CEM Workshop Lectures (10/11): Numerical Modeling of Radar Absorbing Materials

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Numerical Modeling Of Radar
Absorbing Materials
RAM simulations
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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
 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
 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
 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
 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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Effective Medium Theory
 Importance of microstructure:
 The simplest level of information that can be quantified - volume fraction of each constituent
 The effective properties in general cannot be obtained using simple mixture rules
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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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 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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   
 1211
12111
1
1

 


Af
AffAf

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 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 lete(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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 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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 Filler prolate spheroid aspect ratio = 1 (spherical)
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 Filler prolate spheroid aspect ratio = 1.5
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 Filler prolate spheroid aspect ratio = 2
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 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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 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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 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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CEM Based Calculations

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CEM Based Calculations

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Methodology
 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
 If the distribution is truly random average properties must be isotropic
 In isotropic case the following expression can also be used
 How to check whether distribution is truly random?
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   
   





 N
i
ii
N
i
iiii
1
1
rErE
rErE 


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Methodology
 If the distribution is truly random average properties must be isotropic
 In isotropic case the following expression can also be used
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   
   





 N
i
ii
N
i
iiii
1
1
rErE
rErE 


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Methodology
 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





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Example of Simulation
Comparison of full wave analysis with effective medium theories
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+91 72760 31511
Abhishek Jain
contact@zeusnumerix.com
Thank You !

CEM Workshop Lectures (10/11): Numerical Modeling of Radar Absorbing Materials

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CEM Workshop Lectures (10/11): Numerical Modeling of Radar Absorbing Materials