Finite Volume Time Domain Technique (FVTD) Computations for ...
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Finite Volume Time Domain (FVTD) Computations for Electromagnetic
Scattering
Prof. A. ChatterjeeDepartment Of Aerospace Engineering
IIT Bombay.
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Outline Of PresentationPart I : Introduction to the finite volume time domain technique
Maxwell Equation in Conservative formFinite volume method for conservation lawsNumerical SchemesVolume grid generationBoundary ConditionsValidation
PEC sphereAlmondOgivelossy spherelossy cone sphere
Part II: ApplicationsRCS prediction of low observable aircraft Configurations
Intake B2F117
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Maxwell’s curl Equations with losses:
Constitutive Relations
Maxwell Equation (differential form)
HEB
t
EJHD
it
HBED
j
j
r
r
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3D Maxwell’s Equations in Conservative form:
where,
Maxwell Equation in Conservative form
shgfu zyxt uuu )()()(
ziz
yiy
xix
z
y
x
x
y
x
y
x
z
x
z
y
z
y
z
z
y
x
z
y
x
EJ
EJEJ
H
HH
s
B
B
D
D
h
B
BD
D
g
BB
DD
f
D
DDB
BB
u
;
0/
/0/
/
;
/0
//
0/
;
//0
//
0
;
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Numerical Technique
Finite Volume Time Domain (FVTD) technique.
Higher Order Characteristic based technique for spatial discretization based on the Essentially Non-Oscillatory (ENO) method.
Multi-stage Runge-Kutta time integration.
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Numerical FormulationMaxwell’s Equations (Operator form)
Decomposition of Total Field
suL )(
iii
iss
suLS
SsuL
)(
)(
sisi ssuuL )(
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Finite Volume FrameworkConservative form can be written as
where
integrating over an arbitrary control volume,
isz
sy
sx
sst Ssuhugufu )()()(
iz
iy
ix
iit
i suhugufuS )()()(
dVt
VddVss
dVt
Vd
i
v
v
i
v
is
s
v
v
s
)]([)(
)]([
uFu
uFu
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discretized form for jth cell,
application of divergence theorem gives,
dVussdSnt
Vd
v
it
is
s
isv
s
)(ˆ)]()([ uFuF
u
)~
~~()}ˆ)]()(({[~
1 dtd
ssVSndt
dV
iji
jsjj
M
mjm
issj
j
uuFuF
u
Higher order characteristic based technique
Runge Kutta time stepping
3D domain divided into hexahedral cells
Cell centred formulation
Finite Volume Framework ….contd
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Boundary ConditionsOn Perfectly Conducting (PEC) surface
Total tangential electric field,
Total normal magnetic field,
Far-field boundaries
Characteristic boundary conditions (zero scattered field)
.0En
.0 Bn
Numerical Boundary Treatment
Surface of the body is a Perfect Electric Conductor (PEC)
Tangential component of electric field zero at surface, i.e,
In scattered field formulation, implemented as (Einc known)
Similarly the boundary condition for the magnetic field,
Field values in the ghost cells computed by extrapolating the scattered field values from the interior
Normal components of the electric fields and tangential components of the magnetic fields in the ghost cells taken identical to those on the surface of the conductor
.0En
.0 Bn
incscat EnEn
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Boundary Conditions ….cont.Numerical Boundary Treatment
In the far field, characteristic based boundary treatments are applied
Scattered field is taken as zero
Fluxes are decomposed along the characteristic directions normal to the cell faces; for the ghost cells outside the domain, incoming characteristic fluxes for the scattered field are taken to be zero
Methodology Time domain computations for sinusoidal steady state
Complex field in frequency domain from time history of solution using
Fourier Transform
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Finite Volume Formulation for Conservation Laws
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IntroductionAim: To solve a set of governing equations describing a set of conservation laws in the integral form in a specified domain with prescribed boundary conditions
Equation of Conservation Laws e.g. In Fluid Mechanics
Mass conservationMomentum conservationEnergy conservation
+ Equation of state as a closure equation
e.g. In ElectromagneticsMaxwell’s Curl Equations
e.g. In MagnetohydrodynamicsNavier Stokes Equations & Maxwell’s Equations
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Conservation EquationConsider generic conservation equation for a conserved variable u, and assume that the corresponding flux vector known
Integrating over an arbitrary volume
where,
shgfu zyxt uuu )()()(
kujuiu ˆ)h(ˆ)g(ˆ)f(F
n̂
dVsdSnt
Vd
vs
v
)(ˆ)]([ uF
u
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Conservation Equation for The conservation equation for in a general form:
VV SdVqdSSVdV
t
Rate of changeof in a control
volume
Convective flux acrossthe surface
Source orsink term
Here : = 1 Mass conservation equation = u,v,w Momentum conservation equation = Energy equation
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Finite Volume Method for Conservation Laws
Starting point: Integral form of conservation equation.
Domain covered by finite number of contiguous control volumes (CV)
Conservation equation is applied to each CV
Computational node : Center of each CV, where the Variable values are calculated.
Interpolation is used to calculate variable values of CV surfaces in terms of nodal (CV center) values.
Results in an algebraic equation for each CV per equation
Methodology
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Computational Domain for Finite Volume Method
Control Volume
Surface Grid
Volume Grid
Computational domain divided into finite control Volumes
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Finite Volume DiscretizationsGeometry CreationDomain Discretization: Domain is subdivided into a finite number of small control volumes (CV) by a grid which defines the control volume boundariesGrid Terminology
Node-based finite volume scheme: u stored at vertexCell-based finite volume scheme: u stored at cell centroid
n̂
Typical CV with the notations
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Finite Volume Method … contd.The generic conservation equation for a conserved variable u, reproduced here is applied to each CV
Net flux through CV boundary faces =
f = component of the convective flux vector in the direction normal to the CV face
Note: CVs should not overlap
Domain volume should be equal to the sum of volumes of all the CVs
Each CV face is unique to two CVs which lie on either side of it
dVsdSnt
Vd
vs
v
)(ˆ)]([ uF
u
SdSfk
numS
f
V
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Finite Volume Method … contd.
To evaluate surface integral exactly one
should know integrand f everywhere on “S”
The above information is not available, only nodal (CV center) values of ‘u’ are calculated. Approximation must be introduced
Integral is approximated in terms of the variable values at one or more locations on the cell face.
Cell face values are approximated in terms of nodal CV center values
SdSfk
numS
f
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Finite Volume Method … contd.Global Conservation:
Integral conservation equation is applied to each CV
Sum equations for all CVs
Global conservation is obtained since surface integral over CV faces cancel out
Advantage Dis-advantage
Simplest method
Complex geometries
Scheme is conservative
Most popular amongst CFD workers
( > 95%)
Three levels of approximations
• Interpolation
• Differentiation
• Integration
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Numerical SchemesDiffer by how numerical flux fnum is evaluated at cell face and by how time integration is performed
Differ in spatial and temporal order of accuracy
Upwind (Characteristic based) Schemes:
Flux Splitting Schemes, Riemann/Godunov Solvers
Steger Warming, Van Leer flux vector splitting
Roe type solvers
Central Difference based Schemes:
Lax-Wendroff Scheme
Jameson’s scheme
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Time Stepping
Space and Time discretization combined
e.g. Lax Wendroff Scheme
Taylor Series expansion in Time
Space and Time seperated
Set of ODE’s obtained after space discretization
March in time with Runge-Kutta method
0)( uRdtdu
SV
uR
f1)(
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ENO (Essentially Non-Oscillatory)Scheme Higher order accuracy
Non-oscillatory resolution of discontinuities (adaptive stenciling)
Models discontinuous changes in material properties
Numerical flux at right-hand face of jth cell with rth-order accurate ENO scheme
The smoothest possible stencil
),.....,,( 21 kjrkjrkjk xxxS
1
01,2/1
r
llkrj
rlkj ff
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Validation for Standard Test Case
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ValidationMetallic Sphere:
Volume Grid – O-O Topology, Single Block (50×45×20 cells)Frequency for analysis = 0.09 GHz ( Electric Size = 1.4660 )
Volume Grid Discretization (O-O Topology)
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Bistatic RCS (dB) for Metallic Sphere at 0.09 GHz
E-plane h-plane
Validation ….contd Metallic Sphere:
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Validation - Metallic Ogive Metallic Ogive:
Electro-Magnetic Code Consortium (EMCC) benchmark target
Geometry – Half angle 22.62 degrees, length 5”, thickness 1”
Volume Grid - O-H topology, single block (40×40×138 cells)
Monostatic RCS at 1.18 GHz (Electric size 2π) for vertical polarization
Excellent agreement with experimental results
Maximum RCS of approx -20 dBsm = 10-2 m2
Minimum RCS of approx -60 dBsm = 10-6 m2
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Validation ….contd Metallic Ogive:
Rendered OgiveSurface Grid
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Ogive Volume Grid Cross-Section
Validation ….contd
Ogive – Volume Grid and Surface Currents
Metallic Ogive:
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Validation ….contd Metallic Ogive:
Ogive Monostatic RCS Plot (1.18 GHz, VV Polarization)
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Validation ….contdAlmond:
Electro-Magnetic Code Consortium (EMCC) benchmark target
Geometry – length 9.936”
Volume Grid - O-O topology, single block (15×121×125 cells)
Monostatic RCS at 1.19 GHz (Electric size 2π) for vertical polarization
Excellent agreement with experimental results
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Validation ….contdAlmond:
Rendered AlmondSurface Grid
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Validation ….contd Almond:
Almond – Surface Currents
Angle of incidence = 0 deg. Angle of incidence = 90 deg.
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Validation ….contd Almond:
Almond Monostatic RCS Plot (1.19 GHz, VV Polarization)
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Validation ….contd PEC Sphere with Non-Lossy coating:
PEC ka = 2.6858 Coating (t / λ) = 0.05, ka1 = 3.0, ε‘ = 3.0 and 4.0, μ‘ = 1.0 Volume Grid O-O Topology, Single Block (64×45×32 cells)
Bistatic RCS for Sphere with Discontinuous Nonlossy Coating (Backscatter at 180 degree)
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Validation ….contd PEC Sphere with Lossy dielectric coating:
PEC ka = 1.5 Coating (t / λ) = 0.05, εr = 3.0 – j4.0, μr = 5.0 – j6.0 Volume Grid O-O Topology, Single Block (64×48×32 cells)
Monostatic RCS Sphere with Lossy Coating
For different orders of accuracy For different discretization
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Validation ….contd PEC Cone Sphere with Lossy Coating:
Geometry : vertex angle 90 degree, sphere diameter 0.955 λ Coating (t / λ) = 0.01, εr = 3.0 – j4.0, μr = 5.0 – j6.0 Volume Grid O-O Topology, Single Block (80×40×38 cells)
Bistatic RCS for cone-sphere with lossy Coating (Backscatter at 180 degree)
E-plane h-plane
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Radar Cross Section of Low Observable Aircraft Configurations
Applications: Industrial Problems
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Industrial Problems
RCS Analysis of Engine intake configurations
B2 “Advanced Technology Bomber”
F-117 “Nighthawk”
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RCS of Low Observable Aircraft Configurations
Introduction
RCS, low-observability, air intake configurations
Intake geometries and grid generation
Results
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RCS Analysis of Engine Intake configurationsIntroduction:
Low-observability: low back-scatter for near-axial incident illumination as well as low
returns over a broad angular region
Low-observables characterized by greater contribution of traveling and creeping waves to the
Radar Cross-Section (RCS)
Definition of RCS: the area of an isotropic reflector returning the same power per solid angle
as the given body. At far field, it is proportional to the ratio of the power received from the
target to the power incident on the target.
Engine Intake Configurations studied
B2 “Advanced Technology Bomber” and F-117 “Nighthawk” chosen for present study:
Low-observable (stealthy) aircraft configurations (RCS of -40 and -25 dBsm respectively)
Fine geometric details not available due to military sensitivity
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RCS of some military aircraftsAircraft RCS RCS RCS
[dBsm] [m2] [ft2]
F-15 Eagle 26 405 4,358
F-4 Phantom II 20 100 1,076
B-52 Stratofortress 20 99.5 1,071
Su-27 12 15 161.4
B-1A 10 10 107.6
F-16 Fighting Falcon 7 5 53.82
B-1B Lancer 0.09 1.02 10.98
F-18E/F Super Hornet 0 1 10.76
BGM-109 Tomahawk -13 0.05 0.538
SR-71 Blackbird -18.5 0.014 0.15
F-22 Raptor -22 0.0065 0.07
F-117 Nighthawk -25 0.003 0.03
B-2 Spirit -40 0.0001 0.01
Boeing Bird of Prey -70 0.0000001 0.000008
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Estimation of RCSComputation of first time-domain numerical solution proposed by Yee in 1966 (FDTD,
second order in space & time); followed by other FDTD-based algorithms
FVTD-based schemes adopted to handle more complicated geometries
Hyperbolicity of the Maxwell's equations in their conservative form exploited by
characteristic-based algorithms
Drawbacks:
Requirement of large computational resources (both processor speeds and memory)
Lack of theoretical estimates on grid-fineness and minimum distance to the far-field
Similar studies:
RCS of VFY218 (Conceptual aircraft): approx. 15 dBsm @ 100 MHz, nose-on
incidence (monostatic) Not a low-observable
RCS of F117 (Stealth Fighter): approx. -20 dBsm @ 215.38 MHz, nose-on incidence
(monostatic)
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RCS Calculation
Above algorithm used to compute the total fields at the surface of the PEC. Equivalent surface currents are given by
Far-zone transform used to calculate the scattered fields Esc and Hsc at infinite distance and RCS is calculated as
where R is taken to be a sufficiently large number (100,000)
2
22
||||4
inc
sc
EER
En̂Ms
Hn̂Js (equivalent electric current)
(equivalent magnetic current)
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RCS ANALYSIS OF ENGINE INTAKE CONFIGURATIONS
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RCS And Air Intake configurations
Width of air intake duct vis-à-vis wavelength of electromagnetic radiation
Sr-71 air intake
Grilled air intake of f117
Position / profile of intake duct
s-shaped wing mounted intake duct of b-2
Wing blended intake duct of yf-23
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Intake centre Body
SR-71
CENTRE BODY
ANNULAR REGION
DUCT WALL
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INTAKE GRILL
F-117 AIRCRAFT
SOFT
FACEHARD
FACE
GRILL
ENG. FACE
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B-2 BOMBER
S-shaped duct with coated wall
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YF-23 ADVANCED TACTICAL FIGHTER
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RCS Analysis of Engine Intake configurations
Representative models of engine intake cavities considered are
Straight cylindrical cavity, open from one end and the other end
Terminated by a flat plate
Terminated by a hub and a flat plate
Terminated by a hub with a set of straight rotor blades and a flat plate
Hub of cylindrical shape
Hub of sphere-cylindrical shape
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k
i
j
2 λ
4 λ
Multiblock representation of a straight cylindrical cavity terminated by a plate
PLATED END
ƒ = 15 GHz λ = 20 mm es = 4 π Contd ..
20 λ
OPEN END
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Bistatic RCS: Straight Cylindrical Cavity with a Flat Plate Termination
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Monostatic RCS: Straight Cylindrical Cavity with a Flat Plate Termination
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Front view
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Rear view (angle of irradiation φ=28°)
Φ
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λ
2 λ
4 λ λ
HUB (SPHERE-CYLINDER)
λ = 20 mm es = 4 π
i
j
k
PLATED END
OPEN END
Straight cylindrical cavity with a hub & plate termination
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Bistatic RCS: Straight Cylindrical Cavity with different terminations and that of a Hub
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FRONT VIEW
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Straight cylindrical cavity with a hub blade & plate termination
6λ
ƒ=6 GHz λ=50 mm es = 6π
45°
2.5°
OPEN END BLADE HUBj
i
4
5
6
78
9
10
11
PLATED END
4λ
6λ
3λ
k
VIEW-OPEN END
i
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One block
VOL. DISCRETIZATION OF BLOCKS 4-11 (REPRESENTING REGION BETWEEN THE BLADES)
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Bistatic RCS: Straight Cylindrical Cavity with a Hub, Blades and Plate termination
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Monostatic RCS: Straight Cylindrical Cavity with a Hub, Blades and Plate termination
8GHz, Vertical-Polarization
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Monostatic RCS: Straight Cylindrical Cavity with a Hub, Blades and Plate termination
8GHz, Horizontal-Polarization
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Surface Current Distribution on Straight Cylindrical Cavity with Hub, Blades and Plate terminations
6 GHz, Horizontal Polarization, Φ=0°
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Surface Current Distribution on Straight Cylindrical Cavity with Hub, Blades and Plate terminations
8 GHz, Vertical Polarization, Φ=0°
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Straight cylindrical cavity with a hub blade & plate termination
6λ
6λ3λ
ƒ=6 GHz λ=50 mm es = 6π
45°
2.5°
OPEN END BLADE HUBj
i
k4
5
6
78
9
10
11
PLATED END
4.5 λ
VIEW-OPEN END
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Surface discretization of the complete domain
1
2
3
4 -11
12 13
14
15
Resolution = 20 grid pts/λ
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Comparative Monostatic RCS: Intake cavities with Cylindrical and Sphere-cylindrical Hub configurations
8GHz Vertical Polarization
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Side view of the cavity (φ =0)
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Front view of the cavity
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B2 – Advanced Technology Bomber Geometry and Grid Generation:
Geometrical details taken from public literature - 3-view diagram from Jane's All the World's Aircraft
Lack of fine details due to military sensitivity; photographs from the internet used to reconstruct the geometry
Basic dimensions: 20.9 m length, 5.1 m height and 52.43 m wingspan
Shape of aircraft contributes to its low-observability: blended wing-body, buried engines, lack of external tanks or weapons, etc
Surface grid: 25 2-D blocks, 45,840 nodes, formed by piece-wise bilinear surfaces
Airfoil section used for the wing: E180 (used in a radio-controlled model of the aircraft)
Average cell dimensions: 0.9 mm in longitudinal direction and 1.7 mm in span-wise direction
Volume grid: 52 blocks, approx. 1.5 million cells, nodes clustered near the surface
Wingspan of the grid: 293.9 mm
Minimum wavelength allowable on present grid (satisfying Nyquist's sampling criterion) is 1.8 mm (electric size 1025.4), corresponding to a wavelength of 0.321 m (frequency of approx 1 GHz) for the full scale aircraft
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B2 – Advanced Technology Bomber
Image of the B2 in flight
Rendered Image of B2 Surface Grid
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B2 – Advanced Technology Bomber B2 Surface Grid (close view):
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Surface Currents on B2 – Nose-on incidence @ 300 MHz, VV polarization
B2 – Advanced Technology Bomber
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Surface Currents on B2 – Broadside incidence @ 300 MHz, VV polarization
B2 – Advanced Technology Bomber
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B2 – Advanced Technology Bomber
Surface Currents on B2 – Nose-on incidence @ 300 MHz, HH polarization
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Surface Currents on B2 – Broadside incidence @ 300 MHz, HH polarization
B2 – Advanced Technology Bomber
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Bi-static RCS Plot @ 50 MHz (VV)
B2 – Advanced Technology Bomber
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Bi-static RCS Plot @ 300 MHz (VV)
B2 – Advanced Technology Bomber
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Bi-static RCS Plot @ 300 MHz (HH)
B2 – Advanced Technology Bomber
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Monostatic RCS Plot @ 300 MHz
B2 – Advanced Technology Bomber
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F-117 “Nighthawk”
Rendered Image of F-117 Surface Grid
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Surface Currents at 68 degrees angle of incidence (HH polarization)
Surface Currents at 90 degrees angle of incidence (VV polarization)
F-117 Surface Currents:
F-117 “Nighthawk”
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F-117 “Nighthawk”
Bi-static RCS Plot @ 280 MHz (VV)
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Acknowledgement
Anuj Shrimal Narendra Deore
Manoj Vaghela Debojyoti Ghosh
Wing Cdr. A. Bhattacharya Sandip Jadhav
IITZeus Grid Generator Prof. G.R.Shevare and his Team
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Thank you