An Enhanced Genetic Algorithm for Structural Topology Optimization

Transcription

1 An Enhanced Genetic Algorithm for Structural Topology Optimization S.Y. Wang a, M.Y. Wang a,, and K. Tai b a Department of Automation & Computer-Aided Engineering The Chinese University of Hong Kong Shatin, NT, Hong Kong b Centre for Advanced Numerical Engineering Simulations School of Mechanical and Production Engineering Nanyang Technological University, 50 Nanyang Ave, Singapore Abstract Genetic Algorithms (GAs) have become a popular optimization tool for many areas of research and topology optimization an effective design tool for obtaining efficient and lighter structures. In this paper, a versatile, robust and enhanced genetic algorithm (GA) is proposed for structural topology optimization by using problem-specific knowledge. The original discrete black-and-white (0-1) problem is directly solved by using a bit-array representation method. To address the related pronounced connectivity issue effectively, the four-neighborhood connectivity is used to suppress the occurrence of checkerboard patterns. A simpler version of the perimeter control approach is developed to obtain a well-posed problem and the total number of hinges of each individual is explicitly penalized to achieve a hinge-free design. To handle the problem of representation degeneracy effectively, a recessive gene technique is applied to viable topologies while unusable topologies are penalized in a hierarchical manner. An efficient FEMbased function evaluation method is developed to reduce the computational cost. A dynamic penalty method is presented for the GA to convert the constrained optimization problem into an unconstrained problem without the possible degeneracy. With all these enhancements and appropriate choice of the GA operators, the present GA can achieve significant improvements in evolving into near-optimum solutions and viable topologies with checkerboard free, mesh independent and hinge-free characteristics. Numerical results show that the present GA can be more efficient and robust than the conventional GAs in solving the structural topology optimization problems of minimum compliance design, minimum weight design and optimal compliant mechanisms design. It is suggested that the present enhanced GA using problemspecific knowledge can be a powerful global search tool for structural topology optimization. KEY WORDS: topology optimization; genetic algorithms; bit-array representation; connectivity analysis; black and white design; hinge-free design Correspondence to: M.Y. Wang, Department of Automation and Computer-Aided Engineering, The Chinese University of Hong Kong, Shatin, NT, Hong Kong. Tel.: ; Fax: yuwang@acae.cuhk.edu.hk (M.Y. Wang). 1

2 1. Introduction Structural topology optimization has become an effective design tool for obtaining efficient and lighter structures since the pioneer work of Michell [1] and the seminal work of Bendsøe and Kikuchi [2]. It has proven to be a powerful technique for conceptual design and it is also of considerable practical interest in the fact that it can achieve far greater savings than the conventional sizing or shape optimization [3]. Topology optimization is even regarded as the best method for solving the structural optimal design problem and for producing the best overall structure [4]. Continuum structural topology optimization as a generalized shape optimization problem for higher volume fractions [3] has received extensive attention and experienced considerable progress over the past few years. Up to now, various families of structural topology optimization methods have been well developed [5, 6]. One of the most established families of methods is the one based on the homogenization approach first proposed by Bendsøe and Kikuchi [2], in which the structural form is represented by microstructures with voids and the material throughout the structure is redistributed by using an optimality criteria procedure. However, the evaluation of optimal microstructures and their orientations is usually cumbersome and numerically complicated [7]. As an important alternative approach within this family, the power-law approach [7], which is also called the SIMP (Solid Isotropic Microstructure with Penalization) method [8] and originally introduced by Bendsøe [9], has got a fairly general acceptance in recent years [3] because of its computational efficiency and conceptual simplicity. However, it does not directly attack the original 0-1 problem [10] and thus tends to converge to a local optimal topology with blurry boundary or undesirable checkerboard patterns [3,6,10,11] 2

3 or converge to an infeasible solution to the original 0-1 problem [12]. Since the problem of non-existence of solutions (ill-posedness) is not resolved, priori restrictions on the admissible design configurations such as a perimeter constraint, a gradient constraint or with filtering techniques [7, 10] must be introduced [13]. Another recognized family of structural optimization methods is the one based on the Evolutionary Structural Optimization (ESO) approach proposed by Xie and Steven [14], in which the material in a design domain which is not structurally active is considered as inefficiently used and can thus be slowly removed [15]. However, it is only an intuitive method without a proof of optimality and it may also easily lead to a non-optimal design [3, 16, 17] and more recently it has been recommended that ESO should be applied to structural problems having pin-jointed connections since ESO appears to produce truss-like topologies [4]. For structural topology optimization problems, an emerging family of methods is the one based on the optimization of implicit interfaces, which are used to define the exterior and interior boundaries of a structural topology, such as the level-set methods in [18 21] and implicit functions regularization method in [22]. However, since the interface is moving across the elements of a fixed mesh, accurate finite element analysis (FEA) with remeshing would become too cubersome and the less accurate but efficient FEA procedures without remeshing adopted in the literature [19 22] may become inappropriate. In topology optimization, usually there are many solutions such as one global and many local minima to a given problem. All those afore-mentioned families of methods cannot perform a global search and thus do not necessarily converge to the global optimal solution for the given objective function and constraints [3, 17]. Furthermore, the final solution depends on the given initial design [10] and different solutions to the same problem with the same discretization and optimization method by using different start- 3

4 ing solutions may be readily obtained. Instead of searching for a local optimum, one may want to find the globally best in the feasible region. For many real world problems only the absolutely best (extremum) is good enough [23] and thus a global optimization method is generally more desirable. It is well known that the GAs, which are based on the Darwinian survival-of-the-fittest principle to mimic natural biological evolution, are a stochastic global optimization method. Since the seminal work of Holland [24] and the comprehensive study of Goldberg [25], GAs have become an increasingly popular optimization tool for many areas of research. More recently, GAs have been gradually recognized as a powerful and robust stochastic global search method for structural topology optimization [6, 11, 26 37]. Sandgren et al. [26] are among the first researchers to develop a GA-based approach for optimal topology design of continuum structures. In their work, the design domain was discretized into small elements, where each element contains material or void and thus no intermediate densities are allowed. This is a typical bit-array (binary-string) representation method [27,28], in which a bit-array or binary-string is used to define the design variables and can be directly mapped into the design domain discretized by a fixed regular mesh, where each of the small elements contains either material or void. In the work of Sandgren et al. [26], with this representation method, the GA was then used to determine the optimal configuration of material and void within the design domain such that the structure s weight is minimized subject to displacement and stress constraints. Hence, the original 0-1 optimization problem was attacked directly by using a bit-array representation method and a genetic algorithm. This bit-array representation method has been widely adopted since it is an intuitive and straightforward method to represent the structural topology for the optimal topology design problems using the GAs [36]. 4

5 This GA-based approach has been extended by Jakiela and his co-workers [27, 31, 38], Schoenauer and his co-workers, [28, 29, 36], Fanjoy and Crossley [32, 33], and, more recently, by Wang and Tai [6]. Jakiela and his co-workers [27, 31, 38] extended the work of Sandgren et al. [26] by addressing such problems as cantilevered plate topologies of high discretization, techniques for obtaining finely discretized topologies, techniques for obtaining families of highly fit designs and a variety of different structural design fitness functions. Schoenauer and his co-workers [28, 29, 36] demonstrated the advantages of using this natural representation and presented two specific two-dimensional crossover operators to alleviate the strong geometrical bias against vertical blocks in one-dimensional crossover operations. Furthermore, the disadvantages in computational cost [29] and representation degeneracy [28, 36] were also addressed. Fanjoy and Crossley [32, 33] investigated the use of a GA for topology design of planar cross-sections under bending and torsion. Four crossover methods were examined and a chromosome mask was used to enforce connectivity. Wang and Tai [6] further studied the problem of representation degeneracy of this bit-array representation method and proposed a connectivity handling approach based on image processing and dynamic penalty to alleviate this problem. Furthermore, a uniform initialization method was demonstrated as effective in evolving structurally connected topologies and avoiding possible failure of the GAs as mentioned in [36]. Although all these extensions can well prevent checkerboard patterns by exploiting a connectivity restriction, the other numerical instabilities [10] in structural topology optimization such as mesh dependency and one-node connections still exist. More importantly, the issue of design connectivity analysis, which affects the computational efficiency significantly, has not been fully solved. It is well known that design connectivity analysis is an important issue in struc- 5

6 tural topology design optimization using the GAs based on a bit-array representation method [6, 32] since structural topologies must be well connected to possess the loadcarrying capacity while the bit-array representation cannot guarantee this requirement for connectivity. However, this issue has not been fully addressed yet. In the connectivity analysis model presented by Jakiela and his co-workers [27, 31, 38], all the solid elements in the design domain which are not connected (whether directly or indirectly via other elements) to a seed element are switched to void. Since unconnected designs (individuals) are discarded only, the representation for unconnected designs becomes heavily degenerate [6, 28] and thus it cannot drive the GA search towards topologies combining higher structural performance and fewer disconnected material elements [38] and the computational cost turns out to be much high, as stated in [31]. Representation degeneracy is accepted to be generally bad for the GA optimization since the application from the genotypic space into the phenotypic space is not injective [6, 28]. In the work of Fanjoy and Crossley [32,33], connectivity is enforced by using a chromosome mask, in which only selective portions of the chromosome that correspond to connected locations are expressed in the design. Analogous to recessive genes, parts of a chromosome not expressed in a parent design are kept and could become connected in one of the offspring designs after the crossover operation. It was illustrated that this handling approach outperforms other approaches such as ignoring connectivity, discarding unconnected designs and penalty for unconnected elements in evolving connected planar cross-sections since it may lead to a broader search of the design space [32]. However, representation degeneracy still exists since only some parts of a chromosome are used in the estimation of the objective function. Furthermore, it cannot drive disconnected topologies to evolve into connected topologies to speed the GA search. In the connectivity analysis model 6

7 developed by Schoenauer [28], the representation degeneracy was handled by accounting for the unconnected material in the fitness function, but penalizing the total area of the unusable material only would not drive the formation of fewer disconnected objects. Since various combinations of disconnected objects may have the same total area, this handling approach also has the problem of representation degeneracy. Hence, it cannot favor the topologies with fewer unusable objects and the computational cost may still be high [28, 29]. As a whole, all the afore-mentioned design connectivity handling approaches cannot well bias the formation of connected topologies from randomly generated disconnected topologies through GA evolution [6] since numerous invalid designs would have the same null or poor fitness value after the FEA is performed by assuming a small Young s modulus for the void elements [27, 28, 31, 32, 38]. The efficiency of the GAs would thus be greatly reduced for solving practical problems. As reported in [28], an inappropriate connectivity handling approach has led to the failure of a GA for topology optimization of a long narrow design domain, because there may be no connected individuals existing in the population in the early generations and without the bias the GA itself was degenerated to a random search and was thus unable to arrive at a solution within reasonable time [32]. In the connectivity analysis model proposed by Wang and Tai [6], the problem of representation degeneracy was greatly alleviated by biasing the GA search towards the topologies with higher structural performance, less unusable material and fewer separate objects in the design domain. The total number of separate objects and the total area of unusable objects are penalized in a hierarchical manner for invalid designs. However, representation degeneracy may still exist when a number of invalid designs have the same total area but different sizes of unusable objects. Furthermore, the computational effort to evolve into a structurally connected topology 7

8 may become unnecessarily too much since the recessive gene technique [32] was not introduced such that a disconnected topology with a structurally connected part cannot be taken as a viable structure. The objective of this study is to develop an enhanced GA for structural topology optimization using the bit-array representation method. To address the connectivity issue effectively, the four-neighborhood connectivity originally developed for image processing is used. A restriction on the minimum size of internal holes of viable topologies is imposed to obtain a well-posed problem for topology optimization. The total number of one-node connected hinges of each connected topology is explicitly penalized to achieve hinge-free designs. In the design connectivity analysis, a recessive gene technique is adopted and structurally disconnected topologies without viable components are penalized in a hierarchical manner. A modified version of the dynamic penalty method is presented for the GA to convert the constrained optimization into unconstrained optimization. The present GA is finally applied into solving the structural topology optimization problems of minimum compliance design, minimum weight design and optimal compliant mechanisms design. 2. Principle of Genetic Algorithms GAs are efficient and generally applicable global search procedures based on a stochastic approach which relies on the Darwinian survival-of-fittest principle [24, 39]. GAs operate on a population of potential solutions to produce possibly better and better approximations to the optimal solution through evolution [25]. The population is a set of chromosomes and the basic GA operators are selection, crossover and mutation. 8

9 At each generation, a new set of approximations is created by the process of selecting individuals and breeding them together using crossover and mutation operators which are conceptually borrowed from natural genetics. Hopefully, this process leads to the evolution of better individuals with near-optimum solutions over time [6]. Generally, GAs perform well in finding areas of interest even in a complex, real-world scene. While a GA may never produce the absolutely best solution (global optimum), it is mathematically likely to get very close to it by using a fraction of the computational requirements of an exhaustive deterministic search. The advantages of GAs would include not only the global nature of the search process, but also the indifference to system specific information [40], especially the derivative information, the versatility of application [31], the ease with which heuristics can be incorporated in optimization [41], the capability of learning and adapting to changes over time, the implicitly parallel directed random exploration of the search space [42], and the ability to accommodate discrete variables in the search process [24]. However, GAs are usually computationally expensive. There may be a significant increase in the number of function evaluations required to attain an optimal solution [41]. GAs may even become unrealistic to handle largescale design problems due to the prohibitive computation [30, 31, 36, 41]. Nevertheless, the computational efficiency of the GAs can be definitely improved. It has been well recognized that the use of micro-gas can increase the search efficiency significantly in problems involving a large number of candidates [41, 43, 44]. Furthermore, to use parallel implementations of the GAs, which can integrate large numbers of processors and significantly reduce the computational time of many practical applications, has become one of the most promising choice because of the parallel nature of the GAs and the availability of parallel computing hardware [25, 42, 45]. 9

10 In the present study, instead of using a general method such as a parallel GA or a micro-ga, improvements of the GAs in the robustness and efficiency are achieved by using the problem-specific knowledge. It has been reported by many researchers [6, 11, 25, 40] that incorporating problem-specific knowledge leads to a more advantageous genetic algorithm. In the present GA for structural topology optimization, the robustness and efficiency of the GA are enhanced by using an image-processing-based connectivity analysis, the well-posedness of the problem, a hierarchical constraint violation penalty method, and an efficient strategy in the FEA, as well as the appropriately chosen GA operators. It will be illustrated that by using this enhanced GA, for relatively high volume fractions, even coarse meshes can give a good indication of shape and topology and a good estimate of the optimal solution, similar to the topology optimization using the SIMP as stated in [10]. Note that for comparatively small volume fractions the layout theory and truss topology methods may be more applicable for structural topology optimization than the continuum structural topology optimization methods [3, 10]. 3. Implementation of the Enhanced GA 3.1. Image-Processing-Based Connectivity Analysis In this design connectivity analysis, the fixed discretized design domain in topology optimization is regarded as a black-and-white digital image. Each element is analogously considered as one pixel and its color is represented by the binary design variable, where white is void and black is solid material. Similar image-processing-based handling approach for continuous structural topology optimization has been proposed by 10

11 Sigmund [46] to eliminate mesh dependency and checkerboard problems [47], though connectivity analysis was not involved. In image processing [48], either a 4-neighborhood connectivity, where only vertical and horizontal directions can be followed, or a 8-neighborhood connectivity, where horizontal, vertical and diagonal directions are allowed, can be used, as shown in Fig. 2. To determine the design connectivity more effectively, the 4-neighborhood connectivity is employed since the undesirable patches of checkerboard patterns, within which the density of the material assigned to contiguous finite elements (FEs) varies in a periodic fashion similar to a checkerboard consisting of alternating solid and void elements [10], will not be considered as appropriately connected in the 4-neighborhood connectivity and can thus be totally eliminated in a connected topology. In image processing, region identification is necessary for region description [48]. In the present study, connected component labeling, which labels each region with a unique (integer) number, is used for region identification. With this labeling, the number of connected regions and their relative areas can be readily obtained with a simple inspection of the labeled image s histogram. Figure 2 shows that the present region identification can not only label the connected components of a topology, but also identify the structurally connected component, which has the load bearing capacity when assuming that the left end of the design domain is fixed and a concentrated load is applied at the centre of the right free end, as shown in Fig. 2(d). A recessive gene technique [32] is here further developed to deal with the disconnected topology with a load bearing connected component, which is defined as a viable topology in the present study. When such a component is identified, all the other components are switched off to void, and the original topology is degenerated into a structural topology to perform the structural analysis and objective 11

12 function evaluation. It should be noted that parts of a bit-array representation chromosome not expressed in the parent structural topology are not modified and could thus become useful in one of the offspring structural topologies after the crossover operation. Generally, this handling approach may lead to a broader search of the design space [32] and improve the computational efficiency. Due to the stochastic nature of the GAs, it is very common to produce topologies without a load bearing connected component. As discussed in detail in [6], for a moderately large problem, it is very often to see that there is no structurally connected component which has the load bearing capacity in the population in the early generations due to the random initialization. For individuals without a structurally connected component, to prevent the GA from degenerating into a random search and to improve the computation efficiency by driving the structurally disconnected components to evolve into structurally connected components, a penalty function on the design connectivity violation is defined as follows: viol(x) c = Γ d n d + Γ a à d + Γ m à m (1) where x is the bit-array design variable vector, viol(x) c the penalty function on the design connectivity violation, n the total number of structurally disconnected components, à d the total area of structurally disconnected components, à m the minimum area of structurally disconnected components, Γ d, Γ a and Γ m the corresponding penalty parameters, respectively, which are usually chosen in a hierarchical manner such as Γ d Γ a Γ m in order to favor the evolution into a single structurally connected component effectively. To alleviate the problem of representation degeneracy [28], both the total area and minimum area of disconnected components are here used. 12

13 3.2. Well-posedness of the Problem As observed by many researchers [6, 27, 30, 36], the bit-array representation method has the strong limitation of mesh dependency, which makes the final solution sensitive to the fineness of the FE mesh discretization and the final solution with more details and smaller holes obtained from a finer mesh may create a problem from a manufacturers point of view. This limitation was not directly resolved in the GA-based topology optimization methods in [6,27,30,36]. Generally, in topology optimization, mesh dependency is a typical numerical instability, as discussed in detail in [10]. The original infinite dimensional topology optimization problem which the discretized problem approximates may lack a solution in its general continuum setting [10] since the optimization problem is ill-posed and the set of feasible designs is not closed [49]. The approaches to generate mesh-independent solutions are to reduce the space of admissible designs by adding priori restrictions either globally or locally [7, 10], such as the perimeter control approach [50 52], the mesh independent sensitivity filtering method [46], the density slope method [53], and the nonlinear bilateral filtering method [13]. Among them, the perimeter control approach [50 52] can be efficient to control the number and size of holes and hence favors the manufacturability of the design. However, to approximate fluctuations in the design variables can be quite difficult and the choice of the bounding value of the perimeter constraint can be rather tricky [10]. In the present study, a simpler version of the perimeter control approach is developed. It uses only the minimum area of internal holes of structurally connected components in viable topologies, rather than the sum of the lengths/areas of all inner and outer boundaries [10, 51], to enforce a restriction such that the minimum size of internal holes is independent of the mesh 13

14 size and thus the mesh-dependent small holes will be prevented in the feasible designs. It should be noted in the bit-array representation all the areas of internal holes can be obtained easily by using the present connected component labeling approach based on image processing. The size of small internal holes that can be produced in the design domain is thus constrained and existence of solutions to the perimeter controlled topology optimization can be assured, as discussed in [10]. To maintain this perimeter control to achieve the well-posedness of the problem, a penalty function is defined as follows: viol(x) s = Γ s (A 0 Ãs(x)) (2) where viol(x) s is the penalty function on this perimeter control violation, A 0 a predefined value to restrict the minimum area of internal holes, Ã s (x) the minimum area of the internal holes and Γ s the corresponding penalty parameter. In continuum topology optimization, another serious numerical instability is the occurrence of checkerboard patterns in the final topologies. As afore-mentioned, checkerboard patterns can be effectively prevented and eliminated by adopting a 4-neighborhood connectivity. However, moment-free one-node connected hinges cannot be prevented in the final designs by this strategy. Since the stress in a sharp hinge would approach infinity and the structure would break, techniques to avoid them should be developed. According to Bendsøe and Sigmund [10], the existence of one-node hinges in optimal topologies is a widespread phenomenon in structural topology optimization and only some specially designed schemes such as the one in [54] can prevent the non-physical one-node connected hinges. In the present study, an explicit constraint is added to achieve hinge-free optimal designs and the corresponding violation function can be de- 14

15 fined as follows: viol(x) h = Γ h n h (3) where viol(x) h is the penalty function on the moment-free one-node connected hinges, n h the total number of hinges and Γ h corresponding penalty parameter Efficient Evaluation of the Objective Function In this study, a finite element method (FEM) based on the bilinear rectangular finite elements [7] is used to evaluate the objective function. In GA-based topology optimization methods [27, 28], a common strategy was often adopted that void elements are assigned a very small Young s modulus to keep them inactive and to avoid the singularity, just as the one using in homogenization-based topology optimization methods [2, 9, 10]. Since the void elements are not actually killed in this strategy, computational cost would be unnecessarily too large. In the present study, a different strategy in which the void elements are totally eliminated in the FEA is developed. All the degrees of freedom related to the void elements only are eliminated and thus the local stiffness matrices of void elements are not involved in the assembled global stiffness. The finite element analysis is performed for individuals with usable topologies only, rather than for all the individuals no matter whether they are structurally viable or not as in [27,28,36]. Since the present image-processing-based connectivity can guarantee that usable topologies are structurally connected, the problem of singularity does not exist. Hence, the computational efficiency in the evaluation of objective function can be significantly improved due to the large reduction of the total number of degrees of freedom in the global stiffness equation for structurally viable topologies and the skip of FEA for the structurally 15

16 unusable topologies Artificial Unconstrained Optimization Generally, the single-objective constrained optimization problem can be written as follows: Minimize: f(x), x Ω subject to g i (x) 0, i = 1, 2,..., I (4) h j (x) = 0, j = 1, 2,..., J where x is the solution vector in the design domain Ω, f(x) the objective function, g i (x) the i-th inequality constraint function, I the number of inequality constraints, h j (x) the j-th equality constraint function, and J the number of equality constraints. In this study, only structural topology optimization problems with a single objective are considered. In the field of structural topology optimization, viable topologies must have the load bearing capacity and thus the connectivity requirement should be introduced an a constraint. Furthermore, as afore-mentioned, other constraints such as the minimum size of internal holes and hinge-free designs should also be taken into account. Considering the fact that all the other constraints such as the constraints on the volume fraction or the maximum displacement can be taken as inequality constraints, the general single-objective constrained optimization problem in Eq. (4) can be re-written 16

17 as Minimize: f(x), x Ω subject to g i (x) 0, i = 1, 2,..., I n c 1 = 0 (5) A 0 Ãs(x) 0 n h = 0 where n c is the total number of structurally connected components corresponding to the bit-array design variable x in the design domain Ω discretized by a fixed regular mesh. It is well known that the GAs are usually designed for unconstrained optimization only [25]. In order to tackle the constrained optimization, the constrained optimization problem shown in Eq. (5) has to be converted into an artificial unconstrained optimization one by adopting a constraint handling approach. Since GAs are generic search methods, most applications of GAs to constrained optimization problems have used the penalty function constraint handling approach [29,36,44,55,56], but the problem of how to set appropriate values for the penalty parameters in order to always obtain feasible best individuals in the population may arise. It should be stressed that inappropriate choices may lead the convergence to infeasible solutions. In the present study, a constraint handling method proposed by Deb [55], which is also based on the penalty function approach, is further developed to address this issue. The main idea of this method is to use a tournament selection operator and to apply a set of criteria to decide the selection process as follows: 1. Any feasible solution is preferred to any infeasible solution. 2. Between two feasible solutions, the one having better objective function value is preferred. 17

18 3. Between two infeasible solutions, the one having smaller constraint violation is preferred. Based on these criteria, an artificial unconstrained objective function of the constrained optimization shown in Eq. (5) is defined as f(x) F (x) = f + viol(x) if x F otherwise (6) where F (x) is the artificial unconstrained objective function, F the feasible region of the fixed design domain Ω, f the objective function value of the worst feasible solution in the population, and viol(x) the summation of all the violated constraint function values. It can be seen from Eq. (6) that this approach is based on the penalty function approach, but it does not require any penalty parameter to enforce that infeasible solutions are always with worse objective function values than feasible ones, while the conventional penalty function approaches usually have the difficulty to set the right values of penalty parameters to fulfill this goal. Hence, this method can be more efficient and robust. If there are no feasible individuals existing in the population, which is often seen in the early generations of the GA, a pre-defined worst value which can guarantee that any infeasible solution is worse than any feasible one, rather than 0 as recommended in Deb s method [55], is assigned to f in order to allow the GA to use an elitist strategy [25]. In Deb s method, since it is assumed f = 0 for this case, it cannot guarantee that any feasible solution is preferred to any infeasible solution when some most highly fit but infeasible individuals of the population are passed on to the next generation without being altered by genetic operators. Furthermore, due to the FE discretization, some objective functions such as the weight can only take discrete values. Since many different feasible topologies may have 18

19 the same weight value, the problem of degeneracy arises. For minimum weight design problems, to alleviate this degeneracy, the objective function F (x) defined in Eq. (6) is modified with a bias towards the stiffer one if both topologies are feasible and have the same weight, which can be written as f(x) Γ f (D lim D(x) max ) F (x) = f + viol(x) if x F otherwise (7) where the problem dependent penalty parameter Γ f is set a small value Γ f = 10 6 in the present study, D(x) max the maximal displacement under the loading, and D lim the prescribed limit on displacement. Since the stiffer one between two feasible solutions with the same objective function value will be assigned a better objective function value, its rank-based fitness will be higher and the probability to produce better feasible offspring will be higher. It should also be noted that if Γ f < 0, the bias will be set to the weaker, and if Γ f = 0 no bias will be imposed, just as shown in Eq. (6). According to the previous discussion, the summation of all the violated constraint function values viol(x) can be written as 0 if x F viol(x) = viol(x) p else if x C (8) viol(x) c + viol(x) s + viol(x) h otherwise where C is the set of structurally viable topologies, viol(x) p the summation of all the violated constraint functions related to the performance of a structurally viable topology, and viol(x) c, viol(x) s, viol(x) h given by Eqs. (1), (2), (3), respectively. Since design connectivity is an important issue that must be handled properly in the structural topology design to ensure the success of the GA, all the violations should be penalized in a hierarchical manner such that the GA search towards topologies with viable components 19

20 are more preferred to guide rapid evolution from unusable topologies into structurally viable topologies GA Operations Chromosome Representation As afore-mentioned, in the present study, the bit-array representation method is adopted to define the topology in a design domain discretized by a fixed regular mesh, in which each discrete design variable can be either 0 or 1. This is a straightforward and natural representation method and the decoding step is almost eliminated [32] Rank-Based Fitness Assignment Rank-based fitness assignment usually behaves in a robust manner. In the present work, Baker s linear ranking algorithm [57] with a selective pressure of 2 is used to ensure that no single individual generate an excessive number of offspring. The fitness of each individual in the population is defined as F (x i ) = 2 (n i 1) N ind 1, (9) where F (x i ) is the fitness of the i-th individual, n i the position of the i-th individual in the individuals rank based on the values of the artificial unconstrained objective function F (x), and N ind the population size Elitist Strategy In the present study, an elitist strategy (elitism) [25] is also employed to ensure that one or more of the most fit individuals in the population propagate through successive 20

21 generations. Elitism usually brings about a more rapid convergence of the population and also improves the chances of locating the optimal individual Selection Method Selection is the process of determining the number of times that a particular individual is chosen for reproduction. In this study, the SUS (Stochastic Universal Sampling) method [58] is adopted since SUS can ensure a selection closer to what is deserved than the roulette wheel selection and has the advantage of minimizing chance fluctuations Crossover Method Crossover is the main GA operator to produce new individuals that have some parts of both parent s genetic material. The uniform crossover method [59] is adopted in the present work since it makes every locus a potential crossover point so that any form of associated bias is reduced Mutation Method Mutation is usually used as a background GA operator to enforce a random move in the neighborhood of a given point in the design domain. A binary mutation method in the Breeder Genetic Algorithm (BGA) [60], which flips the value of each bit with a low mutation rate, is adopted to test more often in the neighborhood of the given point Stopping Criteria of the GA It is usually difficult to formally specify convergence criteria of the GA because of its stochastic nature. The stopping criterion used here is to terminate the GA after a 21

22 prespecified maximum number of generations, or if no improvement has been observed over a prespecified number of generations, whichever is encountered first. 4. Results and Discussion In this section, the present enhanced GA is applied to typical structural topology design optimization problems such as minimum compliance, minimum weight and optimal compliant mechanism to demonstrate its efficiency and versatility. Unless otherwise stated, the following settings are used in the numerical experiments presented below: standard GA evolution with a population size of 100, a fraction of 0.1 of the population are considered elites and a generation gap of 0.9 and a mutation rate of 0.01; all runs are stopped after 500 generations or if no improvement has been obtained over 150 generations (whichever occurs first); all the runs are carried out using MATLAB; all the numbers of function evaluations are referred to the numbers of evaluations of the structurally viable topologies using the FEA based on Sigmund s corresponding MATLAB codes in [7, 10], in which planar stress rectangular elements are used; all the results obtained for each problem are based on 20 independent runs of the GA; and all the CPU time is based on a desktop computer with an Intel Pentium IV processor of 3.2 GHz clock speed. In the comparative topology optimization using the SIMP method [10], it is assumed that the penalization power is 3.0 and the filter size 1.2 (divided by element size) [7]. As for the material properties of the plates, it is assumed that the Young s elasticity modulus E = 1, thickness of the plate t = 1 and Poisson s ratio ν = 0.3. For the penalty parameters, it is also assumed that Γ d = nelx nely, where nelx and nely are the number of elements in the horizontal and vertical directions, respectively, 22

23 Γ a = nelx nely, and Γ m = Γ s = Γ h = Minimum Compliance Design The minimum compliance topology optimization can be expressed as Minimize: C (x), x Ω subject to V (x)/v 0 f n c 1 = 0 (10) A 0 Ãs(x) 0 n h = 0 where C(x) is the compliance of the topology, V (x) and V 0 the material volume and the design domain volume, respectively, and f the prescribed volume fraction. The minimum compliance design problem, as shown in Fig. 3, is a 2 1 cantilever beam with the left boundary fixed with support and a unit point force applied vertically downward at halfheight of the right boundary. The mesh size is and thus nelx = 24, nely = 12. It is also assumed that f = 0.5 and A 0 = 3 (divided by element size). The performance comparison using different connectivity handling approaches only is shown in Fig. 4. On the average (due to the stochastic nature of the GAs), the present connectivity handling approach as shown in Eq. (1) outperforms Wang and Tai s approach [6] as well as Fanjoy and Crossley s approach [32] in the convergence speed of the objective function F (x) in terms of number of generations as shown in Fig. 4(a) and the ratio of viable topologies in the population during the GA evolution as shown in Fig. 4(b). Wang and Tai s approach [6] performs well in the early generation due to its bias towards topologies with fewer components. However, since a viable topology is referred to a topology with a single structurally connected component only and the recessive gene 23

24 technique [32] was not introduced, the search space will be greatly limited and thus after the early generations its performance is outperformed by Fanjoy and Crossley s approach. Due to the random initialization, there may be no viable topologies in the initial population as shown in Fig. 4(b). Since no bias is imposed towards unusable topologies in Fanjoy and Crossley s approach, the GA itself may be degenerated to a random search to find a viable topology in the early generations and therefore its performance becomes the worst in the early generations. A further numerical comparison is shown in Table 1. It can be seen that the present approach generates largest number of viable and feasible topologies, while Wang and Tai s approach produces the smallest and also requires the smallest amount of CPU time due to the fact that the FEA, usually the most time consuming part in GAs, is needed for feasible topologies only for this example with a volume constraint. The comparison of final optimal topologies is shown in Fig. 5. It can be seen that the final optimal topologies are very similar but not exactly the same due to the existence of small holes and moment-free one-node hinges, which are not addressed directly in these connectivity handling approaches. Figure 6 displays the performance comparison using different combinations of the constraint handling schemes shown in Eqs. (1), (2) and (3). On the average, among the four possible combinations, the one with the connectivity handling viol(x) c only achieves a fastest convergence speed due to the fewest constraints involved. However, it converges with a lower ratio of viable topologies (87.75%) than any of the other combinations (about 93.25%) due to the degeneracy resulting from the existence of small holes and one-node hinges in viable topologies. Since the additional introduction of constraints on one-node hinges would remove all internal holes with one-node hinges, which benefits the control on the size and number of internal holes, the combination 24

25 viol(x) c +viol(x) s without constraint of hinges converges slowest among the four combinations. Table 2 displays the numerical comparison using different combinations of the constraint handling scheme, in which the notation (...) is referred to the combination using an inefficient FEA procedure in which a small Young s modulus E = E is assigned to void elements as described in [27,28]. It can be seen that this FEA procedure is quite inefficient since it needs more CPU time (about 1 times more) per generation than the proposed FEA-based evaluation of the objective function, though the averaged final solutions are almost the same. Hence, the present evaluation of the objective function is accurate and efficient. Due to the constraint on the size of internal holes, the total number of internal holes is significantly reduced. The final optimal topologies using different combinations of the constraint handling schemes are shown in Fig. 7. It can be seen that all the final topologies are similar to some extent but the proposed combination viol(x) c +viol(x) s +viol(x) h performs best to achieve an optimal hinge-free design without mesh-dependent small holes. It would be interesting to compare the present global search method with a local search method: the popular SIMP method [3, 10]. Figure 8 displays two different final topologies using the SIMP method with a uniform distribution initialization and with or without filtering. It can be seen that in SIMP the checkerboard patterns in the final topology are apparent without using filtering (or adding other restrictions) but a linear filtering technique [7] with an appropriately chosen filter size can eliminate the checkerboard patterns effectively. Comparing Figs. 8(b) with 7(d), it can be seen that both final topologies are quite similar and the one resulting from the SIMP method may suffer from its blurry boundary due to the smoothing of the linear filter across the edges [13]. The blurry boundary phenomenon may cause difficulties in boundary 25

26 identification and design realization in a post-processing step which is necessary for shape recovery from the optimization solution. It should be noted that the objective of most topology optimization is to generate black-and-white design with distinct solids and voids. In this sense, the present GA-based topology optimization, which can only produce black-and-white designs, has its definite advantage. However, the computational cost of the present GA may be prohibitive compared with that of the SIMP method. The comparison of the convergence speed using the present GA and the SIMP method is shown in Fig. 9. It should be noted that an inefficient FEA procedure is used in the SIMP method [7,10] so that the CPU cost per FEA can be larger. Anyway, the present GA needs larger amount of CPU time to reach the convergence due to its independency of the initialization and parallel implementations of the GAs, though not implemented in the present study, can reduce the computational time significantly in general [25,42,45]. On the other hand, the local search SIMP method, though computationally effective, needs a good initial guess to converge to a good solution. Since a good initial guess is generally unavailable for real-world complex problems, a combination of both the local search SIMP method and the global search GA to incorporate their advantages can be a promising approach for finding near-optimum solutions, however, it is out of the range of the present study. It has been shown that for the volume fraction f = 0.5 a good indication of the optimal topology has been achieved by using a coarse mesh by both the present GA and the SIMP method. Furthermore, for a higher volume fraction of f = 0.7, as shown in Figs. 10(a) and 10(c), the present GA can still obtain a good black-andwhite solution, while the SIMP method generates a final topology with more blurry boundary and thus a finer mesh and more computational cost are needed to produce 26

27 a better solution (an acceptable minimum filter window size has been used). In this sense, the present GA is robust. A convergence speed comparison is shown in Figs. 10(c) and 10(d). Again, the local search SIMP method demonstrates its much higher computational efficiency than the present GA Minimum Weight Design The minimum weight optimal topology design problem can be expressed as Minimize: W (x), x Ω subject to D(x) max D lim 0 n c 1 = 0 (11) A 0 Ãs(x) 0 n h = 0 where W (x) is the weight of the topology. The minimum weight optimal topology design problem in [36], as shown in Fig. 11, is a 1 2 cantilever beam with the left boundary fixed with support and a unit point force applied vertically downward at half-height of the right boundary. The mesh size is and thus nelx = 10, nely = 20. It is also assumed that D lim = 20 and A 0 = 1 (divided by element size). The unconstrained objective function F (x) defined in Eq. (7) is adopted for this example to alleviate the problem of degeneracy in feasible solutions which have the same weight value but different topologies, as afore-mentioned. The performance comparison using different suggested bias approaches is shown in Fig. 12. The present approach with a bias towards the stiffer topologies as shown in Eq. (6) converges fastest while the one with a bias towards the weaker ones converges slowest since the stiffer topologies would be most preferred and the weaker ones least desirable for this problem. It can also be 27

28 seen that without the bias the speed of convergence becomes slower than the one with a bias towards the stiffer due to the existence of degeneracy. Figure 13 displays a comparison of different resulting final topologies. It can be seen that the present approach achieves a best final topology with a minimum weight of 0.2, which is identical to the one shown in [36] using a different representation method (Voronoi-bar representation) and a maximum number of generations of 2,000. Hence, the accuracy and efficiency of the present approach is justified Optimal Compliant Mechanism Design This example aims to find the optimal design of a basic compliant mechanism problem, a displacement converter as shown in Fig. 14(a), using the present GA-based topology optimization. In the design of compliant mechanisms using the FE-based continuum structural topology optimization methods, it is well known that there is a strong tendency to generate de facto hinges into the final designs, making them functionally similar to rigid-body mechanisms [61, 62]. Such de facto hinge zones are typically artifacts of the FE model used in the numerical analysis [10, 62] and one of the major methods to eliminate them is to introduce an artificial spring model, as shown in Fig. 14(b), which has been adopted by many researchers [61, 62]. The objective of this optimization problem is to maximize the displacement u out on a workpiece modeled by a spring with a stiffness k out under the action of an input actuator modeled by a spring with a stiffness k in and a force f in. This optimal compliant mechanism design problem can be expressed 28