Applied Mechanics and Materials Vols. 110-116 (2012) pp 3422-3428 Online available since 2011/Oct/24 at www.scientific.net © (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.110-116.3422
Imperialist Competitive Algorithm for Multiobjective optimization of Ellipsoidal Head of Pressure Vessel Behzad Abdia, Hamid Mozafarib, Amran Ayobc, Roya Kohandeld a,b,c
Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310, Johor, Malaysia d
Faculty of Civil Engineering, Universiti Teknologi Malaysia, 81310, Johor, Malaysia a
[email protected], b
[email protected], c
[email protected], d
[email protected]
Keyword: Imperialist competitive algorithm; Ellipsoidal Head, Optimization; Multiobjective
Abstract. This work devoted to an ellipsoidal head of pressure vessel under internal pressure load. The analysis is aimed at finding an optimum weight of ellipsoidal head of pressure vessel due to maximum working pressure that ensures its full charge with stresses by using imperialist competitive algorithm and genetic algorithm. In head of pressure vessel the region of its joint with the cylindrical shell is loaded with shear force and bending moments. The load causes high bending stresses in the region of the joint. Therefore, imperialist competitive algorithm was used here to find the optimum shape of a head with minimum weight and maximum working pressure which the shear force and the bending moment moved toward zero. Two different size ellipsoidal head examples are selected and studied. The imperialist competitive algorithm results are compared with the genetic algorithm results. Introduction Ellipsoidal head of pressure vessel are used in many industries, e.g., missile, oil, chemical, air force and so on. This type of head disturbs the membrane state of stresses in pressure vessel. Spence and Tooth [1] worked on the problem of strength and stability of this type of head. Magnucki and Monczak [2] found out the minimal wall thickness of a circular cylindrical tank with ellipsoidal head. Strength and Optimization of Thin-walled Vessels was presented by Magnucki [3] that showed the disturbance of strength is mainly as the result of the discontinuity of the main meridian radius of curvature at the region of joint of head to the cylindrical shell. Elin and Haritinov [4] worked on Determination of optimal size of thin-walled vessels by using the calculus of variations. Minimum weight of internally pressurized domes subject to plastic load failure was presented by Blachut [5]. Wilczynski used the finite element method to define the shape of fully stressed heads of pressure vessels [6]. Błachut and Galletly [7] investigated the influence of local shape imperfections on the elastic buckling of torispheres and hemispheres. Ke ˛dziora and Kubiak [8], using FEM, numerically calculated the stress distribution in pressure tanks. Also, Magnucki and Lewinski defined optimal shapes of heads of cylindrical pressure vessels under strength and geometric constraints [9]. Atashpaz-Gargari and Lucas [10] introduced the Imperialist Competitive Algorithm (ICA) this algorithm is a new socio–politically motivated global search strategy that has recently been introduced for dealing with different optimization task. This evolutionary optimization strategy has shown great performance in both convergence rate and better global optima achievement [11]. Imperialist competitive algorithm and genetic algorithm are used here to find the optimum shape of a head with minimum weight which the shear force and the bending moment moved toward zero. And finally, the results of both algorithms are compared together From the theory of boundary disturbance, the practical application of which is presented in the monogragh [3], the results is that at the point of contact of both shells their main meridian radius should be equal. Hence, the curvature ( 1/ R1 ) of the meridional profile of the head at the point of contact with the circular cylindrical shell should be zero [9]. The main curvature radius of the center surface of an ellipsoidal head are (Figure 1)
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Applied Mechanics and Materials Vols. 110-116
R1 = a 2 / b (1 + k sin 2 θ )
−3/ 2
,
R2 = a 2 / b (1 + k sin 2 θ )
−1/ 2
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(1)
where k = ( a / b) −1 2
,
0 ≤θ ≤π /2
(2)
Figure 1: Discontinuity of the curvature [9]
Problem statement In ellipsoidal head of pressure vessel the region of the joint with the cylindrical shell is affected with shear force and bending moments because of boundary disturbance. The amount of shear force and bending moment increase to high bending stresses in the area of the joint. The aim of this study is finding a optimum shape and weight of ellipsoidal head meridian for which the shear force and the bending moment equal zero and only the membrane stresses appear [9]. The meridian profile of which is shaped in the form of circular arc OA and a special curve AB (Figure 2). The shape of the curve and the arc R0 should be chosen by imperialist competitive algorithm and genetic algorithm so as the equivalent stresses of the head do not exceed the equivalent stresses in the cylindrical shell. The weight of a full ellipsoidal head of pressure vessel is: 1/ m
Weight = 2πγ t (a 2 m + 2a mb m ) / 3
,
m = 1.6075
(3)
where a, b are major and minor axis and t is the thickness of ellipsoidal head and γ is the specific gravity. Due to the ASME design codes working pressure of ellipsoidal head of pressure vessel with thickness t is given by: Pw = 2 SEt /( Dk + 0.2t )
(4)
where D = diameter of the shell to which the head is attached, E = joint efficiency, S = allowable stress, and K = stress intensity factor. In this study the joint efficiency assumed to be equal to 1 and K is given by the following expression: 2 K = 1/ 6 2 + ( a / b )
(5)
where a and b are the semi-major and semi-minor axes of the ellipse. Longitudinal stress σ θ and circumferential stress σ ϕ in the revolution of the shell are:
σ θ = 1/ 2( R2 p / t ) , σ ϕ = (1 − R2 / 2 R1 ) R2 p / t And the equivalent stress σ eq in the ellipsoidal head of pressure vessel is: σ eqhead = σ θ2 − σ θ σ ϕ + σ ϕ2 = ( R2 p / 2t ) 3 (1 − R2 / R1 ) + ( R2 / R1 )
(6) 2
(7)
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where R1 and R2 are main meridian and parallel radius of the middle surface of the head, t is the thickness of the head and cylindrical shell and p is internal uniform pressure.
Figure 2: The meridian profile of ellipsoidal head And the equivalent stress in the cylindrical shell is: σ eqcylinder = ap 3 / 2t
(8)
where a is the radius of the cylindrical shell. The stresses constraint of head is:
σ eqhead ≤ σ eqcylinder
(9)
By substituting Eq. (5) and (6) into the Eq. (7) obtains:
R2 /(a 3) 3 (1 − R2 / R1 ) + ( R2 / R1 ) ≤ 1 2
(10)
Mathematical formulation An internally pressurized ellipsoidal head of pressure vessel is optimized for the minimum weight with maximum working pressure multiobjective satisfying specified design requirement. The following optimization problem can then be formulated. Cost function. The purpose of multi-objective optimization is different from that of singleobjective optimization: in the latter, the goal is to find the best solution, which is the design that minimizes (or maximizes) the objective function. In contrast, in multiobjective optimization there is no single solution that minimizes (maximizes) all the objective functions. Indeed, the objective functions often conflict, as a design that decreases one objective will increase another . The interaction between the objective functions gives rise to a set of compromise solutions called Pareto set. A solution belongs to the Pareto set if there is no other design such that all the objective functions are lower at the same time. The designer will then need to use additional information to prioritize the objective functions in order to choose between the elements of the Pareto set. In this paper the Pareto set is generated by optimizing a convex combination of the two objectives, weight of ellipsoidal head, w and working pressure Pw . F ( x) = αW + (1 − α ) Pw
(11) For a series of values of the multiplier α , Several values α of were chosen successively and the combined objective function was minimized using a single objective optimizer based on imperialist competitive algorithm and genetic algorithm. If the Pareto set is convex, this procedure yields points that belong to the Pareto set. [11]. Design Variables. The dimensions of the ellipsoidal head are selected as the independent design variables. These variables are: The thickness of ellipsoidal head, x1 = t and the ratio of minor axis to the major axis, x2 = K = b / a . Therefore, the vector x will be given as xopt = ( t , K )
(12)
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Constraints. The lower and upper bounds are imposed on all the design variables. Based on the standard of practical design, the constraints Gi are represented by the following: The equivalent stresses constraint:
G1 = R2 /(a 3) 3 (1 − R2 / R1 ) + ( R2 / R1 ) − 1 ≤ 0 2
(13)
The lower bound on design variables: G2 = ( x Li − xi ) /( xUi − x Li ) ≤ 0
(14)
Where xi are design variables ( i = 1, 2 ) , x
Ui
is the upper bound value of design variable xi and x Li
is the lower bound value of design variables xi .The upper bound on design variables: G3 = ( xi − xUi ) /( xUi − x Li ) ≤ 0
(15)
Imperialist Competitive Algorithm In computer science, Imperialist Competitive Algorithm (ICA) [11] is a computational method that is used to solve optimization problems of different types. Like most of the methods in the area of evolutionary computation, ICA does not need the gradient of the function in its optimization process. From a specific point of view, ICA can be thought of as the social counterpart of genetic algorithms (GAs). ICA is the mathematical model and the computer simulation of human social evolution, while GAs is based on the biological evolution of species. This algorithm starts with an initial population. Each individual of the population is called a country. Countries are the counterpart of chromosomes in GAs. After evaluating the cost function at each of countries, some of the best of them (in optimization terminology, countries with the least cost) are selected to form the initial empires by controlling the other countries (colonies). All the colonies are divided among the mentioned initial imperialists based on their power. The power of each country, the counterpart of fitness value in the GA, is inversely proportional to its cost. The initial imperialist states together with their colonies form the initial empires. After forming initial empires, the evolution begins. The colonies in each of the empires start moving toward their relevant imperialist country. This movement is a simple model of assimilation policy which was pursued by some of the imperialist states. Besides assimilation, revolution is another operator of this algorithm. Revolution occurs in some of the colonies by making random changes to their position in the socio-political axis. The total power of an empire depends on both the power of the imperialist country and the power of its colonies. Imperialistic Competition is another step of the algorithm. All empires try to take the possession of colonies of other empires and control them. The imperialistic competition gradually brings about a decrease in the power of weaker empires and an increase in the power of more powerful ones. The imperialistic competition is modeled by picking some (usually one) of one of the colonies of the weakest empire and making a competition among all empires to possess these (this) colonies.The above steps continue until a stop condition is satisfied by reaching to an acceptable suboptimal solution [12-14]. Model description Pressure vessels are generally formed by using cylindrical shell and concial shell for central section and hemispherical, ellipsoidal, and torispherical shell for head domes. To reference models of ellipsoidal dome end shell of pressure vessel were considered and studied. Both models have the same elastic modulus of 207.0 GN / m 2 and the same poison’s ratio of o.3 and the same specific gravity of 7800 kg / m3 . The fixed major axis a for model 1 is 162.5 cm and 233.0 cm for model 2. The principal
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dimensions for two reference models are listed in Table 1. Due to the previous discussions, the range of the ratio of minor axis to major axis K = b / a is selected as 0.21 ≤ K ≤ 0.95 and the range of thickness as 50 ≤ D / t ≤ 150 . Both models are under Maximum working Pressure that are calculated based on ASME design code. Table 1: The principal design data of reference head Ellipsoidal head
Model 1
Model 2
Length of major axis, a ratio of minor axis to major axis, K = b / a Thickness, t Weight, W Maximum working pressure, Pw
1.625 m 0.61 0.032 m 3.1206 kN
2.33 m 0.53 0.045 m 8.4729 kN
34.67 MN / m 2
28.68 MN / m 2
Numerical results and discussion The optimal comparison between the results of first model of ellipsoidal head and the reference model is showed in Table 1. Also the optimal comparison between the results of second model of ellipsoidal head and the reference model is showed in Table 2 and Table 3. The comparisons between reference model and optimal models are displayed in Figure 3. Table 2: The numerical comparison table of the model 1 Ellipsoidal head
Length of major axis, a ratio of minor axis to major axis, K = b / a Thickness, t Weight, W Maximum working pressure, Pw
Model 1 Ref. head
Opt. head (GA)
Opt. head (ICA)
1.625 m 0.61 0.032 m 3.1206 kN
1.625 m 0.646 0.028 m 2.808 kN
34.67 MN / m 2
32.35 MN / m 2
1.625 m 0.642 0.0280 m 2.800kN 32.13 MN / m 2
Table 3: The numerical comparison table of the model 2 Ellipsoidal head
Length of major axis, a ratio of minor axis to major axis, K = b / a Thickness, t Weight, W Maximum working pressure, Pw
Ref. head
Model 2 Opt. head (GA)
Opt. head (ICA)
2.33 m
2.33 m
2.33 m
0.53 0.045 m 8.4729 kN 28.68 MN / m 2
0.5114 0.0404 m 7.49 kN
0.504 0.04 m 7.38 kN
24.59 MN / m2
23.88 MN / m2
For both reference models and both optimum models of ellipsoidal head, the behavior of equivalent stresses over the ellipsoidal head are studied and showed in figure 4. If thickness is kept constant, the influences of ratio K = b / a to the equivalent stress of both optimum models are studied. The results are displayed in Figure 5.
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Model 1 Model 2 Figure 3: The head configuration comparison of both models From Tables 3-4 and Figure 3, the ratios of K are found to be 0.642 for optimum model 1 and 0.53 for optimum model 2 while the ratios of K are chosen as 0.61 and 0.53 for model 1 and model 2. The maximum working pressure are 32.13 MN / m 2 and 23.88 MN / m2 for optimum models, and 34.67 MN / m2 , 28.68 MN / m2 for reference models therefore by using imperialist competitive algorithm we found a optimum ellipsoidal head that weight reduced 10.27% while the working pressure reduced 7.3% for model 1 and for model 2, the weight reduced 12.9% while the working pressure reduced 16.7% . Due to the Tables 3-4, it can be seen that the results of imperialist competitive algorithm are very close to the results of genetic algorithm.. The reason for that is we set the preassigned equivalent stress bound and select the minimum weight and maximum working pressure types of multiobjective function.
Figure 4: The behavior of equivalent stresses over the ellipsoidal head
Figure 5: The behavior of equivalent stresses via the slope K = b / a over the ellipsoidal head with various The equivalent stress behaviors over the ellipsoidal head for both optimum and reference models are showed in figure 4. from This figure, it can been see that in model one the maximum equivalent stress are happened in the region of joint to the cylindrical shell for both optimum and reference
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model 1 and the maximum amount of equivalent stress are happened in the top of the head for model 2 that the reason for that is in model one we found a optimum head that is bigger than the reference model but in model 2 we found an optimum model that is smaller than reference models. From Figure 5 that are showed the behavior of equivalent stress over the ellipsoidal head in various K = b / a ( 0.1 ≤ K ≤ 0.99 ) for both models of optimum ellipsoidal head, it can been see that the maximum equivalent stress is happened in the region of joint to the cylindrical shell and the amount of equivalent stress increase by increasing the amount of K = b / a . In other words, small ellipsoidal head have the small equivalent stresses in the region of joint to the cylindrical shell. Conclusions In this study, the imperialist competitive algorithm (ICA) and genetic algorithm (GA) are used to find the optimum shape design of fully stressed ellipsoidal head of pressure vessel. The results of imperialist competitive algorithm are very close to the results of genetic algorithm, therefore it can be mentioned that the imperialist competitive algorithm (ICA) can be used in mechanical engineering optimization problem. Also, in this study, the effects of shape of ellipsoidal head of pressure vessel on equivalent stress are studied. Due to the this study, the region of the joint of ellipsoidal head with the cylindrical shell are found as critical region and in this are the amount of equivalent stress are bigger that the other area.
References [1]. J. Spence, A. S. Tooth, “Pressure vessels design concepts and principles”, Second Seminar of Modern design Methods of Pressure Vessels. Lodz, University of technology, 1995. [2]. K. Magnucki, T. Monczak. ”Determination of minimal wall thickness of circular cylindrical tank with ellipsoidal heads” Archive of Mech Engineering 1998; XLV (2): 73–85. [3]. K. Magnucki, “Strength and Optimization of Thin-walled Vessels” (inPolish) Warszawa / Poznan: Science Publishers PWN, 1998. [4]. V. D. Elin, V. I. Haritonov, “Determination of optimal size of thin-walled vessels”, Raboty po mechanike sploshnich sredov.Tula 1977: 107-112 (in Russian). [5]. J. Błachut, “Minimum weight of internally pressurized domes subject to plastic load failure”. Thin-Walled Structures 1997; 27(2): 127–46. [6]. B.Wilczyn˜ski, “Optimal shape design of LPG tanks”. In: 37th Symposium ‘Modelling in Mechanics’, Gliwice: Silesian Technical University, 1996; 2:239–245. [7]. J. Błachut, G. D. Galletly, “Influence of local imperfections on the Collapse strength of domed end closures”, Proc Inst Mech Engrs 1993; 207:197–207. [8]. S. Kedziora, T. Kubiak., “Application of FEM for calculation of stresses and strains in pressure tanks”. Dozor Techniczny, 1999; 4:83 – 5 . (inPolish). [9]. K. Magnucki, J. Lewinski, “Fully stressed head of a pressure vessel”., Thin-Walled Struct 2000; 38: 167–78. [10] K. Deb (2001), “Multi-Objective Optimization Using Evolutionary Algorithms”, UK Wiley, Chichester. [11]. L.Grosset, S.Venkataraman and R.T. Haftka, ” Genetic optimization of two-material composite laminates”, Annual Technical Conference, June 25-30, 2001 [12]. Zhang, Yang; Wang, Yong; Peng, Cheng (2009). "Improved Imperialist Competitive Algorithm for Constrained Optimization". Computer Science-Technology and Applications, IFCSTA. [13]. R. Rajabioun, E. Atashpaz-Gargari, and C. Lucas, “Colonial Competitive Algorithm as a Tool for Nash Equilibrium Point Achievement,” Lecture notes in computer science, 5073, 680-695. 2008
Mechanical and Aerospace Engineering 10.4028/www.scientific.net/AMM.110-116
Imperialist Competitive Algorithm for Multiobjective Optimization of Ellipsoidal Head of Pressure Vessel 10.4028/www.scientific.net/AMM.110-116.3422
