331
1 INTRODUCTION
The detection of small vessels with insignificant radar
scattering cross section (RCS) is of crucial importance
for navigation safety when navigating in coastal
waters with dense maritime traffic such as near
tourist destinations with large tourist populations
during summer season. This is especially true for
indented coasts with numerous islands and deep
coves such as Croatian coast. With these
circumstances, the ability to detect small vessels at
short ranges determines safety of navigation,
especially during night and in bad weather
conditions. Situation is additionally worsened by the
typical material composition of these small vessels
since plastic, wood and rubber do not contribute to
RCS as much as metal parts. For that reason, this
paper concerns with the numerical computation of
RCS of the small rubber boat.
Radar installations on ships navigating these
waters vary from small radars installed on small
vessels with typical power output of 4kW (e.g.
Garmin) to 25kW marine radars typically installed on
larger passenger vessels such as ferry-boats. Many
ship captains have reported the inability to reliably
detect fast approaching rubber boat with radar which,
additionally, is difficult to distinguish from sea
clutter. Thus, in this paper, we attempt not to improve
radar itself but we rather numerically investigate the
possibility of finding the optimal height of radar
installation above the sea line to maximize the
possibility of radar detection of rubber boat.
Edge Element Calculation of Radar Cross Section of
Small Maritime Targets with Respect to Height of Radar
Antenna
H. Dodig,
S. Vukša & P. Vidan
University of Split, Split, Croatia
M.
Bukljaš
University of Zagreb, Zagreb, Croatia
ABSTRACT: From the aspect of navigational safety and collision avoidance it is very important to be able to
detect small maritime targets such as buoys and small boats. Ship's radar is supposed to detect these types of
targets, however the ability of radar to detect such targets depends on several factors. The most important
factors affecting the detection probability of small maritime targets are height of the antenna installation on the
ship and radar cross section of the target. The methods of computation radar cross section are diverse and
complicated, however, in this paper we apply our previously published numerical method for the RCS
computation which had proven to be very accurate. Physically to find RCS of the target one has to find the
solution of electromagnetic scattering problem. The numerical method relies on the combination of finite edge
volume elements and finite edge boundary elements to obtain the solution of Maxwell equations. The radiation
pattern of ships radar antenna is the source of excitation for the numerical method. At the end of the paper the
RCS of small maritime targets as the function of antenna height is shown. These results can be used as a
parameter in radar design, as well as the guideline for the height of installation of the ship's radar antenna
above the sea.
http://www.transnav.eu
the
International Journal
on Marine Navigation
and Safety of Sea Transportation
Volu
me 14
Number 2
June 2020
DOI:
10.12716/1001.14.02.08
332
Numerical methods available for RCS computation
generally fall in one of three distinct categories: ray-
tracing methods, physical optics methods and full
wave methods. Ray tracing methods are often
complicated because multibouncing of rays need to be
taken into account (Liu, 2012). Furthermore, ray
tracing methods do not take into account the
changing electromagnetic properties of the materials
used for interior of the vessel. Methods such as
physical optics (PO) and physical theory of diffraction
(PTD) can compute RCS with acceptable error,
however these methods are not well suited for
accounting the changing material properties in the
interior of the ship.
Thus, if we were to account for changing material
properties inside the vessel we can use one of the
following: FDTD (finite difference method), some
hybrid combination of method of moments (MoM)
with finite element method (FEM). However, these
methods can only provide near field solution of
electromagnetic scattering problem and they need to
be subjected to near-to-far field transformation
(NTFFT) in order to compute RCS which is
cumbersome procedure (Taflove, 2005).
To avoid NTFFT transform, in this research we use
our own previously published method for RCS
computation based on hybrid BEM/FEM with edge
elements (Dodig, 2017). This method first finds the
near field solution and from these electromagnetic
field values, using our own RCS equation, we
compute RCS directly from near field values. Thus,
the NTFFT transformation is avoided and sometimes
this approach produces better results (e.g. the case of
RCS computation at interior resonance frequencies,
see. Dodig, 2017).
The results of the numerical computation of RCS
are presented in section 5, where RCS is expressed as
the function of the angle between the line of sight
connecting radar antenna and rubber boat and the sea
level. It is shown that detection probability of the
rubber boat is very angle dependent for vertical
polarization and that in order to maximize the
detection probability of rubber boat at certain distance
the radar antenna should be placed at some definite
height above the sea level.
2 NEAR FIELD COMPUTATION WITH EDGE
ELEMENT HYBRID BEM/FEM
To compute the RCS of radar target the necessary step
is the computation of backscattered electric and
magnetic field at the exterior boundary of the
computational problem shown in figure. This
computed backscattered EM field is the near field
solution of EM scattering problem and, as such does
not represent the far field data necessary for RCS
computation.
In order to obtain the near field solution one needs
to obtain the solution of general 3D electromagnetic
scattering problem. This general 3D scattering
problem is shown in figure 1, where incident electric
and magnetic fields are denoted as
i
E
and
i
H
,
backscattered fields are denoted
S
E
and
S
H
and
interior fields are denoted
int
E
and
int
H
. Interior
and backscattered electromagnetic fields are the fields
we wish to compute, while the incident electric and
magnetic fields are known, and in the case of the
computation of ship’s RCS they come from radar
antenna. Because the electromagnetic properties of
materials (
,,
rr
µσ
) change inside computational
domain
V
, we need to use computational method
that can take these changes into account.
The method of computational electromagnetism
that can take into account the change of these
electromagnetic properties is hybrid BEM-FEM,
which is the combination of boundary element
method (BEM) and finite element method (FEM), and
the method is thoroughly described in ref. Dodig
2012-2014. Electric field exterior to computational
boundary
V∂
, shown in figure 1, can be described
by Stratton-Chu electric field integral equation (EFIE)
which in its time harmonic form can be written as
(e.g. Stratton 1939):
Figure 1. Outline of EM scattering problem. Volume of
computational domain is denoted
V
and the artificial
boundary is denoted
V∂
. Fields
i
E
and
i
H
are
incident to
V∂
while
S
E
and
S
H
are backscattered
fields.
( )
( )
' ' ''
''
1
ext i ext ext
VV
ext
V
E E i dS H G dS E
G dS n H G
i
α ωµ
σω
∂∂
∂
=− × + ××
∇ − ∇⋅ × ∇
′ ′′
+
′
∮∮
∮
(1)
For interior fields, that is for electromagnetic fields
inside computational volume
V
shown in figure 1,
the time harmonic Faraday’s law takes the following
mathematical form:
''
EH
int int
i
ωµ
∇× =−
′
(2)
and time harmonic Maxwell-Ampere equation is
given by:
( )
''
HE
int int
i
σω
′
∇× = +
(3)
333
Taking the curl of Equation 3 and combining with
Equation 2 yields the following differential equation:
''
1
H H0
σ iω
int int
i
ωµ
∇× ∇ × + =
′
+
(4)
With computational methods for electrostatics the
unknown fields
'
E
int
and
'
H
int
are usually
approximated with nodal approximating functions.
However, that is not appropriate for full wave
methods. With full wave methods we use edge
element approximating functions in order to preserve
the continuity of tangential components of electric
and magnetic fields (see Dodig 2017 for details). Edge
elements approximate electric and magnetic fields
using vector approximating functions
i
w
as:
1
E
n
int i i i
i
we
δ
=
=
∑
(5)
1
H
n
int i i i
i
wh
δ
=
=
∑
(6)
where
n
is the number of edges on the element,
i
e
and
i
h
are unknown coefficients associated with
each edge of the element. Vector approximating
functions
k
w
are associated with
th
k
edge of the
element by the following relation:
k i j ji
w NN NN=∇−∇
(7)
where
i
N
and
j
N
are nodal approximating
functions associated with nodes of the element
(Nedelec, 1980).
Due to physical jump conditions of electric and
magnetic fields at the interface between two materials
with different electromagnetic properties, all the
exterior fields in Equation 1 i.e.
'
E
ext
and
'
ext
H
, can
be replaced by interior fields
'
E
int
and
'
int
H
. This is
due to tangential continuity of electric and magnetic
fields across the boundary where material properties
change. With these conditions, Equation 1 and
Equation 4 can be coupled and combining with
Equations 5 – 7 the following system of equations is
obtained:
0
0
00
bI
b
v
GH e e
DF Fh
F Fh
−
−=
(8)
where
I
e
are edge element coefficients computed
from incident field
i
E
,
b
e
and
b
h
are edge
element coefficients associated with edges at the
artificial boundary
V∂
and
v
h
are edge element
coefficeints associated with interior of the
computational domain
V
. The hybrid BEM-FEM
method of numerical computation of near
electromagnetic field was rigorously tested over the
period of several years in various physical settings
(Dodig 2012-2017 and Cvetković 2017).
3 NUMERICAL METHOD OF RADAR CROSS
SECTION COMPUTATION
Radar cross section is computed directly from the
edge element coefficients computed in previous
section. These coefficients are associated with near
electric and magnetic field, that is, the near
electromagnetic field can be reconstructed from edge
element coefficients (Dodig 2017). To compute radar
cross section, one has to convert the near field to far
field. Previously, this transformation from near field
to far field was achieved by the employment of some
elaborate and computationally expensive numerical
methods (Taflove 2005). These methods are
collectively known as Near-to-Far-Field-
Transformation (NFFT).
Figure 2. RCS plot of the metallic sphere coated with
dielectric layer with relative permittivity
4
r
=
. This
calculation was performed using EFIE formulation and
hybrid BEM/FEM at resonant frequency of 300 MHz and
was compared with Mie series solution.
The necessity for NFFT can be circumvented
completely and far field can be computed directly
from edge element coefficients by the application of
our previously published computational technique
(Dodig 2017). Radar cross scattering section
σ
is
defined as ratio of backscattered and incident field:
2
*
22
2
*
lim 4 lim 4
S
SS
rr
II
i
E
EE
rr
EE
E
σπ π
→∞ →∞
⋅
= =
⋅
(9)
where
*
S
E
represents the complex conjugate of
backscattered vector field
S
E
. It was shown in ref.
Dodig 2017 that this backscattered field can be written
in compact form as:
4
ik r
SS
e
EF
r
π
−
=
(10)
where complex vector
S
F
can elegantly be
computed from edge element coefficients as the sum:
334
( )
( )
( )
3
'
11
Δ
3
'
11
Δ
'
3
'
11
Δ
'
'
n
'
σ iω
i
i
i
N
ikr e
S bj j j
ij
N
ikr e
bj j j
ij
N
Sj
ikr e
bj
ij
F e i h e dS w
ik e e dS w
w
e ik h e dS e
ρ
ρ
ρ
ρ
ρρ
ωµ δ
δ
⋅
= =
⋅
= =
⋅
= =
=− ×+
××
∇⋅ ×
−
′
+
∑∑
∫
∑∑
∫
∑∑
∫
(11)
Computation of
( )
S
Fe
ρ
from known boundary
edge coefficients
bj
e
and
bj
h
is fast, and if
necessary, the line integrals in Equation 11 can be
solved analytically to further improve the speed and
accuracy of RCS calculation. Equations 9 – 11 are well
tested on canonical models, were compared with Mie
series analytical solutions and were tested in the case
of dielectrically coated PEC sphere (e.g. Dodig 2017)
where it has been shown that accurate results can be
obtained even at resonance frequencies as shown in
figure 2.
4 PHYSICAL AND GEOMETRICAL MODEL OF
RUBBER BOAT
From the standpoint of collision avoidance and from
standpoint of early detection of small targets in
military missions, the rubber boat is considered as the
radar target of choice. The rubber boat model used for
numerical RCS computation is the model of small
service boat usually attached to Croatian Navy ships
for the support of some small scale off-ship military
missions. The rubber boat was subjected to series of
3D laser measurements to accurately capture the
geometry of the boat, as shown in figure 3.
Figure 3. Photo of the rubber boat at laser measurement site.
The relevant geometric features of the boat are captured at
spatial points marked with green dots.
Geometry capture software produced the set of 3D
points and set of linear triangles from laser
measurements, conveniently given in the form of STL
(stereolitography) file. However, to produce the mesh
of good quality with Ansys ICEM software the
parasolid or similar input file is required. For that
purpose, Geomagic software was used to convert
from STL file to Parasolid x_t format and then the air
and water volume surrounding the boat were added
with Siemens SolidEdge CAD software as shown in
figure 4.
CAD model of the rubber boat was loaded into
Ansys ICEM mesher to produce the tetrahedral mesh
shown in figure 5, where the volume of the air was
removed to enhance the visibility of interior elements.
Figure 4. CAD model of the geometry for RCS computation.
Upper half of the volume consists of air and the lower half
of the volume consists of seawater. Rubber boat is partially
immersed in water and partially immersed in air.
Final computational models consists of 285,064
tetrahedral elements and of 1,870 triangular elements
used to model the boundary of the computational
model.
Figure 5. The mesh of the model shown in figure 4 with air
volume removed. The model consists of 286,934 elements
where 285,064 tetrahedral elements were used to model the
interior of computational domain and 1,870 elements were
used to model the boundary of computational domain.
Electrical parameters
σ
,
r
,
r
µ
of the seawater,
air, rubber and plastic were compiled from various
sources from literature (e.g. Talley 2011, Garazza
2011) and these are shown in table 1.
Table 1. Electrical properties
σ
,
r
,
r
µ
of the materials
used for RCS computation compiled from various sources.
_______________________________________________
Physical
σ
r
r
μ
property [S/m]
_______________________________________________
Seawater 0.00 88.00 1.00
Air 0.00 1.00 1.00
Rubber 0.00 2.50 1.00
Plastic 3.00 3.00 1.00
_______________________________________________
335
5 RESULTS OF RCS COMPUTATION
Physical setting for RCS computation is shown in
figure 6, where the rubber boat is at distance
l
from
the ship and radar antenna is at height
h
above the
sea level. The angle between the line connecting the
radar antenna and rubber boat and the sea level is
θ
.
Furthermore, it is assumed that sea is at calm state.
Horizontal polarization (HP) and vertical polarization
(VP) of the radar EM wave are shown in the same
figure and
k
indicates the direction of propagation
of radar EM wave.
Figure 6. The rubber boat is at horizontal distance
l
from
the radar antenna and the radar antenna is at height
h
above sea level. Angle subtended between the line
connecting the antenna an boat and between sea level is
θ
.
To find the radar cross section
S
σ
of the sea
without the rubber boat we have first performed the
series of calculations for the physical setup shown in
figure 6 but without rubber boat. The results of
numerical radar cross section calculation of
S
σ
for
various angles
θ
is shown in figure 7 for both HP
and VP.
Figure 7. Radar cross sections
S
σ
of the empty sea at
calm state without rubber boat for various angles
θ
are shown for both HP and VP.
Then the same series of numerical calculations is
performed for the configuration shown in figure 6
with the rubber boat included. This time, the radar
cross section
σ
is the total cross section that
includes reflections from both the sea and from
rubber boat. These results are shown in figure 8.
Figure 8. Radar cross section
σ
of the rubber boat and the
sea at calm state as the function of angle
θ
is shown for
both HP and VP.
Figure 9. The ratio of cross section
/
S
σσ
is shown as the
function of angle
θ
is shown for both HP and VP. The sea
state is considered to be calm.
The reason why these calculations were performed
separately (without rubber boat and with rubber boat)
is to find how much radar cross section of the rubber
boat distinguishes itself from the radar cross section
of the sea. The ratio
/
S
σσ
shows how much radar
cross section of the rubber boat is above the radar
cross section
S
σ
of the empty sea and is shown in
figure 9.
Figure 10. Magnitude and vector plot of near electric field
with air omitted for
3
θ
=
, horizontal polarization (HP).
336
6 CONCLUSION
RCS of the rubber boat is computed for physical
setting shown in figure 6 for various angles
θ
and
for both vertical and horizontal polarization of the
radar EM wave. For numerical computation of near
field we have used hybrid BEM/FEM with edge
elements. The results of near field computation were
used as input for our own RCS computational method
which uses this values directly without the need for
NTFFT transformation. This method falls in the class
of the full wave methods and it can account for the
change of electric and magnetic properties of
materials inside the computational domain. An
example of near field computation for
3
θ
=
and
horizontal polarization is shown in figure 10.
To be able to distinguish the radar cross section of
the sea and of the rubber boat we have first computed
the RCS of the sea patch without the rubber boat and
these results are shown in figure 7. Then we have
computed total RCS of both rubber boat and sea as
shown in figure 8. The measure of how much rubber
boat is distinguished from the RCS of the sea itself is
shown in figure 9. This measure is expressed as
simple ratio between RCS of the sea and RCS of both
sea and rubber boat.
It should be noted that from figure 9 it follows that
the detectability of the rubber boat differs for vertical
polarization (VP) and horizontal polarization (HP).
From figure 9 it follows that detectability of rubber
boat for horizontal polarization (HP) is
approximately constant for the range of angles
θ
.
However, for vertical polarization the detectability of
rubber boat varies significantly with angle
θ
.
Furthermore, the capabilities of our RCS
computational software are currently limited with
two limitations: the size of computational model (the
number of unknowns) and with computational time.
Currently, an effort is underway to address these
issues so that much larger ship models could be used
for RCS computation.
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