Computers and Structures 84 (2006) 2065–2080
www.elsevier.com/locate/compstruc
Multi-objective optimization of fiber reinforced composite laminates
for strength, stiffness and minimal mass
Jacob L. Pelletier, Senthil S. Vel
*
Mechanical Engineering Department, University of Maine, 5711 Boardman Hall, Orono, ME 04469, USA
Received 6 July 2005; accepted 1 June 2006
Available online 12 September 2006
Abstract
We present a methodology for the multi-objective optimization of laminated composite materials that is based on an integer-coded
genetic algorithm. The fiber orientations and fiber volume fractions of the laminae are chosen as the primary optimization variables.
Simplified micromechanics equations are used to estimate the stiffnesses and strength of each lamina using the fiber volume fraction
and material properties of the matrix and fibers. The lamina stresses for thin composite coupons subjected to force and/or moment resultants are determined using the classical lamination theory and the first-ply failure strength is computed using the Tsai–Wu failure criterion. A multi-objective genetic algorithm is used to obtain Pareto-optimal designs for two model problems having multiple,
conflicting, objectives. The objectives of the first model problem are to maximize the load carrying capacity and minimize the mass
of a graphite/epoxy laminate that is subjected to biaxial moments. In the second model problem, the objectives are to maximize the axial
and hoop rigidities and minimize the mass of a graphite/epoxy cylindrical pressure vessel subject to the constraint that the failure pressure be greater than a prescribed value.
2006 Elsevier Ltd. All rights reserved.
Keywords: Genetic algorithm; Combinatorial optimization; Laminated composite materials; Stacking sequence; Composite pressure vessel
1. Introduction
Fiber reinforced composites are extensively used
in many modern engineering applications due to their ability to improve structural performance. Examples include
lightweight, strong and rigid aircraft frames, composite
drive shafts and suspension components, sports equipment,
pressure vessels and high-speed flywheels with superior
energy storage capabilities. A reason for the widespread
use of laminated composite materials is their inherent tailorability, which enables them to meet specific design
objectives for a given application. Several techniques have
been reported in the literature for the optimization of
laminated composite materials. Venkataraman and Haftka
*
Corresponding author. Tel.: +1 207 581 2777; fax: +1 207 581 2379.
E-mail address: [email protected] (S.S. Vel).
0045-7949/$ - see front matter 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compstruc.2006.06.001
[1] describe genetic algorithms as the most popular method
for the combinatorial optimization of laminate stacking
sequence. Genetic algorithms (GAs) are contemporary
search techniques developed by Holland [2], that mimic
the evolutionary principles and chromosomal processing
in natural genetics. A GA begins its search with a population of random individuals. Each member of the population possesses a chromosome which encodes, in some
fashion, certain characteristics of the individual. In the
present case, an individual member of the population corresponds to a particular laminate design and its chromosome consists of the fiber orientations, fiber volume
fractions and lamina thicknesses. The algorithm systematically analyzes each individual in the population of designs
according to set specifications and assigns it a fitness rating
which reflects the designer’s goals. This fitness rating is
then used to identify the structural designs that perform
better than others, thereby enabling the genetic algorithm
2066
J.L. Pelletier, S.S. Vel / Computers and Structures 84 (2006) 2065–2080
to determine the designs that are weak and must be eliminated using the reproduction operator. The remaining,
more desirable genetic material is then utilized to create a
new population of individuals. This is performed by applying two more operators similar to natural genetic processes, namely gene crossover and gene mutation [3]. The
process is iterated over many generations in order to obtain
optimal designs. The evolutionary technique provides
major benefits over traditional gradient based optimization
routines, such as nominal insensitivity to problem complexity and the ability to seek out global rather than local
optima.
Several researchers have utilized GAs for the singleobjective optimization of composite laminates. For
example, Le Riche and Haftka [4] and Nagendra et al. [5]
optimized the stacking sequence of composite plates to
maximize the buckling load. A strategy for the optimal
design of composite grid-stiffened cylinders subjected to
global and local buckling constraints and strength constraints was proposed by Jaunky et al. [6]. Soremekun
et al. [7] utilized a GA with generalized elitist selection to
maximize the bending/twisting coupling of a laminated
cantilever beam. Rajendran and Vijayarangan [8] have
demonstrated that it is possible to obtain a significant
reduction in weight of a mono-leaf composite spring compared to a seven-leaf steel spring. Park et al. [9] analyzed
symmetric composite laminates using a shear deformation
theory and the Tsai–Hill failure criterion to obtain optimal
designs of symmetric composite laminates subject to various loading and boundary conditions. Messager et al.
[10] employed an analytical model of shell buckling coupled to a GA to determine optimized stacking sequences
for underwater cylindrical composite structures.
The design of practical composite structures often
require the maximization or minimization of multiple,
often conflicting, objectives. Many researchers have tackled complex multi-objective optimization problems by
scalarizing the multiple objective functions into a single
objective using a weight vector [11–14]. A disadvantage
of this approach is that the resulting optimal lamination
scheme depends on the chosen weight vector. In contrast,
the principles of true multi-objective optimization do not
require the specification of the relative importance of the
objective functions a priori. In general, a multi-objective
optimization algorithm yields a set of optimal solutions,
instead of a single optimal solution [15]. The reason for
the optimality of many solutions is that no one solution
can be considered better than any other with respect to
the objective functions. These optimal solutions are
known as Pareto-optimal solutions [16,15]. The primary
goals of a multi-criteria optimization algorithm are to
guide the search towards the global Pareto-optimal front
and to maintain population diversity in the Pareto-optimal solutions. Upon completion of the optimization
procedure, the designer can view the manner by which
the Pareto-optimal solutions are distributed in the
performance space, perform trade-off studies and choose
the most suitable solution based on higher level
information.
In regards to laminate design, Costa et al. [17] have
investigated the multi-objective optimization of laminates
with isotropic plies for compliance, cost, mass and thickness. In the present paper, an improved methodology for
the multi-objective optimization of fiber reinforced composite materials for strength, stiffness and minimal mass
via the layerwise tailoring of fiber orientations and fiber
volume fractions is described. The orthotropic stiffnesses
and strengths of each lamina are estimated using Chamis’
simplified micromechanics equations [18]. The classical
lamination theory is utilized to determine the lamina stresses for thin laminates subjected to force and/or moment
resultants and the first-ply failure load is obtained using
the Tsai–Wu failure criterion. An integer-coded multiobjective genetic algorithm, based on the elitist non-dominated sorting genetic algorithm (NSGA-II) [15,19], is
implemented to obtain Pareto-optimal designs for conflicting objectives. A novel feature of our proposed methodology is the incorporation of an automatic termination
criterion for multi-objective genetic algorithms. It keeps
track of the number of new designs that are added to a
historical archive of non-dominated individuals and
terminates the algorithm when it reaches the point of
diminishing returns. Numerical results are presented for
two model problems having various conflicting objectives.
In the first model problem, the aim is to maximize the failure load while minimizing the mass of a graphite/epoxy
laminate subjected to biaxial moments. The second model
problem explores the optimal design of a graphite/epoxy
cylindrical pressure vessel for the three objectives of mass,
hoop rigidity and axial rigidity, while subject to the constraint that the failure pressure be greater than a prescribed
value.
2. Formulation
A rectangular Cartesian coordinate system x, y and z is
used to describe the infinitesimal deformations of a N-layer
laminated composite material, shown in Fig. 1, in the
unstressed reference configuration. The total thickness of
the laminate is H and the bottom and top surfaces are
located at z = H/2 and H/2, respectively. Lamina n consists of a macroscopically homogeneous fiber-reinforced
composite material with fiber volume fraction V(n), extending from z(n1) to z(n) in the z-direction. The principal fiber
direction is oriented at an angle of /(n) to the x-axis. A layerwise principal material coordinate system, denoted by 1,
2 and 3, is employed to analyze lamina failure. The 1 and 2
axes are aligned parallel and perpendicular to the fiberdirection in the x–y plane, respectively, and the 3-axis is
aligned parallel to the global z-axis. In the present formulation, the fiber volume fractions V(n), the fiber orientations
/(n), the thicknesses h(n) = z(n) z(n1) of each lamina, and
the total number of laminae N, are treated as optimization
variables.
J.L. Pelletier, S.S. Vel / Computers and Structures 84 (2006) 2065–2080
2067
Ny
My
Myx
N yx
Mxy
z,3
Mx
2
Nx
N xy
y
φ
1
Mxy
Nx
x
Nxy
Mx
Nyx
My
Myx
Ny
H/2
z(N)
(n)
(n) z
h
z (n-1)
z (1)
z (0)
(N)
φ(N),V (N)
(n)
φ(n),V (n)
(2)
(1)
φ(1),V (1)
..
.
..
.
0
-H/2
Fig. 1. Representation of a thin laminated composite shell subjected to force and moment resultants, with a close-up view of the shell’s cross-section.
2.1. Laminate analysis
The infinitesimal deformation of thin orthotropic laminates is analyzed using the classical lamination theory
(e.g. see [20,21]). The constitutive equations relating stresses to the midsurface strains and curvatures in the global
x, y and z coordinate system for lamina n are
9
8 9ðnÞ 2
3ðnÞ 8 0
0
>
>
Q11 Q12 Q16
=
=
< rx >
< ex þ zjx >
6
7
e0y þ zj0y ;
ð1Þ
ry
¼ 4 Q12 Q22 Q26 5
>
>
>
;
: >
: 0
0 ;
sxy
cxy þ zjxy
Q16 Q26 Q66
ðnÞ
ðnÞ
where rðnÞ
x and ry are the normal stresses, sxy is the shear
0
0
0
stress, ex , ey and cxy are the midsurface strains, j0x , j0y and
ðnÞ
j0xy are the curvatures and Qij are the off-axis reduced stiffnesses of the lamina [20]. The force and moment resultants
per unit width of the laminate are defined in terms of the
stresses as follows:
R H =2
fN x ; N y ; N xy g ¼ H =2 frx ; ry ; sxy g dz;
ð2Þ
R H =2
fM x ; M y ; M xy g ¼ H =2 frx ; ry ; sxy gz dz:
Substitution of (1) into (2) yields the following matrix relations that relate the force and moment resultants to the
midsurface strains and curvatures,
8
Nx
>
>
>
>
N
>
y
>
>
<N
9
>
>
>
>
>
>
>
=
2
A11
6 A12
6
6A
xy
6 16
¼6
>
>
6 B11
M
x >
>
>
>
6
>
>
>
>
M > 4 B12
>
>
;
: y>
M xy
B16
A12
A22
A26
B12
B22
B26
A16
A26
A66
B16
B26
B66
B11
B12
B16
D11
D12
D16
B12
B22
B26
D12
D22
D26
38 e0 9
B16 >
x >
>
>
>
0 >
>
>
e
7
>
B26 7>
y >
>
>
>
>
=
< 0 >
B66 7
7 cxy
;
7
D16 7>
j0x >
>
>
>
7>
> 0>
>
>
D26 5>
>
> jy >
>
;
: 0 >
D66
jxy
ð3Þ
where Aij are the extensional stiffnesses, Dij the bending
stiffness and Bij the bending-extensional coupling stiffnesses, which are defined in terms of the off-axis reduced
stiffnesses as
Aij ¼
N
X
ðnÞ
Qij ½zðnÞ zðn1Þ ;
n¼1
Bij ¼
N
1X
ðnÞ
2
2
Q ½ðzðnÞ Þ ðzðn1Þ Þ ;
2 n¼1 ij
Dij ¼
N
1X
ðnÞ
3
3
Q ½ðzðnÞ Þ ðzðn1Þ Þ ;
3 n¼1 ij
ð4Þ
and indices i and j range over 1, 2, 6. The six-by-six matrix
in (3) is called the laminate stiffness matrix or ABD
matrix. Eq. (3) can be inverted to obtain the mid-surface
2068
J.L. Pelletier, S.S. Vel / Computers and Structures 84 (2006) 2065–2080
strains and curvatures
resultants
8 0 9 2
ex >
>
a11 a12
>
>
> 0 >
>
>
6
>
>
e
>
a12 a22
y >
>
>
> 6
>
>
= 6
< c0 >
6 a16 a26
xy
¼6
6b
0 >
>
b21
11
>
> 6
> jx >
>
>
6
>
>
0
>
>
4
b
b22
j
>
> y >
12
>
>
;
: 0 >
b16 b26
jxy
in terms of the force and moment
a16
b11
b12
a26
b21
b22
a66
b61
b61
d 11
b62
d 12
b62
b66
d 12
d 16
d 22
d 26
9
38
b16 > N x >
>
>
>
>
> Ny >
>
b26 7
>
>
7>
>
>
>
7>
<
b66 7 N xy =
7
;
d 16 7
>
> Mx >
7>
>
>
>
7>
>
>
d 26 5>
>
> My >
>
>
;
:
M xy
d 66
ð5Þ
where the abd matrix in (5) is the inverse of the ABD matrix. For simple loading cases where the force and moment
resultants are prescribed, the mid-surface strains and curvatures are computed using (5) and the stresses rxðnÞ ; ryðnÞ
and sðnÞ
xy in the global coordinate system are determined
ðnÞ
ðnÞ
ðnÞ
using (1). The stresses r1 ; r2 and s12 in the principal
material coordinate system are obtained by a tensor
transformation,
8 9ðnÞ 2
38 9ðnÞ
>
cos2 /ðnÞ
sin2 /ðnÞ
2sin /ðnÞ cos /ðnÞ >
=
=
< r1 >
< rx >
7
6
r2
¼4
:
sin2 /ðnÞ
cos2 /ðnÞ
2sin /ðnÞ cos /ðnÞ 5 ry
>
>
>
;
: ;
: >
sxy
s12
sin /ðnÞ cos/ðnÞ sin /ðnÞ cos/ðnÞ cos2 /ðnÞ sin2 /ðnÞ
ð6Þ
When comparing the stiffnesses of different laminates, especially symmetric laminates that are subjected to inplane
loading, it is often convenient to define the effective extensional modulus Ex , the effective extensional modulus Ey ,
and the effective shear modulus Gxy of the laminate, as
follows:
Ex ¼
1
;
a11 H
Ey ¼
1
;
a22 H
Gxy ¼
1
:
a66 H
ð7Þ
The areal mass density qs, which is the mass of a composite
laminate per unit surface area, is defined as
qs ¼
Z
H=2
q dz ¼
H =2
N
X
½.f V ðnÞ þ .m ð1 V ðnÞ ÞhðnÞ ;
ð8Þ
n¼1
where .f and .m are the densities of the fiber and matrix,
respectively. The units of the areal mass density qs is kg/m2.
2.2. Tsai–Wu failure criterion
One of the goals of this study is to maximize the failure
load of composite laminates. We use the Tsai–Wu [22]
criterion to analyze the failure of laminated composite
materials. The Tsai–Wu failure condition for the twodimensional stress state considered here, is
F ðr1 ; r2 ; s12 Þ ¼ F 1 r1 þ F 2 r2 þ F 11 r21 þ F 22 r22
þ F 66 s212 þ 2F 12 r1 r2 ¼ 1;
where
ð9Þ
1
1
1
þ C;
F 11 ¼ T C ;
T
r1 r 1
r1 r1
1
1
1
F 22 ¼ T C ;
F2 ¼ T þ C;
r2 r 2
r2 r2
1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
F 12 ¼ F 11 F 22 ;
F 66 ¼ F 2 :
2
ðs12 Þ
F1 ¼
ð10Þ
Here rT1 is the tensile strength in the fiber direction, rC1 is
the compressive strength in the fiber direction, rT2 is the tensile strength perpendicular to the fiber direction, rC2 is the
compressive strength perpendicular to the fiber direction
and sF12 is the inplane shear strength. In order to obtain a
safe design, the stresses at every point in the laminate must
lie within the Tsai–Wu failure envelope. That is, the following inequality:
F ðr1 ; r2 ; s12 Þ < 1
ð11Þ
must be satisfied for all layers at all positions within the
laminate.
2.3. Effective properties of fiber reinforced composite
materials
Several homogenization methods are available in the literature for estimating the effective properties of continuous
fiber reinforced orthotropic materials from their constituent properties (e.g. see [23]). Examples include the simplified strength of materials approach of Chamis [18], the
cylindrical assemblage model of Hashin and Rosen [24],
and the method of cells by Aboudi [25]. We use the simplified method of Chamis [18] due to its ease of implementation and computational efficiency. This method assumes
that the matrix phase is reinforced with, and ideally bonded
to, a periodic array of square fibers [26]. Using three
dimensional finite element simulations of realistic microstructures, Chamis has demonstrated that the simplified
micromechanics model can estimate the effective elastic
properties E1, E2, G12 and m12 over a wide range of fiber
volume fractions V. In addition, Chamis has extended his
method to estimate the strengths rT1 , rC1 , rT2 , rC2 and sF12 of
fiber reinforced composite laminae from the strength properties of the fiber and matrix constituents and the fiber volume fraction. Chamis has shown that the computed
strengths give good correlation with experimental data.
3. Multi-objective optimization of laminates using genetic
algorithms
3.1. Formulation of the optimization problem
In our implementation of the laminate optimization
algorithm, we choose the maximum number of laminae
in our laminate a priori to be Nmax. A laminate design, x,
is represented by a real-valued array which consists of the
fiber volume fractions, V(n), fiber orientations, /(n), and
thicknesses, h(n), of the laminae
J.L. Pelletier, S.S. Vel / Computers and Structures 84 (2006) 2065–2080
x ¼ ffV ð1Þ ; V ð2Þ ; . . . ; V ðN max Þ g; f/ð1Þ ; /ð2Þ ; . . . ; /ðN max Þ g;
fhð1Þ ; hð2Þ ; . . . ; hðN max Þ gg:
V ðnÞ ¼
vðnÞ
ðV max V min Þ þ V min ;
NV
/ðnÞ ¼
hðnÞ
ð/ /min Þ þ /min ;
N / max
hðnÞ ¼
gðnÞ
ðhmax hmin Þ þ hmin :
Nh
ð12Þ
Thus, there are 3Nmax decision variables. It is possible for
some of the laminae to have zero thicknesses, in which
case the corresponding volume fractions, fiber orientations and thicknesses are simply deleted from the array
x. If there are Nzero laminae that have zero thickness, then
the laminate consists of N = Nmax Nzero laminae of
finite thicknesses.
A multi-objective optimization problem, which has a
number of objective functions that need to be maximized,
is stated in the following form:
2069
ð15Þ
Thus, the fiber orientations are incremented in multiples of
D/ = (/max /min)/N/. In cases where it is not possible to
find an integer N/ that yields the desired incrementation
angle D/, a lookup table can be used to to map the integers
h(n) to a set of pre-defined fiber orientations /(n).
3.3. The genetic algorithm
Find
x
Maximize
F k ðxÞ;
Subject to
gm ðxÞ 6 0;
k ¼ 1; 2; . . . ; K
ð13Þ
m ¼ 1; 2; . . . ; M;
where Fk(x) are the K objective functions and gm(x) are the
M constraints. Specific objective functions and constraints
will be considered in Section 4. In problems that involve
one or more objective functions that need to be minimized,
only those objective functions are multiplied by 1 to
transform the problem into one in which all the objective
functions are maximized. Equality constraints, although
not explicitly stated, can be handled by converting them
to inequality constraints [27].
3.2. Genetic coding
Genetic algorithms directly manipulate strings of decision variables to generate improved designs via the crossover and mutation operators. It is common to use binary
coding to represent the decision variables. In a binary
coded GA of string length b, the incrementation of a decision variable is DX = (Xmax Xmin)/(2b 1). For example,
when a 5-digit binary string is used to represent fiber orientations ranging from 0 to 180, the incrementation is 180/
(25 1) = 5.8065. Thus, binary representation may not be
suitable for representing fiber orientations commonly used
in practical engineering applications. In an attempt to comply with manufacturing constraints, we specify the incrementation of each decision variable through the use of
integer coding. Integer coding has been previously used
for laminate optimization by Le Riche and Haftka [4]. In
our implementation of NSGA-II for the optimal design
of laminates, we use integers to represent the individual
decision variables as follows:
x ¼ ffvð1Þ ; vð2Þ ; . . . ; vðN max Þ g; fhð1Þ ; hð2Þ ; . . . ; hðN max Þ g;
fgð1Þ ; gð2Þ ; . . . ; gðN max Þ gg;
ð14Þ
where v(n), h(n) and g(n) are integers ranging from 0 to NV, 0
to N/ and 0 to Nh, respectively, and the transformations
from integer coded values to the decision variables are,
The Pareto-optimal designs are obtained using an integer-coded version of the non-dominated sorting genetic
algorithm [15] (NSGA-II). The NSGA-II algorithm is modified to include an archive of the historically non-dominated
individuals, Ht. A schematic of the process that is used to
update of the parent population, Pt, child population, Qt,
and historical archive of non-dominated solutions, Ht, from
generation t to t + 1 is shown in Fig. 2. Details about each
step of the multi-objective genetic algorithm are described
below in Sections 3.3.1–3.3.6. First, the parent population,
Pt, and child population, Qt, each consisting of S individuals, are combined to form a population, Rt. The objective
functions and constraint violations of each individual in
Rt are computed and they are non-dominated sorted and
ranked. The archive of non-dominated solutions Ht is
updated to include the better ranked individuals, and the
number of new individuals that have been added to the
archive is counted. If the moving average (over a fixed number of generations) of the number of new solutions that
have been discovered is less than a prescribed value, the
algorithm is assumed to have reached a point of diminishing
returns. In this case, the historical non-dominated solutions, Ht+1, are reported as the Pareto-optimal solutions
and the algorithm is terminated. Otherwise, a crowded distance sort of the individuals is performed within each rank
of Rt and a controlled elitist selection process is used to
form an updated parent population, Pt+1. Subsequently,
an intermediate mating pool is obtained from the parent
population, Pt+1, using a crowded tournament selection
operation and the child population, Qt+1, is generated by
crossover and mutation. The process is iterated over several
generations until the termination criterion is satisfied. It
should be noted that in the present work, the terms individual and design are used interchangeably.
3.3.1. Non-dominated sorting
The parent population Pt and offspring population Qt
are combined to create Rt = Pt [ Qt where t denotes the
generation number. The combined population Rt is sorted
according to non-constrain-dominance and the individuals
are ranked. Following the definition by Deb [15], an
J.L. Pelletier, S.S. Vel / Computers and Structures 84 (2006) 2065–2080
Crowded
Distance Sort
r1
START
r2
r3
r4
Initialize
P0 and Q 0
r5
Pt
}
Controlled Elist
Selection of Parents
}
V
Non-constraindominated Sort
}
V
Controlled NSGA-II
Evolutionary
Search Cycle
}
}
}
Qt
R t+1
Pt+1
VV V V V
2070
Q t+1
}
}
Rt
Mating Pool via
Tournament Selection
Sub-loop to determine
termination of
NSGA-II Search Routine
Children via
Crossover and Mutation
False
Update Historical
Archive with r1
If ( t > G)
&
t< ε
V
Ht+1
Ht
True
Report on Archive
&
STOP
Determine non-constraindominated set of (Ht U r1 )
and compare to Ht
Fig. 2. Schematic of the controlled elitist NSGA II multi-objective genetic algorithm with termination criteria based on a non-constrain-dominated
historical archive.
individual x(i) 2 Rt is said to constrain-dominate an individual x(j) 2 Rt, if any of the following conditions are true:
ð1Þ
xðiÞ and xðjÞ are feasible; with
ðaÞ xðiÞ is no worse than xðjÞ in all objectives; and
ðbÞ xðiÞ is strictly better than xðjÞ in at least
one objective;
ð2Þ
xðiÞ is feasible while individual xðjÞ is not;
ð3Þ
xðiÞ and xðjÞ are both infeasible; but xðiÞ
has a smaller constraint violation:
ð16Þ
Here, the constraint violation CðxÞ of an individual x is
defined to be equal to the sum of the violated constraint
function values [27],
M
X
CðxÞ ¼
Hðgm ðxÞÞgm ðxÞ;
ð17Þ
m¼1
where H is the Heaviside step function. The concept of
constrain-domination enables us to compare two individuals in problems that have multiple objectives and constraints, since if x(i) constrain-dominates x(j), then x(i) is
better than x(j). If none of the three conditions in (16) are
true, then x(i) does not constrain-dominate x(j). Perhaps
most easily visualized in the case of two objective functions, Fig. 3(a) provides a graphical depiction of the dominance relation where two objectives are to be maximized.
Shaded regions represent the line of sight of a particular
individual. When comparing two feasible individuals x(i)
and x(j), x(j) is considered to be dominated by x(i) if x(j) is
able to see x(i). Conversely, feasible individuals that lack
any other feasible individuals in their line of sight are found
to be to be non-dominated. When both solutions are infeasible, the concept is extended to define constrain-domination based on the magnitude of constraint violation. In
Fig. 3(a), x(7) is dominated by x(3) and x(8), since x(3) is
strictly better than x(7) in both objectives; and x(8) is equal
(no worse) in objective 2, while it is strictly better in objective 1. Although x(5) and x(10), are within the line of sight of
x(7), they are infeasible and therefore dominated in all cases
by feasible individuals. The non-constrain-dominated set
consisting of x(9), x(3), x(8) and x(2), are designated to be
of rank 1, denoted by r(1) = {x(9), x(3), x(8), x(2)}. The rank
1 individuals are temporarily disregarded from the population and the non-constrain-dominated solutions of the
remaining population are found and designated as the
non-constrain-dominated set of rank 2. In Fig. 3(a),
J.L. Pelletier, S.S. Vel / Computers and Structures 84 (2006) 2065–2080
(a)
Maximize
by two neighbors, they are assigned the largest crowding
distances
Feasible Individual
Infeasible Individual
ðq;kÞ
x(10)
ðq;kÞ
w1
(9)
x
(3)
x
F2 (x)
x(7)
x(5)
ðq;kÞ
wJ
x(8)
x(1)
x(4)
(2)
x
Maximize
F1 (x)
(b)
¼1þ
¼1þ
F k ðs1
ðq;kÞ
Þ F k ðs2
ðq;kÞ
F k ðs1 Þ
ðq;kÞ
F k ðsJ 1 Þ
ðq;kÞ
F k ðs1 Þ
Þ
;
ðq;kÞ
F k ðsJ Þ
ðq;kÞ
F k ðsJ Þ
:
ðq;kÞ
F k ðsJ Þ
ð19Þ
In all cases we record which individual, x(i), corresponds to
ðq;kÞ
a particular sorted individual sj
and its distance wjðq;kÞ .
(i)
The crowding distance W(x ) of individual x(i), can then
be defined as a summation of the crowding distances along
each of the K objective function axes,
K
X
wðq;kÞ
;
ð20Þ
WðxðiÞ Þ ¼
j
r (1)
(2)
x(6) r
2071
Maximize
k¼1
x(11)
where we ensure that summation occurs over crowding distances wðq;kÞ
corresponding to x(i) only. This is graphically
j
depicted in Fig. 3(b) for two objective functions, where
the crowding distance of an individual is the perimeter of
the rectangle with its nearest neighbors at diagonally opposite corners.
x(15)
x(12)
x(14)
F2 (x)
x(13)
r (q)
Maximize
F1 (x)
Fig. 3. (a) Depiction of the constrain dominance relation for the
simultaneous maximization of two objectives and (b) depiction of the
crowding distance metric for non-constrain-dominated front r(q) under
bi-objective optimization.
r(2) = {x(7), x(1), x(4), x(6)}. This procedure is continued until
the entire population is classified into various subpopulations r(q) of rank q. Infeasible solutions are ranked according to the magnitude of their constraint violation.
3.3.2. Crowding distance metric
One of the goals of a multi-objective GA is to ensure
population diversity in the non-dominated set. This is
achieved by giving preference to individuals that are more
evenly spaced (i.e., less crowded) in the objective space.
Each individual of a ranked subpopulation is given a
crowding distance based on its closeness to adjacent neighbors with equal rank, in the objective space. Subpopulations r(q) having identical rank q, are sorted in descending
order according to each objective function Fk and denoted
by s(q,k). For the number of individuals, J = js(q,k)j, in the
sorted subpopulation s(q,k), each individual has a sorted
order, j, with respect to the objective function Fk. We
assign a crowding distance for each objective function Fk
based on the metric,
ðq;kÞ
wðq;kÞ
j
¼
ðq;kÞ
F k ðsjþ1 Þ F k ðsj1 Þ
ðq;kÞ
F k ðs1
ðq;kÞ
Þ F k ðsJ
Þ
for j ¼ 2; 3; . . . ; J 1;
ðkÞ
ð18Þ
where wðkÞ
j refers to the distance of individual sj , to its two
ðkÞ
ðkÞ
closest neighbors sjþ1 and sj1 in the kth objective. Since
extreme values in each objective are not directly bordered
3.3.3. Controlled elitism sort
Crowded distance sorted subpopulations are now joined
via a controlled elitism sort procedure to form the updated
parent population, Pt+1. To preserve diversity, we control
the effects of elitism by choosing the number of individuals
from each subpopulation r(q) according to the geometric
distribution
1 c q1
Sq ¼ S
c ;
ð21Þ
1 cw
to form a parent search population, Pt+1, of size S, where
0 < c < 1. Note that Sq is the number of individuals taken
from non-dominated subpopulation r(q), c is a parameter
that governs the shape of the geometric distribution and
w is the total number of ranked non-dominated subpopulations that comprise the parent population. By controlling
the degree of elitism, we ease selection pressure in order
to preserve the diversity of the population. This method,
has been shown to provide improved convergence to the
true Pareto-optimal front when compared to a standard
elitist NSGA-II [28].
3.3.4. Tournament selection
Now that the parent population, Pt+1, has been defined,
the next step in the process is the crowded tournament
selection routine, where each individual competes in
exactly two tournaments with randomly selected individuals, a procedure which imitates survival of the fittest in nature. Individual x(i) is said to win a tournament with
individual x(j) if any of the following conditions are true:
ð1Þ xðiÞ has a smaller ði:e:; betterÞ rank than xðjÞ ;
ð2Þ xðiÞ and xðjÞ have the same rank; but xðiÞ
has a large crowding distance
that is WðxðiÞ Þ > WðxðjÞ Þ:
ð22Þ
2072
J.L. Pelletier, S.S. Vel / Computers and Structures 84 (2006) 2065–2080
In this manner, we generate an intermediate mating pool
from the parent population, that has a higher occurrence
of better ranked and less crowded solutions. Inclusion of
the crowded distance metric enables the genetic algorithm
to seek out a well distributed Pareto-optimal front. For
example, in the case of Fig. 3(b), all individuals have the
same rank, although if x(14) were to compete with x(15),
x(15) would win since it has a larger crowding distance.
3.3.5. Crossover and mutation
Uniform crossover and random uniform mutation are
employed to obtain the child population, Qt+1. An integer
representation such as that shown in (14) may be directly
manipulated by these operators. As shown by Fig. 4(a),
the integer-based uniform crossover operator takes two
distinct parent individuals and interchanges each decision
variable with a probability, 0 < pc 6 0.5. The crossover
probability controls the degree to which children are similar to their parents. Furthermore, laminate parameters are
naturally grouped together such that the fiber orientation
variables do not interact with the volume fractions or
thickness variables. Following crossover, the mutation
operator changes each of the children’s integer coded decision variable with a mutation probability, pm, from its cur-
(a)
rent value to a random integer between 0 and NV, N/ or
Nh, depending on whether the decision variable selected
for mutation corresponds to the volume fraction, fiber orientation or lamina thickness, respectively. This process is
depicted in Fig. 4(b).
3.3.6. Termination criterion
Assigning a termination criterion to a multi-objective
GAs in more difficult than single-objective GAs, mainly
due to difficulty in specifying a formal convergence criterion. Usually, the algorithm is stopped after a prescribed
number of generations. This approach has the disadvantage that it may be either prematurely terminated or the
number of generations may be much more than necessary.
In the present work, we keep track of the number of new
designs that are added to a historical archive of non-dominated individuals and terminate the algorithm when it
reaches the point of diminishing returns. The historical
non-dominated set, Ht, is updated as
Htþ1 ¼ NDðr1 [ Ht Þ;
ð23Þ
where ND denotes the non-constrain-domination operator which extracts the rank 1 individuals from a set. The
number of new individuals, Pt+1, that have been added
to the historical non-dominated set Ht+1, is
Ptþ1 ¼ jHtþ1 n Ht j;
ð24Þ
where n denotes the difference between two sets. In other
words, Pt+1 is the number of new non-dominated individuals that have been evolved by the current generation’s
search population, as compared to the historical nondominated set at the previous generation and it provides
a metric for the improvement of the non-dominated set.
Furthermore, we also monitor how these new individuals,
Pt+1, affect the average crowding distance of the historical
non-constrain-dominated set from the previous to the current generation. The average crowding distance, W, is
defined as follows:
jHtþ1 j
1 X
Wtþ1 ¼
WðxðjÞ Þ;
ð25Þ
jHtþ1 j j¼1
(b)
Fig. 4. (a) Example operation of the uniform crossover genetic operator
used to generate two offsprings from parents A and B, (b) example
operation of the uniform mutation genetic operator on offspring A.
where x(j) 2 Ht+1. If new individuals are added to the historical non-constrain-dominated set, yet the change in
average crowding distance is negligible, then the non-dominated set really has not changed by any significant
amount. Therefore, we define the number of new, less
crowded, individuals evolved, Ht+1, as follows:
(
Ptþ1 if 1 WWtþ1 > d;
Htþ1 ¼
ð26Þ
t
0;
otherwise;
where d is the tolerance on minimum improvement in average crowding distance. With this definition, we terminate
the multi-objective GA if the moving average, Kt+1, of
the new, less crowded, individuals is smaller than a predefined constant, , over G generations. That is,
G
1 X
Ktþ1 ¼
Htþ1s < :
ð27Þ
G s¼0
J.L. Pelletier, S.S. Vel / Computers and Structures 84 (2006) 2065–2080
The moving average metric provides an intuitive measure of the average number of new individuals that are
evolved over G generations. If the average number of
new individuals is less than, , the historical non-dominated
set, Ht+1, is reported to be Pareto-optimal and the genetic
algorithm is terminated. While this termination criterion
provides a method for determining when the NSGA-II
search routine has stagnated, it does not provide information as to how close the final NSGA-II Pareto-optimal
designs are to the true Pareto-optimal front. Nevertheless,
it provides an automatic method for terminating the
NSGA-II algorithm when successive generations are less
likely to produce a significant improvement to the historical non-constrain-dominated set. The method proposed
here extends the excellent search capabilities of the controlled NSGA-II to allow the identification of more individuals on the Pareto-optimal front than there are
individuals in the search population, S, since the size of
the historical non-constrain-dominated archive may be
arbitrary.
4. Numerical results and discussion
In this section we present the results of optimal design
studies performed on two model problems. The thickness
h(n) of each lamina can either be zero or h, where h is a constant value. That is, hmin = 0, hmax = h and Nh = 1 in (15).
Optimal values are sought for the volume fractions, the
fiber orientations and the lamina thicknesses. The GA
parameters were tuned by studying the convergence of
the algorithm for the model problems. A search population
consisting of S = 150 individuals, crossover probability
pc = 0.25 and gene mutation probability pm = 0.116 are
used. This results in an average of 3.5 mutated decision
variables per individual per generation. It was found that
controlled elitism, with c = 0.5 in (21), enhanced the ability
for the algorithm to seek out the entire Pareto-optimal
front. In consideration of the overall accuracy of the search
versus computation time, the termination criterion parameters are chosen to be G = 100, = 0.01 and d = 0.005.
That is, the algorithm is terminated when it is unable
to find a single, new, historically non-dominated, less
crowded solution over a span of 100 generations that
changes the average crowding distance of the historical
archive by 0.5%. In the model problems, we consider
graphite fiber reinforced epoxy laminates. The mechanical
properties of the constituent materials are provided in
Table 1.
4.1. Model problem I: stacking sequence optimization of a
laminate subjected to biaxial moments
In the first model problem, we investigate the multiobjective optimization of laminated composite coupons
subjected to biaxial moments. The maximum number of
laminae is limited to Nmax = 10. The thickness h(n) of each
lamina can either be zero or 0.45 mm. The fiber orienta-
2073
Table 1
Material properties
E1 (GPa)
E2 (GPa)
G12 (GPa)
m12
Sut (MPa)
Suc (MPa)
Sus (MPa)
q (kg/m3)
Graphite fiber [18]
IMHS epoxy [18]
220.0
13.70
8.960
0.25
2415.0
2070.0
–
1772
3.447
3.447
1.276
0.350
103.0
241.0
89.60
1210
tions vary from /min =90 to /max = 90 in 5 increments
(i.e., N/ = 36). The laminate is subjected to the following
biaxial moment resultants:
1
ð28Þ
M x ¼ M;
M y ¼ M:
2
The force resultants Nx, Ny and Nxy and twisting moment
resultant Mxy are zero. The stresses in the principal material coordinate system r1(z), r2(z) and s12(z) for each lamina are obtained as linear functions of M using (5), (1) and
(6). The stresses are substituted into the Tsai–Wu failure
criterion (9) and the resulting quadratic equation is solved
to obtain the positive and negative failure values M+ and
M for a specified location. The first-ply failure moment
Mf of the laminate is defined as the smallest magnitude
of the failure value over the entire laminate thickness. That
is,
Mf ¼
min
z2½H =2;H =2
½minðjM j; jM þ jÞ:
ð29Þ
We seek to maximize the first-ply failure moment Mf,
which is a measure of the flexural strength of a laminate,
and minimize the areal mass density, qs. There are no constraint equations gm(x).
First, we perform the multi-objective optimization with
the volume fraction of graphite fibers fixed at V(n) = 0.75.
The moving average of the new, less crowded, individuals,
Kt, is shown in Fig. 5(a) for increasing number of generations. The algorithm terminates in 107 generations. The
non-dominated solutions at generations t = 0, 1, 10, 50
and 107 are shown in Fig. 5(b). The 10 non-dominated
solutions at generation 107 are the numerically obtained
Pareto-optimal designs. Next, we allow the graphite fiber
volume fraction to vary from Vmin = 0.10 to Vmax = 0.75
in 0.05 increments for each individual lamina. In this case,
the multi-objective GA takes a total of 1624 generations to
terminate. The moving average, Kt, is shown in Fig. 5(c).
Compared to the case with constant volume fraction, the
multi-objective GA is able to find more non-dominated
solutions due the layerwise tailorability of the volume fractions. The algorithm yields a total of 274 Pareto optimal
designs at generation 1624. The non-dominated sets at
intermediate generations are shown in Fig. 5(d). The evolution of the parent population, Pt, for intermediate generations is shown in Fig. 6. As it progresses, the algorithm
2074
J.L. Pelletier, S.S. Vel / Computers and Structures 84 (2006) 2065–2080
(b)
(a) 0.20
4000
t=0
t=1
t = 10
t = 50
t = 107
Constant V
0.16
Mf (Nm/m)
3000
V
t
0.12
0.08
1000
0.04
0
2000
0
100
101
102
103
104
105
106
107
0
1
t
(c)
6
Variable V
(d)
Mf (Nm/m)
t
V
2
3
4
5
4
5
ρs (kg/m2)
6
7
8
6
7
8
t=0
t=1
t = 10
t = 100
t = 1624
3000
3
3
4000
5
4
2
2000
2
1000
1
0
0
250
500
1000
750
1250 1500
1750
0
0
t
1
ρs (kg/m2)
Fig. 5. (a, c) The moving average Kt of new, less crowded, individuals evolved with increasing generations for model problem I, and (b, d) corresponding
non-constrain-dominated sets.
improves the performance of the designs in the objective
space and yields a diverse non-constrain-dominated set.
The Pareto-optimal fronts for the fixed and variable volume fraction cases are superposed in Fig. 7. As is evident
from the Pareto-optimal fronts, by tailoring the volume
fraction of each lamina individually, the algorithm is able
to find designs that are of the same areal mass density as
the optimal designs with constant volume fractions, but
can withstand higher flexural loads. This is achieved by
judiciously increasing the fiber volume of the heavier but
stronger graphite fibers in the outer layers where the bending stresses are larger. The lamination schemes for the
designs which are labeled in Fig. 7 and their corresponding
failure moments, Mf, and areal mass densities, qs, are listed
in Table 2. It is noted that the variable volume fraction
design n has a failure moment that is 1.1% higher than
the constant volume fraction design j although it 4.5%
lighter. It is noted that the reduction in mass is because
design n has smaller fiber volume fraction near the mid-surface than on the top and bottom surfaces.
Through-the-thickness plot of the Tsai–Wu function
F(r1, r2, s12) corresponding to laminate design k is shown
in Fig. 8(a) corresponding to a failure load of Mf =
864.255 N m/m. As is evident, the no failure condition
F(r1, r2, s12) < 1 is satisfied at all locations. Through-the-
thickness plots of stresses r1, r2 and s12 in the principle material coordinate system are shown in Fig. 8(b)–(d). It is
observed that the magnitude of r1 exceeds the strength rC1
at the top surface and r2 is slightly larger than the strength
rT2 at the bottom surface, although the Tsai–Wu failure criterion F(r1,r2,s12) < 1 is satisfied at those locations.
4.2. Model problem II: optimization of a thin walled pressure
vessel for mass, strength and stiffness
In the second optimization problem, we increase the
dimension of the performance space to conduct tri-objective optimization studies of a thin walled pressure vessel.
We consider a symmetrically laminated pressure vessel of
radius R = 1 m, composed of graphite fiber reinforced
epoxy and subjected to an internal pressure p, as depicted
in Fig. 9. The force resultants, calculated via considerations
of static equilibrium, are
1
N x ¼ pR;
2
N y ¼ pR;
N xy ¼ 0:
ð30Þ
The moment resultants are identically zero. The Tsai–Wu
failure criterion yields a quadratic equation in p, which is
used to determine the positive and negative pressures pþ
ðnÞ
and p
ðnÞ , respectively, that would cause the nth lamina to
J.L. Pelletier, S.S. Vel / Computers and Structures 84 (2006) 2065–2080
2075
4000
t=1
t=0
t=5
Mf (Nm/m)
3000
2000
1000
0
0
1
2
4
3
6
5
ρs (kg/m2)
7
8
ρs (kg/m2)
t = 50
t = 100
Mf (Nm/m)
t = 10
ρs (kg/m2)
ρs (kg/m2)
ρs (kg/m2)
t = 500
ρs (kg/m2)
t = 1624
Mf (Nm/m)
t = 1000
ρs (kg/m2)
ρs (kg/m2)
ρs (kg/m2)
Fig. 6. Time lapse sequence showing the evolution of the parent population Pt in the objective space for model problem I.
sure, pf, of the laminated pressure vessel is determined by
the smallest positive value of pþ
ðnÞ ,
4000
n
Variable V
3000
Mf (Nm/m)
j
pf ¼ min pþ
ðnÞ :
Constant V
n2½1;N i
h
m
2000
g
l
k
f
1000
e
b
a
0
0
1
c
2
d
3
4
5
6
7
8
ρs (kg/m2)
Fig. 7. Pareto-optimal designs for maximum failure moment and minimum areal mass density, as discussed in model problem I.
fail. We ignore p
ðnÞ since the pressure vessel is subjected to
only positive internal pressures. The first-ply failure pres-
ð31Þ
The thickness of the graphite/epoxy laminae can either be
zero or 0.45 mm and the fiber volume fraction of all laminae
are kept constant at V(n) = 0.75. The corresponding material properties are E1 = 165.862 GPa, E2 = 9.79619 GPa,
G12 = 4.96043 GPa, m12 = 0.275, rT1 ¼ 1811:25 MPa, rC1 ¼
1064:35 MPa, rT2 ¼ 94:0562 MPa, rC2 ¼ 220:073 MPa,
sF12 ¼ 80:6854 MPa. The fiber orientations vary from
/min = 90 to /max = 90 in 15 increments. The maximum number of laminae is limited to Nmax = 20, although
only one half of the laminate needs to be designed since it is
symmetric.
A multi-objective optimization of the pressure vessel is
performed for three objective functions with the goal of
maximizing the failure pressure pf, maximizing the hoop
rigidity Ey H and minimizing the areal mass density qs. This
problem is relevant to the design of stiff, light weight fuel
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J.L. Pelletier, S.S. Vel / Computers and Structures 84 (2006) 2065–2080
Table 2
Model problem I: Pareto-optimal designs for flexure coupons having maximum strength and minimum mass as shown in Fig. 7
Design
Fiber volume fractions
Fiber orientations (deg.)
qs (kg/m2)
Mf (N m/m)
a
b
c
d
e
f
g
h
i
j
k
l
m
n
[0.75]
[0.75]2
[0.75]3
[0.75]4
[0.75]5
[0.75]6
[0.75]7
[0.75]8
[0.75]9
[0.75]10
[0.75/0.75/0.20/0.75/0.75]
[0.75/0.75/0.10/0.10/0.75/0.75]
[0.75/0.75/0.10/0.10/0.10/0.75/0.75]
[0.75/0.75/0.75/0.70/0.20/0.10/0.70/0.75/0.75/0.75]
[0]
[35/35]
[90/0/0]
[25/55/40/35]
[25/55/50/40/35]
[20/70/25/70/30/35]
[20/70/15/80/65/30/35]
[20/70/10/30/65/35/30/35]
[70/0/10/25/70/70/40/10/35]
[10/70/50/10/25/70/70/40/10/35]
[25/55/20/65/20]
[30/45/45/55/60/20]
[30/45/25/20/0/60/20]
[20/25/80/60/20/80/50/20/60/20]
0.7342
1.4684
2.2025
2.9367
3.6709
4.4051
5.1392
5.8734
6.6076
7.3418
3.5318
4.0763
4.6461
7.0130
7.07281
51.9458
120.038
503.275
852.294
1296.28
1754.07
2313.47
2884.69
3534.94
864.255
1259.04
1624.56
3574.18
(b) 1/2
z/H
z /H
(a) 1/2
0
-1/2
0
0.20
0.40
0.60
0.80
0
-1/2
-2000
1.00
σ1
σ1T
σ1C
-1000
F(σ1, σ2, τ12)
σ2
σ2T
σ2C
0
-1/2
(d) 1/2
z/H
z /H
(c) 1/2
-200
-100
0
MPa
100
200
0
MPa
1000
2000
τ12
F
τ12
F
−τ12
0
-1/2
-100
-50
0
MPa
50
100
Fig. 8. Through-the-thickness variation of (a) Tsai–Wu failure function and (b, c, d) normal and shear stresses for laminate design k, shown in Fig. 7.
tanks containing compressed gas. The moving average of
the new, less crowded, individuals, Kt, is shown in Fig. 10
for increasing number of generations. The algorithm identifies a total of 161 Pareto-optimal designs in 493 generations. Since our algorithm maintains a historical archive
of non-constrain-dominated solutions, we obtain more
Pareto-optimal designs than the size of the search population. Fig. 11(a) depicts the Pareto-optimal front in three
dimensions. The lamination schemes and corresponding
performance of the selected designs that are labelled in
Fig. 11 are given in Table 3. The projected views of the
Pareto-optimal front onto the pf Ey H , pf qs and
Ey H qs planes are depicted in Fig. 11(b), (c) and (d),
respectively. The Pareto-optimal curves enable us to perform trade-off studies. For example, designs b and c have
identical mass, and the hoop rigidity of c is 4.45% smaller
than that of b. However, the failure pressure of design c is
268% larger than that of b.
In the next part of the analysis, we seek to optimize the
stiffness and mass of a composite pressure vessel such that
the failure pressure is greater than a prescribed value of
2.5 MPa, (i.e. g1 = 1 pf/(2.5 · 106) < 0) and perform a
multi-objective optimization to maximize the axial rigidity
Ex H , maximize the hoop rigidity Ey H and minimize the
J.L. Pelletier, S.S. Vel / Computers and Structures 84 (2006) 2065–2080
2077
3.5
closed rigid end cap
3.0
2.5
2.0
V
t
z
y
1.5
φ
1.0
x
0.5
0
0
p
100
200
300
t
400
500
Fig. 10. The moving average Kt of new, less crowded, individuals evolved
with increasing generations for the tri-objective optimization of a thinwalled laminated pressure vessel according to maximum failure pressure,
maximum hoop rigidity, and minimum areal mass density, as discussed in
model problem III.
R
internal pressure
closed rigid end cap
Fig. 9. Illustration of a thin-walled laminated pressure vessel, with closed
rigid end caps, subjected to internal pressure.
(a)10
(b) 10
i
h
i
j
j
g
8
areal mass density qs. Fig. 12(a) shows the resulting Paretooptimal front in three dimensions. A total of 277 Paretooptimal laminate designs were obtained in 194 generations.
8
h
e
6
e
pf (MPa)
p
f
d
c
4
2
0
0
b
500
Ey H
f
a
6
d
2
15
1000
f
ρs
5
0
g
8
f
h
Ey H (MN/m)
c d
b
a
1000
b
a
5
10
ρs (kg/m2)
j
c
i
500
e
d
f
0
g
h
2
0
1500
1000
500
(d) 1500
i
e
pf (MPa)
0
Ey H (MN/m)
j
4
b a
10
10
6
c
4
1500 0
(c)
g
15
0
0
5
10
15
ρs (kg/m2)
Fig. 11. Pareto-optimal designs for a thin-walled laminated pressure vessel for maximum failure pressure, maximum hoop rigidity and minimum areal
mass density.
J.L. Pelletier, S.S. Vel / Computers and Structures 84 (2006) 2065–2080
Table 3
Model problem II: selected Pareto-optimal designs for the tri-objective
optimization of a pressure vessel for maximum hoop rigidity and
maximum failure pressure while simultaneously minimizing mass as
shown in Fig. 11
Design
Fiber orientations
(deg.)
qs
(kg/m2)
pf
(MPa)
Ey H
ðM N=mÞ
a
b
c
d
e
f
g
h
i
j
[904]s
[90/75/60/90]s
[0/903]s
[45/60/60/60]s
[903/15/30/90]s
[9010]s
[0/75/903/45/903]s
[904/0/903/0]s
[90/30/90/30/15/903/15/90]s
[905/0/902/02]s
5.8734
5.8734
5.8734
5.8734
8.8101
14.6835
13.2152
13.2152
14.6835
14.6835
0.754433
0.954566
3.51275
3.83157
5.20814
1.88608
5.59817
7.43345
9.59726
9.47640
597.102
479.518
458.154
161.089
617.161
1492.79
1069.48
1066.00
939.277
1075.03
Lamination schemes corresponding to some designs on the
Pareto-optimal front are listed in Table 4. The projected
views of the Pareto-optimal front are shown in
Fig. 12(b)–(d). Each local front provides a continuous
trade-off between hoop and axial rigidity for fixed areal
mass density as exemplified by designs e, f and g, all three
of which consist of the same number of layers. Among all
feasible 16 layer laminates, e has the largest hoop stiffness
but the smallest axial stiffness, g has the largest axial stiffness but the smallest hoop stiffness, whereas laminate f
(a)
1500
d
c
Ey H
A methodology is presented for the multi-objective optimization of laminated composite materials. The fiber orientations, fiber volume fractions and thicknesses are chosen
as the optimization variables. The layerwise material properties are estimated using simplified micromechanics equations. An integer-coded version of the NSGA-II multiobjective genetic algorithm has been extended to include
an archive of the historical non-constrain-dominated set,
which is updated at each generation. This archive is used
to accumulate a larger number of Pareto-optimal designs
than the search population size employed by a traditional
NSGA-II type algorithm. The historical archive is used in
conjunction with a crowded distance metric to obtain a termination criterion that automatically stops the algorithm
when the moving average of the number of new, less
crowded, non-constrain-dominated designs added to the
historical non-constrain-dominated set falls below a specified threshold value.
In the first model problem, the layerwise fiber orientations and volume fractions are tailored to maximize the
load carrying capacity and minimize the mass of laminated
graphite/epoxy coupons subjected to biaxial moments. The
1500
h
i
f
e
i
j
b
500
k
a
0
0
g
1000
f
j
a
10
1000
ρs
5
0
1500 0
d
500
15
500
Ex H
5. Conclusions
(b)
e h
1000
has intermediate values for both the axial and hoop
stiffnesses.
Ey H (MN/m)
2078
b
0
c
g
k
1500
1000
500
Ex H (MN/m)
(c)
1500
h
e
(d)
i
1500
k
j
g
d
c
b
500
Ey H (MN/m)
Ey H (MN/m)
f
1000
a
g
0
0
5
ρs
10
(kg/m2)
i
1000
c
500
b
a
j
k
15
0
0
d f
e h
10
5
ρs
15
(kg/m2)
Fig. 12. Pareto-optimal designs for a thin-walled laminated pressure vessel for maximum hoop rigidity, maximum axial rigidity, and minimum surface
density, subject to the constraint that the failure pressure be greater than 2.5 MPa.
J.L. Pelletier, S.S. Vel / Computers and Structures 84 (2006) 2065–2080
2079
Table 4
Model problem II: selected Pareto-optimal designs for the tri-objective optimization of a pressure vessel for maximum axial and hoop rigidity while
minimizing mass, subject to the constraint pf > 2.5 MPa, as shown in Fig. 12
Design
Fiber orientations (deg.)
qs (kg/m2)
Ex H ðMN=mÞ
Ey H ðMN=mÞ
pf (MPa)
a
b
c
d
e
f
g
h
i
j
k
[0/902]s
[90/0/902/0]s
[0/902/02/90]s
[902/02/902/0]s
[903/0/904]s
[0/45/0/90/75/0/902]s
[902/06]s
[904/0/904]s
[903/02/90/02/902]s
[90/0/90/07]s
[02/90/0/45/15/04]s
4.405
7.341
8.810
10.278
11.746
11.746
11.746
13.215
14.6835
14.6835
14.6835
167.485
326.142
475.956
484.796
211.580
512.544
916.309
220.397
652.284
1215.67
1218.78
308.428
467.094
475.956
625.751
1056.68
646.534
352.614
1206.24
934.188
370.252
251.602
2.8779
4.6843
5.0573
6.4098
4.6768
6.9146
3.8100
4.9023
9.3686
3.9270
2.5031
Pareto-optimal designs for graphite/epoxy coupons with
layerwise variable volume fractions are shown to be superior to coupons that have fixed graphite volume fraction
of 0.75 throughout. This is due to the fact that the algorithm is able to evolve efficient structures with a spatial variation of the fiber volume fraction. In the second model
problem, we increase the complexity of our search tool to
optimize a laminated pressure vessel according to the three
objectives of failure pressure, stiffness and areal mass density. It is found that nonlinearities in the shape of the Pareto-optimal front enables us to perform trade-off studies
when choosing a particular design.
In summary, the results demonstrate the effectiveness of
the proposed methodology for the multi-objective optimization of composite materials. When coupled with more
sophisticated analytic or numeric laminate analysis procedures and greater computational resources, such a methodology will provide engineers with a useful tool for designing
superior laminated composite structures.
Acknowledgement
The support provided by the US National Science
Foundation through grant DMI-0423485 is gratefully
acknowledged.
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