Performance evaluation of a two stage adaptive genetic algorithm (TSAGA) in structural topology optimization

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Applied Soft Computing 8 (2008) 1607–1624 www.elsevier.com/locate/asoc

Performance evaluation of a two stage adaptive genetic algorithm (TSAGA) in structural topology optimization R. Balamurugan, C.V. Ramakrishnan *, Nidur Singh Department of Applied Mechanics, Indian Institute of Technology, New Delhi, India Received 2 January 2006; received in revised form 29 May 2007; accepted 3 October 2007 Available online 16 January 2008

Abstract Genetic algorithm with island and adaptive features has been used for reaching the global optimal solution in the context of structural topology optimization. A two stage adaptive genetic algorithm (TSAGA) involving a self-adaptive island genetic algorithm (SAIGA) for the first stage and adaptive techniques in the second stage is proposed for the use in bit-array represented topology optimization. The first stage, consisting a number of island runs each starting with a different set of random population and searching for better designs in different peaks, helps the algorithm in performing an extensive global search. After the completion of island runs the initial population for the second stage is formed from the best members of each island that provides greater variety and potential for faster improvement and is run for a predefined number of generations. In this second stage the genetic parameters and operators are dynamically adapted with the progress of optimization process in such a way as to increase the convergence rate while maintaining the diversity in population. The results obtained on several single and multiple loading case problems have been compared with other GA and non-GA-based approaches, and the efficiency and effectiveness of the proposed methodology in reaching the global optimal solution is demonstrated. # 2008 Published by Elsevier B.V. Keywords: Topology optimization; Genetic algorithm; Adaptive genetic algorithm; Island genetic algorithm; Global optimal solution

1. Introduction Structural topology optimization aims at maximizing the stiffness of the structure subject to a volume constraint or at minimization of material volume subject constraint on deflection under different loading conditions. Among the various attempts on solving the problem, the method developed by Bendsoe and Kikuchi [1] has been more popular in which the design domain is filled with composite materials with periodic microstructures and size/orientation of the voids present in the microstructures is considered as design variables. The effective material properties of the composites are evaluated through homogenization theory [2] and the optimum distribution of the material is obtained using an Optimality Criteria (OC) technique. The power law approach [3,4] uses directly the element relative densities as design variables and an empirical formula called power law to relate the density with the elastic modulus without the need for homogenization. These

* Corresponding author. Tel.: +91 11 26591221; fax: +91 11 26581119. E-mail address: [email protected] (C.V. Ramakrishnan). 1568-4946/$ – see front matter # 2008 Published by Elsevier B.V. doi:10.1016/j.asoc.2007.10.022

approaches have been highly researched [5–7], and computationally effective but these approaches have limitations and are so far limited to linear elasticity problems only [8]. Xie and Steven [9] have proposed evolutionary structural optimization (ESO) procedure in which the material is iteratively removed from the low stress region and is added in the highly stressed region. This method is similar to the fully stressed design approach for skeletal structures. In ESO the decisions to remove/add elements are based on intuitive rules and lack any mathematical formulation and hence the solution does not guarantee optimum. The method has also limitations in solving practical problems involving multiple loading cases [10]. Moreover, these approaches cannot perform a global search and thus often converge to local optimal solutions [11–13]. Genetic algorithm gives a general approach for solving the structural topology optimization problem including multiple loading cases, non-linear problems [8,14], for different kind of objective functions [15], etc. Since the GA-based method require only first order function evaluations the implementation methodology has been found to be straight forward for all kind of mechanical models unlike other OC [1,16–18] or non-linear programming (NLP) [19] based methods that require complex

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design sensitivity calculations. The main drawback of the GAbased topology optimization process has been the large number of function evaluations needed to arrive at the optimum solution. The progress is very slow especially in the final stages and most of the times it does not appear to be reaching the expected global optimal solutions [20,21]. Hamda and Schoenauer [22] have proposed adaptive representation techniques based on Voronoi cells to improve the performance of GA-based topology optimization process and has demonstrated its performance over benchmark problems including 3D. But this method is computationally very expensive and is reported to be applicable only for weight minimization problems. Recently, Wang and Tai [23] have demonstrated the use of Deb’s constraint handling approach [24] for improving the performance of GA-based bit-array represented topology optimization process. The present work is an attempt in this direction on improving the performance of GA-based topology optimization process by incorporating appropriate adaptive techniques in GA to suit the nature of the bit-array represented topology optimization problem. 2. Literature on adaptive genetic algorithm Adaptation of evolutionary algorithms to the nature of the environment has been investigated in detail [25,26] and is reported to be an essential feature that makes the algorithms closer to natural evolution. It has been further pointed out that GA will be accurate and efficient only if the genetic parameters are appropriately adapted to suit the nature of the problem failing which the process may terminate at local optima. The adaptive parameters have also to be depending on the type of crossover and mutation methods. The difficulties present in manual adaptation have been overcome through many adaptive GAs (AGAs) that can dynamically adjust the parameter values [27,28]. Theoretical aspects and implementation methodologies of various self-adaptive procedures have been discussed and compared in detail [29–31]. In problems where the optimal population size is not known parallel searches with different population sizes have been proposed expecting one or more of them would yield good results [32,33]. But it is not realistic to conduct large number of searches in parallel throughout the optimization process in which most of them would perform inefficiently. It is even more difficult to accept these kinds of approaches in problems like topology optimization where objective function evaluation is computationally costlier and adjusting one or two parameters alone does not give the desirable improvement. To overcome the issue of early convergence of GAs to near by optimal solutions various scaling techniques and diversity preserving mechanisms such as large mutation rate, inclusion of niching operator, simulated binary crossover (SBX) [34] and others have been proposed. For adapting crossover rate and mutation rate of individuals a procedure based on the 2D lattice space location of the individuals has been proposed [35] in which an effort has also been made to maintain diversity in the population by limiting the number of individuals in each lattice space. The control map approach proposed by Goldberg and

Deb [36] to compute effective value ranges of parameters seeks for a detailed analytical problem formulation depending on the type of problem attempted and does not present an general approach to solve all kind of mechanical models of topology optimization. Different adaptive techniques have been proposed for making GAs to perform efficiently in noisy environments where the fitness function values drastically changes for almost identical individuals [37–39]. Murata et al. [40] have proposed an Agent Oriented SelfAdaptive Genetic Algorithm (A-SAGA) that combines metaGA [41] and GA with distributed environment search scheme [42] where the distributed environment GA is a kind of parallel GA. This A-SAGA performs well once the algorithm is trained according to the nature of the problem in hand and appropriate genetic parameters are identified. The training of the algorithm is equivalent to solving the complete optimization problem and becomes computationally very costly. To overcome this drawback Takashima et al. [43,44] have proposed a selfadaptive island GA (SAIGA) procedure for adapting multiple genetic parameters in which a number of island runs are executed in parallel that reaches very close to global optimal solutions. But the algorithm does not efficiently utilize the genetic characteristics of the individuals obtained from different islands (only the elite individuals are allowed to immigrate between islands). The procedure executes all the islands in parallel through out the optimization process, which is computationally very expensive, and also most of them would become redundant and ineffective after the initial stage in the context of topology optimization. 3. Present work In structural optimization it is extremely important to have a robust algorithm to find global optimal solutions considering its area of application such as the aerospace industry. The ability of a methodology to reach global optimal solutions depends on generality of the representation scheme used to define a topology design. The bit-array representation scheme, in which, the design domain is discretized into a finite number of rectangular blocks and 1/0 (solid/void) fill approach is used to find out the optimal material distribution, is more general to represent all possible topologies. In the present work a two stage adaptive genetic algorithm (TSAGA) has been proposed for bit-array-based structural topology optimization and detailed studies have been carried out in evaluating the performance of the algorithm in reaching the global optimal solution. The method consists of two stages in which the first stage involves the self-adaptive island GA that helps in searching for global optimal solutions, while the second stage consists adaptive techniques that are designed to increase the convergence rate while maintaining the diversity in population. Thus the advantages of the SAIGA in searching for global optimal solutions and adaptive procedures in accelerating the convergence have been blended together to form an efficient and effective algorithm for the use in bit-array represented topology optimization. The results obtained on several single and multiple loading case benchmark problems have been

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compared with simple genetic algorithm (SGA), self-adaptive genetic algorithm (SAIGA), adaptive GA procedures, and the efficiency and effectiveness of the proposed methodology in searching for global optimal solutions is demonstrated. The present work is the expansion of the methodology presented by the authors [45]. The results have also been compared with some of the other approaches found in the literature.

and has compared its performance with other schemes. In the present study one-block, two-block and uniform crossover schemes have been used with a user-defined probability of 50, 40 and 10% respectively based on which a particular scheme will be considered for the crossover operation. A crossover probability of 0.9 is used failing which the designs will be duplicated and skip the crossover operation.

4. Genetic algorithm for structural topology optimization

5.2. Mutation

In bit-array represented topology optimization the design domain is discretized and is represented by an indicator function x(x) that takes a value either 1 for solid fill or 0 for void fill. The variables are the values of the indicator function x(x) and are randomly generated to form the initial population. These initial designs undergo finite element analysis for fitness calculation. GA operates on these designs represented as strings (Fig. 1) through its genetic operators reproduction, crossover and mutation to produce new set of designs (offsprings) for the subsequent generation. 5. Simple GA The following crossover and mutation operators have been used in the present work for topology optimization with simple GA. 5.1. Block crossover It is essential that specific genetic parameters have to be used according to the problem at hand for better progress of GAbased optimization processes. Kane [46] has explained how one-dimensional crossover schemes like one- and two-point crossovers result in geometrical bias. A number of twodimensional crossover schemes, like two-block, three-block and diagonal crossover schemes have been developed and tested to be useful in bit-array-based topology optimization. Swaminathan [21] has developed a one-block crossover scheme

The standard flip-bit mutation in which the value of bits of the string are changed from ‘0’ to ‘1’ or ‘1’ to ‘0’ if the mutation probability ( pm) test is passed has been used. It is used along with crossover and duplication operators. A fixed low mutation probability of 0.01 is used. 5.3. Selection scheme Remainder stochastic sampling without replacement has been used as the selection scheme for mating pool formation. Swaminathan [21] has developed two new mating pair selection schemes, namely monogamy and polygamy based on the hypothesis that mating strings with good fitness values are more likely to yield better offspring than mating a good string with a bad one. In monogamy, the best member mates with the second best, the third with the fourth and so on. In polygamy, the best member mates with the second best, the third, the fourth and so on; the second best with the third, the fourth; and so on. These mating pair selection schemes have been used along with the standard crossbreed method in which the pairs are selected at random. The mating pair schemes are chosen based on predefined probabilities and in the present work monogamy and polygamy have been put under one category with a probability of 75% and crossbreed 25%. Among the monogamy and polygamy the priority is given to the former with 90% probability. Piecewise linear fitness scaling procedure [47] has been used which helps to avoid over dominance of best member in the initial stages and to initiate healthy competition between near equal members in the later stages.

Fig. 1. Mapping of topology into the string. (a) Topology design; (b) bit-array; (c) string.

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5.4. Elitism In each generation a user-defined number of good parents replace as many unfit offspring, which is a kind of elitism, employed in the present work. When such elitism is not used, there is frequent loss of highly fit members and the resulting performance shows fluctuations and slow convergence [21]. 6. Self-adaptive island GA Self-Adaptive Island GA is a kind of parallel GA in which a number of GAs executed in parallel/separate are searching for solution of a problem from different peaks. Two levels of GA called low- and high-level GA have been used. The low-level GA is an island GA, which consists a number of islands each searching for the solution to the given problem with different values of genetic parameters (such as population size, crossover probability, mutation probability, etc.) called parameter vector. After the islands are run for a predefined number of generations (called era), some of the best members are allowed to immigrate between the islands. Then the high-level GA evaluates the performance of parameter vector of each island through their parameter fitness values. These parameter fitness values are the cumulative increases in fitness of elite individuals of the respective islands except the increase caused by immigration. Based on these parameter fitness values the parameter vector for the subsequent era of each island is evaluated. Thus the task of the high-level GA is to create new parameter vectors for the use of low-level GAs in the subsequent era. This process is repeated for a predefined number of eras or until some convergence criteria is satisfied as similar to that of SGA. More detailed information and pseudo code can be found in Ref. [43]. In the present study the island GA procedure has been implemented in a single workstation. Five island GAs (lowlevel GAs) are executed in parallel each starting with different random initial population taking the bit-array of the topology optimization problem as variables. The parameter vector (vi ¼ fni ; pc ; pm g) consists of population size, crossover rate and mutation rate, and is created at random for the initial era of five generations. The following lower and upper limits of (50, 150), (0.3, 0.95), and (0.01, 0.08) have been considered for population size, crossover rate and mutation rate respectively. At the end of the era the parameter fitness values for each island GA is computed, which is the increase in fitness values (i.e., the decrease in the compliance + penalty values) of the elite designs evolved in that era. Some of the worst designs (3% of the islands population) are replaced with best elites of other islands and immigration is achieved. The parameter fitness values are written to an output file called pvfit. The high-level GA program remains idle and waits until the parameter fitness values of all the island GAs are written to the pvfit file. Then the parameter vectors of island GAs are decoded into binary form by the high-level GA and are concatenated to form a binary string. A 5 bit string has been used to represent each of the variables and thus the total string length for the high-level GA operation is taken as 15. The high-level GA is then executed for

one generation to produce the offsprings (i.e., new parameter vectors) for the next era of low-level GAs. Tournament selection has been used in the work of Takashima et al. [43] but in present work the selection schemes discussed in Section 5.3 has been used in both levels of GAs. One-block and uniform crossover schemes have been used for low- and high-level GAs respectively. Simple bit-flip mutation method has been used in both levels of GAs. The following mapping rule has been used for encoding the binary string into real values for the use of lowlevel GAs.     1  DV DV LL  þ  UL (1) 2ðSL1Þ 2ðSL1Þ where LL and UL are the lower and upper limits, DV is the decoded value and SL is the total string length. All the low-level GAs are run for 100 eras unless otherwise specified. 7. Adaptive GA Exploration efficiency of GAs largely depends on the genetic parameter values and a procedure for adapting crossover schemes, crossover rate and mutation rate is presented in this section. To cope with the nature of complex problems such as topology optimization these parameters have to be automatically adjusted for better results. The problem needs appropriate selection of these parameters and also a population of greater variety and potential for faster improvements. The adaptive genetic algorithm incorporates several such features in SGA methodology for topology optimization problem. Since the adaptive GA features are extensively used in the TSAGA also, this is explained in detail below. 7.1. Adaptive crossover Crossover is the procedure where in selected parent string is broken into segments and some of these segments are exchanged with corresponding segments of another parent string. The segments of each string are chosen at random. 7.1.1. Adaptation in crossover rate Crossover is performed on the parent string based on a crossover probability ( pc). In the present adaptive GA procedure this probability is varied with respect to the fitness value of the designs. Each design is assigned a crossover probability calculated from its fitness as follows, pc ¼ k 1 

fitness max: fitness

(2)

where k1 is the crossover coefficient and is taken as 0.95. The good designs get 80–95% crossover probability whereas the designs having poor fitness value will get less than about 50% crossover probability. The bad designs are mostly failed to make it for crossover and undergo complete duplication with possible mutation of all the genes. To understand the proposed adaptive crossover strategy let us consider a group of 6 designs that are assumed to have different fitness values (Table 1). In

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Table 1 Adaptation of crossover and mutation probabilities Design no.

Fitness

Rank

SGA crossover probability ( pc)

SGA mutation probability ( pm)

TSAGA crossover probability ( pc)

Swapped fitness

TSAGA mutation probability ( pm)

1 2 3 4 5 6

6 18 14 3 20 11

5 2 3 6 1 4

0.9 0.9 0.9 0.9 0.9 0.9

0.01 0.01 0.01 0.01 0.01 0.01

0.285 0.855 0.6675 0.1425 0.95 0.5225

18 6 11 20 3 14

0.045 0.015 0.0275 0.05 0.0075 0.035

SGA the crossover probabilities are fixed as 0.9 for all designs. In TSAGA the good designs (Best Design No. 5 gets a high crossover probability of 0.95) get relatively high crossover probabilities as they are expected to produce good offsprings. The bad designs (Worst Design No. 4 gets only 0.1425 crossover probability) get relatively low crossover probabilities, as they are not expected to produce any better designs. It can be seen from Table 1 that bad designs get low crossover probabilities of less than 0.5 and thus most of them fail to make it for crossover, and will undergo complete duplication with possible inversion of all the genes with relatively high adaptive mutation rate (discussed in Section 7.2) whereas SGA or SAIGA do not distinguish between good and bad designs. This type of biased crossover allows renewing the worse individuals and increasing the diversity in the population and thus favouring the exploration of the domain in greater detail without dispersing their genetic patrimony. 7.1.2. Adaptation in crossover schemes Three types of basic crossover schemes one-block, twoblock and uniform each with a user-defined probability have been employed in SGA. In the present adaptive GA the designs are allowed to choose between these three crossover schemes based on their performance in the previous era. During the initial phase of the genetic process all the three types of crossover schemes are assigned equal probability of selection. After the initial phase the designs are associated with a crossover index (0, 1 or 2 for single block, two-block and uniform crossover schemes respectively) based on the type of crossover it has undergone. At the end of the first era (an era consists of 5 generations as that of SAIGA) the crossover schemes are ranked. The best crossover type is the one corresponding to the crossover index of the best design of the population. Similarly the second best crossover type is identified from that of the second best design and the one left out gets the third place. No crossover type is allowed to get more than one place thus avoiding over dominance of the best crossover type and to ensure the second and third place for the second best and the least performed crossover schemes respectively. Then the probabilities of 60, 30, and 10% (or any other user-defined probabilities) are assigned to the first, second and third best crossover schemes respectively based on which the designs will undergo crossover operation. The ranking of the crossover schemes is done after every generation in which improvement in designs is found and the newly ranked schemes will become active in the successive era.

7.2. Adaptive mutation Mutation is an important operator for bringing variety into the population and also to make some effective minor changes in the design. The population-based mutation operator defined in Ref. [48] aims at preventing the premature convergence of the population. The probability of flipping a given bit is adjusted by considering all bits in the same position in the whole population. More precisely, the probability pi for a given bit i to be flipped depends on the mean value mi of this bit over the population. If this mi takes a uniform value (mi = 0 or 1), the probability to mutate is set to a high value pmax, on the opposite it is set to a low value pmin if there is about the same proportion of 0s and 1s. The probability of flipping a bit is then a parabolic function of the mean value mi between these points. This operator thus imposes high values of mutation rate at positions that have already converged. Kane [46] has proposed a boundary mutation method for topology optimization, in which the elements present along the inner and outer boundary of the topology are selected for mutation. The boundary mutation is defined such that boundary bits, i.e., bits having one edge on the boundary of the connected component of the shape, are given higher probability ( pmax) to be flipped than the other bits ( pmin). It has been reported that the population-based mutation performed slightly better than the standard bit-flip mutation. The boundary mutation has only been used in the end of the evolution process to fine-tune the solution. In the present adaptive GA each design is associated with its own mutation probability, which is determined by the relative fitness value of the other designs in the population. The fitness of the good designs is swapped with bad ones (best with worst, second best with second worst and so on) and the mutation rate is calculated from the swapped fitness value. pm ¼ k 2 

swapped fitness max: fitness

(3)

where k2 is the mutation coefficient that takes the value 0.05. This adaptive mutation strategy can be easily understood by looking at Table 1 in which it is shown that the best designs get relatively low mutation probabilities (Best Design No. 5 gets only 0.0075 mutation probability) while the bad designs get relatively high mutation probabilities (Worst Design No. 4 gets 0.05 mutation probability). Thus the best members and the worst members influence each other’s mutation rates, i.e., the mutation rate of the worst members increases as the best

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Fig. 2. Group selection procedure.

member gets better and better and similarly the mutation rate of the best members decreases as more number of worse members generated. This kind of adaptive mutation increases the diversity in population by allowing possible inversion of all genes of the worst designs while increasing the convergence speed by keeping the best members mutation rate as low as possible. A group selection procedure has been applied to categorize the entire population in two groups as good and bad as described by Pezeshk and Camp [30]. About 75% of the designs are placed under the good group while the rest of the designs form the bad group as indicated in Fig. 2. During the final stage, the good members are not allowed to have the mutation rate below a certain limit ( plm) and when the mutation rate goes below the limit it is taken to the value of plm and this procedure plays an important role in avoiding premature convergence of good designs. This final stage for using this plm limit is considered to be reached when there is no improvement in the designs for a predefined number of consecutive eras, which is taken as 10 in the present work. 7.3. Accelerated evolutionary era (AEE) At the end of each era, the improvement in the designs are evaluated, by comparing the fitness value of the current best design with the fitness value of the best design of the first generation of the era. If there is no improvement, that is, the fitness value of the best designs remain unchanged during an era, the crossover rate and the ranking of crossover schemes are

randomly changed in the subsequent era, which we call as accelerated evolutionary era. These eras are executed especially to bring variety into the population and are found to be more effective. During this AEE, more number of copies of the best member is allowed to enter into the mating pool and a high mutation probability of 0.08 is fixed for all members of all generations of the era. The crossover rate and the crossover ranking at which the design has improved during this period will be selected for successive eras. If there is no improvement then the old values, which had been used in the previous era will be retained for forthcoming eras. The accelerated evolutionary eras are executed only at an interval of a predefined number of eras (20 in the present study). 8. Two stage adaptive GA The two stage GA comprises the island and adaptive GA features explained above which are effectively used to explore the design space in greater detail. 8.1. Stage 1 The first stage consists a number of sample runs (Ni) similar to island runs each starting with a different set of randomly generated initial designs, thus favouring the search for better designs from different peaks. The parameter vector considered for each island consists of the population size ni and the coefficients used in adaptive procedure for crossover and mutation (k1i, k2i respectively) and an inversion coefficient k3i. The inversion coefficient is used to increase the mutation probability of the designs falling into the bad category in the group selection procedure described in Section 7.2. During the first era these values are randomly created between the limits given below for each island i. The lower and upper limits for the genetic parameters are, 50  ni  150 for ni ; 0:8  k1i < 0:95 for k1i ; 0:01  k2i < 0:05 for k2i ; and 1  k3i  10 for k3i : After the end of an era the parameter fitness values are evaluated based on which the high-level GA produces new parameter vectors for the subsequent era of low-level GAs as described in SAIGA. The genetic operators (crossover and mutation) and selection procedures are same as that of the SAIGA. Each island is run for a predefined number of eras (max_eras1) or until the parameter vector converges to a predefined convergence criteria. 8.2. Stage 2

Fig. 3. IGA block.

The second stage of TSAGA starts after all the island runs are completed. Unlike SAIGA where only the elite designs are immigrated between islands, more number of good designs is selected from each island to form the initial population for the second stage. The number of designs that will be selected from

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Fig. 4. Flowchart for TSAGA.

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each island (P2i) is determined as follows, P2i ¼

P2 Ni

(4)

where P2 is the population size for the second stage and Ni is the number of islands. The initial population formed for the second stage provides opportunity for the designs from different islands to share their genetic information and offers great potential for effective and efficient exploration of the design space. The genetic parameter vector (V = {n1, k1, k2, k3}) at which the best design of all the islands evolved is selected for the first era of this stage. All the adaptive techniques described in Section 7 (adaptive crossover and mutation rates, AEE) have been incorporated and elitism is also used. This second stage is run until the convergence or a predefined number of eras (max_eras2). The island GA module is schematically shown in Fig. 3. The flowchart for the TSAGA procedure is shown in Fig. 4. 9. Numerical results Two types of topology optimization problems for (i) minimum work design and (ii) minimum weight design have been studied and the results are compared with other approaches. These are the commonly employed cost functions in the context of practical structural optimization. If only a single loading case is considered, minimization of volume subject to a displacement constraint is the dual of minimization of compliance for a prescribed volume. Under multiple loading conditions it is appropriate to deal with the primal problem of minimization of volume subject to different displacement constraints. Each optimization block is meshed with four elements to reduce the effect of numerical instability on using linear elements [49]. A constraint on solid perimeter of the design is imposed to overcome the ill-posed nature of the topology optimization problem [21,50]. The Young’s modulus and Poisson’s ratio for the solid elements are 2.1  105 N/mm2 and 0.3 respectively where as for void elements 1 N/mm2 and 0.03 have been assigned unless otherwise specified. In the studies conducted using SGA and AGA, all symmetry problems are run for 1000 generations with a population size of 50 and the problems for which symmetry assumption is not made 2000 generations with a population size of 100 has been considered unless otherwise specified. For studies on SAIGA five islands have been executed in parallel and each is run up to 50 eras for symmetry problems and 100 eras for problems without symmetry assumption. In the first stage of the TSAGA five islands have been used and each one is run up to 5 eras for symmetry problems, 10 eras in problems where symmetry assumption is not made. The second stage is run for 75 and 350 eras respectively for symmetry and non-symmetry cases. An elite fraction of 0.1 is used in SGA, AGA and second stage of TSAGA to make sure that the best design of a generation propagates to the next generation unchanged replacing the worst one. The lower limit on the mutation rate ( plm) of the best

designs in the second stage of TSAGA is set as 0.001. Six and eleven independent runs have been carried out for symmetry and non-symmetry problems respectively in each case and the best run among them has been considered for discussion. For the problems considered, the principal computing effort is on account of FE calculations. Thus the number of FE calculations is a good indicator of CPU time. For a typical 2D problem with about 1000 degrees of freedom the CPU time for one analysis was approximately 0.125 s with a 3.2 GHz Intel Pentium Xeon System with 2GB RAM. 9.1. Minimum work design problems The minimum work design (minimum compliance) problems can be expressed as, min Wð¯xÞ ¼ f T u¯ ; subject to g1 ¼

V  Va 0 n1

g2 ¼

P  Pa 0 n2

(5)

where n1, n2 are scaling constants, V, P are respectively the volume and total perimeter of a design, and Va, Pa are the corresponding allowable values. f¯ denotes the nodal load vector and u¯ is the vector of global nodal displacements in the FE formulation. The constraints g1 and g2 are the normalized constraints on volume and perimeter respectively. A constraint on total solid perimeter of the design with an allowable limit of 100 (for 10  10 mesh) and 200 (for 20  20 mesh) is added to reduce the complexity of the topology optimization problem. The constraint optimization problem (5) has been transformed to an unconstrained one using the exterior penalty function formulation where a fitness function is minimized. Detailed information about the problem formulation can be found in Ref. [21]. Two cantilever plates 1  1 (Fig. 5) and 2  1 with left boundary fixed with support and a vertical download load applied at the center point of the right boundary edge have been studied. The problem has been studied for different mesh sizes.

Fig. 5. 1  1 cantilever plate problem.

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Fig. 6. Results of various GA approaches. (a) SGA, workdone = 7.797; (b) SAIGA, workdone = 7.593; (c) AGA, workdone = 7.652.

struggling with some local optimal solution and the progress is very slow. The progress of AGA has been very good but it could not reach the global optimal solution. SAIGA has been able to reach the global optimal solution rendered by TSAGA but the performance of the later has been far superior. The progress of SAIGA is slow in the initial stages, which is expected as it searches for appropriate genetic parameters initially but shows good progress in the later stages once the optimum parameters are identified. The SAIGA has reached global optimal solution after 47,200 FE evaluations whereas TSAGA has been able to reach it within 19,000 FE evaluations thereby reducing the computational effort by a fraction of about 1/2.5.

Fig. 7. TSAGA result, workdone = 7.593.

9.1.1. 1  1 cantilever plate problem (10  10 mesh) The proposed TSAGA has been tested on a 10  10 finite discretization and the performance is compared with other GAs. The final designs obtained using SGA, SAIGA and AGA are shown in Fig. 6 with workdone values 7.797, 7.593 and 7.652 respectively. The result obtained through TSAGA is shown in Fig. 7. The performance of the TSAGA (Fig. 8) has been better than all the other GAs. SGA is found to be

Fig. 8. Performances of SGA and TSAGA.

9.1.2. 1  1 cantilever plate problem (20  10 assuming symmetry) The problem is studied with a finer discretization of 20  20 optimization blocks. Symmetry is assumed by considering only the top half (20  10) of the blocks as variables. For finite element analysis the complete model has been considered. The results are mapped on to a 60  60 FE mesh for the accurate comparison of the results. A design with three interior holes (Fig. 9(a)) is obtained through SGA with a work of 9.06. The results obtained through SAIGA and AGA are shown in Fig. 9(b and c) whose workdone values are 8.63 and 8.59 respectively. Fig. 10 shows the design obtained through power law approach [51] with a workdone value of 9.448. The optimum design (Fig. 11) obtained through the present TSAGA is the stiffest with a workdone value of 8.493 and renders 6.7, 1.6, 1.1, and 11.2% improvement in minimum compliance values over SGA, SAIGA, AGA and power law approaches respectively. It can also be observed that the design obtained by the power law approach contains lot of small holes because of the filtering scheme used in their work to overcome gray images where as the designs obtained by the GA-based approaches are very clean. The optimality criteria-based power law approach is computationally more efficient taking only 50 iterations. But they are local search algorithms that work with a single starting point and often end up with local optimal solutions as in the present case and are not suitable for more complex problems involving multiple constraints which requires non-linear programming methods whereas the global search algorithms like GA work with a number of randomly generated points and converges to global optimal solutions. Among the GA approaches studied in the present work the proposed TSAGA reaches its final solution in a more efficient way than the SGA

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Fig. 9. Results of various GA approaches. (a) SGA, workdone = 9.06; (b) SAIGA, workdone = 8.63; (c) AGA, workdone = 8.59.

Fig. 10. Power law approach workdone = 9.448.

and SAIGA approaches (Fig. 12). The AGA shows good convergence rate but again converges to a local optimal solution. The performance SGA has been the worse among all the approaches. 9.1.3. 1  1 cantilever plate problem (20  20 without assuming symmetry) The problem has also been studied without considering the symmetry on the finer mesh. SGA is run for 3000 generations with a population size of 100 and the final design (Fig. 13(a)) still contains floating masses and single element holes. The

Fig. 11. TSAGA result workdone = 8.493.

results are evaluated on common 60  60 FE mesh for accurate comparison. The workdone values of the results that correspond to SGA, SAIGA, and AGA are 10.201, 8.66, and 9.22 respectively (Fig. 13). The design (Fig. 14) obtained using the TSAGA approach is having better stiffness than the other approaches and gives 19.1, 1.1, and 12.5% improvement in minimum compliance values than SGA, SAIGA and AGA solutions respectively. The performance of the TSAGA clearly outperforms all other GA approaches considered for studies in the present work (Fig. 15). Again the performance of SAIGA is slower in the initial stages but gathers momentum in the later stages and gets near to the optimal solution given by TSAGA. The AGA is slightly more efficient than TSAGA but it is not able to finetune the solution for better stiffness. Again the performance of SGA has been the worst. It should also be noted that in all the three studies conducted on the 1  1 plate problem (10  10 mesh, 20  20 mesh symmetry, 20  20 mesh without symmetry) the final designs rendered by TSAGA are exactly similar in topology and thus always converges to the global optimal solutions. On the other hand the SGA and AGA often converge to local optimal topologies. SAIGA has also rendered similar global optimal topologies in all the three runs but is less efficient than the TSAGA.

Fig. 12. Performance comparison.

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Fig. 13. Results of various GA approaches. (a) SGA, workdone = 10.201; (b) SAIGA, workdone = 8.66; (c) AGA, workdone = 9.220.

Fig. 16. 2  1 cantilever plate problem.

Fig. 14. TSAGA result, workdone = 8.568.

9.1.4. 2  1 cantilever plate problem The problem is studied on a 24  12 discretization for a volume fraction of 50% and a unit point force is applied as in Ref. [23] (Fig. 16). The same material properties of Et = 1,

v ¼ 0:3 and symmetry assumptions have been considered for the purpose of comparing the results, where E is the Young’s modulus, t is thickness of the plate and v is the Poisson’s ratio. Same 24  12 mesh has been used for FE analysis for comparison of results with other approaches. The results obtained using SGA, SAIGA and AGA are shown in Fig. 17 along with their workdone values. Fig. 18(a and b) shows the optimum design reported by power law approach [51] and bit-array GA [23] that have workdone values of 74.118 and 65.26 respectively. The optimum design obtained

Fig. 15. Performance comparison.

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Fig. 17. Results of various GA approaches. (a) SGA, workdone = 67.73; (b) SAIGA, workdone = 65.272; (c) AGA, workdone = 67.4025.

Fig. 18. Results reported in literature. (a) Power law approach, workdone = 74.118; (b) Wang’s bit-array GA workdone = 65.26.

Fig. 19. TSAGA result, workdone = 64.811.

by the present TSAGA approach is the best among them (Fig. 19) having a workdone value of 64.811 and thus offering 4.5, 0.7, and 3.9% improvement in the workdone values over SGA, SAIGA and AGA respectively. The TSAGA result is very much accurate than the power law approach with 14.4% further improvement in the minimum compliance value. Against the Wang’s bit-array GA the present TSAGA has given 0.69% improvement in the final workdone value. It can be clearly observed from Fig. 20 that the TSAGA has been better than SGA and SAIGA and has reached the global

Fig. 20. Performance comparison.

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Fig. 21. Results of various GA approaches. (a) SGA, weight = 0.345; (b) IGA, weight = 0.335; (c) AGA, weight = 0.335.

optimal solution with only 42300 FE evaluations. As in the previous cases the AGA has been more efficient but could not reach the global optimal solution. Even though Wang’s bitarray GA and power law approaches are computationally efficient than the present TSAGA-based bit-array method they converge to local optimal solutions. It should also be noted that the resulting topology of the power law approach displays a blurred boundary with gray elements even though a sensitivity filtering scheme [51] is used where as the GA-based approaches render distinct black and white designs. The result of the Wang’s bit-array GA approach contains many single element holes that are not practical whereas the present TSAGA approach has offered a practical global optimal solution that is completely free from noise. 9.2. Minimum weight design problems The minimum weight design problems subject to constraints on displacements can be formulated as, min V subject to gj ¼

W j  W ja  0; nj

gj ¼

P  Pa  0; nj

j ¼ 1; . . . nl

j ¼ nl þ 1

9.2.1. The 2  1 cantilever plate problem The design domain is discretized into a mesh of 20  10 and a unit point force applied vertically downward at the midpoint of the right side boundary with left side boundary fixed as in Ref. [52]. A maximal displacement of 220 has been taken as the allowable limit. The material properties of Et = 1, v ¼ 0:3 have been considered, where E is the Young’s modulus, t is thickness of the plate and v is the Poisson’s ratio. Fig. 21 shows the results obtained for the problem using SGA, SAIGA and AGA with 0345, 0.335 and 0.335 respectively. The result reported using the Voronoi bar representation method is shown in Fig. 22 and has a weight of 0.33. The design obtained using the present TSAGA approach with bit-array representation us shown in Fig. 23 with a workdone value of 0.34. The progress of SGA, SAIGA, and AGA has been compared with TSAGA in Fig. 24. The results obtained using the SAIGA and AGA for this problem are slightly better than TSAGA. In terms of computational efficiency AGA has been superior to all other approaches and TSAGA has been found to be efficient than SAIGA and SGA. The Voronoi bar representation method has given slightly better result than TSAGA but it has taken 1,60,000 FE evaluations whereas the present TSAGA approach needed only 99,800 FE evaluations to reach their final designs respectively. Moreover it has also been reported that the Voronoi bar method is not applicable for minimum work design problems where as the present TSAGA-based bit-array approach has given competitive results for both type of problems.

(6)

where njs are scaling constants and ‘nl’ is the number of independent load cases. V, P, Wj are volume, total perimeter, workdone of a design for jth loading case respectively and Pa, Wja are the corresponding allowable values.

9.2.2. Multiple loading case The TSAGA has also been tested on a slightly difficult problem with the design space containing many local and alternate optimal solutions. The multiple loading problem shown in Fig. 25 of size 1  1 has been considered for this investigation. Three point loads P1, P2, P3 of 100 units each

Fig. 22. Voronoi bar representation weight = 0.33.

Fig. 23. TSAGA result weight = 0.34.

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Fig. 24. Performance comparison.

Fig. 25. Multiple loading problem.

have been applied as independent cases (Fig. 25) and the left side boundary of the plate is fixed at the corners. The problem is to find the optimum design with minimum volume subject to constraints on the allowable maximum displacements corre-

sponding to each load and allowable solid perimeter (1 1 0). The maximum nodal displacement of a design is compared with the allowable displacement to see whether the displacement constraint is violated by the design. The allowable displacement values for the load cases P1, P2 and P3 are taken as 0.27, 0.24 and 0.3 respectively. The results obtained for the problem using SGA, SAIGA and AGA are shown in Fig. 26 with weights of 0.3375, 0.3075 and 0.3175. Fig. 27 shows the optimum design obtained using the TSAGA approach and has a weight of 0.2825, which is an improvement of 19.4, 8.8 and 12.3% in the minimum weight value over the SGA, SAIGA and AGA approaches. It can also be seen that the results of SGA and AGA contains many single element holes (even hanging masses in SGA result) whereas the results of SAIGA and TSAGA almost free of such noises. In terms of computational efficiency the present TSAGA has been far superior (Fig. 28) to all the three other approaches and has reached its final solution after 1,89,800 FE evaluations. The performance of the TSAGA in this complex problem is

Fig. 26. Results of various GA approaches. (a) SGA, weight = 0.3375; (b) SAIGA, weight = 0.315; (c) AGA, weight = 0.3175.

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value of 8.417 is obtained after 3,78,000 FE evaluations. The design in Fig. 30 is reported in the work of Belblidia et al. [7] with a workdone value 8.435. It is important to note that the symmetry assumption is not made in the present work and thus the problem is more complex demonstrating the scalability of the algorithm. It is also observed that the optimum designs obtained for all the studies conducted for this problem with different mesh sizes (10  10, 20  20 symmetry, 20  20 without symmetry, 30  30 without symmetry) are identical, which demonstrates the robustness of the algorithm in converging to global optimal solutions. 9.4. 3D problem

Fig. 27. TSAGA result, weight = 0.2825.

extremely good in comparison to other GAs that are struggling against different local optimal solutions. 9.3. Performance on fine discretization The performance of TSAGA has been studied on a finer discretization of 30  30 for the 1  1 cantilever plate problem without assuming symmetry and the problem consists 900 variables The first stage of TSAGA run is carried out with a population size of 100, 5 islands and each island run is carried out for 20 eras. The second stage is run for 700 eras nonsymmetry cases. Optimum design (Fig. 29) with a workdone

The TSAGA has also been implemented for even more complex 3D applications. The problem consists of a threedimensional domain as shown in Fig. 31 which is under the action of a uniform pressure load on one face and supported at the corners representing the design of a four legged stool. Taking advantage of symmetry only one quarter of the problem is considered for optimization. Minimization of work is considered as the objective with a constraint on volume ratio of .125. The whole domain is discretized into 8  8  8 mesh. The first stage is carried out with a population size of 100. Five island runs are performed and each island run is carried out for 20 eras. The second stage is run for 2700 eras. Optimum design (Fig. 32) with a workdone value of 268.72 is obtained after 2,65,600 FE evaluations. The constraints on volume and surface have been handled for the present using a penalty approach. The results are very practical but no comparisons with other solutions are available.

Fig. 28. Performance comparison.

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Fig. 32. Solution 268.72 N mm.

10. Conclusions Fig. 29. Result of fine discretization of 30  30 mesh, work = 8.417.

Fig. 30. 30  30 mesh, work = 8.435.

Fig. 31. Problem definition of 3D problem.

It is known that genetic algorithm offers a general approach for solving all kind of topology optimization problems including non-linear field problems. But most often SGA has been found to be converging to local optimal solutions and computationally very expensive. The proposed TSAGA overcomes both these drawbacks and always leads to global optimal solutions within half of the computational effort needed for SGA. Studies using advanced GAs such as SAIGA and AGA have also been carried out. SAIGA reaches very close to the global optimal solutions in many cases but its progress is very slow since it executes many islands in parallel and requires more number of FE evaluations. AGA is efficient but mostly converges to local optimal solutions. The proposed TSAGA has shown the best performance among all of them and it is also observed that, as the complexity of the problem increases the improvement in performance of TSAGA over the other GAs is more significant as indicated in the case of multiple loading problem where the TSAGA’s performance is far superior to others. In comparison with the power law approach, as an evolutionary algorithm TSAGA requires more number of function evaluations but always produces global optimum solutions. Wang’s bit-array GA approach requires less computational effort than the TSAGA approach but this also converges to a local optimal solution that contains many single element holes whereas the TSAGA has offered a global optimal solution that is completely free of noise. The result of the Voronoi bar representation method is slightly better than the TSAGA approach but the method is computationally more expensive than TSAGA. Apart from guaranteed convergence to global optima the present TSAGA has been computationally more efficient than SGA and SAIGA. The two-phase approach [53] based on the use of digital image processing techniques (such as thinning or medial axis transformation algorithm) for identifying the skeleton convergence of bit-array represented topology optimization problem and the use of a hybrid scheme combining GA and NLP has been tested to be more successful that reduces the computational effort by a factor of 6 along with TSAGA. Further research is needed to render it nearly competitive to gradient-based NLP approaches.

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References [1] M.P. Bendsoe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Eng. 71 (1988) 197–224. [2] A. Bensousson, J.L. Lions, G. Papanicolaou, Asymptotic analysis for periodic structures, North-Holland, 1978. [3] M.P. Bendsoe, Optimal shape design as a material distribution problem, Struct. Optim. 1 (1989) 193–202. [4] G.I.N. Rozvany, M. Zhou, T. Birker, Generalized shape optimization without homogenization, Struct. Optim. 4 (1992) 250–254. [5] G. Allaire, Shape Optimization by the Homogenization Method, Springer-Verlag, 2001. [6] L. Ambrosio, G. Buttazzo, An optimal design problem with perimeter penalization, Calc. Var. Partial Diff. Equat. 1 (1993) 55–69. [7] F. Belblidia, J.E.B. Lee, S. Rechak, E. Hinton, Topology optimization of plate structures using a single- or three-layered artificial material model, Adv. Eng. Softw. 32 (2001) 159–168. [8] C. Kane, M. Schoenauer, Topological optimum design using genetic algorithms, Control Cybern. 25 (5) (1996) 1059–1088. [9] Y.M. Xie, G.P. Steven, A simple evolutionary procedure for structural optimization, Comput. Struct. 49 (5) (1993) 885–896. [10] P. Tanskanen, The evolutionary structural optimization method: theoretical aspects, Comput. Methods Appl. Mech. Eng. 191 (2002) 5485–5498. [11] G.I.N. Rozvany, Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics, Struct. Multidiscip. Optim. 21 (2) (2001) 90–108. [12] M. Zhou, G.I.N. Rozvany, On the validity of ESO type methods in topology optimization, Struct. Multidiscip. Optim. 21 (2001) 80–83. [13] C.V. Ramakrishnan, R. Balamurugan, N. Swaminathan, Local and alternate optima in structural topology and shape optimization, in: Proceedings of International Conference on Computational Engineering and Sciences ICCES’05, Indian Institute of Technology Madras, Chennai, India, 2005, pp. 871–876. [14] R. Balamurugan, C.V. Ramakrishnan, N. Swaminathan, Integrated optimal design of structures under multiple loads for topology and shape using genetic algorithm, Eng. Comput. 23 (1) (2006) 57–83. [15] M.J. Jakiela, C. Chapman, J. Duda, A. Adewuya, K. Saitou, Continuum structural topology design with genetic algorithms, Comput. Methods Appl. Mech. Eng. 186 (2000) 339–356. [16] K. Suzuki, N. Kikuchi, A homogenization method for shape and topology optimization, Comput. Methods Appl. Mech. Eng. 93 (1991) 291–318. [17] M.P. Bendsoe, O. Sigmund, Topology Optimization: Theory, Methods and Applications, Springer-Verlag, 2004. [18] B. Hassani, E. Hinton, A review of homogenization and topology optimization, Comput. Struct. 69 (1998) 707–717 (I—homogenization theory for media with periodic structure) 719–738 (II—analytical and numerical solution of homogenization equations) 739–756 (III—topology optimization using optimality criteria). [19] M. Stolpe, K. Svanberg, Modeling topology optimization problems as linear mixed 0-1 programs, Int. J. Numer. Methods Eng. 57 (2003) 723– 739. [20] N. Swaminathan, C.V. Ramakrishnan, R. Balamurugan, Optimum topology design of products using genetic algorithms, in: International Conference on Numerical Methods in Industrial Forming Processes, Columbus, USA, 2004. [21] N. Swaminathan, Topology optimization of structures using genetic algorithms, Ph.D. Thesis, Indian Institute of Technology, Delhi, India, 2005. [22] H. Hamda, M. Schoenauer, Adaptive techniques for evolutionary topology optimum design, in: Parmee (Ed.), Evolutionary Design and Manufacture, 2000, 123–136. [23] S.Y. Wang, K. Tai, Structural topology design optimization using genetic algorithms with a bit-array representation, Comput. Methods Appl. Mech. Eng. 194 (2005) 3749–3770. [24] K. Deb, An efficient constraint handling method for genetic algorithms, Comput. Methods Appl. Mech. Eng. 186 (2–4) (2000) 311–338.

1623

[25] K. Deb, M. Goyal, A robust optimization procedure for mechanical component design based on genetic adaptive search, Trans. ASME: J. Mech. Des. 120 (2) (1998) 162–164. [26] K. Deb, M. Goyal, A combined adaptive search (GeneAS) for engineering design, Comput. Sci. Inform. 26 (4) (1996) 30–45. [27] T. Back, Self-adaptation in genetic algorithms, in: P.B. FJ Varela (Ed.), Proceedings of the First European Conference on Artificial Life, 1992, pp. 263–271. [28] E. Kee, A. Airey, W. Cye, An adaptive genetic algorithm, in: Proceedings of the Genetic and Evolutionary Computation Conference, San Francisco, Morgan Kaufmann Publishers, (2001), pp. 391–397. [29] H.G. Beyer, Toward a theory of evolution strategies: self-adaptation, Evolut. Comput. 3 (3) (1996) 311–347. [30] S. Pezeshk, C.V. Camp, State of the art on the use of Genetic Algorithms in design of steel structures, in: S. Burns (Ed.), Recent Advances in Optimal Structural Design, American Society of Civil Engineers, 2002. [31] N. Saravanan, D.B. Fogel, K.M. Nelson, A comparison of methods for self adaptation in evolutionary algorithms, BioSystems 36 (1995) 157–166. [32] F.G. Lobo, The parameter-less genetic algorithm: rational and automated parameter selection for simplified genetic algorithm operation, Ph.D. Thesis, University of Lisbon, Portugal, 2000. [33] R. Hinterding, Z. Michalewicz, T.C. Peachey, Self-adaptive genetic algorithm for numeric functions, in: Proceedings of the Fourth Conference on Parallel Problem Solving from Nature, 1996, pp. 420–429. [34] K. Deb, H.G. Beyer, Self-adaptive genetic algorithm with simulated binary crossover, Technical Report No. CI-61/99, Department of Computer Science/XI, University of Dortmund, Dortmund, Germany, 1999. [35] T. Krink, R.K. Ursem, Parameter control using the agent based patchwork model, in: Proceedings of the Congress on Evolutionary Computation, 2000, pp. 77–83. [36] D. Goldberg, K. Deb, A comparative analysis of selection schemes used in genetic algorithms, in: J.E.R. Gregory (Ed.), Foundations of Genetic Algorithms, Morgan Kaufmann Publishers, 1991, pp. 69–93. [37] H.G. Beyer, Evolutionary algorithms in noisy environments: theoretical issues and guidelines for practice, Comput. Methods Appl. Mech. Eng. 186 (2000) 239–267. [38] B.L. Miller, D.E. Goldberg, Genetic algorithms, selection schemes, and the varying effects of noise, Evolut. Comput. 4 (2) (1997) 113–131. [39] J.M. Fitzpatrick, J.J. Grefenstette, Genetic algorithms in noisy environments, in: P. Langley (Ed.), Machine Learning: Special issue on Genetic Algorithms, vol. 3, Kluwer Academic Publishers, Dordrecht, 1988, pp. 101–120. [40] Y. Murata, N. Shibata, K. Yasumoto, M. Ito, Agent oriented self adaptive genetic algorithm, in: Proceedings of Communications and Computer Networks (CCN), Cambridge, USA, November 4–6, (2002), pp. 348–353. [41] R. Weinberg, Computer simulation of a living cell. Dissertations Abstracts International 31(9) (1970) 5312B. [42] M. Miki, K. Hiroyasu, M. Kaneko, I. Hatanaka, A parallel genetic algorithm with distributed environment scheme, in: Proceedings of IEEE Systems, Man and Cybernetics Conference SMC’99, 1999, pp. 695–700. [43] E. Takashima, Y. Murata, N. Shibata, M. Ito, Self adaptive island genetic algorithm, in: Proceedings of Congress on Evolutionary Computation, 2003, pp. 1072–1079. [44] E. Takashima, Y. Murata, N. Shibata, M. Ito, Techniques to improve exploration efficiency of parallel self adaptive genetic algorithms by dispensing synchronization, in: Proceedings of the Fifth International Conference on Simulated Evolution And Learning (SEAL2004), 2004. [45] C.V. Ramakrishnan, R. Balamurugan, A two-stage adaptive genetic algorithm for structural topology optimization. Compilation of Abstracts, in: K.J. Bathe (Ed.), Third MIT Conference on Computational Fluid and Solid Mechanics, MIT, Cambridge, USA, (2005), p. 311. [46] C. Kane, Algorithmes genetiques et optimization topologique. Ph.D. Thesis, University of Paris, 1995. [47] D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, MA, USA, 2001. [48] D.L. Calloway, Using a genetic algorithm to design binary phase-only filters for pattern recognition, in: R.K. Belew, L.B. Booker (Eds.),

1624

R. Balamurugan et al. / Applied Soft Computing 8 (2008) 1607–1624

Proceedings of the Fourth International Conference on Genetic Algorithms, Morgan Kaufmann, 1991. [49] C.S. Jog, R.B. Haber, Stability of finite element models for distributedparameter optimization and topology design, Comput. Methods Appl. Mech. Eng. 130 (1996) 203–226. [50] R.B. Haber, C.S. Jog, M.P. Bendsoe, Variable-topology shape optimization with a control on perimeter, in: B.J. Gilmore, D.A. Hoeltzel, D. Dutta, H.A. Escheneuer (Eds.), Advances in Design Automation, ASME DE, American Society of Mechanical Engineers, New York 69(2) (1994) 261–272.

[51] O. Sigmund, A 99-line topology optimization code written in MATLAB, Struct. Multidiscip. Optim. 21 (2) (2001) 120–127. [52] H. Hamda, F. Jouve, E. Lutton, M. Schoenauer, M. Sebag, Compact unstructured representations for evolutionary design, Appl. Intell. 16 (2) (2002) 139–155. [53] R. Balamurugan, C.V. Ramakrishnan, N. Swaminathan, A two phase approach based on skeleton convergence and geometric representation for structural topology optimization using GA, Comput. Struct., submitted for publication.