smart-structures-and-materials-2017

250 A. Sohouli et al.

move limit or trust region strategy. This leads to the solution of complex

optimization problems in an efficient manner. The direct method [24] and the

conservative Method of Moving Asymptotes (MMA) [25] are two classical

approximations in SAO.

In this study, an exponential approximation method is used which exhibits a

better computational performance when compared to the MMA for minimum

compliance optimization problems [26]. This approximation method is modified for

Discrete Material Optimization (DMO) where the algorithm searches for the new

feasible designs vector Xk + 1 such that the new objective function follows

C0ðXk + 1Þ < C0ðXkÞ while satisfying the constraint function. Thus, the approxima-

tion to the objective is defined as: h a

ÀÁ xi i = xki
Ck0̂ ðXÞ C0 Xk ∑in= 1 xki xki ∂C0
= + − 1 a ∂xi xi ð13:9Þ

αkj ≤ xj ≤ βjk

where n is the number of design variables, X is the vector of the design variables,
Ĉk0ðXÞ is the approximated compliance, k in the number of iteration, and αjk and βjk
are the move limits. The parameter a is an exponential parameter to ensure that the
approximation is convex and it is linked to the penalty parameter with a = − 2p,
and a is updated by increasing the penalization coefficient. Then, the constrained
optimization problem Eq. (13.5) is solved by using the Augmented Lagrangian

Method (ALM) [27], in the form:

LALM ðX, λm, rm, βw, rwÞ ! !2

= Ĉ0k ðXÞ + na gkm, iðxÞ, − λm, i na gkm, iðxÞ, − λm, i
2rm
∑ λm, imax + rm ∑ max 2rm

i=1 i=1

np np

+ ∑ βw, jgkw, jðxÞ + rw ∑ gkw, jðxÞ2

j=1 j=1

ð13:10Þ

where the rm and rw are penalty factors for the manufacturing and weight con-
straints, and na and np are the number of adjacency manufacturing constraints and
the number of patches. The λm and βw are the updating penalties for the manu-
facturing and weight constraints based on the estimation of the Lagrange multipliers
in the ALM scheme. The ALM function LALM is solved using L-BFGS-B [28] as
the optimization solver, which is a limited-memory quasi-Newton code for
bound-constrained optimization. The constraint functions are not approximated and
the exact calculation is taken in the SAO scheme. The optimization strategy used
here is not the complete dual method used in the classical SAO scheme such as the
MMA. The weight of design variables is initially taken to be equal to 1 n̸ c to avoid
any skewing towards the candidate material at the beginning of the optimization
process. The design variables are bounded by a move limits strategy, while the
penalization coefficient is continually increasing in the optimization process.

13 Design Optimization and Reliability Analysis … 251

13.4.1 Move Limit Strategy

A move limit strategy is used based on the MMA method to bound the values of the
design variables at each sub-problem used in the SAO scheme. As done in the the
MMA, Ujkand Ljk, the upper and lower asymptotes, are defined at the first iteration:

ð13:11Þ
Ljk = xjk − s xmj ax − xjmin


Ljk = xkj − s xmj ax − xmj in

0.1
s=

1+p

For k ≥ 2 these asymptotes are changed to: ð13:12Þ


Lkj = xjk − s xkj − 1 − Lkj − 1


Ujk = xkj + s Ujk − 1 − xkj − 1

To speed up the convergence, s is selected to be s = 0.7 if the sign of xkj − xkj − 1
and xkj − 1 − xkj − 2 are opposite and s = 1.2 if they have the same sign. The move
limits, αkj and βjk, are obtained as follows:

ð13:13Þ
αik = max xjmin, Ljk + μ xjk − Ljk


βik = min xmj ax, Ujk − μ Ujk − xkj

where μ is 0.5.

13.4.2 Penalty Continuation Scheme

In order to obtain the discrete material optimization, the penalization coefficient is
continually increasing during the optimization process. Therefore, a continuation
scheme is employed in the Discrete Material Optimization such that the penaliza-
tion coefficient is gradually increased from 1 to 4 in increments of 0.1 with ten
optimization iterations to penalize the intermediate densities and to slowly increase
the non-convexity of the problem during optimization process.

252 A. Sohouli et al.

13.5 Reliability Analysis

The probability of failure is usually predicted by using limit state function defined
as follows:

GðwðXÞ, YÞ = δall − δðwðXÞ, YÞ ð13:14Þ

where X is deterministic variables and Y is random variables. G(.) is the limit state
function which Gð.Þ > 0 indicates safe region. δall and δð.Þ are the allowable per-
formance variable and performance variable, respectively. The failure probability is
obtained by:

ZZ

Pf = Pr½GðwðXÞ, YÞ ≤ 0Š = . . . fY ðyÞdy ð13:15Þ

GðwðXÞ, YÞ ≤ 0

where fY ðyÞ is the joint probability density function (PDF) of random variables
Y and Pr[.] is the probability operator. Approximate methods have been proposed
to evaluate the integral in Eq. (13.15). Sampling methods, such as the Monte Carlo
simulation (MCS), importance sampling and Latin hypercube sampling, are used to
compute the above integral. Optimization-based methods such as the first order
reliability methods (FORM) are another proposed procedure to compute
Eq. (13.15). The original space of random variables is usually transformed to the
standard normal space to make it easier to approximate the probability of failure.
The standardized form of a normal variable Y is defined as:

U = Y − μ ð13:16Þ
σ

where μ and σ are the mean value and the standard deviation of the random
variables, respectively.

13.5.1 Monte Carlo Simulation (MCS) Approach

This approach relies on statistical analysis of the results for repeated random
sampling [29]. The MCS is applicable when an analytical solution is not obtainable
and it has been widely used in probability and reliability analysis due to simple
process and high precision. The probability of failure in MCS is calculated by:

Pf = Pr½GðwðXÞ, YÞ < 0Š = Nf ð13:17Þ
N

where N is the number of sampling points and Nf is the number of sampling points
inside the failure region.

13 Design Optimization and Reliability Analysis … 253

13.5.2 First Order Reliability Method (FORM)

The FORM-based analysis is represented as the first-order Taylor series expansion
of the limit-state function (Eq. 13.14) at the Most Probable Point (MPP) on the
limit-state surface [30].

The distance from the origin of the standard space to the closest point on the
transformed limit-state function indicates the reliability index. The point on the
surface of limit–state is the Most Probable Point (MPP). This point has the most
significant contribution to the nominal failure probability. The MPP is calculated
much like an optimization problem:

β = minðUT UÞ12; ð13:18Þ
UϵGðUÞ = 0

where U and GðUÞ are the random variables and limit-state function in the stan-
dardized normal space, respectively. The interpretation of reliability index, the
MPP, limit-state surface and failure space in the standardized normal space are
shown in Fig. 13.2.

MPP

Failure Space
True Limit-State Function
Linearized Limit-State Function

Fig. 13.2 The interpretation of reliability index, the MPP, limit-state surface and failure space in
the standardized normal space

254 A. Sohouli et al.

The probability of failure is defined by using the reliability index as follows:

Pf = Pr½GðwðXÞ, YÞ < 0Š = Φð − βÞ ð13:19Þ
where Φð.Þ is the standard normal cumulative distribution function.

13.5.3 Stochastic Response Surface Method

Response Surface Method (RSM) is used in structural reliability analysis as a
surrogate method to construct the limit state function, since the true limit state
function is usually implicit between the response and random variables. Therefore,
the limit state function is approximated to reduce the computational cost by the
simple functions at the neighborhood of the design points. In this work, the RSM
approximates the true limit state function by an explicit quadratic polynomial with
mixed terms as follows:

Gð̃ YÞ = ã + Nr 2 m cq Nr Yipiq ð13:20Þ

∑ ∑ bijYij + ∑ ∏

i=1 j=1 q=1 i=1

The equation is the second order formulation in [31]; where Gð̃ YÞ is the
approximated limit state function, Nr is the number of random variables, m is the
number of mixed terms. ã, bij and cq are obtained using the least square method.
Also, piq is determined as the power of a variable in a mixed term and it should not
be more than one, and the total order of the mixed terms should not be larger than

two. The sample points are chosen as follows:

Y = μ ± 3σ ð13:21Þ

13.6 Results and Discussion

A cantilever rectangle plate with dimensional ratio (length/width) of one and two
are investigated in this study. Different patches and different candidate numbers are
investigated to show the improvement of the structure stiffness. The finite element
analysis is carried out using ABAQUS© and, the DMO and the reliability analysis
and optimization processes are implemented in MATLAB©.

Two load cases are investigated and the stiffness of the structure is optimized
using the DMO approach. The first case is under uniform pressure and the second
case is under concentrated load. The simulation is carried out for single layer
material and for in-plane stress analysis. The material for the optimization analysis

13 Design Optimization and Reliability Analysis … 255

Fig. 13.3 The optimal fiber
orientations of the curved
fiber composite under uniform
pressure a Dimension ratio of
one, b Dimension ratio of two

is the unidirectional fiber carbon composite and the shifting orientation is consid-
ered one, two and three for 4, 8 and 12 candidate orientations, respectively.

The optimal fiber orientations for the curved fiber composite for the first case are
shown in Fig. 13.3. The optimal fiber orientation for the straight fiber composite is
45° for dimension ratio of one, while the optimal orientations for curved fiber
composite for 10 patches and 12 candidates are [−30°; −30°; −30°; −22.5°; 22.5°;
67.5°; 67.5°; 75°; 75°; 90°]. In case of a dimensional ratio of two, the optimal
orientation of the straight fiber composite is 0° and the optimal orientations of
curved fiber composite for 10 patches and 12 candidates are [0°; 0°; 0°; 0°; 0°; 0°;
45°; 67.5°; 67.5°; 75°].

The comparison of normalized compliance for the different number of patches

and candidates is given in Table 13.1. The compliance values are normalized by the
compliance value of the design with one patch or the optimal straight fiber com-
posite to show the structural improvement of the new design. Therefore, the lower

value represents a better structural performance. It can be seen that the structural

performance is improved by increasing the number patches and candidates. The
improvement of the plate for dimensional ratio of one is more significant (16%
improvement over dimensional ratio of two). The optimal fiber orientations for both

256 A. Sohouli et al.

Table 13.1 The comparison Dimension ratio 1 2
of normalized compliance of
the optimal designs of the Number of 4 8 12 4 8 12
plate under uniform pressure candidates

Number of 1 1.00 0.99 0.99 1 1 1
patches 2 0.97 0.97 0.97 0.98 0.90 0.98

4 0.95 0.95 0.90 0.97 0.98 0.98

10 0.95 0.86 0.84 0.96 0.96 0.96

dimensional ratios are shown in Fig. 13.3, where the discrete orientation of fibers in
each patch is converted to continuous tow paths as described earlier.

The optimal designs and improvement of compliance for the second load case
are shown in Fig. 13.4 and Table 13.2. The optimal orientations for the curved fiber
composite and the dimension ratio of one are [0°; 0°; 0°; 0°; −22.5°; −22.5°; −30°;
−45°; −60°; −90°], while the optimal fiber angle for the straight fiber is −45°. In
case of dimension ratio of two, the optimal orientations of the curved fiber com-
posite are [0°; 0°; 0°; 0°; 0°; 0°; 0°; −22.5°; −30°; −67.5°] and the optimal fiber
orientation of the straight fiber composite is 0°. In this case, it can be seen that the
improvement of compliance is more than the first load case.

The improvement of the compliance is increased by 37% for both dimensional
ratios for the concentrated load. It is noted that the curved fiber composites are more
effective for the case when concentrated stresses are present.

Fig. 13.4 The optimal fiber
orientations of the curved
fiber composite under
concentrated load
a Dimension ratio of one,
b Dimension ratio of two

13 Design Optimization and Reliability Analysis … 257

Table 13.2 The comparison Dimension ratio 1 2
of normalized compliance of
the optimal designs of the Number of 4 8 12 4 8 12
plate under concentrated load candidates

Number of 1 1.00 1.00 1.00 1 1 1
patches 2 0.89 0.77 0.72 0.86 0.86 0.86

4 0.68 0.63 0.63 0.74 0.74 0.74

10 0.68 0.63 0.61 0.64 0.62 0.62

13.6.1 Reliability Analysis Results

The reliability analysis of the second load case is carried out to compare the
probability of failure of curved and straight fiber composites with different standard
deviations to consider the effect of gaps and overlaps in the composite laminate.

The true limit state function for the optimal design, obtained using the Discrete
Material Optimization, is approximated using the RSM. In the next step, the reli-
ability analysis of the structure is estimated using either FORM or MCS. In this
study, two types of limit state function are investigated: the tip deflection and the
first-ply failure limit state.

The random variables are assumed to be independent and normally distributed
and these represent the material properties of carbon fiber reinforced polymer
composite. The length of plate for dimensional ratio of one is 0.5 m and the
thickness of plate is 0.1 mm. The applied load on the plate is 400 N.

13.6.1.1 Reliability Analysis Using Tip Deflection Limit State

The deflection at the tip is used for the first limit state function where the allowable
performance variable or allowable tip deflection is 1.4 and 2.75 mm for the
dimensional ratio of one and two, respectively. The limit state function is defined in
the form given by Eq. (13.14). The reliability analysis is performed using 3 random
variables given in Table 13.3 for different standard deviations of material proper-
ties. The random variables for the longitudinal modulus are normally distributed
and these are shown in Fig. 13.5.

As an example, the histogram of the tip deflection limit state function for
dimensional ratio of two and Coefficient of Variation (CV) of 0.2 is shown in
Fig. 13.6 and the kernel distribution fit is constructed for this histogram.

Table 13.3 The random variables in the analysis of the tip deflection limit state

No. Random variables Symbols Mean value Distribution types

1 Longitudinal modulus (Pa) E11 142.00 × 109 Normal
10.30 × 109 Normal
2 Transverse modulus (Pa) E22 Normal
7.20 × 109
3 In-plane shear modulus (Pa) G12

258 A. Sohouli et al.

Fig. 13.5 The histogram of
the longitudinal modulus as
one of the random variables

Fig. 13.6 The histogram of
the tip deflection limit state
function in the standardized
normal space at dimension
ratio of two and CV of 0.2

The failure probability of the optimal straight fiber composite is compared with
optimal curved fiber composite and it is shown in Fig. 13.7. It can be seen that
failure probability of the optimal curved fiber composite for the high standard
deviation (CV = 0.35) presents a lower probability of failure compared to the
straight fiber composite with the low standard deviation (CV = 0.1). Here, the
curvilinear and the straight fiber composites are assumed to have the same material
properties.

The sensitivities of reliability analysis are evaluated based on the correlation
coefficients between all random input variables and the random output parameter.
The sensitivity analysis is carried out using the Spearman rank coefficient of cor-
relation approach for this limit state. The comparison of sensitivity analysis of the
optimal curved fiber composite and the straight fiber composite is shown in
Fig. 13.8 for dimensional ratio of two and CV of 0.2.

13 Design Optimization and Reliability Analysis … 259

Fig. 13.7 Probability of Failure Probability 1 Optimal straight fiber- Ratio=2 0.35
Failure of deflection 0.8 Optimal curved fiber- Ratio=2
state-limit for different 0.6 Optimal straight fiber- Ratio=1
standard deviations 0.4 Optimal curved fiber- Ratio=1
0.2
0.15 0.2 0.25 0.3
0 Standard Deviation
0.1

Fig. 13.8 Spearman’s Rank 0.9 Curved Fiber
Order correlation coefficients 0.7 Straight Fiber
of the optimal curved fiber 0.5
composite for the tip 0.3 E22
deflection state limit at 0.1
dimension ratio of 2 and CV -0.1
of 0.2

E11 G12

These correlations explain the strength of the relationship between the stochastic
quantities and range from −1 to +1. A correlation of −1 describes perfect negative
relation and +1 a strong positive relation. A correlation close to 0 indicates no or
weak relation. It is apparent from Fig. 13.8 that the longitudinal stiffness E11 and
the shear modulus G12 have a strong positive relation with the tip deflection limit
state function.

The reliability analysis of the optimal curved fiber composite is carried out using
FORM for dimension ratio of two and CV of 0.2. The design point or Most
Probable Point (MPP) is given in Table 13.4 for this case.

13.6.1.2 Reliability Analysis Using First-Ply Failure Limit State

In this section, the reliability analysis of the plate under concentrated load is per-
formed using the first-ply failure limit state function. The limit state function is

Table 13.4 Comparison of failure probability and Most Probable Point (MPP) of the optimal
curved fiber for dimension ratio of two and CV of 0.2

Failure probability Most probable point (MPP)

MCS FORM E11 E22 G12
1.23 × 1011 9.95 × 109 6.04 × 109
0.12 0.15

260 A. Sohouli et al.

defined here by means of the Tasi-Wu [32] failure criterion using the following
equation:

GðYÞ = minðλÞ − 1 ð13:22Þ

where λ is the Tsai-Wu factor which can be evaluated as follows:

λ2 À + f22 σ22 + f66τ122 + 2f12 σ1σ2 Á + λðf1 σ1 + f2σ 2Þ − 1 = 0
f11σ21
,1 1 1 ,1 1 1 ,1
f11 = f1 = R1T − R1C , f22 = f2 = R2T − R2C , f66 = ð13:23Þ
R1T R1C R2T R2C R212

f12 ≅ − 1 ðf11f22Þ0.5
2

where σ1, σ2,τ12 are stress components referred to the system (1–2). R1T, R2T,
R1C and R2C are the ultimate tensile and compressive strength in longitudinal and
transverse direction and R12 is the shear strength. The Tsai-Wu factor is evaluated at
each element integration point and the composite is safe if the minimal Tsai-Wu

factor of all elements is bigger than a unit value. The number of random variables

for this limit state is 8 as shown in Table 13.5.
The histogram of the first ply failure limit state function for a dimensional ratio

of two and standard deviation of 0.2 is shown in Fig. 13.9. The histogram is shown

also with a Kernel estimator curve.
The comparison of failure probability of the optimal straight and curved fiber

composite is shown in Fig. 13.10. It can be seen from Fig. 13.10 the failure
probability of optimal curved fiber composite for both dimensional ratios is almost
zero, while the figures for the optimal straight fiber are close to one. It is noted that
the plate is optimized for the stiffness.

The design point of the optimal curved fiber composite is given in Table 13.6 for
the dimension ratio of 2 and the standard deviation of 0.2.

Table 13.5 The random variables in the analysis of the first-ply failure limit state

No. Random variables Symbols Mean value Distribution

types

1 Longitudinal modulus (Pa) E11 142.00 × 109 Normal
10.30 × 109 Normal
2 Transverse modulus (Pa) E22 7.20 × 109 Normal
2.00 × 109 Normal
3 In-plane shear modulus (Pa) G12 4.00 × 107 Normal
1.50 × 109 Normal
4 Ultimate longitudinal tensile strength (Pa) R1T

5 Ultimate transverse tensile strength (Pa) R2T

6 Ultimate longitudinal compressive R1C
strength (Pa)

7 Ultimate transverse compressive strength R2C 1.50 × 108 Normal
(Pa)

8 In-plane shear strength (Pa) R12 8.00 × 107 Normal

13 Design Optimization and Reliability Analysis … 261

Fig. 13.9 The histogram of
the first ply limit state
function in the standardized
normal space at dimension
ratio of two and CV of 0.2

Fig. 13.10 Probability of Failure Probability 1
failure of first-ply state limit 0.8
for different CV 0.6
0.4
0.2 Optimal straight fiber- Ratio=2 0.35
Optimal curved fiber- Ratio=2
0 Optimal curved fiber- Ratio=1
0.1 Optimal curved fiber- Ratio=1

0.15 0.2 0.25 0.3

Standard Deviation

The sensitivity analysis of the optimal curved fiber composite is compared with
the straight fiber composite for the dimensional ratio of two and CV of 0.2 and it is
shown in Fig. 13.11.

It can be observed in Fig. 13.11 that the in-plane shear strength, ultimate
transverse tensile and compressive strength have strong positive relations with the
first ply state limit function for both types of composite, whereas the ultimate
longitudinal tensile and compressive strength (material direction) have a very weak
relation. Longitudinal stiffness has a moderate role on the limit state function. It can
also be seen in Fig. 13.11 that the transverse modulus for straight fiber composite
has a very weak negative relation, while it has a moderate negative relation for the
curved fiber composite.

Table 13.6 Comparison of failure probability and Most Probable Point (MPP) of the optimal curved fiber for dimension ratio of two and standard deviation of 262 A. Sohouli et al.
0.2

Failure Most probable point (MPP)
probability
E11 E22 G12 R1T R2T R1C R2C R12
MCS FORM 1.34 × 1011 1.03 × 1010 7.71 × 109 2.00 × 109 1.53 × 109 4.04 × 107 1.53 × 108 3.59 × 107
0.004 0.003

13 Design Optimization and Reliability Analysis … 263

0.9
0.7 Curved fiber Straight fiber

0.5
0.3
0.1

-0.1

-0.3

-0.5 E11 E22 G12 R1T R1C R2T R2C R12

Fig. 13.11 Comparison of Spearman’s Rank Order correlation coefficients for the first ply state
limit at dimension ratio of 1 and CV of 0.2

13.7 Concluding Remarks

In this study, a computational framework was developed and applied to the design
optimization of curved fiber composites to maximize the stiffness of plates. The
rectangular plate was partitioned into small patches with manufacturing constraints

to ensure minimization of gaps and overlaps. These are common defects in variable
composite structures manufactured using the automated fiber placement techniques

It has been shown that the structural performance can significant improve when
using variable stiffness composites based on curvilinear fibers. The designs were
improved by increasing the number of patches and the curvilinear fibers are more
effective in the presence of stress concentrations.

The reliability of the optimized curvilinear fiber composite was compared to the
conventional straight fiber composite. The reliability analysis was carried out using
varying standard deviations to account for the effect of gaps and overlaps in the
composite laminate and the probability of failure of curved fiber was compared to
the conventional straight fiber composites.

The probability of failure was analyzed for the tip deflection limit state and it was
shown that the curved fiber composites are more reliable even with a high standard
deviation of random variables compared the straight fiber with a low standard
deviation.

For the first ply failure limit state, it was observed that the probability of failure
of the optimal curved fiber composite is also much less than the straight fiber
composite, while maximizing the stiffness of the structure. The optimal curvilinear
fibers composite laminates allow tailoring of the stiffness and the capability to better
adjust stress concentrations in composite structures.

264 A. Sohouli et al.

Acknowledgements The authors acknowledge the Graduate Fellowship from the Fundação para
a Ciencia e Tecnologia through the MIT-Portugal program and also the MITACS graduate visiting
student internship grant at the University of Victoria.

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Chapter 14

Robust Multi-objective Evolutionary
Optimization-Based Inverse Identification
of Three-Dimensional Elastic Behaviour
of Multilayer Unidirectional Fibre
Composites

Mohsen Hamdi and Ayech Benjeddou

Abstract This chapter focuses on mixed finite element (FE)-experimental fre-
quency based inverse identification of three-dimensional (3D) elastic behaviour of
multilayer unidirectional (UD) carbon fibre reinforced plastic (CFRP) composites
using a robust multi-objective evolutionary optimization procedure. This combines
numerical sensitivity analysis through FE design of experiments, response surfaces
methodology-based meta-modelling and a non-sorting genetic algorithm of second
generation. All identifiable 3D elastic behaviours are considered as well as
uncertainties of material properties. The sensitivity analyses show that the four
engineering constants that describe the two-dimensional (2D) elasticity are domi-
nant for the considered freely vibrating UD CFRP thin plate. Differently from its
manufacturer’s assumption as quasi-isotropic and its 2D inverse identification as
orthotropic, this sample’s 3D elastic behaviour is identified as transversely isotropic
before sensitivity analyses, but orthotropic after the latter, i.e. using the four
dominant in-plane engineering constants during the multi-objective optimization.

Abbreviations

2D Two-dimensional
3D Three-dimensional
ANOVA Analysis of variance
CFRP Carbon fibre reinforced plastic
DoE Design of experiments
err Error

M. Hamdi 267
Institut Supérieur des Sciences Appliquées et de Technologie,
2112 Gafsa-Zarroug, Tunisia

A. Benjeddou (✉)

Institut Supérieur de Mécanique de Paris, 3 rue Fernand Hainaut,
93400 Saint Ouen, France
e-mail: [email protected]

© Springer International Publishing Switzerland 2017
A.L. Araujo and C.A. Mota Soares (eds.), Smart Structures and Materials,
Computational Methods in Applied Sciences 43,
DOI 10.1007/978-3-319-44507-6_14

268 M. Hamdi and A. Benjeddou

FE Finite element
I Isotropic
MSE Mean square error as defined in (28b)
NSGA II Non-sorting genetic algorithm of second generation
orth Orthotropic
QI Quasi-isotropic
QTI Quasi-transversely isotropic
R2 Determination factor as defined in (28a)
RSM Response surface methodology
TI Transversely isotropic
UD Uni-directional

14.1 Introduction

Multilayer unidirectional (UD) carbon fibre reinforced polymer (CFRP) composites
are widely used in manufacturing lightweight, resistant and high performant sport,
space and aeronautic structures. Their design for safety is nowadays done using
finite element (FE) analysis that requires knowing material constants associated to a
chosen elastic behaviour. While two-dimensional (2D) models are desirable,
three-dimensional (3D) ones are often unavoidable. In this case, 3D material elastic
behaviour should be considered. However, composites’ manufacturers provide only
partial material data that are insufficient for conducting 3D FE simulations. Simi-
larly, only few material constants can be obtained directly from normalized static
tests and coupons. Therefore, various inverse identification techniques were
developed for characterizing elastic properties of composite structures [1] among
which the mixed FE-experimental vibration-based inverse identification technique
appears to be of practical choice [2].

In vibration-based inverse identification technique, frequencies are the most used
as inputs [3–9] although modal shapes can be additionally considered for enhancing
the identification’s accuracy of transverse shear moduli and in-plane Poisson’s ratio
[10, 11]. The corresponding optimization problem may be solved using nonlinear
mathematical programming [3–6], gradient-based [7–11] or hybrid
genetic-simulated annealing [12] algorithms. Design of experiments (DoE) and
response surface methodology (RSM) have been also proposed to solve the inverse
identification problem [7, 8]; in this case, the FE analysis is performed only for the
DoE reference points. Different types of error norms were used within single-
objective optimization-based elastic parameters identification; this includes non-
weighted sum of squared relative deviations of first few measured and scaled FE
squared radial frequencies [3, 7], weighted sum of squared relative deviations of
first few measured and FE eigenvalues [4–6], non-weighted sum of squared relative
deviations of first few measured and FE squared frequencies [8], deviations of first

14 Robust Multi-objective Evolutionary Optimization-Based Inverse … 269

few measured and FE normalized frequencies [9], relative deviations of first few
measured and FE radial frequencies [10, 11] and non-weighted sum of squared
relative deviations of first few measured and FE frequencies [12]. Thus,
multi-objective optimization was not considered.

Since almost all above mentioned contributions used two-dimensional (2D)

shear resistant plate FE models under plane-stress assumption, the outputs of the
identification are the corresponding few engineering constants [3–8, 10, 11]. The
rare contributions that have considered 3D FE models have even reduced their
identifications to fewer outputs. Indeed, focus has been made in [9] on the iden-
tification of in-plane Young’s and shear moduli (only three constants) using a 3D

®

FE model within the commercial code ABAQUS , while additional behaviour
symmetry and simplification assumption of all Poisson’s ratios equality have been
considered in [12] so that only the in-plane four engineering constants can be used
for 3D FE simulations within the commercial code ANSYS®. Thus, the 3D full set
of engineering constants, of number nine, six or five for orthotropic (orth),
quasi-transversely isotropic (QTI) or transversely isotropic (TI) composites, has not
been identified.

The mixed numerical-experimental frequency-based identification technique
cumulates errors originating from modal testing and FE simulations. To alleviate
effects of experimental frequencies errors on the identification outputs, weights can
be used in the optimization single-objective functions either for first frequency
model-test updating [3] or for expressing confidence levels in experimental fre-
quencies [4–6]. Alternatively, in [10, 11], ±30% variation on each elastic property
of the assumed orthotropic material has been taken into account in the test database.

Similarly, 1% deviation of the experimental frequency data was assumed in [13]

within a Bayesian framework which normally considers the input experimental

frequencies uncertain [14]. Two-steps procedures have been also suggested to cope

with missing modes or modes order [12] and to consider the different sensitivities of
the Young’s/shear moduli and Poisson’s ratios to frequency variations [9]. On the
other hand, model errors are of various types (geometrical simplifications,
misalignment of lamina angles from considered stacking sequence, kinematics
order, etc.) and have significant influence on input numerical data and identification
material constants outputs [8]. They have then to be reduced as possible.

As discussed above, the common practice in vibration-based mixed
numerical-experimental inverse identification appears to a priori assume a com-
posite material elastic behaviour and an equivalent single-layer shear plate model so
that a reduced number of engineering constants can be identified; that is, the focus
is on the identification of four material parameters for classical laminate theory and
five or six engineering constants for shear deformation ones depending, respec-
tively, on the assumed composite transversely isotropic [3, 7, 12] or orthotropic
[4–11, 13, 14] material symmetry. Thus, composites’ manufacturers assumed
quasi-transversely isotropic elastic behaviour was not considered.

With reference to above literature review, this chapter focuses not on composites

elastic material constants characterization but on their actual (none a priori
assumed) elastic behaviour frequency-based mixed numerical-experimental inverse

270 M. Hamdi and A. Benjeddou

identification using a multi-objective evolutionary optimization procedure [15, 16].
In order to minimize experimental and modelling errors, multilayer 3D FE simu-
lation and DoE based on complete factorial plans that consider usually observed
engineering constants measurements errors margins of ±20% variation on the
nominal (initial) design variables of all identifiable elastic behaviours (orth, QTI,
TI) are considered. For reducing computational cost, a priori sensitivity analyses are
conducted in order to explore retaining only the most influent design parameters
and first-order polynomial (also used in [8]) RSM-based numerical frequency meta-

®

models are used instead of the ANSYS 3D FE simulations within the retained non-
sorting genetic algorithm of second generation (NSGA II) for the evolutionary
multi-objective optimization of the frequency error norm; the latter is here defined
as the relative deviation of the meta-model estimated and experimental frequencies.
The most common form of RSM meta-models is a priori known low (first or
second)-order polynomials which coefficients have to be determined (tuned) using
nonlinear least square fitting [7]. However, unknown approximation shape func-
tions can be also considered, but this requires specific softwares [7] or algorithms.
Here, the first eight modes of a freely vibrating 16-plies laminated CFRP composite
plate of symmetric stacking sequence [90/45/0/-45]2S [17] are retained. A higher
modes number generally enhances the transverse shear moduli sensitivity to fre-
quency, thus enhancing their identification accuracy, but this is limited by modal
testing excitation and sensing used equipments. Thus, in [3–6], this number ranges
from 6 to 13 modes, while in [7, 8], 10 and 16 or 19 modes were used, respectively.
Besides, in [9–11], the modes number ranges from 7 to 16, whereas in [12, 13], the
retained number ranges from 7 to 14.

In the following, identifiable 3D elastic behaviours of composites are first
introduced; then, the robust multi-objective combined RSM-based meta-models and
evolutionary (NSGA II) optimization-based inverse identification methodology is
briefly presented; next, the latter is applied for inversely identifying the elastic
behaviour of a multilayer UD CFRP composite plate without and with considering
the DoE-based a priori sensitivity analyses results. Summary conclusions are finally
given as a closer of this chapter.

14.2 Identifiable 3D Elastic Behaviours of Composites

The 3D elastic behaviour of a composite is theoretically anisotropic; however, in
practice this behaviour is orthotropic and is described by nine engineering constants
(3 Young’s moduli E1, E2, E3; 3 shear moduli G12, G13, G23; 3 major ν12, ν13, ν23 or
3 minor ν21, ν31, ν32 Poisson’s ratios) which are defined in the following compli-
ance elastic matrix [S], of components Spq (p, q = 1, …, 6), that relates the lin-
earized Green-Lagrange strain vector {ε}, of components εp, to the Cauchy stress
one {σ}, of components σq, via this generalized Hooks law that is written in the
standard Voigt two-indices matrix notation

14 Robust Multi-objective Evolutionary Optimization-Based Inverse … 271

8 92 1 E̸ 1 38 9
<>>>>>> ε1 =>>>>>> − ν12 E̸ 1 − ν21 E̸ 2 − ν31 E̸ 3 5777777>>>>>>>>>><>>: σ1 >>>>>>=
>>>>>>: ε2 >>>>>;> = 4666666 − ν13 E̸ 1 1 E̸ 2 − ν32 E̸ 3 σ2 >>>>;>>
ε3 σ3
ε4 − ν23 E̸ 2 1 E̸ 3 σ4
ε5 σ5
ε6 1 G̸ 23 σ6

1 G̸ 13

1 G̸ 12

ð14:1Þ

The symmetry of the compliance matrix provides this relation between the
major, νij (i ≠ j; i, j = 1, 2, 3), and, minor νji, Poisson’s ratios

νij = νji ð14:2Þ
Ei Ej

Besides, the material stability requires these constraints on the engineering
constants

E1, E2,pE3ffiffi,ffiffiGffiffiffi1ffiffi2ffiffi, G13, G23p≻ffiffi0ffiffiffi,ffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi
jν12j ≺ E1 E̸ 2, jν13j ≺ E1 E̸ 3, jν23j ≺ E2 E̸ 3 ð14:3Þ

1 − ν12ν21 − ν23ν32 − ν13ν31 − 2ν21ν32ν13 ≻ 0

Combining (14.2) with Poisson’s ratios constraints in second line of (14.3)
provides, alternatively to the latter, these restrictions on the minor Poisson’s ratios

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi
jν21j ≺ E2 E̸ 1, jν31j ≺ E3 E̸ 1, jν32j ≺ E3 E̸ 2 ð14:4Þ

The orthotropic elastic behaviour of a composite material is then here defined by
the following xorth set of nine engineering constants subjected to restrictions (14.3)

xorth = ½E1, E2, E3, G12, G13, G23, ν12, ν13, ν23Š ð14:5Þ

A special sub-class of orthotropy is the quasi-transversely isotropy which pre-
sents symmetry of UD material directions 2 and 3 that constitute a plane of isotropy
so that these symmetry relations between the engineering constants occur

E3 = E2, G13 = G12, ν13 = ν12 ð14:6Þ
Consequently, the following symmetry relations also hold

ν31 = ν21, ν32 = ν23 ð14:7Þ

Therefore, the quasi-transversely isotropic elastic behaviour of a composite
material is here defined by the following xQTI set of six engineering constants

272 M. Hamdi and A. Benjeddou

xQTI = ½E1, E2, G12, G23, ν12, ν23Š ð14:8Þ

Subjected to restrictions (14.3) and (14.4) which simplify thanks to (14.2), (14.6)
and (14.7), respectively, to the following constraints on the identified engineering
constants

E1, E2,pG1ffiffi2ffiffi,ffiffiGffiffiffiffi2ffi3 ≻ 0, ð14:9Þ
jν12j ≺ E1 E̸ 2 , jν23j ≺ 1 ð14:10Þ
1 − 2ν12ν21 − ðν23Þ2 − 2ν21ν12ν23 ≻ 0

pffiffiffiffiffiffiffiffiffiffiffi
jν21j ≺ E2 E̸ 1

Further, a particular case of the QTI elastic behaviour is the transversely iso-
tropic one which is characterized by this additional constraint that relates the
transverse shear modulus G23 and Poisson’s ratio ν23

G23 = E2 ð14:11Þ
2ð1 + ν23Þ

Hence, the transversely isotropic elastic behaviour of an UD composite material
is here defined by the following xTI set of five engineering constants

xTI = ½E1, E2, G12, ν12, ν23Š ð14:12Þ

Under the restrictions (14.9) and (14.10) where the moduli constraint line of
(14.9) simplifies, thanks to (14.11), to the following one

E1, E2, G12 ≻ 0 ð14:13Þ

Alternatively, the TI elastic behaviour of an UD composite material may be
defined by the engineering constants E1, E2, G12, ν12, G23 satisfying (14.9) and
(14.10). This is for example the choice made in commercial FE code ABAQUS®
but for an isotropic plane 1-2 (instead of the here considered 2-3 one).

An originality of the present approach is that the symmetry relations (14.6) and
(14.7) for the QTI elastic behaviour or (14.6) and (14.7) and constraint (14.11) for
the TI one are simply implemented in the corresponding complete factorial plans for
the DoE 3D FE simulations. Alternatively, these relations as well as the outputs
restrictions (14.9), (14.10) and (14.13) can be implemented in the optimization
algorithm (here the NSGA II) leading to a constrained optimization procedure.
Here, restrictions (14.3), (14.9) and (14.13) on multi-objective optimization outputs
are simply verified a posteriori.

Worthy to notice also that there is a common confusion about the quasi-isotropic
(QI) elastic behaviour; indeed, this vocabulary introduced by composite manufac-
turers and adopted in some literature contributions is in fact the previously defined
QTI elastic behaviour. Indeed, theoretically, the QI elastic behaviour is a particular

14 Robust Multi-objective Evolutionary Optimization-Based Inverse … 273

case of the TI one; it is met when the UD composite material directions 1 and 2 are
equivalent so that they define an additional isotropic plane. In this case, indices can
be dropped and (14.12) reduces to this xQI three parameters set

xQI = ½E, G, νŠ ð14:14Þ

Subjected to the following restrictions that result from (14.9), (14.10) and
(14.13) after dropping corresponding material constants indices

E, G ≻ 0, ð14:15Þ

jνj ≺ 1
1 − 3ν2 − 2ν3 ≻ 0

Further, a particular case of the QI elastic behaviour is the classical isotropic
(I) one for which this constraint which links the shear modulus to Poisson’s ratio
holds similarly to (14.11)

E ð14:16Þ
G = 2ð1 + νÞ

Hence, (14.14) reduces to this xI two parameters set ð14:17Þ
xI = ½E, νŠ

Under restrictions (14.15) which moduli constraint simplifies, thanks to (14.16), to

E≻0 ð14:18Þ

The present work focuses only on the first three (orth, QTI and TI) elastic
behaviours inverse identification because the last two ones (QI and I) are not met in
the investigated UD composites.

For purpose of completeness and comparison with 2D plate/shell or
elasticity-based identifications, plane-stress reduced constitutive equations are also
given. Thus, when the normal transverse stress is nullified (σ3 = 0), relation (14.1)
reduces to

8 92 38 9
< ε1 = 1 E̸ 1 − ν21 E̸ 2 0 < σ1 =
;=4 1 E̸ 2 5
: ε2 − ν12 E̸ 1 0 0 : σ2 ; ð14:19Þ
ε6 0 G̸ 12 σ6
1

ε4 = ð1 G̸ 23Þσ4, ε5 = ð1 G̸ 13Þσ5 ð14:20Þ

where, the normal transverse strain ε3 = ð − ν13 E̸ 1Þσ1 + ð − ν23 E̸ 2Þσ2 is not nil but
its product with the dual normal transverse stress component is nil since σ3 = 0;
hence, it does not intervene in the strain energy of the problem formulation.

274 M. Hamdi and A. Benjeddou

Relations (14.19) and (14.20) can be used for defining the 2D composite
shear-resistant plate (SP) elastic behaviour which is characterized by a six
parameters set

xSP = ½E1, E2, G12, G13, G23, ν12Š ð14:21Þ

Under these stability constraints that can be derived from first two ones of (14.3)
as

E1, E2, pG1ffiffi2ffiffi,ffiffiGffiffiffi1ffiffi3, G23 ≻ 0, ð14:22Þ
jν12j ≺ E1 E̸ 2

However, for 2D classical (shear-less) laminated plate/shell theory or plane
elasticity (PE), the elastic behaviour is described only by the in-plane constitutive
Eq. (14.19) which are characterized by this four parameters set

xPE = ½E1, E2, G12, ν12Š ð14:23Þ
Under restrictions (14.22) which first line reduces to

E1, E2, G12 ≻ 0 ð14:24Þ

It’s worthy to notice that the 2D elastic engineering constants restrictions (14.22)
or (14.24) were rarely considered in the analysed literature. Nevertheless, alterna-
tively to (14.22), these inequalities were formulated in [3, 7] as constraints of the
error minimization problem

E1 E̸ 2 ≻p1ffiffi,ffiffiffiGffiffiffiffi1ffi2ffi E̸ 1 ≻ 0, G13 E̸ 1 ≻ 0, G23 E̸ 1 ≻ 0 ð14:25Þ
jν12j − E1 E̸ 2 ≺ 0

From which it can be remarked that the four last inequalities are equivalent to the

corresponding ones in (14.22), provided that E1 > 0 (missing constraint), while the
first one has no similar in (14.22). Nevertheless, physically this additional constraint
makes sense since the Young’s modulus along the fibre direction is usually greater
than that in the perpendicular direction. Also, above missing restriction was added
as E1 E̸ 01 ≻ 0 in [5] in term of the longitudinal Young’s modulus initial guess E01.

It is well known that FE formulations implement the stiffness matrix [C] rather

than the compliance one. Besides, under plane-stress assumption, the former is

reduced so that its non-nil components are after inversing (14.19) and (14.20)

C̄ 11 = E1 ð̸ 1 − ν12ν21Þ, C̄ 22 = E2 ð̸ 1 − ν12ν21Þ, C̄ 12 = ν12C̄ 22 ð14:26Þ
C̄ 44 = G23, C̄ 55 = G13, C̄ 66 = G12

14 Robust Multi-objective Evolutionary Optimization-Based Inverse … 275

Which are sometimes used as design variables of the inverse identification [7,
13]. Hence, engineering constants of (14.21) are obtained only after
post-processing.

In [12], while moduli restrictions, as in (14.24), have been well observed, the
Poisson’s ratio one, as in the second line of (14.22), was incorrectly considered since it
was considered only positive. Besides, engineering constants were defined from
effective stiffness parameters of a plate modelled using classical lamination theory

EEG1212===111222dddeetteððtDDðDÞÞÞÀÀ̸̸ DDÀ̸ D12121DD1D666622−−−ÀDDD21226621ÁÁ2hhÁh33,,3, ð14:27Þ

ν12 = ðD12D66 − D16D26Þ ̸ D22D66 − D226 Á

where, h is the plate thickness and det(D) is the determinant of the effective bending
stiffness matrix [D].

14.3 Robust Multi-objective Evolutionary
Optimization-Based Inverse Identification
Methodology

Following the authors earlier researches on smart structures electromechanical
updating [15] and bonded patch effective piezoelectric/dielectric behaviour identifi-
cation [16], the robust multi-objective optimization-based mixed numerical-
experimental frequency-based inverse identification shown in Fig. 14.1 is used for
each elastic behaviour (orth, QTI, TI) in order to determine the corresponding
engineering constants before and after a sensitivity analysis assessing their variations
effects on the first eight frequencies of a free multilayer UD composite plate.

The proposed methodology first step is the estimation of the design parameters set
initial guess. Here, in order to accelerate the convergence, the initial engineering
constants 3D values are completed from 2D inversely identified ones for a similar
composite material plate coupon [18] using TI symmetry relations (14.6) for the
missing constants (transverse Young’s modulus and Poisson’s ratios); for simplicity,
this choice is kept the same for the three identifiable elastic behaviours. The second
step is the ANSYS® 3D FE simulations of the DoE defined by complete factorial plans
of size mn, where n is the number of design variables (here the 9, 6 or 5 engineering
constants of orth, QTI or TI elastic behaviour, respectively) and m is their number of
levels (here 2, corresponding to ±20% engineering constants uncertainties usually
observed from composite materials datasheets). The third step is the optional
numerical sensitivity analysis that aims to assess the elastic parameters variations
effects on the considered frequencies using the classical analysis of variance
(ANOVA) statistical approach. Therefore, for a design variable experiment, a given
parameter’s effect results from the scalar cross product of mn-sized vectors containing

276 M. Hamdi and A. Benjeddou

Elastic behaviour engineering constants
set xorth,QTI,TI initial guess

DoE 3D FE simulated
frequencies f num

Sensitivity analysis of f num
to engineering constants

RSM meta-modelling estimated frequencies f mme

Experimental ε εmin
frequencies f exp

Error norms ε (f ,mme f )exp
minimization by NSGA II

multi-objective optimization

ε ≤ εmin
Elastic behaviour engineering constants

set xorth,QTI,TI identified

Fig. 14.1 Elastic behaviour robust multi-objective inverse identification methodology

the non-dimensional design variables values (−1, 1) and corresponding design
experiment frequency values (in Hz); for all sensitivity analyses, the parameters

effects are then in Hz. This step allows eventually reducing the number of design

variables by keeping only the most influent ones, hence reducing the size of the DoE

factorial plans and corresponding 3D FE simulations. The fourth step uses the latter
for the considered frequencies meta-modelling (estimations) via a first-order poly-
nomial RSM; the resulting frequency meta-models are validated using the following
statistical determination factor (R2) that measures the models fit goodness and should
be as close as possible to unity (0 < R2 ≤ 1) and mean square error (MSE) that
should not overpass 0.09028 value in order to satisfy the six sigma criterion

N N

∑ ðyi − yî Þ2 ∑ ðyi − yî Þ2

R2 = 1 − i=1 , MSE = i=1 × σ2yi ð14:28a; bÞ

N ðyi − yī Þ2 yī

∑

i=1

where, N is the number of measures (simulations); yi, yî , yī are the exact response,
estimated one, and mean of the exact responses, while σyi is the exact responses
standard deviation.

14 Robust Multi-objective Evolutionary Optimization-Based Inverse … 277

Population initialization: Gen = 0

Front = 1

Populations classified? No

Yes Non-dominated individuals

Gen+1 Frequency RSM meta-models

Fitness

Selection, crossing, mutation Sharing
Front+1
Yes
Gen <max ?
No
Stop

Fig. 14.2 NSGA II and frequency RSM meta-models combination

The fifth step is the minimisation of the experimental (taken from [17]) and
estimated (meta-modelled) eight frequencies error (relative deviation) norms with
the NSGA II shown in Fig. 14.2 and having parametrically fixed characteristics of
selection probability Ps = 0.8, crossing probability Pc = 0.5, mutation probability
Pm = 0.01, initial population size = 30, and generation number = 50; its conver-
gence provides the considered elastic behaviour engineering constants set as
multi-objective optimization outputs.

The above presented robust evolutionary multi-objective frequency-based
inverse identification is conducted for the identifiable three (orth, QTI, TI) beha-
viours; then, the resulting engineering constants are used as inputs for final 3D FE
simulations; next, the residual errors of the resulting numerical frequencies with
reference to the experimental ones [17] are evaluated and compared in order to
identify the elastic behaviour of the investigated multilayer UD CFRP composite
plate. The retained elastic behaviour is that corresponding to the minimum opti-
mized residual errors for maximum number of the considered eight frequencies.
This is also the way how to retain a single optimal solution from multiple ones.

278 M. Hamdi and A. Benjeddou

14.4 Multilayer UD CFRP Composite Plate Elastic
Behaviour Inverse Identification

The previously outlined elastic behaviour robust evolutionary multi-objective

optimization-based mixed numerical-experimental frequency-based inverse identi-
fication methodology is now applied to a free multilayer UD CFRP composite

coupon, denoted plate C; the latter was cut by its manufacturer, with nominal
in-plane dimensions of 200 × 300 mm2, from a larger panel having dimensions of
1560 × 1060 × 4.16 mm3 and mass density of 1512 kg/m3. However, its labo-
ratory measurements provided 3D dimensions of 200.3 × 300 × 4.2 mm3 and
mass density of 1521 kg/m3. This geometry and mass variability was also noticed

for two other coupons (cut from the same panel), denoted plate A and plate B and of

same in-plane nominal dimensions as plate C, since they have measured average
thickness of 4.10 and 4.13 mm and mass density of 1537 and 1543 kg/m3 [6],

respectively. The manufacturer indicated that the panel is made of IMS/977-2

epoxy-based CFRP composite material of 16 plies symmetric stacking sequence of

[90/45/0/45]2S (0° angle refers to the width/x-axis direction) and having these ply
elastic properties (supposed to be those of a QI elastic behaviour [6, 18])

E1 = 160 GPa, E2 = 8.60 GPa, G12 = 4.85 GPa, ν12 = 0.32 ð14:29Þ
E3 = E2, G13 = G12, ν13 = ν12
G23 = ð2 3̸ Þ * G12, ν23 = 0.40 (estimation)

It can be noticed that (14.29) mid-line symmetry relations are those of a QTI
elastic behaviour as defined in (14.6); however, using the (14.29) last line relation
provides G23 = 3.23 GPa, while the TI symmetry relation given in (14.11) gives a
4.95% lower value of G23 = 3.07 GPa. Hence, this acceptable relative difference
suggests that the multilayer UD epoxy-based CFRP composite panel has a rather TI
elastic behaviour than the assumed QI one [6, 18].

The manufacturer data (14.29) were used for a preliminary 3D FE analysis using
ANSYS® with a 2400 elements (17303 nodes) regular mesh of 40 × 60 × 1
SOLID 191 quadratic (20 nodes) elements along the plate C width, length and
thickness. The resulting first eight numerical frequencies beyond those of rigid
body modes are given in Table 14.1 together with experimental ones for the three
plates. Table 14.1 indicates that the manufacturer material data (14.29) are not
satisfactory as the minimum error with reference to test [17] of plate C is beyond
9%. This motivates the present interest for inversely identifying this plate elastic
behaviour.

It’s worthy to mention that experimental modal analyses of plates A and B used
small hammer exciter and scanning laser sensor [6, 18], while those of plate C
(present) used hammer exciter and point laser sensor [17, 19]. Thus, the results
differences between the three plates are mainly due to above mentioned dimensions
and mass density variability. Besides, all plates benefitted from two test campaigns
although only one results’ set is available for plate A which inverse identification,

14 Robust Multi-objective Evolutionary Optimization-Based Inverse … 279

Table 14.1 First eight frequencies (Hz) of free multilayer UD CFRP epoxy-based plates

Plate Analysis 1 2 3 4 5 6 7 8
A Test [18] 230.8 316.0 558.7 580.5 725.1 900.4 1075.9 1173.0
B Test [18] 229.4 311.9 556.2 581.9 725.6 893.1 1072.5 1171.2
Test [6] 229.4 312.5 557.2 582.5 726.3 893.4 1073.4 1170.6
C Test [19] 228.95 301.35 524.04 580.07 727.3 874.6 1070.3 1108
Test [17] 229.3 310.9 554.6 581.0 724.7 889.1 1068.4 1163.7
3D FE 250.86 348.85 610.38 637.41 796.24 992.85 1176.9 1289.3
(% err) (9.4) (12.21) (10.06) (9.71) (9.87) (11.67) (10.16) (10.76)

Table 14.2 2D identified engineering constants of plates A and B and initial guess for plate C

Plate Moduli E1 E2 E3 G12 G13 G23 ν12 ν13 ν23
(GPa) (GPa) (GPa) (GPa) (GPa) (GPa) 0.58 − −
A Friedmann
et al. [18] 136.2 6.9 − 6.1 4.1 2.2
Araújo et al.
[6] 136.2 7.3 − 6.0 5.2 1.0 0.55 − −

B Friedmann 129.8 10.9 − 5.7 4.4 3.2 0.37 − −
et al. [18]
Araújo et al. 130.8 10.6 − 5.6 4.2 3.0 0.36 − −
[6]
129.8 10.9 10.9 5.7 4.4 3.2 0.37 0.37 0.55
C Guess

as shown in Table 14.2, was judged unsatisfactory. Indeed, plates A and B test
results were used for gradient optimization-based mixed 2D plate FE-experimental
frequency (12 and 13 modes retained for plates A and B, respectively) plane-stress
engineering constants inverse identification [6, 18]. Therefore, as only plate B
elastic engineering constants were judged satisfactory, they were completed from
[18] using TI symmetry relations (14.6) and (14.11) (see last line of Table 14.2) for
their later use as design variables initial guess (nominal values) for the present
NSGA II optimization-based mixed 3D solid FE-experimental frequency (8 modes
retained) elastic behaviour inverse identification of present plate C.

Table 14.2 indicates that increasing modes number from 12 [18] to 13 [6] for 2D
inverse identification of plate A leads to different properties (see bold values).
However, this was not the case for plate B which showed comparable results; this
motivated retaining its values in [18] as a basis for defining plate C initial guess.
Nevertheless, different plane-stress 2D engineering constants were obtained for
plates A and B (of different mass density and thickness) due to the sensitivity of the
FE-modal inverse identification to different experimental inputs, here frequencies
(see Table 14.1). It can be also observed from the obtained engineering constants of
Table 14.2 that plates A and B identified elastic behaviours are orthotropic while
they were assumed initially QI (correctly QTI as discussed earlier). This provides
another motivation for conducting plate C elastic behaviour inverse identification.

280 M. Hamdi and A. Benjeddou

Table 14.3 3D FE first eight frequencies (Hz) of free plate C with initially guessed material data

Mode 1 2 3 4 5 6 7 8
3D FE 235.3 319.6 569.82 595.3 741.7 913.3 1098.3 1198.5
Err (%)
2.6 2.8 2.7 2.5 2.3 2.7 2.8 3.0

Using the Plate C initial material data (Table 14.2) for another preliminary 3D
FE simulation, with the same model as above, provided results (see Appendix A for
mode shapes) shown in Table 14.3 that devise by 3 the test [17] -3D FE model
relative deviation (err ≤ 3%).

14.4.1 Identifiable Three-Dimensional Elastic Behaviours
Analyses

According to above Sect. 14.2, the plate C has three identifiable elastic behaviours
with orthotropic, quasi-transversely isotropic and transversely isotropic symmetry.
They are first analysed here prior to their respective sensitivity analyses. Thus, let’s
consider the orthotropic elastic behaviour which identification problem has 9 design
parameters (14.5) with two levels each as they vary in the design space of ±20%

around their nominal values given in last line of Table 14.2.
Considering first a complete factorial plan, the DoE has a size of 29 = 512 3D

FE simulations that are too numerous to be listed here. Then, applying a first-order
polynomial RSM for each frequency of beyond six rigid body eight first modes
provides the meta-model estimated frequencies listed in Table 14.4 together with
their validation statistical measures. As all determination factors R2 and MSE are,

respectively, close to unity and much less than threshold value of 0.09028, the eight

frequency meta-models can be considered as statistically validated. It can be

noticed also from Table 14.4 that seventh and eighth frequency meta-models do not
depend on the transverse Poisson’s ratio ν13, while their dependence on the in-plane
shear modulus G12 is much greater than for the other frequencies.

The multi-objective (i.e., multi-modal: here 8 modes, thus 8 objective functions)

minimization problem, to be solved using the NSGA II, for identifying the nine

parameters orthotropic elastic behaviour can be written as

Min εiðxÞjx ∈ = f mi meðxÞ − fei xp ; i = 1, ...,8 ð14:30Þ
exp
xorth f i

The NSGA II multi-objective optimized engineering constants of the orthotropic
elastic behaviour are listed in the second line of Table 14.5. After simple calcu-
lations, it is proved that restrictions (14.3) and (14.4) are fully satisfied a posteriori.
The corresponding 3D FE frequencies and their residual errors with reference to
[17] are listed in second and third lines of Table 14.6 which show considerable

14 Robust Multi-objective Evolutionary Optimization-Based Inverse … 281

Table 14.4 Orthotropic plate C RSM first-order polynomial meta-model estimated frequencies

First eight frequency (Hz) meta-models R2 MSE
0.9995 5.4162E-4
f1mme = 234.6292 + 17.5244E1 + 1.1206E2 + 0.0487E3 + 4.5914G12 0.9999 8.4960 E-5
+ 0.2187G13 + 0.1599G23 − 0.5862ν12 − 0.0071ν13 + 0.0222ν23 0.9997 3.4249 E-4
0.9996 3.7688 E-4
f2mme = 318.4446 + 28.7192E1 + 1.1900E2 + 0.0438E3 + 2.0555G12 0.9997 2.9947 E-4
+ 0.0797G13 + 0.0592G23 + 0.0417ν12 − 0.0054ν13 + 0.0188ν23 0.9998 2.0942 E-4
0.9998 2.2607 E-4
f3mme = 568.0805 + 44.2801E1 + 3.3158E2 + 0.0986E3 + 8.5144G12 0.9996 3.6097 E-4
+ 0.6417G13 + 0.4701G23 − 0.5302ν12 − 0.0138ν13 + 0.0446ν23

f4mme = 593.55 + 47.8775E1 + 8.3746E2 + 0.0697E3 + 3.158G12
+ 0.2227G13 + 0.1616G23 + 2.1108ν12 − 0.0103ν13 + 0.0323ν23

f5mme = 739.5024 + 58.7591E1 + 5.2962E2 + 0.1409E3 + 9.1079G12
+ 0.7538G13 + 0.5512G23 − 0.8074ν12 − 0.0196ν13 + 0.0639ν23

f6mme = 909.8253 + 80.4177E1 + 6.6211E2 + 0.064E3 + 3.294G12
+ 0.8467G13 + 0.629G23 + 3.1607ν12 − 0.0092ν13 + 0.0294ν23

fm7 me = 1094.5 + 86.2E1 + 8.3E2 + 0.2E3 + 13G12
+ 1.7G13 + 1.2G23 + 0.6ν12 + 0.1ν23

f8mme = 1194.7 + 95.1E1 + 6.5E2 + 0.2E3 + 15.1G12
+ 2.1G13 + 1.6G23 − 0.6ν12 + 0.1ν23

Table 14.5 Plate C NSGA II optimized 3D elastic behaviours engineering constants

Elastic E1 E2 E3 G12 G13 G23 ν12 ν13 ν23
behaviour (GPa) (GPa) (GPa) (GPa) (GPa) (GPa)
0.37 0.44 0.59
Orth 120.1 11.6 9.6 5.9 4.3 3.8 0.33 0.33 0.44
QTI 120.6 11.3 11.3 5.7 5.7 3.6 0.38 0.38 0.58
TI 121.04 11.51 11.51 5.68 5.68 3.64

enhanced test [17]-model correlations (residual err ≤ 0.44%) compared to those in
Table 14.3.

Let’s now consider the particular case of QTI elastic behaviour which identifi-
cation problem has 6 design parameters (14.8) with two levels each as they vary

also in the design space of ±20% around their nominal values given in last line of
Table 14.2. Considering first a complete factorial plan that includes the QTI
symmetry relations (14.6) as constraints, the DoE has now a reduced size of
26 = 64 3D FE simulations that are still numerous to be listed here. Then, applying
a first-order polynomial RSM for each frequency of the eight first modes beyond the
six rigid body ones, provides the meta-model estimated frequencies listed in

Table 14.7 together with their validation statistical measures. Again all determi-
nation factors R2 and MSE are, respectively, close to unity and much less than

threshold value of 0.09028 so that all frequency meta-models are statistically

validated.

282 M. Hamdi and A. Benjeddou

Table 14.6 3D FE first eight frequencies (Hz) of free plate C with optimized material data

Elastic 1 2 3 4 5 6 7 8
behaviour
Orth 229.9 309.54 555.93 580.42 723.18 885.91 1071.4 1168.3
Residual err (%) 0.26 −0.44 0.24 −0.10 −0.21 −0.36 0.28 0.40
QTI 309.65 578.84 723.21 885.18
Residual err (%) 229.84 −0.40 555.82 −0.37 −0.21 −0.44 1071.1 1169.0
TI 0.24 310.24 0.22 581.86 724.04 889.28 0.25 0.46
Residual err (%) −0.21 −0.09
229.77 556.39 0.15 0.02 1073.5 1170.7
0.20 0.32 0.48 0.60

Table 14.7 QTI plate C RSM first-order polynomial meta-model estimated frequencies

First eight frequency (Hz) meta-models R2 MSE
0.9995 4.6552 E-4
f1mme = 234.9675 + 17.5672E1 + 1.1859E2 + 4.8206G12 0.9999 7.8178 E-5
+ 0.1253G23 − 0.5834ν12 + 0.0369ν23 0.9997 2.6347 E-4
0.9996 3.5577 E-4
f2mme = 318.5919 + 28.7416E1 + 1.2462E2 + 2.1322G12 0.9998 2.3854 E-4
+ 0.0441G23 + 0.0456ν12 + 0.03ν23 0.9998 1.8335 E-4
0.9999 1.3514 E-4
f3mme = 569.0214 + 44.417E1 + 3.4573E2 + 9.1783G12 0.9998 2.2610 E-4
+ 0.362G23 − 0.522ν12 + 0.0723ν23

fm4 me = 593.9008 + 47.9383E1 + 8.4733E2 + 3.3711G12
+ 0.1192G23 + 2.1198ν13 + 0.0520ν23

f5mme = 740.632 + 58.9352E1 + 5.4842E2 + 9.8836G12
+ 0.4208G23 − 0.8008ν13 + 0.1042ν23

fm6 me = 910.9509 + 80.6772E1 + 6.7253E2 + 4.1075G12
+ 0.4659G23 + 3.1762ν12 + 0.0456ν23

f7mme = 1096.8 + 86.6E1 + 8.5E2 + 14.6G12
+ 0.9G23 + 0.6ν13 + 0.1ν23

fm8 me = 1197.7 + 95.6E1 + 6.8E2 + 17.3G12
+ 1.2G23 − 0.6ν12 + 0.1ν23

The multi-objective (multi-modal) minimization problem, to be solved using the
NSGA II, for identifying the six parameters QTI elastic behaviour is similar to
(14.30) and has the following expression

Min εiðxÞjx ∈ xQTI = f mme ðxÞ − f iexp ; i = 1, ...,8 ð14:31Þ
i f iexp

The NSGA II multi-objective optimized engineering constants of the QTI elastic
behaviour are listed in the third line of Table 14.5. After elementary calculations, it
is proved that restrictions (14.9) and (14.10) are fully satisfied a posteriori. Besides,
the corresponding 3D FE frequencies and their residual errors with reference to

14 Robust Multi-objective Evolutionary Optimization-Based Inverse … 283

[17], as listed in fourth and fifth lines of Table 14.6, show test [17] -model cor-
relations in the same range (err ≤ 0.46%) as in the earlier case.

Finally, consider the sub-class of TI elastic behaviour which identification
problem has 5 design parameters (14.12) with two levels each as they vary in the

same design space of ±20% around their nominal values given in last line of
Table 14.2. Considering first a complete factorial plan that includes the TI sym-
metry relations (14.6) and (14.11) as constraints, the DoE has now a further reduced
size of 25 = 32 3D FE simulations that are listed in Appendix B. Then, applying a
first-order polynomial RSM for each frequency of the eight first modes beyond the
six rigid body ones provides the meta-model estimated frequencies listed in

Table 14.8 together with their validation statistical measures. Here also, all deter-
mination factors R2 and MSE are, respectively, close to unity and much less than

threshold value of 0.09028 so that all frequency meta-models are statistically

validated.

The multi-objective (multi-modal) minimization problem, to be solved using the
NSGA II, for identifying the five parameters TI elastic behaviour is similar to
(14.30) and (14.31) and has the following expression

Min εiðxÞjx ∈ xTI = f mme ðxÞ − f ei xp ; i = 1, ...,8 ð14:32Þ
i
exp
f i

The NSGA II multi-objective optimized engineering constants of the TI elastic
behaviour are listed in the fourth line of Table 14.5. After few calculations, it is

Table 14.8 TI plate C RSM first-order polynomial meta-model estimated frequencies

First eight frequency (Hz) meta-models R2 MSE
0.9995 4.685 E-4
f1mme = 235.03 + 17.572E1 + 1.3125E2 + 4.8062G12 0.9999 7.35 E-5
− 0.58813ν12 − 0.013125ν23 0.9997 2.474 E-4
0.9994 3.347 E-4
f2mme = 318.61 + 28.746E1 + 1.2912E2 + 2.1281G12 0.9996 2.203 E-4
+ 0.045625ν12 + 0.015ν23 0.9998 1.55 E-4
0.9998 1.181 E-4
fm3 me = 569.21 + 44.448E1 + 3.8359E2 + 9.1459G12 0.9998 1.881 E-4
− 0.52219ν12 − 0.059062ν23

fm4 me = 593.96 + 47.952E1 + 8.5972E2 + 3.3553G12
+ 2.1203ν12 + 0.0096875ν23

fm5 me = 740.84 + 58.976E1 + 5.9225E2 + 9.8456G12
− 0.80125ν12 − 0.048125ν23

fm6 me = 911.19 + 80.734E1 + 7.2087E2 + 4.0525G12
+ 3.1806ν12 − 0.11937ν23

f7mme = 1097.3 + 86.745E1 + 9.4572E2 + 14.551G12
+ 0.57031ν12 − 0.21406ν23

fm8 me = 1198.3 + 95.681E1 + 8.05E2 + 17.194G12
− 0.625ν12 − 0.2875ν23

284 M. Hamdi and A. Benjeddou

proved that restrictions (14.9), (14.10) and (14.13) are satisfied a posteriori.
Besides, the corresponding 3D FE frequencies and their residual errors with ref-
erence to [17], as listed in sixth and seventh lines of Table 14.6, show better test
[17] -model correlations for four (in bold, 2 for orth or QTI) of the eight fre-
quencies. Hence, prior to sensitivity analyses, the plate elastic behaviour can be
identified as TI.

14.4.2 A Priori Sensitivity-Based Identifiable Behaviours
Analyses

In this sub-section, the plate C elastic behaviour is re-identified after a priori ana-
lysing retained eight frequencies sensitivity to engineering constants variations
of ± 20% with regards to nominal (initial) values. The aim is to investigate the
possibility to reduce the design variables, hence meta-models number to be used
within the NSGA II-based multi-objective optimization.

First, orthotropic elastic behaviour nine engineering constants effects on the plate
C beyond rigid body first eight frequencies are given in Fig. 14.3. The latter shows
that: (i) transverse parameters have no significant effects; (ii) in-plane and shear
moduli are the most influent, in particular the Young’s modulus along the fibre
direction; (iii) in-plane Poisson’s ratio has smaller effect but should be kept.

Therefore, considering a priori this sensitivity analysis result, the nine design
parameters of the orthotropic elastic behaviour are now reduced to the dominant
four in-plane ones while the others are maintained in their nominal values in the

Fig. 14.3 Orthotropic elastic behaviour engineering constants effects on the first eight frequencies

14 Robust Multi-objective Evolutionary Optimization-Based Inverse … 285

Table 14.9 Dominant constants-based orthotropic plate C meta-model estimated frequencies

First eight frequency (Hz) meta-models R2 MSE
0.9995 4.2193 E-4
f mme = 234.5325 + 17.5412E1 + 1.1412E2 + 4.5950G12 − 0.56ν12 0.9999 5.2770 E-5
1 0.9997 2.5553 E-4
0.9997 3.2642 E-4
f mme = 318.3456 + 28.7331E1 + 1.2069E2 + 2.0594G12 + 0.0631ν12 0.9998 2.0684 E-4
2 0.9998 1.7484 E-4
0.9998 1.4401 E-4
f mme = 567.9181 + 44.3219E1 + 3.3556E2 + 8.5244G12 − 0.4769ν12 0.9997 2.5791 E-4
3

f mme = 593.4081 + 47.9044E1 + 8.4056E2 + 3.1606G12 + 2.1481ν12
4

f mme = 739.2506 + 58.8194E1 + 5.3531E2 + 9.1206G12 − 0.7331ν12
5

f mme = 909.8025 + 80.4700E1 + 6.6500E2 + 3.2987G12 + 3.1963ν12
6

f mme = 1094.3 + 86.3E1 + 8.3E2 + 13G12 + 0.6ν12
7

f mme = 1194.6 + 95.2E1 + 6.6E2 + 15.1G12 − 0.5ν12
8

reduced DoE complete factorial plan (of size 24 = 16 instead of 512) that is pro-
vided in Appendix C. Consequently, corresponding frequency meta-models have
now the polynomials of Table 14.9. Using the latter within the NSGA II
multi-objective optimization provides the orthotropic dominant constants shown in
second line of Table 14.10 that result in frequencies and their residual errors
(slightly higher as err ≤ 0.45%), with regards to test [17] and using the nominal
values for non-optimized 3D engineering constants, given in second and third lines
of Table 14.11. After evaluating in-plane moduli ratio, it is proved that restrictions
(14.22) and (14.24) on optimized in-plane engineering constants are fully satisfied
a posteriori.

Second, QTI elastic behaviour six engineering constants effects on place C
beyond rigid body first eight frequencies are shown in Fig. 14.4. The same com-
ments as the earlier case can be made here; that is, the four in-plane and shear

Table 14.10 Plate C NSGA II optimized dominant engineering constants

Elastic E1 (GPa) E2 (GPa) G12 (GPa) ν12
behaviour
121.6 10.6 5.8 0.38
Orth 122.62 10.61 5.76 0.44
QTI 123.12 10.42 5.42 0.32
TI

Table 14.11 3D FE first eight frequencies (Hz) of free plate C with optimized dominant constants

Elastic 1 2 3 4 5 6 7 8
behaviour
Orth 229.97 310.52 556.15 579.43 723.15 887.52 1071.2 1168.9
Residual err (%) 0.29 −0.12 0.28 −0.27 −0.21 −0.18 0.26 0.45
QTI 311.88 583.5 725.85 894.46
Residual err (%) 230.38 −0.31 558.18 −0.43 −0.15 −0.6 1077.1 1174.1
TI 0.47 311.49 0.64 579.26 724.41 889.26 0.81 0.89
Residual err (%) −0.19 −0.29 −0.04
229.93 556.54 0.01 1072.4 1171.4
0.27 0.35 0.37 0.66

286 M. Hamdi and A. Benjeddou

Fig. 14.4 QTI elastic behaviour engineering constants effects on the first eight frequencies

moduli and Poisson’s ratio dominate also the QTI elastic behaviour-based plate C
response.

Therefore, similarly, considering a priori this sensitivity analysis result, the six
design parameters of the QTI elastic behaviour are also reduced to the dominant
in-plane four ones while the others are maintained in their nominal values in the
reduced DoE complete factorial plan (of size 24 = 16 instead of 64) that is not
provided here for space saving reason. The corresponding frequency meta-models
are listed in Table 14.12. Using the latter within the NSGA II multi-objective
optimization provides the QTI dominant constants shown in third line of
Table 14.10 that result in the frequencies and their residual errors (now higher as
err ≤ 0.89%), with regards to test [17] and using the nominal values for
non-optimized 3D engineering constants, given in fourth and fifth lines of
Table 14.11. After evaluating the in-plane moduli ratio, it is proved that restrictions

Table 14.12 Dominant constants-based QTI plate C meta-model estimated frequencies

First eight frequency (Hz) meta-models R2 MSE
0.9996 3.4725 E-4
f mme = 234.9119 + 17.5731E1 + 1.1719E2 + 4.8194G12 − 0.5406ν12 1 4.5605 E-5
1 0.9998 1.8673 E-4
0.9997 3.1918 E-4
f mme = 318.5394 + 28.7481E1 + 1.2344E2 + 2.1331G12 + 0.0831ν12 0.9998 1.5352 E-4
e 0.9998 1.6508 E-4
0.9999 7.7169 E-5
f mme = 568.9169 + 44.4344E1 + 3.4306E2 + 9.1744G12 − 0.4369ν12 0.9998 1.6149 E-4
3

f mme = 593.8206 + 47.9506E1 + 8.4544E2 + 3.3681G12 + 2.1794ν12
4

f mme = 740.4788 + 58.9600E1 + 5.4450E2 + 9.8775G12 − 0.6787ν12
5

f mme = 910.9056 + 80.6969E1 + 6.7119E2 + 4.0994G12 + 3.2356ν12
6

f mme = 1096.7 + 86.7E1 + 8.5E2 + 14.6G12 + 0.7ν12
7

f mme = 1197.5 + 95.6E1 + 6.7E2 + 17.3G12 − 0.5ν12
8

14 Robust Multi-objective Evolutionary Optimization-Based Inverse … 287

Fig. 14.5 TI elastic behaviour engineering constants effects on the first eight frequencies

(14.22) and (14.24) on optimized in-plane engineering constants are fully satisfied
a posteriori.

Third, TI elastic behaviour five engineering constants effects on plate C beyond
rigid body first eight frequencies are shown in Fig. 14.5. Same comments as for the
earlier two cases are valid here also; hence, the four in-plane and shear moduli and
Poisson’s ratio dominate the TI elastic behaviour-based plate C response.

Thus, considering a priori this sensitivity analysis result, the five design param-
eters of the TI elastic behaviour are reduced to dominant four in-plane ones while the
others are maintained in their nominal values in the reduced DoE complete factorial
plan (of size 24 = 16 instead of 32). The resulting frequency meta-models are listed
in Table 14.13. Their use within the NSGA II multi-objective optimization provides
the TI dominant constants shown in fourth line of Table 14.10 resulting in

Table 14.13 Dominant constants-based TI plate C meta-model estimated frequencies

First eight frequency (Hz) meta-models R2 MSE
0.9996 3.4738 E-4
f mme = 234.97 + 17.582E1 + 1.3037E2 + 4.8112G12 − 0.54125ν12 0.9999 4.3175 E-5
1 0.9998 1.7955 E-4
0.9997 3.0323 E-4
f mme = 318.56 + 28.752E1 + 1.2794E2 + 2.1294G12 + 0.081875ν12 0.9998 1.4191 E-4
2 0.9998 1.442 E-4
0.9999 6.7529 E-5
f mme = 569.07 + 44.46E1 + 3.81E2 + 9.15G12 − 0.435ν12 0.9999 1.3464 E-4
3

f mme = 593.87 + 47.961E1 + 8.5787E2 + 3.3575G12 + 2.1812ν12
4

f mme = 740.65 + 58.994E1 + 5.8812E2 + 9.85G12 − 0.67875ν12
5

f mme = 911.1 + 80.739E1 + 7.1912E2 + 4.055G12 + 3.235ν12
6

f mme = 1097.1 + 86.754E1 + 9.4287E2 + 14.554G12 + 0.72ν12
7

f mme = 1198 + 95.7E1 + 8E2 + 17.2G12 − 0.4625ν12
8

288 M. Hamdi and A. Benjeddou

frequencies and their residual errors (in the same range as err ≤ 0.66%), with
regards to test [17] and using the nominal values for non-optimized 3D engineering
constants, given in sixth and seventh lines of Table 14.11. After evaluating the
in-plane moduli ratio, it is proved that restrictions (14.22) and (14.24) on optimized
in-plane engineering constants are fully satisfied a posteriori.

Using Table 14.11 for comparing the dominant constants-based orth, QTI and TI
elastic behaviours, it is clear that the orthotropic behaviour provides better test [17]
-model correlations for five (in bold, 2 for TI but none for QTI) of the eight
frequencies. Hence, after sensitivity analyses, the plate elastic behaviour can be
identified as orthotropic. This four in-plane constants-based identification result is
coherent with 2D thin plate one.

14.5 Summary Conclusions

Mixed finite element (FE)-experimental frequency based inverse identification of
three-dimensional (3D) elastic behaviour of multilayer unidirectional (UD) carbon
fibre reinforced plastic (CFRP) composites is reached using a robust multi-objective
evolutionary optimization procedure. This combines numerical sensitivity analysis
through FE complete factorial plan design of experiments (DoE), first-order poly-
nomial response surfaces methodology (RSM)-based meta-modelling and a
non-sorting genetic algorithm of second generation (NSGA II). All identifiable 3D
elastic behaviours were considered as well as ±20% uncertainties of nominal
material properties. The sensitivity analyses showed that the four engineering
constants that describe the two-dimensional (2D) elasticity are dominant for the
considered freely vibrating UD CFRP multilayer composite thin plate. Differently
from its manufacturer’s assumption as quasi-isotropic and its 2D inverse identifi-
cation as orthotropic, this sample’s 3D elastic behaviour was identified a priori to
the sensitivity analysis as transversely isotropic. However, after the sensitivity
analysis, the elastic behaviour was identified as orthotropic in conformity with 2D
identifications. Both identified behaviours are to be considered as effective ones as
they were obtained through model-test correlation approach and as they represent
the behaviours of the composite plies after their adhesive stacking. That is, the
elastic behaviour inverse identification considers the bonding effect although the FE
simulations ignore it.

It’s worthy to stress that the presented approach contributes originally with the
integration of the identifiable elastic behaviours symmetry relations in the DoE,
hence in the RSM-based meta-models. Also, the restrictions on the identifiable
elastic behaviours optimized parameters were found to be fully satisfied although
resulting from non-constrained optimization. Another originality is that the inverse
identification is based on a robust evolutionary (NSGA II) multi-objective opti-
mization procedure. Finally, as the above identifiable elastic behaviours are for the
identical composite plies and not the whole (homogenized) multilayer composite,
using the dominant four parameters is enough since the former are always very thin.

14 Robust Multi-objective Evolutionary Optimization-Based Inverse … 289

Hence, there is no need to use two-steps identifications, either through separating
the moduli and Poisson’s ratios identifications or through using thin and thick
coupons to get accurate in-plane constants and transverse ones, respectively.

Acknowledgement The second author gratefully acknowledges the support of the Austrian
Comet K2 Linz Centre of Mechatronics (LCM).

Appendix A

The plate C eight modal shapes, computed using Table 14.2 nominal data, are
shown in Fig. 14.6, where the numbers between brackets indicate number of
zero-crossing along width (x-axis) and length (y-axis) directions.

Mode shape 1 (1,1) : 235.3 Hz Mode shape 2 (0, 2) : 319.6 Hz

Mode shape 3 (1, 2) : 569.82 Hz Mode shape 4 (2,0) : 595.3 Hz

Mode shape 5 (2,1) : 741.7 Hz Mode shape 6 (0,3) : 913.3 Hz

Mode shape 7 (1,3) : 1098.3 Hz Mode shape 8 (2, 2) : 1198.5 Hz

Fig. 14.6 Beyond rigid body first eight modal shapes of plate C (using Table 14.2 nominal data)

Table 14.14 DoE complete factorial plan (frequency in Hz) for TI elastic behavior analyses 290 M. Hamdi and A. Benjeddou

N° E1 E2 G12 ν12 ν23 F1 F2 F3 F4 F5 F6 F7 F8
665.37 815.54 984.51 1075.9
1 −1 −1 −1 −1 −1 211.3 286 511.18 530.99 785.21 979.13 1270.6
678.29 829.75 1160.2 1093.3
2 1 −1 −1 −1 −1 247.17 344.09 601.59 629.16 796.82 991.3 1004.2 1286.2
686.36 823.77 1178.8 1112.3
3 −1 1 −1 −1 −1 214.3 288.75 519.46 548.67 803.77 987.12 1015.1 1303.1
698.87 837.77 1188.3 1129.1
4 1 1 −1 −1 −1 249.91 346.4 609.22 644.18 814.91 998.83 1033.9 1317.7
664.85 821.52 1205.8 1075.9
5 −1 −1 1 −1 −1 221.56 290.47 530.62 538.2 784.82 983.85 1270.4
677.17 838.81 986.6 1093.3
6 1 −1 1 −1 −1 256.17 348 618.76 635.49 796.03 998.44 1162 1285.9
686.02 829.93 1006.7 1112.7
7 −1 1 1 −1 −1 224.44 293.22 538.66 555.7 803.48 991.95 1181.1 1303.1
698.01 847.11 1017.5 1129.6
8 1 1 1 −1 −1 258.79 350.28 626.06 650.33 814.27 1006.1 1190.2 1317.8
666.14 815.73 1036.8 1076.6
9 −1 −1 −1 1 −1 210.57 286.35 510.88 535.17 785.65 979.04 1208.3 1270.6
679.65 830.19 1094.6
10 1 −1 −1 1 −1 246.55 344.33 601.35 632.59 797.67 991.37 985.15 1286.6
687.18 824.05 1160.3 1113.1
11 −1 1 −1 1 −1 213.14 289.16 518.75 554.83 804.31 987.2 1005.5 1303.5
700.25 838.27 1179.3 1130.6
12 1 1 −1 1 −1 248.95 346.71 608.69 649.26 815.85 999.06 1015.9 1318.5
1188.6 (continued)
13 −1 −1 1 1 −1 220.91 290.96 530.5 542.39 1035.3
1206.6
14 1 −1 1 1 −1 255.61 348.32 618.62 638.93

15 −1 1 1 1 −1 223.39 293.84 538.21 561.9

16 1 1 1 1 −1 257.89 350.71 625.69 655.44

17 −1 −1 −1 −1 1 211.58 286.3 511.67 531.43

18 1 −1 −1 −1 1 247.35 344.3 601.85 629.46

19 −1 1 −1 −1 1 214.77 289.23 520.35 549.39

20 1 1 −1 −1 1 250.24 346.76 609.75 644.68

21 −1 −1 1 −1 1 221.85 290.76 531.15 538.66

22 1 −1 1 −1 1 256.38 348.22 619.1 635.82

23 −1 1 1 −1 1 224.91 293.68 539.56 556.43

24 1 1 1 −1 1 259.13 350.64 626.67 650.87

Table 14.14 (continued) 14 Robust Multi-objective Evolutionary Optimization-Based Inverse …

N° E1 E2 G12 ν12 ν23 F1 F2 F3 F4 F5 F6 F7 F8

25 −1 −1 −1 1 1 210.25 286.11 510.16 534.76 663.89 820.91 985.21 1074.3

26 1 −1 −1 1 1 246.24 344.1 600.61 632.18 783.84 983.09 1160.4 1268.5

27 −1 1 −1 1 1 212.7 288.81 517.76 554.25 675.84 838.01 1004.8 1091.1

28 1 1 −1 1 1 248.52 346.38 607.69 648.7 794.71 997.49 1179 1283.4

29 −1 −1 1 1 1 220.62 290.74 529.88 542.02 685.18 829.45 1016.3 1111.3

30 1 −1 1 1 1 255.33 348.12 618 638.58 802.65 991.39 1189 1301.6

31 −1 1 1 1 1 222.99 293.51 537.33 561.37 696.8 846.45 1035.2 1127.7

32 1 1 1 1 1 257.37 350.41 624.82 654.94 813.1 1005.4 1206.6 1315.7

291

292 M. Hamdi and A. Benjeddou

Appendix B

Table 14.14 lists the numerical DoE complete factorial plan of size 25 = 32 for the
TI behaviour sensitivity analysis and RSM meta-modelling. Notice that this
behaviour’s symmetry relations (14.6) and (14.11) are coded within the DoE.

Appendix C

Table 14.15 lists the DoE complete factorial plan of size 24 = 16 for dominant
constants-based orthotropic behaviour sensitivity analysis and RSM
meta-modelling.

Table 14.15 DoE complete factorial plan (frequency in Hz) for dominant constants orthotropic
behavior analyses

N° E1 E2 G12 ν12 F1 F2 F3 F4 F5 F6 F7 F8
1 −1 −1 −1 −1 211.31 286.01 511.19 531.00 665.39 815.55 984.55 1076.0
2 1 −1 −1 −1 247.17 344.09 601.61 629.17 785.24 979.15 1160.2 1270.7
3 −1 1 −1 −1 214.04 288.67 518.72 548.44 677.46 828.87 1002.4 1090.9
4 1 1 −1 −1 249.56 346.27 608.16 643.81 795.57 989.83 1176.0 1282.6
5 −1 −1 1 −1 221.18 290.36 529.55 537.90 685.13 822.59 1012.5 1108.8
6 1 −1 1 −1 255.65 347.81 617.23 634.99 801.94 985.11 1184.3 1298.0
7 −1 1 1 −1 223.85 293.05 536.99 555.22 696.99 835.95 1029.8 1123.8
8 1 1 1 −1 257.98 350.00 623.71 649.55 812.15 995.80 1199.8 1310.0
9 −1 −1 −1 1 210.32 286.13 510.38 534.82 664.13 821.20 985.78 1075.0
10 1 −1 −1 1 246.34 344.14 600.93 632.29 784.22 983.58 1161.3 1269.7
11 −1 1 −1 1 212.50 288.72 517.22 554.05 675.22 837.38 1003.5 1089.4
12 1 1 −1 1 248.26 346.26 606.94 648.40 793.81 996.47 1177.1 1281.0
13 −1 −1 1 1 220.29 290.63 528.94 541.73 684.08 828.40 1014.0 1108.2
14 1 −1 1 1 254.87 347.94 616.67 638.12 801.05 989.64 1185.5 1297.2
15 −1 1 1 1 222.44 293.33 535.78 560.87 695.05 844.72 1031.5 1122.8
16 1 1 1 1 256.76 350.12 622.67 654.17 810.58 1002.6 1201.1 1308.7

14 Robust Multi-objective Evolutionary Optimization-Based Inverse … 293

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