Multiobjective optimization of composite cylindrical shells for strength and frequency using genetic algorithm and neural networks

1 Introduction

The development of the composite materials technology and reduction of their costs has speeded up the application of these materials as proper replacements of metal alloys. An unbelievable increase in composite materials application indicates the fact that extensive efforts have been assigned to analysis and design and development of composite structures. One of these composite structures includes cylindrical shells, the stacking sequence optimization of which is of interest for many researchers. The earlier works in this field have been done by Hayashi [1]. Chen and Liew [2] have studied the buckling behavior of functionally graded plates analytically by considering only one term in the nonlinear strain field.

Chen and Bert undertook the problem of composite structures optimization from the view of buckling strength [3]. Fukunaga et al. developed a method called “layer parameter” to optimize the stiffness and natural frequency and published the method in two papers [4, 5]. Recently, Abouhamze and Shakeri [6] have implemented multiobjective optimization of vibration and buckling of cylindrical panels using genetic algorithm (GA) and the artificial neural network (ANN). ANN has found extensive applications in different branches of science and engineering for linear and nonlinear problems, among which the earlier work of the authors [7] can be mentioned. Investigations have also been made to optimize the crash response of structures using neural networks [8, 9]. As another example of artificial intelligence application in the structural optimization approach, Bisagni and Lanzi [10] have used neural networks in the GA to optimize the response of composite stiffened panels in the buckling and postbuckling stages up to collapse of the structure.

In the optimization of composite structures, the laminated composite materials are usually used and design variables in this process include fiber orientation, total thickness of the layers, and the number of layers. The parameters involved in the optimization of composite structures are usually known to be weight, cost, strength, vibration properties, stress, and buckling. Variation in number or thickness of the layers, weight, and cost as well as variation in fiber orientation will certainly affect the strength of the structure. However, in the optimization problems concerning static and dynamic loads and vibration properties, stress and buckling are usually assumed to be constant and fiber orientations in each layer are varied within a particular range. This process of optimizations is called layers stacking optimization. In this process, an objective function is minimized or maximized using a specific method. An efficient method for optimization is a procedure in which, in addition to getting appropriate results, the process is done in a minimum time and calculations. However, the speed of convergence to find the best solutions is also of great importance. GA is an ideal optimization algorithm for these kinds of problems. GA as an optimization calculation algorithm searches the different areas of the solution efficiently by considering a set of points in the solution space in each part of calculation. Although GA is a powerful technique for search and optimization problems with discrete variables, Gantovnic et al. [11] have tried to find a suitable algorithm for a GA with memory that can work with discrete and continuous variables simultaneously.

One of the defects of GA is to be computationally expensive in solving the optimization problem. Because of this and also the random nature of this algorithm in solving the optimization problems, the process of optimization would be slow. This research has tried to address this problem by means of a very powerful way using ANN while keeping all the other useful properties of the method; eventually, the optimal solutions including fiber orientations and also the final values of objective function have been obtained.

2 Problem formulations

The cylindrical shell of M layers and length of L is shown in Figure 1, where the fibers make angle θ with x axis. By changing the fiber orientations with respect to the principle axis, the mechanical properties of the material would change. Obtaining the optimal angles of the fibers with respect to the principle axis is of interest, in which the structure would have the best properties from the view of strength and frequency. One of these composite structures are shells that in its own type are very important and practical. The aim of this paper is to obtain the optimal orientations of the fibers with respect to the x axis in a four- and six-layer cylindrical shells in which the strength and the first natural frequency are maximized simultaneously by the hybrid method of GA and ANN.

Figure 1

Laminated cylindrical shell of M layers.

3 Modeling

To model the frequency and strength objective functions, training data sets of neural network are required. The needed data sets for training and testing the neural network are prepared by writing the codes incorporating the equations derived in the previous section. In multiobjective optimization, herein two-objective optimization, two separate neural networks for strength and frequency should be designed. The structures of the trained ANNs for four-layer shells are presented in Tables 2 and 3.

In this modeling, a back-propagation neural network algorithm of three layers (input layer, one hidden layers, and output layer), known as Leverberg-Marquart (LM), has been used [13] using training functions available in MATLAB 7.0 [14].

The mean square error for neural network training has been considered as 10^-7. Because the shell is made up of four layers, the number of neurons in the input layer is equal to four. The input neurons are considered as fiber orientation and the output neurons are considered as strength and frequency.

4 Optimization

For strength calculation, the inverse form of Tsai-Hill yield criterion as Equation (15) was used:

where X,Y,Z are yield stresses at three principal directions of the composite material and R,S,T are shear stresses.

The calculated stresses are obtained by the following transform:

where m=sin(θ), n=cos(θ), and θ are the fiber orientation with respect to the main axes of the shell. The stresses in three principal directions are attained in each layer by using Equations (14) and (15) and will be introduced into yield criterion. Then, the strength is maximized by maximizing Q. For two-objective optimization, first, the fitness functions of strength and frequency should be combined using weight coefficients. To do this, the frequency and strength should become dimensionless. The combination of dimensionless strength and frequency using weight coefficients is as in Equation (17):

where W₁ and W₂ are weight coefficients and Q and w* are strength and frequency functions, respectively.

According to Equation (17) in two-objective optimization, if W₁=0, then the optimization would change to a single-objective optimization of strength, and if W₁=1, then the optimization would change to a single-objective optimization of natural frequency. GA utilizes the objective functions modeled by neural network instead of analytical solution. Here, for the weight coefficients of W₁=W₂=0.5, initially, the maximum fitness function is calculated for and then the fitness function increase and optimal solutions are compared with corresponding situation of weight coefficient when the GA uses the analytical solution, and once the correctness of neural network application in GA is ensured, fitness function and optimal angles of fibers in composite shell layers will be calculated for other values of weight coefficients, and the optimal fiber orientations of laminates in composite shell will be obtained.

According to Figure 2, the best solution for two-objective function for a four-layer shell in GA and analytical solution for W₁=0.5 is 5.21. The stacking associated with this solution is

Figure 2

Comparison of objective function increase by GA and ANN and analytical solution for four-layer shell in w=0.5.

θ=[75° 70°-65° 50°].

The solution obtained in GA and ANN is 5.29 for which the stacking is as follows:

θ=[75° 45°-55° 20°].

Comparing the solutions obtained from these two methods, it can be seen that using ANN in GA instead of the analytical solution is acceptable with a small error (∼1.53%), but the use of ANN increases the optimization speed significantly. The increase of objective function for other values of weight coefficients is presented in Figures 3–8.

Figure 3

Increase of objective function by GA and application of ANN for four-layer shell in w=0.

Figure 4

Increase of objective function by GA and application of ANN for four-layer shell in w=0.2.

Figure 5

Increase of objective function by GA and application of ANN for four-layer shell in w=0.4.

Figure 6

Increase of objective function by GA and application of ANN for four-layer shell in w=0.6.

Figure 7

Increase of objective function by GA and application of ANN for four-layer shell in w=0.8.

Figure 8

Increase of objective function by GA and application of ANN for four-layer shell in w=1.

Finally, the optimization results and optimal orientations of the fibers for the four-layer shell are presented in Table 4.

The same procedure is applied to six-layer shell and the objective function and corresponding optimal orientations are obtained. According to Figure 9, the best solution of objective function for a six-layer shell in GA and analytical solution for W₁=0.5 is equal to 6.37 and the result obtained in GA and ANN is equal to 6.48.

Figure 9

Increase of objective function by GA and application of ANN for six-layer shell in w=0.5.

Comparison of the results obtained by these two methods also shows that application of ANN in GA instead of the analytical solution is acceptable with a small error (∼1.72%).

For the six-layer shell, the optimization results are also obtained for different weight coefficients, where the optimal solutions and the corresponding orientations are presented in Table 5.

According to the optimization results obtained for four- and six-layer shells, it is considered that the stacking sequences are different for different weight coefficients. In w=0.1 for which the two-objective optimization turns into single-objective optimization of frequency and strength, the fitness function has its highest value, and in w=0.2, 0.4, 0.6, and 0.8, the optimal fitness function has lower values.

The results of the best fitness function for four- and six-layer shells are presented in Figure 10 for comparison. It can be noted from the figure that the fitness function for six-layer shell has higher values compared to its corresponding weight coefficients in four-layer shell, that is, increasing the number of layers improves the optimal solution.

Figure 10

Variations of fitness function with weight coefficient W₁ increase.

5 Conclusion

Comparison of fitness function progress when GA uses the analytical solution and when it uses ANN instead of analytical solution shows that, in the case of ANN application, although there is a small error, the optimization speed increases significantly, and it can be said that, in this way, one of the major problems of the GA method, which is high processing time, is resolved while maintaining its other advantages.

The results of two-objective optimization show that the strength function is much more sensitive to the angle variations, and because the fiber angle increment is considered as 5° in programming, with a small change, the value of this function changes significantly. Hence, the layers stacking with various weight coefficients is closer to the considered stacking in the strength objective function.

By increasing the number of layers in two-objective optimization, the optimal solution also increases, and by changing the weight coefficients, the number of population generation in GA also varies, so for W₁=0.1 for which the two-objective optimization turns into single-objective optimization, the minimum number of population generation will be reached. The reason is that near the coefficients of 0.1, the members of the population decrease, and at 0.1, half of the members are removed. Therefore, searching in fewer members needs less time and thus fewer numbers of population generations.

Increase in number of layers leads to increased number of population generation to reach the optimal solution and this increased number of population generation is due to increased members of original sample, which was selected for a population generation. The optimal solution varies by change of W₁ until in W₁=0.6, the optimal fitness function will have its minimum value. Eventually, different fiber orientations have been obtained as optimal solution, which can be used in design and production of cylindrical shells.