Post on 26-Apr-2023
Proceedings of the Fourth International Conference on Mathematical and Computational Applications
June 11-13, 2013. Manisa, Turkey, pp.221-230
OPTIMAL DESIGN OF STEEL TRUSSES USING STOCHASTIC SEARCH
TECHNIQUES
Serdar Carbas1 and Oguzhan Hasancebi
2
1 Middle East Technical University, Department of Engineering Sciences,
06800, Ankara, Turkey, 2 Middle East Technical University, Department of Civil Engineering,
06800, Ankara, Turkey
carbas@metu.edu.tr
Abstract- The aim of this study is to present the efficiency of stochastic search
techniques and their applications to the optimum design of steel truss structures.
Suitable profiles for each element in a truss have to be selected from a given selection of
standard profiles. Stochastic particle swarm optimizer, harmony search optimization and
genetic algorithms are used as a solution method. These stochastic optimization
algorithms are widely used and suitable methods for hard combinatorial problems. The
comparison of algorithms is done using example problems. The optimum design
problems of these structures are formulated considering the design limitations imposed
by ASD-AISC (Allowable Stress Design Code of American Institute of Steel
Institution).
Key Words- Stochastic Search Techniques; Particle Swarm Optimizer; Harmony
Search Optimization; Genetic Algorithms
1. INTRODUCTION
Optimization has been expanding in all directions at an astonishing rate during the
last few decades. New algorithms and theoretical techniques have been developed; the
diffusion into other disciplines has proceeded at a rapid pace. At the same time, one of
the most striking trends in optimization is the constantly increasing emphasis on the
interdisciplinary nature of the field [1]. There are great advantages to recognizing or
formulating an engineering problem as an optimization problem. The most basic
advantage is that the problem can then be solved, very reliably and efficiently, using
stochastic optimization methods. There are also theoretical or conceptual advantages of
formulating an engineering problem as an optimization problem [2]. Development of
optimization algorithms made it possible for engineers to analyze a structure in much
more detail. Stochastic search techniques based algorithms, which are borrowing their
characteristics from biological methodologies and other natural phenomena, are widely
used in structural engineering to obtain the optimal designs of complex structures [3].
The main idea behind the skeleton of these algorithms is that a structure is analyzed by
dividing it into structural element groups and solving the equations of mechanics for
each of those element groups separately. With the increase of the research and
applications based on stochastic search techniques [4,5], constraint handling used in
metaheuristic computation techniques has been a hot topic in both academic and
engineering fields [6-10].
222 S. Carbas and O. Hasancebi
min max
min max
min max
1,2,...,
1,2,...,
0 1,2,...,
1,2,...,
i
i
b
i i
i
i m
i n
i nc
A A A i ng
1
( ) . .n
i i iiW x A L
In this study, the performance of the optimum structural design algorithms based
on particle swarm optimizer [11], harmony search optimization [12] and simple genetic
algorithms [13] is examined on the optimum design of real size plane trusses. For this
purpose, a simple truss structure and a shallow truss structure are designed using each of
the mentioned algorithms and their performances are compared.
2. OPTIMUM DESIGN PROCEDURE OF STEEL TRUSSES
The general formulation of the weight minimization problem for a truss structure is
formulated as in Eqns. (1) and (2).
minimize (1)
subject to
(2)
where W({x}) = weight of the structure; n = number of members making up the
structure; m= number of nodes; nc = number of elements subjected to compression; ng
= number of groups (number of design variables); الi = material density of member i; Li
= length of member i; Ai = cross-sectional area of member i chosen between Amin and
Amax; min = lower bound and max = upper bound; ζi and δi = stress and nodal
deflection, respectively; b
i =allowable buckling stress in member i when it is subjected
to compression. The design limitations of the optimization problem are imposed
according to ASD-AISC [14].
3. STOCHASTIC SEARCH TECHNIQUES
The stochastic search techniques inspire from the nature such as survival of the
fittest, immune system or cooling of molten metals through annealing to develop a
numerical optimization algorithm. These methods are offbeat stochastic search and
optimization methods used to find the optimum solution of combinatorial optimization
problems. Having not needed to the gradient information of the objective function and
constraints, they have been come in to vogue nowadays. The particle swarm optimizer,
harmony search optimization, and simple genetic algorithms are selected among others
from literature to be used as solution tools in this study. As can be understood from their
names each technique mimics one particular phenomenon that exists in the nature [15].
3.1 Particle Swarm Optimizer (PSO)
The particle swarm optimization method is one of the stochastic random search
methods that is developed by Eberhart and Kennedy [16], inspired by social behavior of
bird flocking or fish schooling. This behavior is concerned with grouping by social
forces that depend on both the memory of each individual as well as the knowledge
gained by the swarm. The phenomenon behind this behavior is called swarm
Optimal Design of Steel Trusses Using Stochastic Search Techniques 223
intelligence. The steps of the algorithm are outlined in the following as given in [17,
18]:
Step 1. Initializing Particles: A swarm consists of a predefined number of particles
referred to as swarm size. Each particle incorporates two sets of components; a position
(design) vector and a velocity vector. The position vector retains the values (positions)
of design variables, while the velocity vector is used to vary these positions during the
search.
Step 2. Evaluating Particles: All particles are analyzed, and their objective function
values are calculated.
Step 3. Updating the Particles’ Best and the Global Best: A particle‟s best position
(the best design with minimum objective function) thus far is referred to as particle‟s
best and is stored separately for each particle in a vector. On the other hand, the best
feasible position located by any particle since the beginning of the process is called the
global best position, and it is stored in a vector. At the current iteration, both the
particles‟ bests and the global best are updated.
Step 4. Updating a Particle’s Velocity Vector: The velocity vector of each particle is
updated considering the particle‟s current position, the particle‟s best position and
global best position.
During numerical applications to structural optimization problems, it has been found
that all the particles in the swarm are eventually dragged to the position identified by the
global best. When this happens, the current and best positions of all the particles
become identical to the global best, resulting in almost zero velocity vectors. Since
particles cannot fly any more in such a case, the search gets stuck in a very poor design
point. A reformulation is therefore considered in the present study, where an additional
velocity term is defined and added as in Eqns. (3-4) to give each particle a random
move in certain directions in the close neighborhood of its current position.
t
Nr
t
IBrc
t
IGrcwvv
s
i
k
i
k
i
k
i
k
ik
i
k
i
3
)()(
22
)()(
11
)()1( (3)
1 1/ 2 or 0 1/ 2i d dif r N if r N (4)
In Eqns. (3-4), r3 is a random number between 0 and 1; Ns is the number of steel
sections in the profile list; and i is 0-1 Heaviside function implemented by sampling a
random number r between 0 and 1, Nd is design variables. Probabilistically, Eqn. (3)
implies that in every two iterations only one design variable in a particle is changed to a
new position with random velocity term. The reformulated equation has been observed
to eliminate the aforementioned drawback and greatly improve the efficiency of the
technique.
Step 5. Updating a Particle’s Position Vector: Next, the position vector of each
particle is updated with the updated velocity vector, which is rounded to nearest integer
value for discrete variables.
Step 6. Termination: The steps 2 through 5 are repeated in the same way for a
predefined number of iterations.
224 S. Carbas and O. Hasancebi
3.2 Harmony Search Optimization (HSO) The harmony search method is first introduced by Geem and Kim [19]. This
presented stochastic technique is based on the musical performance process that takes
place when a musician searches for a better state of harmony [20]. In the process of
musical production, a musician chooses and brings together number of different notes
from the whole notes and then plays these with a musical instrument to find out whether
it gives a pleasing harmony. Similarly, a candidate solution is generated in the optimum
design process by modifying some of the decision variables to find optimum solution.
Step 1. Initialization of Harmony Memory Matrix: A harmony memory matrix is
generated and initialized at first. It incorporates a specified number of solutions referred
to as harmony size.
Step 2. Evaluation of Harmony Memory Matrix: Solutions are then analyzed, and
their objective function values are calculated.
Step 3. Improvizing a New Harmony: A new harmony is improvised (generated) by
selecting each design variable from either harmony memory or the entire discrete set.
The probability that a design variable is selected from the harmony memory is
controlled by a parameter called harmony memory considering rate ( hmcr ). If a design
variable attains its value from harmony memory, it is checked whether this value should
be pitch-adjusted or not (par).
Step 4. Update of Harmony Matrix: After generating the new harmony vector, its
objective function value is calculated. If this value is better (lower) than that of the
worst harmony vector in the harmony memory, it is then included in the matrix while
the worst one is discarded out of the matrix.
Step 5. Termination: The steps 3 and 4 are repeated until a pre-assigned maximum
number of cycle is reached.
3.3 Genetic Algorithms (GAs) Theory of GAs method depends on the principle of Darwin‟s theory of survival
fittest. This can be summarized that any individual animal or plant which succeeds in
reproducing itself is "fit" and will contribute to survival of its species, not just the
"fittest" ones, though some of the population will be better adapted to the circumstances
than others [21, 22]. The fundamentals of this algorithm are outlined below [13].
Step 1. Initial Population: In simple GAs, a design variable is not represented by its
actual design value; instead it is encoded as a binary string of finite length. A coded
design variable consists of zeros and ones, such as 001001, and is referred to as a
substring in GA terminology. If there are design variables in all, then such substrings
are joined together to form a complete string (chromosome or individual) that represents
a potential design to a problem at hand.
Step 2. Encoding: For each individual decoding is carried out to map all substrings to
some integer values representing the sequence numbers of standard steel sections in a
given profile list.
Step 3. Evaluation and Fitness: Once an individual is decoded, analysis is carried out
to obtain its structural response under external loads, and an objective function value is
assigned to it. After all the individuals are evaluated in this manner, each individual is
assigned a fitness score, which indicates the merit of the individual with respect to
overall population. In the present study, an inverse transformation function (Eqn. 5) is
Optimal Design of Steel Trusses Using Stochastic Search Techniques 225
used to calculate the fitness score kf of an individual k such that the objective function
value k of the individual is proportioned to the maximum ( max ) in the population.
k
kf
max (5)
The individuals‟ fitnesses calculated from Eqn. (5) need to be scaled to eliminate the
dominance of highly fit individuals during selection process. Eqn. (6) shows the fitness
scaling function used for this purpose.
ave
avemax
avemaxave
avemax )(
)(
)(
)1(f
ff
fsff
ff
sff cc
kk
(6)
In Eqn. (6), maxf , minf and avef are the maximum, minimum and average fitnesses
in the population, respectively; cs is a real valued scaling factor (typically taken as 2.0);
and kf is the scaled fitness score of the k-th individual. By virtue of fitness scaling, the
ratio of maximum scaled fitness to average scaled fitness is set to a value around cs .
Step 4. Selection and Reproduction: Selection is carried out next, where individuals
of high fitness scores are selected and reproduced, while the least fit ones are eliminated
to create an intermediate population called mating pool. Theoretically, the number of
copies (reproductive trials) which an individual is represented with in the mating pool is
determined in accordance with fitness proportionate selection scheme (Eqn. 7).
avef
f kk
(7)
In Eqn. (7), avef denotes the average scaled fitness of the population; and k
represents the number of reproductive trials allocated to an individual. Eqn. (7) is not
directly used due to possible round-off errors; instead its algorithmic realization is
conducted by the roulette wheel selection approach. In this approach, firstly the
individuals take up slots on a simulated roulette wheel in connection to their scaled
fitnesses. The roulette is then spun times to select and reproduce the individuals for
mating pool.
Step 5. Crossover: The selected and reproduced individuals are mated randomly to
form 2/ pairs, and crossover is separately implemented between each pair according
to a preset probability value called crossover probability. When applied to a pair,
crossover swaps the genetic information between the mating (parent) individuals to
produce two new child individuals. A number of different strategies have been devised
to implement this task. In the present study, a two-point crossover approach is used,
where mating individuals are cut at two randomly selected crossover sites, and a design
exchange is facilitated by swapping either the inner portion falling between the
crossover sites or the two outer portions.
Step 6. Mutation: Mutation is applied on the genes of child individuals by randomly
altering a gene of 1 to 0 or vice versa based on a specified gene-wise mutation
probability.
Step 7. Termination: The child (new) population replaces the parent one and the steps
2 through 6 are repeated until a maximum number of generations.
226 S. Carbas and O. Hasancebi
4. DESIGN EXAMPLES
In order to present the effectiveness of the stochastic search techniques described
above are encoded in three abovementioned optimization algorithms. Two different
design examples, a 46-bar simple truss structure and a 46-bar plane shallow truss
structure, are used to test and compare their numerical performances on optimum design
of real size truss structures. The number of structural analysis is taken as 50,000 for
design examples to make sure that all the stochastic techniques are given the equal
opportunity to reach the global optimum. The following material properties of the steel
are used in all design examples; modulus of elasticity (E) 29000ksi (203893.6MPa)
and yield stress ( yF ) 36ksi (253.1MPa).
4.1. 46-Bar Simple Plane Truss As a first design example, a 46-bar simple plane truss shown in Figure 1 is
selected [23]. This truss has three-span with a total length of 33m. The truss is to be
optimized for minimum weight with the cross-sectional areas of the members being the
design variables. The members of the structure are grouped into 4 independent design
variables considering the upper chords members as first group, the lower chord
members as second group, the braces as third group and the column members as fourth
group. The grouping of members is shown in Figure 1. A single point design load
through x-axis at the mid-point of lower chord as well seven point design loads through
y-axis are applied at lower chord as shown in Figure 1. A discrete set of 137 economical
standard steel sections selected from W-shape profile list based on area and radii of
gyration properties is used to size the variables. The stress and stability limitations of
the members are calculated according to the provisions of ASD-AISC [14]. In addition,
the displacements of all nodes in any direction are restricted to a maximum value of
0.254 cm. The minimum weights reached by each stochastic search technique and
section designations attained for each member group for 46-bar simple truss structure is
given in Table 1. The optimum solution of this problem is reached as 35723.73 lb
(16204.01 kg) by both PSO and HSO. The GAs has obtained the minimum structure
weight as 35815.09 lb (16245.45 kg). Figure 3-(a) displays the variation of feasible best
designs attained by stochastic optimization techniques.
Figure 1. 46- Bar Simple Plane Truss Structure
11 x 3 m = 33 m
4
m
48 kN
48 kN
48 kN
48 kN
48 kN
48
kN
48 kN
48 kN
1
2
3
4
3
3
4
1
2
x
y
Optimal Design of Steel Trusses Using Stochastic Search Techniques 227
Table 1. Optimal designs of 46-bar simple plane truss
Table 2. Nodal coordinates for the left side of 46-bar shallow truss structure
4.2. 46-Bar Shallow Truss
The shallow truss containing 46 elements and 25 nodes is considered as the
second truss example. This structure represents a typical crane or robotic manipulator
[24]. Symmetry is to be enforced so that the 23 design variables define the cross-
sectional areas of all bars of the structure. The node coordinates of the left part of the
truss is demonstrated in Table 2. Figure 2 shows the geometry and the 23 element
groups. The all nodes of the truss in the x and y directions are subjected to the
displacement limits of 0.254cm. The allowable cross-sectional areas in this example are
selected from W-shape profile list with 137 economical standard steel sections. The
structure is to be designed for loads of 13.34, 53.38, and 13.34 kN applied in the
negative y-direction at nodes 7, 13 and 19, respectively. The optimum design weights
obtained by each technique for 46-bar shallow truss and the pipe section designations
attained for each member group are given in Table 3. For this structure, PSO technique
gives the least weight, which is 10599.52 lb (4807.86 kg). This design is considered to
be the optimum solution of this problem. But HSO and GAs methods obtained the
optimum design weights as 10601.58 lb (4808.79 kg) and 10675.79 lb (4842.46 kg),
respectively. Noticeable is that GAs once more has given the heaviest design. The
design history graph is shown in Figure 3-(b).
Figure 2. 46-Bar Shallow Truss Structure
Size
variables
PSO HSO GAs
Ready
Section
Area, in2
(cm2)
Ready
Section
Area, in2
(cm2)
Ready
Section
Area, in2
(cm2)
1 W16x67 19.7 (127.09) W16x67 19.7 (127.09) W12x65 19.1 (123.23)
2 W12x45 13.2 (85.16) W12x45 13.2 (85.16) W12x45 13.2 (85.16)
3 W10x54 15.8 (101.94) W10x54 15.8 (101.94) W12x50 14.7 (94.84)
4 W21x62 18.3 (118.06) W21x62 18.3 (118.06) W12x65 19.1 (123.23)
Weight 35723.73 lb
(16204.01 kg)
35723.73 lb
(16204.01 kg)
35815.09 lb
(16245.45 kg)
Node X (m) Y (m) Node X (m) Y (m)
1 0.00 0.00 8 3.92 0.24
2 0.28 0.13 9 4.62 0.61
3 0.25 0.18 10 5.38 0.38
4 1.01 0.05 11 6.07 0.76
5 1.70 0.32 12 6.10 0.46
6 2.46 0.09 13 6.35 0.635
7 3.16 0.47
12.702 m
0.635 m
13.34 kN 13.34 kN
53.38 kN
1
2
3
4
5
6
7
8
9
10
11
12
13 23
22
21
20
19
18
17
16 15
14
13
12
11
10
9
8
7
6
5
4 3
1
2
x
y
228 S. Carbas and O. Hasancebi
10000
12500
15000
17500
20000
0 10000 20000 30000 40000 50000
PSO HSO
GAs
35000
37500
40000
42500
0 10000 20000 30000 40000 50000
PSO HSO
GAs
(a) 46-bar simple plane truss (b) 46-bar shallow truss
Figure 3. The design history graph obtained with stochastic optimization techniques
for (a) 46-bar simple plane truss, (b) 46-bar shallow truss.
5. CONCLUSIONS
The highly promising outcome of this research suggests that the stochastic
numerical optimization techniques that are inspired by imitating biological principals
provide efficient and robust techniques for obtaining the solution of discrete optimum
design problem of steel trusses. These techniques use one of the natural phenomena to
generate a numerical optimization technique that are quite suitable for obtaining the
solution of combinatorial optimum design problems. For example, genetic algorithms
are based on the principle of survival of the fittest while harmony search optimization
imitates the improvisation process of a musician seeking a pleasing harmony. The
design examples show that these techniques are more powerful than the ones based on
mathematical programming methods in solving the discrete optimum design problem of
steel trusses.
Table 3. Optimal designs of 46-bar shallow truss
Size variables
PSO HSO GAs
Ready Section
Area, in2 (cm2)
Ready Section
Area, in2 (cm2)
Ready Section
Area, in2 (cm2)
1 W33x130 38.3 (127.09) W33x130 38.3 (127.09) W30x124 36.5 (127.09)
2 W14x145 42.7 (85.16) W36x150 44.2 (85.16) W24x112 47.7 (85.16) 3 W14x68 20.0 (101.94) W21x68 20.0 (101.94) W18x55 16.2 (101.94)
4 W27x114 33.5 (118.06) W10x112 32.9 (118.06) W30x108 31.7 (118.06)
5 W6x9 2.68 (127.09) W6x9 2.68 (127.09) W6x9 2.68 (127.09) 6 W30x124 36.5 (85.16) W30x124 36.5 (85.16) W27x117 34.4 (85.16)
7 W6x9 2.68 (101.94) W6x9 2.68 (101.94) W6x9 2.68 (101.94)
8 W24x104 30.6 (118.06) W27x102 30.0 (118.06) W18x106 31.2 (118.06) 9 W6x9 2.68 (127.09) W6x9 2.68 (127.09) W6x9 2.68 (127.09)
10 W30x132 38.9 (85.16) W30x132 38.9 (85.16) W12x136 39.9 (85.16)
11 W6x9 2.68 (101.94) W6x9 2.68 (101.94) W6x9 2.68 (101.94) 12 W14x90 26.5 (118.06) W12x87 25.6 (118.06) W12x106 31.1 (118.06)
13 W6x9 2.68 (127.09) W6x9 2.68 (127.09) W6x9 2.68 (127.09)
14 W36x135 39.7 (85.16) W12x136 39.9 (85.16) W30x124 36.5 (85.16) 15 W6x9 2.68 (101.94) W6x9 2.68 (101.94) W6x9 2.68 (101.94)
16 W21x101 29.8 (118.06) W27x102 30.0 (118.06) W27x102 30.0 (118.06)
17 W6x9 2.68 (127.09) W6x9 2.68 (127.09) W6x9 2.68 (127.09) 18 W21x122 35.9 (85.16) W30x124 36.5 (85.16) W24x131 38.5 (85.16)
19 W6x9 2.68 (101.94) W6x9 2.68 (101.94) W6x9 2.68 (101.94)
20 W30x116 34.2 (118.06) W30x116 34.2 (118.06) W33x118 34.7 (118.06) 21 W8x67 19.7 (127.09) W18x65 19.1 (127.09) W18x60 17.6 (127.09)
22 W36x135 39.7 (85.16) W14x145 42.7 (85.16) W14x132 38.8 (85.16)
23 W36x135 39.7 (101.94) W12x136 39.9 (101.94) W36x135 39.7 (101.94)
Weight 10599.52 lb (4807.86 kg) 10601.58 lb (4808.79 kg) 10675.79 lb (4842.46 kg)
Number of Analysis
Bes
t F
easi
ble
Des
ign
(lb
)
Number of Analysis
Bes
t F
easi
ble
Des
ign
(lb
)
Optimal Design of Steel Trusses Using Stochastic Search Techniques 229
6. REFERENCES
1. C. Pearce and E. Hunt Springer (Eds.), Optimization: Structure and Applications,
Springer Optimization and Its Applications 32, New York, 2009.
2. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press,
Cambridge, 2004.
3. O. Hasancebi and S. Carbas, Ant Colony Search Method In Practical Structural
Optimization, International Journal Of Optimization In Civil Engineering 1, 1,
91-105,2011.
4. Z. Michalewicz, A survey of constraint handling techniques in evolutionary
computation methods, Proceedings of the fourth Annual Conference on Evolutionary
Programming, San Diego, 135–155, 1-3 March 1995.
5. C.A.C. Coello, Theoretical and numerical constraint handling techniques used with
evolutionary algorithms: a survey of the state of the art, Computer Methods in Applied
Mechanics and Engineering 191, 11-12, 1245–1287, 2002.
6. S. Carbas and M.P. Saka, Optimum Topology Design of Various Geometrically
Nonlinear Latticed Domes Using Improved Harmony Search Method, Structural and
Multidisciplinary Optimization 45, 3, 377-399, 2012.
7. M.P. Saka, Optimum Geometry Design of Geodesic Domes Using Harmony search
Algorithm, Advances in Structural Engineering 10, 6, 595-606, 2007.
8. S. Carbas and M.P. Saka, Optimum Design of Single Layer Network Domes Using
Harmony Search Method, Asian Journal of Civil Engineering 10, 1, 97-112, 2009.
9. O. Hasancebi, S. Carbas, E. Dogan, F. Erdal, and M.P. Saka, Performance evaluation
of metaheuristic search techniques in the optimum design of real size pin jointed
structures, Computers and Structures 87, 5-6, 284-302, 2009.
10.O. Hasancebi, S. Carbas, E. Dogan, F. Erdal, and M.P. Saka, Comparison of non-
deterministic search techniques in the optimum design of real size steel frames,
Computers and Structures 88, 17-18, 1033-1048, 2010.
11. J. Kennedy and R. Eberhart, Particle Swarm Optimization, IEEE International
Conference on Neural Networks, Perth, 1942-1948, 27 November-1 December 1995.
12. Z.W. Geem and J.H. Kim, A New Heuristic Optimization Algorithm: Harmony
Search, Simulation 76, 2, 60-68, 2001.
13. D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning,
Addison-Wesley Longman Publishing Co., Boston, 1989
14. Manual of Steel Construction, Allowable Stress Design, 9th
edition, AISC, American
Institutes of Steel Construction, Chicago, 1989.
15. M.P. Saka and O. Hasancebi, Adaptive Harmony Search Algorithm for Design Code
Optimization of Steel Structures, In: Z.W. Geem (Ed.), Harmony Search Algorithms for
Structural Design Optimization, Studies in Computational Intelligence, Springer-Verlag
23, 79-120, 2009.
16. J. Kennedy and R.C. Eberhart, A discrete binary version of the particle swarm
algorithm, Proceedings of the World Multi-Conference on Systemic, Cybernetics and
Informatics, Piscataway, 4104-4109, 1997.
17. P. Fourie and A. Groenwold, The Particle Swarm Optimization Algorithm in Size
and Shape Optimization, Structural and Multidisciplinary Optimization 23, 4, 259-267,
2002.
230 S. Carbas and O. Hasancebi
18. R.E. Perez and K. Behdinan, Particle Swarm Approach for Structural Design
Optimization, Computers and Structures 85, 19-20, 1579-1588, 2007.
19. K.S. Lee and Z.W. Geem, A New Structural Optimization Method Based on the
Harmony Search Algorithm, Computers and Structures 82, 9-10, 781-798, 2004.
20. K.S. Lee and Z.W. Geem, A new meta-heuristic algorithm for continuous
engineering optimization: harmony search theory and practice, Computer Methods in
Applied Mechanics and Engineering 194, 36-38, 3902-3933, 2002.
21. Y.M. Xie and G.P. Steven, Evolutionary Structural Optimization, Springer, 1997.
22. E.C. Scott, Evolution vs. Creationism: An Introduction, Greenwood Press, Westport,
Connecticut, 2009.
23. A. Joghataie and M. Ghasemi, Fuzzy Multistage Optimization of Large-Scale
Trusses, Journal of Structural Engineering 127, 11, 1338-1347, 2001.
24. A. Suleman and R. Sedaghati, Benchmark case studies in optimization of
geometrically nonlinear structures, Structural and Multidisciplinary Optimization 30, 4,
273-296, 2005.