Faculty of Civil Engineering, Universiti Teknologi Malaysia 81310 UTM Skudai, Johor, Malaysia E-mail: [email protected]
Abstract
This paper presents an application of the imperialist competitive algorithm (ICA) in optimization of composite structures design. The recently introduced algorithm has proven its excellent capabilities, such as faster convergence and better global optimum achievement. In this paper, imperialist competitive algorithm (ICA) is used to demonstrate its application in finding the optimal design of laminated composite structures due to the various failure criteria. The proposed method can be used in various multi-objective optimization problems. The effectiveness of the proposed method, in comparison to Genetic Algorithm (GA), is proven through solving several examples of composite structure problems. Keywords: Imperialist Competitive Algorithm (ICA), Genetic Algorithm, Laminated
Composite Structure, Failure Criteria. 1. Introduction Laminated composite materials find a wide range of applications in structural design, particularly in the field of automotive, aerospace and marine engineering. This is primarily due to the high specific strength and stiffness values with minimum weight that these type of materials offer. Although
Imperialist Competitive Algorithm and its Application in Optimization of Laminated Composite Structures 175
composite materials are attractive replacement for metallic materials for many structural applications, the design and analysis is more complex than those of metallic structures.
Composite materials are usually understood as the combination of two or more materials on a microscopic scale to form a useful third material [1]. Laminated composites are usually designed due to the designer’s needs by choosing the thickness, orientation and number of laminae. The thickness and orientation of the laminae are usually limited to some set values due to manufacturing limitations. Searching for the optimum solution in laminated composite structures is a discrete optimization problem. Evolutionary algorithms, such as GA, PSO and SA were suggested in the past decades for solving optimization problems in different fields. Also, these methods have been recently applied to laminated composite problems. The genetic algorithm has been widely used to find the optimal design of laminated composite structures due to the ability of its algorithm to solve this kind of optimization problems [2-13]. The advantages of the use of GAs include the following: (i) they do not require gradient information and can be applied to problems where the gradient is hard to obtain or simply does not exist; (ii) if correctly tuned, they do not get stuck in local minima; (iii) they can be applied to nonsmooth or discontinuous functions; and (iv) they furnish a set of optimal solutions instead of a single one, thus giving the designer a set of options. On the other hand, the use of GAs has a number of known disadvantages, which include the following: (i) they require the tuning of many parameters by trial and error to maximize efficiency; (ii) the a priori estimation of their performance is an open mathematical problem; and (iii) an extremely large number of evaluations of the objective function are required to achieve optimization, which can make the use of GAs economically. The basic parts of a GA are: the variable coding, the selection method, the genetic operators and the how the constraints are handled [14].
In computer science, the Imperialist Competitive Algorithm (ICA) [15] is a computational method that is used to solve many types of optimization problems. Like most of the methods in the field of evolutionary computation, ICA does not need the gradient of the function in its optimization process.
Specifically, ICA can be thought of as the social counterpart of genetic algorithms (GAs). ICA is the mathematical model and computer simulation of human social evolution, while GAs is based on the biological evolution of species.
Figure 1 shows the flowchart of the Imperialist Competitive Algorithm. This algorithm starts with an initial population. Each individual of the population is called a country. Countries are the counterpart of chromosomes in GAs. After evaluating the cost function of each country, some of the best of them (in optimization terminology, countries with the least cost) are selected to form the initial empires by controlling the other countries (colonies). All the colonies are divided among the initial imperialists based on their power. The power of each country, the counterpart of fitness value in the GA, is inversely proportional to its cost. The initial imperialist form states together with their colonies to form the initial empires.
176 Behzad Abdi, Hamid Mozafari, Amran Ayob and Roya Kohandel
Figure 1: Flowchart of the Imperialist Competitive Algorithm (ICA). [17]
Start Is there an empire with no colonies
Eliminate this empire
Stop condition satisfied
END
Assimilate colonies
Exchange the positions of that imperialist and the colony
Is there a colony in an empire which has lower cost than that of the imperialist
Compute the total cost of all empires
Imperialistic Competition
Yes
Yes
Yes
NoNo
No
Initialize the empires
Revolve some colonies
Unite Similar Empires
After forming initial empires, the evolution begins. The colonies in each empire start moving toward their relevant imperialist country. This movement is a simple model of assimilation policy which was pursued by some of the imperialist states. Besides assimilation, revolution is another operator of this algorithm. Revolution occurs in some of the colonies by making random changes to their position in the socio-political axis. The total power of an empire depends on both the power of the imperialist country and the power of its colonies.
Imperialistic Competition is another step of the algorithm. All empires try to take the possession of colonies of other empires and control them. The imperialistic competition gradually brings about a decrease in the power of weaker empires and an increase in the power of more powerful ones. The imperialistic competition is modeled by picking some (usually one) of the colonies of the weakest empire and making a competition among all empires to possess these (this) colonies. The above steps continue until a stop condition is satisfied by reaching to an acceptable suboptimal solution [16-20].
One of the important issues in the optimization problems is the constraints handling. In the optimization of laminated composite structures one usual constraint is the symmetry and balance of the laminate. In general, it is taken into account using a data structure strategy which contains of writing code only for half of laminate and considering that each stack of the laminate is made of two laminate with the same orientation but opposite signs [3, 6]. Data structure, repair strategy and penalty functions are the most common ways to treat constraints [4, 6 and 9].
The structural failure constraints in optimization of laminated composite structure are often handled by using penalty function. The different forms of construction of the objective function with the penalty terms were studied by Le riche and Haftka [6]. Several failure criteria were studied by Lopes and Luerson [14] and by Narayana and et al [21].
In optimization of laminated composite structures, two types of objective (cost) function have been proposed. When considering the strength of the structure, the buckling load [2, 4 and 22], strength, stability and the stiffness in one direction are maximized [23]. When considering the benefit of cost, material cost, manufacturing cost and weight are minimized [24].
In this paper, the imperialist competitive algorithm is introduced and subsequently its application in optimizing laminated composite structures. The rest of this paper is organized to introduce Tsai – Wu (TW) and maximum stress failure criterion (MS) that used as constraints in the optimization of composite structure. The results of imperialist competitive algorithm are compared with the previous study which had used the genetic algorithm for optimization.
Imperialist Competitive Algorithm and its Application in Optimization of Laminated Composite Structures 177
2. Formulation of Imperialist Competitive algorithm 2.1. Creation of Initial Empires
The goal of optimization is to find an optimal solution in terms of the variables of the problem. An array of variable values to be optimized is formed. In the genetic algorithm, this array is called “chromosome”, but in ICA the term “country” is used. In an varN -dimensional optimization problem, a country is a var1 N× array. This array is defined as follow,
var1 2 3[ , , , , ]NCountry p p p p= … (1) where ip is the variables to be optimized. The variable values in the country are represented as floating point numbers. Each variable in the country can be interpreted as a socio-political characteristic of a country. From this point of view, all algorithms search for the best country with the best combination of socio-political characteristics such as culture, language, economical policy, and even religion. From optimization point of view this leads to finding the optimal solution of the problem, Figure 2 shows the interpretation of country using some of socio-political characteristics.
Figure 2: The candidate solutions of the problem, called country. [15]
1 2 3[ , , , . . . , ]
v a rNc o u n tr y p p p p=
Language Economical PolicyCulture …..… Religion
The cost of a country is found by evaluating the cost function f at variables
var1 2 3( , , ,..., )Np p p p . So,
var1 2 3( ) ( , , ,..., )Ncost f country f p p p p= = (2) To start the optimization algorithm, initial countries of size CountryN are produced. impN of the
most powerful countries to form the empires are selected. The remaining colN of the initial countries will be the colonies each of which belongs to an empire.
To form the initial empires, the colonies are divided among imperialists based on their power. That is, the initial number of colonies of an empire should be directly proportionate to its power. To proportionally divide the colonies among imperialists, the normalized cost of an imperialist is defined by
max{ }n n iiC c c= − (3)
where nc is the cost of the thn imperialist and nC is its normalized cost. Having the normalized cost of all imperialists, the normalized power of each imperialist is defined by
1
imp
nn N
ii
CpC
=
=
∑ (4)
The initial colonies are divided among empires based on their power. Then the initial number of colonies of the thn empire will be
. . { . }n n colN C round p N= (5)
178 Behzad Abdi, Hamid Mozafari, Amran Ayob and Roya Kohandel
where . .nN C is the initial number of colonies of the nth empire and colN is the total number of initial colonies. To divide the colonies, . .nN C of the colonies are randomly chosen and given to the
thn imperialist. These colonies along with the nth imperialist form the thn empire. Figure 3 shows the initial empires. As shown in this figure, bigger empires have greater number of colonies while weaker ones have less. In this figure, Imperialist 1 has formed the most powerful empire and consequently has the greatest number of colonies.
Figure 3: Generation of initial empires. [15]
2.2. Assimilation: Movement of Colonies toward the Imperialist
Pursuing assimilation policy, the imperialist states tried to absorb their colonies and make them a part of themselves. More precisely, the imperialist states made their colonies to move toward themselves along different socio-political axis such as culture, language and religion. This process is modeled by moving all of the colonies toward the imperialist along different optimization axis, as shown in Figure 4. Considering a two-dimensional optimization problem, in this figure the colony is absorbed by the imperialist in the culture and language axes. Then the colony will get closer to the imperialist in these axes. Continuation of assimilation will cause all the colonies to be fully assimilated into the imperialist.
Figure 4 shows a colony moving toward the imperialist by units. The new position of the colony is shown in a darker color. The direction of the movement is the vector from the colony to the imperialist. In this figure x is a random variable with uniform (or any proper) distribution. Then
Figure 4: Movement of colonies toward their relevant imperialist. [15]
( )0x ~ U ,β d× (6) where β is a number greater than 1 and d is the distance between the colony and the imperialist state. The condition 1β > causes the colonies to get closer to the imperialist state from both sides.
Imperialist Competitive Algorithm and its Application in Optimization of Laminated Composite Structures 179
Assimilating the colonies by the imperialist states does not result in direct movement of the colonies toward the imperialist. That is, the direction of movement is not necessarily the vector from colony to the imperialist. To model this fact and to increase the ability of searching more area around the imperialist, a random amount of deviation is added to the direction of movement. Figure 5 shows the new direction. In this figure θ is a parameter with uniform (or any proper) distribution. Then
( ),θ ~ U -γ γ (7) where γ is a parameter that adjusts the deviation from the original direction. Nevertheless the values of β and γ are arbitrary, and in many implementations a value of about 2 for β and about / 4π for γ results in good convergence of countries to the global minimum.
Figure 5: Movement of colonies toward their relevant imperialist in a randomly deviated direction. [15]
2.3. Revolution; A Sudden Change in Socio-Political Characteristics of a Country
Revolution is a fundamental change in power or organizational structures that takes place in a relatively short period of time. That is, instead of being assimilated by an imperialist, the colony randomly changes its position in the socio-political axis. Figure 6 shows the revolution in Culture-Language axis. The revolution increases the exploration of the algorithm and prevents the early convergence of countries to local minimums. The revolution rate in the algorithm indicates the percentage of colonies in each colony which will randomly change their position. A very high value of revolution decreases the exploitation power of algorithm and can reduce its convergence rate. In the present simulations the revolution rate is 0.3 that is 30 percent of colonies in the empires change their positions randomly.
Figure 6: Revolution; a sudden change in socio-political characteristics of a country. [15]
180 Behzad Abdi, Hamid Mozafari, Amran Ayob and Roya Kohandel
2.4. Exchanging Positions of the Imperialist and a Colony
While moving toward the imperialist, a colony might reach a position with lower cost than the imperialist. In this case, the imperialist and the colony change their positions. Then the algorithm will continue with the imperialist in the new position and the colonies will be assimilated by the imperialist in its new position. Figure 7.a depicts the position exchange between a colony and the imperialist. In this figure the best colony of the empire is shown in a darker color. This colony has a lower cost than the imperialist. Figure 7.b shows the empire after exchanging position between the imperialist and the colony.
Figure 7: Exchanging the positions of a colony and the imperialist. [15]
(a) Before (b) After 2.5. Uniting Similar Empires
In the movement of colonies and imperialists toward the global minimum of the problem some imperialists might move to similar positions. If the distance between two imperialists becomes less than a threshold distance, they both will form a new empire which is a combination of these empires. All the colonies of two empires become the colonies of the new empire and the new imperialist will be in the position of one of the two imperialists. Figure 8 shows the uniting process of two empires before uniting and resulting from uniting two empires, respectively.
Figure 8: The uniting process of two empires. [15]
(a) Before (b) After 2.6. Total Power of an Empire
The total power of an empire is mainly contributed by the power of imperialist country. However the power of the colonies of an empire has a negligible effect, on the total power of that empire. This fact is modeled by defining the total cost of an empire by
. . ( )n nT C Cost imprialist mean{Cost(colonies of empire )}ξ= + (8)
Imperialist Competitive Algorithm and its Application in Optimization of Laminated Composite Structures 181
where . .nT C is the total cost of the thn empire and ξ is a positive small number. A small value for ξ causes the total power of the empire to be determined by just the imperialist and increasing it will increase the role of the colonies in determining the total power of an empire. The value of 0.1 for ξ has shown good results in most of the implementations. 2.7. Imperialistic Competition
All empires try to take possession of colonies of other empires and control them. The imperialistic competition gradually brings about a decrease in the power of weaker empires and an increase in the power of more powerful ones. The imperialistic competition is modeled by just picking some (usually one) of the weakest colonies of the weakest empire and making a competition among all empires to possess these (this) colonies. Figure 9 shows a bigger picture of each empires taking possession of the mentioned colonies. In other words, these colonies will not definitely be possessed by the most powerful empires, but these empires will be more likely to possess them.
Figure 9: Imperialistic competition. [15]
To start the competition, first a colony of the weakest empire is chosen and then the possession probability of each empire is found. The possession probability pp is proportionate to the total power of the empire. The normalized total cost of an empire is simply obtained by
. . . . . max{ . . }n n iiN T C T C T C= − (9)
where, . .nT C and . . nN T C are the total cost and the normalized total cost of thn empire, respectively. Having the normalized total cost, the possession probability of each empire is given by
1
. . .
. . .n imp
np N
ii
N T CpN T C
=
=
∑ (10)
To divide the mentioned colonies among empires vector P is formed as following
1 2 3, , ,...,
Nimpp p p pp p p p⎡ ⎤=⎣ ⎦
P (11)
Then the vector R with the same size as P whose elements are uniformly distributed random numbers is created,
1 2 3
1 2 3, , , ...,
, , , ..., (0,1)
Nimp
impN
R r r r r
r r r r U
⎡ ⎤=⎣ ⎦ (12)
182 Behzad Abdi, Hamid Mozafari, Amran Ayob and Roya Kohandel
Then vector D is formed by subtracting R from P
1 2 3
1 2 3
1 2 3
, , , ...,
, , , ... ,
im p
N im pim p
N
p p p p N
D D D D
p r p r p r p r
⎡ ⎤= ⎣ ⎦⎡ ⎤− − − −⎣ ⎦
D = P - R
= (13)
Referring to vector D the mentioned colony (colonies) is handed to an empire whose relevant index in D is maximized.
The process of selecting an empire is similar to the ones used in selecting parents in genetic algorithm. But this method of selection is much faster because it is not required to calculate the cumulative distribution function and the selection is based only on the values of probabilities. Hence, the fast process of selecting the empires will increase its execution speed.
The main steps of ICA is summarized in the pseudo-code given in Figure 10. The continuation of the mentioned steps will cause the countries to converge to the global minimum of the cost function. Different criteria can be used to stop the algorithm. One method is to use a number of maximum iteration of the algorithm, called maximum decades, to stop the algorithm. Another stop criterion to end the imperialistic competition is when there is only one empire. On the other hand, the algorithm can be stopped when its best solution in different decades cannot be improved for some consecutive decades.
Figure 10: Pseudo code of the Imperialistic Competitive Algorithm. [15]
3. Review of Composite Material Failure Criteria Two failure theories that are commonly used in the design of laminated composite structures are described briefly below: 3.1. Maximum Stress Failure Criterion
Due to this theory, a lamina fails when any of the stress components LLσ , TTσ and LTτ reaches its corresponding strength value [22, 27].
LL T TT C
TT T TT C
LT LT
X or XY or Y
S or S
σ σσ στ τ
≥ ≥≥ ≥≥ + ≥ −
(14)
where LLσ , TTσ and LTτ are longitudinal, transverse, shear stress along material axes ,L T , respectively and , ,T C TX X Y and CY are tensile and compressive strengths in X Y− direction.
1) Select some random points on the function and initialize the empires. 2) Move the colonies toward their relevant imperialist (Assimilation). 3) Randomly change the position of some colonies (Revolution). 4) If there is a colony in an empire which has lower cost than the imperialist, exchange the positions of that colony
and the imperialist. 5) Unite the similar empires. 6) Compute the total cost of all empires. 7) Pick the weakest colony (colonies) from the weakest empires and give it (them) to one of the empires
(Imperialistic competition). 8) Eliminate the powerless empires. 9) If stop conditions satisfied, stop, if not go to 2.
Imperialist Competitive Algorithm and its Application in Optimization of Laminated Composite Structures 183
3.2. Tsai – Wu Failure Criterion
This theory is a quadratic failure criterion, which is most extensively used in the design of laminated composite structures [21, 27]. This failure criterion states that the lamina fails when the following condition is satisfied,
1 . 1,2, ,6ij i j i iF F i jσ σ σ+ ≥ = … (15) where ,i ijF F are strength parameters and ,i jσ σ are stress components. In principal material
coordinates ( ),L T , for the case of a lamina, this failure criterion becomes 2 2 22 1LL LL LT LL TT TT TT SS LT L LL T TTF F F F F Fσ σ σ σ σ σ σ+ + + + + ≥ (16)
4. Numerical Results Two different optimization problems are solved by imperialist competitive algorithm and the results are compared with genetic algorithm results. These problems are the same as in the examples in Ref. [14] that are solved by using genetic algorithm. The first example presents a validation of the proposed imperialist competitive algorithm and second example presents an optimization problem under the constraint of two different failure criteria. 4.1. Example 1 – Optimization of Weight Under Failure Mode Criteria of Maximum Strain and
Buckling
The main purpose of this example is to validate the imperialist competitive algorithm (ICA) by comparing it to a well known laminated composite optimization procedure [6]. The purpose of this example is to find the minimal weight of a laminated composite plate under the constraints of laminate symmetry and balance, maximum number of contiguous plies with the same orientation as well as the maximum strain and buckling. The allowable orientation angle values are 20 , 45± and 90± degrees. Therefore, the optimization problem can be described as [14]
where ,k nθ are the orientation of each stack of the laminate and the total number of stacks, respectively. As mentioned, each stack is composed of two layers to guarantee balance.
Fig. 11 shows a rectangular plate, with simply supported at edges and subjected to compressive in- plane loads per unit length xN and yN . The thickness of each layer is 0.127 mm and the length and width of plate are 0.508m and 0.127m respectively. The classical lamination theory and the linear elastic buckling analysis are used in this study [1]. The buckling load factor bucklingλ represents the failure buckling load divided by the applied load, and is calculated as [29],
where ijD s are coefficients of the laminate bending stiffness matrix, r and s determine the amount of half waves in the x − and y − direction, respectively. a , b are length and width of plate, respectively.
184 Behzad Abdi, Hamid Mozafari, Amran Ayob and Roya Kohandel
Figure 11: Laminated composite plate subjected to compressive loads [14].
The critical normal strain failure factor strainλ is described as [14]
1 2 12
1 2 12
m in m in , ,ua ua ua
strain k k kkf f fS S Sε ε γλε ε γ
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟=
⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
(1)
Where , 1,2uai iε = and 12
uaγ are the allowable strains. , 1, 2ki iε = and 12
uaγ are the strains in the principal direction of the thk lamina and fS is a safety factor. Therefore, the critical failure factor crλ is the smallest between bucklingλ and strainλ .
The elastic material properties of the layers are 1 127,550E MPa= , 2 13,030E MPa= ,
12 6410G MPa= and Poisson’s ratio 12 0.3v = . Also, the ultimate strains are 1 0.008uaε = , 2 0.029uaε = ,
12 0.015uaγ = and the safety factor is 1.5fS = . The plate is studied under three different loading conditions. The first load case has
2277 /xN N mm= and 284.6 /yN N mm= , the second load case has 2189 /xN N mm= and 547.3 /yN N mm= and the third load case has 1716 /xN N mm= and 858.1 /yN N mm= .
The main purpose of this study is to compare the performance of the imperialist competitive algorithm with genetic algorithm and then comparing the results of this algorithm with those of [6]. “Price of the search” and “reliability” are two terms that are used in this study to evaluate the performance of the imperialist competitive algorithm. The reliability of an algorithm is the probability it has of finding a practical optimum which in turn is defined, in this study, as an optimal weight design with the critical failure factor crλ with in 0.1% of crλ of the global optimum. The price of the search is the number of analysis necessary to reach 80% reliability, i.e., to have 80% probability of finding a practical optimum [14].
Table 1 shows the optimal design found in Ref. [6] where all the practical optimum information can be found in detail. Table 1: Optimal designs for each load case of Example 1([6, 14])
Load case Optimal design Failure mode Number of practical optima 1 [ ]5 4 4 2 24 5 / 0 / 4 5 / 0 / 9 0 / 0
The results obtained by the imperialist competitive algorithm (ICA) formulated in this study
and their comparison with the results of Refs. [6, 14] are listed in Table 2. The imperialist competitive algorithm shows good performance and achieves results that are as good as the results of Refs. [6,14]. In load case one, the algorithm of Ref. [6] is still better regarding the search price, while the imperialist competitive algorithm gives the same results as the algorithm of Ref. [14]. The imperialist competitive algorithm has obtains better results regarding search price and reliability in second and third load cases.
Imperialist Competitive Algorithm and its Application in Optimization of Laminated Composite Structures 185
Table 2: Price and reliability for Example 1. comparison with Refs [6,14].
4.3. Example 2 – Optimization of Material Cost of a Laminate Under Ply Failure and Weight
Constraints
In this example, the material cost optimization of a hybrid laminated composite plate under ply failure and weight constraints is described. The layers considered in this example are carbon – epoxy (CE) and glass – epoxy (GE). Carbon – epoxy is lighter and stronger, while glass – epoxy has a cost advantages in terms of lower price per square meter. For example, the price per square meter of glass – epoxy is about 8 times less than carbon – epoxy. As in Example 1, the laminate is subjected to symmetry and balance constraints as well as the first ply failure criteria. Also, a maximum weight constraint is used in this study. Therefore, the optimization problem can be described as [14],
{ } { }{ }
2 2find : , , , 0 , 45, 90
, , 1m inim ize : m aterial cost
first p ly failure constraintsubject to :
m axim um w eight: 70
k k k
k
m at n
m at G E C E k n
N
θ θ ∈ ±
∈ = −
⎧⎨⎩
(20)
Using the above description about the material cost, each CE and GE layers is assumed to cost 8 and 1 monetary unit (m.u.), respectively. The elastic properties of the CE layers are
1 116,600E MPa= , 2 7673E MPa= , 12 4173G MPa= and Poisson’s ratio 12 0.27v = and mass density 31605 /kg mρ = and the elastic properties of GE layers are 1 37,600E MPa= , 2 9584E MPa= ,
12 4081G MPa= and Poisson’s ratio 12 0.26v = and mass density 31903 /kg mρ = . the failure properties of the both laminas are showed in Table 3. The thickness of each layer is 0.1mm and the length and width of the plate are 1.0 m . The plate is under in-plane fixed applied loads ( 2000 /xN N mm= and 2000 /yN N mm= − )[14]. Table 3: Strength properties of Glass – Epoxy (GE) and Carbon – Epoxy (CE)
Glass – Epoxy (GE)
TX 1134 MPa CY 150 MPa 1Tε 0.0302 fmσ 1.3
CX 1031MPa 21S 75MPa 1Cε 0.0295 ( )||p +
⊥ 0.3
TY 54 MPa 1fE 72,000 MPa 12fv 0.22 ( )||p −
⊥ 0.25
Carbon – Epoxy (CE)
TX 2062 MPa CY 240 MPa 1Tε 0.0175 fmσ 1.1
CX 1701MPa 21S 105 MPa 1Cε 0.014 ( )||p +
⊥ 0.3
TY 70 MPa 1fE 230,000 MPa 12fv 0.23 ( )||p −
⊥ 0.25
The optimization results of the imperialist competitive algorithm and the optimal results of Ref.
[14] are listed in Table 4.
186 Behzad Abdi, Hamid Mozafari, Amran Ayob and Roya Kohandel
Table 4: Optimal material cost and stacking sequence of the laminate for different criteria
Imperialist competitive Algorithm (ICA) Material cost and weight Stacking Reliability Search FC Cost (m.u.) Weight (N) TW 208 68.23 ( ) ( )2 26 4
0 9 0C E G E
s⎡ ⎤⎣ ⎦ 0.09
MS 148 63.11 ( ) ( )2 24 50 9 0C E G E
s⎡ ⎤⎣ ⎦ 0.08
Reference [14] Material cost and weight Stacking Reliability Search FC Cost (m.u.) Weight (N) TW 208 68.23 ( ) ( )2 26 4
0 9 0C E G E
s⎡ ⎤⎣ ⎦ 0.09
MS 148 63.11 ( ) ( )2 24 50 9 0C E G E
s⎡ ⎤⎣ ⎦
0.07
It is interesting to mention that the optimum results obtained by using the imperialist
competitive algorithm are same as in Ref. [14]. From Table 4, all layers with an orientation of 90o are made of GE, while those with an orientation of 0o are made of CE. Also, as in weight minimization, each failure criterion yielded a different optimum. From the results of this study and from comparisons with Ref. [14], when optimizing laminated composite structures, the choice of a failure criterion corresponding to the real behaviour of the structure is crucial for both economy and safety.
5. Conclusion In this study, two different optimization algorithms (Genetic algorithm and imperialist competitive algorithm) are used in optimization of laminated composite plates. The main objective of design optimization in aerospace composite structures is to minimize the weight of the laminate for a given loading. Weight minimization and the material cost minimization are two different objectives that were considered in this study. The Imperialist competitive algorithm Showed good performance in this kind of optimization problems and compared satisfactorily genetic algorithm. Also, two failure criterions (maximum stress failure and Tsai – Wu failure) are used in this study. This study shows that the optimal weight of laminated composite structures depend on the choice of the failure criterion as well as the load conditions and it can be mentioned that there is not direct relation between optimal weight and failure criterion.
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