Computing the Radar Cross Section of a Perfectly Conducting Sphere

This classic benchmark problem in computational electromagnetics is about computing the monostatic radar cross section (RCS) of a perfectly conducting sphere in free space, illuminated by a linearly polarized plane wave. The RCS is computed for sphere radius to free space wavelength ratios ranging from 0.1 to 0.8 and is compared to an exact analytical solution. This region represents the lower half of a transition zone between a long wavelength asymptotic solution, “Rayleigh scattering,” and a short wavelength asymptotic solution, “Geometrical Optics.” The transition zone is known as the “Mie region” after the originator of the exact solution. A mesh convergence study is performed for the first scattering resonance at a sphere radius to free space wavelength ratio of approximately 0.16364.

Model Setup

Geometry

Due to symmetry, it is sufficient to model only one quarter of the sphere. Figure 1 shows the geometry and boundary conditions.

Figure 1: The computational domain for computing the RCS of a PEC sphere in free space. Due to symmetry, it is sufficient to model one quarter of the sphere.

The geometry consists of two concentric spherical shells. The innermost shell, adjacent to the sphere, represents the free space domain, and the second shell represents a perfectly matched layer (PML) region that is used to provide an approximately reflection free termination of the, in reality unbounded, free space domain.

Equation

The model is set up and solved using a frequency domain formulation for the scattered electric field. The incident plane wave travels in the positive x direction, with the electric field polarized along the z-axis. The governing frequency domain equation can be written in the form

where the scattered electric field Esc is the dependent variable and the incident electric field Ei= (0,0,Ez), with

The equation is discretized using second order edge elements (also known as vector elements, Nedelec elements, or curl-conforming elements). It is well known that in order to resolve the wave field, one should strive for 10 or more discretization points per wavelength. The combination of using second-order elements and 8 elements per wavelength fulfills this criterion with some margin. To respect the geometry, a mesh that is somewhat finer for the longest wavelengths is required on the surface of the scatterer. A maximum element size of half the radius is used on those boundaries. The PML region requires special meshing as described under the section Perfectly Matched Layer below.

Boundary conditions

The sphere has perfect electric conductor (PEC) boundaries. The PEC boundary condition

sets the tangential component of the electric field to zero. It is used for the modeling of lossless metallic surfaces or as a symmetry type boundary condition. It imposes symmetry for magnetic fields and “magnetic currents” and antisymmetry for electric fields and electric currents.

PEC boundary conditions and perfect magnetic conductor (PMC) boundary conditions apply on the symmetry planes used to subdivide the sphere model.

The PMC boundary condition

sets the tangential component of the magnetic field and thus also the surface current density to zero. On external boundaries, this can be interpreted as a “high surface impedance” boundary condition or used as a symmetry type boundary condition. It imposes symmetry for electric fields and electric currents and antisymmetry for magnetic fields and “magnetic currents.”

Perfectly Matched Layer

The PML region, the second concentric shell around the sphere, provides an approximately reflection free termination of the computational domain by applying a complex-valued coordinate stretching in the radial (outward) direction. For good accuracy, there should be at least five elements through the thickness of the PML. This condition is usually most efficiently met by using a swept mesh so that the effective element quality becomes insensitive to the scaling in the radial direction. The mesh used in this example is shown in Figure 2. It consists of a free tetrahedral mesh around the sphere and a swept mesh in the PML domain.

Figure 2: A free tetrahedral mesh is used in the free-space region around the sphere, and a swept mesh is used in the PML region.

The free space region around the sphere is defined to be the far-field domain. This specifies that a near-field to far-field calculation is done on the boundary of this domain, which takes the computed electric fields around the sphere and uses the Stratton-Chu equation to find the scattered electric field infinitely far away from the origin.

In 3D, this is:

For scattering problems, the far field in COMSOL is identical to what in physics is known as the “scattering amplitude.”

The radiating or scattering object is located in the vicinity of the origin, while the far-field point p is taken at infinity but with a well-defined angular position

In the above formulas,

•

E and H are the fields on the “aperture” — the surface S enclosing the sphere.

•

r0 is the unit vector pointing from the origin to the field point p. If the field points lie on a spherical surface S', r0 is the unit normal to S'.

•

n is the unit normal to the surface S.

•

η is the wave impedance:

•

k is the wave number.

•

λ is the wavelength.

•

r is the radius vector (not a unit vector) of the surface S.

•

Ep is the calculated far field in the direction from the origin toward point p.

The unit vector r0 can be interpreted as the direction defined by the angular position

and Ep is the far field in this direction.

Results and Discussion

Figure 3 compares the simulation result for the RCS with the analytic solution computed using the scattered component of the electric field from this example and equation 11-247 in Ref. 1. As the figure shows, there is good agreement between the analytic solution determined in this manner and the finite-element model.

Figure 3: Comparison of the analytic solution and the COMSOL Multiphysics model of the RCS of a PEC sphere in free space.

Mesh Convergence

For the wavelength corresponding to the first maximum in the RCS plot in Figure 3, a mesh convergence study is performed to validate that the model converges toward a unique solution when refining the mesh isotropically. The model is solved in a parametric sweep over the number of mesh elements per wavelength. In the PML, the mesh density is not changed in the radial outward direction (that is, in the sweep direction for the swept mesh). The PML is resolved by 5 element layers in this direction which is sufficient to resolve the exponential damping in the radial direction. Thus the error contribution from the PML is not expected to decrease by adding more element layers. The main error contribution from the PML is due to the fact that it is not perfectly absorbing because of finite thickness and damping rather than mesh density. Thus, it is expected to give a contribution to the error in the computed RCS that does not decrease when refining the mesh.

Figure 4 shows the mesh convergence. The displayed error is the difference between the RCS from the finite element model and the exact solution from equation 11-247 in Ref. 1. As mentioned, the PML is expected to yield an error contribution which cannot be eliminated by refining the mesh. As there is no sign of stagnation in the convergence plot, this error contribution must be smaller than 0.1%. The RCS plot in Figure 3 corresponds to 8 elements per wavelength, that is a relative error of about 3% at the wavelength of the maximum in the RCS versus wavelength curve.

Figure 4: Mesh convergence for the difference in backscattering (monostatic) RCS between the COMSOL model and the exact solution.

Reference

1. C.A. Balanis, Advanced Engineering Electromagnetics, John Wiley & Sons, 1989.

RF_Module/Verification_Examples/rcs_sphere

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_lda	0.5	0.5	Sphere radius in wavelengths
r0	5[cm]	0.05 m	Sphere radius
lda	r0/r_lda	0.1 m	Wavelength
k0	2*pi/lda	62.832 1/m	Wavenumber
f0	c_const/lda	2.9979E9 1/s	Frequency
t_air	lda/2	0.05 m	Thickness of air around sphere
t_pml	lda/2	0.05 m	Thickness of PML
h_size	8	8	Number of elements per wavelength
E0	1[V/m]	1 V/m	Incident field magnitude

Geometry 1

First, create a sphere with two layer definitions. The outermost layer represents the PMLs and the core represents the PEC sphere for RCS analysis. The median layer is the air domain.

Sphere 1 (sph1)

In the toolbar, click

In the window for , locate the section.

In the text field, type r0+t_air+t_pml.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (m)
Layer 1	t_pml
Layer 2	t_air

Click

Definitions

Add a view with a different angle of perspective.

In the window, under right-click and choose .

Camera

Change only the sign of y in the Position and Up Vector sections:

In the window, expand the node, then click .

In the window for , locate the section.

In the text field, type 1.871.

Locate the section. In the text field, type -0.412.

Click

Choose wireframe rendering to get a better view of the interior parts.

Click the

button in the toolbar.

Geometry 1

Due to the symmetry of the structure, it is sufficient to model only one quarter of the sphere. Delete the domains which are not part of the modeling domain.

Delete Entities 1 (del1)

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

On the object , select Domains 1–3, 5–7, and 9–15 only.

Click

Definitions

After removing unnecessary domains, change the view to the first view definition which gives a better angle showing all layers.

Click the

button in the toolbar.

This is the modeling domain for RCS analysis.

Electromagnetic Waves, Frequency Domain (emw)

Now set up the physics. You will solve the model for the scattered field, which requires background electric field (E-field) information. The background plane wave is traveling in the positive x direction, with the electric field polarized along the z-axis. The default boundary condition is perfect electric conductor, which applies to all exterior boundaries including the boundaries perpendicular to the background E-field polarization.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Specify the Eb vector as


0	x
0	y
E0exp(-jk0*x)	z

Choose wireframe rendering in the current view to get a better view of the interior parts.

Click the

button in the toolbar.

Scattering Boundary Condition 1

In the toolbar, click

and choose .

Select Boundaries 3 and 14 only.

Definitions

The outermost domains from the center of the sphere are the PMLs.

Perfectly Matched Layer 1 (pml1)

In the toolbar, click

Select Domains 1 and 4 only.

In the window for , locate the section.

From the list, choose .

Electromagnetic Waves, Frequency Domain (emw)

Set PMC on the boundaries parallel to the background E-field polarization.

Perfect Magnetic Conductor 1

In the toolbar, click

and choose .

Select Boundaries 1, 4, 9, and 12 only.

Far-Field Domain 1

In the toolbar, click

and choose .

Far-Field Calculation 1

In the window, expand the node, then click .

In the window for , locate the section.

Select the check box.

From the list, choose .

Materials

Next, assign material properties. Use air for all domains.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Mesh 1

Use a tetrahedral mesh for the air domains.

Free Tetrahedral 1

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Select Domains 2 and 3 only.

The maximum mesh size is at most 0.2 wavelengths in free space. In this model, use 0.125 wavelengths.

Size

In the window, click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type lda/h_size.

In the text field, type lda/h_size.

Use a swept mesh for the PML domains.

Swept 1

In the toolbar, click

Distribution 1

Right-click and choose .

Compare the mesh with that shown in Figure 2.

Study 1

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
r_lda (Sphere radius in wavelengths)	range(0.1,0.025,0.8)

Step 1: Frequency Domain

In the window, click .

In the window for , locate the section.

In the text field, type f0.

In the window, click .

In the window for , locate the section.

Clear the check box.

In the toolbar, click

Results

Follow the instructions below to reproduce the plot in Figure 3. First, show the computed RCS values using square markers.

1D Plot Group 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Click to expand the section. From the list, choose .

Find the subsection. Clear the check box.

Clear the check box.

Find the subsection. In the text field, type RCS Calculation.

Locate the section. Select the check box.

In the associated text field, type Sphere radius in wavelengths (a/\lambda0).

Select the check box.

In the associated text field, type Normalized monostatic RCS (\sigma3-D/\pi a2).

Locate the section. Select the check box.

Point Graph 1

Right-click and choose .

Select Point 2 only.

In the window for , locate the section.

In the text field, type emw.bRCS3D/(pi*r0^2).

Locate the section. From the list, choose .

Click to expand the section. Find the subsection. From the list, choose .

Find the subsection. From the list, choose .

From the list, choose .

In the toolbar, click

The observed RCS graph pattern is oscillatory in the Mie region.

Next, proceed to perform the mesh convergence study at the first resonance in the Mie region.

Start by extending the parameter list with the resonant radius and the associated theoretical RCS value.

Global Definitions

Parameters 1

Add the two rows at the end and change the first line according to.

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
r_lda	r1	0.16364	Sphere radius in wavelengths
r0	5[cm]	0.05 m	Sphere radius
lda	r0/r_lda	0.30556 m	Wavelength
k0	2*pi/lda	20.563 1/m	Wavenumber
f0	c_const/lda	9.8114E8 1/s	Frequency
t_air	lda/2	0.15278 m	Thickness of air around sphere
t_pml	lda/2	0.15278 m	Thickness of PML
h_size	8	8	Number of elements per wavelength
E0	1[V/m]	1 V/m	Incident field magnitude
r1	0.16363636363636364	0.16364	Relative radius at 1st resonance
RCS1	3.6549540474068576	3.655	RCS at 1st resonance