FINDING THE ELECTROMAGNETIC HOMOGENOUS
EQUIVALENT OF THE COMPOSITE MATERIAL USING
GLOBAL OPTIMIZATION TECHNIQUES TO SOLVE THE
INVERSE PROBLEM
Jana Jilková
Dept. of Radar Technology, University of Defence, Kounicova 65, 612 00, Brno, Czech Republic
Keywords: Genetic algorithms, Multi-objective optimization, Composite material, Homogeneous isotropic equivalent
material.
Abstract: The equivalent homogeneous isotropic replacement is computationally efficient way how to enable
simulation of the large numeric models of the composite aircrafts for the purpose of precertification
electromagnetic compatibility tests to estimate the level of their resistance against the lightning. The paper
presents application of two global optimization methods to find an appropriate electromagnetic equivalent
of the composite material by homogeneous material to reduce CPU demands of the numeric models of elec-
trically large airplanes for the purpose of electromagnetic compatibility simulations. First, the inverse
problem is specified using the already known scattering parameters of the composite material. Afterwards,
the global optimization method is applied to find the equivalent with such a value of the complex
electromagnetic permittivity to have the impact on the electromagnetic field propagation as close as possible
to the original composite material.
1 INTRODUCTION
The demands on the important operational cost
reduction of the airplanes by important decrease of
the fuel consumption of today’s airplanes is
nowadays being solved by the aircraft designers
replace replacing the fully metallic skin of airplanes
with composite materials.
Composite material has by its nature lower
shielding effectiveness than the fully metallic skin of
airplanes. As the airplanes operate in the
environment, which is disturbed by large number of
electromagnetic interferences, as nature as artificial,
is necessary to know shielding effectiveness of the
composite structure.
While the airplanes are frequently used to deliver
many people or important cargo, each type of newly
designed airplane has to be tested to prove its
resistance against lightning. While each testing
attempt is very expensive and the whole testing
procedure has to be repeated in case of insufficient
shielding, the numeric simulation to discover the
weak parts is a natural choice. Unfortunately
modelling the composite structures directly causes
the mesh cells to increase, just at the composite
parts. (D'Amore and Sarto, 2000). This is naturally
caused by multilayer structure, tiny structure of the
composites and high frequency range, on which the
EMC values are prescribed by standards.
Composite materials used as construction
materials for aircrafts are most often multilayer
epoxy or polyester resin reinforced by carbon fibres
(carbon fibre reinforced composites CFRC, Von
Klemperer, 2009).
To increase the mechanical toughness and the
electromagnetic shielding, it is possible to introduce
a metallic grid between reinforced epoxy resin layers
(see Figure 1). Unfortunately due to the complicated
structure of the composite materials and as it
contains lossy materials, the numeric models are
becoming even more complicated and more CPU-
time demanding (Sarto, 2002).
One of the approaches to reduce the
computational complexity is to replace the
multilayer structure in numerical model with a
homogeneous isotropic equivalent providing the
same or very close behaviour in electromagnetic
field as the original composite structure. The
324
Jilková J..
FINDING THE ELECTROMAGNETIC HOMOGENOUS EQUIVALENT OF THE COMPOSITE MATERIAL USING GLOBAL OPTIMIZATION TECHNI-
QUES TO SOLVE THE INVERSE PROBLEM.
DOI: 10.5220/0003674903240328
In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (ECTA-2011), pages 324-328
ISBN: 978-989-8425-83-6
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)
homogeneous equivalent in numerical models can be
covered by more sparse mesh net leading to the
lower CPU demands causing the CPU time of the
simulation are reduced. To solve these issues, the
global optimization methods were applied to find
such a value of the complex permittivity of a
homogeneous dielectric layer that exhibits similar
frequency responses of scattering parameters as the
composite material.
In the first part the 3D model of composite
material simulated in commercial full-wave CST
Microwave Studio (CST MWS) is described and its
scattering parameters are calculated to be used as the
input value for the optimization process.
In the second part the principle of the inverse
problem used for the determination of complex
permittivity of the homogeneous material is
described.
In the third part, the design of the global
optimization methods is described. The
implementation details together with the impact of
the selected method using single representative of
single-objective and one of the multi-objective
approach is discussed.
2 SCATTERING PARAMETERS
OF COMPOSITE MATERIAL
The polymer composite supported by carbon fibres
is reported to be the most frequently used material
for aircraft construction. To increase its mechanical
robustness and electromagnetic shielding level the
metallic grid is frequently introduced into the
composite. From the perspective of electromagnetic
parameters, conductive carbon fibres are strongly
anisotropic so they have different parameters in
various axes. Practically this increases the numeric
model complexity even more.
The three-dimensional numeric model of such
composite material was created in a commercial
modelling program CST Microwave Studio (see
Figure 1) consisting of polymer matrix with
thickness of 2 mm, with relative permittivity 4 and
conductivity 0 S/m and a carbon fibre matrix with 35
m in diameter, conductivity set to 10
4
S/m along
the fibre 50 S/m across fibre and relative
permittivity set to 2.
The model consisted of two such layers of
polymer-carbon composite surrounding a copper
grid with wire 0.3 mm in diameter and eye size
3x1.5 mm.
The whole structure was placed to the standard
R100 waveguide and simulated for normal incidence
of TM
01
wave. This arrangement was particularly
selected so that the model could have been easily
verified experimentally by direct measurement.
Figure 1: Detailed three dimensional model of carbon fibre
reinforced composite (CFRC) simulated in CST
Microwave Studio.
The high computational demands (from 3 day up
to one week depending on the hardware) on the
simulation solver are not only caused by the
multilayer thin structure of the model, small eye size
of metallic grid and carbon fibres matrix or by
anisotropic nature of the carbon fibres but also by
quite huge frequency range, required for the
precertification test.
Figure 2: Scattering parameters of composite material: red
line – reflection coefficient, blue line – transmission
coefficient (simulated in CST MWS).
This wide frequency range also has a negative
impact to the overall error of the numeric model.
This challenge was described in the paper (Jilková
and Raida, 2009). The output of the simulation – set
of so called scattering parameters of scattering
parameters (see Figure 2) – served as the input
values to the optimization process.
FINDING THE ELECTROMAGNETIC HOMOGENOUS EQUIVALENT OF THE COMPOSITE MATERIAL USING
GLOBAL OPTIMIZATION TECHNIQUES TO SOLVE THE INVERSE PROBLEM
325
3 PRINCIPLE OF INVERSE
PROBLEM
To find the homogeneous equivalent material two
particular members of the global optimization
methods was used. The inverse problem is specified
as finding such a frequency dependency of the
complex permittivity of a homogeneous dielectric
layer that exhibits similar frequency responses of
both scattering parameters as the composite material
(s
11
reflection coefficient and s
21
transmission
coefficient). Replacement principle is shown on
Figure 3.
Figure 3: Principle of inverse problem.
To achieve be able to apply the global
optimization method on this problem, the 2D
problem of the homogeneous dielectrics was created
which was further easier to simulate because of
lower computational demands being therefore
convenient for the optimization. As for the 3D
model, the output of the 2D model consisted of both
scattering parameters that were afterwards compared
with those obtained from 3D model. The input to the
2D model was formed by the the real part of the
relative permittivity and a loss tangent parameter.
These two can be used to calculate so called
complex permittivity that completely characterizes
any lossy dielectric material. The model was created
in COMSOL Multiphysics. For the purpose of the
global optimization, the dielectric constant was
permitted to vary within the interval [1; 1000] and
the loss tangent was set to vary within [0; 1000].
The complex permittivity acquired from the
optimization of the 2D model was finally verified in
the 3D model of the equivalent in CST MWS to see
if the scattering parameters match in the whole
frequency spectra.
4 GLOBAL OPTIMIZATION
Why should the global optimization methods be
used to find an equivalent homogeneous material?
An usual way how to find properties of equivalent
material is to derive its properties using the
analytical description of its electromagnetic
properties.
This involves expressing the characteristics of
the material in Maxwell equations which may
become very complicated when it comes for
complex multilayer lossy anisotropic structure. The
usual way how to cope with it is to introduce some
approximation and simplifications which can be
limiting in many ways.
Also the analytic solution may change heavily if
some parameters of the model change (eg.
introducing nonzero conductivity of the dielectric to
take into account its non-zero loss). On the other
hand using the optimization methods in connection
with numerical models enables finding equivalent
material of much broader range of composite
structures since the only requirement for the
structure is that it has to be possible to reliably
simulate.
The principle stays the same for simple layer
composite, for multilayer composite structure, for
single carbon fibres or for knitted carbon armature.
The method is even convenient for composite
structures with metallic grid.
For the entire frequency range, a single
evaluation of the relative permittivity and a single
evaluation of the loss tangent are required (see Table
1). The fitness function is of the form:
F
1
(x) = [ S
11
,
ref
- S
11
,
o
pt
(x)]
-1
/
2
(1)
ECTA 2011 - International Conference on Evolutionary Computation Theory and Applications
326
F
2
(x) = [ S
21
,
ref
– S
21
,
o
pt
(x)]
-1/2
(2)
Here, S
11,ref
is the computed value of the reflection
coefficient of the realistic model of the composite
material at the frequency range. Then, S
11,opt
(x) is the
reflection coefficient of the equivalent homogeneous
material computed in COMSOL Multiphysics for
the vector of state variables x = [
r
, tan
].
Similarly, S
21,ref
is the computed transmission
coefficient of the realistic model of the composite
material, S
21,opt
(x) is the transmission coefficient of
the equivalent homogeneous material computed in
COMSOLl Multiphysics for the state vector x = [
r
,
tan
]
T
.
The equivalent material was searched for one
particular composite material using two different
optimization approaches – a singleobjective and a
multiobjective. For the single-objective solution, the
fitness functions (1) and (2) are weighted by
weighting coefficients w
1
= 1 and w
2
= 1 and
summed:
F(x) = w
1
F
1
(x) + w
2
F
2
(x) (3)
4.1 Single-objective Optimization
For single-objective optimization a classical global
optimization representative – a genetic algorithm
was selected. Genetic Algorithms (GA) are members
of stochastic global optimization methods and are
based on the Darwinian Theory of the evolution of
species. The values of the control parameters were
chosen according to recommendations published in
(Rahmat-Samii and Michielssen, 1999).
The population consisted of 50 individuals, the
accuracy of binary coding was set to 0.001. Indivi-
duals for the next population were selected by the
tournament operator. Probability of the multi-point
crossover was set to 70 % and probability of the
multi-point mutation was set to 6 %. The elitist
strategy was applied. The optimization was set to
minimize the optimized problem.
Time needed for a single simulation run and then
calculation of the complex permittivity value using
single-objective genetic algorithm was
approximately 12 hours (see Table 1). Because of
time needed for the calculation of the criteria
function, the entire optimization cycle was repeated
only one hundred times.
The values of complex permittivity obtained
from the optimization run were then set to 3D model
to compare the reflection parameters with the model
of the original composite. The error of calculation
was at maximum 0.45% on the whole frequency
range of 15-40 GHz.
In Table 1 one chosen value of complex
permittivity from the set of the optimization run is
shown. On Figure 4, the scattering parameters for
this value of the complex permittivity of the
homogeneous replacement are shown in comparison
with scattering parameters of the original composite.
As it can be seen on Figure 4 good match was
achieved.
4.2 Multi-objective Optimization
Niched-Pareto Genetic Algorithms, by Horn et al., is
a multi-objective GA based on the non-domination
concept. NPGA uses the binary tournament selection
(Deb, 2001).
The population consisted of 50 individuals,
mutation probability was set to 6%, the crossover
probability was 70% and the selected accuracy of
coding is 0.001.
The result of a single optimization run is formed
by a set of results corresponding to all the criterions
(Pareto front). The calculation needed to evaluate
one Pareto front was about half of the time needed
for single-objective optimization (see Table 1), the
optimization run was repeated hundred times.
Figure 4: Reflection coefficients of replacements achieved
by single-objective genetic algorithms (blue) in
comparison with the detailed 3D composite model (green).
After an inverse implementation of the values of
complex permittivity of replacements from all
simulation runs to the 3D model, all the optimized
results were simulated to do the comparison with the
original composite material results.
In Table 1 it is shown, that maximal error of the
replacement on frequency range 15 – 40 GHz is less
then 0.45 %.
On Figure 5 a single choice value of the complex
permittivity of the replacement compared with
reflection coefficient of the original composite
material is shown.
FINDING THE ELECTROMAGNETIC HOMOGENOUS EQUIVALENT OF THE COMPOSITE MATERIAL USING
GLOBAL OPTIMIZATION TECHNIQUES TO SOLVE THE INVERSE PROBLEM
327
Table 1: Comparison of optimization methods, minimum
and maximum accuracy reached for frequency range 15-
40 GHz.
Optimization ε
r
[-] tg δ [-] time [h] Accuracy [%]
GA 36.64 549.34 12 0.16-0.45
NPGA 80.27 726.73 6 0.06-0.45
Figure 5: Reflection coefficients of replacements achieved
by multi-objective NPGA in comparison with originally
reflection coefficient of composite material. Green linie:
composite material, red linie: NPGA replacement.
Simulated in CST MWS.
5 CONCLUSIONS
For the comparison, the single-objective
optimization (reflection coefficient considered) and
the multi-objective optimization (both the reflection
coefficient and the transmission one) were tested.
In order to compute the minima of objective
functions, genetic algorithms and Niched-Pareto
genetic algorithm were applied to the problem. First,
the single-objective approach exhibits a very good
functionality. Second, the substitute synthesized by
the multi-objective approach shows a very good
agreement too.
The most significant advantage of the multi-
objective approach is the speed of the simulation.
Time needed to one simulation for NPGA was one
half compared to single-objective approaches.
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