An evolutionary-Based optimization algorithm for truss sizing design

Vietnam Journal of Mechanics, VAST, Vol. 38, No. 4 (2016), pp. 307 – 317 DOI:10.15625/0866-7136/7476 AN EVOLUTIONARY-BASED OPTIMIZATION ALGORITHM FOR TRUSS SIZING DESIGN Pham Hoang Anh National University of Civil Engineering, Hanoi, Vietnam E-mail: anhpham.nuce@gmail.com Received November 28, 2015 Abstract. In this paper, the optimal sizing of truss structures is solved using a novel evolutionary-based optimization algorithm. The efficiency of the proposed method lies in the combination of global search and local search, in which the global move is applied for a set of random solutions whereas the local move is performed on the other solutions in the search population. Three truss sizing benchmark problems with discrete variables are used to examine the performance of the proposed algorithm. Objective functions of the optimization problems are minimum weights of the whole truss structures and constraints are stress in members and displacement at nodes. Here, the constraints and objective func- tion are treated separately so that both function and constraint evaluations can be saved. The results show that the new algorithm can find optimal solution effectively and it is competitive with some recent metaheuristic algorithms in terms of number of structural analyses required. Keywords: Structural optimization, evolutionary-based optimization, metaheuristics, truss structure, sizing optimization. 1. INTRODUCTION Truss optimization is one of the most popular design problems and has been an ex- tensive research area both in modeling and development of optimization methods. Often the weight of truss structure is to be minimized subject to stress, displacement, and/or natural frequency constraints. This optimization task is in general difficult to solve be- cause of non-linear constraints and non-convex feasible region. This means that the con- vergence of traditional gradient-based optimization methods cannot be ensured [1]. Metaheuristics such as genetic algorithms, particle swarm algorithms and evolu- tion algorithms have been increasingly proposed as alternative techniques for optimiza- tion of truss structures [2]. Recently developed metaheuristics have shown good perfor- mance, for example, are: harmony search algorithm (SAHS) [3], teaching-learning-based optimization (TLBO) algorithm [4, 5], chaotic swarming of particle (CSP) algorithm [6], colliding bodies optimization (CBO) [7, 8], flower pollination algorithm (FPA) [9], water c© 2016 Vietnam Academy of Science and Technology 308 Pham Hoang Anh cycle algorithm (WCA) [10] and mine blast algorithm (MBA) [10, 11], adaptive dimen- sional search (ADS) [12], and some new variants of the well-known differential evolution including adaptive differential evolution algorithm (ADEA) [1], improved constrained differential evolution using discrete variables (D-ICDE) [13], adaptive elitist differential evolution (aeDE) [14] and reliability-based improved constrained differential evolution (SORA-ICDE) [15]. The contribution of this paper is a new evolutionary-based algorithm which is ap- plied to optimal sizing of truss structures. The formulation of the optimization problem and the constraint handling rules are presented in Section 2. In Section 3, the new al- gorithm is described in details. Validation and efficiency of the proposed algorithm in finding optimal structural optimization solutions are given in Section 5, in which three well-known truss structures have been examined and the results have been compared with some state-of-the-art metaheuristics. Conclusions are presented in Section 6. 2. TRUSS SIZING OPTIMIZATION 2.1. Problem formulation In sizing design optimization of truss structures the goal is to find a minimum weight by selecting the cross-sectional areas of structural members such that the final design satisfies strength and serviceability requirements determined by standard design codes. The cross-section areas can be continuous values between the lower bound li and upper bound ui or discrete values form a set S of P given values (often are the cross- sections according to production standards). For a given truss structure, the objective is to find vector of cross-section areas for Nm members of the structure A = [A1, A2, . . . , ANm ] which minimizes the following weight objective function W = Nm ∑ i=1 ρiLiAi , (1) subjected to Nc design constraints gj ≤ g0,j, j = 1, . . . , Nc, where ρi and Li are the ma- terial density and length of the i-th member, respectively; gj are Nc constraint functions (displacement or stress in this paper) and g0,j are allowable values of gj. In this paper, the constraints are modified from the conventional form of con- strained optimization as Eq. (2) and constraint violation is determined by Eq. (3) cj = gj g0,j − 1 ≤ 0, (2) C = Nc ∑ j max{0, cj}. (3) 2.2. Constraint handling To solve the above constrained optimization problem, the constraints and the ob- jective function are treated separately. The following rules are applied: 1. A feasible solution is better than any infeasible one. An evolutionary-based optimization algorithm for truss sizing design 309 2. Between two feasible solutions or two solutions with equal constraint violation, the one having smaller objective function value is better. 3. Between two infeasible solutions, the one having smaller constraint violation is better. These rules were originally suggested by Deb [16] to handle the constrained prob- lem for genetic algorithm, which has been shown successful for other metaheuristics. For handling bound constraints, cutting-off technique [17] is adopted, i.e. the generated vio- lating value is substituted by the bound value, since in many cases the optimum solution is located at one of the bound of a given design variable. 3. PROPOSED EVOLUTIONARY-BASED OPTIMIZATION ALGORITHM Like many other algorithms, the proposed method also uses a population P of NP vectors (individuals) of Nd design variables xk = [xk,1, xk,2, . . . , xk,Nd ] (k = 1, 2, . . . , NP). The population is then restructured by survival individuals evolutionally. The initial population is generated as xk,i = li + rand[0, 1].(ui − li), i = 1, . . . , Nd, (4) where rand[0, 1] is a uniformly distributed random real value in the range [0,1]. At the current generation, each individual xcurrentk of the population is updated us- ing either a global move or a local move. For xcurrentk among p × NP random solutions (1/NP < p < 1), the updating Eq. (5) (global step) is applied, xnewk = x current k + { r(xbest − xcurrentk ), if xcurrentk 6= xbest (r1 − r2)xbest, if xcurrentk ≡ xbest (5) otherwise the updating Eq. (6) (local step) is used. xnewk = x current k + { r(xl − xm), if xl better than xm r(xm − xl), otherwise (6) In Eq. (5) and Eq. (6), xbest is the best found solution in the current population, xl and xm are two different individuals randomly chosen in the population (1 ≤ l 6= m ≤ NP), r, r1, and r2 are vectors of Nd uniformly distributed random numbers within the range [0, 1]. The global move allows a solution to take a random step toward the current best solution, whereas the local move allows a solution move around it current location. If the updated solution xnewk is better than x current k , then x new k becomes the k-th mem- ber in P of the next generation; otherwise, the old individual xcurrentk is retained. The search proceeds until either a convergence criterion is met or the maximum number of iteration/function evaluations is reached. The pseudo code of the proposed algorithm is given in Fig. 1. In the comparison between the two solutions, based on the constraint handling rules in Section 2.2, the following are taken place: 1. If xcurrentk is infeasible: x new k is better than x current k when x new k has smaller constraint violation. Therefore, evaluation of objective function is not necessary. 310 Pham Hoang Anh 2) If current kx is feasible: new kx is better than current kx when new kx is feasible and has smaller objective function value. Thus, in case new kx has equal or larger objective function value, the evaluation of constraint violation of new kx is not necessary. Fig. 1 Pseudo code of the proposed algorithm To adapt the algorithm for problems with discrete variables, the rounding technique can be applied, i.e. each value of a design variable vector, right after created either at initial state or later in the updating, is rounded to the nearest value in the discrete value list. 4. TEST PROBLEMS Three well-known test problems of truss sizing optimization are employed in this paper. The design problems were assigned to minimize the weight of truss subjected to stress and displacement constraints. The design variables are cross-section areas of the truss members which are chosen from a given list of discrete values. 4.1. Ten-bar planar truss The truss layout is illustrated in Fig. 2. The structure is subjected to load P=-100 kips at node 2 and node 4. The design variables are the bar element cross-section areas. The cross-section areas (in 2 ) are chosen from the list: 1.62, 1.80, 1.99, 2.13, 2.38, 2.62, 2.63, 2.88, 2.93, 3.09, 3.13, 3.38, 3.47, 3.55, 3.63, 3.84, 3.87, 3.88, 4.18, 4.22, 4.49, 4.59, 4.80, 4.97, 5.12, 5.74, 7.22, 7.97, 11.5, 13.5, 13.9, 14.2, 15.5, 16.0, 16.9, 18.8, 19.9, 22.0, 22.9, 26.5, 30.0, 33.5. The modulus of elasticity and material density are 10 4 ksi and 0.1 lb/in 3 . The allowable stress is 25 ksi for both tension and compression stress in each member. The allowable displacement of nodes in x and y directions is 2 in. Initialize a population of NP random solutions Evaluate objective function and constraint violation of each solution Set the number of top-ranked solution (or set p) while termination condition not satisfied Select pNP random solutions in the population for k = 1 to NP do if current kx is among pNP selected solutions Generate new solution new kx by global move via Eq. (5) else Generate new solution new kx by local move via Eq. (6) end if If the new solution is better, update it in the next generation end for end while Fig. 1. Pseudo code of the proposed algorithm 2. If xcurrentk is feasible: x new k is better than x current k when x new k is feasible and has smaller objective function value. Thus, in case xnewk has equal or larger objective function value, the evaluation of constraint violation of xnewk is not necessary. To adapt the algorithm for problems with discrete variables, the rounding tech- nique can be applied, i.e. each value of a design variable vector, right after created either at initial state or later in the updating, is rounded to th nearest valu in th discrete value list. 4. TEST PROBLEMS Three well-know test problems of truss sizing optimization are employed in this paper. The design problems were assigned to minimize the weight of truss subjected to stress and displacement constraints. The design variables are cross-section areas of the truss members which are chosen from a given list of discrete values. 4.1. Ten-bar planar truss The truss layout is illustrated in Fig. 2. The structure is subjected to load P = −100 kips at node 2 and node 4. The design variables are the bar element cross-section areas. The cross-section areas (in2) are chosen from the list: 1.62, 1.80, 1.99, 2.13, 2.38, 2.62, 2.63, 2.88, 2.93, 3.09, 3.13, 3.38, 3.47, 3.55, 3.63, 3.84, 3.87, 3.88, 4.18, 4.22, 4.49, 4.59, 4.80, 4.97, 5.12, 5.74, 7.22, 7.97, 11.5, 13.5, 13.9, 14.2, 15.5, 16.0, 16.9, 18.8, 19.9, 22.0, 22.9, 26.5, 30.0, 33.5. The modulus of elasticity and material density are 104 ksi and 0.1 lb/in3. The allowable stress is 25 ksi for both tension and compression stress in each member. The allowable displacement of nodes in x and y directions is 2 in. 4.2. Twenty-five-bar space truss The truss layout is illustrated in Fig. 3. The material density equals to 0.1 lb/in3 and modulus of elasticity equals to 104 ksi. The cross-section areas divided into eight An evolutionary-based optimization algorithm for truss sizing design 311 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 360 in 360 in 360 in Fig. 2. Ten-bar truss layout -100 -50 0 50 100 -100 -50 0 50 100 0 20 40 60 80 100 120 140 160 180 200 9 24 20 16 5 6 17 2 4 8 11 7 13 23 4 5 3 21 1 18 2 6 25 12 9 10 14 8 10 1 3 19 15 22 7 Fig. 3. Twenty-five-bar truss layout member groups. The displacement constraints require that the maximum displacements at nodes 1 and 2 be limited to 0.35 in, in both the x and y directions. The constraints for stress are 40 ksi for both tension and compression stress in each member. The loading data is listed in Tab. 1. The cross-section areas (in2) are chosen from the list: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.8, 3.0, 3.2, 3.4. Table 1. Loading conditions for 25-bar truss Condition Node Fx (kips) Fy (kips) Fz (kips) 1 1 1 -10 -10 2 0 -10 -10 3 0.5 6 0.6 4.3. Seventy-two-bar space truss The truss layout is depicted in Fig. 4. The cross-section areas are divided into six- teen member groups and are chosen from the list of 25 options: 0.1, 0.2, . . . , 2.5 in2. The constraints involve a maximum allowable displacement of 0.25 in at the nodes 1 to 16 along the x and y directions, and a maximum allowable stress in each member of 25 ksi. The density of the material is 0.1 lb/in3and the modulus of elasticity is equal to 104ksi. Two load cases are given in Tab. 2. The proposed algorithm was used to solve each problem with 20 optimization runs. Finite-element method was applied to calculate stress in truss members and dis- placements at nodes. The algorithm and analysis program are implement in MATLAB 7 R2012a environment. The parameter setting in the considered optimization problems is: the population size NP = 30, the rate of global search p = 0.2, i.e. using 6 solutions 312 Pham Hoang Anh 2 6 10 14 18 9 5 1 13 17 (a) 0 20 40 60 80 100 120 0 50 100 0 10 20 30 40 50 60 3 7 3 10 8 15 14 9 7 8 4 4 18 9 2 2 6 17 12 6 16 13 11 5 1 5 1 (b) Fig. 4. 72-bar truss layout: (a) numbering scheme; (b) element numbering for first story Table 2. Loading conditions for 72-bar truss Load case Node Fx (kips) Fy (kips) Fz (kips) 1 17 5 5 -5 2 17 0 0 -5 18 0 0 -5 19 0 0 -5 20 0 0 -5 for global search. The termination condition is when the number of iterations exceeds a predefined value, which is 100 for the ten-bar truss and the 25-bar truss problems, and 200 for the 72-bar truss problem. 5. RESULTS 5.1. Ten-bar truss Optimization results obtained by the proposed algorithm are compared with some most recent metaheuristics, including MBA [11], TLBO [5], ADS [12] and aeDE [14]. The statistical results of optimal weight are given in Tab. 3, including the minimum, the me- dian, the mean, the maximum values and the standard deviation. The proposed algo- rithm results in a minimum design weight of 5490.74 lb for the truss, which is the same as results by aeDE, ADS and TLBO and lighter than that of MBA. On average of 20 runs, the mean weight of the proposed algorithm is 5504.0206 lb, which is better than those of ADS and MBA and as good as that of TLBO and aeDE. More importantly, the pro- posed algorithm exhibits improved computational efficiency when compared to aeDE, TLBO and MBA. The average number of structural analyses required by the proposed An evolutionary-based optimization algorithm for truss sizing design 313 algorithm to converge is 1669. In the best run, the proposed algorithm can find the min- imum designed weight (5490.7379 lb) within 1377 truss analyses. In 100 iterations, there are on average approximately 1893 actual calls for constraint evaluation, i.e. about 1107 structural analyses are skipped. Table 3. Comparison on optimal weights (lb) of 10-bar truss Size of members (in2) This paper aeDE [14] ADS [12] TLBO [5] MBA [11] 1 33.5 33.5 33.5 33.5 30.00 2 1.62 1.62 1.62 1.62 1.62 3 22.9 22.9 22.9 22.9 22.90 4 14.2 14.2 14.2 14.2 16.90 5 1.62 1.62 1.62 1.62 1.62 6 1.62 1.62 1.62 1.62 1.62 7 7.97 7.97 7.97 7.97 7.97 8 22.9 22.9 22.9 22.9 22.90 9 22 22 22 22 22.90 10 1.62 1.62 1.62 1.62 1.62 Min. weight (lb) 5490.7379 5490.7379 5490.74 5490.74 5507.758 Median weight (lb) 5494.4676 - - - - Mean weight (lb) 5504.0206 5502.623 5539.97 5503.21 5527.296 Max. weight (lb) 5538.3529 5549.204 5591.43 - 5536.965 Standard deviation 16.0528 20.780 35.86 20.33 11.38 Number of analyses 1377 2380 1000 5183 3600 0 20 40 60 80 100 5000 5500 6000 6500 7000 7500 8000 8500 Number of Iterations W e ig h t (l b ) mean best worst Fig. 5. Typical convergence history of ten-bar truss Fig. 5 shows the typical convergence history of the proposed algorithm for this truss design problem, including the mean of all runs, the best run and the worst run. 314 Pham Hoang Anh 5.2. Twenty-five-bar truss Tab. 4 gives the statistical results of optimum weight of the 25-bar truss by the proposed algorithm over 20 runs. The best truss design developed by the proposed al- gorithm weighs 484.3286 lb, which is lighter than the design by aeDE [14], MBA [11] and TLBO [5]. The proposed algorithm also uses a significantly smaller number of structural analyses than MBA and TLBO. The proposed algorithm requires an average of approxi- mately 1469 structural analyses to converge to a design, while values reported by TLBO and MBA are 4910 and 2150, respectively. In the best run, the proposed algorithm can find the minimum designed weight within 998 truss analyses. In 100 iterations, the algo- rithm skips about 1159 constraint evaluations. Table 4. Comparison on optimal weights (lb) of 25-bar truss Size of members (in2) This paper aeDE [14] TLBO [5] MBA [11] 1 0.1 0.1 0.1 0.1 2-5 0.4 0.3 0.3 0.3 6-9 3.4 3.4 3.4 3.4 10-11 0.1 0.1 0.1 0.1 12-13 2.2 2.1 2.1 2.1 14-17 1 1 1 1 18-21 0.4 0.5 0.5 0.5 2-25 3.4 3.4 3.4 3.4 Min. weight (lb) 484.3286 484.854 484.854 484.854 Median weight (lb) 484.3286 - - - Mean weight (lb) 484.4075 485.014 484.91 484.885 Max. weight (lb) 485.3797 486.100 - 485.048 Standard deviation 0.2572 0.273 0.17 7.2E-02 Number of analyses 998 1440 4910 2150 0 20 40 60 80 100 450 500 550 600 650 700 750 Number of Iterations W e ig h t (l b ) mean best worst Fig. 6. Typical convergence history of twenty-five-bar truss An evolutionary-based optimization algorithm for truss sizing design 315 Fig. 6 shows the typical convergence history of the proposed algorithm for this truss design problem, including the mean of all runs, the best run and the worst run. 5.3. Seventy-two-bar truss The statistical optimization results of the proposed algorithm for 72-bar truss are given in Tab. 5, together with results reported by CBO [8], improved MBA [10], WCA [10] and improved magnetic charged system search (IMCSS) [18]. The best design obtained by the proposed algorithm is 385.5427 lb, which is identical to the results given by the other methods. However, the proposed algorithm has mean optimum weight of 386.5024 lb, which is a little bit higher than that of WCA and IMBA. The proposed algorithm is also more computationally efficient than the other algorithms. On average, approximately 2613 structural analyses are required by the proposed algorithm to converge and the best run requires 2158 structural analyses. The average number of structural analyses skipped in 200 iterations is 2767. Table 5. Comparison on optimal weights (lb) of 72-bar truss Size of members (in2) This paper CBO [8] IMCSS [18] IMBA [10] WCA [10] 1-4 1.9 1.9 2 1.9 1.9 5-12 0.5 0.5 0.5 0.5 0.5 13-16 0.1 0.1 0.1 0.1 0.1 17-18 0.1 0.1 0.1 0.1 0.1 19-22 1.4 1.4 1.3 1.4 1.4 23-30 0.5 0.5 0.5 0.5 0.5 31-34 0.1 0.1 0.1 0.1 0.1 35-36 0.1 0.1 0.1 0.1 0.1 37-40 0.5 0.5 0.5 0.5 0.5 41-48 0.5 0.5 0.5 0.5 0.5 49-52 0.1 0.1 0.1 0.1 0.1 53-54 0.1 0.1 0.1 0.1 0.1 55-58 0.2 0.2 0.2 0.2 0.2 59- 66 0.6 0.6 0.6 0.6 0.6 67-70 0.4 0.4 0.4 0.4 0.4 71-72 0.6 0.6 0.6 0.6 0.6 Min. weight (lb) 385.5427 385.54 385.54 385.542 385.542 Median weight (lb) 386.5368 - - - - Mean weight (lb) 386.5024 401 - 385.765 385.842 Max. weight (lb) 387.9427 460.98 - 387.942 386.800 Standard deviation 0.9965 16.99 - 0.41 0.55 Number of analyses 2158 4500 3625 5750 3200 316 Pham Hoang Anh Fig. 7 shows the typical convergence history of the proposed algorithm for this truss design problem, including the mean of all runs, the best run and the worst run. 0 50 100 150 200 300 400 500 600 700 800 900 Number of Iterations W e ig h t (l b ) mean best worst Fig. 7. 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