Simultaneous optimization of elastic constants of laminated composites using artificial bee colony algorithm
Abstract
Multifunctional application of laminated composites requires multiobjective optimization of their characteristics. In this paper, the ply angles of laminated composites are determined in order to maximize the effective inplane elastic constants simultaneously. These constants are determined by considering a representative small element of the laminated composite and imposing the conditions of uniformity of outofplane stresses and inplane strains at orthotropic layer interfaces. Then, by combining the artificial bee colony algorithm and various multiobjective optimization methods, the optimal ply angles and the corresponding cooptimized constants are determined. The correctness and accuracy of the method is verified not only by providing a comparison with the existing results but also by solving a known problem. The results are presented and discussed, considering different multiobjective optimization problems. The results show that the increasing number of layers in the problem of simultaneous optimization of Young’s moduli not only does result in finding more Pareto optimal solutions in the feasible objective region but also shifts the Pareto frontier toward the utopia point. However, when the shear modulus optimization is engaged in the problem, using more than two layers only leads to obtaining more Pareto optimal answers.
Keywords
Laminated composites Ply angles Effective elastic constants Multiobjective optimization Artificial bee colony algorithm1 Introduction
Laminated composites have been widely used in structural design due to low weight and high strength and stiffness. Therefore, the improvement and optimum design of them are important from the practical point of view [1]. In this regard, technological developments have resulted in reducing some drawbacks and enhancing mechanical performance of composites [2]. For example, a great improvement in the mechanical properties can be achieved by making noncovalently functionalized reduced graphene oxide reinforced polynanocomposites as discussed by Wang et al. [3, 4]. For analyzing and optimizing laminated composites or sandwich panels with laminated composite facesheets, it is much easier to specify the effective elastic constants of the laminate. These constants are defined as the elastic constants of a hypothetical homogeneous material equivalent to the laminate. In general, the determination of the material elastic constants can be accomplished by performing tensile testing. But, laminated composites, due to their anisotropic nature, need more tests which can be quite tedious and timeconsuming [5]. There are some other experimental methods in this area such as modal analysis and ultrasonic evaluation [6, 7]. However, in these methods, the preparation and testing of the standard specimens are difficult and expensive. Thus, the analytical approaches have come to attention over recent years. In these methods, the overall constants are predicted, considering the thickness, the mechanical properties, and the orientation of each laminae. One of the best methods in analytical determination of the elastic constants of the equivalent homogeneous material has been presented by Chou et al. [8]. They obtained these constants by considering a representative small element of the laminated medium and imposing the conditions of uniformity of the outofplane stresses and the inplane strains at the layers interfaces. This method has become a basis for further research on various issues related to laminated composites [9, 10].
Optimization and enhancement of composites has always been a matter of interest due to the important role of them in different structures. The minimization of total cost and final weight and maximization of elastic properties, material strength, erosion resistance, toughness properties, and fundamental natural frequencies are a number of these cases. For example, Lv et al. [11] have recently recommended an auspicious method to improve particle erosion resistance and delamination toughness of laminated composites. In this regard, they interleaved the composite layers with lightweight thermoplastic polyurethane nonwoven fabrics. The structural design and optimization of laminated composites typically include many design variables among which ply angle is of great importance due to its practicality [12]. Working on this variable is still welcomed in stateoftheart technology. For example, Dong and Davies [13] have recently studied the ply orientation effect on the transverse properties of hybrid composites made of glass and carbon fibers. There is a thorough study of both contemporary design optimization techniques and the mechanics of composite laminates [14]. Due to the large computational cost of gradient methods and discrete nature of composite related problems, the idea of using natureinspired metaheuristics algorithms in optimization of laminated composite structures has come to attention over last two decades. A lot of research has confirmed the applicability of these algorithms in optimizing composites [15, 16]. For example, Muc and Gurba [17] evaluated the applicability of genetic algorithms in combination with the numerical computation of objective functions of laminates.
Sometimes, multifunctional application of laminated composites requires multiobjective optimization of them. Conductive polymer nanocomposites are good examples of this multifunctional application since they can perform the roles of antistatic materials as well as electromagnetic interference shielding and they can be used as sensors and conductors as discussed by Liu et al. [18]. Extensive recent published research confirms the need of multiobjective optimization of composites [19, 20, 21, 22, 23]. For instance, Park et al. [24] simultaneously optimized both the mechanical characteristics and the manufacturing total cost of laminated plates, from the early stage of design. In fact, they sought the stacking sequence that minimized the maximum deflection and the time of molding, by means of genetic algorithms. As another example, Omkar et al. [25] formulated a problem with multiple objectives of minimizing weight and the total cost of the composite component to achieve a specified strength. Their primary optimization variables were the number of layers, stacking sequence, and thickness of each layer. Some research into multiobjective optimization of laminated composites requires a profound understanding of multiobjective concepts. A study by Bloomfield et al. [26] is a case in point. By searching within a prespecified set of likely ply angles, they introduced a twostep technique to obtain the full feasible regions of optimization problem. Firstly, by means of a convex hall, they separately sought the boundary of feasible solutions related to the outofplane and inplane properties as well as those related to coupling. Next, they applied an algebraic definition considering nonlinearity to relate variables to each other. There are thorough reviews of the design optimization approaches of structures made of composite parts in which the improvement techniques are studied in two parts: constant stiffness design and variable stiffness designs. According to these reviews, the problem of optimizing laminated composite structures is often nonlinear, nonconvex, multimodal, and multidimensional and might be expressed by both discrete and continuous variables [27, 28].
To the best knowledge of the authors, no research effort has been devoted so far to find the composite ply angles, which simultaneously maximize the elastic constants of laminated composites. Such research would be viable by combining different multiobjective optimization methods and constitutive laws of laminated composites. There are strong grounds for considering this problem indispensable. For one thing, laminated composites are often required to provide desirable stiffness in more than one direction. For another, visualization of the Pareto frontier helps designers identify the best solution based on their preference. In this research, by combining artificial bee colony algorithm and various multiobjective optimization methods and using Chou et al.’s relations for predicting effective elastic constants of the laminated composites, the optimal ply angles and their corresponding Pareto optimal frontier in the objective region have been determined in order to maximize the effective inplane elastic constants simultaneously.
2 The effective elastic constants of a laminate

The laminate is made of orthotropic plies bonded together and the principal material axis of the each ply is oriented arbitrarily in planes parallel to x–yplane.

The layer thickness is constant and much smaller than the laminate length or width.

The displacements are small compared with the laminate thickness.

Each ply obeys Hooke’s law.

Plies are bonded together perfectly.

There is no interfacial slip.
In order to achieve the constitutive relation of the equivalent medium in terms of the mechanical elastic properties of the individual composite layers, a number of relations between the strains and stresses of each layer and those of the hypothetical equivalent medium will be assumed [8].
Now that all components of the effective stiffness matrix are determined, the effective elastic constants such as Young’s moduli E_{x} and E_{y} or the shear modulus G_{xy} can easily be extracted from this matrix using common related formulae [29].
3 Artificial bee colony algorithm
It should be noted that there are some papers in literature on the performance of ABC algorithm compared to other populationbased algorithms. For example, Karaboga et al. [31, 32, 33] used ABC algorithm for optimization of a large number of wellknown test functions and compared the performance of ABC algorithm with that of particle swarm optimization algorithm (PSO), differential evolution algorithm, and genetic algorithm. Their results showed that the ABC performs better than or at least similar to other algorithms while it enjoys the advantage of using fewer control parameters.
4 Multiobjective optimization
The characteristic of a multiobjective optimization problem is that there is no unique solution for the problem, but a set of good solutions, which have equal worth from a mathematical point of view, are identified. These answers are known as Pareto optimal solutions. Usually, only one answer is selected. Therefore, in a multiobjective optimization problem, there are at least two missions, which must be accomplished well: finding Pareto optimal answers and making decision to select an answer among them. Decisionmaker is a person who has the preferable information about the importance of the objective functions. The main concept in defining solutions for multiobjective optimization problems is that of Pareto optimality. A design solution x* in the feasible design region S is Pareto optimal if there is no other solution x in the set S that deteriorate at least one objective function without improving another one. An objective solution f* in the feasible objective region is called Pareto optimal if its corresponding design solution in the feasible design region is Pareto optimal. The Pareto optimal set can denote to all vectors in the design region or in the objective region. The utopia solution fº is the one obtained by optimizing each objective function independently. It is approximately impossible that one design solution simultaneously optimizes all the objective functions. Thus, the utopia point exists only in the objective region and, in general, it is not attainable [34].
Here, w is a weight vector that is usually set by the designer in a way that \( {\sum}_{i=1}^k{w}_i=1 \) and w_{i} > 0. The weights can be used in two ways. Either the designer can set w before the problem is solved, based on his preferences, or he systematically change weights to find different Pareto optimal solutions. Although this approach is straightforward, and selecting positive weights guarantee its Pareto optimality, there are a few concerns with it. Firstly, even very careful selection of weights does not certainly guarantee that the final solution will be acceptable; thus, the problem may need to be resolved. The next issue is that this method leaves out points on nonconvex portions of the Pareto optimal set. The last difficulty with the weighted sum method is that the results usually have not even distribution through the Pareto optimal set.
Whereas the weighted sum method always yields Pareto optimal points but may miss certain points when the weights are varied, this method can provide all the Pareto optimal points (the complete Pareto optimal set). However, it may provide nonPareto optimal points as well.
Solutions using the global criterion formulation depend on the values of both w and p. Generally, p is proportional to the amount of emphasis placed on optimizing the function with the largest difference between f_{i} and \( {f}_i^{{}^{\circ}} \). In this paper, p is considered two, which is the commonest value for this parameter and leads to solutions that are called compromise solutions [34]. In addition, someone may prefer to omit 1/p since this relation with and without it theoretically provide similar solutions. This method gives a clear explanation of minimizing the distance from the utopia vector and gives a general formulation that allows multiple parameters to be set to reflect preferences. Also, it always provides a Pareto optimal solution [34].
It should be noted that unlike the weighted sum method, both WGC and weighted Tchebycheff methods are able to generate points on nonconvex Pareto frontier [34, 35]. One can find the mathematical proof for why WGC method is able to find points on nonconvex Pareto frontier in Ref. [35] (note that the weighted Tchebycheff method shown in Eq. (16) is the limit of Eq. (17) when p → ∞). As a brief explanation, one can say that a multiobjective method has the ability to find solutions on the nonconvex Pareto frontier only if it includes some parameters via which the function’s curvature can be manipulated [35].
5 Problem statement
It should be noted that because only inplane deformation is considered in this paper, the stacking sequences like [0/90/0/90/0/90/0/90] and [0/0/0/0/90/90/90/90] lead to the same effective inplane elastic characteristics under the situation. Therefore, the variable used in optimization problems is not the stacking sequence of layers, but the combination of different layers (Ply angles). This is an advantage because it gives designers the privilege of changing stacking sequence according to their requirements or considering other criteria (such as laminate strength, buckling load, and natural frequencies).
6 Results and discussion
Comparison between different methods for the problem of composite boxbeam optimization
Variables  ABC  PSO [39]  RCGA [39]  

Discretization  Discretization  Discretization  
I  II  III  I  II  III  I  II  III  
Breadth (mm)  96.27  97.03  97.03  96.01  97.03  97.03  93.73  110.49  110.74 
Depth (mm)  58.93  59.94  56.64  58.93  59.69  56.64  57.66  65.53  67.31 
θ_{1} (°)  30  45  45  30  45  45  40  90  90 
θ_{2} (°)  50  90  0  50  90  0  50  90  90 
θ_{3} (°)  80  15  45  80  15  45  0  45  45 
θ_{4} (°)  20  75  90  20  75  90  40  75  90 
θ_{5} (°)  70  45  45  70  45  45  90  90  90 
Objective function  0.15  0.79  4.5  0.15  0.78  4.5  0.25  2.55  5.08 
Average error (%)  0.09  0.52  4.13  0.09  0.51  4.13  0.19  1.85  4.60 
Results of singleobjective optimization of the elastic constants
Objective function  E_{x} (GPa)  G_{xy} (GPa)  E_{y} (GPa)  Ply angles (°) 

E _{ x}  109.17  5.48  8.47  [(0)_{8}] 
E _{ y}  8.47  5.48  109.17  [(90)_{8}] 
G _{ xy}  18.6  28.43  18.6  [(±45)_{4}] 
Results of multiobjective optimization of the elastic constants using weighted sum, WGC, and the Tchebycheff methods
Multiobjective optimization method  Considered objective functions  E_{x} (GPa)  E_{y} (GPa)  G_{xy} (GPa)  Best ply angles (°) 

Weighted sum  E_{x}, E_{y}  59.07  59.07  5.48  [(0/90)_{4}] 
Weighted sum  E_{x}, G_{xy}  28.6  13.94  26.74  [(± 40)_{2}/(± 35)_{2}] 
Weighted sum  E_{y}, G_{xy}  13.94  28.6  26.74  [(± 50)_{2}/(± 55)_{2}] 
Weighted sum  E_{x}, E_{y}, G_{xy}  31.45  31.45  23.34  [± 25/± 65/± 40/± 50] 
WGC  E_{x}, E_{y}  59.07  59.07  5.48  [(0/90)_{4}] 
WGC  E _{ x} , G _{ xy}  61.34  12.17  18.55  [30/− 40/− 15/− 20/10/35/± 25] 
WGC  E _{ y} , G _{ xy}  12.17  61.34  18.55  [60/− 50/− 75/− 70/80/55/± 65] 
WGC  E _{ x} , E _{ y} , G _{ xy}  43.70  43.70  16.95  [− 45/− 75/− 30/0/60/15/45/90] 
The weighted Tchebycheff  E _{ x} , E _{ y}  59.07  59.07  5.48  [(0/90)_{4}] 
The weighted Tchebycheff  E _{ x} , G _{ xy}  63.53  12.03  18.02  [− 10/− 20/− 35/(20)_{3}/− 30/40] 
The weighted Tchebycheff  E _{ y} , G _{ xy}  12.03  63.53  18.02  [− 80/− 70/− 55/(70)_{3}/− 60/50] 
The weighted Tchebycheff  E _{ x} , E _{ y} , G _{ xy}  47.68  47.68  14.43  [− 85/± 45/70/− 80/10/− 20/5] 
Again, as seen in Fig. 5 b, the Pareto optimal frontier is well determined. As in optimization of E_{x} and E_{y}, a similar straight line in this problem can be created by connecting two points. The first point coordinates are the initial value of ply properties E_{1} and G_{12} (109.17, 5.48). However, the other point coordinates are the value of E_{x} and G_{xy} obtained in singleobjective optimization of G_{xy} (18.60, 28.43). This is because the principal stresses for pure shear load on 0° plies are not along the material axis but along ± 45° axes. Graphs like in Fig. 5 deliver a great perception of the problem to designers. For example, by looking at Fig. 5 b, the designer will know that the maximum available value for shear modulus G_{xy} cannot exceed 14 GPa when the value of elasticity modulus E_{x} is 80 GPa.
7 Conclusions
This research was concerned with simultaneous optimization of laminated composite elastic constants. Hence, the ply angles of laminated composites were sought in order to maximize the elastic constants simultaneously by using a combination of artificial bee colony algorithm and various multiobjective optimization methods. The method was verified in three ways: Reformulating and solving a rather similar existing problem, solving a problem that its answer was known, and finding the Pareto optimal frontier in the feasible objective region. It was observed that multiobjective methods like weighted global criterion and weighted Tchebycheff are successful in finding the Pareto optimal frontier of such optimization problems. Besides, the artificial bee colony algorithm finds the global optimum very fast. During simultaneous optimization of E_{x}, E_{y}, and G_{xy}, it was detected that the obtained angles can be paired in a way that the absolute value of their difference becomes 90°. In simultaneous optimization of E_{x} and E_{y}, a stair shape Pareto optimal frontier was observed. The number of stair steps increased as more number of layers had been used. In fact, increasing the number of layers not only does result in more Pareto optimal answers in the feasible objective region, but also shifts the Pareto frontier toward the utopia point. Finally, the stair shape approached to a straight line as the number of layers increased excessively. In simultaneous optimization of E_{x} and G_{xy}, using more than two layers only leads to better determination of the Pareto frontier. The results showed that the current method is very quick and successful in determining the best ply angles and the Pareto optimal frontier regardless of the problem dimension and provides a thorough knowledge of the subject. Therefore, every designer who has the preferable information about the importance of the objective functions can use this method easily and confidently.
Notes
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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