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MULTIMODAL SIZE, SHAPE, AND TOPOLOGY OPTIMISATION OF TRUSS
STRUCTURES USING THE FIREFLY ALGORITHM
Leandro Fleck FADEL MIGUEL1(*), Rafael Holdorf LOPEZ2 and Letícia Fleck Fadel MIGUEL3
1Prof. Dr., Department of Civil Engineering, Federal University of Santa Catarina, Florianópolis, Brazil. leandro.miguel@ufsc.br
2Prof. Dr., Department of Civil Engineering, Federal University of Santa Catarina, Florianópolis, Brazil. rafaelholdorf@gmail.com
3Profª. Drª., Department of Mechanical Engineering, Federal University of Rio Grande do Sul, Porto Alegre, Brazil. letffm@ufrgs.br
ABSTRACT:
This paper presents an efficient single-stage Firefly-based algorithm (FA) to simultaneously optimise the size,
shape and topology of truss structures. The optimisation problem uses the minimisation of structural weight as its
objective function and imposes displacement, stress and kinematic stability constraints. Unstable and singular
topologies are disregarded as possible solutions by checking the positive definiteness of the stiffness matrix.
Because cross-sectional areas are usually defined by discrete values in practice due to manufacturing limitations,
the optimisation algorithm must assess a mixed-variable optimisation problem that includes both discrete and
continuous variables at the same time. The effectiveness of the FA at solving this type of optimisation problem is
demonstrated with benchmark problems, the results for which are better than those reported in the literature and
obtained with lower computational costs, emphasising the capabilities of the proposed methodology. In addition,
the procedure is capable of providing multiple optima and near-optimal solutions in each run, providing a set of
possible designs at the end of the optimisation process.
KEYWORDS: Truss size, shape and topology optimisation; Firefly algorithm; Multimodal optimisation.
__________________________________________________________________________________________
Leandro Fleck FADEL MIGUEL(*) and Rafael Holdorf LOPEZ, Federal University of Santa Catarina, Department of Civil
Engineering, Rua João Pio Duarte da Silva, CEP 88040-970, Florianópolis, SC, Brazil. (e-mail: leandro.miguel@ufsc.br and
rafaelholdorf@gmail.com).
Letícia Fleck Fadel MIGUEL, Federal University of Rio Grande do Sul, Department of Mechanical Engineering, Av. Sarmento
Leite 425, 2º andar, CEP 90050-170, Porto Alegre, RS, Brazil. (e-mail: letffm@ufrgs.br).
(*) Corresponding author
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1. INTRODUCTION
Methods for the size optimisation of truss structures, in which the member areas are taken as design variables,
are fully established in the literature (Hemp [1], Pedersen [2], Adeli and Kamal [3]). However, it is well-known
that better results can be achieved when size, shape, and topology optimisation are performed simultaneously
(Dominguez et al. [4]). In this case, the problem generally begins with a ground structure to determine the best
element topology, and truss geometry can also be altered by taking nodal coordinates as design variables (Saka
[5]).
Many structural design variables are chosen based on discrete values due to manufacturing constraints. Thus, the
size, shape and topology optimisation of truss structures becomes a mixed variable optimisation problem, in that
it deals simultaneously with discrete and continuous design variables (Šilih et al. [6]). Such problems are usually
nonconvex by nature (Achtziger [7], Torii et al. [8] and [9]), and therefore must be solved by optimisation
methods capable of handling this type of problem.
Metaheuristic algorithms are well suited to solving such optimisation problems. Known advantages of these
algorithms include the following: (i) they do not require gradient information and can be applied to problems in
which the gradient is difficult to obtain or simply does not exist; (ii) they do not become stuck in local minima if
correctly tuned; (iii) they can be applied to non-smooth or discontinuous functions; (iv) they furnish a set of
optimal solutions instead of a single solution, giving the designer a set of options from which to choose; and (v)
they can be easily employed to solve mixed variable optimisation problems.
Genetic algorithms (GA) have been applied within this context for truss optimisation by several research studies.
The initial emphasis on GA algorithms in the literature can be explained by the fact that they were one of the first
developed heuristic algorithms, and have been successfully applied in different engineering fields, such as
scheduling problems (Kuan et al. [10] and Ng et al. [11]), pipe network optimisation (Morley et al. [12]), and
laminated composite structures (Le Riche and Haftka [13], Lopez et al. [14] and [15]), to name just a few. Size
optimisation has been performed by Goldberg and Samtani [16] and Rajeev and Krishnamoorthy [17], while size
and shape optimisation has been employed by Wu and Chow [18], Galante [19], Soh and Yang [20] and
Kelesoglu [21]. Fixed shapes with size and topology optimisation have been studied by Hajela et al. [22] and
Sakamoto and Oda [23], while Grierson and Pak [24], Rajan [25], Hajela et al. [26], Shrestha and Ghaboussi
[27], Deb and Gulati [28] and Tang et al. [29] have studied size, shape, and topology optimisation. However, the
use of GA has presented some drawbacks linked mainly to the long computational time required when dealing
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with large computational models. Hence, alternate approaches have been developed to reduce the computational
demand.
Several researchers have developed and improved robust search techniques in the last decade that simulate the
paradigm of a biological, chemical, or social system and are known as metaheuristic algorithms. This alternative
can be efficient at dealing with the challenges that traditional heuristic optimisation algorithms (e.g., GA or SA)
have faced for years, being particularly suitable to achieving fast and accurate solutions in the field of structural
optimisation (Miguel and Fadel Miguel [30]). For example, the truss optimisation problem has been solved using
alternate heuristic methods such as the simulated annealing (SA) [31], ant colony optimisation (ACO) [32], the
harmony search (HS) [33] and the cuckoo search algorithm (CS) [34]. For a comprehensive review of the
optimisation of truss structures using metaheuristics, the reader is referred to Saka [35], Saka [36], Lamberti and
Pappalettere [37] and the references therein.
The particular emphasis in more recent research (Balling et al. [38], von Buelow [39], Winslow et al. [40],
Martini [41]) has been on so-called multimodal optimisation, which implies that multiple locally optimal or near-
optimal solutions can be identified in each run of an algorithm. The ability to furnish the designer with a set of
options is very attractive because it is impossible to account for construction aspects or aesthetics in the fitness
functions. The alternatives presented by multimodal optimisation allow engineers to evaluate those
characteristics based on external priorities.
In this context, the Firefly algorithm (FA) developed recently by Yang [42] has proved to be more accurate and
efficient than well-established heuristic algorithms such as the GA and Particle Swarm Optimisation (PSO). FA
is capable of effectively and simultaneously finding the global as well as local optima and is particularly suitable
for parallel implementation. Several researchers have focused their attention on solving optimisation problems
using FA in a growing number of papers (Horng and Liou [43], Horng [44], Yang et al. [45], Chandrasekaran
and Simon [46], Gandomi et al. [47], Fateen et al. [48], Coelho and Mariani [49], Sayadi et al. [50]). However,
its implementation in the field of structural optimisation is still fairly recent and requires a substantial amount of
further study (Gandomi et al. [51], Siamak et al. [52]).
This paper employs the FA in the simultaneous optimisation of size, shape, and topology in truss structures.
Unstable and singular topologies are disregarded as possible solutions by checking the positive definiteness of
the stiffness matrix. The procedure is capable of providing multiple optimal and near-optimal solutions in each
run, presenting a set of options at the end of the design stage. The effectiveness of the FA is demonstrated
through a selection of benchmark problems. Section 2 presents the formulation of the optimisation problem and a
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description of the FA is given in Section 3. A series of numerical examples is presented in Section 4, while
Section 5 presents the main conclusions of the study.
2. PROBLEM FORMULATION
The ground structure approach is followed in the proposed methodology. This scheme, initially proposed by
Dorn et al. [53], starts with a universal truss containing all (or almost all) possible member connections n
among all nodes q in the structure. The topology optimisation procedure is then employed to discard the
unnecessary members. In other words, the algorithm chooses, from all possible members n, the m members
that remain in the structure for which nm . Simultaneously, the algorithm performs the size optimisation of
the truss by changing the cross-sectional area ( m
A) of the remaining structural members and the shape
optimisation by modifying the nodal coordinates of the nodes q’ considered as design variables ( 'q
ξ). This
optimisation procedure seeks the minimum structural weight of the truss subjected to stress, displacement and
kinematic stability constraints. For convenience of notation, the design variables A and ξ are grouped into the
vector
'11 ,,,,, qm
AA
x. Thus, the optimisation problem can be posed as:
Find x
(1)
Minimise
m
j
jjj AW
1
ρξx
Subject to
',...,1,5
,...,1,4
,...,1,0:
0:3
,...,1,02
1
maxmin
maxmin
max
qiG
mjAAAG
mjncompressio
ortensionG
qkG
stablellykinematicaistrussG
iii
jjj
c
jj
t
jj
kk
x
x
x
in which W is the structural weight, m is the number of members in the design, ρ is the specific weight of the
bar material, is the length of each bar (which is a function of the nodal coordinates) and G is the set of
constraints. k
and max
k
are the displacement and maximum allowable displacement at node k, respectively,
j
is the stress of the jth bar, t
j
and c
j
are respectively the maximum allowable stress in tension and
compression of the jth bar, min
j
A and max
j
A are the lower and upper bounds of the cross-sectional area of the jth
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bar, respectively, and min
i
and max
i
are the lower and upper bounds of the allowable movement of the ith
design variable nodal coordinate, respectively.
The displacement constraints are formulated by considering that the deflection in a specified coordinate direction
of a node must be lower than an allowable displacement chosen by the designer. Likewise, the stress in the
element must be lower than the allowable stress in tension and compression of the material. Furthermore, the
supports and nodes carrying loads cannot be disregarded in the final solution, which is, in fact, based on the
classical concept of basic and non-basic nodes. Finally, kinematic stability is achieved by checking the positive
definiteness of the stiffness matrix. If buckling constraints are included in the formulation, the allowable stress in
compression is the most critical case between the maximum stress in compression of the material and the critical
stress given by the Euler’s equation (see Eq.(6) in section 4.4.3).
The following scheme is applied to address the constraints in this optimisation problem. First, the feasibility of
the solution is checked (constraint G1 in Eq. (1)). In other words, if the truss is not kinematically stable, it is
discarded and a new solution is attempted. Thus, the member forces and node displacements are calculated only
if this first check is verified. Next, if one or more of the displacement and/or stress constraints are violated
(constraints G2 and G3 in Eq. (1)), a penalty t
P is added to the objective function of the current design. In this
case, the penalty magnitude is proportional to the violation, and takes the form:
m
j
i
j
i
jj
q
kk
kk
thP
11 max
max
x
x
x, (2)
in which
2
,
stands for the absolute value, h is a positive parameter, i is equal to t if
the member is in tension and i is equal to c if the member is in compression . Finally, constraints G4 and G5 are
addressed by a coding approach. These bounds are imposed by not sampling infeasible designs in the computer
code. In general, two classical approaches have been employed in the literature to address the simultaneous
optimisation of size, topology and geometry of trusses. The first alternative attempts to perform a single-stage
procedure, in which all variables including topology, shape and size are determined together. The second
alternative is a two-stage procedure. The main goal is to search for a set of stable topologies, after which the best
size and shape are determined. However, because the problems are not linearly separable, it is impossible to
always provide a global optimal design. Thus, a principal goal of this paper is to achieve better performance by
incorporating the advantages of FA with a single-stage procedure.
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A specific member is eliminated from the ground structure following the criteria proposed by Deb and Gulati
[28]. The cross-sectional area of a member is compared to a user-defined small critical cross-sectional area θ. If
the member area is smaller than θ, this element is eliminated from the ground structure. This method defines how
different topologies can be obtained in a continuous optimisation procedure. Note that the value of θ and the
lower (Amin) and upper (Amax) bounds of the cross-sectional areas must be determined by considering the
probability of a specific element to be absent from the final solution. For example, if Amin = -Amax and the critical
cross-sectional area θ is almost zero, the probability of any member being present in the final structure is
approximately 50 %. In discrete optimisation, the user defines the number of zero-force bars that are added to the
available profile list to generate a reasonable probability of eliminating an element from the ground structure.
3. FIREFLY ALGORITHM (FA)
The Firefly Algorithm (FA) is a very recent heuristic optimisation algorithm developed by Yang [42] and is
inspired by the flashing behaviour of fireflies. According to Yang [42], FA optimisation has three idealised rules.
(a) All fireflies are unisex, so that one firefly is attracted to other fireflies regardless of their sex.
(b) Attractiveness is proportional to brightness, so for any two flashing fireflies, the less bright firefly will move
towards the brighter firefly. Both attractiveness and brightness decrease as the distance between fireflies
increases. If there is no firefly brighter than a particular firefly, that firefly will move randomly.
(c) The brightness of a firefly is affected or determined by the landscape of the objective function.
Based on these three rules, the basic steps of the FA can be summarised as the pseudo-code shown in Fig. 1
(Yang [42]).
There are two essential components to FA: the variation of light intensity and the formulation of attractiveness.
The latter is assumed to be determined by the brightness of the firefly, which in turn is related to the objective
function of the problem under study.
As light intensity and attractiveness decrease and the distance from the source increases, the variation of light
intensity and attractiveness should be a monotonically decreasing function. For example, the light intensity can
be:
2
0
ij
r
ij
Ir Ie
(3)
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in which the light absorption coefficient
is a parameter of the FA and rij, is the distance between fireflies i and
j at xi and xj, respectively, which can be defined as the Cartesian distance ij i j
rxx. Because a firefly’s
attractiveness is proportional to the light intensity seen by other fireflies, it can be defined by:
2
0
ij
r
ij
re
(4)
in which 0
is the attractiveness at r = 0. Finally, the probability of a firefly i being attracted to another, more
attractive (brighter) firefly j is determined by:
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0,,
ij
rtt t t
ijiiiii
e
xxxεxxx (5)
where t is the generation number, i
ε is a random vector (e.g., the standard Gaussian random vector in which the
mean is 0 and the standard deviation is 1) and
is the randomisation parameter. The first term on the right-
hand side of Eq. (5) represents the attraction between the fireflies and the second term is the random movement.
In other words, Eq. (5) shows that a firefly will be attracted to brighter or more attractive fireflies and also move
randomly. Eq. (5) indicates that the user must set parameters 0
,
,
and the distribution of i
ε to apply the
FA, and also shows that there are two limit cases when
is small or large.
a) If
approaches zero, the attractive and brightness are constants, and consequently, a firefly can be seen by all
other fireflies. In this case, the FA reverts to the PSO.
b) f
approaches infinity, the attractiveness and brightness approach zero, and all fireflies are short-sighted or
fly in a foggy environment, moving randomly. In this case, the FA reverts to the pure random search algorithm.
Hence, the FA generally corresponds to the situation falling between these two limit cases. The next section
describes the use of a numerical analysis to demonstrate the effectiveness of the FA at solving truss optimisation
problems.
4. NUMERICAL EXAMPLES
Standard test problems are useful for checking optimisation algorithms. The benchmark examples given in this
section have been widely used for this purpose. Due to the stochastic nature of the FA, the final result can vary
depending on the seed used for the random number generation. Yet there is no established benchmark criterion in
the literature to evaluate the performance of metaheuristics in size, topology and shape optimisation of trusses.
For instance, Hajela et al. [26] and Tang et al. [29] have not mentioned the number of times the search was
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repeated and have presented only the best results obtained. Martini [41] has performed 10 optimisation runs of
4000 design cycles each. Moreover, some technical papers have not explained the adopted strategy in detail
(Rajan [25], Deb and Gulati [28]) and only the optimal results are furnished. With the goal of providing a
statistical basis for further comparison, this paper presents the results of over 100 runs for each example. In this
way, the average values and standard deviations are presented along with the optimal results. The problems are
presented in increasing order of complexity, and the following set of parameters are used in all examples:
h = 1.108,01
, 1
, 0.5
and a value of i
ε that follows a uniform distribution between -0.5 and 0.5.
4.1. Eleven-bar truss example
This truss is often used as a benchmark problem and has been applied in most studies in this field. The ground
structure of this example is shown in Fig. 2 and the design parameters are given in Table 1. Two independent
studies are performed: (i) size and topology optimisation and (ii) simultaneous size, shape and topology
optimisation.
4.1.1. Size and topology optimisation
Several researchers have studied this problem using GA, but have applied two different sets of discrete design
variables. Rajan [25] and Tang et al. [29] have adopted cross-sectional areas from a set of 32 discrete values,
whereas Hajela et al. [26] and Deb and Gulati [28] have allowed the cross-sectional areas to vary within the
range of 0.0-30.0 in2 at increments of 1.0 in2. Because the best known solution in the literature is 4912.15 lb,
found by Deb and Gulati [28], the present paper attempts to reproduce the conditions of that study.
Although Hajela et al. [26] and Deb and Gulati [28] have studied the same problem, each study has employed a
different approach to the solution. Hajela et al. [26] have performed the optimisation in two steps. The goal of
the first step is to search for kinematically stable topologies while disregarding the constraints. These topologies
are then used as seeds to find the best cross-sectional member size. In contrast, Deb and Gulati [28] have
conducted size and topology optimisation in only one step using a population size P = 220 and G = 225
generations, resulting in a total of 49500 objective function evaluations (OFE).
This paper intends to achieve the same accuracy as that of Deb and Gulati [28] while providing the designer with
several optimal or near-optimal structural designs. In addition, the authors aspire to achieve these goals with
fewer function evaluations than reference [28]. In other words, the authors employ n = 10 fireflies and S = 3000
searches, resulting in 30000 OFE.
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The FA approach is capable of finding 46 feasible topologies (from an average of over 100 simulations), which
demonstrates the robustness of the multimodal procedure because the number of different topologies generated is
one measure of its performance. In addition, the range of alternative solutions achieves very similar results when
compared to the best solution of each run, another indication of the multimodal procedure’s effectiveness.
Table 2 presents the normalised mean weight for the top five topologies, in which the results are again an
average over 100 runs (the coefficient of variation is inside the parentheses). Fig. 3 shows three of the topologies
typically generated during an optimisation run. Note that Topology A achieves the best weight. Fig. 3(a) shows
that the best topology consists of only six members and five nodes.
The cross-sectional areas and corresponding truss weights for the best designs over 100 runs obtained by this
study and by Hajela et al. [26] and Deb and Gulati [28] are shown in Table 3 and the convergence history is of
the best topology is shown in Fig. 4. The mean value achieved in the present study is equal to 4919.77 lb and the
coefficient of variation is 0.11 %.
The y-displacement approaches its limit for the nodes carrying loads, reaching 99.95 % (intersection of members
6 and 9) and 99.85 % (intersection of members 2, 4, and 6) of the allowable 2 in displacement. Thus, the
proposed optimisation scheme is capable of reproducing the weight obtained by Deb and Gulati [28], which has
been the best result reported in the literature so far.
4.1.2. Size, shape and topology optimisation
This problem has been studied by Rajan [25] and Balling et al. [38], who have employed the GA, and Martini
[41], who has employed the Harmony Search (HS) optimisation algorithm. Shape is optimised by allowing the
vertical coordinates of the three superior nodes to moves between 180 in and 1000 in, considering the origin in
the intersection of members 1, 2, and 3. Because the nodal coordinates are continuous and the cross-sectional
areas are taken from a set of 32 discrete variables (Rajan [25]), the problem is a mixed variable optimisation
problem in that it addresses integer and continuous design variables simultaneously.
Rajan [25] has performed a unimodal study of this problem with a population size P = 40 and number of
generations G = 96, while Balling et al. [38] has carried out a multimodal analysis with P = 1000 and G = 500,
resulting in a total of 500000 OFE. The latter has not eliminated the possibility of unstable topologies, allowing
the algorithm to determine mechanisms. Martini [41] has employed 75 evaluations on the initialisation of
harmony memory over 4000 design cycles, resulting in 4075 OFE in a multimodal analysis. The present paper
uses n = 10 fireflies and S = 5000 searches.
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The multimodal procedure is capable of finding 62 feasible topologies (the average over 100 simulations).
Meanwhile, Balling et al. [38] has found 26 feasible topologies, even though they do not eliminate the possibility
of mechanisms, and Martini [41] has found less than 10 solutions. Table 4 presents the normalised weights of the
top five topologies, in which the results are again an average over 100 runs (the coefficient of variation is inside
the parentheses). Once again, the range of alternative solutions is very close to the best solution of each run.
Despite the lower number of OFE, the best result over 100 runs in this study weighs 2705 lb, which is slightly
better than the optimum weight determined by Balling et al. [38] (2736 lb) and Rajan [25] (3254 lb). Martini [41]
has also presented a less desirable result, as expected because the number of OFE is much lower. The mean value
achieved in this work is equal to 2893.45 lb and the coefficient of variation is 2.12 %. Thus, the proposed
scheme provides better results than the cited references. Fig. 5 shows three of the topologies typically generated
during the optimisation run, of which Topology A corresponds to the best weight. The cross-sectional areas,
tension members and corresponding weight of Topology A are listed in Table 5.
Fig. 5(a) shows that the best topology is again composed of only six members and five nodes, and the
convergence history for this case is shown in Fig. 6. The y-displacement approaches its limit for the nodes
carrying loads, reaching 99.99 % (intersection of members 6 and 9) and 99.85 % (intersection of members 2, 4,
and 6) of the allowable 2 in displacement. From the results presented in this section, the authors can conclude
that the proposed optimisation scheme is capable not only of reproducing the best results reported in the
literature but also of improving the best design solution for this problem.
4.2. Thirty-nine-bar two-tiered truss example
The second benchmark example is the single-span 39-bar, 12-node, simply supported, two-tiered ground
structure shown in Fig. 7. This structure has been studied before by Deb and Gulati [28], who have demonstrated
one of the best GA procedures in the literature. Thus, it is interesting to test this problem again with the proposed
scheme and compare the results to those of the previous study.
Two independent studies are again performed: (i) size and topology optimisation and (ii) simultaneous size,
shape and topology optimisation. However, the cross-sectional areas are treated as continuous variables in this
case to properly compare the results to those in the literature. The overlapping members are shown laterally
dislocated in the figure for visual clarity. Because the lateral symmetry around member 19 is assumed, the
number of variables is reduced to 21. The allowable strength is 20 ksi and the material properties and maximum
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allowable deflection are the same as those in the previous problem. The continuous cross-sectional areas vary
between -0.225 and 2.25 in2 and a critical area of ε = 0.05 in2 is considered.
4.2.1. Size and topology optimisation
The procedure for this example consists of the optimisation of 21 continuous variables. Deb and Gulati [28] have
found two optimised topologies. The first, corresponding to a simulation run with a population size of 630,
consists of only 17 members and 10 nodes and produces a truss with a final weight of 198 lb. The second
optimised truss weighing 196.546 lb is obtained by increasing the population size to 840. The resulting structure
contains 19 members and 11 nodes, with a different topology than the first option, although both have a very
similar total weight. It is impossible to know the total number of evaluations performed because the number of
generations is not mentioned.
While Deb and Gulati [28] have performed a unimodal analysis, the current study intends to achieve the same
accuracy while attempting to obtain multimodal benefits--that is, to provide a set of high-quality options. Due to
the great complexity of this optimisation problem and the large number of feasible topologies, the algorithm is
truncated to propose only the top 100 topologies. This value, besides being sufficient for the purposes of the
study, also lowers the required computational time. Table 6 presents the normalised weights of the top five
topologies, the results of which are again an average over 100 runs. As in the previous examples, the range of
alternative solutions has results that are very close to the best solution obtained from each run.
The cross-sectional areas, tension members and corresponding truss weights obtained in this example are listed
in Table 7. Fig. 8 shows three of the topologies typically generated during the optimisation run. Note that
Topology A, corresponding to the best weight, contains 9 nodes and 15 members, which is less than both
topologies presented by Deb and Gulati [28]. A typical convergence history of the best topology for this problem
is shown in Fig. 9. The best result over 100 runs has a weight of 193.5472 lb, which is considerably lower than
the solutions in the references mentioned above. The mean value obtained using the FA approach is equal to
221.68 lb and the coefficient of variation is 12.9 %. Thus, the proposed scheme once again obtains the best result
found in the literature to the best of the authors’ knowledge.
4.2.2. Size, shape and topology optimisation
To carry out simultaneous size, shape and topology optimisation, selected nodal coordinates are taken as design
variables in addition to the 21 cross sections. Assuming symmetry and considering that constrained and load-
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carrying nodes must remain fixed and the highest node at the centre of the structure does not move laterally, it is
possible to reduce the number of nodal coordinates to 7. Each of these 7 nodes is allowed to move (-120, 120) in
from its original position.
The optimised configuration proposed by Deb and Gulati [28] has been obtained using a population size
P = 1680 and 300 generations, resulting in 504000 OFE. The resulting truss requires 15 members and nine nodes
and weighs 192.19 lb. As due to the complexity of this optimisation problem and the large number of feasible
topologies, the algorithm is again truncated to propose only the top 100 topologies. Table 8 presents the
normalised weights of the top five topologies, the results of which are again an average over 100 runs. As in the
other cases, the range of alternative solutions is very close to the best solution from each run.
The study uses n = 10 fireflies and S = 5000 searches, resulting in 50000 OFE, which represents less than 10% of
the computational effort of the work of Deb and Gulati [28]. Fig. 10 shows three of the topologies typically
generated during the optimisation run. Note that Topology A, corresponding to the best weight, contains 8 nodes
and 13 members, which is less than in the topology produced by Deb and Gulati [28]. A typical convergence
history of the best topology for this example is shown in Fig. 11, while the cross-sectional areas, tension
members and corresponding truss weights obtained in this study are listed in Table 9. The best result over 100
runs achieved by the FA has a weight of 191.304 lb, which is once again better than that of the reference
solution. The mean value achieved by the FA optimisation scheme is equal to 207.34 lb and the coefficient of
variation is 5.3 %.
4.3. Twenty-five-bar 3D truss example
This 3D truss is also often used as a benchmark problem. The ground structure is shown in Fig. 12 and the details
of the loading and member groupings are given in Tables 10 and 11. The allowable strength is 40 ksi in tension
and compression, the material properties (modulus of elasticity and weight density) are the same as in the
previous problem and the maximum allowable deflection is 0.35 in in any direction for each node. The cross-
sectional areas are chosen from a set D =(0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6,
1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.8, 3.0, 3.2, 3.4) in.
Two independent studies are performed: (i) size and shape optimisation and (ii) simultaneous size, shape and
topology optimisation.
4.3.1. Size and shape optimisation
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Past researchers have studied this problem using the GA, including Tang et al. [29], Rajeev and Krishnamoorthy
[17] and Wu and Chow [18]. The x-, y- and z-coordinates of nodes 3, 4, 5 and 6 and the x- and y-coordinates of
nodes 7, 8, 9 and 10 are taken as design variables, while nodes 1 and 2 remains unchanged. Because double
symmetry is required in both the x-z and y-z planes, the problem includes eight size and five configuration
variables. The side constraints for the configuration variables are 20 in ≤ x4 ≤ 60 in, 40 in ≤ x8 ≤ 80 in,
40 in ≤ y4 ≤ 80 in, 100 in ≤ y8 ≤ 140 in and 90 in ≤ z4 ≤ 130 in. Because the nodal coordinates are continuous and
the cross-sectional areas are taken from a set of 30 discrete variables, this problem is also a mixed variable
optimisation problem in that it deals simultaneously with integer and continuous design variables.
The three different GA codings proposed by Tang et al. [29] have been carried out using a population size P = 40
and 150 generations, resulting in 6000 OFE. To properly compare the FA performance to this GA study, the
present study uses n = 10 fireflies and S = 600 searches, resulting in the same 60000 OFE. The best results over
100 runs achieved by the FA have a weight of 118.83 lb, which is better than that of the reference, as observed in
Table 12. The mean value obtained with the FA optimisation scheme is equal to 132.3 lb and the coefficient of
variation is 5.5 %. The convergence history of the best topology is shown in Fig. 13.
4.3.2. Size, shape and topology optimisation
In addition to the eight discrete size and five continuous configuration variables, all eight member groups are
considered as topology variables. Thus, this is once again a mixed variable optimisation problem.
The previously cited references have performed unimodal procedures, while the present paper intends to achieve
multimodal benefits. The scheme is capable of finding 11 feasible topologies (an average over 100 simulations).
Table 13 presents the normalised weights of the top five topologies, in which the results are again an average
over 100 runs.
Table 14 shows the results of the present study compared to those in the literature. Note that the best solution
over 100 runs obtained by the present algorithm (116.58 lb) is better than that obtained by three of the four
different GA approaches and slightly worse than the fourth (1.5%). The mean value achieved in the present study
is equal to 139.16 lb and the coefficient of variation is 8.0 %. Fig. 14 shows three of the topologies typically
generated during the optimisation run, and the convergence history of the best topology is shown in Fig. 15.
4.4. Fifteen-bar planar truss example
14
This fifteen-bar planar truss was also studied by [18] and [29] and it was adopted here to verify the capability of
the FA multimodal optimisation approach under Euler’s buckling as a constraint. The ground structure is
illustrated in Fig. 16, showing a vertical tip load of 10000 lb applied on node 8. The allowable strength is 25 ksi
and the material properties (modulus of elasticity and weight density) are the same as in the previous examples.
The x- and y- coordinates of nodes 2, 3, 6 and 7 and the y-coordinate of nodes 4 and 8 are taken as design
variables. However, nodes 6 and 7 are constrained to have the same x-coordinates of the nodes 2 and 3,
respectively. Thus, the problem includes fifteen size and eight configuration variables (x2=x6, x3=x7, y2, y3, y4, y6,
y7, y8).
The cross-sectional areas are chosen from a set D = (0.111, 0.141, 0.174, 0.220, 0.270, 0.287, 0.347, 0.440,
0.539, 0.954, 1.081, 1.174, 1.333, 1.488, 1.764, 2.142, 2.697, 2.800, 3.131, 3.565, 3.813, 4.805, 5.952, 6.572,
7.192, 8.525, 9.300, 10.850, 13.330, 14.290, 17.170, 19.180) in. The side constraints for the configuration
variables are 100 in ≤ x2 ≤ 140 in, 220 in ≤ x3 ≤ 260 in, 100 in ≤ y2 ≤ 140 in, 100 in ≤ y3 ≤ 140 in,
50 in ≤ y4 ≤ 90 in, -20 in ≤ y6 ≤ 20 in, -20 in ≤ y7 ≤ 20 in, and 20 in ≤ y8 ≤ 60 in.
Four independent studies are performed: (i) size and shape optimisation without buckling constraints, (ii)
simultaneous size, shape and topology optimisation without buckling constraints, (iii) size and shape
optimisation with buckling constraints and, (iv) simultaneous size, shape and topology optimisation with
buckling constraints.
In the cases (i) and (ii) the allowable strength is 25 ksi for tension and compression.
4.4.1. Size and shape optimisation without buckling constraints
Past researchers have studied this problem using the GA, including Tang et al. [29] and Wu and Chow [18].
Because the nodal coordinates are continuous and the cross-sectional areas are taken from a set of 30 discrete
variables, this problem is also a mixed variable optimisation problem since it deals simultaneously with integer
and continuous design variables.
The three different GA proposed by Tang et al. [29] have been carried out using a population size P = 40 and 200
generations, resulting in 8000 OFE. To properly compare the FA performance to this GA study, the present study
uses n = 10 fireflies and S = 800 searches, resulting in the same 8000 OFE. The best results over 100 runs
achieved by the FA have a weight of 75.5 lb, which is better than that of the reference, as observed in Table 15.
The mean value obtained with the FA optimisation scheme is equal to 82.64 lb and the coefficient of variation is
2.96 %. The convergence history of the best topology is shown in Fig. 17.
15
4.4.2. Size, shape and topology optimisation without buckling constraints
In addition to the eight discrete size and five continuous configuration variables, all fifteen member groups are
considered as topology variables. Thus, this is once again a mixed variable optimisation problem.
The previously cited references have performed unimodal procedures, while the present paper intends to achieve
multimodal benefits. The scheme is capable of finding 75 feasible topologies (an average over 100 simulations).
Table 16 presents the normalised weights of the top five topologies, in which the results are again an average
over 100 runs.
Table 17 shows the results of the present study compared to those in the literature. Note that the best solution
over 100 runs obtained by the present algorithm (74.33 lb) is once again better than the GA approaches. The
mean value achieved in the present study is equal to 84.45 lb and the coefficient of variation is 6.49 %. Fig. 18
shows three of the topologies typically generated during the optimisation run, and the convergence history of the
best topology is shown in Fig. 19.
4.4.3. Size and shape optimisation with buckling constraints
Wu and Chow [18] have studied this problem using the GA and buckling constraints. In this study the member
stresses are constrained to be below the Euler buckling stress:
2
8
100
i
i
cr L
EA
. (6)
Wu and Chow [18] carried out the optimisation using a population size P = 30. However, they did not mention
the maximum number of generations. Aiming to maintain the parameters of the previous problem, the present
study uses n = 10 fireflies and S = 800 searches, resulting in the same 8000 OFE. The best results over 100 runs
achieved by the FA weights 138.06 lb, which is better than reached by the reference, as observed in Table 18.
The mean value obtained with the FA optimisation scheme is equal to 154.21 lb and the coefficient of variation
is 3.85 %. The convergence history of the best topology is shown in Fig. 20.
4.4.4. Size, shape and topology optimisation without buckling constraints
In addition to the eight discrete size and five continuous configuration variables, all fifteen member groups are
considered as topology variables. Eq. (6) is also adopted. Because this problem has not been studied before, the
results obtained could not be compared to the literature.
16
Once again, the present paper intends to achieve multimodal benefits. The scheme is capable of finding 58
feasible topologies (an average over 100 simulations). Table 19 presents the normalised weights of the top five
topologies, in which the results are again an average over 100 runs.
The best results over 100 runs achieved by the FA have a weight of 125.229 lb, as observed in Table 20. The
mean value achieved in the present study is equal to 152.61 lb and the coefficient of variation is 8.69 %. Fig. 21
shows three of the topologies typically generated during the optimisation run, and the convergence history of the
best topology is shown in Fig. 22.
4.5. Further comments
In this section, we compare the results and computational cost of the optimization of four different truss
structures, totalizing ten optimisation cases. The summary of this comparison is: in seven cases the FA
performed better (3 cases with 10% of computational cost and 5 cases with similar computational cost), in one
case the FA performed similar (with 10% of the computational cost), in only one case the FA achieved the
second best weight (with similar computational cost) and in one case the FA was not compared because the
example was not taken from the literature. To make this comparison even clearer for the reader, it is detailed in
Table 21. These results demonstrate the effectiveness of the FA to deal with optimization of truss structures.
5. CONCLUSIONS
This paper employs the Firefly algorithm (FA) in the simultaneous optimisation of the size, shape, and topology
of truss structures in a single-stage procedure. The results show that the approach is especially suited to mixed
variable optimisation problems, which is a typical scenario for such a problem.
It is also shown that the presence of mechanisms can be disregarded as possible solutions by checking the
positive definiteness of the stiffness matrix. Furthermore, the study performs a multimodal optimisation in that
the procedure is capable of providing multiple optimal or near-optimal solutions in each run, resulting in a set of
designs at the end of the optimisation stage. This capability is very attractive in practice because it is not always
possible to account for constructional aspects or aesthetics in the fitness functions, but the multimodal
optimisation allows the engineer to evaluate those options considering external requirements.
The effectiveness of the FA at solving the simultaneous size, shape and topology optimisation of trusses is
demonstrated through several benchmark problems, the results of which are similar or even better than those
17
reported in the literature, with lower computational costs. These examples emphasise the capabilities of the
proposed methodology in this field.
6. ACKNOWLEDGMENTS
The authors gratefully acknowledge the financial support of CNPq and CAPES.
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22
FIGURE CAPTIONS
Figure 1: Pseudo-code of the Firefly algorithm (adapted from Yang, [42]).
Figure 2: Eleven-bar truss benchmark example.
Figure 3: Samples of selected topologies determined for the size and topology eleven-bar truss optimisation
problem.
Figure 4: Convergence history for the size and topology eleven-bar truss optimisation problem.
Figure 5: Samples of selected topologies determined for the size, shape and topology eleven-bar truss
optimisation problem.
Figure 6: Convergence history for the size, shape and topology eleven-bar truss optimisation problem.
Figure 7: Two-tiered truss, thirty-nine-member benchmark example.
Figure 8: Samples of selected topologies determined for the size and topology two-tiered truss optimisation
problem.
Figure 9: Convergence history for the size and topology two-tiered truss optimisation problem.
Figure 10: Samples of selected topologies determined for the size, shape, and topology two-tiered truss
optimisation problem.
Figure 11: Convergence history for the size, shape, and topology two-tiered truss optimisation problem.
Figure 12: Twenty-five 3D truss benchmark example.
Figure 13: Convergence history for the size and shape twenty-five 3D truss optimisation problem.
Figure 14: Samples of selected topologies determined for the size, shape, and topology twenty-five 3D truss
optimisation problem.
Figure 15: Convergence history for the size, shape, and topology twenty-five 3D truss optimisation problem.
1
begin
Objective function f(x), x = (x1, ..., xd)T
Generate initial population of fireflies xi (i = 1, 2, ..., n)
Light intensity Ii at xi is determined by f(xi)
Define light absorption coefficient
while (t < MaxGeneration)
for i = 1 : n all n fireflies
for j = 1 : d loop over all d dimensions
if (Ii < Ij), Move firefly i towards j; end if
Vary attractiveness with distance r via exp[−
r]
Evaluate new solutions and update light intensity
end for j
end for i
Rank the fireflies and find the current global best
end while
Post-process results and visualization
end
Figure 1: Pseudo-code of the Firefly algorithm (adapted from Yang, [42])
1
2
34
5
6
7
89
10
11
360 in 360 in
360 in
1000000lb 1000000lb
Figure 2: Eleven-bar truss benchmark example
2
Figure 3: Samples of selected topologies determined for the size and topology eleven-bar truss optimisation
problem
Topology A
Topology B
Topology C
3
01000 2000 3000
4000
5000
6000
7000
8000
9000
10000
Generations
Weight (lb)
Figure 4: Convergence history for the size and topology eleven-bar truss optimisation problem
4
Topology A
Topology B
Topology C
5
Figure 5: Samples of selected topologies determined for the size, shape and topology eleven-bar truss
optimisation problem
01000 2000 3000 4000 5000
2000
2500
3000
3500
4000
4500
5000
Generations
Weight (lb)
Figure 6: Convergence history for the size, shape and topology eleven-bar truss optimisation problem
1 2
3
4
5
6
7
8
910
11
12
13
14
15
16
17
18 19
20
21
2223
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
120in
120in
20000lb20000lb20000lb
120in 120in 120in 120in
Figure 7: Two-tiered truss, thirty-nine-member benchmark example
6
Figure 8: Samples of selected topologies determined for the size and topology two-tiered truss optimisation
problem
Topology A
Topology B
Topology C
7
01000 2000 3000 4000 5000
100
200
300
400
500
600
Generations
Weight (lb)
Figure 9: Convergence history for the size and topology two-tiered truss optimisation problem
8
Figure 10: Samples of selected topologies determined for the size, shape, and topology two-tiered truss
optimisation problem
Topology A
Topology B
Topology C
9
01000 2000 3000 4000 5000
0
100
200
300
400
500
600
700
800
Generations
Weight (lb)
Figure 11: Convergence history for the size, shape, and topology two-tiered truss optimisation problem
9
5
2
8
4
6
10
1
3
7
xy
z
200 in
200 in
100 in
100 in
75 in
75 in
Figure 12: Twenty-five 3D truss benchmark example
10
0100 200 300 400 500 600
0
100
200
300
400
500
600
Generations
Weight (lb)
Figure 13: Convergence history for the size and shape twenty-five 3D truss optimisation problem
11
Figure 14: Samples of selected topologies determined for the size, shape, and topology twenty-five 3D truss
optimisation problem
25
21
20
16
7
22
4
5
17
6
9
14
2
3
24
8
15
19
18
23
25
21
20
16
7
22
4
5
17
6
1
9
14
2
3
24
8
15
19
18
23
25
21
16
20
7
13
22
4
5
17
6
1
9
14
2
3
24
12
8
19
15
18
23
Topology A Topology B
Topology C
12
0100 200 300 400 500 600
0
100
200
300
400
500
600
700
800
Generations
Weight (lb)
Figure 15: Convergence history for the size, shape, and topology twenty-five 3D truss optimisation problem
1 2 3
4 5 6
789
10 11 12 13 14 15
360in
120in
P
Figure 16: Fifteen-bar planar truss benchmark example
13
0200 400 600 800
0
100
200
300
400
500
600
Generations
Weight (lb)
Figure 17: Convergence history for the size and shape fifteen planar truss optimisation problem without buckling
constraints
14
Figure 18: Samples of selected topologies determined for the size, shape, and topology fifteen planar truss
optimisation problem without buckling constraints
12
3
45
6
79
10 11 12 13
14
15
12
3
45
6
789
10 11 12 13
14
15
12
3
45
6
9
10 11 12 13
14
15
Topology A
Topology B
Topology C
15
0200 400 600 800
0
200
400
600
800
Generations
Weight (lb)
Figure 19: Convergence history for the size, shape and topology fifteen planar truss optimisation problem
without buckling constraints
0200 400 600 800
0
200
400
600
800
1000
Generations
Weight (lb)
Figure 20: Convergence history for the size and shape fifteen planar truss optimisation problem with buckling
constraints
16
Figure 21: Samples of selected topologies determined for the size, shape, and topology fifteen planar truss
optimisation problem with buckling constraints
12
3
45
6
7 8 9
10 13 14
1 2
3
45
6
789
10 12
13 14
1 2
3
45
6
78
9
10 11
13 14
Topology C
Topology B
Topology A
17
0200 400 600 800
100
200
300
400
500
600
700
800
Generations
Weight (lb)
Figure 22: Convergence history for the size, shape and topology fifteen planar truss optimisation problem with
buckling constraints
1
Table 1: Design parameters for the eleven-bar truss benchmark example
Design parameter Value
Modulus of elasticity 104 ksi
Weigh density 0.1 lb/in3
Allowable stress in tension ( t
) 25 ksi
Allowable stress in compression ( c
j
) 25 ksi
Allowable y-displacement 2 in
Table 2: Normalized weigh for the top five topologies of the size and topology eleven-bar truss example
Normalized weigh
1st 2
nd 3
rd 4
th 5
th
FA 1.000 1.038
(1.5%)
1.088
(1.3%)
1.101
(0.9%)
1.117
(1.2%)
Table 3: Member areas of the optimised eleven-bar truss benchmark example
Member
Number
Member Areas (in2)
FA Deb and Gulati [28] Hajela et al. [26]
2 24 24 24
3 20 20 21
4 6 6 6
5 30 30 28
6 16 16 16
9 21 21 22
Weight (lb) 4912.85 4912.85 4942.7
2
Table 4: Normalized weigh for the top five topologies of the size, shape, and topology eleven-bar truss example
Normalized weigh
1st 2
nd 3
rd 4
th 5
th
FA 1.000 1.028
(1.4%)
1.072
(2.8%)
1.102
(3.5%)
1.144
(3.4%)
Table 5: Cross-sectional areas and tension members for the size, shape, and topology eleven-bar truss
optimisation problem
Member
Number Member Areas (in2) Stress (ksi)
2 11.50 10.3
3 2.88 10.0
4 5.74 19.1
5 11.50 10.4
6 7.22 10.2
9 13.50 9.2
Weight (lb) 2705.16
Table 6: Normalized weigh for the top five topologies of the size, and topology thirty-nine-bar two-tiered truss
example
Normalized weigh
1st 2
nd 3
rd 4
th 5
th
FA 1.000 1.024
(2.6%)
1.034
(2.4%)
1.049
(2.4%)
1.062
(2.6%)
3
Table 7: Cross-sectional areas and tension members for the size, and topology thirty-nine-bar two-tiered truss
optimisation problem
Member
Number
FA Deb and Gulati [28]
Member Areas (in2) Stress (ksi) Member Areas (in2)
1 0.0500 0 -
2 0.7524 19.93514 0.751
3 - - 0.051
5 1.5001 19.99902 1.502
7 - - 0.052
8 0.2504 19.96994 0.251
9 - - 0.051
10 1.0647 19.92404 1.061
11 1.0612 19.98894 1.063
14 0.5604 19.95146 0.559
21 1.0016 19.96714 1.005
22 0.0500 0 -
23 0.7524 19.93514 0.751
24 - - 0.051
26 1.5001 19.99902 1.502
28 - - 0.052
29 0.2504 19.96994 0.251
30 - - 0.051
31 1.0647 19.92404 1.061
32 1.0612 19.98894 1.063
35 0.5604 19.95146 0.559
Weight (lb) 193.5472 196.546
Table 8: Normalized weigh for the top five topologies of the size, shape, and topology thirty-nine-bar two-tiered
truss example
Normalized weigh
1st 2
nd 3
rd 4
th 5
th
FA 1.000 1.025
(2.2%)
1.031
(2.3%)
1.044
(2.4%)
1.050
(2.4%)
4
Table 9: Cross-sectional areas and tension members for the size, shape, and topology thirty-nine-bar two-tiered
truss optimisation problem
Member
Number
Present study
Member Areas (in2) Stress (ksi)
1 0.29465559 19.97512
2 1.06482113 19.71634
4 1.19137888 19.98304
8 1.25548701 19.96446
14 0.0505146 19.0254
15 1.53242606 19.94986
19 0.96093992 19.80172
22 0.29465559 19.97512
23 1.06482113 19.71634
25 1.19137888 19.98304
29 1.25548701 19.96446
35 0.0505146 19.0254
36 1.53242606 19.94986
Weight (lb) 191.304
Table 10: Loading of twenty-five bar 3D truss
Node Px (lb) Py (lb) Pz (lb)
1 1000 -10000 -10000
2 0 -10000 -10000
3 500 0 0
6 600 0 0
5
Table 11: Nodes co-ordinates and member grouping of twenty-five bar 3D truss
Node x(in) y(in) z(in) Group Member (end nodes)
1 -37.5 0 200 A1 1(1,2)
2 37.5 0 200 A2 2(1,4), 3(2,3), 4(1,5), 5(2,6)
3 -37.5 37.5 100 A3 6(2,5), 7(2,4), 8(1,3), 9(1,6)
4 37.5 37.5 100 A4 10(3,6), 11(4,5)
5 37.5 -37.5 100 A5 12(3,4), 13(5,6)
6 -37.5 -37.5 100 A6 14(3,10), 15(6,7), 16(4,9), 17(5,8)
7 -100 100 0 A7 18(3,8), 19(4,7), 20(6,9), 21(5,10)
8 100 100 0 A8 22(3,7), 23(4,8), 24(5,9), 25(6,10)
9 100 -100 0
10 -100 -100 0
Table 12: Optimum size and shape solution for the twenty-five bar 3D truss problem
Design
variables Ref. [17] Ref. [18] Tang et al. [29] FA
GA 1 GA 2 GA 3
A1 0.1 0.1 0.1 0.2 0.1 0.1
A2 1.8 0.2 0.2 0.1 0.1 0.1
A3 2.3 1.1 0.9 1 1.1 0.9
A4 0.2 0.2 0.2 0.1 0.1 0.1
A5 0.1 0.3 0.2 0.1 0.1 0.1
A6 0.8 0.1 0.1 0.1 0.2 0.1
A7 1.8 0.2 0.2 0.1 0.2 0.1
A8 3 0.9 1.1 1.1 0.7 1
X4 41.07 24.87 39.327 35.47 37.32
Y4 53.47 62.39 61.296 60.37 55.74
Z4 124.6 117.88 115.906 129.07 126.62
X8 50.8 42.36 65.477 45.06 50.14
Y8 131.48 129.46 135.905 137.04 136.40
Weight (lb) 546.01 136.2 136.09 130.2 124.94 118.83
Max stress
(lb/in2) 15589.7 15834.32 16043.5 18228.6 18830.23
Max
displac (in) 0.347 0.3486 0.3395 0.35 0.35
6
Table 13: Normalized weigh for the top five topologies of the size, shape, and topology twenty-five 3D truss
example
Normalized weigh
1st 2
nd 3
rd 4
th 5
th
FA 1.000 1.150
(15.3%)
1.488
(25.8%)
1.721
(33.6%)
2.150
(28.62%)
Table 14: Optimum size, shape and topology solution for the twenty-five 3D truss problem
Design
variables Ref. [18] Tang et al. [29] FA
GA 1 GA 2 GA 3
A1 0.1 0 0 0 0
A2 0.2 0.1 0.1 0.1 0.1
A3 1.1 1 0.9 0.9 1.1
A4 0.2 0 0 0 0
A5 0.3 0 0 0 0
A6 0.1 0.1 0.1 0.1 0.1
A7 0.2 0.1 0.1 0.1 0.1
A8 0.9 1.1 1 1.1 0.9
X4 41.07 38.83 39.91 39.51 38.50
Y4 53.47 50.62 61.99 70.18 64.35
Z4 124.6 126.55 118.23 105.16 112.87
X8 50.8 50.37 53.13 55.15 49.13
Y8 131.48 125.63 138.49 136.27 134.94
Weight (lb) 136.2 120.88 114.74 118.73 116.58
Max stress
(lb/in2) 15589.66 18840.45 17353.01 21240.45 19791.08
Max
displac (in) 0.347 0.35 0.35 0.3494 0.35
7
Table 15: Optimum size and shape solution for the fifteen planar truss problem without buckling constraints
Design
variables Ref. [18] Tang et al. [29] FA
GA 1 GA 2 GA 3
A1 1.174 1.081 0.954 1.081 0.954
A2 0.954 0.954 0.954 0.539 0.539
A3 0.44 0.111 0.111 0.287 0.220
A4 1.333 1.174 1.174 0.954 0.954
A5 0.954 0.539 2.697 0.954 0.539
A6 0.174 0.539 0.539 0.22 0.220
A7 0.44 0.954 0.111 0.111 0.111
A8 0.44 0.22 0.111 0.111 0.111
A9 1.081 0.539 0.111 0.287 0.287
A10 1.333 0.287 0.539 0.22 0.440
A11 0.174 0.539 0.111 0.44 0.440
A12 0.174 0.111 0.111 0.44 0.220
A13 0.347 0.287 0.539 0.111 0.220
A14 0.347 0.539 0.539 0.22 0.270
A15 0.44 0.27 0.111 0.347 0.220
X2 123.189 134.94 139.667 133.612 114.967
X3 231.595 252.439 220.678 234.752 247.040
Y2 107.189 125.894 115.733 100.449 125.919
Y3 119.175 100.371 106.469 104.738 111.067
Y4 60.462 80.724 53.01 73.762 58.298
Y6 -16.728 10.388 16.362 10.067 -17.564
Y7 15.565 0.503 12.259 -1.339 -5.821
Y8 36.645 28.517 43.689 50.402 31.465
Weight (lb) 120.528 106.007 100.327 79.82 75.55
Table 16: Normalized weigh for the top five topologies of the size, shape, and topology fifteen planar truss
without buckling example
Normalized weigh
1st 2
nd 3
rd 4
th 5
th
FA 1.000 1.121
(8.68%)
1.348
(20.35%)
1.472
(23.18%)
1.575
(24.86%)
8
Table 17: Optimum size, shape and topology solution for the fifteen planar truss problem without buckling
constraints
Design
variables
Tang et al. [29] FA
GA 1 GA 2 GA 3
A1 0.954 1.081 1.081 0.954
A2 0.954 0.954 0.539 0.539
A3 0 0 0 0.141
A4 1.333 1.081 1.081 0.954
A5 0.44 0.539 0.954 0.539
A6 0.44 0.539 0.44 0.287
A7 0 0 0 0.141
A8 0.111 0.111 0.141 0.000
A9 0 0 0 3.813
A10 0.539 0.347 0.27 0.440
A11 0 0.27 0.27 0.440
A12 0.111 0.111 0.539 0.220
A13 0.539 0.539 0.141 0.220
A14 0.539 0.539 0.44 0.347
A15 0 0 0 0.141
X2 139.15 110.7 111.85 112.027
X3 255.03 225.52 242.45 247.076
Y2 110.74 124.76 104.02 137.514
Y3 104.07 114.67 109.22 116.776
Y4 — — — 50.162
Y6 5.58 8.26 -10.82 -10.905
Y7 4.5 -0.56 -11.13 -3.179
Y8 24.59 36.77 48.84 48.825
Weight (lb) 80.59 78.64 77.84 74.33
9
Table 18: Optimum size and shape solution for the fifteen planar truss problem with buckling constraints
Design
variables
Wu and Chow [18] FA
(8000 OFE)
FA
(50000 OFE)
Size
optimisation
Size and
shape
A1 2.8 5.952 1.174
A2 2.8 1.764 1.081
A3 4.805 1.488 0.440
A4 2.697 5.952 1.764
A5 2.697 2.142 1.488
A6 1.488 1.488 1.081
A7 1.488 2.8 0.111
A8 0.111 1.333 0.220
A9 2.697 0.347 1.488
A10 2.697 1.081 0.174
A11 1.333 1.333 0.220
A12 2.8 1.081 0.270
A13 1.333 1.333 1.333
A14 1.488 1.764 0.287
A15 3.565 1.764 1.333
X2 — 118.891 109.559
X3 — 222.569 224.866
Y2 — 108.02 103.312
Y3 — 124.068 100.435
Y4 — 54.835 51.576
Y6 — 0.347 17.060
Y7 — 15.516 19.022
Y8 — 34.711 48.846
Weight
(lb) 483.279 402.357 138.068
10
Table 19: Normalized weigh for the top five topologies of the size, shape, and topology fifteen planar truss with
buckling example
Normalized weigh
1st 2
nd 3
rd 4
th 5
th
FA 1.000 1.281
(8.76%)
1.663
(18.03%)
1.936
(21.71%)
2.318
(24.16%)
Table 20: Optimum size, shape and topology solution for the fifteen planar truss problem with buckling
constraints
Design
variables
FA
(8000 OFE)
FA
(50000 OFE)
A1 0.954
A2 0.954
A3 0.111
A4 2.142
A5 1.081
A6 1.333
A7 0.111
A8 0.220
A9 1.174
A10 0.440
A11 0.000
A12 0.000
A13 1.764
A14 0.539
A15 0.000
X2 131.996
X3 233.985
Y2 115.401
Y3 116.888
Y4 53.065
Y6 13.937
Y7 11.605
Y8 51.895
Weight
(lb) 125.229
11
Table 21: Summary of the results
Example
Deb and
Gulati [28]
Balling et
al. [38]
Tang et al.
[29]
Wu and
Chow [18] FA
Weight (lb) 4.1.1 4912.85 4912.85
OFE (49500) (30000)
Weight (lb) 4.1.2 2736 2705.16
OFE (500000) (50000)
Weight (lb) 4.2.1 196.54 193.55
OFE - (50000)
Weight (lb) 4.2.2 192.19 191.30
OFE (504000) (50000)
Weight (lb) 4.3.1 124.94 118.83
OFE (6000) (6000)
Weight (lb) 4.3.2 114.74 116.58
OFE (6000) (6000)
Weight (lb) 4.4.1 79.82 75.55
OFE (8000) (8000)
Weight (lb) 4.4.2 77.84 74.33
OFE (8000) (8000)
Weight (lb) 4.4.3 402.36 138.07
OFE - (8000)
Weight (lb) 4.4.4 125.23
OFE (8000)
