Composite Plate Stiffness Multicriteria Optimization Using Lamination Parameters 2015 Composite Structures

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Composite plate stiffness multicriteria optimization using laminationparameters

Thiago Assis Dutra ⇑, Sérgio Frascino Müller de Almeida

Instituto Tecnológico de Aeronáutica – ITA, Mechanical Engineering Department, 12.228-900 São José dos Campos, SP, Brazil

a r t i c l e i n f o

Article history:

Available online 18 July 2015

Keywords:

Composite plates

Stiffness optimization

Lamination parameters

a b s t r a c t

Lightweight structures are currently largely applied in several areas and their use is growing every day. Inthis context, composite materials are key parts to achieve design requirements due to their high potential

for optimization. The lamination parameters play an important role as design variables in composite lam-

inates layer optimization. This is due to the fact that the stiffness matrix linear with respect to them. This

work aims at presenting an optimization method based upon a quadratic metamodel used to estimate the

objective function. An analytical formulation to obtain the derivatives of the objective functions with

respect to the lamination parameters is presented. The allowable region of the lamination parameters

is also analyzed and laminate databases are applied in order to avoid problems of defining the boundaries

of the allowable region. A composite plate subjected combined bending and torsion loads is optimized

and the results are presented and discussed in terms of practical design of aeronautical structures.

2015 Elsevier Ltd. All rights reserved.

1. Introduction

Lightweight structures are currently largely applied in several

areas (aeronautical, automotive, oil and gas, sports, biomedicine)

and their use is growing every day. Thus, the aim is to replace con-

ventional materials by composite materials in high performance

structures. Composite materials present a high potential for opti-

mization because their anisotropic nature and the number of

design variables (fiber orientation in each layer). However, the

design of composite structures also involve geometric, load

requirements, and manufacturing constraints. The stiffness opti-

mization using orientation angles as design variables can be a dif-

ficult task due to the fact that the stiffness matrix is highly

non-linear with respect to the layer orientation angles.

In view of the issues related to the orientations angles as design

variables, optimization processes using lamination parameters

have increased over the last years. The great advantage of this

approach is related to the linearity of the stiffness matrix with

respect to the lamination parameters.

The lamination parameters were introduced by [1] as functions

of stacking sequence and ply thickness and orientation. Five stiff-

ness invariants (four of them independents) and 12 lamination

parameters were introduced in order to express the matrix ½ A, ½B

and ½D from the Classical Lamination Plate Theory. Tsai andHahn [2] considered the formulation based on the dimensionless

lamination parameters more useful because that the invariants

can be easily computed from explicit equations and the stiffness

matrix is linear with respect to them. In addition, all these dimen-

sionless lamination parameters belong to ½1; þ1 interval since

they are functions of sine and cosine of ply orientation.

However using lamination parameters as design variables in

optimization processes presents some constraints. The difficult to

obtain the ply orientations from a set lamination parameters is a

very important issue. In addition, there are mathematical relations

between the lamination parameters that result in an allowable

region for them. If these constraints are not accounted for a set

of optimum lamination parameters may not characterize a real

laminate.

A buckling load optimization method for a rectangular plate

under shear loads was presented in [3]. Only symmetric laminates

were accounted. In this work, the assumed in-plane boundary con-

ditions are such that buckling load does not depend of the plane

stiffness. Therefore, only four lamination parameters (related to

½D matrix) were considered as design variables. The boundaries

of the allowable regions for the out-of-plane lamination parame-

ters were obtained for orthotropic and non-orthotropic laminates.

For the orthotropic allowable region, the optimized solution

resulted angle-ply laminates. On the other hand, unidirectional

laminates were found as optima for the non-orthotropic allowable

region.

http://dx.doi.org/10.1016/j.compstruct.2015.07.029

0263-8223/ 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.

E-mail address: [email protected] (T.A. Dutra).

Composite Structures 133 (2015) 166–177

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Another buckling load optimization process applied to cylin-

drical shells was presented in [4]. Symmetric and balanced lami-

nates were considered. The coupling terms were neglected in

order to reduce the design variables. The invariants were slightly

modified from the original formulation. The derivatives of the

objective function were obtained by the central finite difference

method. Six different initial laminates were used for each load

condition. All initial laminates converged to same optimumlaminate.

In view of the difficult to obtain the ply orientations from a set

of lamination parameters, [5] presented a method to solve this

issue. Their work assumes general symmetric laminates of eight

plies maximum. This specific configuration allows establishing

some relations from the allowable region. Nevertheless a general

solution was not obtained for laminates with more than eight

plies.

A method based on two lamination parameters as design vari-

ables in order to optimize the buckling load of laminated compos-

ite plates under uniaxial loads was presented in [6]. A closed form

analytical solution was proposed and the optimal points were

obtained from geometric relations between the allowable region

and the objective function. These assumptions facilitate the allow-

able region to be expressed in mathematical terms.

The transverse shear lamination parameters were firstly intro-

duced in [7]. Two new stiffness invariants related to the transverse

shear were introduced. It was demonstrated that the new two lam-

ination parameters were identical to the in-plane lamination

parameters provided that no hybrid laminate is assumed.

An optimization method based on the plies orientation was

introduced in [8]. The algorithm uses the plies orientation gradi-

ents which makes the problem very ill conditioned. In this case,

the solution might present local optimal points. In order to avoid

this, the algorithm searches a similar laminate that presents the

same set of lamination parameters. The optimization process is fin-

ished when no other best laminate is found. In this case the opti-

mal point is obtained but is not efficient in a computational

point of view.Diaconu et al. [9] introduced a complex method to obtain the

allowable region of the lamination parameters for a general lami-

nate. It was assumed that each ply had the same thickness and

same material properties. The problem with the presented method

was the fact that the total number of plies affects the allowable

region was not accounted for.

Liu and Haftka [10] presented a genetic algorithm procedure in

order to optimize a representative aircraft wing. The ply orienta-

tions were restricted to 0, 45, 45 and 90 (symmetric and bal-

anced). In addition the number of plies for each orientation was

specified and consequently a hexagonal domain was obtained.

The optimal points were found but the objective function was

computed too many times making the process not efficient.

A clamped composite plate under flutter was presented in [11].Once again, genetic algorithm using lamination parameters as

design variables was implemented to obtain the optimal laminate.

The total number of plies was restricted to eight enabling the use

of the method introduced in [5]. The original genetic algorithm

used in [5] was modified resulting in a more efficient method.

However, this gain in efficiency was possible since only eight plies

laminates was considered. This approach would be not efficient

from the computational point of view for more than eight plies.

Bloomfield et al. [12] proposed a method to express the allow-

able region very similar to [9]. It consists in obtaining the allowable

regions separately and the use a non-linear algebraic identity to

relate them. This method presents some limitations and can be

considered mathematically very complex to be applied on opti-

mization algorithms.

In order to find a robust and efficient method of optimization

using lamination parameters, [13] introduced an alternative for-

mulation to the lamination parameters. A linear metamodel with

respect to the lamination parameters was used. This metamodel

aims at estimating the objective function in an efficient way. It

was proposed using laminate databases comprised by lamination

parameters for laminates with different number of plies and orien-

tations to describe the allowable region. Thus all points in the data-base belong to the allowable region and consequently are

associated to a real laminate. These laminate databases are created

one-time-only and any design guideline can be adhered to them

during their construction. Therefore, the optimal solution automat-

ically will comply with all the required guidelines.

A novel formulation of the lamination parameters was firstly

introduced in [14] in order to deal with hybrid laminates. A proce-

dure for pre and post buckling optimization using lamination

parameters and laminate databases was described. The critical

buckling load can be also obtained. In this novel formulation, 18

lamination parameters were introduced for matrices ½ A, ½B and

½D. This novel formulation was a very important step for a new

approach using lamination parameters. The gradients of the objec-

tive function were obtained by finite difference method.

A sensitivity analysis of the transverse shear was presented in

[15]. A thick composite plate under bending load was used as a

case study. Two different composite materials were tested. A

new set of transverse shear lamination parameters was introduced.

Only two parameters were introduced in [7]. In this case, it was

introduced a set of three lamination parameters related to the

transverse shear. Their most important contribution was to intro-

duce a novel formulation where the gradients of a given objective

function could be obtained analytically. This is a very important

step in the computational point of view mainly when a quadratic

(or higher order) metamodel is used.

The proposed method by [13] was implemented in [16] to opti-

mize composite plates subject to buckling and small mass impact.

The formulation of general lamination parameters was described

and simplified to composite plates of a single material. A novel for-mulation was introduced in order to construct laminate databases

comprised by laminates with a large total number of plies. This

highly increases the efficiency of the algorithm because the total

number of laminates in database can be minimized without affect-

ing the result. Several filters were applied to the databases result-

ing only laminates that meet previous requirements. Similar to

[14], the gradients of the objective function were obtained numer-

ically by the finite difference method.

In view of the discussed aspects and characteristics of using

lamination parameters as design variables in optimization pro-

cesses, this work aims at introducing a multicriteria optimization

in which more than one objective function is accounted for in

the solution of the problem. In order to accomplish that, lamina-

tion parameters are used as design variables and combination of laminate databases are performed. Also, gradients of the objective

functions are analytically obtained and a second order metamodel

is described.

2. Problem formulation

2.1. Lamination parameters

The constitutive law for composite materials based on the

First-order Shear Deformation Theory (FSDT) can be written as

[19]:

fN g

fM

g ¼

½ A ½B

½B

½D

femg

fjg ð1Þ

T.A. Dutra, S.F.M. de Almeida / Composite Structures 133 (2015) 166–177 167



fV g ¼ v½ A ct

0

ð2Þ

where fN g, fM g and fV g are the resultants forces, moments and

transverse shearing forces respectively. femg, fjg and fct 0g are the

strains and curvatures at mid-plane and the transverse shear strains

respectively.The shearcorrection factor is represented byv in Eq.(2).

In order to define the generalized lamination parameters, con-

sider that [14]:

U iREF ¼ maxðU ik Þ ð3Þ

/ik¼

U ikU iREF

ð4Þ

where i ¼ E ;G;Dc ;mc ;at ;Dt and k is the laminate ply. From these

definitions, the lamination parameters related to ½ A, ½B and ½D

matrices can be written as:

n AG;E ;D;c 1;m;c 2

n o ¼

Xn

k¼1

1

T ð z k z k1Þ uf gk

nBG;E ;D;c 1;m;c 2

n o ¼

Xn

k¼1

2

T 2 ð z 2k z 2k1

Þfugk

nDG;E ;D;c 1;m;c 2

n o ¼

Xn

k¼1

4T 3

ð z 3k z 3k1Þfugk

ð5Þ

where

fugk ¼ /E k/Gk

/Dkcos2hk /c 1k

sin2hk /mkcos4hk /c 2k

sin4hk

T

ð6Þ

If all the plies of laminate have the same material properties, it

means that U ik ¼ U iREF

. Thus, fug can be rewritten as:

fugk ¼ 1 1 cos 2hk sin 2hk cos 4hk sin 4hkf gT ð7Þ

In a similar approach, the lamination parameters related to ½ A can

be defined as:

n A

a;t 1;t 2

n o ¼

Xn

k¼1

1

T ð z k z k1Þ

/at k

/Dt kcos2hk

/Dt k sin2hk

8><>:

9>=>; ð8Þ

Similarly, if all plies of the laminate have the same properties, the

lamination parameters related to the transverse shear can be writ-

ten as:

n A

a;t 1;t 2

n o ¼

Xn

k¼1

1

T ð z k z k1Þ

1

cos 2hk

sin 2hk

8><>:

9>=>; ð9Þ

Now, defining the matrix ½nr with r ¼ A; B; D as:

nr ½ ¼

nr E 0 n

r D

0 nr m 0

nr

E 0 nr D 0 n

r

m 00 nr

G 0 0 nr m 0

0 0 0 nr c

1

2 0 nr

c 2

0 0 0 nr c

1

2 0 nr

c 2

nr E 2n

r G 0 0 nr

m 0

266666666664

377777777775ð10Þ

the six independent components of ½ A, ½B and ½D matrices can be

defined as:

f Ag ¼ T n Ah i

fU g

fBg ¼ T 2

4 n

B

fU g

fDg ¼ T 3

12 ½nDfU g

ð11Þ

where

fU g ¼ U E REF U GREF U Dc REF U Dc REF U mc REF U mc REF f g

T ð12Þ

Similarly, the three independent components of ½ A matrix can be

defined in function of the generalized lamination parameters as:

f A

g ¼

n A

a n A

t 10

n A

a n A

t 1 0

0 0 n A

t 2

2664 3775U at REF

U Dt REF

U Dt REF

8><>: 9>=>; ð13Þ

It can be observed that, for a single material laminate, the lam-

ination parameters of the transverse shear in Eq. (9) are identical to

those in Eq. (7) (as shown in [7]). The stiffness invariants in Eqs.

(12) and (13) can be defined in terms of material engineering elas-

tic constants as:

U E ¼ 3E 1 þ ð3 þ 2m12ÞE 2

8ð1 m12m21Þ þ

G12

2

U G ¼ E 1 þ ð1 2m12ÞE 2

8ð1 m12m21Þ þ

G12

2

U Dc ¼ E 1 E 2

2ð1 m12m21Þ

U tc ¼ E 1 þ ð1 2m12ÞE 2

8ð1 m12m21Þ

G12

2

U at ¼ G13 þ G23

2

U Dt ¼ G13 G23

2

ð14Þ

2.2. Finite element modeling

In a linear elastic problem formulated with the finite element

method, the objective is to solve:

½Kfqg ¼ ff g ð15Þ

where ½K is the global stiffness matrix, ff g is the global vector of forces and moments and fqg represent the displacements and rota-

tions of the degrees of freedom. The global stiffness matrix ½K in Eq.

(15) is an assembly of the stiffness matrices of each element which

can be expressed as:

½Ke ¼ 1

2

Z V

½BT ½D½Bdv ð16Þ

where ½B contains the derivative of the shape functions and ½D is a

constitutive matrix composed by ½ A, ½B, ½D and ½ A matrices:

½D ¼

½ A ½B ½0

½B ½D ½0

½0 ½0 ½ A

2

64

3

75ð17Þ

Similarly the global vector of forces is an assembly of the nodal

forces of each element which can be defined as:

ff eg ¼

Z A

½UT fF gdA: ð18Þ

3. Laminate database

3.1. Allowable region

The dependency between lamination parameters, which defines

the allowable region, was investigated in [9,12]. One presented

very complex relations, leading to lack of efficiency on the opti-

mization algorithms, while the other has limited applicability. Inboth studies, it is difficult to obtain the correlation between

168 T.A. Dutra, S.F.M. de Almeida / Composite Structures 133 (2015) 166–177



lamination parameters and ply orientation. A method to obtain ply

orientation from lamination parameters is presented in [5], for a

restricted number of plies.

The evaluation of allowable regions contributes to understand

the lamination parameters not only as an approach for optimiza-

tion, but also as design variables that are subjected to constraints

in the domain ½1; 1 to correspond to a feasible laminate.

Fig. 1 shows an example of the allowable region for a generallaminate with 6 layers orientated with angles multiples of 5. In

this case, even with very small angle increments – compared to

typical laminates increments of 45 –, the allowable region domain

is clearly discrete.

It may be observed from Fig. 1 that the domain of lamination

parameters is not convex neither continuous in the allowable

region. The domain is denser for a larger number of plies, but it

would be continuous and convex only if an infinite number of plies

was considered. To exemplify the increase of the number of points

in the allowable region, Fig. 2 presents the domain for a general

laminate with 12 layers orientated with angles multiples of 11.25.

Therefore, finding values for the lamination parameters that

minimizes the objective function is not enough to solve a laminate

optimization problem. It is important to verify if the parameters

found correspond to a point in the allowable region, i.e., to a feasi-

ble laminate. Ferreira et al. [13] proposed an optimization method

in which the evaluated points come from a previously determined

laminate database, detailed on the next subsection.

3.2. Laminate database

The proposed laminate database correlates plies orientation to

the lamination parameters, defining the set of lamination parame-

ters for each considered laminate. This method assures that the

optimization algorithm will select only valid points of the allow-

able region.

A database is created for each number of plies, and it is fulfilled

with a large number of feasible laminates. Rules based on best

practices for laminates design may be applied to reduce the num-ber of laminates in the database, as well as to select the most suit-

able laminates to be associated with the allowable region domain,

contributing to the efficiency of the optimization algorithm. For

instance, a database may be constructed only with balanced lami-

nates in order to eliminate coupling between in-plane extension

and shear strains (n Ac 1

¼ n Ac 2

¼ 0), or with a maximum value for

parameters nDc 1

and nDc 2

to minimize bending and torsion coupling.

Another practice that may be adopted is to establish a maximum

number for identical consecutive unidirectional plies.

It is worth emphasizing that some design rules, such as mini-mum quantity of consecutive unidirectional plies in a specific ori-

entation [17] or a proportion between the quantity of plies in two

or more different pre-defined orientations [18], may result in sig-

nificant constraints on the allowable region, thus reducing the

design space. However, any set of design guidelines may be easily

used to filter the laminate databases.

The laminate databases are organized in 0.2 width cells with the

aim of increasing the search efficiency. The cells are identified by

the position of parameters n AD

, n Am , nD

D e nD

m . Considering the domain

½1; 1 for the lamination parameters, there are 10 cells for each

parameter, thus 10,000 cells for each database. These four param-

eters were chosen to identify the cells because they are generally

the most relevant ones for the optimization process, since it is

often convenient to avoid extension-bending coupling (using sym-metric laminates, so that ½B ¼ ½0) and coupling between in-plane

and out-of-plane behavior (minimizing n Ac 1

, n Ac 2

, nDc 1

and nDc 1

). The

out of plane stiffness matrix ½ A parameters are also omitted

because the laminates are frequently made of a single material,

and in this case they will correspond to the in-plane stiffness

matrix ½ A parameters.

Thereby, a given point of the database may be defined as a func-

tion of n AD

, n Am , nD

D and nD

m :

P ðn A0

D ; n

A0

m ; nD0

D ; n

D0

m Þ ¼ 1000n A0

D þ 100n

A0

m þ 10nD0

D þ n

D0

m þ 1 ð19Þ

with

ni0

j ¼ int

ni j þ 1

lcel !

ð20

Þ

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

n = 6 layers

ξD

∆

ξ D ν

Fig. 1. Allowable region for lamination parameters nDD and nDm , considering a generallaminate with 6 layers orientated with angles multiples of 5.

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

n = 12 layers

ξD

∆

ξ D ν

Fig. 2. Allowable region for lamination parameters nDD

and nDm , considering a general

laminate with 12 layers orientated with angles multiples of 11.25.


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where int( x) is the integer part of real variable x. For example, a

laminate with parameters n AD

¼ 0:321, n Am ¼ 0:587, nD

D ¼ 0:08

and nDm ¼ 0:123 will belong to cell 3245. As there may be more

than one laminate in each cell, additional information about the

laminate is stored, with the intent to correlate the set of parameters

to its physical representation. Several empty cells may also exist, as

they may be located out of the allowable region.

A laminate database with about 40,000 sets of laminationparameters is considered to represent adequately their corre-

sponding allowable region. As the number of sets varies exponen-

tially with respect to the number of plies, the use of laminate

databases is originally inefficient for a large number of plies. This

problem is eliminated for symmetric laminates with the method

proposed in [16], in which laminates from two databases with ade-

quate sizes are combined using simple mathematical manipulation

of lamination parameters, creating a laminate with large number

of plies, as illustrated in Fig. 3.

The method requires that laminate 2 has an even number of

plies and laminates 1 and 2 are both symmetric. The resulting lam-

ination parameters are thus defined as:

n A

LC i ¼ n1

n1 þ n2n A

1i þ n2

n1 þ n2n A

2i ð21Þ

nDLC

i ¼ n3

1

ðn1 þ n2Þ3 n

D1

i þ n3

2

ðn1 þ n2Þ3 n

D2

i þ 3 n2

1n2

ðn1 þ n2Þ3 n

A2

i

þ 6 n1n

22

ðn1 þ n2Þ3 n

Bup

2

i ð22Þ

where i ¼ ðD; c 1;m; c 2Þ, n ALC

i and nDLC

i are the lamination parameters of

the combined laminate, n A1

i and nD1

i are the lamination parameters

of laminate 1, n A2

i and nD2

i are the laminate parameters of laminate

2 and nBup

2

i corresponds to the lamination parameters of the coupling

stiffness matrix ½B considering only the upper part of laminate 2.

The number of plies of laminates 1 and 2 are represented respec-tively by n1 and n2.

A target database must be then defined in order to drive the

choice of combined laminates. In [16] this database is created so

that the lamination parameters sets are uniformly distributed in

the cells (ideally 81 sets per cell). In the present work, the database

construction is based on a uniform distribution of the lamination

parameters throughout the allowable region, using the relations

between the lamination parameters presented by [7]. The domain

of lamination parameters nD and nm for this target laminate data-

base is shown in Fig. 4.

Table 1 presents the laminate databases used in this paper. The

original algorithm was employed for databases with up to 10 lay-

ers. For more than 10 layers, combinations of existent databases

were used. All databases were created using routines implementedin FORTRAN 77 .

4. Metamodel

4.1. Objective function

The optimization study presented herein refers to flexibility

problems. In this case, the objective function to be minimized is

the maximum displacement. Eqs. (5), (6) and (8) show that, from21 lamination parameters that constitute the stiffness matrices,

+ =

Laminate 1n1 layers

Laminate 2n2 layers

Combined Laminaten1 + n2 layers

Fig. 3. Symmetric laminates combination [16].

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Objective Laminate

ξ∆

ξ ν

Fig. 4. Domain of lamination parameters for the target laminate database.

Table 1

Laminates databases configurations.

Number of Layers Dh Total Laminates

6 3 29791

7 7.5 28651

8 7.5 28651

9 11.25 59049

10 11.25 59049

11 11.25 48296

12 11.25 3652213 11.25 38105

14 11.25 48039

16 11.25 46809

20 15 47895

32 15 54364

40 15 54994




only 14 depend on the fibers orientation. Considering only single

material symmetric laminates, the optimization problem degrees

of freedom is reduced to 8 independent parameters.

Linear and quadratic metamodels, based on Taylor series, are

presented in [16]. The estimated objective function in terms of

the lamination parameters components nsi; j and a known value of

the objective function F ðfnsg0Þ is given by:

F̂ ðfnsgÞ ¼ F ðfnsg0

Þ þXnpi¼1

@ F

@ nsi

Dnsi þ

1

2

Xnpi¼1

Xnp j¼1

@ 2F

@ nsi @ ns

j

DnsiDn

s j ð23Þ

where np is the number of lamination parameters. In the present

work, the lamination parameters vector considered is

fnsg ¼ f n AD n A

c 1n Am n A

c 2nDD nD

c 1nDm nD

c 2g

T .

To obtain the gradients of the objective function through a finite

differences method in the quadratic metamodel adopted in [14]

and [16], it is necessary to disturb the lamination parameters in

pairs, which is computationally expensive. Considering the eight

lamination parameters shown above, it is necessary to know the

objective function value in 144 points inside the allowable region.

Aiming to avoid this, the present work makes use of explicit calcu-

lation of the gradients, as in [15].Assuming the objective function to be minimized as a combina-

tion of the nodal displacements q:

fqðfnsgÞg ¼ fqðfnsg0

Þg þXnpi¼1

@ fqg

@ nsi

Dnsi þ

1

2

Xnpi¼1

Xnp j¼1

@ 2fqg

@ nsi @ ns

j

DnsiDn

s j

ð24Þ

it is necessary to consider its gradients with respect to the lamina-

tion parameters. Differentiation of Eq. (15) with respect to the lam-

ination parameters results in

@ ½K

@ nsi

fqg þ ½K @ fqg

@ nsi

¼ f0g ð25Þ

From this expression the first derivative of q is:

@ fqg

@ nsi

¼ ½K1 @ ½K

@ nsi

fqg ð26Þ

Differentiating Eq. (25) and taking into account the linear rela-

tion between the stiffness matrix ½K and the lamination parame-

ters, the second derivative is obtained:

@ 2fqg

@ nsi@ ns

j

¼ ½K1 @ ½K

@ nsi

@ fqg

@ ns j

þ @ ½K

@ ns j

@ fqg

@ nsi

! ð27Þ

The global stiffness matrix ½K derivatives are obtained by dif-

ferentiating the expression in Eq. (16) for the stiffness matrix of

each element:

@ ½K e

@ nsi ¼

Z 1

1

Z 1

1 ½Bðn;gÞT @ ½D

@ nsi ½Bðn;gÞj J jdndg ð28Þ

The operations on the stiffness matrix are made only once for

each initial point. The advantage of this formulation over the

numerical derivatives is the capacity of estimating the neighbor

points from a single known value of the objective function.

Unlike buckling problems, flexibility problems have highly non-

linear relations between the objective function and lamination

parameters, so that the estimations found in the metamodel are

not very well-behaved for large increments.

The estimations of the objective function also present trunca-

tion errors compared to the finite elements solution.

Nevertheless, these errors are relatively small and they do not

affect the optimization result.

The numerical efficiency may be further increased by notingthat @ ½K=@ nsi is constant since global matrix ½K is linear in terms

of lamination parameters. Therefore, the global matrix can be com-

puted as:

½K ¼Xnpi¼1

@ ½K

@ nsi

nsi ð29Þ

Notice that the derivatives @ ½K=@ nsi need to be computed a sin-

gle time for each problem since they are constants. Therefore, all

numerical integrations and assembly operations involved in thecomputations of these derivatives needs to be computed only np

times (number of the lamination parameters involved). This fea-

ture dramatically reduces the time for computing the global matrix

using Eq. (29).

4.2. Optimization algorithm

The optimization algorithm implemented was developed by

[13] and later used in [16], with the intent to be simple, robust

and reliable. For a given start point, the objective function is calcu-

lated by finite elements modeling and its gradients are calculated.

The objective function is then estimated for each point in the lam-

inate database that is located within a given distance d from the

initial point. A local optimum is determined and becomes thenew start point. The procedure is repeated until the local optimum

obtained is the start point itself.

Distance d is a non-Euclidean measure defined by [13] that

allows taking into account the sensitivity of the objective function

with respect to each of the lamination parameters:

d ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiXnpi¼1

xi nsi

v

nsi

p

2

v uut ð30Þ

where nsi

v

are the lamination parameters of the neighboring point,

nsi

p

are the lamination parameters of the start point, np is the num-

ber of lamination parameters and xi is given by:

xi ¼j @

F @ nsi j ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPnp

i¼1@ F @ ns

i

2

r ð31Þ

Distance d must be carefully chosen to avoid inefficiency of the

algorithm and large truncation errors due to nonlinearities.

The optimization algorithm can be summarized in the following

steps:

- Selection of a start point in the laminate database

- Calculation of the objective function and its gradients in this

point.

- Estimation of the objective function for neighbor points within

a distance from the start point

- Determination of a local optimum and assignment of this pointas new start point.

- If the optimal point is equal to the start point, the current point

is a global optimum. Else, the algorithm is repeated until the

global optimum is found.

5. Laminated plate under bending and torsion loads

Consider the plate and its dimensions presented in Fig. 5. Those

dimensions were adopted in order to have an 10 1 aspect ratio

between length and width. They provide it a mechanical behavior

similar to a beam. The mechanical properties are listed in Table 2

and the boundary conditions are presented in Table 3.

A bicubic lagrangian plate element with 16 nodes was used in

the finite element modeling. The nodal displacements are: u, v ,w, b x and b y. All finite element modeling subroutines were


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implemented in Fortran 77. This element uses a high degree of the

polynomial interpolation functions thus minimizing the shear

locking. A mesh convergence analysis was performed for two dif-

ferent load cases as presented in Table 4. The mesh convergence

results are presented in Table 5.

In view of the obtained results for the mesh convergence anal-

ysis, it was adopted a mesh grid with 10 elements that gives a

1 1 dimensional aspect ratio for the element. This configuration

is used for all the studies in this work. Fig. 6 presents the mesh grid

and its respective nodes.

5.1. Algorithm validation

In order to validate the optimization algorithm, a very complete

study was performed. As seen in Chapter 4, the metamodel is based

on the quadratic estimate of the objective function viewing that

the linear approximation does not give satisfactory results. This

shall be numerically demonstrated in what follows. Thus, the esti-

mate of the transversal displacement and the rotations at the

extremity of the plate were compared to those obtained by the

finite element model. Two different laminates were considered:

½06 and ½45= 45=45s. The neighborhood for each one of these

laminates was analyzed. The mechanical properties are listed in

Table 2 and the boundary conditions are presented in Table 3.

The applied load cases are listed in Table 6. For the laminate ½06

and its neighbors the load case 1 was applied. For the laminate

½45= 45=45s

and its neighbors the load case 2 was applied.

(See Figs. 7 and 8 for the path covered by the optimization algo-

rithm in both cases)

The results obtained by linear and quadratic approximation for

each laminate are presented in Tables 7 and 8. It can be observed

from Tables 7 and 8 that linear metamodel provides an acceptable

estimate for the transversal displacements. However the linear

metamodel solution does not correspond to that obtained by finite

element modeling. On the other hand, the quadratic metamodel

could provide good solutions for both analysis.

An important aspect is also inferred from Tables 7 and 8: the

search distance. The careless choice of the search distance for both

metamodels may lead the algorithm to inaccurate estimates of the

480

48

Fig. 5. Laminated plate geometry. Dimensions in [mm].

Table 2

Mechanical properties of a typical unidirectional carbon/epoxy AS4 3501-6 [20].

Mechanical property Value

Longitudinal Modulus, E1 [GPa] 147.0

Transverse in-plane Modulus, E2 [GPa] 10.3

Poisson ratio, m 12 0.27

In-plane shear Modulus, G12 [GPa] 7.0

Out-of-plane Shear Modulus, G13 [GPa] 7.0

Out-of-plane Shear Modulus, G23 [GPa] 3.7

Ply Thickness, t [mm] 0.22

Table 3

Applied boundary conditions.

Nodal coordinate [mm] Boundary condition

x y

0 0 u ¼ v ¼ w ¼ b x ¼ b y ¼ 0

0 16 u ¼ w ¼ b x ¼ b y ¼ 0

0 32 u ¼ w ¼ b x ¼ b y ¼ 0

0 48 u ¼ w ¼ b x ¼ b y ¼ 0

Table 4

Applied load cases in the convergence analysis.

Nodal Coordinate

[mm]

Case 1 [N ] Case 2 [N ]

x y

480 0 F z ¼ 1 F z ¼ 3

480 48 F z ¼ 1 F z ¼ 3

Table 5

Mesh convergence results.

Number of Elements Element Dimension [mm] Case 1 Case 2

Width Height w ½mm Difference (%) w y Difference (%)

4 40 16 54.479 0.097 0.2511 3.38

8 20 16 54.495 0.068 0.2527 2.77

10 16 16 54.499 0.061 0.2534 2.50

20 8 16 54.532 – 0.2599 –

y

x

Fig. 6. Laminated plate mesh grid.

Table 6

Nodal load cases.

Nodal Coordinates

[mm]

Bending [N ] Torsion [N ]

x y

480 0 F z ¼ 1 F z ¼ 3

480 16 F z ¼ 1 F z ¼ 1

480 32 F z ¼ 1 F z ¼ 1

480 48 F z ¼ 1 F z ¼ 3




objective function and consequently may not find the global opti-

mal design.

5.2. Search distance

From the previous results it can be observed that a maximum

search distance d for the quadratic metamodel is dmax ¼ 0:5. For

the linear metamodel this maximum distance reduces todmax < 0:3. The maximum values is known but correct choice for

the distance shall consider other issues, for example, the number

of computations of the metamodels and the finite element model.

Thus an optimization study for a 6 layers laminate was performed

in order to find the best distance d such as 0:1 6 d 6 0:5 for the tor-

sion load case as listed in Table 6. Linear and quadratic metamodels

were analyzed and their solutions compared to each other. The

mechanical properties are listed in Table 2 and the boundary condi-

tions are presented in Table 3. A laminate with the most number of

±45 layers was expected as the optimum. Dueto theincreasein the

search path, the laminate with most number of plies at 90 was

selectedas initial point. The results are presented in Tables9 and 10.

It can be verified that only for small distances the linear meta-

model is able to achieve the correct solution and consequently it is

neither efficient nor robust. It requires several computations of the

metamodel and the objective function. However, the quadratic

metamodel was able to find the optimum laminate for all evalu-

ated search distances. Given that using large distances, the algo-

rithm requires a very small computational cost, the quadratic

metamodel with the search distance d ¼ 0:5 was chosen corrobo-

rating that linear metamodels are not indicated for stiffness opti-

mization problems.

5.3. Initial points

The final validation performed was to confirm that the algo-

rithm was able to arrive at same optimum independent on the ini-

tial point. Thus an optimization study several laminates with

different initial points was performed. The mechanical propertiesare listed in Table 2 and the boundary conditions are presented

in Table 3. The applied load cases are listed in Table 6. The results

are presented in Tables 11 and 12.

From the previous results it is observed that the algorithm

always arrives at the global optimum independently on the initial

point. Thus it can be confirmed that the algorithm is robust, reli-

able and efficient. The next section presents the optimization anal-

ysis laminated plates under combined bending/torsion loads.

6. Multicriteria optimization

Complex structures (i.e. an aircraft wing or fuselage), are sub-

jected to complex loads. In this sense, a real stiffness optimization

problem may have more than one objective function.Consequently, the optimum point is the one that best meets their

relationship. With the aim of propose a multicriteria optimization

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

Laminate 6 layers − d = 0.5

ξD

∆

ξ D ν

Fig. 7. Path covered by the optimization algorithm for a 6 layers laminate subjected

to pure bending with the laminate 90½ 6 as initial point.

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Laminate 6 layers − d = 0.5

ξD∆

ξ D ν

Fig. 8. Path covered by the optimization algorithm for a 6 layers laminate subjected

to pure torsion with the laminate ½906 as initial point.

Table 7

Linear and quadratic estimate for the transversal displacement w 124 obtained by the metamodel and compared to the finite element solution. Initial transversal displacement for

the laminate ½06 is w 124 ¼ 108:9 mm.

Neighbor laminate d FEM/ Linear estimate Quadratic estimate

w124 ½mm w124 ½mm Difference (%) w124 ½mm Difference (%)

½3= 12= 69 s 0.118 116.9 116.4 0.4 116.8 0.1

½6=15=72s 0.169 120.3 119.0 1.0 120.2 0.1

½3= 18= 66 s 0.181 122.0 120.4 1.3 121.9 0.1

½9= 12= 90s 0.206 121.0 115.3 5.0 122.1 0.9

½0=60= 3 s 0.453 141.6 133.5 6.0 140.5 0.7


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method, this work have adopted a combined bending and torsion

load according to Table 13.

In order to perform the multicriteria optimization method, it

was defined a strategy based on weights for the objective

functions. The transversal displacements of nodes 121 and 124

were chosen as sub-objective functions according to Eqs. (32)

and (33). The average transversal displacements and rotations

are obtained and a primary objective function is defined and used

by the optimization algorithm.

F P ¼ F flexP flex þ F tor P tor ð32Þ

where

F flex ¼ w124þw121

2

F tor ¼ w124w121

2

ð33Þ

and w124 is the transversal displacement of node 124, w121 is the

transversal displacement of node 121, P flex is the weight for the

sub-objective function related to the transversal displacements

and P tor is the weight for the sub-objective function related to the

rotations at plate extremity. Due to the plate aspect ratio between

length and width, the transversal displacements at plate edge can

be considered quasi-linear in y direction. So, Eq. (33) is valid.This

work proposed two distinct analyses: one minimizing simultane-

ously the transversal displacements and rotations and other mini-

mizing the rotations. Thus, a laminated with mechanical

properties listed in Table 2 and boundary conditions presented in

Table 3 was applied. The applied load cases are listed in Table 13.

Based on the previous results the selected initial point is the closest

to the quasi-isotropic (n A;B;DD;c 1 ;m;c 2

¼ 0). This reduces the calls to the

finite element model and to the metamodel. In order to minimize

simultaneously the transversal displacements and rotations,

applied weights are:

P flex ¼ 1

P tor ¼ 10 ð34Þ

Table 8

Linear and quadratic estimate for the rotation w124 obtained by the metamodel and compared to the finite element solution. Initial rotation for the laminate ½45= 45=45s is

w124 ¼ 0:0607.

Neighbor laminate d FEM Linear estimate Quadratic estimate

w124 w124 Difference (%) w124 Difference (%)

½51= 51= 51s 0.119 0.0629 0.0628 0.1 0.0629 0.0

½54= 51= 72s 0.217 0.0664 0.0657 1.1 0.0664 0.0

½57= 57= 39s

0.312 0.0695 0.0679 2.3 0.0692 0.4

½33= 60= 66 s 0.433 0.1118 0.0761 46.9 0.1019 9.7

½57=45=90s 0.780 0.0791 0.0445 77.5 0.1069 26.1

Table 9

Results for the optimization of a 6 layers laminate under torsion loads with ½906 as

initial point using the linear metamodel.

Linear metamodel

Distance

d

Calls to

MEF

Calls to

metamodel

Searched

cells

Optimum laminate

0.1 26 69,809 2328 ½45= 45= 45s

0.2 17 49,154 1691 ½45= 48= 48s

0.3 26 85,047 3024 ½39=51=6s

0.4 27 81,185 2714 ½48

= 42

= 42

s0.5 46 14,5787 5330 ½57= 33= 33s

Table 10

Results for the optimization of a 6 layers laminate under torsion loads with ½906 as

initial point using the quadratic metamodel.

Quadratic metamodel

Distance

d

Calls to

MEF

Calls to

metamodel

Searched

cells

Optimum laminate

0.1 23 60,597 2037 ½45= 45= 45s

0.2 12 32,203 1077 ½45= 45= 45s

0.3 10 28,062 963 ½45= 45= 45s

0.4 10 27,921 956 ½45= 45= 45s

0.5 10 27,921 956 ½45

= 45

= 45

s

Table 11

Results obtained for the optimization of generic laminates (with 6, 16 and 40 layers) using different initial points. The laminates are subjected to pure bending loads.

Initial point Number of layers Distance d Calls to MEF Calls to metamodel Searched cells

P1 6 0.5 20 42,334 1508

P2 6 0.5 12 31,125 1036

P3 6 0.5 6 15,448 603

P4 16 0.5 11 21,279 1017

P5 16 0.5 11 33,484 1219

P6 16 0.5 6 29,875 937

P7 40 0.5 12 24,467 1018

P8 40 0.5 11 30,973 945

P9 40 0.5 6 38,369 941

P1 – ½906 .

P2 – ½45= 45= 45 s .

P3 – ½30=90= 30s .

P4 – ½9016 .

P5 – ½45=ð45 Þ2=ð45Þ2=ð45Þ2=45s .

P6 – ½11:25=0=ð56:25 Þ2=78:75=33:75= 22:5= 56:25s .

P7 – ½9040.

P8 – ½ð45= 45Þ2=ð45=45Þ2=ð45Þ22s .

P9 – ½75= 60=45=ð45Þ2=30=15=ð0Þ2=90=45= 30=45=75=ð0Þ2=90= 30= 45=45s .




This choice is related to the magnitude of these sub-objective

functions. Tables 14 and 15 present the results obtained for multi-criteria optimization for different laminates considering the

weights in Eq. (34).

For the multicriteria optimization aiming to minimize the rota-

tions at plate extremity, the weights are presented in Eq. (35).

Tables 16 and 17 present the result obtained for this study mini-

mizing the rotations.

P flex ¼ 1

P tor ¼ 100 ð35Þ

Table 13

Nodal load case for a combined bending and torsion load.

Nodal Coord. [ mm] Combined bending-torsion load [N ]

x y

480 0 F z ¼ 7

480 16 F z ¼ 3

480 32 F z ¼ 1

480 48 F z ¼ 5

Table 14

Results for multicriteria optimization minimizing transversal displacements and rotations at plate extremity.

Number of layers Distance d Calls to MEF Calls to metamodel Searched cells

6 0.5 8 23,764 946

7 0.5 7 22,114 1015

8 0.5 6 20,862 1102

9 0.5 5 40,793 1059

10 0.5 4 37,193 960

11 0.5 5 32,123 874

12 0.5 5 26,719 911

13 0.5 6 28,307 896

14 0.5 4 30,440 729

16 0.5 4 26,111 753

20 0.5 4 27,897 776

32 0.5 5 37,080 854

40 0.5 4 34,278 746

Table 15

Optimum laminates for multicriteria optimization minimizing transversal displacements and rotations at plate extremity.

Number of layers Optimum laminate

6 ½18=3=63s

7 ½15=7:5=90=37:5=90=7:5= 15

8 ½0= 22:5=60=7:5s

9 ½0= 22:5=56:25=ð11:25Þ4=56:25= 22:5=0

10 ½0= 22:5=56:25=11:25=78:75s

11 ½22:5=33:75=78:75=ð0Þ2= 22:5=ð0 Þ2=78:75=33:75= 22:5

12 ½0= 22:5=78:75=ð0Þ2= 22:5s

13 ½0=67:5= 11:25=78:75=ð0Þ2= 22:5=ð0Þ2=78:75= 11:25=67:5=0

14 ½0= 67:5=90=22:5= 22:5= 11:25= 22:5s

16 ½0=67:5=90= 78:75=22:5= 22:5= 11:25= 22:5s

20 ½ð0Þ6=75=0= 15= 30s

32 ½30=0=ð90Þ2=15=ð30 Þ2=45=ð0Þ3= 15=ð0Þ3= 30s

40 ½0=ð30Þ3= 75=45= 30=45=ð45Þ2=ð0 Þ6=75=0= 15= 30s

Table 12

Results obtained for the optimization of generic laminates (with 6, 16 and 40 layers) using different initial points. The laminates are subjected to pure torsion loads.

Initial point Number of layers Distance d Calls to MEF Calls to metamodel Searched cells

P1 6 0.5 10 27,921 956

P2 6 0.5 6 21,221 878

P3 6 0.5 11 32,258 1103

P4 16 0.5 11 37,222 1336

P5 16 0.5 5 29,462 843

P6 16 0.5 10 43,173 1443P7 40 0.5 10 52,323 1327

P8 40 0.5 5 37,166 731

P9 40 0.5 10 49,541 1244

P1 – ½90 6 .

P2 – ½30=90= 30s;.

P3 – ½06 .

P4 – ½9016 .

P5 – ½11:25=0=ð56:25Þ2=78:75=33:75= 22:5= 56:25s .

P6 – ½016 .

P7 – ½9040 .

P8 – ½75= 60=45=ð45Þ2=30=15=ð0Þ2=90=45= 30=45=75=ð0Þ2=90= 30= 45=45s .

P9 – ½040 .


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7. Conclusion

This paper presents a robust, effective and efficient method for

the optimization of stiffness in composite laminate plates, based

on the use of lamination parameters stored in laminate databases.A formulation that enables the use of hybrid laminates is adopted

for the lamination parameters. The use of laminate databases

assures that any optimal set of parameters found corresponds to

a physical laminate.

An analytical formulation is proposed for obtaining the gradi-

ents of the objective function, rather than the finite differences

method approach adopted in previous works. This increases the

performance of the algorithm by eliminating the need of knowing

numerous values of the objective function. However it requires

operations on the stiffness matrix, which must be available. In this

work, the stiffness matrix is easily taken from the finite element

models, which are totally implemented in FORTRAN 77 routines.

The combination of databases proposed for laminates with large

number of plies also differs slightly from the one used in previousworks, aiming to better distribute the lamination parameters

throughout the allowable region.

The investigation on the influence of the search distance upon

the optimization algorithm performance led to the determination

of an optimal search distance for the analyzed cases.

Furthermore, it was demonstrated that the linear metamodel is

not adequate to optimization in flexibility problems, as it requires

very restrict search distances to converge to an optimal result,

reducing significantly the efficiency of the algorithm.

The influence of the start point selection upon the algorithm

performance was also studied. As expected, its influence is limited

to the number of iterations of the algorithm, which increases for

start points farther from the optimal solution. In all cases, the same

optimal solution was found for a given loading case and optimiza-

tion criterion, independently of the start point.

The proposed multicriteria optimization strategy consists on

establishing weights for the components of displacement (in this

case transverse displacement and rotation) to compose the objec-

tive function. As expected, the optimal solutions found for com-

bined loading cases were between the optimal solutions for thesingle loading cases

It was highlighted that the design criteria must be evaluated for

each loading case before the definition of the weights for each

component of the objective function, otherwise the solution may

not be adequate. One option is to verify the proportion between

the allowable values for each component of the displacement.

Additionally, design criteria and constraints may be applied as

rules for the laminate databases construction. This may be consid-

ered as one of the greatest advantages observed for the use of lam-

inate databases in optimization problems. Besides enhancing the

performance of the algorithm, eliminating the options with unde-

sired characteristics, it aids to realize the impact of very restrictive

design rules on the optimization effectiveness.

Acknowledgements

This work was funded by Brazilian agencies CNPq (Grants

133954/2013-7 and 303799/2010-2) and FAPESP (Grants

2006/61257-5 and 2008/57866-1).

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Table 16

Results for multicriteria optimization minimizing priority rotations at plate extremity.

Number of layers Distance d Calls to MEF Calls to metamodel Searched cells

6 0.5 5 14,027 690

7 0.5 9 34,196 1726

8 0.5 5 19,382 1071

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Table 17

Optimum laminates for multicriteria optimization minimizing priority rotations at plate extremity.

N umbe r of laye rs Op timum laminat e

6 ½60= 18=15s

7 ½22:5=7:5=60=15=60=7:5= 22:5

8 ½30=7:5=90=22:5 s

9 ½22:5=45=90=11:25=67:5=11:25=90=45= 22:5 10 ½22:5=67:5=11:25=22:5=67:5s

11 ½22:5=0=ð56:25Þ2= 22:5= 11:25= 22:5=ð56:25Þ2=0=22:5

12 ½78:75=ð0Þ2=11:25= 33:75= 22:5s

13 0=11:25=22:5= 56:25=67:5= 22:5= 11:25= 22:5=67:5= 56:25=22:5=11:25=0½

14 ½0=ð78:75Þ2=56:25= 22:5=0= 22:5 s

16 ½11:25=78:75= 45=56:25=90=0=ð22:5Þ2s

20 ½75= 45=45= 60= 30=15=90=ð15Þ2= 30s

32 0=15= 45= 30=90= 15=90= 75=75= 60= 15= 30=0=45= 15= 30½ s

40 45= 45=45=ð15Þ2=0=90= 60= 45=45=15= 15=60= 45= 60=75=ð15Þ2=90= 15

s


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Composite Plate Stiffness Multicriteria Optimization Using Lamination Parameters 2015 Composite Structures