Optimization of geometrical parameters in a specific composite lattice ...

Journal of Mechanical Science and Technology 30 (4) (2016) 1763~1771 www.springerlink.com/content/1738-494x(Print)/1976-3824(Online)

DOI 10.1007/s12206-016-0332-1

Optimization of geometrical parameters in a specific composite lattice structure using neural networks and ABC algorithm† M. SadeghYazdi1, S. A. Latifi Rostami2,* and A. Kolahdooz3 1

Department of Mechanical Engineering, Babol University of Technology, Babol, Iran 2 Faculty of Mechanical Engineering, University of Semnan, Semnan, Iran 3 Young Researchers and Elite Club, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran (Manuscript Received April 24, 2015; Revised November 18, 2015; Accepted December 17, 2015) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract Due to their light weights and high load carrying capacities, composite structures are widely used in various industrial applications especially in aerospace industry. Strength to weight ratio is known to be as one of the most critical design parameters in these structures. In this paper, geometrical parameters of composite lattice structures are optimized to obtain the desired strength to weight ratio using finite element method, neural networks and ABC algorithm. At first, the finite element model is validated by experimental results and neural network is employed as the fitness function. The ABC algorithm is also applied to achieve the optimized strength to weight ratio. The results obtained from PSO algorithm on the basis of neural network have shown reasonable agreement with those of the finite element simulation. Increasing the thickness of the outer shell causes the structural strength-to-weight ratio to rise by 50 percent. The next effective parameter is reduction of rib angle which provides an increase of 30 percent in strength-to-weight ratio. Although Stiffeners (ribs) have a major role in load carrying, increasing the rib thickness causes the structural weight to rise. Thus compared with the two previous parameters, they do not have a significant effect on the strength of structures. Keywords: Composite lattice structure; Finite element; Neural network; ABC algorithm ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction Structural efficiency is a primary concern in today’s aerospace and aircraft industries. This brings about the need for strong and light weight components. Due to their high specific strength, cylindrical structures made of fiber reinforced polymers find wide application in these areas. Aircraft fuselage, and launch vehicle fuel tanks are some of the many applications of these structures in aerospace and aircraft industries. Stiffening ribs are the main load carrying parts of the lattice structures [1]. A quick review of literature reveals that studies on these structures are mainly categorized into analytical, experimental and finite element studies. Cylindrical and conical anisogrid composite shells are highly efficient and extensively have been used in various structural applications, such as rocket inter stages, pay load adapters for space craft launchers, fuselage components for aerial vehicles and components of the deployable space antennas. Buckling under various loading conditions is one of the typical modes of failure of the anisogrid composite lattice *

Corresponding author. Tel.: +98 9118617941, Fax.: +98 1134571616 E-mail address: [email protected], [email protected] † Recommended by Associate Editor Kyeongsik Woo © KSME & Springer 2016

shells. The case of buckling of the lattice cylindrical shells under axial load is of significant practical interest and has been the most extensively studied and reported in the existing publications. The history of the lattice structures, review of the founding studies and analysis of design approaches and fabrication techniques were introduced in detail by Vasiliev et al. [2, 3]. Slinchenko and Verijenko [4] performed analysis of composite isogrid cylindrical lattice shells using the smeared stiffness approach. In their work, constitutive equations are developed and the expressions for components of stress and strain tensors are derived for the shells of revolution with different lattice patterns. Numerical verification of the mathematical models is performed. The advantage of their approach is the possibility to calculate structural stress resultants without performing finite element analysis which allows achieving higher computational efficiency. Wodesenbet et al. [5] used analytical, experimental and FE methods for analyzing anisogrid stiffened composite cylindrical shells with specific cellular shape. They obtained numerical results of failure loads and modes of buckling by creating a three-dimensional finite element model based on the distribution of the unit cell and verified answers with those reported in the literature. Totaro and Gurdal [6] proposed an optimization

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method for composite lattice shell structures under axially compressive loads using the continuum model. Morozov et al. [7] studied the buckling behavior of anisogrid composite lattice cylindrical shells under axial compression, transverse bending and torsional loading. They used finite-element method to investigate the buckling of lattice shells with cutouts. The effect of various geometrical parameters such as the length of the shells, the number of helical ribs and angles of orientation of these ribs on the buckling behavior of lattice structures was examined. Hillburger et al. [8] examined a cylindrical composite structure with reinforced cutout under the effect of axial compressive load and compared the strength of the structure without the reinforcement of the cutout. Nonlinear analysis was used to investigate the nonlinear behavior of shells, which predicted statically stable and dynamically unstable behaviors of buckling. Elastic instability of composite lattice structures for aeronautic applications under external hydrostatic force was analyzed with finite element method and experimental by Frulloni et al. [9]. They used 8-noded layered shell elements available in ANSYS. Fan et al. [10] used the same FE package to investigate the uniaxial buckling strength of periodic lattice composites. Craig et al. [11] studied optimization of buckling characteristics of lattice composite shells with geometric defects incorporating Karhunen-Love based geometrical imperfections. Rahbar Ranji et al. [12] utilize a semi-analytical technique to study bending behavior of cylindrical panels with different boundary conditions under general distributed loading. In this paper, the solution of the partial differential equations was reduced to an iterative sequential solution of a double set of ordinary differential equations using extended Kantorovich method. Vasiliev et al. [13] reviewed recent Russian experiences in the development and application of composite grid structures. They studied developments in application of the composite lattice structures in the aerospace industry. Lattice structures of cylindrical and conical forms were examined. In their study, information about manufacturing processes, design, analysis and mechanical properties of lattice composites in the aerospace industry was presented. Buragohain and Velmurugan [14] carried out the buckling analysis of composite hexagonal lattice cylindrical shells. An energy-based Smeared stiffener model (SSM) is developed to obtain equivalent stiffness coefficients. Using the equivalent stiffness coefficients, Ritz buckling analysis was carried out. Extensive finite element modeling covering different representative sizes have been carried out. SSM is validated by comparing the estimated buckling loads. Variation of material properties of rib unidirectional composites from those of normal unidirectional composites is accounted in the energy formulations. Hence, optimization of geometrical parameters to achieve the optimum strength to weight is very important in Composite lattice structure. Abdessalem Jarraya et al. [15] searching about the validation of a recently proposed hexahedral solidshell finite element in the buckling analysis of a laminated

Fig. 1. The unit cell of a lattice structure.

composite plate with delamination. The object is to study the buckling behavior of structures with delamination using the Enhanced assumed strain (EAS) solid shell element with 5, 7 and 9 parameters. The EAS three-dimensional finite element formulation presented in this paper is free from shear locking and leads to accurate results for distorted element shapes. Mirdamadi et al. [16] investigate the free vibration of multidirectional functionally graded circular and annular plates using a semianalytical/ numerical method, called state spacebased differential quadrature method. Three-dimensional elasticity equations are derived for multi-directional functionally graded plates and a solution is given by the semianalytical/numerical method. Meshless collocations utilizing Gaussian and Multiquadric radial basis functions for the stability analysis of orthotropic and cross ply laminated composite plates subjected to thermal and mechanical loading are presented by Sandeep Singh et al. [17]. In this study, the geometrical parameters of a specific composite lattice structure are optimized using finite element simulation, neural networks and ABC algorithm to reach the highest strength to weight ratio. Thickness and angle of the rib and the shell thickness are considered as design variables. First, finite element models are developed and validated for the composite structure through comparison with the experimental results. Then, several simulations are performed in training the neural network. Neural network was used as the fitness function. Finally, the optimized variables were achieved by using ABC algorithm based on the neural network.

2. Problem description Anisogrid (Anisotropic grid) composite lattice cylindrical shells are composed of curvilinear helical ribs made of a unidirectional composite material having high specific strength and stiffness. A helical rib deviates by an angle from the cylinder generator line. The geometrical modeling of the lattice shell structure is based on four input variables, i.e. the overall length, L, the overall diameter, D, the helical angle and the number of ribs, n. The lattice structure may be represented by only one helical direction. Hence, the model is generated using a single typical unit cell only, as shown in Fig. 1. Generally, each lattice structure is made by repeating several unit cells. The strength of a composite lattice structure is directly related to these cells. In lattice composite shells that are often made with curved ribs, the dimensions including width, thickness, number of ribs, their distance from each other and angular alignment

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Table 1. Geometrical parameters of the composite cylindrical shell. Number of ribs 6

Inner radius

Outer radius

75 (mm) 81 (mm)

(a)

Table 2. Mechanical properties of the FE model.

Shell diameter

Shell thickness

Sample length

Gyz

Gxy,Gxz

νyz

νxy,νxz

Ez , Ey

Ex

Rib

0.3(GPa)

1(GPa)

0.15

0.25

1.9(GPa)

7(GPa)

162 (mm)

0.8 (mm)

300 (mm)

Shell

0.3(GPa)

1(GPa)

0.15

0.25

1.9(GPa)

6(GPa)

(b)

Fig. 2. (a) Fabricated structure; (b) composite structure.

relative to the shell axis are design outputs. Experimental studies on the behavior of cylindrical lattice structures under various loading conditions revealed that buckling of these structures is the major factor in their failure [7]. On the other hand, the weight of structures especially those used in aerospace structures has been considered as a very important factor in the design and manufacture process. Geometrical parameters of composite structures play an important role in their final buckling load. In this study, both thickness and angle of ribs and the shell thickness (i.e. thickness of outer shell) are considered as effective geometrical parameters in composite structures (shown in Fig. 2). Optimization techniques are employed to optimize geometrical parameters of composite structures to achieve the highest strength to weight ratio.

3. Finite element model Fig. 3(a) shows a composite lattice cylindrical shell made of six ribs with orientation angles of ±30º to the cylinder axis. The fiber angle of outer shell is equal to ±14º with respect to the radial direction (Perpendicular to longitudinal direction) [18]. Geometrical parameters used in the simulation analysis are shown in Table 1. Akbari Alashti et al. [18] described the manufacturing method of test samples, experimental buckling analysis and finite element evaluation of the buckling load considering two cases of cylinders with and without the rib defect. The main goal of the present paper is to develop techniques to optimize geometrical parameters of the composite lattice structure studied in their research work. Thickness and angle of the rib and the shell thickness are considered as the design variables. In the analysis of the anisogrid composite lattice shell, radius and height of this structure are used as the input variables [3, 6]. In this paper behavior of the structure under buckling condition is simulated. ANSYS12 suite of program is used for the finite element modeling of the structure as shown in Fig. 3(b). SOLID191 and SHELL99 type of elements are used for the

(a)

(b)

Fig. 3. (a) Manufactured composite lattice shell; (b) FE model of the shell.

linear analysis in order to achieve the buckling modes and element types SOLID46 and SHELL91 are used for nonlinear analysis to achieve the final buckling loads. To prevent edge distortion in the shell and consequently creation of virtual modes of buckling, force was applied on a rigid flange attached to the edge of the cylinder on upper end. The flange was created by first locating a node at the center of the circular edge of upper end of the cylinder. Then a mass element with very low mass and moment of inertia was located on this node. Then the degrees of freedom of terminal nodes of the model and this node were coupled together and the force was applied on the mass element’s node. This technique provides uniform distribution of load on boundary nodes as well as preventing undesired defects on the edge. The method selected for extracting the buckling loads was linear or eigenvalue buckling method. In this method the eigenvalue problem of Eq. (1) is solved to obtain the eigenvalues and eigenvectors. In this equation K ¢ and K ¢¢ are flexural and initial stress stiffness matrices respectively and N and D are eigenvalues and eigenvectors, respectively. (ëéK ¢ûù + N i ëéK ¢¢ûù) ëéD ûù = 0 .

(1)

In this equation K ¢ is the standard flexural stiffness matrix of the structure while K ¢¢ takes into account the effects of in plane loading on the stiffness of the structure. The Lanczos Iteration Method is used by the software to solve this problem. In this analysis, the axial pressure loading is increased incrementally until the structure can no longer resist even the slightest addition in the applied load. Mechanical properties of the shell used in the finite element analysis were obtained experimentally, as shown in Table 2. The finite element analysis is used to study and simulate the behavior of composite structures as an alternative method to time-consuming and costly experimental tests. Hence, results

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Table 3. The range of parameters of the composite structure. Parameters max min

Number of Rib thickness rib (mm) 16 4

Rib angle

Shell thickness (mm)

2

10o

0.3

9

o

1.5

80

Fig. 5. Behavior of honeybee foraging for nectar. Fig. 4. Geometrical parameters of composite structure under study.

5. Artificial bee colony (ABC) obtained from the FE analysis are used to generate new data for training the neural network instead of experimental tests. The parameters which are considered in this paper are shown in Fig. 4. By different combinations of these parameters, structures with various geometric shapes are made. Thus, the final buckling load and the weight could be obtained for each of the structure by FE simulation. The range of each parameter is shown in Table 3.

4. Neural network Neural networks are considered as a classification of intelligent systems that have excellent capability of learning the relationship between input-output mapping from a given data set without any knowledge or assumptions about the statistical distribution of data. In general, a neural network is characterized by the following three major components: a) The computational characteristics of each unit, for example, activation function; b) the network architecture; and c) the learning algorithm to train the network. The smallest unit of information processing in neuronal networks called neuron. Two or more neurons can be combined to create a layer. A specific network can be composed of several layers. In this study, backpropagation learning algorithm is used to train multi layers network. Back-propagation is an approximate steepest descent algorithm in which the performance index is the mean square error [19]. In order to improve generalization of network, Bayesian regularization is used [20]. It minimizes a combination of squared errors and weights, and then determines the correct combination so as to produce a network that generalizes well. The purpose of using neural network in this study is prediction of the structural behavior relative to the geometric variables. In other words, the designed neural network is used as fitness function for ABC algorithm.

Artificial bee colony (ABC) is a relatively new member of swarm intelligence. ABC tries to model the natural behavior of real honey bees in food foraging [21]. The mission is implemented by all members of the colony, by efficient division of labor and role transforming. Each bee performs one of following three kinds of roles: Employed bees (EB), Onlooker bees (OB) and Scout bees (SB) [22]. They could transform from one role to another in different phases of foraging. The flow of nectar collection is as follow [23]: (1) In the initial phase, there are only some SB and OB in the colony. SB are sent out to search for potential nectar source, and OB wait near the hive for being recruited. If any SB finds a nectar source, it will transform into EB. (2) EB collect some nectar and go back to the hive and then dance with different forms to share information of the source with OB. Diverse forms of dance represent different quality of nectar source. (3) Each OB estimates quality of the nectar sources found by all EB, then follows one EB to the corresponding source. All OB choose EB according to some probability. Better sources (More nectar) are more attractive (with larger probability to be selected) to OB. (4) Once every source is exhausted, the corresponding EB will abandon them, transform into SB and search for new source. By repeating this process, the bee colony assigns more members to collect the better source. Typical behavior of honey bee foraging is shown in Fig. 5.

6. Result and discussions 6.1 Development and validation of the FE model As previously mentioned, the manufactured sample is tested under an axial load to determine its final buckling load. Fig. 6 depicts results of the experimental and finite element analysis

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Table 4. Strength to weight ratio for the lattice shell with different geometrical parameters. Number

Arib

Trib (mm)

Tshell (mm)

Ratio (kN/kg)

1

20

3

0.5

40.58382

2

20

3

0.75

48.48299

3

32

3

1.5

113.6102

;

;

;

;

;

;

;

;

;

;

99

70

5.6

1.5

80.7573

100

70

7

0.5

17.2464

Fig. 8. Structure of ANN used in the investigation.

different shapes. Finite element simulations are carried out using combinations of different values of the variables and results are shown in Table 4. Geometrical parameters of this table are shown in Fig. 4. 6.2 Neural network design

Fig. 6. Buckling mode of the composite lattice shell.

In this study, MATLAB R13a programming tool is employed to design and run the neural network. The Bayesian regulation back propagation is defined in a network training function that is called trainbr. In this method, two groups of data are used. First, training data is used to calculate weight and bias matrices. Then testing data is used to evaluate the performance of designed neural networks. Before training the neural networks, all variables should be normalized, i.e. obtain values between [0 1] by the following relation: Vnor = (Vi -Vmin) / (Vmax -Vmin)

Fig. 7. Variation of end shortening with the applied force, FE versus experimental result.

carried out on the composite cylindrical sample [18]. As shown in the Fig. 6, buckling modes that obtained by the finite element model are in very good agreement with those of the experimental test. Furthermore as seen Fig. 6 local buckling mode causes failure of the structure. In addition, the ultimate buckling load is an important factor studied in the analysis of composite lattice structures. Variation of displacement versus the applied force for the manufactured composite structure is utilized to obtain the final buckling loads as shown in Fig. 7 [18]. Considering the good agreement between the simulation re results and experimental data as seen in Figs. 6 and 7, it can be concluded not only the fabricated model is reliable but also the finite element model can be developed to determine the strength to weight ratio for the cylindrical lattice shell with

(2)

where Vnor is the normalized variable, Vi is the value of a certain variable, Vmax and Vmin are the maximum and minimum values of the independent variable, respectively. This was modeled by using the data of Table 4. Out of 100 numbers of data obtained from the simulation, about 25% were considered as network test data and the remaining were considered as training data. In this stage, various networks were created and trained with different layers and neurons. Networks that had mean squared error less than 0.0001 (MSE < 0.0001) and less MAE to respond to the test data, were selected as better networks. Fig. 8 shows the best structure of ANN network obtained. The results show the neural network predicted values versus the ones predicted by ANSYS corresponding to training and test data. 6.3 Optimization In the ABC algorithm, position of the nectar source is presented by the coordinate in D-dimensional space. It is the solution vector X of some special problem and the quality of the nectar source is presented by the objective function F(X) of this problem. The initial solutions are generated by Eq. (2) in (j) (j) principle where Xi is the jth element of the ith solution. Φ is a uniformly distributed random real number in the range of [0, 1]. (j)

(j)

(j)

(j)

(j)

Xi = LB + Φ ( UB - LB )

(3)

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Table 5. Setting parameters of the ABC algorithm. Population number

Triggering threshold

Maximum cycle number

Dimension of the problem

150

3

500

4

Table 6. Determined optimum results. Strength-toStrength-toweight ratio of weight ratio of FEM(kN/kg) ANN(kN/kg) 114.187

(a)

115.11

Shell thickness (mm)

Rib angle

Rib thickness (mm)

1.5

20o

3

[23]. In each cycle, the best solution vector stored by the system is: prob(i ) = Fitness (i )

å

PN 2 i =1

Fitness (i ) .

(5)

In this paper, the rib thickness, the rib angle and the shell thickness are considered as the variables. Input parameters in ABC algorithm is shown in Table 5. Finally, the greatest strength-to-weight ratio of composite structure is obtained. After obtaining the results, 3-D FEM models that were validated previously, are simulated using composite structure with the geometrical parameters found by the optimization algorithms. It is observed that the relative percentage error of the result obtained by the FE simulation against that of the ANN model was about 0.8%. The optimized parameters of the composite structure obtained by ABC are shown in Table 6.

(b)

6.4 Sensitivity analysis (c) Fig. 9. BPNN outputs versus target values in (a) Training data; (b) all data; (c) testing data.

where UB (j) and LB (j) are the maximum and minimum values of the independent variable, respectively. Each EB randomly modifies single element Xi (j) of the source i by the Eq. (3). Then fitness of the two solutions, i.e. before and after modification is estimated and the best value will be saved in its memory. (j)

(j)

(j)

(j)

X i( j ) = Xi +λ ( Xi - Xk )

(4)

where X i( j ) is the corresponding new element of the solution after modification and λ (j) is the uniformly distributed random real number in the range of [-1 1]. By using roulette wheel selection mechanism, an onlooker bee chooses a food source with the probability Eq. (4) and produces a new source in selected food source site by Eq. (3)

In this paper, the effect of three important design parameters on the strength-to-weight ratio of composite structures is investigated. An important point to be noted is that the effect of each parameter on the specific target is different with the effect of that element when all parameters are simultaneously considered. When the effect of all parameters is simultaneously considered, the effect of one parameter may be so large that it impresses the effect of other parameters on the target. In this section, at first the effect of each parameter is separately investigated and then the contribution of each factor on the strength-to-weight ratio of the structure is studied. The trained neural network was used to study the structural behavior in the face of changing any of the parameters. So, two of the parameters under study are assumed constant and variation of the strength-to-weight ratio through variation of the third parameter is found as shown in Fig. 10. The results show that the ultimate load increases with the increase in the shell thickness. When the thickness of the shell increases, the mechanical behavior of the cylindrical shell changes and the shell will have a greater effect on the load bearing capacity of the shell. It should be noted that there is an

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Table 7. The percentage contribution of each parameter. Parameter

SA

ST

PA

Rib thickness

4.6987e+04

5.8660e+04

80.1

Shell thickness

9.6202e+03

5.8660e+04

16.4

Rib angle

2.0531e+03

5.8660e+04

3.5

(a)

(a)

(b)

(b) Fig. 11. Buckling mode for Different rib angle: (a) 20º; (b) 30º. (c) Fig. 10. The effect of: (a) Rib angle; (b) rib thickness; (c) shell thickness on the strength-to-weight ratio.

optimum amount for increasing the thickness and may reduce the amount of the special load (Strength-to-weight ratio) because of increase in weight is more than the ultimate load. Since the applied load is axial, as the rib angle gets closer to the axial direction of the cylinder, the shell becomes more reinforced and the buckling load increases. Because the operation of ribs in this structures such as columns in building. But it should be noted that reduction of rib angle causes buckling modes becomes critical which will have a disruptive effect for the structure. The effect of reducing the rib angle is shown in Fig. 11. When the rib thickness increases, the buckling load of the shell increases and since in this structure, the rib carries the maximum percentage of the load, the buckling mode of the

shell tends to buckling mode shapes of column. On the other hand, as the cross-sectional area increases, the structure weight will increase as well. This weight increase caused the special buckling load to reduce. Furthermore, as the rib thickness increases, the flexibility of the shell reduces and its response becomes similar to a barrel shape structure. The percentage contribution of each parameter is obtained by the following relation [24]: PA = SA×(ST /100)

(6)

where SA is the sum of squares of the factor and ST is the total sum of squares of a factor. The contribution of each parameter is shown in Table 7. The results show that the shell thickness has the highest influence on the strength to weight ratio of composite lattice cylindrical shells. Thus, the shell thickness is an appropriate

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parameter to achieve an optimum strength-to-weight ratio of a composite structure. If this parameter is considered as a design variable, the role of other important parameters such as rib angles can be ignored.

7. Conclusion In This study, the effect of geometric parameters on the strength-to-weight ratio of composite lattice cylindrical shells is investigated. The rib thickness, rib angle and shell thickness are considered as variables. At first, finite element models are verified with experimental results. Then, required data for training the neural network is provided by the FE model. Designed neural network can predict the strength-to-weight ratio of structures in terms of its geometric parameters with good accuracy. Next, by using ABC algorithm, Values of input parameters leading to maximum strength-to-weight ratio of composite structure are obtained. The following results are the main conclusions: As it is expected, ultimate load increases as the thickness of outer shell increases. However, the weight of the shell may increases as a result. It is found that as the rib angle reduces, the buckling load of the structure increases. But, reduction of rib angle causes buckling modes becomes critical. Furthermore, as the rib thickness increases, the structure becomes stronger and the buckling critical load increases. But the effect of increasing the rib thickness on structure's weight is so greater than structure's final buckling load. Despite the fact that the weight increases as the shell thickness increases, but after a certain thickness, the overall buckling force increases that overcomes the increasing of the weight and then resulting in an increase of the strength-toweight ratio of structure. The shell thickness is the most important design parameter in comparison with others. The effect of the shell thickness is so dominating that the impact of other parameters such as the rib angle and rib thickness can be overlooked.

Nomenclature-----------------------------------------------------------------------νxy Ex Gxy

: Poisson’s ratio in xy plane : Modulus of elasticity in X direction : Modulus of rigidity in xy plane

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M. Sadegh Yazdi is a Ph.D. student at the Department of Mechanical Engineering of Babol University of Technology. He holds a BA in mechanical engineering 2009 by the University of Mazandaran, and he received in 2011 the title of Master Degree in the field of Hydroforming in mechanical structures and optimization methods, having studied at the Babol University of Technology. He continued this field in his doctoral thesis. S. A. Latifi Rostami is a Ph.D. student at Faculty of Mechanical Engineering of University of Semnan. He holds a BA in Computer Science since 2009 by the University of Mazandaran, and he received in 2012 the title of Master Degree in the field of isogrid lattice structure and having studied at the Babol University of Technology. His doctoral thesis is focused on artificial intelligence, specifically in the field of combinatorial optimization, studying and developing heuristics and metaheuristics solving routing problems.