Issue 
Int. J. Simul. Multidisci. Des. Optim.
Volume 10, 2019



Article Number  A14  
Number of page(s)  16  
DOI  https://doi.org/10.1051/smdo/2019014  
Published online  23 September 2019 
Research Article
Optimizing the buckling characteristics and weight of functionally graded circular plates using the multiobjective Pareto archived simulated annealing algorithm (PASA)
^{1}
Department of Mechanical Engineering, Islamic Azad University, Khomeinishar Branch, Khomeinishar, Iran
^{2}
Department of Mechanical Engineering, Islamic Azad University, Najafabad Branch, Najafabad, Iran
^{*} email: zh_farhat@yahoo.com; farhatnia@iaukhsh.ac.ir
Received:
27
August
2018
Accepted:
27
August
2019
In this study for the first time, weight and critical buckling load in two kinds of functionally graded (FG) circular plates, namely, aluminum–alumina of (Al/Al_{2}O_{3}) and aluminum–zirconia (Al/ZnO_{2}), are optimized using multiobjective Pareto archived simulated annealing algorithm (PASA). Material properties are assumed to vary with the power law in terms of the volume fractions of the constituent in two forms of symmetric and asymmetric with respect to the middle surface. The plate is subjected to uniform radial load and is considered for two boundary conditions, namely, simply supported and clamped edges. Aim at obtaining the Pareto archive is to achieve simultaneously the maximum buckling and the minimum weight concerning with proposed constraints. The parameters include the radius, thickness and volume fraction that the certain range is intended individually. The constraints are presented in form of the ratio of thickness to radius in category of the thin plates as well as the critical buckling stress being in the elastic range. Proposed simulated annealing algorithm is coded in MATLAB to obtain optimal nondominated solution.
Key words: Critical buckling load / functionally graded materials / multiobjective optimization / simulated annealing algorithm / Pareto archive
© F. Farhatnia et al., published by EDP Sciences, 2019
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Nowadays, the development of intelligent systems inspired the nature, is one of the most popular areas of artificial intelligence. Methods such as simulated annealing algorithms, genetic algorithms, ant colony optimization, are of the issues that human beings could achieve them by inspiring nature. On the other hand, plates are structures that are commonly used in military, petrochemical, aerospace industries, such as those found in turbine drive system, wall of pressure vessels, nuclear reactors and storage tanks floor. In recent years, functionally graded materials are one of the efficient and modern composite that are used in the abovementioned industry, especially for using in the high temperature environments.
These kinds of materials are advanced nonhomogeneous microstructure composites whose mechanical properties change smoothly and continuously from one surface to another. One of the most important phenomena in the plates, which causes the inefficiency in performance, is the buckling of the plate. In thin plates due to their low thickness, the membrane stiffness of the plate is much higher than its bending stiffness. Thus, it saves a large amount of membrane strain energy without causing much deformation. If the same amount of membrane energy is stored in the form of bending energy in the plate, it will be much deformed. Therefore, if the membrane strain energy stored in the plate is converted to bending strain energy under certain conditions, the plate is severely damaged that this phenomenon is called the buckling. Study of buckling behavior of plates have always been considered as one of the most important subjects in the structure analysis. Brush and Almorth [1] have comprehensively analyzed the buckling problems of columns, plates and shells and studied various methods for formulating the governing nonlinear equilibrium equations in buckling. In addition, many researchers have provided analytical and numerical solution for study of the buckling behavior of circular and annular FG plates based on thin, first and higher order shear deformation theories [2–20].
In general, the optimization methods has been developed to improve the response to engineering demands in the modern industrial world; objective functions are selected depending to the performance criteria in engineering. For instance including weight loss, reduction of residual stresses, increase of vibration frequency, improvement of failure resistance, and increase of buckling load. So far, many studies have been carried out to optimize the behavior of the plates. Osaka et al. [21] examined the optimization of thickness profile of solid and annular plates in order to maximize critical buckling load using the finite element method and taking into account the shear deformation effects based on the firstorder shear deformation. Ootao et al. [22] optimized the fraction volume index of FG plates by the neural network in the desired temperature range. In the other work, Ootao et al. [23] optimized the fraction volume index of FG hollow cylinder under thermal loading using neural networks method. Goupee and Vel [24] used the elementfree Galerkin method and the genetic algorithm for twodimensional optimization of the fraction volume index of functionally graded materials in two separate model problems. In one model, they minimized the residual stress by cooling down a FGM component (Ni/Al_{2}O_{3}) and in another one, the weight of the Al/zirconia (FGM components) was optimized by restriction of the effective stress and maximum temperature experienced by metal constituent. Mozaffari et al. [25] optimized the critical buckling load of the FGM rectangular plate using the Imperialist Competitive Algorithm. Fereidoon et al. [26] used particle swarmbased algorithms to optimize the volume fraction of functionally graded materials. JuinDer Duh and Brown [27] showed in multioptimization problems, the solutions suggested by the knowledgeinformed approach are more effective in approximation the set of Pareto optimal solution than those generated by the standard Pareto simulated annealing. Şahin and Turkbey [28] obtained Pareto solution for multiobjective facility layout using simulated annealing algorithm.
As abovementioned, buckling behavior of plates has been received the great deal of attention in scientific publications and is always considered as one of the most important subjects in structural analysis. Thus, the scholars have been paid attention to the optimization of buckling behavior. In the present study, twoobjective optimization of buckling of the plates has been studied. The aim is to obtain the maximum critical buckling load according to the existing constraints with the approach of achieving the minimum weight. Thickening the plate to increase the critical buckling load, beside increasing the weight of the plate and consequently, more transverse loading, as well as increasing the cost of consumables materials. Therefore, it is important to determine the appropriate thickness of the plate and appropriate composition of volumetric fraction of ceramics and metal can achieve to the optimal bending rigidity required to maximize critical loads. Herein, the simulated annealing algorithm (SA) is exploited as the optimization method. The rationale behind SA extends to the annealing process of physical systems applied in thermodynamics. In this process, a physical system initially at a highenergy state is gradually cooled down until its minimum energy level is reached [29]. The idea is establishing a direct analogy between minimizing the energy level of a physical system and lowering the cost of an objective function. SA algorithm takes CPU time less than genetic algorithm (GA), due to find the optimal solution using pointbypoint iteration rather than a search over a population of individuals [30]. The unique feature of SA accepts to move towards a worse solution during the search, therefore it prevents this method to trap in the local optima [31].
The aim of multiobjective optimization is to find a set of compromise solutions with different tradeoffs among criteria, also known as Pareto optimal set; when this set is plotted in the objective space, it is called the Pareto front [32]. Serafini [33] presented the first version of the multiobjective simulation algorithm, was similar to the simulation algorithm of single objective optimization. The difference with the main algorithm was in the method of acceptance criteria. He investigated different alternative criteria to increase the probability of nondominated optimal solutions that achieved to a special rule to obtain the nondominated Pareto solutions. Suman [34] presented four simulated annealing based multiobjective algorithm and showed they are robust with algorithmic parameters and are capable of generating a large number of optimal solution. Suppapitnarm et al. [35] presented a novel implementation of the simulated annealing algorithm designed to explore the tradeoff between multiple objectives. They suggested a new acceptance probability formulation based on proposition multiple temperature for multiple objectives in an annealing schedule. Czyzak and Jaszkiewicz [36] presented a new algorithm for multiobjective simulated annealing based on the Pareto optimal that is called PSA. Ulungu et al. [37] proposed and tested a multiobjective simulation algorithm (MOSA) to solve problems. They used a cumulative weight function to evaluate the objective function. The algorithm started with an initial solution, but a massive nondominated solutions was obtained during the algorithm implementation. Akbulut and Fazil [38] optimized the thickness of a composite plate by simulated annealing algorithm (SA). They considered orientation of the resin and the number of sublayers in each layer as the design variables and predicted failure. Moreover, Gutjahr and Pichler [39] used multidimensional optimization for studying noncomputational methods as well as computational modeling to solve the problems. Recently, Rangaiah [40] described the process of engineering systems using multiobjective optimization procedure in his book. Chen et al. [41] compared the simulated annealing algorithm by using the tabusearch algorithms through study of advanced programming system for filtering color of TFTLCD screens. Deb [42] provided a brief introduction and research focus to the field of multiobjective optimization based on evolutionary algorithm (EO). He remarked the multiobjective optimization methods give rise to set of Paretooptimal solution, which by using the modified EO, the preferred region is provided on the Paretooptimal front instead of complete front. In the engineering design community, size, shape, and topology optimization procedures are three classes of extensively studied design methodologies with assumed materials. Tang et al. [43] concerned with simultaneous designs of material selection and geometry optimization under static and thermal loads in a framework of multiobjective optimization of tracking the Pareto curve. Study of literature review reveals that this work is the first attempt to investigate the optimization of material composition and geometrical parameters of a FG circular plate by utilizing of a multiobjective simulation annealing algorithm using Pareto archive. The present paper is structured as follows. In Section 2, we give the expressions for the present problem, Section 3 is devoted to introduce Pareto archived simulated annealing algorithm. In Section 4, we present the numerical example for validation of the exploited algorithm. Finally, in Section 5, the performance of the proposed algorithm is investigated by studying the optimization in buckling and weight for two kinds of FG circular plate namely, aluminum–alumina of (Al/Al_{2}O_{3}) and aluminum–zirconia (Al/ZnO_{2}).
2 Definition of the problem
A circular solid FG plate with radius R and thickness h under a radially uniform pressure loading is considered. The mechanical properties are assumed to be varied in the direction of the thickness. Figure 1 shows the schematic variation of the properties for symmetric and asymmetric FG plates that they can be defined as follows:(1a) (1b)where P(z), P_{m} and P_{c} denote the mechanical properties for the FG plate, the metal and the ceramic, respectively. f_{c} denotes the volume fraction of the ceramic and varies between zero and one. Mechanical properties include elasticity modulus, density and yield stress of the FG material. Moreover, h and k are the thickness and the FG volume fraction index, respectively.
As the critical buckling load and the weight of FG plates are depended to geometrical parameters and the volume fraction index of k, the purpose of this study is to determine the optimal values of these variables, so that the buckling load can be maximized, whereas the value of the weight is minimized. The relations for determination of the weight of solid circular plate and critical buckling load are as follows.
The weight of circular FG plate:(2)
The buckling load of circular plate [4]:
 (A)
 (B)
In the above equations, R and D are the radius and the bending rigidity of the plate, respectively. For the asymmetric and symmetric FG plates, the bending rigidity is defined as follows:
 (A)
 (B)
In order to deal with problem of optimization of material and geometrical parameters to minimize weight and maximize critical buckling load, the Zenergy function is defined to estimate the metropolis criterion [44]:(7)
In the above relation, f _{1} and f _{2} stand for log(w) and log(P _{ cr }), respectively. β denotes the weighting coefficient of the weight of plate, therefore (1−β) evaluate the weighting coefficient of the critical buckling load. Noteworthy is that due to be not equal the scale of both objective functions, the logarithm of the two objective functions is utilized. Herein, β equal to 0.5, which means that both weight and buckling coefficient containing the identical significance for designer to make a decision among the optimal solutions according to the design task and its constraints. The aforementioned function is introduced as the sum of the two objective functions, and the critical buckling load is indicated by a negative sign in the energy function because the characteristics of the two objective functions is not the same, one function (plate weight) must be minimized and another one (critical buckling load) would be maximized.
The constraints are expressed as: 0.1 < R < 1, 0 < k < 100, 0.0125 < < 0.1 and P _{ cr } < σ _{ Y } h, where σ _{ Y } is the mean yield stress. The latter constrain means that the design of the optimum plate is remained in the elastic region; to this end, critical buckling load should be less than σ _{ Y } h. The problem is formulated as:
By assuming that the ceramic constituent obeys linear elastic behavior, whereas the metal constituent achieves to the yield stress σ _{ Ym }, with respect to the rule of volumetric mixture, the value of the mean yield stress at any arbitrary point is determined as follows [45]:(8)
It should be noted that the parameter q in equation (8) is determined numerically by the finite element micromechanics mode. Selecting a value of 500 GPa for q is appropriate in a wide range of volume fraction of FG plates with ceramicmetal composition [45]. Herein, the mechanical properties of metal–ceramic constituents of two kinds of FG plate, namely aluminum–zirconia (Al/ZnO_{2}) and aluminum–alumina (Al/Al_{2}O_{3}) are given in Table 1.
Fig. 1
Profile of variation of the properties of FG plates of (a) symmetric distribution, (b) asymmetrical distribution. 
3 Pareto archive of multiobjective annealing simulation algorithm
It is noteworthy to mention that contrary to the singleobjective optimization problem, for the multiobjective optimization problem, there is a set of nondominated solution called Pareto optimal set. Simulated annealing (SA) is a stochastic search method, which initiates the physical annealing of solid for finding solution to combinatorial of annealing of solids [28]. This algorithm is one of the few algorithms that have explicit strategies to avoid local minima. For this reason, the solutions of worse quality than the current solution is accepted in order to escape from local minima [46]. The basic steps involved in the PASA algorithm for the present problem are as follows:
 (a)
Define the simulated annealing parameters, such as: the initial temperature, cooling factor, weighting coefficients for two objective function.
 (b)
Generate the initial solution in the prescribed range (0.1 < R < 1, 0 < k < 100 and 0.0125 < < 0.1) and evaluating the objective functions; if it is feasible, adding into the Pareto archive.
 (c)
Store the initial solution as the current solution.
 (d)
Store the initial solution as the optimal solution in the problem.
 (e)
Generate a new solution in the neighborhood of the current solution randomly and evaluating the objective function and check the constraint of P _{ cr } < σ _{ Y } h. If it is not satisfied with the aforementioned constraint, algorithm generates a new point and perform the step (b) again.
 (f)
Evaluate the objective functions. If the candidate solution dominates any point in the archive, the algorithm is proceeded by step h.
 (g)
If the new solution is dominated by any points of the archive, evaluate the energy function based on the equation (7). The energy function is determined to check the Metropolis acceptance criteria and to progress the algorithm. The probability of acceptance is obtained according to the relation of u < e ^{−∇Z/T }, where u is a random number between (0 and 1), ∇Z is the energy difference for the new and the current solutions, and T is also the temperature of the algorithm. If Metropolis is not satisfied, go to step i.
 (h)
The Pareto archive is updated, that means: the current solution is discarded and the new solution is added.
 (i)
The stop condition of inner loop is checked.
 (j)
If the stop condition is met, decrease temperature as much as the inner loop. In the otherwise, go to step b.
 (k)
The stopping criteria of the outer loop is considered. If it is met, the algorithm is finished. If the stopping criteria are not fulfilled, the algorithm goes to step b.
According with the aforementioned relationship, when the temperature is high, most of points are accepted so that the algorithm can search the entire target space and, with the progress of the algorithm, the probability of accepting the points that worsen the situation is reduced, and only the points that cause improvement of the algorithm are accepted.
In the first step, after entering the initial parameters, initially the algorithm starts with a primary random point in the feasible range and evaluates this point, then the initial point is saved as the current point of the problem to Pareto archive as well as the optimal point. The parameters used in the simulation annealing algorithm are included of the initial temperature, the final temperature, the predefined number of iterations, the length of decrement, the relationship between the step of reducing the temperature and the initial solution. Herein, the initial temperature is determined by getting the average of 50 energy functions selected randomly . Moreover, according to the stopping criteria of the algorithm, the final temperature is obtained using the equation of T _{ f } = 0.01T _{ i }. The number of iterations for the current temperature level chosen to be 200. Using the relation of the cooling function (T _{ k+1} = αT _{ k }), the temperature is decremented, in which α is the cooling coefficient, which is considered between zero and one. In this paper, α is chosen to be 0.98. The flowchart of the present method is described in Appendix A.
4 Providing a numerical example for validation
In this section, two objective functions are minimized multiobjective Pareto archived simulated annealing algorithm and compared the results with genetic algorithm using uniform genetic algorithm (UGA), hybrid genetic algorithm (HGA) [47–48]. The test objective functions are defined as follows [49]:(9a) (9b)So that ‒3 < x < 3, ‒5 < y < 5. In addition, Table 2 shows the number of generated solutions and the number of Pareto solutions using by different algorithms.
As shown in Table 2, the number of solutions obtained from the Pareto archive is more than the total evaluated solutions by the HGA algorithm. According to the results, the scattering of the results of the simulation annealing algorithm is greater, the optimal Pareto solution are searched in greater space.
Performance comparison of the present method with HGA and UGA
5 Results
Herein, for the present problem, three approaches are represented. The results are demonstrated for the two metal ceramic constituents of FG plate, as: aluminum–zirconia (Al/ZnO_{2}) and aluminum–alumina (Al/Al_{2}O_{3}).
5.1 The first approach: constant radius
In the first case, the radius is assumed to be constant, and the volume fraction and thickness of the plate is considered as variables that would be optimized. Figures 2–5 show the distribution of all dominated accepted and Pareto optimal solutions (Pareto optimal front) for asymmetric and symmetric FG plates with two boundary conditions, namely, simplysupported and clamped edges for radius of R = 1 m. In all figures, the bright blue and dark blue points indicate all the acceptable dominated and nondominated optimal solutions (Pareto front), respectively.
Figure 2 shows the results of the implementation of algorithm for Al/ZnO_{2} plate with simply supported and clamped edges in the asymmetric and symmetric cases. When the radius of the plate is specified and the aim is to achieve optimum critical buckling load and weight, thus Figure 2 can be used. As sketched, firstly, the graph raises continuously, then a rough trend is occurred due to difference between the properties of two constituents of FG plate. In addition, in Figure 2, the results obtained by the present method are compared with those of Genetic algorithm. As seen, the results indicate a very good agreement between them.
As illustrated in Figure 3, at the left side of boundary of the thickness, the optimal dark points indicate to the maximum volume fraction and the minimum value of the weight of the plate. In Figure 3a, while being far from the left side border, Pareto archive converge to a volume fraction index of 10–50 (that means a FG plate with a higher metal percentage). This trend continues until toward the end of the border on the horizontal axis labeled thickness. These points are not dominated by the greater value of thickness and the volume fraction index, due to the fact that the thickness limitation does not permit to proceed more; hence the volume fraction index tends to be zero to maximize the buckling load and minimize the weight. Those points are not added into Pareto archive if they violate the elasticity relation for calculation critical buckling forces.
The results of the algorithm are presented for Al/Al_{2}O_{3} plate with a simply supported and clamped edges in asymmetric and symmetric cases in Figures 4 and 5. Figure 5a–d shows the corresponding points in Figure 4a–d. As shown in Figures 4 and 5, due to the mechanical properties of aluminum and alumina, as well as aluminum and zirconia, there is no convergence of the algorithm towards pure metal, and the optimal Pareto points are in the form of ceramic and metal mixture and tend towards ceramic properties.
As sketched in Figure 5, at the initiation and the end of the thickness boundary, the volume fraction index converges to the upper boundary of the volume fraction index (pure metal). This is due to the fact that the weight of the plate in the boundary thickness is minimized at the beginning and end of the thickness boundary with high volume fraction index; these points do not violate clause i of the algorithm, therefore are added to the Pareto archive and is not dominated by any point.
In the upper part of Figure 4c and d, some dispersed distribution of Pareto optimal points is seen, this is due to achieving to the upper boundary of the thickness. As the constraint for thickness does not permit to be increased this parameter in the upper bound, the volume fraction index increases and if it does not violate part I of the algorithm, it is added to the Pareto archive due to containing the lowest weight. This can be observed for the corresponding points in diagrams (c) and (d) of Figure 5.
Fig. 2
Critical buckling loadweight diagram, accepted dominated solution, nondominated solution (Pareto archive) and GA solution of the Al/ZnO_{2} plate in R = 1 m for (a) the asymmetricalclamped edge, (b) asymmetricalsimply supported, (c) symmetricalclamped edge, (d) symmetrical simply supported. 
Fig. 3
Thicknessvolume fraction diagram, accepted dominated solution (light blue points) and nondominated solution (dark blue points) of Al/ZnO_{2} plate in R = 1 for (a) the asymmetricalclamped, (b) asymmetricalsimply supported, (c) symmetrical clamped, (d) symmetrical simply supported edge. 
Fig. 4
Critical buckling load weight diagram, accepted dominated solutions and nondominated solution (Pareto archive) of the Al/Al_{2}O_{3} plate in R = 1 (a) asymmetricalclamped, (b) asymmetricalsimply supported, (c) symmetricalclamped, (d) symmetricalsimply supported edge. 
Fig. 5
Thicknessvolume fraction diagram, accepted dominated solution (light blue) and nondominated solution (dark blue) of Al/Al_{2}O_{3} plate in R = 1 (a) the asymmetricalclamped, (b) asymmetricalsimply supported, (c) symmetricalclamped, (d) symmetricalsimply supported edge. 
5.2 Second approach: constant volume fraction index
In this case, the volume fraction index is assumed to be constant, whereas the radius and thickness are variable. The range of variables in this case is 0.0125 < < 0.1 and 0.1 < R < 1. In this approach, the purpose is to estimate an optimum ratio of thickness to radius in the desired volume fraction index. Figures 6 and 7 show the results of an optimal ratio of thickness to radius for (Al/Al_{2}O_{3}) and (Al/ZnO_{2}) FG plates corresponding to the prescribed volume fraction index, respectively.
Through Figures 6 and 7, the volume fraction index is assumed to be taken of 0, 0.5, 2 and 10 for the optimal ratio of thickness to radius. For instance, if the volume fraction index of 10 is selected in Figure 7d, thus would be 0.061824 to be achieved to the maximum critical buckling load and the lowest weight. In the following, Figure 8 is sketched to present the Pareto archive corresponding to Figure 7d for aforementioned quantities of volume fraction index and thickness to radius ratio.
In Figure 8, the brighter area represents all the acceptable points and the darker curved lines represent the Pareto archive of nondominated solution, which are obtained for of 0.061824. As depicted in Figures 7 and 8, as the volume fraction index increases, the critical buckling load decreases; whereas the optimal ratio of thickness to radius increases; which consequently raises the weight of the plate.
Fig. 6
Critical buckling loadweight diagram with regarding to the optimum ratio of thickness to radius for Al/Al_{2}O_{3} plates: (a) the asymmetricalclamped, (b) asymmetrical simply supported, (c) symmetricalclamped, (d) symmetrical simply supported edge. 
Fig. 7
Critical buckling loadweight diagram with regarding to the optimum ratio of thickness to radius for the FG Al/ZnO_{2} plates: (a) asymmetricclamped support, (b) asymmetricalsimply supported, (c) symmetricalclamped, (d) symmetricsimply supported edge. 
Fig. 8
Critical buckling loadweight diagram, accepted dominated solutions and nondominated solutions for symmetrical Al/ZnO_{2} plates with the simply supported edge for volume fraction of 10. 
5.3 Third approach: variable radius, thickness and volume fraction
In this case, the radius, thickness, and volume fraction of metalceramic constituents, are assumed to be varied. For instance, as presented in Tables 3 and 4, the weight of 100 kg is chosen as the target and the closest nondominated points for aluminum–zirconia and aluminum–alumina plates are presented in Tables 2 and 3, respectively.
By running computer code in Matlab for three times, the solutions are tabulated in Tables 3 and 4. As shown, the objective is to obtain the closest nondominated point to the specified weight; in fact, the obtained points containing the maximum buckling load. On the other hand, the aim can be considered as the closest nondominated solution to a specified buckling load. In Tables 5–8, according to the buckling load of 5 × 10^{6} (N/m), the nearest nondominated Pareto point is represented.
The closest optimal Pareto point to the objective function of the weight of 100 kg for Al/ZnO_{2} plate.
The closest optimal Pareto point to the objective function of the weight of 100 kg for Al/Al_{2}O_{3} plate.
The closest Pareto optimal point to the objective function of the buckling load equals to 5 × 10^{6} (N/m) for the asymmetric Al/ZnO_{2} plate.
The closest Pareto optimal point to the objective function of the buckling load equals to 5 × 10^{6} (N/m) for the symmetric Al/ZnO_{2} plate.
The closest Pareto optimal point to the objective function of the buckling load equal to 5 × 10^{6} (N/m) for the asymmetric Al/Al_{2}O_{3} plate.
The closest optimal Pareto point to the objective function of the buckling load equals to 5 × 10^{6} (N/m) for the symmetric Al/Al_{2}O_{3} plate.
6 Conclusion
In the present article, the critical buckling load as well as the weight of the plate was optimized by the multiobjective annealing simulation algorithm using the Pareto archive. The aim is to find a nondominated Pareto archive, so that the critical buckling load is maximized whereas the amount of plate weight is minimized. The optimization of the asymmetric and symmetric functionally graded aluminum–zirconia (Al/ZnO_{2}) and aluminum–alumina (Al/Al_{2}O_{3}) plates under simply supported and clamped edge conditions have been considered. As the mechanical properties of traditional functionally graded materials depend on the volume fraction index of ceramicmetal composition, it is noteworthy to mention that which volume fraction index and thickness can satisfy all the constraints of the problem, and optimize the critical buckling load and weight. Herein, the solution based on the multiobjective annealing simulation algorithm is obtained using programming in MATLAB. The results are proceeded in three approaches and the solutions are extracted in three categories, as exhibited in tables and figures. The solutions are demonstrated as the nondominated Pareto archive as well as the optimal solutions of the problem, In fact, it provides the decisionmaker with a Pareto solution set and permit her or him to make a choice depending on design preferences. The closest and best solution of the Pareto archive solutions can be selected by accessing the Pareto archive and having enough information about it. Herein, the following conclusion can be outlined:
 I.
By assuming that the maximum weight/ minimum critical buckling load is specified, the second approach can be used to obtain optimal state.
 II.
By considering the necessities of the problem and using the third approach, and with the implementation of the algorithm for several times, the solutions based on the other two approaches can be obtained.
 III.
As shown for Al/ZnO_{2} plate, optimization of critical buckling load and weight is achieved from a plate containing a higher volume percentage of metal constituent; whereas, for Al/Al_{2}O_{3} plate, this trend is reverted and is resulted from a plate containing a higher volume percentage of ceramic constituent.
 IV.
The present result is utilized as a new reference for the designers in the field of aerospace and marine structures industries. For future studies, optimization of critical temperature of thermal buckling, critical frequency of vibration of FG plate are interesting subjects to be considered.
Appendix A
The flowchart of Pareto archived Annealing simulation is described as follow:
Figure A.1
Flowchart of present method. 
References
 D.O. Brush, B.O. Almorth, Buckling of bars, plates and shells (McGrawHill, New York, 1975) [Google Scholar]
 S. Majumdar, Buckling of a thin annular plate under uniform compression, AIAA J. 9 , 1701–1707 (1971) [CrossRef] [Google Scholar]
 J.N. Reddy, A.A. Khdeir, Buckling and vibration of laminated composite plate using various plate theories, AIAA J. 27 , 1808–1817 (1989) [CrossRef] [Google Scholar]
 M.M. Najafizadeh, M.R. Eslami, Buckling analysis of circular plates of functionally graded materials under uniform radial compression, Int. J. Mech. Sci. 44 , 2479–2493 (2002) [CrossRef] [Google Scholar]
 J.H. Kang, A.W. Leissa, Exact solutions for the buckling of rectangular plates having linearly varying inplane loading on two opposite simply supported edges, Int. J. Solids Struct. 42 , 4220–4238 (2005) [CrossRef] [Google Scholar]
 M.H. Naei, A. Masoumi, A. Shamekhi, Buckling analysis of circular functionally graded material plate having variable thickness under uniform compression by finiteelement method, Proc. Inst. Mech. Eng. C 221 , 1241–1247 (2007) [CrossRef] [Google Scholar]
 P. Qiao, L.Y. Shan, Explicit local buckling analysis of rotationally restrained composite plates under biaxial loading, Int. J. Struct. Stab. Dyn. 7 , 487–517 (2007) [CrossRef] [Google Scholar]
 M.M. Najafizadeh, H.R. Heydari, An exact solution for buckling of functionally graded circular plates based on higher order shear deformation plate theory under uniform radial compression, Int. J. Mech. Sci. 50 , 603–612 (2008) [CrossRef] [Google Scholar]
 H. Akhavan, Sh. Hosseini Hashemi, H. Rokni, A. Alibeigloo, Sh. Vahabi, Exact solutions for rectangular Mindlin plates under inplane loads resting on Pasternak elastic foundation. Part I: Buckling analysis, Comp. Mater. Sci. 44 , 951–961 (2009) [CrossRef] [Google Scholar]
 H. Koohkan, A. Kimiaeifar, A. Mansourabadi, R. Vaghefi, An analytical approach on the buckling analysis of circular, solid and annular functionally graded thin plates, J. Mech. Eng. 41 , 7–14 (2010) [CrossRef] [Google Scholar]
 F. Farhatnia, A. Golshah, Investigation on buckling of orthotropic circular and annular plates of continuously variable thickness by optimized Ritz method, Int. J. Simul. Multidisci. Des. Optim. 4 , 127–133 (2010) [CrossRef] [Google Scholar]
 A.A. Jafari, S.A. Eftekhari, An efficient mixed methodology for free vibration and buckling analysis of orthotropic rectangular plates, Appl. Math. Comput. 218 , 2670–2692 (2011) [Google Scholar]
 M. Bodaghi, A.R. Saidi, Stability analysis of functionally graded rectangular plates under nonlinearly varying inplane loading resting on elastic foundation, Arch. Appl. Mech. 81 , 765–780 (2011) [CrossRef] [Google Scholar]
 Y. Kiani, M.R. Eslami, An exact solution for thermal buckling of annular FGM plates on an elastic medium, Composites Part B: Eng. 45 , 101–110 (2013) [CrossRef] [Google Scholar]
 M. Latifi, F. Farhatnia, M. Kadkhodaei, Buckling analysis of rectangular functionally graded plates under various edge conditions using Fourier series expansion, Eur. J. Mech. A: Solids 41 , 16–27 (2013) [CrossRef] [Google Scholar]
 H. Yaghoobi, A. Fereidooni, Mechanical and thermal buckling analysis of functionally graded plates resting on elastic foundations: an assessment of a simple refined nthorder shear deformation theory, Composites Part B: Eng. 62 , 11–26 (2014) [CrossRef] [Google Scholar]
 A. Mojahedin, M. Jabbari, Buckling analysis of functionally graded circular plates made of saturated porous materials based on higher order shear deformation theory, ThinWalled Struct. 99 , 83–90 (2016) [CrossRef] [Google Scholar]
 F. Farhatnia, M. GhanbariMobarakeh, S. RasouliJazi, S. Oveissi, Thermal buckling analysis of functionally graded circular plates resting on pasternak elastic foundation via differential transform method, Facta Univ. Ser. Mech. Eng. 15 , 545–563 (2017) [CrossRef] [Google Scholar]
 F. Poodeh, F. Farhatnia, M. Raeesi, Buckling analysis of orthotropic thin rectangular plates subjected to nonlinear inplane distributed loads using generalized differential quadrature method, Int. J. Comput. Methods Eng. Sci. Mech. 19 , 102–116 (2018) [CrossRef] [Google Scholar]
 A.A. Daikh, A. Megueni, Thermal buckling analysis of functionally graded sandwich plates, J. Thermal Stress. 41 , 139–159 (2018) [CrossRef] [Google Scholar]
 M. Osaka, N. Taysi, F. Kolcu, Buckling analysis and shape optimization of elastic variable thickness circular and annular platesI: finite element formulation, Eng. Struct. 25 , 181–192 (2003) [CrossRef] [Google Scholar]
 Y. Ootao, R. Kawamura, Y. Tanigawa, T. Nakamura, Neural network optimization of material composition of a functionally graded material plate at arbitrary temperature range and temperature rise, Arch. Appl. Mech. 68 , 662–676 (1998) [CrossRef] [Google Scholar]
 Y. Ootao, Y. Tanigawa, T. Nakamura, Optimization of material composition of FGM hollow circular cylinder under thermal loading: a neural network approach“, Composites Part B: Eng. 30 , 415–422 (1999) [CrossRef] [Google Scholar]
 A.J. Goupee, S.S. Vel, Twodimensional optimization of material composition of functionally graded materials using meshless analyses and a genetic algorithm, Comput. Methods Appl. Mech. Eng. 195 , 5926–5948 (2006) [CrossRef] [Google Scholar]
 H. Mozafari, A. Ayob, F. Kamali, Optimization of functional graded plates for buckling load by using imperialist competitive algorithm, Proc. Technol. 1 , 144–152 (2012) [CrossRef] [Google Scholar]
 A. Fereidoon, F. Sadri, H. Hemmatian, Functionally graded materials optimization using particle swarmbased algorithms, J. Therm. Stresses 35 , 377–392 (2012) [CrossRef] [Google Scholar]
 J. DerDuh, D.G. Brown, Knowledgeinformed Pareto simulated annealing for multiobjective spatial allocation, Comput. Environ. Urban Syst. 31 , 253–281 (2007) [CrossRef] [Google Scholar]
 R. Şahin, O. Turkbey, A simulated annealing algorithm to find approximate Pareto optimal solutions for the multiobjective facility layout problem, Int. J. Adv. Manufact. 41 , 1003–1018 (2009) [CrossRef] [Google Scholar]
 S. Jafari, M.H. Hojjati, A. Fathi, Classical and modern optimization methods in minimum weight design of elastic rotating disk with variable thickness and density, Int. J. Pressure Vessels Piping 92 , 41–47 (2012) [CrossRef] [Google Scholar]
 Suman, P. Kumar, A survey of simulated annealing as a tool for single and multiobjective optimization, J. Oper. Res. Soc. 57 , 1143–1160 (2006) [CrossRef] [Google Scholar]
 H. Kumar Singh, T. Ray, W. Smith, CPSA constrained Pareto simulated annealing, Inf. Sci. 180 , 2499–2513 (2010) [CrossRef] [Google Scholar]
 F. Luna, P. Isasi, Multiobjective metaheuristics for multidisciplinary engineering applications, Eng. Optim. 44 , 241–242 (2012) [CrossRef] [Google Scholar]
 P. Serafini, Mathematics of Multi Objective Optimization, International Centre for Mechanical Sciences, No. 289 (Springer Verlag, Wien New York, 1985) [Google Scholar]
 Suman, Study of simulated annealing based algorithms for multiobjective optimization of a constrained problem, Comput. Chem. Eng. 28 , 1849–1871 (2004) [CrossRef] [Google Scholar]
 A. Suppapitnarm, A simulated annealing algorithm for multiobjective optimization, Eng. Optim. 33 , 59–85 (2000) [CrossRef] [Google Scholar]
 P.P. Czyzak, A. Jaszkiewicz, Pareto simulated annealing − a metaheuristic technique for multipleobjective combinatorial optimization, J. MultiCriteria Decis. Anal. 7 , 34–47 (1998) [CrossRef] [Google Scholar]
 E.L. Ulungu, J. Teghem, P.H. Fortemps, D. Tuyttens, MOSA method: a tool for solving multiobjective combinatorial optimization problems, J. Multicrit. Decis. Anal. 8 , 221–236 (1999) [CrossRef] [Google Scholar]
 M. Akbulut, O.S. Fazil, Design optimization of laminated composites using a new variant of simulated annealing, Comput. Struct. 89 , 1712–1724 (2011) [CrossRef] [Google Scholar]
 W.J. Gutjahr, A. Pichler, Stochastic multiobjective optimization: a survey on nonscalarizing methods, Ann. Oper. Res. 236 , 475–499 (2016) [CrossRef] [Google Scholar]
 G.P. Rangaiah, MultiObjective Optimization: Techniques and Applications in Chemical Engineering , 2nd edn. (Advances in Process Systems Engineering, World Scientific Publishing, 2017), Vol. 5 [CrossRef] [Google Scholar]
 J.C. Chen, Y.Y. Chen, T.L. Chen, J.Z. Lin, Comparison of simulated annealing and tabusearch algorithms in advanced planning and scheduling systems for TFTLCD colour filter fabs, Int. J. Comput. Integr. Manufactur. 30 , 516–534 (2017) [CrossRef] [Google Scholar]
 K. Deb, Current trends in evolutionary multiobjective optimization, Int. J. Simul. Multidiscipl. Des. Optim. 1 , 1–8 (2007) [CrossRef] [Google Scholar]
 X. Tang, D.H. Bassir, W. Zhang, Shape, sizing optimization and material selection based on mixed variables and genetic algorithm, Optim. Eng. 12 , 111–128 (2011) [CrossRef] [Google Scholar]
 N. Metropolis, A.W. Rosenbluth, M.N. Rosenbloth, A.H. Teller, E. Teller, Equation of state calculation by fast computing machines, J. Chem. Phys. 21, 1087–1092 (1953) [NASA ADS] [CrossRef] [Google Scholar]
 R. Cho, J.H. Choi, A yieldcriteria tailoring of the volume fraction in metalceramic functionally graded material, Eur. J. Mech. A 23 , 271–281 (2004) [CrossRef] [Google Scholar]
 S. Zhan, J. Lin, Z. Zhang, Y. Zhong, Listbased simulated annealing algorithm for traveling salesman problem, Comput. Intell. Neurosci. 2016 , 1–12 (2016) [CrossRef] [Google Scholar]
 Y.W. Leung, Y. Wang, Multiobjective programming using uniform design and genetic algorithm, IEEE Trans. Syst. Man Cybern. C 30 , 293–304 (2000) [CrossRef] [Google Scholar]
 H. Ishibuchi, T. Murata, A multiobjective genetic local search algorithm and its application to flowshop scheduling, IEEE Trans. Syst. Man Cybern. C 28 , 392–403 (1998) [CrossRef] [Google Scholar]
 S.L. Ho, Sh. Yang, H.C. Wong, G. Ni, A simulated annealing algorithm for multiobjective optimizations of electromagnetic devices, IEEE Trans. Magn. 39 , 1285–1288 (2003) [CrossRef] [Google Scholar]
Cite this article as: Fatemeh Farhatnia, S.Ali Eftekhari, Alireza Pakzad, Soheil Oveissi, Optimizing the buckling characteristics and weight of functionally graded circular plates using the multiobjective Pareto archived simulated annealing (PASA) algorithm, Int. J. Simul. Multidisci. Des. Optim. 10, A14 (2019)
All Tables
The closest optimal Pareto point to the objective function of the weight of 100 kg for Al/ZnO_{2} plate.
The closest optimal Pareto point to the objective function of the weight of 100 kg for Al/Al_{2}O_{3} plate.
The closest Pareto optimal point to the objective function of the buckling load equals to 5 × 10^{6} (N/m) for the asymmetric Al/ZnO_{2} plate.
The closest Pareto optimal point to the objective function of the buckling load equals to 5 × 10^{6} (N/m) for the symmetric Al/ZnO_{2} plate.
The closest Pareto optimal point to the objective function of the buckling load equal to 5 × 10^{6} (N/m) for the asymmetric Al/Al_{2}O_{3} plate.
The closest optimal Pareto point to the objective function of the buckling load equals to 5 × 10^{6} (N/m) for the symmetric Al/Al_{2}O_{3} plate.
All Figures
Fig. 1
Profile of variation of the properties of FG plates of (a) symmetric distribution, (b) asymmetrical distribution. 

In the text 
Fig. 2
Critical buckling loadweight diagram, accepted dominated solution, nondominated solution (Pareto archive) and GA solution of the Al/ZnO_{2} plate in R = 1 m for (a) the asymmetricalclamped edge, (b) asymmetricalsimply supported, (c) symmetricalclamped edge, (d) symmetrical simply supported. 

In the text 
Fig. 3
Thicknessvolume fraction diagram, accepted dominated solution (light blue points) and nondominated solution (dark blue points) of Al/ZnO_{2} plate in R = 1 for (a) the asymmetricalclamped, (b) asymmetricalsimply supported, (c) symmetrical clamped, (d) symmetrical simply supported edge. 

In the text 
Fig. 4
Critical buckling load weight diagram, accepted dominated solutions and nondominated solution (Pareto archive) of the Al/Al_{2}O_{3} plate in R = 1 (a) asymmetricalclamped, (b) asymmetricalsimply supported, (c) symmetricalclamped, (d) symmetricalsimply supported edge. 

In the text 
Fig. 5
Thicknessvolume fraction diagram, accepted dominated solution (light blue) and nondominated solution (dark blue) of Al/Al_{2}O_{3} plate in R = 1 (a) the asymmetricalclamped, (b) asymmetricalsimply supported, (c) symmetricalclamped, (d) symmetricalsimply supported edge. 

In the text 
Fig. 6
Critical buckling loadweight diagram with regarding to the optimum ratio of thickness to radius for Al/Al_{2}O_{3} plates: (a) the asymmetricalclamped, (b) asymmetrical simply supported, (c) symmetricalclamped, (d) symmetrical simply supported edge. 

In the text 
Fig. 7
Critical buckling loadweight diagram with regarding to the optimum ratio of thickness to radius for the FG Al/ZnO_{2} plates: (a) asymmetricclamped support, (b) asymmetricalsimply supported, (c) symmetricalclamped, (d) symmetricsimply supported edge. 

In the text 
Fig. 8
Critical buckling loadweight diagram, accepted dominated solutions and nondominated solutions for symmetrical Al/ZnO_{2} plates with the simply supported edge for volume fraction of 10. 

In the text 
Figure A.1
Flowchart of present method. 

In the text 
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