An Enhanced Genetic Algorithm for Structural Topology

Optimization

S.Y. Wang a, M.Y. Wang a, ∗, † and K. Tai b

aDepartment of Automation & Computer-Aided EngineeringThe Chinese University of Hong Kong

Shatin, NT, Hong Kong

bCentre for Advanced Numerical Engineering SimulationsSchool of Mechanical and Production Engineering

Nanyang Technological University, 50 Nanyang Ave, Singapore 639798

Abstract

Genetic Algorithms (GAs) have become a popular optimization tool for many areas ofresearch and topology optimization an effective design tool for obtaining efficient and lighterstructures. In this paper, a versatile, robust and enhanced genetic algorithm (GA) is proposedfor structural topology optimization by using problem-specific knowledge. The original discreteblack-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 con-nectivity is used to suppress the occurrence of checkerboard patterns. A simpler version of theperimeter control approach is developed to obtain a well-posed problem and the total numberof hinges of each individual is explicitly penalized to achieve a hinge-free design. To handle theproblem of representation degeneracy effectively, a recessive gene technique is applied to viabletopologies while unusable topologies are penalized in a hierarchical manner. An efficient FEM-based function evaluation method is developed to reduce the computational cost. A dynamicpenalty method is presented for the GA to convert the constrained optimization problem intoan unconstrained problem without the possible degeneracy. With all these enhancements andappropriate choice of the GA operators, the present GA can achieve significant improvementsin evolving into near-optimum solutions and viable topologies with checkerboard free, meshindependent and hinge-free characteristics. Numerical results show that the present GA canbe more efficient and robust than the conventional GAs in solving the structural topologyoptimization problems of minimum compliance design, minimum weight design and optimalcompliant mechanisms design. It is suggested that the present enhanced GA using problem-specific knowledge can be a powerful global search tool for structural topology optimization.

KEY WORDS: topology optimization; genetic algorithms; bit-array representation; con-nectivity analysis; black and white design; hinge-free design

∗Correspondence to: M.Y. Wang, Department of Automation and Computer-Aided Engineering, The ChineseUniversity of Hong Kong, Shatin, NT, Hong Kong. Tel.: +852-2609-8487; Fax: +852-2603-6002.

†E-mail: yuwang@acae.cuhk.edu.hk (M.Y. Wang).

1

1. Introduction

Structural topology optimization has become an effective design tool for obtaining effi-

cient 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 experi-

enced 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 microstruc-

tures 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

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 Opti-

mization (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 proce-

dures 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 op-

timal 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

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 gradu-

ally 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

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 re-

cently, 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 advan-

tages 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 Cross-

ley [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 con-

nectivity handling approach based on image processing and dynamic penalty to alleviate

this problem. Furthermore, a uniform initialization method was demonstrated as effec-

tive 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

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 load-

carrying capacity while the bit-array representation cannot guarantee this requirement

for connectivity. However, this issue has not been fully addressed yet. In the connec-

tivity 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 ex-

pressed 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 de-

signs 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

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 can-

not 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 gen-

erated 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 assum-

ing 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

may become unnecessarily too much since the recessive gene technique [32] was not in-

troduced 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 pro-

cessing 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 penal-

ized in a hierarchical manner. A modified version of the dynamic penalty method is

presented for the GA to convert the constrained optimization into unconstrained opti-

mization. The present GA is finally applied into solving the structural topology opti-

mization 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 stochas-

tic 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

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 in-

clude 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 appli-

cation [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 expen-

sive. 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 large-

scale 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 par-

allel 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

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 advanta-

geous 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 viola-

tion 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 rela-

tively 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 op-

timization 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 analo-

gously 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 han-

dling approach for continuous structural topology optimization has been proposed by

10

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 hor-

izontal, 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 com-

ponent, 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 iden-

tified, 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

function evaluation. It should be noted that parts of a bit-array representation chromo-

some 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 mod-

erately 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 genera-

tions 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 = Γd nd + ΓaAd + ΓmAm (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 compo-

nents, Ad the total area of structurally disconnected components, Am the minimum

area of structurally disconnected components, Γd, Γa and Γm the corresponding penalty

parameters, respectively, which are usually chosen in a hierarchical manner such as

Γd À Γa À Γm in order to favor the evolution into a single structurally connected com-

ponent effectively. To alleviate the problem of representation degeneracy [28], both the

total area and minimum area of disconnected components are here used.

12

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 opti-

mization 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 di-

mensional topology optimization problem which the discretized problem approximates

may lack a solution in its general continuum setting [10] since the optimization prob-

lem 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 sim-

pler 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

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 do-

main 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 = Γs(A0 − As(x)) (2)

where viol(x)s is the penalty function on this perimeter control violation, A0 a pre-

defined value to restrict the minimum area of internal holes, As(x) the minimum area

of the internal holes and Γs the corresponding penalty parameter.

In continuum topology optimization, another serious numerical instability is the oc-

currence of checkerboard patterns in the final topologies. As afore-mentioned, checker-

board 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

fined as follows:

viol(x)h = Γhnh (3)

where viol(x)h is the penalty function on the moment-free one-node connected hinges,

nh the total number of hinges and Γh corresponding penalty parameter.

3.3. Efficient Evaluation of the Objective Function

In this study, a finite element method (FEM) based on the bilinear rectangular finite ele-

ments [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 com-

putational 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 stiff-

ness equation for structurally viable topologies and the skip of FEA for the structurally

15

unusable topologies.

3.4. Artificial Unconstrained Optimization

Generally, the single-objective constrained optimization problem can be written as fol-

lows:

Minimize: f(x) , x ∈ Ω

subject to gi(x) 6 0 , i = 1, 2, . . . , I

hj(x) = 0 , j = 1, 2, . . . , J

(4)

where x is the solution vector in the design domain Ω, f(x) the objective function, gi(x)

the i-th inequality constraint function, I the number of inequality constraints, hj(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 intro-

duced 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 vol-

ume 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

as

Minimize: f(x) , x ∈ Ω

subject to gi(x) 6 0 , i = 1, 2, . . . , I

nc − 1 = 0

A0 − As(x) 6 0

nh = 0

(5)

where nc 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 optimiza-

tion 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 inappropri-

ate 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

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) if x ∈ F

f ∗ + viol(x) 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

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) =

f(x)− Γf (Dlim −D(x)max) if x ∈ F

f ∗ + viol(x) otherwise

(7)

where the problem dependent penalty parameter Γf is set a small value Γf = 10−6 in

the present study, D(x)max the maximal displacement under the loading, and Dlim 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 Γf < 0, the bias will be set to the weaker,

and if Γf = 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

viol(x) =

0 if x ∈ F

viol(x)p else if x ∈ C

viol(x)c + viol(x)s + viol(x)h otherwise

(8)

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 topol-

ogy, 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 topol-

ogy 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

are more preferred to guide rapid evolution from unusable topologies into structurally

viable topologies.

3.5. GA Operations

3.5.1. 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].

3.5.2. 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 (xi) =2 (ni − 1)

Nind − 1, (9)

where F (xi) is the fitness of the i-th individual, ni the position of the i-th individual in

the individuals rank based on the values of the artificial unconstrained objective function

F (x), and Nind the population size.

3.5.3. 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

generations. Elitism usually brings about a more rapid convergence of the population

and also improves the chances of locating the optimal individual.

3.5.4. 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.

3.5.5. 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.

3.5.6. 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.

3.5.7. 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

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 de-

sign 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 gen-

erations (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 struc-

turally 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 ob-

tained 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 Γd = nelx × nely, where nelx and

nely are the number of elements in the horizontal and vertical directions, respectively,

22

Γa =√nelx× nely, and Γm = Γs = Γh = 1.

4.1. Minimum Compliance Design

The minimum compliance topology optimization can be expressed as

Minimize: C (x) , x ∈ Ω

subject to V (x)/V0 6 f

nc − 1 = 0

A0 − As(x) 6 0

nh = 0

(10)

where C(x) is the compliance of the topology, V (x) and V0 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 half-

height of the right boundary. The mesh size is 24 × 12 and thus nelx = 24, nely = 12.

It is also assumed that f = 0.5 and A0 = 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

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 com-

binations (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

viol(x)c+viol(x)s without constraint of hinges converges slowest among the four combi-

nations. 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∗ = 1×10−5E 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 aver-

aged 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

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 24× 12 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-and-

white 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

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.

4.2. Minimum Weight Design

The minimum weight optimal topology design problem can be expressed as

Minimize: W (x) , x ∈ Ω

subject to D(x)max −Dlim 6 0

nc − 1 = 0

A0 − As(x) 6 0

nh = 0

(11)

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 10 × 20 and thus nelx = 10, nely = 20. It is also

assumed that Dlim = 20 and A0 = 1 (divided by element size).

The unconstrained objective function F (x) defined in Eq. (7) is adopted for this ex-

ample 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

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 com-

parison 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.

4.3. 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 ten-

dency 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 prob-

lem is to maximize the displacement uout on a workpiece modeled by a spring with a

stiffness kout under the action of an input actuator modeled by a spring with a stiffness

kin and a force fin. This optimal compliant mechanism design problem can be expressed

28

as

Maximize: uout(x) , x ∈ Ω

subject to V (x)/V0 6 f

nc − 1 = 0

A0 − As(x) 6 0

nh = 0

(12)

where the volume fraction is assumed to be f = 0.3 for this example. The 1x1 square

domain is discretized by a 40 × 40 mesh and thus nelx = 40, nely = 40. Due to the

symmetry, only a half structure is used in the numerical analysis. It is also assumed

that A0 = 4 if the perimeter control is applied, kin = kout, fin = 2, and the maximum

number of generations is 3,000.

Figure 16 shows the optimal designs given by the SIMP method using the correspond-

ing code for compliant mechanism synthesis developed by Bedsøe and Sigmund [10] with

the initialization of uniform distribution and two different sets of spring stiffness values

kin = kout = 0.1 and kin = kout = 0.01. It can be seen that higher spring stiffness

eliminates the de facto hinges effectively, however, the displacement uout becomes much

smaller (Values in brackets in Fig. 16 are referred to displacements without using the ar-

tificial springs). Hence, appropriate choice of the stiffness values of the artificial springs

to reach a good compromise between hinge-free designs and high complimentary com-

pliances can be a challenging problem, in which a systematically theoretical study has

not been reported yet [62]. Using this spring model with kin = kout = 0.1, the present

GA-based topology optimization method generates optimal final designs as shown in

Fig. 16. It can be seen that the final designs are hinge-free and the present perimeter

control can reduce the total number of internal holes significantly (from 8 to 2). How-

29

ever, it should also be noted that there are some hinge-like solid elements functioning

essentially as one-node hinges in the final designs, similar to the one from the SIMP

method shown in Fig. 15(a). In this sense, the de facto hinges may not be completely

eliminated by using an artificial spring model if the spring stiffness is not too large. The

performance comparison of the SIMP method and the present GA using this artificial

spring model is shown in Fig. 17 and, as expected, the present GA converges to a better

solution with much more computational cost.

Since the present enhanced GA can ensure that the final designs are hinge-free by

using the constraint handling approach as shown in Eq. (3), the artificial spring model

shown in Fig. 14(b) may become unnecessary in the present GA-based topology op-

timization method. By removing the artificial springs in the numerical analysis, the

present analysis model becomes spring-free. Figure 18 displays the optimal designs ob-

tained from the present spring-free model (kin = kout = 0). The totally hinge-free final

designs composed of stiffer and slenderer components are with significantly higher com-

pliance values (more than 10 times higher than the ones shown in Fig. 18) due to the

less amount of materials being used (about one third less than the prescribed volume

fraction value of 0.3). Therefore, different from the minimum compliance design and the

compliant mechanism design using an artificial spring model, optimal designs of compli-

ant mechanisms may not have to use all the prescribed amount of materials due to the

stiffness related with the solid materials. Figure 19 shows the comparison of convergence

speed using the present spring-free model. As expected, the convergence speed can be

significantly affected by the present perimeter control due to the additional restriction.

30

5. Conclusions

In this study, an investigation into structural topology optimization using an enhanced

GA is performed. A bit-array representation method is presented to attack the origi-

nal discrete black-and-white problem directly. Due to this representation, the problem

of design connectivity becomes pronounced and an image-processing-based connectivity

analysis method is developed to resolve this issue effectively. Since the four-neighborhood

connectivity is employed, the problem of the occurrence of checkerboard patterns can

be overcome. A simpler version of the perimeter control approach is implemented to

achieve the well-posedness of the problem. To handle the moment-free one-node hinge

connection problem typically resulting from the FE model, the total number of one-node

hinges is considered as an explicit constraint and thus hinge-free designs can be ensured

in feasible solutions. To resolve the problem of representation degeneracy effectively, a

recessive gene technique is developed for viable topologies with a structurally connected

component while unusable topologies are penalized in a hierarchical manner such that

a broad search space can be maintained and a rapid evolution from unusable topolo-

gies into viable topologies can be obtained. An efficient FEM-based function evaluation

method is proposed to remove the void elements totally to reduce the computational

time. A dynamic penalty method is presented for the GA to convert the constrained

optimization problem into an unconstrained problem without the possibility that infea-

sible individuals may be more preferred than feasible ones in the population and the

degeneracy in the evaluation of the objective function. With all these improvements

and appropriate choice of the GA operators, GAs’ ability in evolving into near-optimum

solutions and viable topologies with checkerboard-free, mesh-independent and hinge-free

31

characteristics is greatly enhanced. Numerical results illustrate the efficiency, versatil-

ity and robustness of the present GA. It is shown that the present enhanced GA using

problem-specific knowledge can be more efficient and robust than the conventional GAs

in solving the minimum compliance design problem, the present artificial objective func-

tion evaluation method with a bias to the stiffer individuals performs better in solving

the minimum weight design problem, and the present GA-based topology optimization

method can achieve hinge-free designs with better complimentary compliances without

using the widely adopted artificial spring model. The performance comparison using the

present GA and the popular SIMP method is also carried out and it is obtained that the

present GA generates better solutions with more computational cost due to the global

search nature. It is suggested that the present enhanced GA using problem-specific

knowledge can be a powerful global search tool for structural topology optimization.

Acknowledgements

This work was partially supported by a Post-doctoral Research Fellowship from the

Singapore-MIT Alliance, the Research Grants Council of Hong Kong SAR (Project No.

CUHK4164/03E), a Post-doctoral Fellowship from the Chinese University of Hong Kong

(No. 03/ENG/12), and the Natural Science Foundation of China (NSFC) (Grants No.

50128503 and No. 50390063), which the authors gratefully acknowledge.

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40

List of Figures

1 Image-processing-based connectivity. . . . . . . . . . . . . . . . . . . . . 422 Region identification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Definition of the design domain of the 2× 1 cantilever beam. . . . . . . . 434 Performance comparison using different connectivity handling approaches.

(solid line: present approach; dashed line: Wang and Tai’s [6]; dotted line:Fanjoy and Crossley’s [32]) . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Comparison of final topologies using different connectivity handling ap-proaches.(a): Present approach, Cmin = 64.410; (b): Fanjoy and Crossley’s [32]),Cmin = 64.294; (c): Wang and Tai’s [6], Cmin = 65.264 [6]; (d): Wang andTai’s [6], C = 68.388 (fewest holes) . . . . . . . . . . . . . . . . . . . . . 44

6 Performance comparison using different combinations of the constrainthandling schemes.(solid line: viol(x)c only; dashed line: viol(x)c+viol(x)s; dotted line:viol(x)c+viol(x)h; dash-dotted line: viol(x)c+viol(x)s+viol(x)h) . . . . 44

7 Comparison of final topologies using different combinations of the con-straint handling schemes.(a): viol(x)c only, Cmin = 64.410; (b): viol(x)c+viol(x)s, Cmin = 67.104;(c): viol(x)c+viol(x)h, Cmin = 65.946; (d): viol(x)c+viol(x)s+viol(x)h,Cmin = 66.902 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

8 Comparison of final topologies using the SIMP method. . . . . . . . . . 459 Comparison of the convergence speed. . . . . . . . . . . . . . . . . . . . 4610 Comparison of the present GA with the SIMP method. . . . . . . . . . 4611 Definition of the design domain of the 1× 2 cantilever beam. . . . . . . . 4712 Comparison of final solutions using different bias approaches.

(solid line: no bias; dashed line: bias to the stiffer; dotted line: bias tothe weaker) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

13 Comparison of the final topologies using different bias approaches.((a): No bias, W (x)min = 0.225; (b) Bias to the stiffer, W (x)min = 0.20;(c): Bias to the weaker, W (x)min = 0.235) . . . . . . . . . . . . . . . . . 48

14 Displacement converter design. . . . . . . . . . . . . . . . . . . . . . . . 4815 Optimal designs using the SIMP method. . . . . . . . . . . . . . . . . . . 4816 Optimal designs using the present GA with an artificial spring model

(kin = kout = 0.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4917 Performance comparison using an artificial spring model (kin = kout = 0.1). 4918 Optimal designs using the present spring-free model (kin = kout = 0). . . 5019 Performance comparison using the present spring-free model (kin = kout =

0).(solid line: A0 = 1; dashed line: A0 = 4) . . . . . . . . . . . . . . . . . . 50

41

0

1

2

3

(a) 4-neighborhood connectivity

1

3

5

7

2

46

0

(b) 8-neighborhood connectivity

Fig. 1. Image-processing-based connectivity.

(a) Original topology

1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

(b) Bit-array representation

2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 0 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 2 2 2 2 2 2 2 1 1 1 0 0 0 0 3 0 0 0 0 0 0 0 0 2 2 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 2 2 2 2 2 0 0 0 0 4 0 0 0 0 0 0 2 2 2 2 2 0 0 0 0 4 4 4 4 4 0 0 0 0 2 2 2 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0

(c) Connected component labeling (d) Structural topology

Fig. 2. Region identification.

42

1

1

2

Fig. 3. Definition of the design domain of the 2× 1 cantilever beam.

0 100 200 300 400 5000

50

100

150

200

Number of generations

Ave

raged

bes

tF

(x)

(a) Best F (x)

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

Number of generations

Ave

rage

rati

oof

via

ble

topolo

gie

s

(b) Viable topologies

Fig. 4. Performance comparison using different connectivity handling approaches.(solid line: present approach; dashed line: Wang and Tai’s [6]; dotted line: Fanjoy and Crossley’s [32])

43

(a) (b)

(c) (d)

Fig. 5. Comparison of final topologies using different connectivity handling approaches.(a): Present approach, Cmin = 64.410; (b): Fanjoy and Crossley’s [32]), Cmin = 64.294; (c): Wang and

Tai’s [6], Cmin = 65.264 [6]; (d): Wang and Tai’s [6], C = 68.388 (fewest holes)

0 100 200 300 400 5000

100

200

300

400

500

Number of generations

Ave

raged

bes

tF

(x)

(a) Best F (x)

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

Number of generations

Ave

rage

rati

oof

via

ble

topolo

gie

s

(b) Viable topologies

Fig. 6. Performance comparison using different combinations of the constraint handling schemes.(solid line: viol(x)c only; dashed line: viol(x)c+viol(x)s; dotted line: viol(x)c+viol(x)h; dash-dotted line:

viol(x)c+viol(x)s+viol(x)h)

44

(a) (b)

(c) (d)

Fig. 7. Comparison of final topologies using different combinations of the constraint handling schemes.(a): viol(x)c only, Cmin = 64.410; (b): viol(x)c+viol(x)s, Cmin = 67.104; (c): viol(x)c+viol(x)h,

Cmin = 65.946; (d): viol(x)c+viol(x)s+viol(x)h, Cmin = 66.902

(a) Without filtering (b) With filtering

Fig. 8. Comparison of final topologies using the SIMP method.

45

0 100 200 300 400 500 60050

100

150

200

250

300

350

Number of generations

Bes

tF

(x)

(a) Present GA

0 5 10 15 20 25 3050

100

150

200

250

300

350

Number of iterations

Com

pli

ance

C(x

)

(b) The SIMP method

Fig. 9. Comparison of the convergence speed.

(a) Optimal topology from the present GA (b) Optimal topology from the SIMP method

0 100 200 300 400 50040

50

60

70

80

90

100

110

120

Number of generations

Bes

tF

(x)

(c) Convergence of the present GA

0 5 10 15 20 25 30 3540

50

60

70

80

90

100

110

120

Number of iterations

Com

pli

ance

C(x

)

(d) Convergence of the SIMP method

Fig. 10. Comparison of the present GA with the SIMP method.

46

1

2

1

Fig. 11. Definition of the design domain of the 1× 2 cantilever beam.

0 100 200 300 400 5000.25

0.3

0.35

0.4

0.45

0.5

0.55

Number of generations

Bes

tF

(x)

(a) Average solution

0 100 200 300 400 5000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Number of generations

Bes

tF

(x)

(b) Best solution

Fig. 12. Comparison of final solutions using different bias approaches.(solid line: no bias; dashed line: bias to the stiffer; dotted line: bias to the weaker)

47

(a) (b) (c)

Fig. 13. Comparison of the final topologies using different bias approaches.((a): No bias, W (x)min = 0.225; (b) Bias to the stiffer, W (x)min = 0.20; (c): Bias to the weaker,

W (x)min = 0.235)

Ac tuator

outu inu

inf

Design demain

(a) Basic problem

outu

inf ink outk

Design demain

(b) Artificial spring model

Fig. 14. Displacement converter design.

(a) kin = kout = 0.1, uout = 0.8235 (254.1626) (b) kin = kout = 0.01, uout = 7.9718 (1538.0689)

Fig. 15. Optimal designs using the SIMP method.

48

(a) A0 = 1, uout = 0.9861 (154.7838) (b) A0 = 4, uout = 0.9398 (181.6142)

Fig. 16. Optimal designs using the present GA with an artificial spring model (kin = kout = 0.1).

0 50 100 150 200−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Number of iterations

uout

(a) The SIMP method

0 500 1000 1500 2000 2500 3000 3500−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Number of generations

uout

A0 = 1A0 = 4

(b) The present GA (best solution)

Fig. 17. Performance comparison using an artificial spring model (kin = kout = 0.1).

49

(a) A0 = 1, uout = 3, 456, V (x)/V0 = 0.19 (b) A0 = 4, uout = 1, 876, V (x)/V0 = 0.20

Fig. 18. Optimal designs using the present spring-free model (kin = kout = 0).

0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

3500

Number of generations

Ave

raged

bes

tF

(x)

(a) Averaged best solution

0 500 1000 1500 2000 2500 3000 35000

500

1000

1500

2000

2500

3000

3500

Number of generations

Bes

tF

(x)

(b) Best solution

Fig. 19. Performance comparison using the present spring-free model (kin = kout = 0).(solid line: A0 = 1; dashed line: A0 = 4)

50

List of Tables

1 Performance comparison using different connectivity handling approaches.(tg: Average CPU time per generation; F : Averaged best F (x); nv: Av-eraged total number of viable topologies; nf : Averaged total number offeasible solutions;) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2 Performance comparison using different combinations of the constrainthandling schemes.(tg: Average CPU time per generation; F : Averaged best F (x); nv:Averaged total number of viable topologies; nf : Averaged total numberof feasible solutions; nh: Averaged total number of internal holes) . . . . 52

51

Table 1. Performance comparison using different connectivity handling approaches.(tg: Average CPU time per generation; F : Averaged best F (x); nv: Averaged total number of viable

topologies; nf : Averaged total number of feasible solutions;)

Connectivity handling approach tg (s) F nv nf

Present 3.3218 68.846 38,217 32,905Wang and Tai’s [6] 1.3243 72.197 17,808 9,358

Fanjoy and Crossley’s [32] 2.8917 68.117 31,560 27,788

Table 2. Performance comparison using different combinations of the constraint handling schemes.(tg: Average CPU time per generation; F : Averaged best F (x); nv: Averaged total number of viable

topologies; nf : Averaged total number of feasible solutions; nh: Averaged total number of internal holes)

Combination tg (s) F nv nf nh

viol(x)c 3.3218 68.846 38,217 32,905 9.7viol(x)c + viol(x)s 2.2466 72.367 37,615 21,408 1.6viol(x)c + viol(x)h 2.4658 71.247 38,527 19,963 5.6

(viol(x)c + viol(x)h)∗ 4.9832 70.883 38,260 19,168 5.4

viol(x)c + viol(x)s + viol(x)h 2.0689 73.046 38,483 16,940 2.7(viol(x)c + viol(x)s + viol(x)h)

∗ 4.7155 73.324 39.027 18,167 2.2

52