OPTIMIZATION OF STEEL PLANE TRUSS MEMBERS CROSS SECTIONS WITH SIMULATED ANNEALING METHOD

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1 OPTIMIZATION OF STEEL PLANE TRUSS MEMBERS CROSS SECTIONS WITH SIMULATED ANNEALING METHOD Snezana Mitrovic College of Civil Engineering and Geodesy, Belgrade, Serbia Goran Cirovic College of Civil Engineering and Geodesy, Belgrade, Serbia This paper presents an optimal design problem analysis, with the Simulated Annealing method. The main algorithm represents seeking for the solution in the search space with the aim to minimize a value of the subject function. A stochastic procedure has been proposed to determine organization rule analog to atomic organization with minimum energy. The strategy which avoids getting trapped at a local minimum point is the main preference of this method over others. A simple and significant tool considering optimal cross sections of plane truss members has been ensured by this method. Also, this work proposes a comparison of our results with the results obtained in other methods application. In addition, the advantage of this method over others has been established. KEYWORDS: simulated annealing, optimization, total weight, steel truss. INTRODUCTION Despite its name, Simulated Annealing has nothing identical either with simulation or with annealing. Simulated Annealing is a technique of problem solving based on analogy with the method of annealing metal in order to increase its strength. When a heated metal cools off very slowly, it forms its structure in the shape of simple (minimal energy) crystal lattice thermo-dynamic equilibrium in compliance with Boltzman s factor of probability e E/k BT for atom configuration during metal melting. In physics, every set of all system atoms is valued by Boltzman s probability factor, where E is configuration energy, T is temperature expressed in 0 C or K, and kb is Boltzman s constant equals kb=1, * J/K. When a metal is forcibly cooled, atoms will form metastable system, non-crystal lattice, irregular and weak structure, with high energy as a consequence of internal forces. For the first time the method of Simulated Annealing was applied in 1953 (Metropolis, 1953) to solve non-linear optimization problems. In his work, Metropolis with his associates developed Monte Carlo method of calculating characteristics of any substance that can be considered as a compound of mutually dependent molecules. This inspired Kirkpatrick (Kirkpatric et al, 1983) to develop an algorithm of Simulated Annealing (SA) that can be applied to solving various optimization problems. The Metropolis procedure is an exact copy of this physical process, which can be used in atom configuration simulating in termodynamic equilibrium at a given temperature. New geometry has been designed as a random atom disposition in each iteration. Energy generated in every new generation is being calculated, as well as the probability that exactly such a geometry will be generated. 165

2 In the analogy between a combinatorial optimization problem and the process of annealing, the states of bodies represent possible solutions of optimization problems, the states of energy correspond to the value of the concerned functions, the state of minimal energy corresponds to optimal problem solution, and accelerated cooling can be observed as local optimum. Customary, the algorithm consists of iterative steps. Every iteration consists of random change of instantaneous solution in order to create a new solution in its neighborhood. The neighborhood is defined by the selection of generational mechanism. The ideas of function and solution are defined in the same way as in Genetic Algorithm methods. From the same method various mechanisms can be developed, like, for example, mutations and convergence. General algorithm consists of solution searching within solution area that minimizes the value of a given function, namely annealing can be considered as a stochastic procedure in defining the organization of atoms with minimal energy. The advantage of this method over others is that there is no probability of getting the solution stuck in the local minimum area. The probability that the value of the requested considered function will be increased can be expressed according to Boltzman-Gibbs disposition: p=exp (-Δf k B / T)...(1) where Δ is a value change of the considered function f, k B is Boltzman s constant, and T is a control parameter called temperature. The counter of random numbers then makes numbers evenly arranged in the interval (0, 1), and if the sample is smaller than p, the step is accepted. Analogous to physical process, temperature T is initially high. That is why the probability referring to acceptance of the step that increases the considered function is also initially high. The temperature is being significantly reduced as the process continues, i.e. the system is cooling slowly. Eventually, the acceptance of the step that increases the considered function becomes exceptionally small. Generally, the temperature is being reduced in compliance with annealing scheduling. When defining annealing scheduling for annealing simulation algorithm four parameters must be defined. These are: the initial temperature, the rule of changing temperature, the number of iterations at each temperature step and the termination criterion. Several different annealing dispositions have been presented in bibliography. Continual and non-monotonous schemes temperature reduction involve very simple annealing strategies (geometric annealing scheme etc). Most frequently applied annealing disposition is exponential cooling. It begins at some initial temperature, T o, and it is being reduced in temperature steps in compliance with T k+1 =αt k ; where 0<α<1. Certainly, a fixed number of steps at each temperature must be adopted before the calculation is continued. The algorithm is terminated either when the temperature achieves certain final value, T f, or when some other termination criterion is met. The value selection of α, T o and T f is completely dependent on the type of problem. Empirical records show that it is most convenient to adopt value 0.95 for a, and the initial probability 0.8. The initial generation is usually adopted by the principle of coincidence. The algorithm presented below is based on application in bibliography (Huang and Arora, 1997): Step 1: Select initial temperature T 0 (the expected global minimum of function) and a visible point x(0). Calculate f(x(0)). Select the integer value L (e.g. the limit of iterations in order to obtain the expected minimal value). Set the iterative counter as K=0 and other counter k=1. 166

3 Step 2: Create a new point x(k) in the neighborhood of the current point. If the point is within the area of the impossible, create a new point until the probability of existence is satisfied. Create the random number z evenly disposed in the interval [0,1]. Calculate f(x(k)) and Δf =f(x(k)) f(x(0)). Step 3: If Δf < 0, as a new best point x(0) arose, set f(x(0)) = f(x(k)) and move to step 4. Invasively, calculate the disposition of probability function (equation 1). If z < p(δf), then x(k)is treated as a new best point and go to step 4. Invasively, return to step 2. Step 4: If k < L, then set k=k+1 and go to step 2. If k > L and if any other termination criterion is satisfied, then stop. Invasively, go to step 5. Step 5: Set K = K+1, k =1; set TK = rtk-1; move to step 2. The authors Huang and Arora then suggest: L has the value of 100, and r= 0,9. In step 2 only one point is generated in the neighborhood in a period of time. Namely, although Simulated Annealing (SA) usually creates project points without the need for information on function or gradients, the matter is not in a mere research of coincidence in the whole research area. In an early stage, a new point can be located distantly from the current point in order to accelerate the process and to avoid drawing into local minimum. After the temperature is lowered, the new point is usually generated in the immediate vicinity with the aim to focus onto the local area. This is controlled by defining the procedure about the size of the steps. In step 2, the newly created point must be possible. To achieve this, some other method can be applied, e.g. the method of penalty functions. The process is done until freezing point is reached. In step 4, the following has been suggested: a) the programme is terminated if the value changes are less than for four successive iterations. B) The programme comes to a hault if I/L < 0.05, where L is a limit, i.e. the number of the possible created points in one iteration, and I is the number of points that satisfy the condition Δ f <0 (see Step 3). C) The programme stops if K achieved the supposed value. The algorithm in the form of a tree is presented, which represents the structure of Simulated Annealing (See Figure 1). 167

4 Figure 1: The structure of Simulated Annealing algorithm PRACTICAL IMPLEMENTATION OF SIMULATED ANNEALING (SA) The solution for the problems with continual project variables should be created in compliance with the formula: x i+1 =x i +Q u...( 2) where n is a vector of random variables within the interval (- 3, + 3), so that each has a median and the unit variance equal to zero, and Q is a matrix which regulates the disposition size by steps. Aimed at making steps with the co-variance matrix S, Q can be found by solving: S=QQ T...(3) according to Cholesky decomposition, for example. S should be changed as the research proceeds in order to involve information about local topography: S i+1 =(1-α) S i + αωx...(4) where matrix X is a measure of the next step co-variance and α dumping constant which controls the rate of information transfer from X to S with weight coefficient ω. The disadvantage of this scheme is that the solution of the equation (3), which must be obtained each time when S is changed, is rather robust. There is an alternative strategy for solution making according to the formula: 168

5 x i+1 =x i +D u...(5) where n is now a vector of random numbers within the interval (-1, 1), and D is the diagonal matrix that defines maximal permitted changes of each variable. After accepting the solution change, D can be defined as: D i+1 =(1-α) D i + αωr...(6) where R is a diagonal matrix of elements that consists of magnitudes of successful changes over each control variable. The change of probability value defined in equation (1) is: p=exp (-δf/ Td)...(7) where d is an average size of streps, and -δf/td is the measure of change efficiency. Since the size of the steps is considered at value calculating p, D need not be ajusted when calculating T. The development of a solution SA algorithm neither requires nor creates some deduced information, it is only necessary to be provided with the considered function for every solution it produces. It is important that problem function evaluation is carried out efficiently. In most cases the procedure can be simply programmed to reject any offered opportunities resulting in constraint violation. Namely, there are two significant circumstances in which this approach cannot be carried out (Busetti, 2002.): If there is any constraint at the system If the possible area defined by constraints is such that it is not possible to connect all the possible solutions without passing through the above mentioned impossible area. In any case, the problem should be transformed into a problem without constrains by designing magnifying considered functions, so that constraints are incorporated into penalty function: f A (x) = f(x) + 1/T w T c V (x)...(8) where w is the vector of non-negative weight coefficients and the vector c V designates magnitudes of any constraint violation. Penalty inversive dependence on temperature is largely directed towards the possible research area as it progresses. Through equations from (1) to (7), annealing scheduling determines the degree of progress. Initial temperature should be high enough to grind completely the system and it should be lowered to the freezing point during algorithm solving. OPTIMAL CONSTRUCTION DESIGNING The construction analysis and designing usually involve highly complex procedures and a great number of variables. As a result, the solution is found through iterative procedure, while initial values are set in the form of variables. Also, the number of analysis steps grows significantly if optimal values are set among all possible alternatives. Describing physical behavior of a construction through mathematic functions, the aim of optimization techniques is achieved, namely, the extreme values of these functions. The studies of axial loaded constructions optimization can be classified in three main categories: a) optimization of member sizes, b) optimization of configuration or geometry and c) optimization of topology or connectivity. Simultaneous optimizations of topology and geometry may be referred to as 169

6 layout optimizations. Optimization of member sizes is adjusted to optimization of construction weight (mass), where the areas of cross-sections of all elements are considered as project variables and coordinates of nodes and lengths of members as fixed. Generally, minimizational problem can be expressed as: min f(x i ) i=1,...n...( 9) in compli- ance with g j (x i ) 0 j=1,...m...(10) h k (x i ) =0 k=1,...l...(11) x i l x i x i u...(12) where f designates the considered function and X=(x 1,x 2,...,x n ) T the vector of design variables. The remaining functions are constrained functions that correspond to inequations with constraints (g), equations with constraints (h) and limit constraints with lower and upper limits designated with l and u, respectively. These functions, which can be solved analytically or numerically, can be linear or nonlinear and can include project variables in explicit or inexplicit form. A very wide range of solutions can be obtained for the problems with continual design variables mathematically defined through equations from (2) to (7), irrespective of whether they are ''small'' construction problems or not. In the text below an example of optimization technique with SA for steel plane truss optimization will be presented (See Figure 2), an example where cross sections are discrete variables that make the solution possible not only from mathematical, but also from practical point of view. Although only a reduced number of accurate cross sections are considered because of economical and aesthetic reasons, the number of combinations is very large (Kripka, 2004). This indicates, if a construction has a certain degree of vagueness, stresses can be redistributed through alteration of relative stiffness of members when the cross section of only one member is altered. Nomenclature: A cross section area, m 2 E elasticity module, KN/m 2 W the objective function, weight in KN L member length, in m F the objective function with penalty, in KN f the objective function, in KN K B Boltzman s constant T body temperature Greek symbols: σ stresses, in MPa ρ specific mass, in KN/m 3 δ displacing of supports, m 170

7 Figure 2: Plain truss with 10 elements This example of plane truss, which is frequently described in bibliography (e.g. Rajeev, Krishnamoorthy, 1992), has been deliberately chosen because its optimization has been carried out in a large number of ways, so that the obtained results can be easily compared. Also, to process this example, namely its optimization, a great number of easily available commercial and free software is at our disposal. 42 commercially available W cross sections collected from the American Steel Construction Institute have been adopted for design variables in order to compare the results (See Table 1). The procedure is also identical with the data that would be collected from a domestic and European standard (I, U, HEA,...). Table 1: Available cross sections areas in cm 2 The above mentioned constructions optimization problem can also be solved successfully through application of algorithm that supports the previous idea of Simulated Annealing method. min f 10,45 11,61 12,84 13,74 15,35 16,90 16,97 18,58 18,90 19,94 20,19 21,81 22,39 22,90 23,42 24,77 24,97 25,03 26,97 27,23 28,97 29,61 30,97 32,06 33,03 37,03 46,58 51,42 74,19 87,10 89,68 91,61 100,00 103,23 109,03 121,29 128,39 141,94 147,74 170,97 193,55 216,13 n i 1 1 2T ( x) W A L gx2...(13) i i 171

8 in compliance with i 1 0 dop. i 1 0 dop....(14) Much more complicated constraint conditions can be conducted upon the concerned function. For example, temperature changes, temperature variations, residual stresses in material, stresses generated at the joint of elements... Analogy can also be applied to constructions loaded to bending. Aimed at computer application, constraints are considered with the support of dynamic penalty technique known as annealing penalty (Michalewicz, Schoennauer, 1996). Similar to optimization techniques, penalty factor has a relatively small initial value, which is significantly increased as the temperature is lowered. Thus, even when a problem starts with impossible solutions, small constraint violation is initially permitted. Input constants are identical as in every other analysis of this example in order to compare the results. Owing to its heuristic nature, the initial point is less significant than in other methods. Output programme results are given comparing the results with other optimization techniques for the same example of the plane truss. THE RESULTS OF THE ANALYSIS AND FINAL CONSIDERATIONS The results are obtained applying software package of Simanneal Simulated Annealing code. The final solution is obtained through 1200 iterations in the process which remind as climbing on a local hill to avoid drawing into local minimum (See Figure 3). Figure 3: Convergence of Simulated Annealing The results are remarkably similar to the results obtained by Kripka in 2004, applying some other computer code for an almost identical problem assumption, but with a little differently adopted parameters (See Table 2). It can be concluded that insignificantly better results are obtained by applying Simulated Annealing algorithm than the results obtained by Simple Genetic Algorithm. Furthermore, this would be reflected on construction cost. Similarly, methods that do not treat constraint in cross sections give seemingly better results. It is dealt 172

9 with seemingly better results because it could be established by a more detailed analysis that construction cost price is significantly increased through the increase of cross sections varieties, especially introducing those that are not commercially available. Table 2: The comparison of plane truss optimization results Author W[kg] A1[cm 2 ] A2 A3 A4 A5 A6 A7 A8 A9 A10 Previous 2491,00 216,13 10,45 147, ,45 10,45 51,42 141,93 141,93 10,45 study Kripka (2004) 2490,00 216,13 10,45 147,74 90,07 10,45 10,45 51,42 147,74 141,93 10,45 Improved method of penalty function (Cai, Theireu, 1993) Genetic algorithm (Rajeev at al, 1992) 2491,00 216,13 10,45 147, ,45 10,45 51,42 141,93 141,93 10, ,00 216,13 10,45 147, ,45 10,45 91,61 128,39 128,39 10,45 Genetic 2534,10 193,55 10,45 147,74 87,10 10,45 10,45 89,69 141,93 141,93 10,45 algorithm (Coelo, 1994) Genetic algorithm (Togan, Daloglu 2006) MATLAB 2564,10 216,13 16,91 141,94 103,23 10,45 10,45 30,96 193,55 128,38 10,45 WITHOUT CONSTRAINTS IN CROSS SECTIONS Penalty 2258,0 186,59 0,65 155,30 90,07 0,65 0,65 49,62 142,52 141,62 3,61 function of double linear segments (Pezeshk, Camp, Chen, 2000)FEAP- GEN GA with method of Lagrange s multipliers (Bieniawski, Kroo, Wolpert,2004) 2332,24 200,00 3,23 158,06 93,55 0,65 0,65 54,84 135,48 132,26 6,45 REFERENCES Bieniawski S.R., Kroo I.M., Wolpert, D.H. (2004.) Discrete, Continuous, and Constrained Optimization Using Collectives, 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, New York Busetti, F. (2002.) Testing for (common) stochastic trends in the presence of structural breaks, Journal of Forecasting Cai, J.B., Thiereut, G. (1993) Discrete optimization of structures using an improved penalty function method, Engineering. Optimization,

10 Coello, C.A. (1994) Discrete Optimization of Trusses using Genetic Algorithm', Expert Systems Applications and Artificial Intelligence Huang, M. W., Arora, J. S. (1997) Optimal Design with Discrete Variables: Some Numerical Experiments, International J. for Numerical Methods in Engineering, 40. Kirkpatrick, S., Gelatt C.D., Vecchi, M.P. (1983) Optimization by Simulated Annealing, Science 220, Kripka, M. (2004) Discrete optimization of trusses by simulated annealing, Journal of the Brazilian Society of Mechanical Sciences and Engineering. Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A. Teller, E. (1953) Equation of State Calculations by Fast Computing Machines, J. Chem. Phys. 21. Michalewicz, Z., Schoennauer, M. (1996) Evolutionary Algorithms for Constrained Parameter Optimization Problems, Evolutionary Computation (MIT Press), 4(1). Pezeshk, S., Camp, C.V., Chen, D. (2000) Design of Nonlinear Framed Structures Using Genetic Optimization, ASCE J. Struct. Engineering. Rajeev, S. and Krishnamoorthy, C.S., (1992) Discrete Optimization of Structures Using Genetic Algorithms'', Journal of Structural Engineering, 118, 5. Togan, V., Daloglu, A., (2006) Optimization of 3D Trusses with Adaptive Approach in Genetic Algortihms, Engineering Structures,

5. Simulated Annealing 5.1 Basic Concepts. Fall 2010 Instructor: Dr. Masoud Yaghini

5. Simulated Annealing 5.1 Basic Concepts Fall 2010 Instructor: Dr. Masoud Yaghini Outline Introduction Real Annealing and Simulated Annealing Metropolis Algorithm Template of SA A Simple Example References