Vibration Optimization of Fiber-Metal Laminated Composite Shallow Shell Panels Using an Adaptive PSO Algorithm

Vibration Optimization of Fiber-Metal Laminated Composite Shallow Shell Panels Using an Adaptive PSO Algorithm

H. Ghashochi-Bargh ^a*, M.H. Sadr ^b

^a Department of Industrial, Mechanical and Aerospace Engineering, Buein Zahra Technical University, Buein Zahra, Qazvin, Iran

^b Aerospace Engineering Department, Amirkabir University of Technology, Hafez Avenue, Tehran, Iran

1. Introduction   

Fiber-metal laminate (FML) composites are laminates composed of alternating layers of reinforced polymeric composites and aluminum alloys in such a way that the aluminum alloy sheets are outer layers that protect the inner composite layers. FMLs’ main attribute is their improved fatigue resistance. Because of their outer aluminum alloy layers, FMLs have improved resistance to impacts and environmental conditions. The use of aluminum alloy layers improves specific stiffness and strength and also results in weight savings in the design of tension-dominated stresses in structural components. These hybrid materials are divided into three groups according to the type of fiber used in the polymeric composite layers, as follows: reinforced with aramid fibers (ARALL), glass fibers (GLARE), and carbon fibers (CARALL) [1, 2]. They have been introduced as structural composite materials for advanced aerospace applications, such as in the Airbus A380 aircraft fuselage, as aircraft lower wing skin, and as the internal parts of airplanes [3]. Figure 1 shows an FML-composite shallow shell panel structure that is discretized by strips. In aerospace applications, vibration can be a problem when the excitation frequency coincides with the resonance frequency, which is essential to maximize the fundamental frequency in laminated cylindrical panels.

Several researchers have reported different studies on optimizing the natural frequency of laminated composite materials. Mateus et al. [4] studied the optimal design of thin, laminated plates and obtained results for the maximum fundamental frequency and minimum elastic strain energy using the finite element method to determine the frequency response. Narita [5] offered a Ritz-based layerwise optimization approach for symmetrical composite plates with respect to fiber orientation. Apalak et al. [6] determined the optimal layer sequences of symmetrical composite plates using a genetic algorithm, artificial neural networks, and the finite element method. The fundamental frequency optimization of laminated composite plates was studied by Ghashochi and Sadr [7, 8] using an elitist genetic algorithm, a particle swarm optimization (PSO) algorithm, and the finite strip method. They also studied the optimal design of FML plates and obtained results for the maximum fundamental frequency using a PSO algorithm [9]. Sumana et al. [10] investigated the effect of fiber orientation on the hydrostatic buckling behavior of a fiber-metal composite cylinder. Moniri Bidgoli and Heidari-Rarani [11] investigated the buckling behavior of an FML circular cylindrical shell under axial compression via both analytical and finite element approaches. They studied the effects of FML parameters such as metal volume fraction (MVF), composite fiber orientation, the stacking sequence of layers, and geometric parameters on the buckling loads. Topal [12] studied the frequency optimization of symmetrically laminated angle-ply annular sector plates using the first-order shear deformation theory and finite element method to investigate the effects of annularity, boundary conditions, and sector angles on the optimal designs. Nazari et al. [13] optimized the maximum natural frequencies of FMLs of cylindrical shells. They used first-order deformation theory and the double Fourier series to solve the free vibration problem.

The objective of the present study was to find the optimum stacking sequence of the inner composite layers of FML shallow shell panels that gives the maximum natural frequency using the adaptive PSO (A-PSO) algorithm and finite strip method. To improve the speed of the optimization process, an adaptive strategy was used in the PSO algorithm. The classical shallow shell theory (Donnell’s formulation) was used for the finite strip formulation of the laminated shallow shell panels. Finally, the effect of different panel aspect ratios, ply angles, number of layers, and boundary conditions on the optimal designs was investigated.

1. Mathematical Modeling

 The classical shallow shell theory (CST) was used to perform the analysis. In the CST, the constitutive equations for a shallow shell panel can be expressed by performing appropriate analytical integration through the uniform thickness, as follows [14]:

where N_x, N_y, and N_xy are the membrane direct and shearing stress resultants per unit length; M_x, M_y, and M_xy are the bending and twisting stress couples per unit length; and A_ij, B_ij, and D_ij are the matrices of stiffness coefficients. These matrices are defined by the following:

and ε and ψ are the in-plane strain vectors at the mid-plane and the curvature strain vector. They are defined by the following:

The panel stiffness coefficients are defined as   

Here, is the transformed reduced stiffness matrix. Due to the symmetry of the lay-up, the coupling between in-plane and out-of-plane force and deformations will not appear (i.e., ).

The strain energy of the structure per unit volume is . Using Equations (1) and (2) and integrating through the thickness of the structure with respect to z gives an expression for the strain energy of a finite strip, which can be put into the following form:

The general expression for the strip kinetic energy is:

where [k] is the strip stiffness matrix, [m] is the strip mass matrix, {d} is a column matrix that contains the strip’s degrees of freedom, and ρ is the mean mass per unit area of the panel. For the whole structure, the total strain energy and kinetic energy are obtained by summations of the corresponding energy components of all strips.

The structural equation of motions can be obtained by applying the Lagrange equations as

The solution of this eigenvalue problem is

where ω are the natural frequencies. The natural frequency is normalized as a frequency parameter in

where the reference bending rigidity is

The optimal design problem can be stated as follows:

where k, Al, and ϴare half of the inner composite layers’ number, the number of aluminum layers, and the optimum ply angles, respectively. The optimal stacking sequences and ply angles were searched for with the A-PSO algorithm.

In the bi-objective optimization of FML panels, the objective functions combined with each other through the weighted summation method. The obtained single objective function was then optimized using A-PSO. To simultaneously maximize the fundamental natural frequency and minimize the weight, the objective function was considered to be in the form f(θ, Al) as a function of laminated angles and the sequence of metal and composite layers, which is defined as

where W1 and W2 are the weighting coefficients that sum the two objective functions to have a single fitness function, ω and σ are the optimum frequency and optimum weight, and ω₀ and σ₀ are the fundamental frequency and weight corresponding to the prescribed stacking sequences [Al/90/0/90]_s [8].

In addition, the bi-objective optimization of FML panels is considered as follows:

where ρ, t, and A are density, thickness, and area of layers, respectively.

The assumed in-plane displacement and out-of-plane displacement in the full-energy semi-analytical finite strip method are

where u_li, u_2i, v_li, and v_2i are the undetermined in-plane nodal displacement parameters; w_li, w_2i, ϴ_li, andϴ_2i are the undetermined, out-of-plane nodal displacement parameters along the edges of the strip; ; and  [8, 9].

1. Adaptive Particle Swarm Optimization Algorithm

PSO is a population-based stochastic optimization technique developed by Eberhart and Kennedy in 1995 [15] inspired by the social behavior of animals, such as fish schooling, insects swarming, and birds flocking. This method is used to search for the global optimum of a wide variety of arbitrary problems. PSO shares many similarities with evolutionary computation techniques such as genetic algorithms (GA). However, unlike GA, PSO has no evolution operators such as crossover and mutation. Compared to GA, PSO is based on social biology, which requires cooperation, while GA is based on competition. The advantages of PSO are its simple structure, its immediate accessibility for practical applications, its ease of implementation, its speed to acquire solutions, and its robustness. In PSO, each single solution is a “bird” in the search space. We call it “particle.” All of the particles have fitness values that are evaluated by the fitness function to be optimized and have velocities that direct the flight of the particles. The particles fly through the problem space by following the currently optimum particles.

PSO is initialized with a group of random particles and then searches for optima by updating generations. In every iteration, each particle is updated by following two "best" values. The first one (pⁱ) is the best position attained by the particle i in the swarm so far. Another best value ( ) is the global best position attained by the swarm at iteration k-1. After finding the two best values, the basic swarm parameters of position and velocity are updated using the following equations at each iteration [16]:

where the superscript i denotes the particle; the subscript k denotes the iteration number; v denotes the velocity; and x, which is a real number, denotes the position. The variables r₁ and r₂ are uniformly distributed random numbers in the interval [-1,1], and c₁ and c₂ are the acceleration constants. In a different reference, it is mentioned that the choice of these constants is problem-dependent. In this work, c₁ = c₂ = 1 was chosen, which gives optimal results in fewer iterations. The results were also rounded to the nearest integer values after optimization. A simple way to understand this updating procedure is described by Hassan, Cohanim, and Weck [17].

Since the initial development of PSO by Kennedy and Eberhart, several variants of this algorithm have been proposed by researchers. The first modification introduced in PSO was the use of an inertia weight parameter in the velocity update equation of the initial PSO, resulting in Equation (13), a PSO model that is now accepted as the global best PSO algorithm [18]. It is as follows:

In Eberhart and Shi’s paper [19], a random value of inertia weight was used to enable the PSO to track the optima in a dynamic environment. It is difficult to predict whether a given time exploration (large values of w) or exploitation (small values of w) would be better in dynamic environments. So, a random value of w is selected to address this problem.

where rand() is a random number in [0,1] and w is then a uniform random variable in the range [0.5,1].

In the papers by Eberhart and Shi [20, 21], a linearly decreasing inertia weight was introduced and was shown to be effective in improving the fine-tuning characteristic of the PSO. In this method, the value of w is linearly decreased from an initial value of w to a final value of w according to the following equation:

where iter is the current iteration of the algorithm, and iter_max is the maximum number of iterations the PSO is allowed to continue. This strategy is very common, and most of the PSO algorithms adjust the value of inertia weight using this updating scheme.

There are other approaches that use decreasing inertia weights [22], such as

In the present study, an adaptive strategy was used in the algorithm. In this strategy, a more diverse set of solutions generate near the best solutions so far until the end of each cycle in the algorithm. The use of an adaptive inertia weight provides a balance between global and local exploration and exploitation, and results in fewer iterations on average to find a sufficiently optimal solution. A parameter is defined to control the amount of generated solutions near the best solutions. In the algorithm, this parameter was assumed to be 20% of the best solutions. Thus, the same number of solutions will always be generated from a solution. As the value of the parameter increases, the PSO search becomes more localized. Thus, using a large value for the parameter may prevent the PSO from finding the global optimum. A selection procedure based on the fitness function picks the best solutions and replaces them in algorithm.

Adaptive inertia weight is updated using the          following equation at each iteration [23]:

where Fitness _min is the minimum value of Fitness and Fitness _max is the maximum value of Fitness in every iteration of the PSO.

Table 1.Test functions used in the paper were unimodal (U), multimodal (M), separable (S), and nonseparable (N).

To evaluate the proposed algorithm (A-PSO), it was compared with three other optimization algorithms. To investigate the performance of the optimization algorithms, ten benchmark test functions were adopted. Table 1 provides a detailed description of these functions.

The performance of the A-PSO is shown in Tables 2–5 in comparison with other algorithms for different parameters. From the tables, it can be inferred that for the ten test problems, A-PSO performed better than ABC and PSO in the case of nine problems. As seen from tables, the results obtained by A-PSO are closer to the results of RABC, whereas RABC performs better than A-PSO for five functions. In addition, it was concluded that using A-PSO provides a much higher convergence and reduces the required CPU time in comparison with the ABC and PSO algorithms.

Table 2.Search result comparisons on ten test functions.

Table 3. The standard deviation of the best solutions on ten test functions.

Table 4. Number of functions evaluated for optimization with different algorithms on ten test functions.

Table 5. Comparison of CPU time for different algorithms on ten test functions.

4.   Results and Discussions

The fundamental frequency of hybrid laminates was maximized for different panel aspect ratios, ply angles, length/R (a/R) ratios, number of layers, and boundary conditions. The degree of panel curvature was taken as a/R = 0 (plate), a/R = 0.2 (relatively shallow), and a/R = 0.5 (shallow) in the examples. A rise in the center was only 3% of the radius R for a shell of a/R = 0.5, and this can be regarded as geometry within the shallow shell theory [25–27]. The laminates were symmetric and made of AS/3501 graphite/epoxy material [28] (inner composite layers) and aluminum alloy 2024-T3 [1] (outer aluminum layers). The material properties of the lamina are given as follows: for the composite layers, E₁ = 138 GPa, E₂ = 8.96 GPa, G₁₂ = 7.1 GPa, and υ₁₂ = 0.3; and for the aluminum layers, E = 72.4 GPa and υ = 0.33. Each of the lamina was assumed to be same thickness.

In Table 6, the optimization of composite laminated panels using the PSO algorithm was validated with those given by Narita [5] through the use of the Ritz-based layerwise optimization method. As seen, the PSO algorithm was successful in predicting the optimal solutions, and higher natural frequencies than those predicted by Narita were achieved for different edge conditions. It can also be seen in Table 7 that there was a very good agreement between the results of the present approach and the published paper by Shooshtari and Razavi [1] for the fundamental frequency of symmetric FML panels (Glare 3). The lay-up and material properties of this hybrid composite (Glare 3) are as follows [1]:

Al(2024-T3)/[0⁰/90⁰]GFRC/Al(2024-T3)/[90⁰/0⁰]GFRC/Al(2024-T3)

Glass fiber–reinforced composite layers (0.2 mm thickness each): E₁ = 55.8979 GPa, E₂ = 13.7293 GPa, G₁₂ = 5.5898 GPa, υ₁₂ = 0.277, and ρ = 2550 kg/m³

Aluminum alloy 2024-T3 layers (0.3 mm thickness each): E = 72.4 GPa, υ = 0.33, and ρ = 2700 kg/m³

The dimensionless fundamental frequency of Glare 3 was obtained by using the following equation:

where I₀=  is the mass moment of inertia.

The accuracy of the A-PSO in predicting the optimal solutions is shown in Figure 2 in comparison with other stacking sequences with various aspect ratios when the length/R = 0.5. The comparisons demonstrate that the panels with the present optimum lay-ups actually give higher fundamental frequencies than panels with other lay-ups. The typical stacking sequences of symmetric 10-layer FML-composite shallow panels were chosen for comparison purposes; namely, [Al/0/0/0/0]_S, [Al/30/-30/30/-30]_S, and [Al/45/-45/45/-45]_S.

The optimal layered sequences and maximal natural frequency parameters of symmetric FML shallow shell panels were searched for using A-PSO for the various combinations of free (F), simply supported (S), and clamped (C) edge conditions. The panel length/width ratios (a/b = 1, 2, 3, 4), length/R (a/R = 0, 0.2, 0.5), and the layer number (n = 6, 10) are given in Tables 8, 9, and 10. The fiber angle of each inner ply in the FML-composite shallow shell panels was changed with a step of Δθ = 1⁰ between -90⁰  θ_n  90⁰.  

Table 6.Comparison of the optimal stacking sequences and natural frequency parameter of symmetric eight-layered composite panels (a/b = 2, increment ).

Table 7.Dimensionless frequencies of rectangular Glare 3–hybrid composite(h/ɑ = 0.01).

Table 8. Optimum solutions for symmetric, fiber-metal laminated, shallow shell–type panels for different a/b ratios and edge conditions when a/R = 0.5.

Table 9. Optimum solutions for symmetric, fiber-metal laminated, shallow shell–type panels for different a/b ratios and edge conditions when a/R = 0.2.

Table 10. Optimum solutions for symmetric, fiber-metal laminated, shallow shell-type panels for different a/b ratios and edge conditions when a/R = 0.

Table 11. Bi-objective optimization results of symmetric, FML, eight-layered panel (a = 1 m, b = 1 m).

As can be seen, the optimal fiber orientations vary from 0⁰ to 90⁰ or 0⁰ to -90⁰ with increases in a/b ratios. The results demonstrate that the edge conditions play an important role in the natural frequency parameter of shallow FMLs. As the number of the clamped panel edges increased, an evident increase in the natural frequency parameters was observed. This can be explained by the fact that the clamped edges provide less degrees of freedom, and the effect of this is to stiffen the shallow panels. Since graphite/epoxy is far less dense than aluminum, the presented optimum FML results with two aluminum layers are optimum in terms of weight, too.

The results of the bi-objective optimization of symmetric, FML, eight-layered panels with simply supported edge conditions are presented in Table 11 with respect to the first natural frequency and the weight for a/b = 1 and a/R = 0. The laminates were made of glass fiber–reinforced composite [1] and aluminum alloy 2024-T3 [1], and the thickness of layers was considered to be 0.25 mm. As inferred from the results, the angle of layers and sequence of metal and composite layers play an important role in the fundamental frequency and weight of panels. As seen, FML panels with outer aluminum layers have the maximum natural frequency and minimum weight in comparison with other stacking sequences.

1. Conclusions

In this study, the fundamental frequency optimization of FML curved panels was studied using the combination of A-PSO and FSM for various panels edge conditions, a/b ratios, a/R ratios, and layer numbers. As inferred from the results, the A-PSO provides a much higher convergence and reduces the required CPU time in comparison with the PSO and ABC algorithms. As seen, the combination of A-PSO and FSM was successful in determining the fundamental frequency and the optimal layering sequences of FML curved panels. It can be noticed from the results that the maximum fundamental frequency and the optimum stacking sequences were substantially influenced by edge conditions, a/b ratios, and a/R ratios. The maximum fundamental frequency and the optimum fiber orientations were not substantially influenced and approach a limiting value with an increasing layer number.

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