More Related Content Similar to CEM Workshop Lectures (10/11): Numerical Modeling of Radar Absorbing Materials (20) More from Abhishek Jain (20) CEM Workshop Lectures (10/11): Numerical Modeling of Radar Absorbing Materials2. 2
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Contents
19-Jul-2019 Numerical Modeling Of Radar Absorbing Materials
Design of radar absorbing materials
Why pure materials do not work as RAM, need for composite materials
Calculation of effective properties of composite medium using various effective
medium models along with examples of simulation
Simulation of random simulation and use of CEM methods in effective medium
theory along with example of simulation
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RADAR Absorbing Materials
19-Jul-2019 Numerical Modeling Of Radar Absorbing Materials
Radar cross section has implications to survivability and mission capability
The materials for reduction of radar cross section rely on magnetic permeability and
electric permittivity, while principles from physical optics are used to design absorber
structure. It is a combination of optics and materials that lead to signature reduction
Advanced techniques are used for absorber optimization
Radar absorbers can be classified as impedance matching or resonant absorbers
Dynamic absorbers should be studied in order to counter frequency agile radars
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Types of RAM
19-Jul-2019 Numerical Modeling Of Radar Absorbing Materials
Large volume RAM is usually resistive carbon loading added to fiberglass hexagonal
cell aircraft structures or other non-conducting components. Fins of resistive
materials can also be added. Thin resistive sheets spaced by foam or aerogel may be
suitable for space craft.
Thin coatings made of only dielectrics and conductors have very limited absorbing
bandwidth, so magnetic materials are used when weight and cost permit, either in
resonant RAM or as non-resonant RAM
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Types of RAM
19-Jul-2019 Numerical Modeling Of Radar Absorbing Materials
Resonant but somewhat 'lossy' materials are applied to the reflecting surfaces of the
target. The thickness of the material corresponds to /4 of the incident radar-wave.
The incident radar energy is reflected from the outside and inside surfaces of the
RAM to create a destructive wave interference pattern. Deviation from the resonant
frequency will cause losses in radar absorption, so this type of RAM is only useful
against radar with a single frequency
Non-resonant magnetic RAM uses ferrite particles suspended in epoxy or paint to
reduce the reflectivity of the surface to incident radar waves. Because the non-
resonant RAM dissipates incident radar energy over a larger surface area, it usually
results in a trivial increase in surface temperature, thus reducing RCS at the cost of
an increase in infrared signature. A major advantage of non-resonant RAM is that it
can be effective over a wide range of frequencies, whereas resonant RAM is limited
to a narrow range of design frequencies.
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Design of RAM
19-Jul-2019 Numerical Modeling Of Radar Absorbing Materials
Requires application of electromagnetic theory and CEM along with Materials
Science and statistical principles (For modelling random heterogenous media)
EM theory requires impedance matching at interface between free space and
coating of RAM on PEC- This is possible only if = for the RAM material
Pure ferrites have ~ 1 and ~ 15, hence desirable to disperse ferrite in low
material (epoxy) for better impedance matching
Computational problem is thus study of dispersion of particles with specified , in
host matrix with different = as a function of volume/ weight fill factor
Need to calculate effective properties of this composite medium
Approximate effective medium theory
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Importance of microstructure:
Various features of the microstructure
Volume fraction
Orientation, size and shape of
inclusions
Spatial distribution of inclusions
Connectivity of phases etc.
Quantitatively described by n point
correlation functions
Cluster formation
Maxwell Garnett (MG)
Percolation
Mainly three models for calculation
of effective properties
Self consistent approximation (SC)
Differential effective medium (DEM)
approximation
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Effective Medium Theory
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Effective Medium Theory
19-Jul-2019 Numerical Modeling Of Radar Absorbing Materials
Є, Є1, and Є2 denote, respectively, the effective permittivity of the composite
material, the permittivity of the matrix with surface fraction f1, and the permittivity
of the inclusion phase with a surface fraction f2. A(0<A<1) is the depolarization
factor which depends on the shape of the inclusions. For disks A= 1/2, for spheres A
= 1/3 .
The most popular mixing laws or EMT are those of Maxwell-Garnett:
Self consistent approximation
Effect of all material outside an inclusion is to produce a homogeneous medium with
effective properties e
Impose the condition that the perturbation to a uniform field is zero on an average
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12111
1
1
Af
AffAf
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Effective Medium Theory
19-Jul-2019 Numerical Modeling Of Radar Absorbing Materials
Differential Effective Medium Approximation (DEM):
Similar to SC approximation (incremental homogenization)
Phase 2 treated as matrix phase with volume fraction v2 and property 2
Phase 1 with volume fraction f1 property 1 is treated as filler
Assume the effective property e (f1) known
Treat e (f1) as host matrix dielectric constant and lete(f1+f1) represent
effective property after a fraction f1 (1- f1 ) has been replaced by inclusion of
phase 1
Using the dilute inclusion formula obtain a differential equation for e with
initial condition e (f1=0) = 2.
Solve the differential equation to get the effective properties
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Effective Medium Theory
19-Jul-2019 Numerical Modeling Of Radar Absorbing Materials
In general Maxwell and DEM approximation fail for dispersion which forms clusters
SC approximation gives better results in such cases
Summary :
Maxwell approximation works only for dilute dispersions
SC works only for dispersion with phase inversion symmetry in which no connectivity exists
between any phase It can consider cluster formation
DEM works well even for connected phase even in high concentration but can not account
for clusters
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Effective Medium Theory
19-Jul-2019 Numerical Modeling Of Radar Absorbing Materials
More accurate values of effective properties can be obtained by computer
simulation by taking into account many particle correlations
Create various random distributions with a fixed volume fraction of fillers and
checking numerically the correlations
Can give better estimates of upper and lower bounds
However, simple model reported above is sufficient for predicting effective
properties for calculating reflection properties of various composites
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Effective Medium Theory
19-Jul-2019 Numerical Modeling Of Radar Absorbing Materials
Statistically based algorithm
Reconstruct the microgeometry
Must allow systems with both arbitrary shapes and arbitrary EM characteristics to be
considered
An initial random configuration of particles in a unit cell is generated
Variants of traditional 2D METROPOLIS sampling scheme adapted to generate
equilibrated sets of realizations in 3D
The basic parameters in this model simulation are the length L of the square
primitive cell side, the number N of hard disks, their diameter D, and their surface
fraction f2
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Effective Medium Theory
19-Jul-2019 Numerical Modeling Of Radar Absorbing Materials
After an initial configuration is generated, one attempts to move randomly the
center-of-mass coordinate of each disk. The new configuration is accepted if the
particle does not overlap with any other particles
This process is repeated until equilibrium is achieved
To minimize boundary effects due to the finite size of the system, periodic boundary
conditions are employed
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Typical equilibrium configurations
(sample realizations) of the two-
phase composite consisting of
circular disks randomly distributed
within a square primitive cell.
The sample packing results from the
sequential algorithm applied to a
binary mixture with a given surface
fraction of disks.
(a) f2=0.1, (b) f2=0.3, (c) f2=0.5, (d)
f2=0.6, (e) f2 =0.7, (f) f2=0.8, (g)
f2=0.82, (h) f2=0.83, and (i) f2=0.85.
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Examples of Equilibrium
Configuration
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Assume cubic shape of inclusion
[Xi, Yj, Zk] coordinates / sites
i = 1,..,Nx
Nx = no. of unit cells along X =
Length of X axis / unit cell dimension along X
axis
j = 1,..,Ny
k = 1,..,Nz
Random nos. along X = ( 3√f * Nx )
f = volume fraction of inclusion
Run random no. generation engine (1+ 3√f *
Ny * 3√f * Nz) times to get 3D profile of
composite
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Alternate Method
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Methodology
19-Jul-2019 Numerical Modeling Of Radar Absorbing Materials
To calculate effective properties of the medium, perform average fields as below (
denotes components)
These averages are defined as follows
where f is any variable. The averaging is carried out over a representative cell as
discussed earlier and ri represents the center of cell i
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H
H
E
E
N
f
dv
dvf
f
N
i
i
V
V
1
rr
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Methodology
19-Jul-2019 Numerical Modeling Of Radar Absorbing Materials
How to check whether distribution is truly random?
In 1 dimension we have the following test: (for random numbers between a and b)
For random discrete medium we can use this method to check randomness
Here xi are the centroids of ith particle (cube or sphere) dispersed randomly with
properties of medium 2. Medium 1 is assumed to be the matrix or remaining cubes.
Similar expressions can be used for y and z components as well
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2
ab
dx
xdx
x b
a
b
a
ab
ab
ndx
dxx
x
nn
b
a
b
a
n
n
11
1
1
2
211 xx
N
x
x
N
i
i
12
121
1 xxnN
xx
N
x
x
nn
N
i
n
i
n