FEM), the boundary element method (BEM) does not necessitate an air-domain enclosing the sphere and incorporating absorbing features.

Results and Discussion

Figure 2 shows correspondence between the computed and analytical values of RCS in the Mie region, with both values scaled by the factor 2πr02, where r0 represents the radius of the sphere.

Figure 2: RCS comparison between the computed and analytical values.

In Figure 3, the computed field solution outside the scattering object is assessed using a grid dataset. Initially, the plot of the metallic surface depicts only half the size of the sphere, which corresponds to the actual computation area. However, by employing a mirror dataset, the entire sphere can be visualized. Additionally, the material appearance subfeature provides a glossy metallic look, enhancing the visual effects.

Figure 3: Adjusted multislice plot of the electric field norm on a grid 3D dataset with a metallic sphere.

Notes About the COMSOL Implementation

This model uses the Electromagnetic Waves, Boundary Elements interface to demonstrate its functionalities. In principle, the Electromagnetic Waves, Frequency Domain interface can be used to model the same structure and achieve the same results. Both BEM and FEM solve the full Maxwell equations but FEM requires a finite simulation domain with volumetric meshing while BEM can model infinite domains and only require boundary meshing. Although the degrees of freedom in a BEM model are generally fewer compared to FEM, the memory and computation time requirements are not necessarily smaller. Therefore, one method could be more efficient than the other, depending on the type of problem to be solved.

Reference

1. J.A. Stratton, Electromagnetic Theory, Adams Press, 2013.

Wave_Optics_Module/Optical_Scattering/rcs_sphere_bem

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > .

Click .

Click

In the tree, select > .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
r	1[um]	1E-6 m	Radius of the sphere
cir	2pir	6.2832E-6 m	Circumference of the sphere
rho	1	1	Ratio of circumference to wavelength

Definitions

To compute the RCS using the analytical expression, the spherical Bessel functions of the first and second kinds and the spherical Hankel function of the second kind need to be defined. Create a node for each function to avoid clustering all the variables in a single node. For the frequency range of interest in this model, truncate the summation in the analytical expression of the RCS at n=6.

Spherical Bessel Function of the First Kind

In the window, expand the > node.

Right-click and choose .

In the window for , type Spherical Bessel Function of the First Kind in the text field.