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Optimization of 3d trusses with adaptive approach in genetic algorithms
Engineering Structures 28 (2006) 1019–1027 www.elsevier.com/locate/engstruct
Optimization of 3d trusses with adaptive approach in genetic algorithms Vedat To˘gan ∗ , Ays¸e T. Dalo˘glu Karadeniz Technical University, Department of Civil Engineering, 61080 Trabzon, Turkey Received 2 February 2005; received in revised form 14 November 2005; accepted 16 November 2005 Available online 18 January 2006
Abstract This paper discusses the adaptive approach in genetic algorithms (GAs). It is tried to show how the adaptive approach affects the performance of GAs, suggesting some improvements in both the penalty function, and mutation and crossover. A strategy is also considered for member grouping to reduce the size of the problem. Some practical design of space truss examples taken from technical literature are optimized by the algorithm suggested in the current work. Design constraints such as displacement, tensile stress and stability given by national specifications are incorporated and the results are compared with the ones obtained by previous studies. It is concluded that the member grouping together with the adaptive approach increase the probability of catching the global solution and enhance the performance of GAs. c 2005 Elsevier Ltd. All rights reserved. Keywords: Adaptive approach; Optimization; Space truss; Genetic algorithms; Structural optimization
1. Introduction Due to the fact that material cost is one of the major factors in the construction of a building, it is preferable to reduce it by minimizing the weight or volume of the structural system. All of the methods used for minimizing the volume or weight intend to achieve an optimum design having a set of design variables under certain design criteria. It is necessary to understand the characteristics of the problem to select an appropriate optimization method for structural design. The important characteristic of structural design optimization is that the solution sought is the global optimal solution [1] and the design variables are discrete and must be chosen from a predetermined set. The genetic algorithm (GA) that differs from other classical optimization in four ways [2] is a part of evolutionary computational technique and probabilistic and global search method. Due to these advantages, the GA has been preferred in wide ranges of optimization problems among researchers [3–9]. In order to apply the genetic algorithm, a population of solutions within a search space is initialized in contrast to the traditional optimization methods that start from a single point solution. The population can be viewed as points in ∗ Corresponding author. Tel.: +90 462 377 2671; fax: +90 462 377 2606.
E-mail address: [email protected] (V. To˘gan). c 2005 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2005.11.007
the search space of all solutions to the optimization problem. Each individual in population has a fitness value defined by a fitness function. Then the artificial evolution processes, called the genetic loop, which mimic natural evolution are applied to produce new candidate solutions. At the end of the process, the newly created generation replaces the previous generation and revolution is repeated until a satisfying solution to the problem is obtained ensuring certain design criteria are satisfied or a maximum number of generations are reached. In this study a new adaptive penalty scheme and adaptive mutation and crossover are proposed. A strategy is also adopted for member grouping. Thus, the intention is to be protected from becoming stuck on a local optimum, and to get close to the global optimum instead. 2. Adaptive penalty scheme It is well-known that the GA is an unconstrained optimization method and it cannot explicitly handle constraints of the optimization problem. Therefore, the objective function of a structural optimization problem involves the penalty function which penalizes the design variables depending on the degree of violation of the constraints. Since there is not a unique way to define the penalty term, different forms of the penalty functions have been considered in the literature [10–15]. In all penalty schemes, the degree of penalty depends on the values
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of the various coefficients treated as pre-defined constants during the calculation of the penalty function. Nanakorn and Meesomklin [1] proposed a new adaptive penalty function that will be able to adjust itself automatically during the evolutionary process in such a way that the desired degree of penalty is always obtained. Erbatur et al. [14] modified the penalty function presented by Joines and Houck [16] by normalizing the constraints. Therefore, the penalty given to the individual becomes very small [14] when the power of the constraints is imposed. Chen and Rajan [15] made some enhancements in the simple GA in order to increase the efficiency, reliability and accuracy of the methodology for codebased design of structures. They proposed an algorithm where the penalty weight is computed automatically and adjusted in an adaptive manner. Thus far the coefficients given to the optimization problem with the GA are pre-defined and values of these coefficients are obtained by trail and error in most cases. The degree of penalty can be controlled by varying the values of the penalty parameters. It is impossible to judiciously select appropriate values for them. Even though, in common practice, one value is used for all coefficients, which significantly simplifies situation, the appropriate value of this one coefficient is still not obvious [1]. One of the main objectives of this work is to propose an adaptive penalty scheme that will be able to adjust itself automatically during the genetic process. Mathematical formulation of the penalty scheme is presented as follows, Φ(X) = F(X)(1 + penalty) g(i ) ≥ gave penalty = (gmax + g(i ))/(gmax − gave ) g(i ) < gave penalty = (gave + g(i ))/(gave − gmin ) penalty = 0 g(i ) = 0 i = 1, . . . , n.
(1) (2a) (2b) (2c)
In Eqs. (1) and (2), X is the vector of design variables, F(X) is the objective function for minimum volume, g(i ) is the i th individual of unconstraint on the structural response, n represents the number of constraints, and gmax , gmin , and gave represent, respectively, maximum, minimum, and average violation value of the generation. Finally, Φ(X) is the modified objective function. The formulation of the unconstrained optimization problem is based on the violations of normalized constraints. The adaptive penalty function given in Eq. (2) does not include a pre-defined coefficient or specified coefficient by trial and error. Also magnitudes of the violations are not characterized by a static rate for both near feasible and infeasible solutions. With the expressions in Eq. (2), instead of penalizing all of the infeasible solutions with the same rate, when the level of the violation of infeasible solution tends to get bigger, the magnitude of the penalty tends to get heavier. Thus the proposed penalty scheme is an adaptive approach and it adjusts itself from individual to individual and from generation to generation. It is known that the GA evolves a population of potential solutions for an optimization problem. The penalty functions presented in technical literature so far do not include the population, whereas, in this study, it is possible to establish a relationship between the penalty scheme and the population.
It is shown in Eq. (2) that the magnitude of the penalty increases as the violation value gets closer to gmax . On the other hand it decreases as the violation value gets closer to gmin . Thus, some infeasible individuals that are close to the feasible region in the search space will not disappear through the penalty scheme, and they will find a chance to survive. This may sustain the capacity of finding the global solution of the design problem of the GA. In the GA, each solution in the population gets a fitness value after the penalty scheme is processed. And then the mating pool is formed of the solutions that have the average fitness value or higher. Solutions collected in the mating pool are imposed in a process known as “Structural Information Exchange”. This process is performed by genetic operators, such as crossover, mutation, and elitism. Without an operator of this type some possibly important regions of the search space may never be explored [12]. Due to their power on GA functioning, various types of genetic operators have been proposed. In addition to a new penalty scheme, adaptive mutation and crossover operators are proposed to obtain a global optimum or to get close to it. 3. Adaptive mutation and crossover The genetic operators are applied to produce new candidate solutions or a better solution than previous one. New genetic operators or some suggestions to improve the previous ones are aimed to increase the performance of the GA. However, both in the simple GA and in improved versions of the GA, the crossover and mutation operators that generally take more attention than the other genetic operators are applied with predefined rates that are imposed on the algorithm by the designer. Although both the choice of mutation ( pm ) and crossover ( pc ) probability critically affect the performance of the GA, there are no fixed probabilities of these parameters. pm and pc are used with a specified interval in the literature. Traditional crossover and mutation operators are based on a randomization mechanism, i.e., generating a cut point, and determining the position of the bit shifted by mutation of the solution. But this is not the case in natural evaluation which is mimicked by the GA. Actually renewing the bits of the solution is dynamic or adaptive, but not random. The slightly modified adaptive probabilities of crossover and mutation given by Srinivas and Patnaik [17] are used in the study to choose the probability of mutation and crossover according to the fitness value of the solutions and to relieve the user. The modified expression for pm and pc are as follows: f ≥ f ave pm = 0.5( f max − f )/( f max − f ave ) f < f ave pm = ( f ave − f )/( f ave − fmin ) f ≥ f ave pc = ( f max − f )/( f max − f ave ) f < f ave . pc = 1.0
(3a) (3b) (3c) (3d)
Here, f is the fitness of an individual, f ave the average fitness value of the population, and f max and f min the maximum and minimum fitness value of the population respectively. f is the larger of the fitness values of the solutions to be crossed.
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Srinivas and Patnaik [17] disrupted solutions with a constant probability of mutation. In this study, solutions were disrupted depending on the fitness value by Eq. (3b). Also, f represents the solution with the lower fitness value in the study, unlike the one in the study by Srinivas and Patnaik [17]. Therefore the crossover will be applied between the pairs having the better fitness values. For adaptive mutation, design variables in an individual are arranged according to the level of violation of the constraints. And then the design variables are renewed by the mutation rate starting with the most violated one. Hence, mutation is not done randomly in the algorithm. It is also possible that information exchange between the pairs can be done with flexible points. Crossover points can be varied from 1 to the string length of the solution by adaptive crossover. With the adaptive approach, the probabilities of mutation and crossover are calculated by multiplying Eq. (3) with the string length of each individual in the generation. Probabilities stating the number of design variables to be disrupted for an individual vary according to the fitness value of the individual. The elitist model is a strategy that carries over the best individual in a generation unchanged into the next generation. This model was used to increase the convergence capacity of the GA. Since adaptive operators explicitly protect the best solutions from disappearing, there is no need to use the elitist model in the algorithm. Value (decimal) encoding is also adopted to represent the design variables in the design process. Generally, the structural members are grouped before the initial design at the beginning of the optimization process. In this study, members are grouped after a preliminary analysis. 4. Member grouping strategies It is not possible to float the areas of all members as design variables if the number of members becomes very large since in the case of GA-based optimization this leads to very large string lengths, which delays convergence and precludes useful exchange of information [13]. Members are grouped to reduce the search space of the design problem. However, member grouping done a priori might not lead to accurate grouping [18]. In the present work, a strategy for member grouping is adopted. To implement this strategy, the same cross-sections are assigned for all the structural members first. Then the analysis of the structure is performed using these initial areas for each load cases. Following the static analysis, the entire range of internal forces is divided into several ranges both for tension and compression members. Each member of the truss is placed into different groups according to the magnitude of the axial force of the member. Both Sudarshan [18] and Krishnamoorthy et al. [13] used the member grouping and re-grouping strategy in the design algorithms. Sudarshan [18] applied the member grouping and re-grouping strategy under the some load combination (e.g. Dead Load + Live Load and Dead Load + Wind Load) and two load cases. Krishnamoorthy et al. [13] applied a member grouping strategy for the single load case for space trusses.
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Fig. 1. 112-bar trussed steel dome.
5. Application examples The optimum design of several space trusses are tested by implemented improvements. A 112-bar steel dome is selected as the first example, and it is taken from the literature so that a comparison can be carried out between the coded program and selected work. Then a roof of a 200-bar and a 244-bar transmission tower is optimized to show the efficiency of the algorithm proposed. 5.1. Example 1 The 112-bar steel dome shown in Fig. 1 was already designed by Saka [19] using an optimality criteria approach. Parameters of the optimization process are taken as they are given in the reference study. While the allowable tensile stress and modulus of elasticity, E, are taken as 165 N/mm2 , 2.1 × 105 N/mm2 respectively, the permissible compressive stress is computed according to AISC-ASD. Areas of pipe sections are considered as continuous design variables by Saka [19]. In this study, the properties of pipe sections are taken from AISC (1983). The space truss is also solved by Erbatur et al. [14] using the GA with the cross-sections taken from Turkish codes. Optimum design of the space truss in Fig. 1 is performed in three different ways. First, the same member groups used by Saka [19] are incorporated in the optimum design. Then, the member groups are fixed at one for tension and one for compression following a preliminary analysis as presented in the study, and optimum design is performed. Finally, the member groups are fixed at one for the members in tension and two for the members in compression and the optimization process is repeated. Table 1 summarizes the results for all three choices above. The values obtained in the current study and
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Fig. 2. Variation of violation and static penalty for the 40th individual.
Fig. 3. Variation of violation and adaptive penalty for the 40th individual. Table 1 Results after optimization for Example 1 Parameter
Value
Population size Number of generation*
60 74
Optimized volume at two groups
562 747 cm3
Optimized volume at two groups after the preliminary analysis
500 479 cm3
Optimized volume at three groups after the preliminary analysis* Maximum deflection
422 884 cm3 19.851 mm (node 17)
Table 2 Comparison of results for Example 1 Volume (cm3 ) Saka (1990) 444 700 Erbatur et al. (2000) 435 724 This study 562 747 500 479 422 884
Design variables (mm2 ) A1 A2 A3 A4 714
778
707
557
667
954 515 412
954 954 690
1096
707
A5
A6
A7
707
523
1120
the ones reported by Saka [19] and Erbatur et al. [14] are presented in Table 2. Since Saka [19] used an optimality criteria approach, none of the standard sections available in practice can be included in the study because the design variables are considered as continuous not discrete. The volume of the truss obtained in the current study is bigger than those obtained by
Saka [19] in the first two optimization processes. However, when three groups for the members are adopted, the volume of the truss gets smaller than the previously reported results by Saka [19] and Erbatur et al. [14]. Fig. 2 shows the variations of the violations for individual, average violation of generations, and penalties in which the values are assigned by the penalty function proposed by Rajeev and Krishnamoorthy [10]. Fig. 3 represents the variations of the violations of individual, average violation of generations, and penalties which the values are obtained by the penalty function adopted in the current study. And the variations of the maximum, minimum, and average violations of the generations are presented in Fig. 4. Studying Figs. 2–4 it can easily be seen that no violation is observed after the 13th generation. It can be seen in Figs. 2–4 that during the optimization process an individual has a violation under the average violation of generation or slightly over. For this individual, values of penalty are assigned by the penalty scheme proposed in this study depending on the violation positions of individuals in the generation. However, the magnitude of the violation is not considered by Rajeev and Krishnamoorthy [10] for any of the individuals in the generation. The values of the penalty assigned to an individual by the penalty scheme that is proposed both by Rajeev and Krishnamoorthy [10], Fig. 2, and in the current study, Fig. 3, are rather different from each other, especially at the 11th generation. The difference between the two penalty values is because of static and adaptive approaches. In the first penalty scheme the value of the penalty assigned to an individual is too heavy when all individuals in the generation are taken into consideration. However, due to the fact that all individuals in the generation are considered by
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Fig. 4. Variation of violation of generation.
Fig. 5. Violation of displacement and stress with number of generations for Example 1.
the adaptive approach, all the penalty values assigned for individuals are balanced. Therefore, this penalty value is lighter than the first one. It is shown in Fig. 5 that displacement constraints are dominant over stress constraints in the design since the displacement constraints are violated while the stress constraints are not. All of these imply that the strategies presented in the current study make the design more efficient and accurate. 5.2. Example 2 The space truss in Fig. 6 has 200 bars. The cross-sections of members are collected into three groups. One of the groups contains the bottom chord members. Diagonals are grouped together as another one, and finally top chord members are collected in the third group. Circular hollow sections given in Turkish specification are adopted for the members of the space truss in the study. The allowable compressive stress for each member is computed depending on the Turkish design code (see Appendix). The top chord joints of the space truss are subjected to vertical loading of 13.5 kN and the displacements of the top chord joints are restricted to 20 mm. The allowable tensile stress is taken as 150 N/mm2 and the modulus of elasticity is 210 kN/mm2. In this example, the optimization of the space truss is first performed by using three groups imposed a priori. Then the optimal volume of the truss is obtained with three groups of members that were assumed as one for tension and two for compression members after a preliminary analysis. Table 3 shows the optimum volume for each trial and the maximum deflection of the truss. Fig. 7 shows how the adaptive mutation and crossover affect the solution depending on the fitness value. It is shown in Fig. 7
Fig. 6. Topology of 200-bar space truss.
that the effect of the probabilities of both adaptive operators gets less as the fitness value of the solution increases. Also, with
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Fig. 7. History for the variation rate of mutation and crossover with generations.
Fig. 8. Violation of displacement and stress with number of generations for Example 2. Table 4 Results for different strategies
Table 3 Results for Example 2 Design variables (mm2 )
Strategy
Design variables (mm2 )
Optimized volume at three groups
818 400 cm3
A1 = 819; A2 = 1552; A3 = 1552
Optimized volume at three groups after the preliminary analysis* Maximum deflection
409 880 cm3
A1 = 306; A2 = 819; A3 = 1552
First Second Third Fourth
A1 A1 A1 A1
Parameter
Value
Population size Number of generation*
60 84
11.351 mm (at the upper chord)
the adaptive operators, the number of iterations is much smaller and the volume of the system is less than those designed by other operators such as traditional crossover and mutation [20]. The variation of violation of the displacements and the stress with the number of generation are plotted in Fig. 8. It can be seen that the deflection constraints are not dominant over stress constraints in the design because the displacement constraints imposed at the top chord joints are not violated or slightly violated while the stress constraints are rather violated, and the optimum volume of the space truss is obtained as the stresses of the members are getting closer to the value of allowable stress. Example 2 is also optimized with and without using the four different strategies in order to show the effect of each strategy. For the first strategy, the penalty function given by Rajeev and Krishnamoorthy [10] is used along with single point crossover and static mutation. The penalty function given in the present study and single point crossover and static mutation are imposed in the second strategy. The penalty function given by Rajeev and Krishnamoorthy [10] and the adaptive
= 819; = 819; = 819; = 819;
A2 A2 A2 A2
= 1552; = 1552; = 1552; = 1552;
A3 A3 A3 A3
= 1552 = 1552 = 1552 = 1552
Number of generation
Opt. Volume (cm3 )
73 56 67 54
818 400 818 400 818 400 818 400
approach in mutation and crossover operators suggested in the present study are included in the third strategy. And finally the improvements in both the penalty function and mutation and crossover suggested in the current study are incorporated in the last one. The results obtained in each step using the strategies as explained above are illustrated in Table 4. Although the same value of the optimum volume is reached for all cases, the number of generations is different for each run. The convergence is obtained at generation 54 when the suggestions proposed in the current study are imposed in the algorithm. None of the designs mentioned above includes the member grouping strategy. If the optimum design is performed with the first strategy above incorporating member grouping strategy as well, the optimum volume is obtained as 409 880 cm3 at generation 93. These show that the improvements presented in the current study are effective and enhance the performance of the GA. 5.3. Example 3 Another space truss, a 244-bar transmission tower shown in Fig. 9, is examined as the final design problem to demonstrate the efficiency of the algorithm. Members of the transmission
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Table 6 Results for Example 3 Design variables (mm2 )
Parameter
Value
Population size Number of generation*
80 174
Optimized volume at 26 groups
920 050 cm3
Opt. Vol. at three groups after the preli. analysis*
1561 445 cm3
A1 = 2812; A2 = 3064; A3 = 7096
Opt. Vol. at four groups after the preli. analysis* Maximum deflection
1377 851 cm3
A1 = 3780; A2 = 2135; A3 = 2696; A4 = 7096
14.969 mm (node 17)
Fig. 9. A 244-bar transmission tower. Table 5 Load cases and displacement bounds for Example 3 Load case
1
2
Joint number 1 2 17 24 25 1 2 17 24 25
Loading (kN) x
z
Displacement limitations (mm) x z
10 10 35 175 175 0 0 0 0 0
−30 −30 −90 −45 −45 −360 −360 −180 −90 −90
45 45 30 30 30 45 45 30 30 30
15 15 15 15 15 15 15 15 15 15
tower are initially collected into 26 groups as given by Saka [19]. The value of the modulus of elasticity is taken as 210 kN/mm2 . The allowable value of 140 N/mm2 is employed for tensile stresses and the formulation of buckling obeying AISC-ASD is considered for compressive stresses. The load cases considered and the bounds imposed on the displacements are taken from the study by Saka [19] and are shown in Table 5. The single angle cross-sections are adopted as design variables and they are picked form AISC. Since space trusses are large structural systems, it is very difficult to optimize the area of individual members if the member groups become very large. Hence, member grouping is adopted to reduce the size of the problem which increases due to very large string length. On the other hand, according
to Sudarshan [18] the member groups assumed in earlier optimization strategies like those developed by Adeli and Kamal [21], and Adeli and Cheng [22] have been assigned a priori and they might not lead to an accurate grouping. For a convenient member grouping, the number of sections in the final set must be as small as possible to make fabrication easier. The designer can specify the maximum number of sections that he wants to have [18]. In the current work, the transmission tower is first designed adopting the groups of members reported by Saka [19]. And then the structure is designed again with the groups of members formed after a preliminary analysis. The structure is designed using three groups at one for tension and two for compression members first, and using four groups at two for tension and two for compression members next. Table 6 summarizes the optimized weight for the best solution obtained from over all runs. The variation of the volume of the tower and the maximum deflection are plotted in Fig. 10, while Fig. 11 illustrates the variation of violation of displacements and stresses with the number of generations. It can be observed that, while 26 groups of members were initially assumed for optimization as given by Saka [19], the final optimized set obtained in the current study consists of only three and four groups subsequently. It is true that the optimal volume reported by Saka [19] is smaller than the optimum volumes obtained in the current study. However, as mentioned previously, the results obtained by Saka [19] do not include any standard section used in the practice and consist of 26 design variables. From the practical point of view, the number of the design variables is too many and this optimization is
Fig. 10. Variation of the volume of truss and maximum displacement with number of generations for Example 3.
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Fig. 11. Violation of displacement and stress with number of generations for Example 3.
not applicable. As mentioned by Sudarshan [18], to make fabrication easy, the designer prefers to work with 5 or 6 different sections. Therefore, the final solution consists of an optimized volume and also the least number of sections to be used. The designs reported in the current work fit rather better compared to the ones reported by Saka [19] for the construction in practice. Studying both Figs. 10 and 11, it can be said that displacement constraints were active during the design process. 6. Observation and conclusions An adaptive improvement is presented to be used to enhance the performance of the GA. If only the traditional genetic parameters such as static penalty function, mutation, and various types of crossover are incorporated for optimum design process, the number of iterations required to reach the optimum solution will be high. Moreover, it might be necessary to run the algorithm many times with different user defined parameters. However, the optimum design problem in the current study is performed without any pre-defined parameters. In the present work, the design problems are also solved with a member grouping formed automatically, unlike the earlier works in which the groups are adopted a priori. It is shown from the design examples that the strategy suggested in the current study gives a practical result especially in the case of large structural systems or when having no idea about member grouping. From the results it is possible to say that the GA is a very effective optimization method and adaptive improvements enhance the performance of the GA. As is well known, the results obtained by the GA do not always guarantee to catch the peak point (global solution) in the search space of the problem. For this reason, one avoids increasing the problem size enormously in order to catch the global solution. However, in spite of this, the solution obtained after the design process is quite close to the near-optimal or optimal solution. The presented algorithm results not only in an optimum volume but also results in a few sections chosen from a pre-determined set that are available in practice. Hence, the solution is feasible and the construction of the structure is easy. Appendix According to Turkish design code (TS648), the permissible compressive stress is calculated as follows:
2π 2 E 5λ2 [1 − (1/2)(λ/λ p )2 ]σ y If λ < λ p , Plastic buckling, σc = n 3 n = 1.5 + 1.2(λ/λ p ) − 0.2(λ/λ p )
If λ > λ p ,
Elastic buckling,
σc =
where σc = allowable compressive stress; σ y = yield stress; E = modulus of elasticity; λ = slenderness ratio; λ p is taken as
2π 2 E/σ y .
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Optimization of 3d trusses with adaptive approach in genetic algorithms Vedat To˘gan ∗ , Ays¸e T. Dalo˘glu Karadeniz Technical University, Department of Civil Engineering, 61080 Trabzon, Turkey Received 2 February 2005; received in revised form 14 November 2005; accepted 16 November 2005 Available online 18 January 2006
Abstract This paper discusses the adaptive approach in genetic algorithms (GAs). It is tried to show how the adaptive approach affects the performance of GAs, suggesting some improvements in both the penalty function, and mutation and crossover. A strategy is also considered for member grouping to reduce the size of the problem. Some practical design of space truss examples taken from technical literature are optimized by the algorithm suggested in the current work. Design constraints such as displacement, tensile stress and stability given by national specifications are incorporated and the results are compared with the ones obtained by previous studies. It is concluded that the member grouping together with the adaptive approach increase the probability of catching the global solution and enhance the performance of GAs. c 2005 Elsevier Ltd. All rights reserved. Keywords: Adaptive approach; Optimization; Space truss; Genetic algorithms; Structural optimization
1. Introduction Due to the fact that material cost is one of the major factors in the construction of a building, it is preferable to reduce it by minimizing the weight or volume of the structural system. All of the methods used for minimizing the volume or weight intend to achieve an optimum design having a set of design variables under certain design criteria. It is necessary to understand the characteristics of the problem to select an appropriate optimization method for structural design. The important characteristic of structural design optimization is that the solution sought is the global optimal solution [1] and the design variables are discrete and must be chosen from a predetermined set. The genetic algorithm (GA) that differs from other classical optimization in four ways [2] is a part of evolutionary computational technique and probabilistic and global search method. Due to these advantages, the GA has been preferred in wide ranges of optimization problems among researchers [3–9]. In order to apply the genetic algorithm, a population of solutions within a search space is initialized in contrast to the traditional optimization methods that start from a single point solution. The population can be viewed as points in ∗ Corresponding author. Tel.: +90 462 377 2671; fax: +90 462 377 2606.
E-mail address: [email protected] (V. To˘gan). c 2005 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2005.11.007
the search space of all solutions to the optimization problem. Each individual in population has a fitness value defined by a fitness function. Then the artificial evolution processes, called the genetic loop, which mimic natural evolution are applied to produce new candidate solutions. At the end of the process, the newly created generation replaces the previous generation and revolution is repeated until a satisfying solution to the problem is obtained ensuring certain design criteria are satisfied or a maximum number of generations are reached. In this study a new adaptive penalty scheme and adaptive mutation and crossover are proposed. A strategy is also adopted for member grouping. Thus, the intention is to be protected from becoming stuck on a local optimum, and to get close to the global optimum instead. 2. Adaptive penalty scheme It is well-known that the GA is an unconstrained optimization method and it cannot explicitly handle constraints of the optimization problem. Therefore, the objective function of a structural optimization problem involves the penalty function which penalizes the design variables depending on the degree of violation of the constraints. Since there is not a unique way to define the penalty term, different forms of the penalty functions have been considered in the literature [10–15]. In all penalty schemes, the degree of penalty depends on the values
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of the various coefficients treated as pre-defined constants during the calculation of the penalty function. Nanakorn and Meesomklin [1] proposed a new adaptive penalty function that will be able to adjust itself automatically during the evolutionary process in such a way that the desired degree of penalty is always obtained. Erbatur et al. [14] modified the penalty function presented by Joines and Houck [16] by normalizing the constraints. Therefore, the penalty given to the individual becomes very small [14] when the power of the constraints is imposed. Chen and Rajan [15] made some enhancements in the simple GA in order to increase the efficiency, reliability and accuracy of the methodology for codebased design of structures. They proposed an algorithm where the penalty weight is computed automatically and adjusted in an adaptive manner. Thus far the coefficients given to the optimization problem with the GA are pre-defined and values of these coefficients are obtained by trail and error in most cases. The degree of penalty can be controlled by varying the values of the penalty parameters. It is impossible to judiciously select appropriate values for them. Even though, in common practice, one value is used for all coefficients, which significantly simplifies situation, the appropriate value of this one coefficient is still not obvious [1]. One of the main objectives of this work is to propose an adaptive penalty scheme that will be able to adjust itself automatically during the genetic process. Mathematical formulation of the penalty scheme is presented as follows, Φ(X) = F(X)(1 + penalty) g(i ) ≥ gave penalty = (gmax + g(i ))/(gmax − gave ) g(i ) < gave penalty = (gave + g(i ))/(gave − gmin ) penalty = 0 g(i ) = 0 i = 1, . . . , n.
(1) (2a) (2b) (2c)
In Eqs. (1) and (2), X is the vector of design variables, F(X) is the objective function for minimum volume, g(i ) is the i th individual of unconstraint on the structural response, n represents the number of constraints, and gmax , gmin , and gave represent, respectively, maximum, minimum, and average violation value of the generation. Finally, Φ(X) is the modified objective function. The formulation of the unconstrained optimization problem is based on the violations of normalized constraints. The adaptive penalty function given in Eq. (2) does not include a pre-defined coefficient or specified coefficient by trial and error. Also magnitudes of the violations are not characterized by a static rate for both near feasible and infeasible solutions. With the expressions in Eq. (2), instead of penalizing all of the infeasible solutions with the same rate, when the level of the violation of infeasible solution tends to get bigger, the magnitude of the penalty tends to get heavier. Thus the proposed penalty scheme is an adaptive approach and it adjusts itself from individual to individual and from generation to generation. It is known that the GA evolves a population of potential solutions for an optimization problem. The penalty functions presented in technical literature so far do not include the population, whereas, in this study, it is possible to establish a relationship between the penalty scheme and the population.
It is shown in Eq. (2) that the magnitude of the penalty increases as the violation value gets closer to gmax . On the other hand it decreases as the violation value gets closer to gmin . Thus, some infeasible individuals that are close to the feasible region in the search space will not disappear through the penalty scheme, and they will find a chance to survive. This may sustain the capacity of finding the global solution of the design problem of the GA. In the GA, each solution in the population gets a fitness value after the penalty scheme is processed. And then the mating pool is formed of the solutions that have the average fitness value or higher. Solutions collected in the mating pool are imposed in a process known as “Structural Information Exchange”. This process is performed by genetic operators, such as crossover, mutation, and elitism. Without an operator of this type some possibly important regions of the search space may never be explored [12]. Due to their power on GA functioning, various types of genetic operators have been proposed. In addition to a new penalty scheme, adaptive mutation and crossover operators are proposed to obtain a global optimum or to get close to it. 3. Adaptive mutation and crossover The genetic operators are applied to produce new candidate solutions or a better solution than previous one. New genetic operators or some suggestions to improve the previous ones are aimed to increase the performance of the GA. However, both in the simple GA and in improved versions of the GA, the crossover and mutation operators that generally take more attention than the other genetic operators are applied with predefined rates that are imposed on the algorithm by the designer. Although both the choice of mutation ( pm ) and crossover ( pc ) probability critically affect the performance of the GA, there are no fixed probabilities of these parameters. pm and pc are used with a specified interval in the literature. Traditional crossover and mutation operators are based on a randomization mechanism, i.e., generating a cut point, and determining the position of the bit shifted by mutation of the solution. But this is not the case in natural evaluation which is mimicked by the GA. Actually renewing the bits of the solution is dynamic or adaptive, but not random. The slightly modified adaptive probabilities of crossover and mutation given by Srinivas and Patnaik [17] are used in the study to choose the probability of mutation and crossover according to the fitness value of the solutions and to relieve the user. The modified expression for pm and pc are as follows: f ≥ f ave pm = 0.5( f max − f )/( f max − f ave ) f < f ave pm = ( f ave − f )/( f ave − fmin ) f ≥ f ave pc = ( f max − f )/( f max − f ave ) f < f ave . pc = 1.0
(3a) (3b) (3c) (3d)
Here, f is the fitness of an individual, f ave the average fitness value of the population, and f max and f min the maximum and minimum fitness value of the population respectively. f is the larger of the fitness values of the solutions to be crossed.
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Srinivas and Patnaik [17] disrupted solutions with a constant probability of mutation. In this study, solutions were disrupted depending on the fitness value by Eq. (3b). Also, f represents the solution with the lower fitness value in the study, unlike the one in the study by Srinivas and Patnaik [17]. Therefore the crossover will be applied between the pairs having the better fitness values. For adaptive mutation, design variables in an individual are arranged according to the level of violation of the constraints. And then the design variables are renewed by the mutation rate starting with the most violated one. Hence, mutation is not done randomly in the algorithm. It is also possible that information exchange between the pairs can be done with flexible points. Crossover points can be varied from 1 to the string length of the solution by adaptive crossover. With the adaptive approach, the probabilities of mutation and crossover are calculated by multiplying Eq. (3) with the string length of each individual in the generation. Probabilities stating the number of design variables to be disrupted for an individual vary according to the fitness value of the individual. The elitist model is a strategy that carries over the best individual in a generation unchanged into the next generation. This model was used to increase the convergence capacity of the GA. Since adaptive operators explicitly protect the best solutions from disappearing, there is no need to use the elitist model in the algorithm. Value (decimal) encoding is also adopted to represent the design variables in the design process. Generally, the structural members are grouped before the initial design at the beginning of the optimization process. In this study, members are grouped after a preliminary analysis. 4. Member grouping strategies It is not possible to float the areas of all members as design variables if the number of members becomes very large since in the case of GA-based optimization this leads to very large string lengths, which delays convergence and precludes useful exchange of information [13]. Members are grouped to reduce the search space of the design problem. However, member grouping done a priori might not lead to accurate grouping [18]. In the present work, a strategy for member grouping is adopted. To implement this strategy, the same cross-sections are assigned for all the structural members first. Then the analysis of the structure is performed using these initial areas for each load cases. Following the static analysis, the entire range of internal forces is divided into several ranges both for tension and compression members. Each member of the truss is placed into different groups according to the magnitude of the axial force of the member. Both Sudarshan [18] and Krishnamoorthy et al. [13] used the member grouping and re-grouping strategy in the design algorithms. Sudarshan [18] applied the member grouping and re-grouping strategy under the some load combination (e.g. Dead Load + Live Load and Dead Load + Wind Load) and two load cases. Krishnamoorthy et al. [13] applied a member grouping strategy for the single load case for space trusses.
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Fig. 1. 112-bar trussed steel dome.
5. Application examples The optimum design of several space trusses are tested by implemented improvements. A 112-bar steel dome is selected as the first example, and it is taken from the literature so that a comparison can be carried out between the coded program and selected work. Then a roof of a 200-bar and a 244-bar transmission tower is optimized to show the efficiency of the algorithm proposed. 5.1. Example 1 The 112-bar steel dome shown in Fig. 1 was already designed by Saka [19] using an optimality criteria approach. Parameters of the optimization process are taken as they are given in the reference study. While the allowable tensile stress and modulus of elasticity, E, are taken as 165 N/mm2 , 2.1 × 105 N/mm2 respectively, the permissible compressive stress is computed according to AISC-ASD. Areas of pipe sections are considered as continuous design variables by Saka [19]. In this study, the properties of pipe sections are taken from AISC (1983). The space truss is also solved by Erbatur et al. [14] using the GA with the cross-sections taken from Turkish codes. Optimum design of the space truss in Fig. 1 is performed in three different ways. First, the same member groups used by Saka [19] are incorporated in the optimum design. Then, the member groups are fixed at one for tension and one for compression following a preliminary analysis as presented in the study, and optimum design is performed. Finally, the member groups are fixed at one for the members in tension and two for the members in compression and the optimization process is repeated. Table 1 summarizes the results for all three choices above. The values obtained in the current study and
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Fig. 2. Variation of violation and static penalty for the 40th individual.
Fig. 3. Variation of violation and adaptive penalty for the 40th individual. Table 1 Results after optimization for Example 1 Parameter
Value
Population size Number of generation*
60 74
Optimized volume at two groups
562 747 cm3
Optimized volume at two groups after the preliminary analysis
500 479 cm3
Optimized volume at three groups after the preliminary analysis* Maximum deflection
422 884 cm3 19.851 mm (node 17)
Table 2 Comparison of results for Example 1 Volume (cm3 ) Saka (1990) 444 700 Erbatur et al. (2000) 435 724 This study 562 747 500 479 422 884
Design variables (mm2 ) A1 A2 A3 A4 714
778
707
557
667
954 515 412
954 954 690
1096
707
A5
A6
A7
707
523
1120
the ones reported by Saka [19] and Erbatur et al. [14] are presented in Table 2. Since Saka [19] used an optimality criteria approach, none of the standard sections available in practice can be included in the study because the design variables are considered as continuous not discrete. The volume of the truss obtained in the current study is bigger than those obtained by
Saka [19] in the first two optimization processes. However, when three groups for the members are adopted, the volume of the truss gets smaller than the previously reported results by Saka [19] and Erbatur et al. [14]. Fig. 2 shows the variations of the violations for individual, average violation of generations, and penalties in which the values are assigned by the penalty function proposed by Rajeev and Krishnamoorthy [10]. Fig. 3 represents the variations of the violations of individual, average violation of generations, and penalties which the values are obtained by the penalty function adopted in the current study. And the variations of the maximum, minimum, and average violations of the generations are presented in Fig. 4. Studying Figs. 2–4 it can easily be seen that no violation is observed after the 13th generation. It can be seen in Figs. 2–4 that during the optimization process an individual has a violation under the average violation of generation or slightly over. For this individual, values of penalty are assigned by the penalty scheme proposed in this study depending on the violation positions of individuals in the generation. However, the magnitude of the violation is not considered by Rajeev and Krishnamoorthy [10] for any of the individuals in the generation. The values of the penalty assigned to an individual by the penalty scheme that is proposed both by Rajeev and Krishnamoorthy [10], Fig. 2, and in the current study, Fig. 3, are rather different from each other, especially at the 11th generation. The difference between the two penalty values is because of static and adaptive approaches. In the first penalty scheme the value of the penalty assigned to an individual is too heavy when all individuals in the generation are taken into consideration. However, due to the fact that all individuals in the generation are considered by
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Fig. 4. Variation of violation of generation.
Fig. 5. Violation of displacement and stress with number of generations for Example 1.
the adaptive approach, all the penalty values assigned for individuals are balanced. Therefore, this penalty value is lighter than the first one. It is shown in Fig. 5 that displacement constraints are dominant over stress constraints in the design since the displacement constraints are violated while the stress constraints are not. All of these imply that the strategies presented in the current study make the design more efficient and accurate. 5.2. Example 2 The space truss in Fig. 6 has 200 bars. The cross-sections of members are collected into three groups. One of the groups contains the bottom chord members. Diagonals are grouped together as another one, and finally top chord members are collected in the third group. Circular hollow sections given in Turkish specification are adopted for the members of the space truss in the study. The allowable compressive stress for each member is computed depending on the Turkish design code (see Appendix). The top chord joints of the space truss are subjected to vertical loading of 13.5 kN and the displacements of the top chord joints are restricted to 20 mm. The allowable tensile stress is taken as 150 N/mm2 and the modulus of elasticity is 210 kN/mm2. In this example, the optimization of the space truss is first performed by using three groups imposed a priori. Then the optimal volume of the truss is obtained with three groups of members that were assumed as one for tension and two for compression members after a preliminary analysis. Table 3 shows the optimum volume for each trial and the maximum deflection of the truss. Fig. 7 shows how the adaptive mutation and crossover affect the solution depending on the fitness value. It is shown in Fig. 7
Fig. 6. Topology of 200-bar space truss.
that the effect of the probabilities of both adaptive operators gets less as the fitness value of the solution increases. Also, with
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Fig. 7. History for the variation rate of mutation and crossover with generations.
Fig. 8. Violation of displacement and stress with number of generations for Example 2. Table 4 Results for different strategies
Table 3 Results for Example 2 Design variables (mm2 )
Strategy
Design variables (mm2 )
Optimized volume at three groups
818 400 cm3
A1 = 819; A2 = 1552; A3 = 1552
Optimized volume at three groups after the preliminary analysis* Maximum deflection
409 880 cm3
A1 = 306; A2 = 819; A3 = 1552
First Second Third Fourth
A1 A1 A1 A1
Parameter
Value
Population size Number of generation*
60 84
11.351 mm (at the upper chord)
the adaptive operators, the number of iterations is much smaller and the volume of the system is less than those designed by other operators such as traditional crossover and mutation [20]. The variation of violation of the displacements and the stress with the number of generation are plotted in Fig. 8. It can be seen that the deflection constraints are not dominant over stress constraints in the design because the displacement constraints imposed at the top chord joints are not violated or slightly violated while the stress constraints are rather violated, and the optimum volume of the space truss is obtained as the stresses of the members are getting closer to the value of allowable stress. Example 2 is also optimized with and without using the four different strategies in order to show the effect of each strategy. For the first strategy, the penalty function given by Rajeev and Krishnamoorthy [10] is used along with single point crossover and static mutation. The penalty function given in the present study and single point crossover and static mutation are imposed in the second strategy. The penalty function given by Rajeev and Krishnamoorthy [10] and the adaptive
= 819; = 819; = 819; = 819;
A2 A2 A2 A2
= 1552; = 1552; = 1552; = 1552;
A3 A3 A3 A3
= 1552 = 1552 = 1552 = 1552
Number of generation
Opt. Volume (cm3 )
73 56 67 54
818 400 818 400 818 400 818 400
approach in mutation and crossover operators suggested in the present study are included in the third strategy. And finally the improvements in both the penalty function and mutation and crossover suggested in the current study are incorporated in the last one. The results obtained in each step using the strategies as explained above are illustrated in Table 4. Although the same value of the optimum volume is reached for all cases, the number of generations is different for each run. The convergence is obtained at generation 54 when the suggestions proposed in the current study are imposed in the algorithm. None of the designs mentioned above includes the member grouping strategy. If the optimum design is performed with the first strategy above incorporating member grouping strategy as well, the optimum volume is obtained as 409 880 cm3 at generation 93. These show that the improvements presented in the current study are effective and enhance the performance of the GA. 5.3. Example 3 Another space truss, a 244-bar transmission tower shown in Fig. 9, is examined as the final design problem to demonstrate the efficiency of the algorithm. Members of the transmission
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Table 6 Results for Example 3 Design variables (mm2 )
Parameter
Value
Population size Number of generation*
80 174
Optimized volume at 26 groups
920 050 cm3
Opt. Vol. at three groups after the preli. analysis*
1561 445 cm3
A1 = 2812; A2 = 3064; A3 = 7096
Opt. Vol. at four groups after the preli. analysis* Maximum deflection
1377 851 cm3
A1 = 3780; A2 = 2135; A3 = 2696; A4 = 7096
14.969 mm (node 17)
Fig. 9. A 244-bar transmission tower. Table 5 Load cases and displacement bounds for Example 3 Load case
1
2
Joint number 1 2 17 24 25 1 2 17 24 25
Loading (kN) x
z
Displacement limitations (mm) x z
10 10 35 175 175 0 0 0 0 0
−30 −30 −90 −45 −45 −360 −360 −180 −90 −90
45 45 30 30 30 45 45 30 30 30
15 15 15 15 15 15 15 15 15 15
tower are initially collected into 26 groups as given by Saka [19]. The value of the modulus of elasticity is taken as 210 kN/mm2 . The allowable value of 140 N/mm2 is employed for tensile stresses and the formulation of buckling obeying AISC-ASD is considered for compressive stresses. The load cases considered and the bounds imposed on the displacements are taken from the study by Saka [19] and are shown in Table 5. The single angle cross-sections are adopted as design variables and they are picked form AISC. Since space trusses are large structural systems, it is very difficult to optimize the area of individual members if the member groups become very large. Hence, member grouping is adopted to reduce the size of the problem which increases due to very large string length. On the other hand, according
to Sudarshan [18] the member groups assumed in earlier optimization strategies like those developed by Adeli and Kamal [21], and Adeli and Cheng [22] have been assigned a priori and they might not lead to an accurate grouping. For a convenient member grouping, the number of sections in the final set must be as small as possible to make fabrication easier. The designer can specify the maximum number of sections that he wants to have [18]. In the current work, the transmission tower is first designed adopting the groups of members reported by Saka [19]. And then the structure is designed again with the groups of members formed after a preliminary analysis. The structure is designed using three groups at one for tension and two for compression members first, and using four groups at two for tension and two for compression members next. Table 6 summarizes the optimized weight for the best solution obtained from over all runs. The variation of the volume of the tower and the maximum deflection are plotted in Fig. 10, while Fig. 11 illustrates the variation of violation of displacements and stresses with the number of generations. It can be observed that, while 26 groups of members were initially assumed for optimization as given by Saka [19], the final optimized set obtained in the current study consists of only three and four groups subsequently. It is true that the optimal volume reported by Saka [19] is smaller than the optimum volumes obtained in the current study. However, as mentioned previously, the results obtained by Saka [19] do not include any standard section used in the practice and consist of 26 design variables. From the practical point of view, the number of the design variables is too many and this optimization is
Fig. 10. Variation of the volume of truss and maximum displacement with number of generations for Example 3.
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Fig. 11. Violation of displacement and stress with number of generations for Example 3.
not applicable. As mentioned by Sudarshan [18], to make fabrication easy, the designer prefers to work with 5 or 6 different sections. Therefore, the final solution consists of an optimized volume and also the least number of sections to be used. The designs reported in the current work fit rather better compared to the ones reported by Saka [19] for the construction in practice. Studying both Figs. 10 and 11, it can be said that displacement constraints were active during the design process. 6. Observation and conclusions An adaptive improvement is presented to be used to enhance the performance of the GA. If only the traditional genetic parameters such as static penalty function, mutation, and various types of crossover are incorporated for optimum design process, the number of iterations required to reach the optimum solution will be high. Moreover, it might be necessary to run the algorithm many times with different user defined parameters. However, the optimum design problem in the current study is performed without any pre-defined parameters. In the present work, the design problems are also solved with a member grouping formed automatically, unlike the earlier works in which the groups are adopted a priori. It is shown from the design examples that the strategy suggested in the current study gives a practical result especially in the case of large structural systems or when having no idea about member grouping. From the results it is possible to say that the GA is a very effective optimization method and adaptive improvements enhance the performance of the GA. As is well known, the results obtained by the GA do not always guarantee to catch the peak point (global solution) in the search space of the problem. For this reason, one avoids increasing the problem size enormously in order to catch the global solution. However, in spite of this, the solution obtained after the design process is quite close to the near-optimal or optimal solution. The presented algorithm results not only in an optimum volume but also results in a few sections chosen from a pre-determined set that are available in practice. Hence, the solution is feasible and the construction of the structure is easy. Appendix According to Turkish design code (TS648), the permissible compressive stress is calculated as follows:
2π 2 E 5λ2 [1 − (1/2)(λ/λ p )2 ]σ y If λ < λ p , Plastic buckling, σc = n 3 n = 1.5 + 1.2(λ/λ p ) − 0.2(λ/λ p )
If λ > λ p ,
Elastic buckling,
σc =
where σc = allowable compressive stress; σ y = yield stress; E = modulus of elasticity; λ = slenderness ratio; λ p is taken as
2π 2 E/σ y .
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