ANT COLONY ALGORITHM APPLIED TO FUNDAMENTAL FREQUENCY MAXIMIZATION OF LAMINATED COMPOSITE CYLINDRICAL SHELLS

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1 ANT COLONY ALGORITHM APPLIED TO FUNDAMENTAL FREQUENCY MAXIMIZATION OF LAMINATED COMPOSITE CYLINDRICAL SHELLS Rubem M. Koide 1 *, Marco A. Luersen 1 ** 1 Laboratório de Mecânica Estrutural (LaMEs), Universidade Tecnológica Federal do Paraná (UTFPR), Curitiba, Brazil, * (rubemkoide@hotmail.com), **(luersen@utfpr.edu.br) Abstract: Ant colony optimization (ACO) is a class of heuristic algorithms proposed to solve optimization problems. The idea was inspired by the behavior of real ants, related to their ability to find the shortest path between the nest and the food source. ACO has been applied successfully to different kinds of problems. This paper deals with optimal stacking sequence design of laminated cylindrical composite shells aiming to maximize their fundamental frequency. The stacking sequence optimization is formulated as a combinatorial problem and is solved using ACO. Cylindrical composite shell structures are important in many applications, especially in aeronautical, aerospace and petroleum/gas industries. The knowledge of the dynamic response of these structures is an important issue in their design. The ply thicknesses are often predetermined and the ply orientations are usually restricted to a small set of angles due to manufacturing constraints. This leads to problems of discrete or stacking-sequence optimization. As this optimization problem is very hard to solve, several techniques have been developed. The ACO algorithm was implemented in MATLAB platform and was linked to a finite element code to compute the structural response. Cylindrical shell without and with a cutout geometry are studied. Fundamental frequency was simulated for both cases and the results were presented. The computational efficiency of the ACO applied to cylindrical shells composite has been presented. The connections between ACO and finite element method was also computed. This approach has been prepared for optimize others complex geometries involving cylindrical composite shells or study another different laminated composite parts. Keywords: Ant colony optimization, Laminated cylindrical shells, Frequency maximization. 1

2 1 Introduction Ant colony optimization (ACO) is a class of heuristic algorithms proposed to solve optimization problems. The idea was inspired by the behavior of real ants, related to their ability to find the shortest path between the nest and the food source. ACO has been applied successfully to different kinds of problems. So, this paper deals with optimal stacking sequence design of laminated cylindrical composite shell. The stacking sequence optimization is formulated as a combinatorial problem and is solved using ACO. The developed ACO algorithm is applied to the problems of maximizing the fundamental frequency of cylindrical shells without and with a cutout. 2 ACO applied to Cylindrical Shells 2.1 Ant Colony Optimization (ACO) Ant colony optimization is a metaheuristic in which a colony of artificial ants cooperates in finding good solutions. This technique is applied to difficult discrete optimization problems. As describe by Dorigo and Stülzle[1], the main procedure of the ACO metaheuristic manages the scheduling of the three components of ACO algorithms via the ScheduleActivities construct: (1) management of the ant s activity, (2) pheromone updating, and (3) daemon actions. The pseudo-code of ACO metaheuristic is describe by Dorigo and Stülzle[1] as Procedure ACOmetaheuristic ScheduleActivities ConstructAntsSolutions UpdatePheromones DaemonActions % optional End-ScheduleActivities End-procedure The ConstructionAntsSolutions is a procedure where the solution of the optimization problem is build. In this way the artificial ants apply a stochastic decision rule in a problem defined by a construction graph. While the solution is being built, based on the pheromone trails and heuristic information, the ants evaluate the solutions for searching the optimal feasible candidate. The UpdatePheromone procedure based on the solution available and it influences the quantity of pheromone. It can be increased or decreased. The deposit of new amount of pheromone increases the probability of a good solution in the next decision. The decrease process is related to pheromone evaporation. It influences in a choice of candidate due to the reduction of pheromone trail for the solution available and in this way the bad candidate is not selected. This process actives the convergence of the algorithm. The DaemonActions procedure can be used as an optional process to optimize the algorithm. The local or global information can be used to decide or influence in a search optimization procedure. 2.2 Laminated cylindrical shells Composite cylindrical shell structures find wide applications in many products. These kinds of shell structures are in general efficient in engineering applications including aircraft, offshore platforms, silos, spacecraft, pressure vessels, pipelines and many others industries. Considering these applications many studies have been developed in this area. Heyliger and Jilani [2] studied free vibration of laminated anisotropic cylindrical shells and presented numerical results for isotropic, anisotropic and orthotropic composite shells. In the case of non-circular cylindrical shells Ganapathi and Haboussi [3] have been developed free vibrations of thick laminated. Poore et al. [4] also presented free vibration of laminated cylindrical shells with a circular cutout. Although the advance in frequency studies, involving complex mathematics analysis, experimental tests and recently numeric methods of the optimization this subject studies are in development. This work addresses the development of an ACO algorithm applied to the lay-up design of composite cylinders. Numerical tests were performed for the lay-up optimization of laminated cylindrical shells with holes aiming maximizing the fundamental frequency. Laminated composite materials consist of stacks of layers, each layer usually composed by a matrix of polymeric material and fibers oriented in a specific direction. Laminated composites give the designer the ability to tailor the material according to the application and the structures formed by these materials present high stiffness/mass and strength/mass ratios. The mechanical properties of composite laminates depend on the material of each layer, the number of layers, the thickness of each layer and the fiber orientations in each layer. The ply thicknesses are often predetermined and the ply orientations are usually restricted to a small set of angles due to manufacturing constraints. This leads to problems of 2

3 discrete or stacking-sequence optimization. As this optimization problem is very hard to solve, several techniques have been developed. Genetic algorithms (GA)have been successfully applied to solve these kinds of problems (e.g., [5]). Recently, a new class of algorithms, the ant colony optimization (ACO), was developed to solve combinatorial optimization problems. ACO was inspired by the observation of the behavior of real ants ([1]). Regarding the optimization of composite cylinders, Hu and Wang [6] studied laminated shells with and without cutouts; Hu and Tsai [7] analyzed fundamental frequencies of laminated cylindrical shells. Tabu embedded simulated annealing (TSA) was computed for optimal stacking sequence design by Rao and Arvind [8]; stacking sequence optimization of laminated cylindrical shells for buckling and free vibration using genetic algorithm and neural networks have been investigated by Gharib and Shakeri [9]. A discrete model for optimal design of thin composite plate-shell maximizing a specified natural frequency using two-level approach was analyzed by MotaSoares et al. [10]. Chern and Chao [11] studied the natural frequencies of laminates by 3-D theory in spherical, cylindrical, plate and curved panels. Amabili [12] compared different shell theories for nonlinear vibrations of laminated circular cylindrical shells.rao and Lakshmi [13] proposed an algorithm for solving multi-objective optimization problem. Hybrid scatter search algorithm was employed to solve stacking sequence optimization of hybrid fiber reinforced composite plate, cylindrical shell and pressure vessel. 2.3 ACO applied to laminated cylindrical shells This work aims at studying the ACO applied to frequency maximization of laminated cylinder composite. ACO takes inspiration from the studies of real ant colonies foraging behavior. The pheromone, a chemical concentration that ants deposit on the ground, their concentrationinfluences the choice of best way probabilistically. Called stigmergy, this kind of behavior is the mechanism that controls ant activity and connects them to take the shortest paths.pheromone matrix is the basic information processed by ACO to find the solutions following the way with more concentration of this substance. Artificial ants are used to construct solutions for laminated composite cylinders, by choosing probabilistically the orientation of the stacks of the laminated. The solutions are built on the past search experience based on the level of pheromone, and the heuristic matrix, which brings the information about the laminated cylindrical problem. The ACO procedure starts with the random selection of fiber orientation for a laminated layup. After the artificial ants have finished building the first laminated configurations, the pheromone is released with evaporation and deposited the new amount of pheromone based on the best solutions found at local and global iterations. The algorithm stops when the maximum number of iterations or function evaluations is reached. The following optimization constraints are considered in this work: first ply failure criterion, a maximum number of four contiguous plies with the same orientation. Also, the plate is considered symmetric and balanced, and the allowable ply orientations are (0/0/+ 45/- 45/90/90). 3 Numerical results The case study considered is an ACO applied to composite cylinder. Ant colony optimization was implemented in Matlab and was linked to a commercial finite element code to compute the frequency response. The ply angles considered for the composite cylinder are 0, ±45, 90. The laminates are considered to be symmetric and balanced. The first step of simulation is defined by ACO. ACO works to compute the ply angles and send to the finite element code. Finite element is processed for fundamental frequency response. This value is returned to ACO to maximize the frequency. In order to validate the results, the first numerical studies were a frequency simulation obtained in finite element code and compared with Poore et al. [4]. Cylindrical with radius of the shell in, length, thickness as specified by Poore et al. [4], and the material properties as psi,,,,. Simply supported (SS3) boundary conditions were adopted along the edges of the shell. Table 1 presents the normalized natural frequency which is defined as Table 1.Normalized natural frequency. Layup Poore et al. [4] Present study (1) 3

4 The fundamental frequencies were computed with ACO linked to finite element method for cylindrical composite without and with cutout as showed below. 3.1 Optimization problem The optimization problem is formulated as Find:,, k=1 to n Maximize: Fundamental frequency (2) Subject to: - Symmetric and balanced laminate - Number of contiguous plies=4 where is the orientation of each stack of the laminate and n=64 is the total number of stacks. Graphite/Epoxi material was used. Their properties are,,, The cylindrical geometries are: radius of the shell, length, thickness Simply supported (SS3) boundary conditions were adopted for simulations along the edges of the shell Cylindrical shell without cutout Firstly, the fundamental frequency was obtained considering all the 64 plies oriented with a same angle ( or. The results are showed in Table 2 and the corresponding cylinder geometry is presented in Fig. 1. Table 2.Fundamental frequency without cutout. Layup Frequency (Hz) In the finite element analysis, the laminated cylindrical shells are modeled by eight-node shell elements with six degrees of freedom per node. An 8-node curved thick shell with reduced integration. The optimized fundamental frequency value for laminated was obtained. The stacking sequence of laminated is presented in Table 3. Table 3.Optimized stacking sequence. 3529,40 (Hz) N The is the fundamental frequency, is the lamina stacking sequence optimization and the N value is the number of objective function evaluations. Table 4. Percentual results without cutout. Layup Frequency (Hz) % , , ,71 The Table 4 presents the comparison between optimized frequency value in percentual and the frequency obtained by plies oriented with a same angle ( or. Besides the gain with ply oriented in 45 is not increased, the percentual for and achieved more than 73,14% and 95,71%, respectively Cylindrical shell with cutout In this case the cylindrical shell geometry is more complex. Fig. 2 shows the cutout geometry and Fig. 3 the cylinder with cutout and the corresponding finite element mesh. The fundamental frequency was obtained considering at first 64 plies oriented with or. The results are showed in Table 5. Table 5.Fundamental frequency with cutout. Layup Frequency (Hz) Fig.1. Cylinder shell without cutout. 4

5 Table 7. Percentual results with cutout. Layup Frequency (Hz) % 1998,70 70, ,00 2, ,20 91,48 The percentual results for composite cylindrical shell with cutout are showed in Table 7. Considering the all plies oriented in or the optimized frequency value achieved more than 70%. Fig. 3 presented the simulation of cylindrical shell composite with cutout for optimization fundamental frequency. Fig.2. Cutout geometry. Laminated cylindrical shells are modeled with eightnode shell elements with six degrees of freedom per node with reduced integration. Fig.3. Optimization cylinder shell with cutout. Fig.3. Cylinder shell with cutout. The optimized frequency value for laminated with cutout was obtained and is showed in Table 6. Table 6.Optimized stacking sequence. 3403,00 (Hz) N The is the fundamental frequency, is the lamina stacking sequence optimization and the N value is the number of objective function evaluations. 4 Conclusions The cylindrical shells tailored with composite material were optimized. Fundamental frequency maximization was solved by ACO and finite element method. The ply angle was simulated for cylinder shell without and with cutout. The numerical results are presented in both cases. The presented optimization class can be applied for cylindrical shells with complex geometries. ACO applied to cylindrical composite was efficient in optimize the lamina stacking sequence of laminated and the link with finite element method can optimize different kinds of laminated composite. References [1] M. Dorigo and T. Stützle Ant colony optimization, MIT. Cambridge, USA,

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