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Advanced Composites and Hybrid Materials

, Volume 2, Issue 3, pp 431–443 | Cite as

Simultaneous optimization of elastic constants of laminated composites using artificial bee colony algorithm

  • Meysam EsmaeeliEmail author
  • Behzad Kazemianfar
  • Mohammad Rahim Nami
Original Research

Abstract

Multifunctional application of laminated composites requires multi-objective optimization of their characteristics. In this paper, the ply angles of laminated composites are determined in order to maximize the effective in-plane elastic constants simultaneously. These constants are determined by considering a representative small element of the laminated composite and imposing the conditions of uniformity of out-of-plane stresses and in-plane strains at orthotropic layer interfaces. Then, by combining the artificial bee colony algorithm and various multi-objective optimization methods, the optimal ply angles and the corresponding co-optimized 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 multi-objective 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.

Graphical abstract

Keywords

Laminated composites Ply angles Effective elastic constants Multi-objective optimization Artificial bee colony algorithm 

1 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 non-covalently 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 time-consuming [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 out-of-plane stresses and the in-plane 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 state-of-the-art 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 nature-inspired 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 multi-objective optimization of them. Conductive polymer nanocomposites are good examples of this multifunctional application since they can perform the roles of anti-static 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 multi-objective 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 multi-objective optimization of laminated composites requires a profound understanding of multi-objective concepts. A study by Bloomfield et al. [26] is a case in point. By searching within a pre-specified set of likely ply angles, they introduced a two-step 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 out-of-plane and in-plane properties as well as those related to coupling. Next, they applied an algebraic definition considering non-linearity 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 non-linear, non-convex, 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 multi-objective 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 multi-objective 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 in-plane elastic constants simultaneously.

2 The effective elastic constants of a laminate

Consider a laminated composite where the material of each layer is orthotropic. A representative element of this laminate is shown in Fig. 1. This descriptive element is supposed to be much smaller than the overall composite. At the same time, a small element of the equivalent homogeneous medium is considered.
Fig. 1

The representative element of a laminated composite

The current procedure is based on the following basic assumptions:
  • 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–y-plane.

  • 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].

The following relation is used to represent the effective constitutive law for an N-layered laminate:
$$ \overline{\sigma_i^{\ast }}=\overline{Q_{ij}^{\ast }}\overline{\varepsilon_j^{\ast }}\kern0.5em \left(i,j=1,2,3,4,5,6\right) $$
(1)
The barred notation is used to represent that the relationship applies in the overall x–y–z-coordinate system of the laminated composite. The asterisk superscript is used to denote the effective laminate quantities. Based upon continuity of displacement at the layer interfaces, the normal strains in the 1 and 2 directions and the shear strain in the 1–2 planes are assumed to be uniform and the same in each layer of the element and equal to the corresponding strains in the equivalent homogeneous medium:
$$ \overline{\varepsilon_i^k}=\overline{\varepsilon_i^{\ast }}\kern0.5em \left(i=1,2,6;k=1,2,...,N\right) $$
(2)
where \( \overline{\varepsilon_i^k} \) denotes the strain in the kth ply of the laminated composite. To guarantee stress continuity across layers interfaces, all ply stress components related to the out-of-plane direction are assumed uniform and equal to the corresponding effective ones in the laminated composite. Mathematically, this is shown as
$$ \overline{\sigma_i^k}=\overline{\sigma_i^{\ast }}\kern0.5em \left(i=3,4,5;k=1,2,...,N\right) $$
(3)
where \( \overline{\sigma_i^k} \) denotes the stress in the kth ply of the laminated composite. All remaining effective laminate strains and stresses are assumed the volume average of all their corresponding layer strain and stress components, respectively. Mathematically, these are expressed as
$$ \overline{\varepsilon_i^{\ast }}=\sum \limits_{k=1}^N{V}^k\overline{\varepsilon_i^k}\kern0.5em \left(i=3,4,5\right) $$
(4)
$$ \overline{\sigma_i^{\ast }}=\sum \limits_{k=1}^N{V}^k\overline{\sigma_i^k}\kern0.5em \left(i=1,2,6\right) $$
(5)
where Vk is the ratio of the undeformed volume of the kth ply over the undeformed volume of the whole laminate. The constitutive law for each ply in the laminated composite is expressed below:
$$ \overline{\sigma_i^k}=\overline{Q_{ij}^k}\overline{\varepsilon_j^k}\kern0.5em \left(i,j=1,2,3,4,5,6;k=1,2,...,N\right) $$
(6)
in which \( \overline{Q_{ij}^k} \) is the transformed stiffness matrix of the kth ply that relates strains to the stresses referred to arbitrary axes. Equations (2)–(6) represent 12N + 6 linear algebraic relations with 12N + 12 unknowns. Solution to them yields the following effective three-dimensional stress-strain constitutive law, which can be applied as an equivalent representation for the laminated composite in which the coefficients in the laminate stiffness matrix are given by [10].
$$ \overline{Q_{ij}^{\ast }}=\sum \limits_{k=1}^N{V}^k\left[\overline{Q_{ij}^k}-\frac{\overline{Q_{i3}^k}\overline{Q_{3j}^k}}{\overline{Q_{33}^k}}+\frac{\overline{Q_{i3}^k}{\sum}_{s=1}^N\frac{V^s\overline{Q_{3j}^s}}{\overline{Q_{33}^s}}}{\overline{Q_{33}^k}{\sum}_{s=1}^N\frac{V^s}{\overline{Q_{33}^s}}}\right]\kern0.5em \left(i,j=1,2,3,6\right) $$
(7)
$$ \overline{Q_{ij}^{\ast }}=\overline{Q_{ji}^{\ast }}=0\kern0.5em \left(i=1,2,3,6;j=4,5\right) $$
(8)
$$ \overline{Q_{ij}^{\ast }}=\frac{\sum_{k=1}^N\frac{V^k}{\varDelta_k}\overline{Q_{ij}^k}}{\sum_{k=1}^N{\sum}_{s=1}^N\frac{V^k{V}^s}{\varDelta_k{\varDelta}_s}\left(\overline{Q_{44}^k}\overline{Q_{55}^s}-\overline{Q_{45}^k}\overline{Q_{54}^s}\right)}\kern0.5em \left(i,j=4,5\right) $$
(9)
where
$$ {\varDelta}_k=\overline{Q_{44}^k}\overline{Q_{55}^k}-\overline{Q_{45}^k}\overline{Q_{54}^k} $$
(10)

Now that all components of the effective stiffness matrix are determined, the effective elastic constants such as Young’s moduli Ex and Ey or the shear modulus Gxy can easily be extracted from this matrix using common related formulae [29].

3 Artificial bee colony algorithm

The intelligence observed in the behavior of honeybee swarm for discovering best food sources is the basis for a nature-inspired optimization algorithm named artificial bee colony (ABC). In this algorithm, there are three groups of bees within the colony of the bees. The first group consists of employed bees that are only responsible for searching in the places that were visited by themselves previously. The second group consists of onlooker bees that decide where to search for food based on signals they received from other bees in the dancing area. The bees in the last group are named scouts and their duty is to perform random exploration in order not to leave out any potential good source [30]. In the ABC algorithm, firstly, the colony of bees is divided into two equal groups of the employed and onlooker bees and each employed bee is assigned only one food source. As the searching cycle continues, a number of employed or onlooker bees become scout and abandon their food source, due to poor quality of the food existing in that place or shortage of its amount. In the ABC algorithm, a possible solution of the optimization problem is characterized by a food source location and the nectar quantity and quality of that place associate with the fitness of the corresponding solution. The number of the employed bees or the onlooker bees is equal to the number of initial solutions in the optimization problem. Firstly, the ABC algorithm is started by producing randomly distributed initial population of SN solutions, where SN denotes the size of population. Each solution xi (i = 1, 2, , SN) is a D-dimensional vector where D is the number of optimization variables. The population is randomly generated within the previously specified boundaries of the variables by [31].
$$ {\mathbf{x}}_i={\mathbf{x}}^{\mathrm{min}}+\operatorname{rand}\left(1,0\right)\left({\mathbf{x}}^{\mathrm{max}}-{\mathbf{x}}^{\mathrm{min}}\right) $$
(11)
After generating initial population, the population of the initial solutions is exposed to a number of repetitive cycles of the search procedures of the three groups of the bees. In fact, an artificial employed bee randomly selects a food source position and produces a local modification on the one existing in her memory as
$$ {\nu}_{i,j}={x}_{i,j}+{\varphi}_{ij}\left({x}_{i,j}-{x}_{k,j}\right) $$
(12)
where j is a random integer in the range [1, D] that specifies the optimization parameter for modification and k ∈ {1, 2, . . ., SN} is a randomly selected index that has to be different from i and denotes the selected neighbor for sharing information. Also, φij is a uniformly distributed real random number in the range [− 1, 1]. If a parameter exceeds its boundary, it is set to its boundary. Only if the nectar amount of the new source is higher than that of the previous one does the bee memorize the new position and forget the old one. If not, she keeps the position of the previous one. Equation (12) indicates that as the difference between the parameters of xi, j and xk, j decreases, the perturbation on the position xi, j decreases, too. Thus, as the search comes near to the optimum solution in the search space, the step length is adaptively reduced. When the search process of all employed bees is completed, the information about the food quality and quantity of each position is shared with the onlooker bees via performing waggle dance in the dance area. After comparing the dancing type of different bees and evaluating the worth of each food source, an onlooker bee selects solution with a probability associated with its worth. Similar to the case of the employed bees, onlooker bees generate a change on the solution in their memory and check the worth of the source. In the ABC algorithm, the process of determining each solution probability value Pi is fulfilled based on each solution fitness using the following relation:
$$ {P}_i=\frac{\mathrm{fitnes}{\mathrm{s}}_i}{\sum_{i=1}^{\mathrm{SN}}\mathrm{fitnes}{\mathrm{s}}_i} $$
(13)
where fitnessi represents the fitness value of the position i assessed by its employed bee, which is relative to the food amount and quality of the food source in the position i. After determining each Pi, the simulation of merit-based selection could be done using different selection methods such as roulette wheel selection, tournament selection, or stochastic universal sampling.
The food source whose food is not good enough is exhausted by the bees and substituted with a new food source by the scouts. In the ABC algorithm, this is simulated by randomly generating a position and replacing it with the abandoned one. In the ABC algorithm, if a position cannot be enhanced more through a prearranged number of cycles named limit, then according to the Eq. (11), that food source is supposed to be abandoned and replaced by a new one [30]. The flowchart of the ABC algorithm used in present research is shown in Fig. 2.
Fig. 2

Flowchart representation of ABC algorithm for solving the problem of simultaneous optimization of elastic constants of laminated composites

It should be noted that there are some papers in literature on the performance of ABC algorithm compared to other population-based algorithms. For example, Karaboga et al. [31, 32, 33] used ABC algorithm for optimization of a large number of well-known 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 Multi-objective optimization

The characteristic of a multi-objective 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 multi-objective 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. Decision-maker is a person who has the preferable information about the importance of the objective functions. The main concept in defining solutions for multi-objective 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].

In the multi-objective optimization problems, it is necessary to define a criterion via which different objective functions value can be compared because they may not have similar units or their magnitude may differ remarkably. This issue is usually resolved by transforming the objective functions in a way that their orders of magnitude become similar. In spite of many methods, the commonest one is to make the objective functions non-dimensional as follows:
$$ {f}_i^{\mathrm{norm}}=\frac{f_i-{f}_i^{{}^{\circ}}}{f_i^{\mathrm{max}}-{f}_i^{{}^{\circ}}} $$
(14)
For determining \( {f}_i^{\mathrm{max}} \), each objective fj needs to be optimized to determine \( {\mathbf{x}}_j^{\ast } \). Then, all objective functions need to be evaluated at \( {\mathbf{x}}_j^{\ast } \). The maximum of all the fi values is \( {f}_i^{\mathrm{max}} \). This procedure also determines the utopia point components \( {f}_i^{{}^{\circ}} \). Most approaches for solving multi-objective optimization problems include formulating all objective functions into a single one. In the present study, three methods is used for solving problems. The commonest approach to multi-objective optimization is the weighted sum method:
$$ {U}_w=\sum \limits_{i=1}^k{w}_i{f}_i $$
(15)

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 wi > 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 non-convex 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.

Weighted min-max method (also called the weighted Tchebycheff method) is another method for solving a multi-objective optimization. This method is formulated as follows:
$$ {U}_w=\max \left\{{w}_i\left({f}_i\left(\mathrm{x}\right)-{f}_i^{{}^{\circ}}\right)\right\} $$
(16)

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 non-Pareto optimal points as well.

Weighted global criterion (WGC) is the last method that is used in this research. The most common weighted global criterion is defined as follows:
$$ {U}_w={\left(\sum \limits_{i=1}^k{\left[{w}_i\left({f}_i\left(\mathrm{x}\right)-{f}_i^{{}^{\circ}}\right)\right]}^p\right)}^{1/p} $$
(17)

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 fi 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 non-convex Pareto frontier [34, 35]. One can find the mathematical proof for why WGC method is able to find points on non-convex 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 multi-objective method has the ability to find solutions on the non-convex Pareto frontier only if it includes some parameters via which the function’s curvature can be manipulated [35].

5 Problem statement

Since angle variation is in the plane of laminated composite, these changes affect the in-plane properties of the laminate. So, only optimization of Ex, Ey, and Gxy is considered in this research. In addition, the out-of-plane properties could be enhanced by other method like 3D weaving, stitching, braiding, and z-pinning [36, 37, 38] which are beyond the scope of this study. The problem could be defined as finding the optimal ply angles, which yield to best possible in-plane effective elastic properties of a laminated composite. Different combinations of these elastic constants make different multi-objective optimization problems as follows:
$$ {\displaystyle \begin{array}{l}\mathrm{f}\mathrm{ind}\kern0.20em \theta =\left({\theta}_1,\dots, {\theta}_{\mathrm{n}}\right)\kern0.20em \mathrm{that}\kern0.20em \operatorname{maximize}\kern0.20em \mathrm{f}\left(\theta \right)=\left[{\mathrm{E}}_{\mathrm{x}}\left(\theta \right),{\mathrm{E}}_{\mathrm{y}}\left(\theta \right)\right]\\ {}\mathrm{subject}\ \mathrm{to}-90\le {\theta}_i\le 90\ \left(\mathrm{i}=1,2,...,\mathrm{n}\right)\end{array}} $$
(18)
$$ {\displaystyle \begin{array}{l}\mathrm{f}\mathrm{ind}\;\theta =\left({\theta}_1,\dots, {\theta}_{\mathrm{n}}\right)\kern0.20em \mathrm{that}\kern0.20em \operatorname{maximize}\kern0.20em \mathrm{f}\left(\theta \right)=\left[{\mathrm{E}}_{\mathrm{x}}\left(\theta \right),{\mathrm{G}}_{\mathrm{x}\mathrm{y}}\left(\theta \right)\right]\\ {}\mathrm{subject}\ \mathrm{to}-90\le {\theta}_i\le 90\ \left(\mathrm{i}=1,2,...,\mathrm{n}\right)\end{array}} $$
(19)
$$ {\displaystyle \begin{array}{l}\mathrm{f}\mathrm{ind}\kern0.37em \theta =\left({\theta}_1,\dots, {\theta}_{\mathrm{n}}\right)\ \mathrm{that} \operatorname {maximize}\ \mathrm{f}\left(\theta \right)=\left[{\mathrm{E}}_{\mathrm{x}}\left(\theta \right),{\mathrm{E}}_{\mathrm{y}}\left(\theta \right),{\mathrm{G}}_{\mathrm{x}\mathrm{y}}\left(\theta \right)\right]\\ {}\mathrm{subject}\ \mathrm{to}-90\le {\theta}_i\le 90\ \;\left(\mathrm{i}=1,2,...,\mathrm{n}\right)\end{array}} $$
(20)

It should be noted that because only in-plane 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 in-plane 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

In order to verify the correctness of the current study results, a research on which the current method can be employed is chosen from the literature for comparison. The problem is to find the optimal geometry and stacking sequence of a composite box-beam such that it satisfies the desired bending and torsional stiffness requirements and also has maximum elastic coupling [39]. Therefore, those normalized objective functions that need to be minimized have been described in the reference paper in a way that reflect the closeness of the bending and torsional stiffness of the composite beam with respect to some preferred values. On the other hand, the elastic coupling that its value reflects the stability of the helicopter rotor in the reference paper has been considered as the objective function that needs to be maximized. It should be noted that a maximization problem could be transformed into a minimization problem or vice versa by multiplying the objective functions by − 1. Three cases of discretization are supposed similar to the reference paper. In discretization I, the ply angles are chosen as integer multiples of 10° between 0 and 90°. In discretizations II and III, the permissible angles are integer multiple of 15° and 45°, respectively. In addition, the breadth and height could only be changed within an upper and a lower bound to create feasible box-beam configuration. In the reference paper, the problem is solved using particle swarm optimization (PSO) technique and is compared with real-coded genetic algorithm (RCGA). Using the current method, the box-beam design problem is exactly reformulated and solved according to assumptions considered in the reference paper [39]. In the ABC algorithm, the parameters are set as follows: number of artificial bees = 20, number of cycles = 300, and limit = 30. The results of both current method and reference method are reported in Table 1. The average error in this table is the average difference between the non-dimensional target stiffness values and non-dimensional actual box-beam stiffness values. Looking at the values for objective functions and average errors, one can see that the ABC-based method provides better solution than RCGA approach but similar solutions to the PSO-based technique, which verifies the correctness of the current method results. According to the reference paper with a population of 20 individuals to search the design space, the PSO and RCGA methods require 356 and 624 generations respectively to find the best solution. Running the current method Matlab code for 10 times, here, on average, only 300 cycles of the ABC algorithm was adequate to achieve the best solution. Hence, from the convergence study, the current method is acceptable too.
Table 1

Comparison between different methods for the problem of composite box-beam 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

θ1 (°)

30

45

45

30

45

45

40

90

90

θ2 (°)

50

90

0

50

90

0

50

90

90

θ3 (°)

80

15

45

80

15

45

0

45

45

θ4 (°)

20

75

90

20

75

90

40

75

90

θ5 (°)

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

Another problem that its answer is known is solved to ensure the accuracy of the results in finding global optimum. This problem is to optimize each of the elastic constants of a laminated composite independently. Solving this problem will also determine the utopia point for subsequent considerations. For this purpose, a reinforced laminated composite plate made of eight layers of unidirectional carbon-epoxy is considered. The in-plane elastic constants of unidirectional carbon-epoxy are E1 = 109.173 GPa, E2 = 8.474 GPa, G12 = 5.479 GPa, and ν12 = 0.27. The results of single-objective optimization of the elastic constants are tabulated in Table 2. As shown in this table, the optimal angle for single-objective maximization of Ex is 0° and the one for Ey is 90°. These two answers were predictable; because the highest modulus of elasticity of a unidirectional composite is always obtained in the principle material coordinate of the laminae, which is parallel to the fibers direction. In the problem of single-objective maximization of Gxy, the largest value for Gxy takes place when all the layers are uniformly and equally distributed in the + 45° and − 45° directions as shown in Table 2. This indicates that off-axis reinforcement is essential for good shear stiffness in unidirectional composites. The shear modulus Gxy becomes maximum for ± 45° because the principal stresses for pure shear load on ± 45° plies are along the material axis. Therefore, this answer was also predictable. The results for individual optimization of each of the elastic constants are in line with the expectations and this verifies the accuracy of the method.
Table 2

Results of single-objective optimization of the elastic constants

Objective function

Ex (GPa)

Gxy (GPa)

Ey (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]

Now, it is time to optimize the elastic constants simultaneously. Again, the reinforced laminated composite plate made of eight layers of unidirectional carbon-epoxy is considered. The results of simultaneously optimizing the elastic constants are given in Table 3 where weighted sum, WGC, and the weighted Tchebycheff methods are used as the multi-objective optimization methods. All the result in this table are obtained by considering equal weights for each of the elastic constants which means all of them have an equal preference. In the ABC algorithm, the parameters are set as follows: number of artificial bees = 50, number of cycles = 3000, and limit = 300. In addition, each variable is supposed to be chosen from the interval (− 90°, 90°) with an angle increment of only 5°. As shown in Table 3, all methods lead to the ply angles of [(0/90)4] in simultaneous maximization of Ex and Ey. This is logical because it reflects the fair distribution of principle directions of the layers in the directions x and y. In simultaneous optimization of Ex and Gxy or Ey and Gxy, some remarks could be mentioned about the absolute value of the objective functions. As seen in Table 3, the weighted global criterion and weighted Tchebycheff methods create more reasonable solutions with respect to the weighted sum method as it seems that the weighted sum method has more focus on optimizing Gxy than Ex or Ey. An interesting point is achieved during simultaneous optimization of Ex, Ey, and Gxy. As given in Table 2, the obtained angles can be paired in a way that the absolute value of their difference becomes 90°. For example, in the Tchebycheff method, this feature is evident in the form of 5° and − 85°, 45° and − 45°, 70° and − 20°, and finally 10° and − 80° angles. It should be noted that the results presented in Table 3 might change if the process of problem solving repeats, because of the features of trigonometric functions and possible symmetry forms, but the key issue is the uniqueness of the optimal objective function values (global maximum).
Table 3

Results of multi-objective optimization of the elastic constants using weighted sum, WGC, and the Tchebycheff methods

Multi-objective optimization method

Considered objective functions

Ex (GPa)

Ey (GPa)

Gxy (GPa)

Best ply angles (°)

Weighted sum

Ex, Ey

59.07

59.07

5.48

[(0/90)4]

Weighted sum

Ex, Gxy

28.6

13.94

26.74

[(± 40)2/(± 35)2]

Weighted sum

Ey, Gxy

13.94

28.6

26.74

[(± 50)2/(± 55)2]

Weighted sum

Ex, Ey, Gxy

31.45

31.45

23.34

[± 25/± 65/± 40/± 50]

WGC

Ex, Ey

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]

The best and average non-dimensional Uw values of each cycle for the problem of simultaneous optimization of Ex, Ey, and Gxy using the weighted Tchebycheff method are shown in Fig. 3. It is evident the best answer of each cycle is continuously optimized until the global optimum is achieved. The incremental jumps seen in the graph of the mean solution reflect the new random search of the scout bees whose previous food source is abandoned due to inability to increase the nectar’s amount.
Fig. 3

The variation of non-dimensional objective function value with respect to the cycle number

In multi-objective optimization problems, determination of the Pareto optimal set in the feasible decision region, which entirely covers the Pareto optimal set in the feasible objective region, is very important. This is because a specified Pareto frontier helps designers to choose among the best possible answers easily, considering existing preferences. This frontier could be found by changing the weights systemically. Therefore, to achieve this goal, the multi-objective optimization problems defined in the Table 3 as well as some similar problems with different number of layers are solved by changing the weights until the Pareto frontier is fully determined. This time, only the WGC and the weighted Tchebycheff methods are used because the weighted sum method cannot obtain points on non-convex portions of the Pareto optimal set, as proved in the literature. In addition, the full feasible objective region is also determined in each problem to make the accuracy of solutions clear because the Pareto frontier is obviously a portion of the feasible objective region boundary. For the problem of simultaneous maximization of Ex and Ey, the feasible objective region and the objective Pareto frontier obtained from current solution method for 6 laminated composites with 1, 2, 3, 4, 8, and ∞ layers of unidirectional carbon-epoxy are shown in Fig. 4. The utopia point is also specified in this figure. Here, the case of one layer composite is only presented for better understanding of changes because some of the multi-objective optimization parameters make sense only for more than one variable problems. As seen in Fig. 4 b–f, the obtained points describe the Pareto optimal frontier very well and have accuracy as well as uniform distribution in the whole frontier. In addition, it is obvious that the Pareto frontier is the closest boundary of the feasible region to the utopia point. The Pareto optimal frontier in simultaneous maximization of Ex and Ey has a stair shape, as seen in Fig. 4. In each problem, the number of stair vertical or horizontal lines is equal to the number of variables (plies) of that problem. For example, the vertical line number of the 4-layer optimization problem is four as seen in Fig. 4 d. The more the number of layers, the more stair steps and the wider feasible objective region. In fact, increasing the number of layers not only does result in better determination of the Pareto frontier in the feasible objective region but also shifts the Pareto frontier toward the utopia point. As the number of layers increases excessively, the Pareto frontier approaches to a straight line as seen in Fig. 4 f. Reading the value of Ex and Ey at the start and end of this line, one can notice that this line can be created by connecting two points. The first point coordinates are the initial values of ply elastic constants E1 and E2 (109.17, 8.47) and the second one coordinates are E2 and E1 (8.47, 109.17). One can say that these two points are also the solutions of single-objective optimization of Ex or Ey. This statement is correct too, because the principal stresses for pure tensile load on 0° plies are along the material axis.
Fig. 4

Graphical representation of the problem of simultaneous maximization of Ex and Ey. a 1-layer laminae, b 2-layer laminate, c 3-layer laminate, d 4-layer laminate, e 8-layer laminate. f Laminate with a large number of layers

For the problem of simultaneous maximization of Ex and Gxy, the feasible objective region, the objective Pareto frontier, and the utopia point for laminated composites with 1 and 8 layers of unidirectional carbon-epoxy are shown in Fig. 5. Here, problems with other different number of layers are not considered because in this problem, the feasible region boundaries for more than one layer laminate are similar. In fact, using more than two layers only leads to better determination of the Pareto frontier in this problem. The reason why more layers lead to better determination of the Pareto frontier can be understood better if we compare single-objective optimization of Gxy and Ex with simultaneous optimization of them. As seen before, for single-objective optimization of Gxy, the optimal ply angles combination is [(±45)n] while that of single-objective optimization of Ex is [(0)n]. Indeed, the optimal ply angles for simultaneous optimization of these two modulus must be a compromise between [(±45)n] and [(0)n]. When there are more layers, there will be more compromise solution because more layer combination is possible. Thus, the more the number of layers, the more the number of Pareto solutions.
Fig. 5

Graphical representation of the problem of simultaneous maximization of Ex and Gxy for laminated composites with a 1 layer and b 8 layers of unidirectional carbon-epoxy

Again, as seen in Fig. 5 b, the Pareto optimal frontier is well determined. As in optimization of Ex and Ey, 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 E1 and G12 (109.17, 5.48). However, the other point coordinates are the value of Ex and Gxy obtained in single-objective optimization of Gxy (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 Gxy cannot exceed 14 GPa when the value of elasticity modulus Ex is 80 GPa.

As there are 3 objective functions in the problem of simultaneous maximization of Ex, Ey, and Gxy, the feasible objective region and the Pareto optimal frontier are 3-dimensional, as shown in Fig. 6. The obtained points describe the Pareto optimal frontier precisely and they are uniformly distributed in the whole frontier. Therefore, the number of objective functions does not weaken the method performance.
Fig. 6

Graphical representation of the problem of simultaneous maximization of Ex, Ey, and Gxy for a laminated composite with 8 layers of unidirectional carbon-epoxy

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 multi-objective 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 multi-objective 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 Ex, Ey, and Gxy, 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 Ex and Ey, 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 Ex and Gxy, 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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mechanical EngineeringShiraz UniversityShirazIran

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