an enhanced genetic algorithm for structural topology optimization
Post on 08-May-2023
0 Views
Preview:
TRANSCRIPT
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.
References
1. A. G. M. Michell, “The limits of economy of material in frame structures,” Philo-
sophical Magazine 8, pp. 589–597, 1904.
32
2. M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design
using a homogenization method,” Computer Methods in Applied Mechanics and
Engineering 71, pp. 197–224, 1988.
3. G. I. N. Rozvany, “Aims, scope, methods, history and unified terminology of
computer-aided topology optimization in structural mechanics,” Structural and Mul-
tidisciplinary Optimization 21(2), pp. 90–108, 2001.
4. P. Tanskanen, “The evolutionary structural optimization method: theoretical as-
pects,” Computer Methods in Applied Mechanics and Engineering 191(47-48),
pp. 5485–5498, 2002.
5. S. Bulman, J. Sienz, and E. Hinton, “Comparisons between algorithms for structural
topology optimization using a series of benchmark studies,” Computers & Structures
79(12), pp. 1203–1218, 2001.
6. S. Y. Wang and K. Tai, “Structural topology design optimization using genetic al-
gorithms with a bit-array representation,” Computer Methods in Applied Mechanics
and Engineering , 2004. in press.
7. O. Sigmund, “A 99 line topology optimization code written in MATLAB,” Structural
and Multidisciplinary Optimization 21(2), pp. 120–127, 2001.
8. G. I. N. Rozvany, M. Zhou, and T. Birker, “Generalized shape optimization without
homogenization,” Strutural Optimization 4, pp. 250–254, 1992.
9. M. P. Bendsøe, “Optimal shape design as a material distribution problem,” Struc-
tural Optimization 1, pp. 193–202, 1989.
10. M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Ap-
plications, Springer-Verlag, Berlin, 2003.
33
11. S. Y. Wang and K. Tai, “Graph representation for evolutionary structural topology
optimization,” Computers & Structures 82(20–21), pp. 1609–1622, 2004.
12. M. Stolpe and K. Svanberg, “On the trajectories of penalization methods for topol-
ogy optimization,” Structural and Multidisciplinary Optimization 21(2), pp. 128–
139, 2001.
13. M. Y. Wang and S. Y. Wang, “Bilateral filtering for structural topology optimiza-
tion,” International Journal for Numerical Methods in Engineering , 2004. accepted
for publication.
14. Y. M. Xie and G. P. Steven, “A simple evolutionary procedure for structural opti-
mization,” Computers & Structures 49(5), pp. 885–896, 1993.
15. Y. M. Xie and G. P. Steven, Evolutionary Structural Optimization, Springer-Verlag
London Limited, UK, 1997.
16. G. I. N. Rozvany, “Stress ratio and compliance based methods in topology op-
timization - A critical review,” Structural and Multidisciplinary Optimization 21,
pp. 109–119, 2001.
17. M. Zhou and G. I. N. Rozvany, “On the validity of ESO type methods in topology
optimization,” Structural and Multidisciplinary Optimization 21(1), pp. 80–83, 2001.
18. J. A. Sethian and A. Wiegmann, “Structural boundary design via level set and
immersed interface methods,” Journal of Computational Physics 163(2), pp. 489–
528, 2000.
19. G. Allaire, F. Jouve, and A.-M. Toader, “Structural optimization using sensitivity
analysis and a level-set method,” Journal of Computational Physics 194, pp. 363–
393, 2004.
34
20. M. Y. Wang, X. Wang, and D. Guo, “A level set method for structural topol-
ogy optimization,” Computer Methods in Applied Mechanics and Engineering 192,
pp. 227–246, 2003.
21. M. Y. Wang and X. Wang, “PDE-driven level sets, shape sensitivity and curvature
flow for structural topology optimization,” Computer Modeling in Engineering &
Sciences 6(4), pp. 373–395, 2004.
22. T. Belytschko, S. P. Xiao, and C. Parimi, “Topology optimization with implicit
functions and regularization,” International Journal for Numerical Methods in En-
gineering 57(8), pp. 1177–1196, 2003.
23. A. Neumaier, “Complete search in continuous global optimization and constraint
satisfaction,” in Acta Numerica 2004, A. Iserles, ed., 13, pp. 271–369, Cambridge
University Press, 2004.
24. J. Holland, Adaptation in natural and artificial systems, University of Michigan
Press, Ann Arbor, MI, 1975.
25. D. E. Goldberg, Genetic Algorithms in search, optimization and machine learning,
Addison-Wesley Publishing, Reading, MA, 1989.
26. E. Sandgren, E. Jensen, and J. Welton, “Topological design of structural components
using genetic optimization methods,” in Sensitivity Analysis and Optimization with
Numerical Methods, S. Saigal and S. Mukherjee, eds., Proceedings of the Winter
Annual Meeting of the American Society of Mechanical Engineers 115, pp. 31–43,
(Dallas, TX, USA), 1990.
27. D. Chapman, K. Saitou, and M. J. Jakiela, “Genetic Algorithms as an approach to
configuration and topology design,” Journal of Mechanical Design 116(4), pp. 1005–
35
1012, 1994.
28. M. Schoenauer, “Shape representation for evolutionary optimization and identifica-
tion in structural mechanics,” in Genetic Algorithms in Engineering and Computer
Science, G. Winter, J. Periaux, M. Galan, and P. Cuesta, eds., pp. 443–464, John
Wiley, (Chichester), 1995.
29. C. Kane and M. Schoenauer, “Topological optimum design using Genetic Algo-
rithms,” Control and Cybernetics 25(5), pp. 1059–1088, 1996.
30. M. Schoenauer, “Shape representations and evolutionary schemes,” in Evolutionary
Programming: Proceedings of the Fifth Annual Conference on Evolutionary Pro-
gramming, pp. 121–129, San Diego, USA, 1996.
31. M. J. Jakiela, C. Chapman, J. Duda, A. Adewuya, and K. Saitou, “Continuum
structural topology design with Genetic Algorithms,” Computer Methods in Applied
Mechanics and Engineering 186(2-4), pp. 339–356, 2000.
32. D. W. Fanjoy and W. A. Crossley, “Topology design of planar cross-sections with
a Genetic Algorithm: Part 1–Overcoming the obstacles,” Enginering Optimization
34(1), pp. 1–12, 2002.
33. D. W. Fanjoy and W. A. Crossley, “Topology design of planar cross-sections with
a Genetic Algorithm: Part 2–Bending, torsion and combined loading applications,”
Enginering Optimization 34(1), pp. 49–64, 2002.
34. K. Tai and T. H. Chee, “Design of structures and compliant mechanisms by evolu-
tionary optimization of morphological representations of topology,” ASME Journal
of Mechanical Design 122(4), pp. 560–566, 2000.
36
35. K. Tai, G. Y. Cui, and T. Ray, “Design synthesis of path generating compliant
mechanisms by evolutionary optimization of topology and shape,” ASME Journal
of Mechanical Design 124(3), pp. 492–500, 2002.
36. H. Hamda, F. Jouve, E. Lutton, M. Schoenauer, and M. Sebag, “Compact unstruc-
tured representations for evolutionary design,” Applied Intelligence 16(2), pp. 139–
155, 2002.
37. S. Y. Wang and K. Tai, “Bar-system representation method for structural topol-
ogy optimization using the genetic algorithms,” Engineering Computations , 2004.
accepted for publication.
38. C. Chapman and M. Jakiela, “Genetic algorithm-based structural topology design
with compliance and topology simplification considerations,” ASME Journal of Me-
chanical Design 118(1), pp. 89–98, 1996.
39. S. Pezeshk and C. V. Camp, “State of the art on the use of genetic algorithms in
design of steel structures,” in Recent Advances in Optimal Structural Design, S. A.
Burns, ed., American Society of Civil Engineers, 2002.
40. S. Y. Woon, L. Y. Tong, O. M. Querin, and G. P. Steven, “Knowledge-based algo-
rithms in fixed-grid GA shape optimization,” International Journal for Numerical
Methods in Engineering 58, pp. 643–660, 2003.
41. J. Ryoo and P. Hajela, “Handling variable string lengths in GA-based structural
topology optimization,” Structural and Multidisciplinary Optimization 26, pp. 318–
325, 2004.
42. E. Cantu-Paz, Efficient and accurate parallel genetic algorithms, Kluwer Academic
Publishers, Boston, MA, 2000.
37
43. K. Krishnakumar, “Micro genetic algorithms for stationary and non-stationary func-
tion optimization,” in SPIE Proceedings of Intelligent Control and Adaptive Systems,
1196, pp. 289–296, (Philadelphia, PA), 1989.
44. N. D. Lagaros, M. Papadrakakis, and G. Kokossalakis, “Structural optimization
using evolutionary algorithms,” Computers and Structures 80(7-8), pp. 571–589,
2002.
45. E. Cantu-Paz and D. E. Goldberg, “On the scalability of parallel genetic algorithms,”
Evolutionary Computation 7(4), pp. 429–449, 1999.
46. O. Sigmund, Design of material structures using topology optimization. PhD thesis,
Technical University of Denmark, 1994.
47. B. Hassani and E. Hinton, Homogenization and Structural Topology Optimization:
Theory, Practice and Software, Springer, London, 1999.
48. B. Jahne, Digital Image Processing - Concepts, Algorithms, and Scientific Applica-
tions, Springer-Verlag, Berlin, 4th ed., 1997.
49. M. Stolpe and K. Svanberg, “Modelling topology optimization problems as linear
mixed 0–1 programs,” International Journal for Numerical Methods in Engineering
57(5), pp. 723–739, 2003.
50. L. Ambrosio and G. Buttazzo, “An optimal design problem with perimeter penal-
ization,” Calculus of Variations and Partial Differential Equations 1, pp. 55–69,
1993.
51. R. B. Haber, C. S. Jog, and M. P. Bendsøe, “A new approach to variable-topology
shape design using a constraint on the perimeter,” Structural Optimization 11(1),
pp. 1–11, 1996.
38
52. J. Petersson, “Some convergence results in perimeter-controlled topology optimiza-
tion,” Computer Methods in Applied Mechanics and Engineering 171, pp. 123–140,
1999.
53. J. Petersson and O. Sigmund, “Slope constrained topology optimization,” Interna-
tional Journal for Numerical Methods in Engineering 41, pp. 1417–1434, 1998.
54. T. A. Poulsen, “A simple scheme to prevent checkerboard patterns and one-node
connected hinges in topology optimization,” Structural and Multidisciplinary Opti-
mization 24(5), pp. 396–399, 2002.
55. K. Deb, “An efficient constraint handling method for Genetic Algorithms,” Com-
puter Methods in Applied Mechanics and Engineering 186(2-4), pp. 311–338, 2000.
56. C. A. Coello Coello and E. M. Montes, “Handling constraints in Genetic Algo-
rithms using dominance-based tournaments,” in Proceedings of the Fifth Interna-
tional Conference on Adaptive Computing Design and Manufacture (ACDM 2002),
5, pp. 273–284, Springer–Verlag, 2002.
57. J. E. Baker, “Adaptive selection methods for Genetic Algorithms,” in Proceedings
of the International Conference on Genetic Algorithms and Their Applications, J. J.
Grefenstette, ed., pp. 101–111, Lawrence Erlbaum Associates, (Pittsburgh, PA,
USA), 1985.
58. J. E. Baker, “Reducing bias and inefficiency in the selection algorithm,” in Proceed-
ings of the Second International Conference on Genetic Algorithms, J. J. Grefen-
stette, ed., pp. 14–21, Lawrence Erlbaum Associates, (Cambridge, MA, USA), 1987.
59. W. M. Spears and K. A. De Jong, “On the virtues of parameterized uniform
crossover,” in Proceedings of the Fourth International Conference on Genetic Algo-
39
rithms, R. Belew and L. Booker, eds., pp. 230–236, Morgan Kaufman, (San Mateo,
CA), 1991.
60. H. Muhlenbein and D. Schlierkamp-Voosen, “The science of breeding and its appli-
cation to the breeder genetic algorithm (BGA),” Evolutionary Computation 1(4),
pp. 335–360, 1994.
61. L. Yin and G. K. Ananthasuresh, “Design of distributed compliant mechanisms,”
Mechanics Based Design of Structures and Machines 31(2), pp. 151–179, 2003.
62. S. Rahmatalla and C. C. Swan, “Sparse monolithic compliant mechanisms using
continuum structural topology optimization,” in Proceedings of DETC04, ASME
2004 Design Engineering Technical Conferences and Computers and Information in
Engineering Conference, (Salt Lake City, Utah, USA), 2004. DETC2004-57453.
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
top related