INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 1, No 2, 2010

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1 Structural Optimization Using a Novel Genetic Algorithm for Rapid Convergence M. A. Al Shihri Civil Engineering Department, Umm Al Qura University,Makkah, Saudi Arabia ABSTRACT A novel evolutionary algorithm based upon genetic algorithm is presented in this paper which is suitable for a general class of structural optimization problems. The algorithm is applicable for discrete and/or continuous type(s) of design variables. Proposed algorithm has been designed such that it converges rapidly to local optima whenever a local optimum solution is nearby. In each generation, the algorithm selects a chromosome from the population that represents a design close to a local optimum. Further, a new set of chromosomes called Single Digit Chromosome (SDC) are generated having all zero bits except for one. Chromosomes are selected from the population and their binary addition and subtraction are performed with the SDCs. Through a number of test examples it is shown that the proposed algorithm is much robust and reliable as compared to the traditional genetic algorithm. Keywords: Genetic Algorithm, Evolutionary Algorithms, Structural Optimization, SGA, NGA 1. Introduction Evolutionary Algorithms have been successfully used in Structural Optimization as is obvious from the literature search. It is a continuing trend to use these types of algorithms as is reflected by the recently publications that show the application of these techniques to a variety of problems in structural design [1 4] with modifications to the algorithm to suit particular problems. An detailed survey of Evolutionary Algorithms (EAs) in the context of structural optimization has been presented in [5] covering most of the published literature in this area. This is a class of algorithms where ideas have been taken from genetics, evolutionary theory and cellular biology. Even the terminologies used in such algorithms have been taken from biological processes. These algorithms have been successfully applied for the last many years to solve engineering problems especially optimization problems. A number of researchers have used EAs to solve structural optimization problems. Based upon the survey reported in [5], three different classes of algorithms have been developed in order to solve structural optimization problems. These are named as evolution strategies (ES), evolutionary programming (EP) and genetic algorithms (GAs). Further, genetic programming has been used to develop computer programs in order to solve computational tasks. Using a combination of the above strategies, some hybrid models 123

2 have also been developed namely CHC, structured GA, breeder GA, messy GA and many others [6 8]. The general class of optimization problems involves discrete and/or continuous types of design variables. However, majority of the genetic algorithms proposed for structural optimisation are variations of the simple GA (SGA) which suit specific class of problems involving discrete design space [9 17]. Hence, these cannot be applied arbitrarily to a structural optimization problem in general. In order to represent continuous type of design variables with sufficient accuracy, the size of the design space will depend upon the number of significant digits used to represent values of the design variables. It is obvious that this would increase the size of the design space. Hence using SGA for structural optimisation problems involving huge design spaces will require extremely large number of analyses. In addition to the enormous computational cost of solving such problems using SGA, it was observed that the reliability of obtaining the global optima is poor. In order to improve the performance, a number of hybrid approaches have been proposed in the literature. However, they all suffer from being complex and sophisticated. Some examples are, hybrid approaches of mixing genetic algorithms with gradient methods [18], combining genetic algorithms with multi layer neural networks to approximate the structural response [19] and space condensation heuristic for progressively reducing the size of the multidimensional design space being searched [20]. In this paper a novel evolutionary algorithm based upon genetic algorithm is presented which is suitable for a general class of structural optimization problems. The main idea is a novel but simple modification to the existing algorithm such that its implementation remains simple and without the need for any gradient information. Structural optimisation problems with complex constraints require high probability of small variations in the value of each design variable so that the algorithm keeps generating improved designs during the optimisation iterations. It was found that one of the main reasons of the poor performance of SGA is the low probability of the occurrence of a small change in a single design variable. This observation led to the proposed novel operation presented in this paper. This operation artificially produces the smallest possible variation in the values of the design variables in a deterministic way within the framework of the stochastic nature of the genetic algorithms. This operation is specially effective in situations where SGA exhibits premature convergence thus keeping the optimisation progressing and producing improved solutions. This new GA implementation has been named as NGA (Novel Genetic Algorithm). NGA has been described in this paper and its performance is compared with SGA for a set of test problems including algebraic functions, structural optimisation problems with closed form solution of structural response and problems involving finite element analysis. The results indicate significantly improved performance at a lower computational cost with much higher reliability. 124

3 2. Mathematical Description of the Problem Let D vector of design variables in a structural optimisation problem, C(D) cost or the objective function to be minimized, and g i (D) the constraints. The generalized structural optimisation problem may then be stated in the usual form as follows: Minimize C(D) (1) Subject to: g i (D) 0; i =1, N g (2) where, d j. L j d j U j j = 1, N d (3) N g the number of inequality constraints and L j, U j the lower and upper bounds on the N d individual design variables The objective function and the constraints g i may be given as algebraic expressions in closed form or may be obtained by finite element analysis. The mathematical problem described above is translated for genetic algorithms by representing a solution in the search space with a bit string of 0 s and 1 s. Such a bit string called a chromosome is obtained by the binary coding of the design variable vector. Each chromosome has an associated fitness value F based on the objective function to be minimized. A higher fitness value indicates a better solution. The fitness value F of a chromosome can be related to C(D) in a number of ways. In this work the simplest form is used as follows: F = C(D) (4) Constraint handling in genetic algorithms has been investigated in [21]. One of the approaches is to keep only the feasible solutions in the population and discard any solutions that happen to be infeasible. This approach has some advantages like easier implementation and higher probability of offspring being born with greater fitness values. The major disadvantage is that it may be difficult to generate the initial population that has only feasible solutions. Also, if many randomly generated solutions are discarded because of being infeasible, the chances of finding the global optimum diminish. Another alternative, used in this work, is to keep the infeasible solutions in the population as they 125

4 are randomly generated but to apply constraint violation penalties. A penalty function that can be applied to a general class of structural optimisation problems, without any tuning required for a specific problem, was determined through numerical experiments as given in the following equation. where F P = F ( g v ) (Max.{ F, 100}) (5) F P the fitness value after applying the constraint violation penalty and g v the violated constraints only. 3. Design Space In a structural optimization problem, the design variables may be continuous, discrete or mixed. Hence, an approach is needed that would represent both types of variables in order to model the design space. As was mentioned in Section 2.1, problems involving continuous design variables, accuracy is an important consideration. Although, the number of digits available to represent a number is always limited, the maximum accuracy that can be achieved using a digital computer is very high. Of course it is impractical to solve an optimisation problem with the maximum possible accuracy on a digital computer as this will make the design space extremely large inhibiting any optimisation. One solution is to solve such problems with a user specified accuracy that would make the optimisation manageable. Once this accuracy is specified the design space in fact becomes discrete with regular increments to the values of the design variables. For example, a design parameter with a minimum value of and a maximum value of may be specified with an accuracy of 0.001, i.e., the computer implementation of these values would take 0.100, 0.101, 0.102,,0.998, 0.999, The accuracy is in fact equivalent to the minimum increment that is applied to the value of a design variable. Thus for any given design variable, labelled as the i th design variable, the user input will consist of the lower bound on the design variable L i, the upper bound U i and the value of the required accuracy α i. This approach unifies the representation of both the continuous and discrete design variables and can be implemented by determining the values of design variables through the number of increments, K i, applied to the lowest value (the lower bound) of a design variable. Thus the value of a design variable d i can be expressed as follows: d i = L i + K i α i (6) With the above incremental representation of both the discrete and continuous design variables as described above, the chromosome coding of design variables is simplified because there is no fractional part. The number of bits to represent a design variable is based on the number of possible values in the given range for a design variable with a specified increment. Let r i be the range given by U i L i. The number of bits N b required to represent the increment is then given by: 126

5 N b = INT(log 10 (r i +1.0) / log 10 (2.0)) + 1 (7) INT in the above equation is an operator that implies conversion of the expression value into integer by truncation. The bit string value gives the number of increments to be added to the lower bound for determining the value of a design variable in the given range. The complete chromosome is a bit string concatenation of the bit strings of all design variables in the problem. A decoding routine converts the bit string into the numerical values for all design variables. 4. NGA Implementation Compared with SGA, there is an additional operation in NGA known as novel crossover. This operation involves a set of artificially created chromosomes called novel chromosomes whose example is shown in Figure 1 where a problem has been shown with four design variables. Design Variable Boundaries Figure 1: Novel chromosomes The novel chromosomes have all bits set to zero except the lowest bit corresponding to one of the individual design variables. In the novel crossover operation, the fittest chromosome is selected in each generation. Binary addition and subtraction are then performed between the selected chromosome and each of the novel chromosomes. As a result, the selected chromosome goes through the smallest possible variation in each design variable at the cost of very few (2N d ) calls to the analysis routine. Similar to SGA, NGA begins with a random generation of initial population. However, the new generation is produced in two phases. First, the novel chromosomes are generated and the novel crossovers take place. Second, the normal crossovers and mutations of SGA take place. The implementation procedure is outlined as follows: 1. Generate a random population of size Np 2. Evaluate the population and obtain fitness scores 3. Perform novel crossovers as follows: a. Based on a pre specified selection criterion, select a chromosome from the current generation called Q 127

6 b. Repeat the following until i = Nd (number of design variables) i. Generate a chromosome O having all bits equal to zero ii. Select the ith design variable and corresponding to this variable mutate the lowest bit of O to 1 and denote the novel chromosome by Pi iii. Produce a child Si by performing binary addition Q + Pi iv. Produce another child Ti by performing binary subtraction Q Pi v. Copy Si and Ti to the current generation c. Identify the 2Nd weakest individuals in the generation and remove them (this would maintain the generation size as Np) 4. Perform normal crossovers for all i as follows: a. Select parents to form mating population b. Perform mutation with a pre specified probability of P M. c. Perform one point crossover with a pre specified probability of P C 5. Follow the following steps to form the population for next generation: a. Get the k best individuals from the previous generation b. Copy the k best individuals into the next generation 6. Compute the new fitness for all individuals 7. Check the convergence criterion. 8. Check for the stopping criterion. If the stopping criterion is met, stop and the current population is the solution. Otherwise go to step 3 5. Simulation Results Optimal designs for structural design problems were obtained using the presented algorithm by integrating it with a with a special finite element analysis program developed for this purpose. However before applying the technique to structural optimization problems the presented algorithm was tested and evaluated on some mathematical problems with known optima and structural optimisation problems that have closed form solutions for the structural response. Results obtained for them using SGA and NGA are compared in the next section. Relative percentage error used in for comparing the two techniques is denoted by Γ and is based on the known optimal solution or a published solution of the problem. The term reliability implies the probability of occurrence of solutions with the error less than a given value for a large number of trial runs starting from different random seeds. The notation Λ(a) is used to indicate the reliability for error less than a%. All problems were solved using a crossover probability P C of 30% that is considered normal for SGA. It was found that NGA is not very sensitive to P C. The mutation probability P M for all examples was set to 0%. This again was done to ensure that only the effect of novel crossover is analysed for comparing SGA with NGA. The results presented are based on a large number of trial runs. For each problem, the number of trial runs was increased gradually and the statistics were collected until a level of confidence was achieved from the consistent values of the average error and the reliability. 128

7 5.1 Example 1 This is a non linear optimisation problem taken from Reference [22] with two relative optima. It is a good problem to test the reliability and the efficiency of the NGA in the continuous domain, since the deterministic search algorithms perform much better on such problems. The function to be minimized is as follows: C(D) =4+4.5d 1 4d 2 +d d 2 2 2d 1 d 2 +d 1 4 2d 1 2 d 2 (8) There are two relative minima as follows: * C 1 (1.941,3.854) = (9) * C 2 ( 1.053,1.028) = (10) The lower and upper bounds for both design variables are specified as 5 and +5. The incremental accuracy is specified as for both variables. This results in a design space size of A population size of 50 gave the best results. The performance of NGA and SGA are compared for this problem in Table 1. Performance Parameters (Average) Table 1: Example 1 (Std. Dev.) Average Error Λ(0.1) Λ(1.0) SGA % 65% 90% NGA % 100% 100% NGA finds the optimum with 100% reliability for error being less than 0.1% at a computational cost of 1100 function evaluations on the average. Thus the average number of analyses (function calls) for NGA is only about 45% of the calls needed for SGA with much smaller standard deviation. The average error for NGA is 0% implying that NGA always converged exactly to the known global optimum with the desired accuracy. 5.2 Example 2 This problem has been taken from [23]. Since this problem has been used to evaluate a stochastic optimisation approach called entropy, it will serve well to evaluate NGA. The problem is stated as follows: Minimize: C(D) = (d 1 2) 2 + ( d 2 1) 2 ; (11) Subject to: 129

8 g 1 (D) = d 1 2 d 2 0 (12) g 2 (D) = d d (13) d 1, d 2 0 (14) The known global optimum for this problem has a function value C * (D) = at D * = [1.000,1.000] T. The lower and upper bounds of 0.0 and 5.0 for both design variables with the incremental accuracy of were specified. For the specified lower and upper bounds and the accuracy of the design variables, the design space size is 0.5x10 6. The performance of NGA for this example was found even better than its performance for the first problem. The results are shown in Table 2. Performance Parameters (Average) Table 2: Example 2 (Std. Dev.) Average Error Λ(0.1) Λ(1.0) SGA % 0% 0% NGA % 100% 100% NGA obtains the global optimum with 100% reliability with a very small population size of 10. SGA s best result for the population size of 10 is and it fails to obtain any solution with errors less than 1% whereas NGA finds the exact answer with 100% reliability. The number of function evaluations for NGA is again about half as many as needed for SGA. 5.3 Example 3 The third example has been selected from [24]. In this structural optimisation problem, it is required to determine the minimum weight design of a narrow rectangular cantilever beam of length L, width b and depth h so as to prevent the failure due to flexure stress σ at the root section and twisting instability. The specific optimisation problem is stated as follows [25]: Minimize W = 240bh ; (15) Subject to: σ σ al = (1.44x 10 7 ) / (bh 2 ) (16) p p cr = hb 3 0 (17) The optimum design can be determined theoretically. It is W * = (for b * = , and h * = ). The design space for the trial runs was specified by: L 1 =0.5, U 1 =4.5, α 1 =0.0001; L 2 =6, U 2 =12, α 2 = The required accuracy for the two design variables is not the same. The design space size is 240x10 6. A population size of 50 was used. Table 3 summarizes the results for this problem. 130

9 Performance Parameters (Average) Table 3: Example 3 (Std. Dev.) Average Error Λ(0.1) Λ(1.0) SGA % 10% 90% NGA % 30% 90% NGA did not obtain 100% reliability for this problem. The reliability of NGA for this problem was not much different from SGA. However, the computational efficiency of NGA is clearly superior to SGA as is obvious from the average number of calls to analysis and the average error. 5.4 Example 4 This is an structural optimisation problem and has been taken from [24]. In this problem, the minimum weight design of an elastic grillage is required. It has been selected as an example because it has a complex design space but yet a small problem with closed form analysis. The 1x1 grillage, shown in Fig. 2, consists of two uniform beams subjected to uniformly distributed loads with cross sectional areas A 1 and A 2. The cross sectional areas of the two beams are the design variables of the optimisation problem. The section modulus and the moment of inertia of both beams are assumed related to the cross sectional area by Equations (18) and (19): z = ( A / 1.48 ) 1.82 (18) I = 1.007(A/1.48) 2.65 (19) Figure. 2: 1x1 grillage optimisation problem The solution of the problem as described in [25] is based on the following equations: Minimize W = A 1 L 1 + A 2 L 2 (20) Subject to: 131

10 g1 = σ (21) g2 = σ (22) g3 = σ (23) g4 = σ (24) g4 = σ (25) g5 = σ (26) The stress components σ 1, σ 2 and σ 3 are non linear functions of the design variables. The theoretically calculated optimum solution is W * = for A * 1 = 23.4, and A * 2 = To obtain the results with the same number of significant digits, the design space was specified as: L 1 =1.5, U 1 =30.0, α 1 =0.01; L 2 =1.0, U 2 =30.0, α 2 =0.1. The results are summarized in Table 4. Table 4: Example 4 Performance Parameters (Average) (Std. Dev.) Average Error Λ(0.1) Λ(1.0) SGA % 50% 90% NGA % 100% 100% For this problem, the reliability for error being less than 0.1% improved drastically when the population size was increased from 50 to 100. For N P = 100, the reliability for error < 0.1% became 100% as shown in the table. 5.5 Example 5 The fifth example has been taken from [26]. The optimum topology of supporting columns of a plane frame as shown in Fig. 3 is to be determined. Figure 2: Support topology optimisation problem 132

11 The details for this problem are not repeated here but it is worth pointing out that it is different from the four examples described above. This problem requires finite element analysis to calculate the structural response. The computer program DOFGA was used. The design variables namely X 1, X 2, X 3, X 4, correspond to the placement of the columns rather than sizing. The objective function minimised is the maximum bending moment in the structure and not the mass. There are 5 linear constraints that impose the minimum distance of 100 cm between the concentrated loads and the columns. SGA failed in most of the trial runs for this problem because of too many infeasible solutions in the initial population. The average error for SGA as shown in Table 5 was 41%. Many of the infeasible solutions had the 2 adjacent nodes like nodes labelled 2 and 4 too close to each other resulting in numerical illconditioning in the finite element analysis. The small variations in the individual design variables due to the novel crossover operation pulled some of the chromosomes out of this ill conditioning and as a result NGA was able to keep moving towards improved designs. Performance Parameters (Average) Table 5: Example 5 (Std. Dev.) Average Error Λ(0.1) Λ(1.0) SGA % 0% 0% NGA % 0% 100% NGA could not produce the same results as produced by the deterministic algorithm of [26] but functioned smoothly and clearly superior to SGA. The average error was 6.7%. The reliability was 100% for 10% error but was 0% for error less than 5%. Considering the large number of function evaluations normally required for SGA, the average of 1147 analyses for the quality of solutions obtained is quite satisfactory. The algorithm failed to obtain the exact optimum because small variations in more than one design variables are required to get to the optimum as is done in deterministic algorithms by forming a search direction and then taking a small step in the search direction. This highlights the fact that NGA, although being a significant improvement over SGA, should not be considered as an alternative to the well developed deterministic algorithms for structural optimisation problems in the continuous design space. 5.6 Example 6 Example 6 involves sizing of the columns of a three dimensional building frame structure. It stems from an ongoing research to utilize NGA in developing a computeraided structural design package which may help a structural designer start from a given floor plan layout and produce the optimum structural design. Such a tool is required because the structural designers usually have to design and analyse the structure for a number of different alternative floor plan layouts [27]. 133

12 The first step in this process is to create the structural model (consisting of numerical data concerning joints and elements) automatically from the floor plan layout data. This task is usually very tedious even with the pre processors available in the commonly used structural analysis software packages. For this purpose an available software package VIP PLANOPT [28] was used. VIP PLANOPT is a pre processor for multi storey building structures. It automatically generates three dimensional structural models from the floor plan layout data. User s input is minimal with a visual interface. Distributed loads for all the beams are generated from the user specified live and dead loads for the slabs. Once the structural model data is generated by VIP PLANOPT, the data is input to DOFGA for structural optimisation. Since VIP PLANOPT produces the data file for several structural analysis software packages including SAP2000 [29], provision was made in DOFGA to accept the input data in S2K format of SAP2000. The optimal floor plan of [27] was used as the starting point for this example. For this floor plan, VIP PLANOPT produced the structural model as shown in Figure 4. Figure 4: Structural model for Example 6 Although, the structural model can be generated for a 100 storey building as easily as for a two storey building using VIP PLANOPT, the example was limited to two floors to keep the size of the structural analysis problem small. The three dimensional building frame structure shown in Figure 4 has 48 joints and 74 beam elements (44 horizontal members and 32 columns). The output of VIP PLANOPT representing this structural model was modified to include the lateral loads on the structure. Artificially large lateral loads were imposed to avoid the trivial designs of all columns being alike. This modified data was input to DOFGA with additional input data related to optimisation. The objective is to select the best column sections for all the 32 columns from a list of available sections so that the mass is minimized. Constraints are imposed to keep the maximum tensile, compressive and shear stresses within the specified allowable limits for the selected sections. All the beam sections are assumed pre specified to cut down the 134

13 size of the optimisation problem so that a large number of trial runs could be made for investigating the performance of NGA. The column sections for this problem were chosen rather arbitrarily as shown in Table 6. The first 5 sections were selected from the lighter sections whereas the other 5 sections were heavier ones. Since there are 32 columns and 10 available sections the design space size for this optimisation problem is Table 6: Column sections for Example 6 Section ID Section W14X2 W14X3 Type 6 0 W14X34 W14X38 W14X43 Section ID Section Type W14X9 0 W14X9 9 W14X10 9 W14X12 0 W14X13 2 The best solutions obtained by NGA and SGA are shown in Table 7. SGA s best solution represents 22.3 % higher cost than the cost of the best solution obtained using NGA. It is interesting to see that NGA was successful in bringing the first sections to the smallest size without violating any constraints. Reliability of a stochastic algorithm is a real measure of its superiority. The reliability of NGA was found drastically better than SGA as is obvious from the data shown in Table 8. NGA exhibited 100% reliability for error being less than 10% for the example. It implies that even a single optimisation run using NGA will yield a solution with less than 10% error. Whereas relying on a single optimisation run using SGA may give a solution with 50% error. Table 7: Optimal column sections for Example 6 Section Type Relative Cost SGA 2,3,5,5,1,2,5,3,4,6,3,3,5,5,2,7, 7,3,3,2,5,4,5,1,7,3,1,2,10,4,9,6 NGA 1,1,1,1,1,1,1,1,2,2,4,3,5,3,1,6, 2,1,1,1,3,2,1,4,2,3,2,3,4,2,3, Table 8: Comparison of reliability for Example 6 Reliabilit Λ(.001 Λ(.10 Λ(.50 Λ(1.00 Λ(2.00 Λ(5.00 Λ(5.00 Λ(0) Λ(0) y ) ) ) ) ) ) ) NGA 2.0% 6.0% 36.0% 62.0% 96.0% % % % % SGA 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 50.0%

14 6. Conclusions In this paper, superiority of the proposed novel genetic algorithm (NGA) over existing genetic algorithm (named SGA) has been established for solving structural optimization problems. Further, implementation of NGA is quite simple and can be easily integrated with the traditional GA. The implementation based on the novel crossover is integrated with a finite element analysis software package to evaluate the performance. NGA was tested on six examples of different types and in all it proved to be clearly superior to SGA. Moreover, it exhibited a very high degree of reliability. This leads to the conclusion that increasing the probability of smallest possible variations in the values of the individual design variables is an important factor in improving the performance of GA. Since NGA was evaluated on six different types of problems and it produced significantly improved results, this also provides a set of benchmark problems to evaluate any further work in this direction. Application of NGA to a three dimensional building frame shows the potential of its application to the real world structural design problems. The novel crossover operation is just an additional bit of improvement that can be used to supplement the techniques presented in other published work towards improving the performance of GA in solving structural optimisation problem. 7. References 1.Farhat, F., Shozo Nakamura, Kazuo Takahashi,2009, Application of genetic algorithm to optimization of buckling restrained braces for seismic upgrading of existing structures, Computers & Structures, 87(1 2), Park, C.H., Woo Il Lee, Woo Suck Han, Alain Vautrin,2008, Improved genetic algorithm for multidisciplinary optimization of composite laminates, Computers & Structures, 86(19 20), Toğan, V, Ayşe T. Daloğlu,2008, An improved genetic algorithm with initial population strategy and self adaptive member grouping, Computers & Structures, 86(11 12), Cheng, J,2007, Hybrid genetic algorithms for structural reliability analysis, Computers & Structures, 85(19 20), Kicinger, R, Tomasz Arciszewski, Kenneth De Jong,2005, Evolutionary computation and structural design: A survey of the state of the art, Computers and Structures, Vol. 83, pp Holland, J.H., Adaptation in natural and artificial systems, University of Michigan Press, Ann Arbor, MI (1975) 136

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