# Fitness Estimation Based Particle Swarm Optimization Algorithm for Layout Design of Truss Structures

Authors: Ayang Xiao; Benli Wang; Chaoli Sun; Shijie Zhang; Zhenguo Yang
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

Due to the fact that vastly different variables and constraints are simultaneously considered, truss layout optimization is a typical difficult constrained mixed-integer nonlinear program. Moreover, the computational cost of truss analysis is often quite expensive. In this paper, a novel fitness estimation based particle swarm optimization algorithm with an adaptive penalty function approach (FEPSO-AP) is proposed to handle this problem. FEPSO-AP adopts a special fitness estimate strategy to evaluate the similar particles in the current population, with the purpose to reduce the computational cost. Further more, a laconic adaptive penalty function is employed by FEPSO-AP, which can handle multiple constraints effectively by making good use of historical iteration information. Four benchmark examples with fixed topologies and up to 44 design dimensions were studied to verify the generality and efficiency of the proposed algorithm. Numerical results of the present work compared with results of other state-of-the-art hybrid algorithms shown in the literature demonstrate that the convergence rate and the solution quality of FEPSO-AP are essentially competitive.


---

## Body

## 1. Introduction

As a typical real world project, truss structural analysis is considered to be computationally expensive [1]. Moreover, truss layout optimization, which solves truss sizing variables (e.g., cross-sectional areas of elements) and shape variables (e.g., coordinates of nodes) simultaneously, is known as a typical multimodal and highly nonlinear problem [2]. The various differences (e.g., physical nature, magnitude, continuity, etc.) between the two types of variables make the truss layout optimization problem a difficult task [3]. Hence, in the past decades, researchers mainly focused on solving truss layout optimization via multilevel methods [4].Vanderplaats and Moses [5] proposed an alternating gradient method to decompose truss layout optimization into a number of subproblems, each of which optimized a subset of the design variables. The subproblems are solved iteratively until a converged optimal solution is found. Zhou [6] used a similar two-level approximation concept to optimize the cross-sectional areas of the members and the coordinates of the joints. Gil and Andreu [7] used fully stressed design method and conjugate gradient method to optimize the sizing and shape parameters of bridges, respectively.Due to the strong coupling between the variables, the search efficiency of multilevel methods is often limited [8]. Aiming for addressing this issue and also benefiting from recent rapid advances in computational power, single-level methods which optimize all design variables simultaneously are becoming popular and competitive. Wang et al. [9] presented an optimality criteria (OC) algorithm for spatial truss layout optimization. Fourie and Groenwold [10] used new operators, namely, the elite velocity and the elite particle, in the standard particle swarm optimization (PSO) algorithm to optimize the truss layout, with the purpose of increasing the probability of migration to regions with high fitness. In fact, various metaheuristic algorithms including simulated annealing (SA) [11], genetic algorithm (GA) [12], charged system search (CSS) [13], and artificial bee colony algorithm (ABC) [14] have been introduced to address truss layout optimization problems.More recently, researchers turn to hybridizing different techniques to further enhance the searching efficiency of metaheuristic algorithms. Lingyun et al. [18] proposed a niche hybrid genetic algorithm to solve the truss shape and sizing optimization in a simple and effective manner. Kaveh and Zolghadr [23] developed a hybridized CSS-BBBC algorithm with trap recognition capability for truss layout optimization. Zuo et al. [24] proposed a hybrid OC-GA approach for fast and global truss layout optimization; Kaveh and Javadi [25] used harmony search and ray optimizer to enhance the PSO algorithm to optimize truss layout under multiple frequency constraints; Liu and Ye [26] designed a genetic simulated annealing algorithm for domes layout optimization. Gholizadeh [17] proposed a hybridized cellular automata and PSO algorithm for truss layout optimization.However, the common weakness of metaheuristic algorithms based structural optimization is that a huge number of structural analyses are required, which is quite time-consuming. In this paper, we propose a new hybridized algorithm, termed FEPSO-AP for truss layout optimization that aims to enhance the optimal efficiency by using the fitness estimations to partly substitute the computationally expensive fitness calculations. The finite element method (FEM) is adopted to evaluate the structural performance. Empirical results demonstrate that the proposed method is highly promising for truss layout optimization.


## 2. Statement of Truss Layout Optimization Problem

The main aim of truss layout optimization can be formulated as follows:(1)
Minimize:
w
(
X
)
=
∑
i
=
1
n
e
ρ
i
·
A
i
·
l
i
,
Subject
to:
g
(
X
)
≤
0
,
where w and g denote the weight of truss structure and the maximum violated structural constraint; X is the vector of all design variables; ρ
i, l
i, and A
i represent the material density, length, and cross-sectional area of the ith truss element, respectively.Different types of constraints might be considered simultaneously depending on the problem to be solved. Four typical design constraints involved in this work can be stated by(2)
g
i
σ
(
X
)
=
σ
i
(
X
)
σ
i
,
all
-
1
≤
0
,
i
=
1,2
,
…
,
n
e
,
g
j
d
(
X
)
=
d
j
(
X
)
d
j
,
all
-
1
≤
0
,
j
=
1,2
,
…
,
n
j
,
g
i
b
(
X
)
=
|
σ
i
(
X
)
|
σ
i
,
e
r
-
1
≤
0
,
if
σ
i
(
X
)
<
0
,
g
k
f
(
X
)
=
f
k
(
X
)
f
k
,
all
-
1
≤
0
,
k
=
1,2
,
…
,
n
k
,
in which g
i
σ is the truss element stress constraint, while σ
i and σ
i
,
all stand for the actual largest stress of the ith truss element and its stress limit, respectively; g
j
d is the nodal displacement constraint, while d
j and d
j
,
all stand for the actual largest nodal displacement of the jth node and its displacement limit, respectively; g
i
b is the local buckling constraint, while σ
i
,
e
r stands for the Euler critical stress of the ith element; g
k
f is the structural natural frequency constraint, whilef
k and f
k
,
all stand for the k
th structural natural frequency and its frequency limit; n
e, n
j, and n
k represent the number of structural elements, nodes, and constrained natural frequencies, respectively.


## 3. FEPSO-AP Algorithm

This section describes the fitness estimation based PSO algorithm with an adaptive penalty function approach (FEPSO-AP) developed in this research. As FEPSO-AP integrates PSO, FE, and AP, this section recalls the basic concept of PSO algorithm, fitness estimation strategy, and adaptive constraint handling approach. Finally, a framework of FEPSO-AP algorithm is presented.


### 3.1. The PSO Algorithm

As one of the most popular metaheuristic algorithms, PSO has found a wide application in real world projects for its structural concision and searching efficiency [17]. In a standard PSO [27], it is assumed that each of the particles has a position and a certain velocity. The position of particle represents a candidate solution to the optimization problem, and the velocity of particle determines the particle’s movement. Hence, the flying of particles can be considered as the swarm searching of design domain.If the particle flies from its current position to the next position, its velocity and position are updated by(3)
v
→
i
(
t
+
1
)
=
ω
v
→
i
(
t
)
+
c
1
r
1
(
p
→
i
(
t
)
-
x
→
i
(
t
)
)
+
c
2
r
2
(
p
→
g
(
t
)
-
x
→
i
(
t
)
)
,
x
→
i
(
t
+
1
)
=
x
→
i
(
t
)
+
v
→
i
(
t
+
1
)
,
in which v
→
i
(
t
) and x
→
i
(
t
) indicate the velocity and position of the ith particle at iteration t; p
→
i
(
t
) and p
→
g
(
t
) represent the historical best position of the ith particle and the global best position of all particles till iteration t, respectively. ω stands for the inertia weight; c
1and c
2 are cognitive and social parameters, respectively. r
1 and r
2 are diagonal matrixes, of which the diagonal elements are uniformly distributed random numbers among the range of (
0,1
).


### 3.2. The Fitness Estimation Strategy

Sun et al. [28] proposed a novel fitness estimation strategy for PSO to solve computationally expensive problems.According to (3), the positions of any two arbitrary particles selected from the swarm can be formulated by
(4)
x
→
i
(
t
+
1
)
=
(
1
+
ω
-
φ
1
-
φ
2
)
x
→
i
(
t
)
+
φ
1
p
→
i
(
t
)
+
φ
2
p
→
g
(
t
)
-
ω
x
→
i
(
t
-
1
)
,
x
→
j
(
t
+
1
)
=
(
1
+
ω
-
φ
1
′
-
φ
2
′
)
x
→
j
(
t
)
+
φ
1
′
p
→
j
(
t
)
+
φ
2
′
p
→
g
(
t
)
-
ω
x
→
j
(
t
-
1
)
,
in which φ
1
=
c
1
r
1, φ
2
=
c
2
r
2, φ
1
′
=
c
1
r
1
′, and φ
2
′
=
c
2
r
2
′. It can be found that the above two equations are both related to the global best position p
→
g
(
t
). Hence, (4) can be abbreviated to
(5)
φ
2
x
→
j
(
t
+
1
)
+
φ
2
′
(
1
+
ω
-
φ
1
-
φ
2
)
x
→
i
(
t
)
+
φ
2
′
φ
1
p
→
i
(
t
)
+
φ
2
ω
x
→
j
(
t
-
1
)
=
φ
2
′
x
→
i
(
t
+
1
)
+
φ
2
(
1
+
ω
-
φ
1
′
-
φ
2
′
)
x
→
j
(
t
)
+
φ
2
φ
1
′
p
→
j
(
t
)
+
φ
2
′
ω
x
→
i
(
t
-
1
)
.Use a virtual positionx
→
ν
(
t
+
1
) to present the value of (5). Supposing the influences of all coefficients are similar, Figure 1 describes the geometrical relationship among the virtual position x
→
ν
(
t
+
1
) and all subitems contained by (5). As shown in Figure 1, the fitness of virtual position x
→
ν
(
t
+
1
) can be estimated by either the fitness of x
→
j
(
t
+
1
), x
→
i
(
t
), p
→
i
(
t
), and x
→
j
(
t
-
1
) or the fitness of x
→
i
(
t
+
1
), x
→
j
(
t
), p
→
j
(
t
), and x
→
i
(
t
-
1
). Hence, if the current fitness value and the historical best fitness of every particle at iteration t and t
-
1 are obtained already, the explicit relationship between the fitness of particle i and j at iteration t
+
1 can be expressed in the following formulations:
(6)
f
e
(
x
→
j
(
t
+
1
)
)
=
d
v
j
(
t
+
1
)
·
{
α
·
[
f
(
x
→
i
(
t
+
1
)
)
d
v
i
(
t
+
1
)
+
f
(
x
→
i
(
t
-
1
)
)
d
v
i
(
t
-
1
)
+
f
(
x
→
j
(
t
)
)
d
v
j
(
t
)
+
f
(
p
→
j
(
t
)
)
d
v
p
j
(
t
)
]
-
f
(
x
→
j
(
t
-
1
)
)
d
v
j
(
t
-
1
)
+
f
(
x
→
i
(
t
)
)
d
v
i
(
t
)
+
f
(
p
→
i
(
t
)
)
d
v
p
i
(
t
)
}
,
(7)
α
=
(
1
/
d
v
j
(
t
+
1
)
)
+
(
1
/
d
v
j
(
t
-
1
)
)
+
(
1
/
d
v
i
(
t
)
)
+
(
1
/
d
v
p
i
(
t
)
)
(
1
/
d
v
i
(
t
+
1
)
)
+
(
1
/
d
v
i
(
t
-
1
)
)
+
(
1
/
d
v
j
(
t
)
)
+
(
1
/
d
v
p
j
(
t
)
)
,
in which d
v
i
(
t
+
1
), d
v
i
(
t
), d
v
i
(
t
-
1
), d
v
j
(
t
+
1
), d
v
j
(
t
), d
v
j
(
t
-
1
), d
v
p
i
(
t
), and d
v
p
j
(
t
) represent the distance between the virtual position x
→
ν
(
t
+
1
) and the real positions x
→
i
(
t
+
1
), x
→
i
(
t
), x
→
i
(
t
-
1
), x
→
j
(
t
+
1
), x
→
j
(
t
), x
→
j
(
t
-
1
), p
→
i
(
t
), and p
→
j
(
t
), respectively.Figure 1
Illustration of the virtual position.


### 3.3. Adaptive Constraint Handling Approach

As mentioned in Section2, truss layout optimization problem is often taken as a multiple constrained optimization problem. Coello Coello [29] pointed out that the constraint handling procedure plays an important role in the exploration and exploitation of constrained optimization problems. Similar to other metaheuristic algorithms, FEPSO is designed for unconstrained problems. Hence, it is necessary to incorporate constraint handling approach into FEPSO to solve truss layout optimization problems.In this work, we use penalty function methods (PEM) to transfer a constrained problem into an unconstrained problem by adding the influence of violated constraints to the initial fitness function. A pseudoobjective function is stated as follows:(8)
Φ
(
X
)
=
f
(
X
)
+
W
×
G
(
X
)
,
where W and G
(
X
) are positive penalty factor and penalty function. The common difficulty existing in standard PEM is to set the constant penalty factor W properly. Runarsson and Yao [30] pointed out that if the penalty factor W turns out to be too large, the searching landscape would be quite rough, and thus qualified exploitation and exploration of the design domain are hard to be achieved; on the contrary, if the penalty factor W is too small, it would be quite possible to lose feasible solutions.As summarized in [29], the common target of adaptive penalty factors is to make a good balance between the objective function and the constraints violation. In this work, we developed a laconic adaptive penalty factor as follows:
(9)
W
=
2
1
-
ρ
,
in which ρ is the ratio of feasible solutions in current population. It can be concluded that if W is too small, the obtained unfeasible solutions will increase the value of ρ, and thus, the algorithm would be pushed to explore more feasible solutions. On the other hand, if the feasible solutions congregate in the population, the value of W will approach zero, and thus a detailed exploitation would be performed.


### 3.4. The Framework of FEPSO-AP Algorithm

The penalty functionG
(
X
) of truss layout optimization problem is determined by
(10)
G
(
X
)
=
w
(
X
)
×
V
(
X
)
,
(11)
V
(
X
)
=
max
⁡
{
0
,
g
(
X
)
}
,
in which X is the structural design vector, V
(
X
) is the violated constraints, and w
(
X
) and g
(
X
) are defined by (1).According to (8)–(10), the pseudoobjective function can be abbreviated to
(12)
Φ
(
X
)
=
w
(
X
)
×
(
1
+
2
1
-
ρ
×
V
(
X
)
)
.To transfer the formulated truss layout optimization problem into an unconstrained optimization problem, the adaptive penalty function, the fitness estimation strategy, and the PSO algorithm are assembled as follows.Step  1. Set the termination condition of optimization and the initial values of ω, c
1, and c
2. Set the iteration times iter
=
1.Step  2. Generate the initial positions X
=
{
x
→
i
(
t
)
} and the initial velocities V
=
{
v
→
i
(
t
)
} of particles population randomly, i
=
1,2
,
3
,
…
,
popsize.Step  3. Calculate the fitness values f
(
x
→
i
(
t
)
) of all particles and update every particle’s historical best position p
→
i
(
t
) and the global best position p
→
g
(
t
) by
(13)
f
(
x
→
i
(
t
)
)
=
Φ
(
x
→
i
(
t
)
)
,
(14)
p
→
i
(
t
)
=
min
⁡
{
f
(
x
→
i
(
t
′
)
)
}
,
t
′
=
1,2
,
3
,
…
,
t
,
p
→
g
(
t
)
=
min
⁡
{
f
(
x
→
i
(
t
)
)
}
,
i
=
1,2
,
3
,
…
,
popsize
.Step  4. Update the velocity v
→
i
(
t
+
1
) and the position x
→
i
(
t
+
1
) of every particle based on (3). Set the iteration times iter
=
iter
+
1.Step  5. Calculate the distances between every pair of particles.(
5.1
)
Select particlei randomly and calculate its fitness value f
(
x
→
i
(
t
+
1
)
) based on (13).
(
5.2
)
Choose a particlej nearby particle i and calculate the virtual position x
→
ν
(
t
+
1
) by using the historical information of particle j based on (5).
(
5.3
)
Ifx
→
ν
(
t
+
1
) does not coincide with x
→
i
(
t
+
1
), x
→
j
(
t
), p
→
j
(
t
), x
→
i
(
t
-
1
), x
→
j
(
t
+
1
), x
→
i
(
t
), p
→
i
(
t
), or x
→
j
(
t
-
1
), estimate the fitness value f
e
(
x
→
j
(
t
+
1
)
) of particle j by (6); otherwise, calculate the fitness value of particle j based on (13).
(
5.4
)
Repeat(
5.1
)
–
(
5.4
) till the fitness values of all particles are evaluated.Step  6. Update every particle’s historical best position p
→
i
(
t
+
1
) and the global best position p
→
g
(
t
+
1
). If p
→
g
(
t
+
1
) is obtained by estimation, recalculate the corresponding particle’s fitness value by (13), and rechoose the global best position p
→
g
(
t
+
1
). Repeat this step till a nonestimated value of p
→
g
(
t
+
1
) is obtained.Step  7. Repeat Steps 4–6 till the termination condition is reached. Output the global best position p
→
g
(
t
+
1
) and its objective value.


## 3.1. The PSO Algorithm

As one of the most popular metaheuristic algorithms, PSO has found a wide application in real world projects for its structural concision and searching efficiency [17]. In a standard PSO [27], it is assumed that each of the particles has a position and a certain velocity. The position of particle represents a candidate solution to the optimization problem, and the velocity of particle determines the particle’s movement. Hence, the flying of particles can be considered as the swarm searching of design domain.If the particle flies from its current position to the next position, its velocity and position are updated by(3)
v
→
i
(
t
+
1
)
=
ω
v
→
i
(
t
)
+
c
1
r
1
(
p
→
i
(
t
)
-
x
→
i
(
t
)
)
+
c
2
r
2
(
p
→
g
(
t
)
-
x
→
i
(
t
)
)
,
x
→
i
(
t
+
1
)
=
x
→
i
(
t
)
+
v
→
i
(
t
+
1
)
,
in which v
→
i
(
t
) and x
→
i
(
t
) indicate the velocity and position of the ith particle at iteration t; p
→
i
(
t
) and p
→
g
(
t
) represent the historical best position of the ith particle and the global best position of all particles till iteration t, respectively. ω stands for the inertia weight; c
1and c
2 are cognitive and social parameters, respectively. r
1 and r
2 are diagonal matrixes, of which the diagonal elements are uniformly distributed random numbers among the range of (
0,1
).


## 3.2. The Fitness Estimation Strategy

Sun et al. [28] proposed a novel fitness estimation strategy for PSO to solve computationally expensive problems.According to (3), the positions of any two arbitrary particles selected from the swarm can be formulated by
(4)
x
→
i
(
t
+
1
)
=
(
1
+
ω
-
φ
1
-
φ
2
)
x
→
i
(
t
)
+
φ
1
p
→
i
(
t
)
+
φ
2
p
→
g
(
t
)
-
ω
x
→
i
(
t
-
1
)
,
x
→
j
(
t
+
1
)
=
(
1
+
ω
-
φ
1
′
-
φ
2
′
)
x
→
j
(
t
)
+
φ
1
′
p
→
j
(
t
)
+
φ
2
′
p
→
g
(
t
)
-
ω
x
→
j
(
t
-
1
)
,
in which φ
1
=
c
1
r
1, φ
2
=
c
2
r
2, φ
1
′
=
c
1
r
1
′, and φ
2
′
=
c
2
r
2
′. It can be found that the above two equations are both related to the global best position p
→
g
(
t
). Hence, (4) can be abbreviated to
(5)
φ
2
x
→
j
(
t
+
1
)
+
φ
2
′
(
1
+
ω
-
φ
1
-
φ
2
)
x
→
i
(
t
)
+
φ
2
′
φ
1
p
→
i
(
t
)
+
φ
2
ω
x
→
j
(
t
-
1
)
=
φ
2
′
x
→
i
(
t
+
1
)
+
φ
2
(
1
+
ω
-
φ
1
′
-
φ
2
′
)
x
→
j
(
t
)
+
φ
2
φ
1
′
p
→
j
(
t
)
+
φ
2
′
ω
x
→
i
(
t
-
1
)
.Use a virtual positionx
→
ν
(
t
+
1
) to present the value of (5). Supposing the influences of all coefficients are similar, Figure 1 describes the geometrical relationship among the virtual position x
→
ν
(
t
+
1
) and all subitems contained by (5). As shown in Figure 1, the fitness of virtual position x
→
ν
(
t
+
1
) can be estimated by either the fitness of x
→
j
(
t
+
1
), x
→
i
(
t
), p
→
i
(
t
), and x
→
j
(
t
-
1
) or the fitness of x
→
i
(
t
+
1
), x
→
j
(
t
), p
→
j
(
t
), and x
→
i
(
t
-
1
). Hence, if the current fitness value and the historical best fitness of every particle at iteration t and t
-
1 are obtained already, the explicit relationship between the fitness of particle i and j at iteration t
+
1 can be expressed in the following formulations:
(6)
f
e
(
x
→
j
(
t
+
1
)
)
=
d
v
j
(
t
+
1
)
·
{
α
·
[
f
(
x
→
i
(
t
+
1
)
)
d
v
i
(
t
+
1
)
+
f
(
x
→
i
(
t
-
1
)
)
d
v
i
(
t
-
1
)
+
f
(
x
→
j
(
t
)
)
d
v
j
(
t
)
+
f
(
p
→
j
(
t
)
)
d
v
p
j
(
t
)
]
-
f
(
x
→
j
(
t
-
1
)
)
d
v
j
(
t
-
1
)
+
f
(
x
→
i
(
t
)
)
d
v
i
(
t
)
+
f
(
p
→
i
(
t
)
)
d
v
p
i
(
t
)
}
,
(7)
α
=
(
1
/
d
v
j
(
t
+
1
)
)
+
(
1
/
d
v
j
(
t
-
1
)
)
+
(
1
/
d
v
i
(
t
)
)
+
(
1
/
d
v
p
i
(
t
)
)
(
1
/
d
v
i
(
t
+
1
)
)
+
(
1
/
d
v
i
(
t
-
1
)
)
+
(
1
/
d
v
j
(
t
)
)
+
(
1
/
d
v
p
j
(
t
)
)
,
in which d
v
i
(
t
+
1
), d
v
i
(
t
), d
v
i
(
t
-
1
), d
v
j
(
t
+
1
), d
v
j
(
t
), d
v
j
(
t
-
1
), d
v
p
i
(
t
), and d
v
p
j
(
t
) represent the distance between the virtual position x
→
ν
(
t
+
1
) and the real positions x
→
i
(
t
+
1
), x
→
i
(
t
), x
→
i
(
t
-
1
), x
→
j
(
t
+
1
), x
→
j
(
t
), x
→
j
(
t
-
1
), p
→
i
(
t
), and p
→
j
(
t
), respectively.Figure 1
Illustration of the virtual position.


## 3.3. Adaptive Constraint Handling Approach

As mentioned in Section2, truss layout optimization problem is often taken as a multiple constrained optimization problem. Coello Coello [29] pointed out that the constraint handling procedure plays an important role in the exploration and exploitation of constrained optimization problems. Similar to other metaheuristic algorithms, FEPSO is designed for unconstrained problems. Hence, it is necessary to incorporate constraint handling approach into FEPSO to solve truss layout optimization problems.In this work, we use penalty function methods (PEM) to transfer a constrained problem into an unconstrained problem by adding the influence of violated constraints to the initial fitness function. A pseudoobjective function is stated as follows:(8)
Φ
(
X
)
=
f
(
X
)
+
W
×
G
(
X
)
,
where W and G
(
X
) are positive penalty factor and penalty function. The common difficulty existing in standard PEM is to set the constant penalty factor W properly. Runarsson and Yao [30] pointed out that if the penalty factor W turns out to be too large, the searching landscape would be quite rough, and thus qualified exploitation and exploration of the design domain are hard to be achieved; on the contrary, if the penalty factor W is too small, it would be quite possible to lose feasible solutions.As summarized in [29], the common target of adaptive penalty factors is to make a good balance between the objective function and the constraints violation. In this work, we developed a laconic adaptive penalty factor as follows:
(9)
W
=
2
1
-
ρ
,
in which ρ is the ratio of feasible solutions in current population. It can be concluded that if W is too small, the obtained unfeasible solutions will increase the value of ρ, and thus, the algorithm would be pushed to explore more feasible solutions. On the other hand, if the feasible solutions congregate in the population, the value of W will approach zero, and thus a detailed exploitation would be performed.


## 3.4. The Framework of FEPSO-AP Algorithm

The penalty functionG
(
X
) of truss layout optimization problem is determined by
(10)
G
(
X
)
=
w
(
X
)
×
V
(
X
)
,
(11)
V
(
X
)
=
max
⁡
{
0
,
g
(
X
)
}
,
in which X is the structural design vector, V
(
X
) is the violated constraints, and w
(
X
) and g
(
X
) are defined by (1).According to (8)–(10), the pseudoobjective function can be abbreviated to
(12)
Φ
(
X
)
=
w
(
X
)
×
(
1
+
2
1
-
ρ
×
V
(
X
)
)
.To transfer the formulated truss layout optimization problem into an unconstrained optimization problem, the adaptive penalty function, the fitness estimation strategy, and the PSO algorithm are assembled as follows.Step  1. Set the termination condition of optimization and the initial values of ω, c
1, and c
2. Set the iteration times iter
=
1.Step  2. Generate the initial positions X
=
{
x
→
i
(
t
)
} and the initial velocities V
=
{
v
→
i
(
t
)
} of particles population randomly, i
=
1,2
,
3
,
…
,
popsize.Step  3. Calculate the fitness values f
(
x
→
i
(
t
)
) of all particles and update every particle’s historical best position p
→
i
(
t
) and the global best position p
→
g
(
t
) by
(13)
f
(
x
→
i
(
t
)
)
=
Φ
(
x
→
i
(
t
)
)
,
(14)
p
→
i
(
t
)
=
min
⁡
{
f
(
x
→
i
(
t
′
)
)
}
,
t
′
=
1,2
,
3
,
…
,
t
,
p
→
g
(
t
)
=
min
⁡
{
f
(
x
→
i
(
t
)
)
}
,
i
=
1,2
,
3
,
…
,
popsize
.Step  4. Update the velocity v
→
i
(
t
+
1
) and the position x
→
i
(
t
+
1
) of every particle based on (3). Set the iteration times iter
=
iter
+
1.Step  5. Calculate the distances between every pair of particles.(
5.1
)
Select particlei randomly and calculate its fitness value f
(
x
→
i
(
t
+
1
)
) based on (13).
(
5.2
)
Choose a particlej nearby particle i and calculate the virtual position x
→
ν
(
t
+
1
) by using the historical information of particle j based on (5).
(
5.3
)
Ifx
→
ν
(
t
+
1
) does not coincide with x
→
i
(
t
+
1
), x
→
j
(
t
), p
→
j
(
t
), x
→
i
(
t
-
1
), x
→
j
(
t
+
1
), x
→
i
(
t
), p
→
i
(
t
), or x
→
j
(
t
-
1
), estimate the fitness value f
e
(
x
→
j
(
t
+
1
)
) of particle j by (6); otherwise, calculate the fitness value of particle j based on (13).
(
5.4
)
Repeat(
5.1
)
–
(
5.4
) till the fitness values of all particles are evaluated.Step  6. Update every particle’s historical best position p
→
i
(
t
+
1
) and the global best position p
→
g
(
t
+
1
). If p
→
g
(
t
+
1
) is obtained by estimation, recalculate the corresponding particle’s fitness value by (13), and rechoose the global best position p
→
g
(
t
+
1
). Repeat this step till a nonestimated value of p
→
g
(
t
+
1
) is obtained.Step  7. Repeat Steps 4–6 till the termination condition is reached. Output the global best position p
→
g
(
t
+
1
) and its objective value.


## 4. Benchmark Examples

The following four benchmark examples have been used to demonstrate the generality and efficiency of the FEPSO-AP algorithm:(i)
a planar 15-bar truss subjected to a single load condition and stress constraints,
(ii)
a spatial 25-bar truss subjected to a single load condition under stress and displacement constraints,
(iii)
a planar 37-bar truss subjected to multiple frequency constraints,
(iv)
a planar 47-bar truss subjected to three load conditions under stress and local buckling constraints.Programs of FEPSO-AP algorithm and structural finite element method (FEM) algorithm are developed by using MATLAB R2013a. A personal computer with a Pentium E5700 processor and 2 GB memory under the Microsoft Windows 7 operating system has been used to run the optimization software.For all benchmarks examined in this study, the FEPSO-AP algorithm parameters are set as the usual constants of standard PSO which are obtained by [31]: ω
=
0.7298, c
1
=
2.05, and c
2
=
2.05. According to the design dimensions of four benchmarks, the population sizes are set as 46, 26, 38, and 88, while the maximum numbers of FEM analyses are set as 4000, 4500, 8000, and 20000, respectively.Twenty-five independent runs are performed with the best one being selected for each problem.


### 4.1. Planar 15-Bar Truss

The original geometry of planar 15-bar truss is shown in Figure2(a). Two nodes (ID: 1 and 5) are totally fixed and the x-coordinates of other two nodes (ID: 4 and 8) are fixed as well. The single loading condition is listed in Table 1.Table 1
Loading condition acting on the planar 15-bar truss.
Case
Node
F
x (kips)
F
y (kips)
1
8
0
−10.0Layout optimization of the planar 15-bar truss.
(a)
Geometry and element definitions of the planar 15-bar truss
(b)
Best solution of the planar 15-bar trussAll design variables are classified into 23 groups:sizing variables:A
i, i
=
1,2
,
…
,
15;
shape variables:x
2
=
x
6; x
3
=
x
7; y
2; y
3; y
4; y
6; y
7; y
8.Material parameters and design constraints are listed in Table2.Table 2
Material parameters, design constraints, and search range of the planar 15-bar truss optimization problem.
Category
Values
Material Parameters
Density
0.1 lb/in3
Modulus of elasticity
1
×
1
0
4 ksi
Constraints
Stress
The allowable elements stress interval:[−25 ksi,  25 ksi]
Search range
Shape variables
100
in.
≤
x
2
≤
140
in.
;
220
in.
≤
x
3
≤
260
in.
;
100
in.
≤
y
2
≤
140
in.
;
100
in.
≤
y
3
≤
140
in.;
50
in.
≤
y
4
≤
90
in.
;
-
20
in.
≤
y
6
≤
20
in.
;
-
20
in.
≤
y
7
≤
20
in.
;
20
in.
≤
y
8
≤
60
in.
;
x
1
,
5
=
0
in.;
y
1
=
120
in.
;
x
4
,
8
=
360
in.
Sizing variables
S = {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.2
A
i
∈
S
,
i
=
1
,
2
,
…
,
15Figure2(b) shows the optimum design found by this work, of which two nodes (ID: 4 and 8) are highly overlapped. Table 3 compares the best design found by this work with those reported in the literature. It can be seen that the results achieved by the proposed algorithm are quite close to the best results reported in the literature.Table 3
Comparison of optimized designs found for the planar 15-bar truss.
No.
Variable
FA [15]
FM-GA [16]
PSO [17]
CPSO [17]
SCPSO [17]
FEPSO-AP
1
A
1
0.954
1.081
0.954
1.174
0.954
1.081
2
A
2
0.539
0.539
1.081
0.539
0.539
0.539
3
A
3
0.220
0.287
0.270
0.347
0.270
0.270
4
A
4
0.954
0.954
1.081
0.954
0.954
0.954
5
A
5
0.539
0.539
0.539
0.954
0.539
0.539
6
A
6
0.220
0.141
0.287
0.141
0.174
0.111
7
A
7
0.111
0.111
0.141
0.141
0.111
0.111
8
A
8
0.111
0.111
0.111
0.111
0.111
0.111
9
A
9
0.287
0.539
0.347
1.174
0.287
0.347
10
A
10
0.440
0.440
0.440
0.141
0.347
0.347
11
A
11
0.440
0.539
0.270
0.440
0.347
0.440
12
A
12
0.220
0.270
0.111
0.440
0.220
0.287
13
A
13
0.220
0.220
0.347
0.141
0.220
0.287
14
A
14
0.270
0.141
0.440
0.141
0.174
0.111
15
A
15
0.220
0.287
0.220
0.347
0.270
0.270
16
x
2
114.9670
101.5775
106.052
102.287
137.222
100.009
17
x
3
247.0400
227.9112
239.025
240.505
259.909
248.078
18
y
2
125.9190
134.7986
130.356
112.584
123.501
131.524
19
y
3
111.0670
128.2206
114.273
108.043
110.002
123.211
20
y
4
58.2980
54.8630
51.987
57.795
59.936
54.077
21
y
6
−17.5640
−16.4484
1.814
−6.430
−5.180
−9.039
22
y
7
−5.8210
−13.3007
9.183
−1.801
4.219
−14.905
23
y
8
31.4650
54.8572
46.909
57.799
57.883
54.084
Best weight (lb)
75.5473
76.6854
82.2344
77.6153
72.6153
74.1673
Maximum displacement (in.)
24.9993
24.9992
24.9999
24.9909
24.9912
24.9999
Number of structural analyses
8,000
8,000
4,500
4,500
4,500
4,000


### 4.2. Spatial 25-Bar Truss

The original geometry of spatial 25-bar truss is shown in Figure3(a). Two nodes (ID: 1 and 2) are totally fixed and the z-coordinates of four nodes (ID: 7, 8, 9. and 10) are fixed as well. The loading condition is listed in Table 4.Table 4
Loading condition acting on the spatial 25-bar truss.
Case
Node
F
x (kips)
F
y (kips)
F
z (kips)
1
1
1.0
−10.0
−10.0
2
0.0
−10.0
−10.0
3
0.5
0.0
0.0
6
0.6
0.0
0.0Layout optimization of the spatial 25-bar truss.
(a)
Geometry and element definitions of the spatial 25-bar truss
(b)
Best solution of the spatial 25-bar trussTo ensure the structural symmetries, all design variables are classified into 13 groups:sizing variables:A
1; A
2
=
A
3
=
A
4
=
A
5; A
6
=
A
7
=
A
8
=
A
9; A
10
=
A
11; A
12
=
A
13; A
14
=
A
15
=
A
16
=
A
17; A
18
=
A
19
=
A
20
=
A
21; A
22
=
A
23
=
A
24
=
A
25;
shape variables:x
4
=
x
5
=
-
x
3
=
-
x
6; x
8
=
x
9
=
-
x
7
=
-
x
10; y
3
=
y
4
=
-
y
5
=
-
y
6; y
7
=
y
8
=
-
y
9
=
-
y
10; z
3
=
z
4
=
z
5
=
z
6.Material parameters and design constraints are listed in Table5.Table 5
Material parameters, design constraints, and search range of the spatial 25-bar truss optimization problem.
Category
Values
Material parameters
Density
0.1 lb/in3
Modulus of elasticity
1
×
1
0
4 ksi
Constraints
Stress
The allowable elements stress interval:[−40 ksi, 40 ksi]
Displacement
The allowable nodal displacement interval:[−0.35 in., 0.35 in.]
Search range
Shape variables
20
in.
≤
x
4
≤
60
in.
;
40
in.
≤
x
8
≤
80
in.
;
40
in.
≤
y
4
≤
80
in.
;
100
in.
≤
y
8
≤
140
in.
;
90
in.
≤
z
4
≤
130
in.
;
-
x
1
=
x
2
=
37.5
in.
;
y
1,2
=
0
in.
;
z
1,2
=
200
in.
Sizing variables
S
=
{
0.1,0.2,0.3
,
…
,
3.2
,
3.3
,
3.4
}
in
.
2
A
i
∈
S
,
i
=
1,2
,
…
,
25Figure3(b) shows the optimum design found by this work. Table 6 compares the best design found by this work with those reported in the literature. It can be concluded that by using the proposed algorithm it is possible to achieve the best feasible results at a low computational cost.Table 6
Comparison of optimized designs found for the spatial 25-bar truss.
No.
Variable
FA [15]
FM-GA [16]
PSO [17]
CPSO [17]
SCPSO [17]
FEPSO-AP
1
A
1
0.1
0.1
0.1
0.3
0.1
0.1
2
A
2
0.1
0.1
0.1
0.1
0.1
0.1
3
A
6
0.9
1.1
1.1
1.0
1.0
1.0
4
A
10
0.1
0.1
0.1
0.1
0.1
0.1
5
A
12
0.1
0.1
0.4
0.1
0.1
0.1
6
A
14
0.1
0.1
0.1
0.1
0.1
0.1
7
A
18
0.1
0.2
0.4
0.2
0.1
0.1
8
A
22
1.0
0.8
0.7
0.9
0.9
0.9
9
x
4
37.3200
33.0487
27.6169
33.4976
36.9520
36.8958
10
y
4
55.7400
53.5663
51.6196
62.3735
54.5786
54.1337
11
z
4
126.6200
129.9092
129.9071
114.5945
129.9758
130.0000
12
x
8
50.1400
43.7826
42.5526
40.0531
51.7317
51.9924
13
y
8
136.4000
136.8381
132.7241
133.6695
139.5316
140.0000
Best weight (lb)
118.83
120.1149
129.2076
123.5403
117.2271
117.3022
Maximum displacement (in.)
0.3500
0.3500
0.3503
0.3505
0.3518
0.3500
Maximum stress (ksi)
18.8302
17.1574
16.4391
15.5913
19.9702
20.0182
Minimum stress (ksi)
−9.4017
−6.4822
−10.9931
−6.4102
−9.4049
−9.4024
Number of structural analyses
6,000
10,000
4,500
4,500
4,500
4,500


### 4.3. Planar 37-Bar Truss

The original geometry of planar 37-bar truss is shown in Figure4(a). Eleven nodes (ID: 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20) are totally fixed and the x-coordinates of all other nine nodes (ID: 3, 5, 7, 9, 11, 13, 15, 17, and 19) are fixed as well. A nonstructural mass of 10 kg is attached to all free nodes on the lower chord.Layout optimization of the planar 37-bar truss.
(a)
Geometry and element definitions of the planar 37-bar truss
(b)
Best solution of the planar 37-bar trussTo ensure the structural symmetric about they-axis, all design variables are classified into 19 groups:sizing variables:A
1
=
A
27; A
2
=
A
26; A
3
=
A
24; A
4
=
A
25; A
5
=
A
23; A
6
=
A
21; A
7
=
A
22; A
8
=
A
20; A
9
=
A
18; A
10
=
A
19; A
11
=
A
17; A
12
=
A
15; A
13
=
A
16; A
14;
shape variables:x
3; x
5; x
7; x
9; y
3
=
y
19; y
5
=
y
17; y
7
=
y
15; y
9
=
y
13; y
11.Material parameters and design constraints are listed in Table7.Table 7
Material parameters, design constraints, and search range of the planar 37-bar truss optimization problem.
Category
Values
Material parameters
Density
7800 kg/m3
Modulus of elasticity
2.1
×
1
0
11 N/m2
Constraints
Natural frequencies
f
1
≥
20 Hz; f
2
≥
40 Hz; f
3
≥
60 Hz;
Search range
Shape variables
x
1
=
0
m
;
x
2,3
=
1
m
;
x
4,5
=
2
m
;
x
6,7
=
3
m
;
x
8,9
=
4
m
;
x
10,11
=
5
m
;
x
12,13
=
6
m
;
x
14,15
=
7
m;
x
16,17
=
8
m
;
x
18,19
=
9
m
;
x
20
=
10
m
;
0
m
≤
y
3,5
,
7,9
,
11
≤
3
m
;
Sizing variables
1
×
1
0
-
4
m
2
≤
A
i
≤
10
×
1
0
-
4
m
2
,
i
=
1,2
,
…
,
14
A
j
=
4
×
1
0
-
3
m
2
,
i
=
28,29
,
…
,
30Figure4(b) shows the optimum design explored by this work. Table 8 compares the best design presented by this work with those reported in the literature. It can be seen that the results achieved by the proposed algorithm are even better than the best results reported in the literature.Table 8
Comparison of optimized designs found for the planar 37-bar truss.
No.
Variable
OC [9]
GA [18]
PSO [19]
RO [20]
FEPSO-AP
1
A
1
3.2508
2.8932
2.6797
3.0124
3.4197
2
A
2
1.2364
1.1201
1.1568
1.0623
0.9766
3
A
3
1.0000
1.0000
2.3476
1.0005
0.8313
4
A
4
2.5386
1.8655
1.7182
2.2647
2.8073
5
A
5
1.3714
1.5962
1.2751
1.6339
1.2997
6
A
6
1.3681
1.2642
1.4819
1.6717
1.6483
7
A
7
2.4290
1.8254
4.6850
2.0591
2.4972
8
A
8
1.6522
2.0009
1.1246
1.6607
1.5379
9
A
9
1.8257
1.9526
2.1214
1.4941
1.7590
10
A
10
2.3022
1.9705
3.8600
2.4737
2.7069
11
A
11
1.3103
1.8294
2.9817
1.5260
1.3046
12
A
12
1.4067
1.2358
1.2021
1.4823
1.4004
13
A
13
2.1896
1.4049
1.2563
2.4148
3.0476
14
A
14
1.0000
1.0000
3.3276
1.0034
0.5947
15
y
3
1.2086
1.1998
0.9637
1.0010
0.8756
16
y
5
1.5788
1.6553
1.3978
1.3909
1.2546
17
y
7
1.6719
1.9652
1.5929
1.5893
1.4446
18
y
9
1.7703
2.0737
1.8812
1.7507
1.5889
19
y
11
1.8502
2.3050
2.0856
1.8336
1.6480
Natural frequencies (Hz)
f
1
20.0850
20.0013
20.0001
20.056
20.020
f
2
42.0743
40.0305
40.0003
40.035
40.022
f
3
62.9383
60.0000
60.0001
60.030
60.233
f
4
74.4539
73.0444
73.0440
74.387
72.137
f
5
90.0576
89.8244
89.8240
85.929
84.065
Best weight (kg)
366.50
368.84
377.20
364.04
362.8812
Number of structural analyses
—
—
20,000
32,000
8,000


### 4.4. Planar 47-Bar Truss

The original geometry of planar 47-bar truss is shown in Figure5(a). Four nodes (ID: 15, 16, 17, and 22) are totally fixed and the y-coordinates of other two nodes (ID: 1 and 2) are fixed as well. The loading conditions are listed in Table 9.Table 9
Loading conditions acting on the planar 47-bar truss.
Case
Node
F
x (kips)
F
y (kips)
1
17 and 22
6.0
−14.0
2
17
6.0
−14.0
3
22
6.0
−14.0Layout optimization of the planar 47-bar truss.
(a)
Geometry and element definitions of the planar 47-bar truss
(b)
Best solution of the planar 47-bar trussTo ensure the structural symmetric about they-axis, all design variables are classified into 44 groups:sizing variables:A
1
=
A
3; A
2
=
A
4; A
5
=
A
6; A
7; A
8
=
A
9; A
10; A
11
=
A
12; A
13
=
A
14; A
15
=
A
16; A
17
=
A
18; A
19
=
A
20; A
21
=
A
22; A
23
=
A
24; A
25
=
A
26; A
27; A
28; A
29
=
A
30; A
31
=
A
32; A
33; A
34
=
A
35; A
36
=
A
37; A
38; A
39
=
A
40; A
41
=
A
42; A
43; A
44
=
A
45; A
46
=
A
47;
shape variables:x
1
=
-
x
2; x
3
=
-
x
4; y
3
=
y
4; x
5
=
-
x
6; y
5
=
y
6; x
7
=
-
x
8; y
7
=
y
8; x
9
=
-
x
10; y
9
=
y
10; x
11
=
-
x
12; y
11
=
y
12; x
13
=
-
x
14; y
13
=
y
14; x
19
=
-
x
20; y
19
=
y
20; x
18
=
-
x
21; y
18
=
y
21.Material parameters and design constraints are listed in Table10.Table 10
Material parameters, design constraints, and search range of the planar 47-bar truss optimization problem.
Category
Values
Material parameters
Density
0.3 lb/in3
Modulus of elasticity
3
×
1
0
4 ksi
Constraints
Stress
The allowable elements stress interval:[−15 ksi, 20 ksi]
Local buckling
|
(
σ
c
)
i
|
≤
β
E
A
i
/
l
i
2
,
i
=
1,2
,
…
,
47
,
β
=
3.96
Search range
Shape variables
0
in.
≤
x
i
≤
120
in.
,
(
i
=
2,4
,
6,8
)
;
-
30
in.
≤
x
j
≤
90
in.
(
j
=
10,12,14,20
)
;
30
in.
≤
x
21
≤
150
in.;
60
in.
≤
y
4
≤
180
in.
;
180
in.
≤
y
6
≤
300
in.
;
300
in.
≤
y
8
≤
420
in.
;
360
in.
≤
y
10
≤
480
in.
;
420
in.
≤
y
12
≤
540
in.
;
480
in.
≤
y
14
≤
600
in.
;
540
in.
≤
y
20
≤
660
in.
;
540
in.
≤
y
21
≤
660
in.
Sizing variables
S
=
{
0.1,0.2,0.3
,
…
,
4.8,4.9,5.0
}
in
.
2
A
i
∈
S
,
i
=
1,2
,
…
,
47Figure5(b) shows the optimum design identified by this work. Table 11 compares the best design of this work with those reported in the literature. It can be seen that by using the proposed algorithm it is possible to achieve better results at a lower computational cost.Table 11
Comparison of optimized designs found for the planar 47-bar truss.
No.
Variable
GA [21]
FSD-ES [22]
PSO [17]
CPSO [17]
SCPSO [17]
FEPSO-AP
1
A
3
2.50
2.70
2.80
2.60
2.50
3.20
2
A
4
2.20
2.50
2.70
2.50
2.50
2.80
3
A
5
0.70
0.70
0.80
0.70
0.80
0.70
4
A
7
0.10
0.10
1.10
0.30
0.10
0.10
5
A
8
1.30
0.90
0.80
1.20
0.70
0.60
6
A
10
1.30
1.10
1.30
1.10
1.40
1.60
7
A
12
1.80
1.80
1.80
1.60
1.70
1.90
8
A
14
0.50
0.70
0.90
0.80
0.80
0.90
9
A
15
0.80
0.90
1.20
1.10
0.90
1.00
10
A
18
1.20
1.30
1.40
1.30
1.30
2.00
11
A
20
0.40
0.30
0.30
0.30
0.30
0.10
12
A
22
1.20
1.10
1.40
0.80
0.90
0.40
13
A
24
0.90
1.00
1.10
1.00
1.00
1.40
14
A
26
1.00
0.90
1.20
1.00
1.10
1.50
15
A
27
3.60
0.80
1.60
0.90
5.00
1.20
16
A
28
0.10
0.10
1.00
0.10
0.10
0.70
17
A
30
2.40
2.70
2.80
2.70
2.50
5.00
18
A
31
1.10
0.80
0.80
0.90
1.00
1.00
19
A
33
0.10
0.10
0.10
0.10
0.10
0.10
20
A
35
2.70
3.00
3.00
3.00
2.80
3.00
21
A
36
0.80
0.90
0.90
1.00
0.90
0.50
22
A
38
0.10
0.00
0.10
0.20
0.10
0.10
23
A
40
2.80
3.20
3.30
3.30
3.00
3.30
24
A
41
1.30
1.00
0.90
0.90
1.00
0.30
25
A
43
0.20
0.10
0.10
0.10
0.10
0.10
26
A
45
3.00
3.30
3.30
3.30
3.20
3.50
27
A
46
1.20
1.10
1.20
1.10
1.20
0.40
28
x
2
114.0000
100.9724
98.8628
99.3630
101.3393
87.7275
29
x
4
97.0000
80.4772
78.6595
83.4439
85.9111
72.4352
30
y
4
125.0000
136.8699
146.7331
126.3845
135.9645
162.6451
31
x
6
76.0000
64.3908
66.5231
69.5148
74.7969
67.2113
32
y
6
261.0000
247.0491
239.0901
218.2013
237.7447
218.2041
33
x
8
69.0000
55.2589
55.6936
58.0004
64.3115
50.6507
34
y
8
316.0000
338.4534
327.7882
322.2272
321.3416
375.4549
35
x
10
56.0000
48.7333
48.8641
51.4015
53.3345
36.6525
36
y
10
414.0000
409.7380
398.6775
401.5626
414.3025
408.7230
37
x
12
50.0000
43.4742
43.1400
46.8605
46.0277
36.9960
38
y
12
463.0000
472.1479
464.7831
458.3021
489.9216
483.4295
39
x
14
54.0000
44.8349
37.8993
46.8885
41.8353
37.9558
40
y
14
524.0000
512.1901
511.0450
527.8575
522.4161
535.7644
41
x
20
1.0000
3.8414
18.2341
16.2354
1.0005
4.6875
42
y
20
587.0000
591.1449
594.0710
610.8496
598.3905
599.7416
43
x
21
99.0000
84.5040
90.9369
98.3239
97.8696
101.4535
44
y
21
631.0000
630.3472
621.3943
624.9580
624.0552
605.4302
Best weight (lb)
1925.7897
1842.6609
1975.8393
1908.8301
1864.0985
1799.7037
Maximum stress (ksi)
19.9528
20.0000
19.0636
19.3351
19.4735
19.9808
Minimum stress (ksi)
−14.9973
−15.0000
−14.9999
−14.9986
−15.0000
−14.9986
Number of buckling elements
0
0
0
0
0
0
Number of structural analyses
100,000
55,802
25,000
25,000
25,000
20,000


## 4.1. Planar 15-Bar Truss

The original geometry of planar 15-bar truss is shown in Figure2(a). Two nodes (ID: 1 and 5) are totally fixed and the x-coordinates of other two nodes (ID: 4 and 8) are fixed as well. The single loading condition is listed in Table 1.Table 1
Loading condition acting on the planar 15-bar truss.
Case
Node
F
x (kips)
F
y (kips)
1
8
0
−10.0Layout optimization of the planar 15-bar truss.
(a)
Geometry and element definitions of the planar 15-bar truss
(b)
Best solution of the planar 15-bar trussAll design variables are classified into 23 groups:sizing variables:A
i, i
=
1,2
,
…
,
15;
shape variables:x
2
=
x
6; x
3
=
x
7; y
2; y
3; y
4; y
6; y
7; y
8.Material parameters and design constraints are listed in Table2.Table 2
Material parameters, design constraints, and search range of the planar 15-bar truss optimization problem.
Category
Values
Material Parameters
Density
0.1 lb/in3
Modulus of elasticity
1
×
1
0
4 ksi
Constraints
Stress
The allowable elements stress interval:[−25 ksi,  25 ksi]
Search range
Shape variables
100
in.
≤
x
2
≤
140
in.
;
220
in.
≤
x
3
≤
260
in.
;
100
in.
≤
y
2
≤
140
in.
;
100
in.
≤
y
3
≤
140
in.;
50
in.
≤
y
4
≤
90
in.
;
-
20
in.
≤
y
6
≤
20
in.
;
-
20
in.
≤
y
7
≤
20
in.
;
20
in.
≤
y
8
≤
60
in.
;
x
1
,
5
=
0
in.;
y
1
=
120
in.
;
x
4
,
8
=
360
in.
Sizing variables
S = {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.2
A
i
∈
S
,
i
=
1
,
2
,
…
,
15Figure2(b) shows the optimum design found by this work, of which two nodes (ID: 4 and 8) are highly overlapped. Table 3 compares the best design found by this work with those reported in the literature. It can be seen that the results achieved by the proposed algorithm are quite close to the best results reported in the literature.Table 3
Comparison of optimized designs found for the planar 15-bar truss.
No.
Variable
FA [15]
FM-GA [16]
PSO [17]
CPSO [17]
SCPSO [17]
FEPSO-AP
1
A
1
0.954
1.081
0.954
1.174
0.954
1.081
2
A
2
0.539
0.539
1.081
0.539
0.539
0.539
3
A
3
0.220
0.287
0.270
0.347
0.270
0.270
4
A
4
0.954
0.954
1.081
0.954
0.954
0.954
5
A
5
0.539
0.539
0.539
0.954
0.539
0.539
6
A
6
0.220
0.141
0.287
0.141
0.174
0.111
7
A
7
0.111
0.111
0.141
0.141
0.111
0.111
8
A
8
0.111
0.111
0.111
0.111
0.111
0.111
9
A
9
0.287
0.539
0.347
1.174
0.287
0.347
10
A
10
0.440
0.440
0.440
0.141
0.347
0.347
11
A
11
0.440
0.539
0.270
0.440
0.347
0.440
12
A
12
0.220
0.270
0.111
0.440
0.220
0.287
13
A
13
0.220
0.220
0.347
0.141
0.220
0.287
14
A
14
0.270
0.141
0.440
0.141
0.174
0.111
15
A
15
0.220
0.287
0.220
0.347
0.270
0.270
16
x
2
114.9670
101.5775
106.052
102.287
137.222
100.009
17
x
3
247.0400
227.9112
239.025
240.505
259.909
248.078
18
y
2
125.9190
134.7986
130.356
112.584
123.501
131.524
19
y
3
111.0670
128.2206
114.273
108.043
110.002
123.211
20
y
4
58.2980
54.8630
51.987
57.795
59.936
54.077
21
y
6
−17.5640
−16.4484
1.814
−6.430
−5.180
−9.039
22
y
7
−5.8210
−13.3007
9.183
−1.801
4.219
−14.905
23
y
8
31.4650
54.8572
46.909
57.799
57.883
54.084
Best weight (lb)
75.5473
76.6854
82.2344
77.6153
72.6153
74.1673
Maximum displacement (in.)
24.9993
24.9992
24.9999
24.9909
24.9912
24.9999
Number of structural analyses
8,000
8,000
4,500
4,500
4,500
4,000


## 4.2. Spatial 25-Bar Truss

The original geometry of spatial 25-bar truss is shown in Figure3(a). Two nodes (ID: 1 and 2) are totally fixed and the z-coordinates of four nodes (ID: 7, 8, 9. and 10) are fixed as well. The loading condition is listed in Table 4.Table 4
Loading condition acting on the spatial 25-bar truss.
Case
Node
F
x (kips)
F
y (kips)
F
z (kips)
1
1
1.0
−10.0
−10.0
2
0.0
−10.0
−10.0
3
0.5
0.0
0.0
6
0.6
0.0
0.0Layout optimization of the spatial 25-bar truss.
(a)
Geometry and element definitions of the spatial 25-bar truss
(b)
Best solution of the spatial 25-bar trussTo ensure the structural symmetries, all design variables are classified into 13 groups:sizing variables:A
1; A
2
=
A
3
=
A
4
=
A
5; A
6
=
A
7
=
A
8
=
A
9; A
10
=
A
11; A
12
=
A
13; A
14
=
A
15
=
A
16
=
A
17; A
18
=
A
19
=
A
20
=
A
21; A
22
=
A
23
=
A
24
=
A
25;
shape variables:x
4
=
x
5
=
-
x
3
=
-
x
6; x
8
=
x
9
=
-
x
7
=
-
x
10; y
3
=
y
4
=
-
y
5
=
-
y
6; y
7
=
y
8
=
-
y
9
=
-
y
10; z
3
=
z
4
=
z
5
=
z
6.Material parameters and design constraints are listed in Table5.Table 5
Material parameters, design constraints, and search range of the spatial 25-bar truss optimization problem.
Category
Values
Material parameters
Density
0.1 lb/in3
Modulus of elasticity
1
×
1
0
4 ksi
Constraints
Stress
The allowable elements stress interval:[−40 ksi, 40 ksi]
Displacement
The allowable nodal displacement interval:[−0.35 in., 0.35 in.]
Search range
Shape variables
20
in.
≤
x
4
≤
60
in.
;
40
in.
≤
x
8
≤
80
in.
;
40
in.
≤
y
4
≤
80
in.
;
100
in.
≤
y
8
≤
140
in.
;
90
in.
≤
z
4
≤
130
in.
;
-
x
1
=
x
2
=
37.5
in.
;
y
1,2
=
0
in.
;
z
1,2
=
200
in.
Sizing variables
S
=
{
0.1,0.2,0.3
,
…
,
3.2
,
3.3
,
3.4
}
in
.
2
A
i
∈
S
,
i
=
1,2
,
…
,
25Figure3(b) shows the optimum design found by this work. Table 6 compares the best design found by this work with those reported in the literature. It can be concluded that by using the proposed algorithm it is possible to achieve the best feasible results at a low computational cost.Table 6
Comparison of optimized designs found for the spatial 25-bar truss.
No.
Variable
FA [15]
FM-GA [16]
PSO [17]
CPSO [17]
SCPSO [17]
FEPSO-AP
1
A
1
0.1
0.1
0.1
0.3
0.1
0.1
2
A
2
0.1
0.1
0.1
0.1
0.1
0.1
3
A
6
0.9
1.1
1.1
1.0
1.0
1.0
4
A
10
0.1
0.1
0.1
0.1
0.1
0.1
5
A
12
0.1
0.1
0.4
0.1
0.1
0.1
6
A
14
0.1
0.1
0.1
0.1
0.1
0.1
7
A
18
0.1
0.2
0.4
0.2
0.1
0.1
8
A
22
1.0
0.8
0.7
0.9
0.9
0.9
9
x
4
37.3200
33.0487
27.6169
33.4976
36.9520
36.8958
10
y
4
55.7400
53.5663
51.6196
62.3735
54.5786
54.1337
11
z
4
126.6200
129.9092
129.9071
114.5945
129.9758
130.0000
12
x
8
50.1400
43.7826
42.5526
40.0531
51.7317
51.9924
13
y
8
136.4000
136.8381
132.7241
133.6695
139.5316
140.0000
Best weight (lb)
118.83
120.1149
129.2076
123.5403
117.2271
117.3022
Maximum displacement (in.)
0.3500
0.3500
0.3503
0.3505
0.3518
0.3500
Maximum stress (ksi)
18.8302
17.1574
16.4391
15.5913
19.9702
20.0182
Minimum stress (ksi)
−9.4017
−6.4822
−10.9931
−6.4102
−9.4049
−9.4024
Number of structural analyses
6,000
10,000
4,500
4,500
4,500
4,500


## 4.3. Planar 37-Bar Truss

The original geometry of planar 37-bar truss is shown in Figure4(a). Eleven nodes (ID: 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, and 20) are totally fixed and the x-coordinates of all other nine nodes (ID: 3, 5, 7, 9, 11, 13, 15, 17, and 19) are fixed as well. A nonstructural mass of 10 kg is attached to all free nodes on the lower chord.Layout optimization of the planar 37-bar truss.
(a)
Geometry and element definitions of the planar 37-bar truss
(b)
Best solution of the planar 37-bar trussTo ensure the structural symmetric about they-axis, all design variables are classified into 19 groups:sizing variables:A
1
=
A
27; A
2
=
A
26; A
3
=
A
24; A
4
=
A
25; A
5
=
A
23; A
6
=
A
21; A
7
=
A
22; A
8
=
A
20; A
9
=
A
18; A
10
=
A
19; A
11
=
A
17; A
12
=
A
15; A
13
=
A
16; A
14;
shape variables:x
3; x
5; x
7; x
9; y
3
=
y
19; y
5
=
y
17; y
7
=
y
15; y
9
=
y
13; y
11.Material parameters and design constraints are listed in Table7.Table 7
Material parameters, design constraints, and search range of the planar 37-bar truss optimization problem.
Category
Values
Material parameters
Density
7800 kg/m3
Modulus of elasticity
2.1
×
1
0
11 N/m2
Constraints
Natural frequencies
f
1
≥
20 Hz; f
2
≥
40 Hz; f
3
≥
60 Hz;
Search range
Shape variables
x
1
=
0
m
;
x
2,3
=
1
m
;
x
4,5
=
2
m
;
x
6,7
=
3
m
;
x
8,9
=
4
m
;
x
10,11
=
5
m
;
x
12,13
=
6
m
;
x
14,15
=
7
m;
x
16,17
=
8
m
;
x
18,19
=
9
m
;
x
20
=
10
m
;
0
m
≤
y
3,5
,
7,9
,
11
≤
3
m
;
Sizing variables
1
×
1
0
-
4
m
2
≤
A
i
≤
10
×
1
0
-
4
m
2
,
i
=
1,2
,
…
,
14
A
j
=
4
×
1
0
-
3
m
2
,
i
=
28,29
,
…
,
30Figure4(b) shows the optimum design explored by this work. Table 8 compares the best design presented by this work with those reported in the literature. It can be seen that the results achieved by the proposed algorithm are even better than the best results reported in the literature.Table 8
Comparison of optimized designs found for the planar 37-bar truss.
No.
Variable
OC [9]
GA [18]
PSO [19]
RO [20]
FEPSO-AP
1
A
1
3.2508
2.8932
2.6797
3.0124
3.4197
2
A
2
1.2364
1.1201
1.1568
1.0623
0.9766
3
A
3
1.0000
1.0000
2.3476
1.0005
0.8313
4
A
4
2.5386
1.8655
1.7182
2.2647
2.8073
5
A
5
1.3714
1.5962
1.2751
1.6339
1.2997
6
A
6
1.3681
1.2642
1.4819
1.6717
1.6483
7
A
7
2.4290
1.8254
4.6850
2.0591
2.4972
8
A
8
1.6522
2.0009
1.1246
1.6607
1.5379
9
A
9
1.8257
1.9526
2.1214
1.4941
1.7590
10
A
10
2.3022
1.9705
3.8600
2.4737
2.7069
11
A
11
1.3103
1.8294
2.9817
1.5260
1.3046
12
A
12
1.4067
1.2358
1.2021
1.4823
1.4004
13
A
13
2.1896
1.4049
1.2563
2.4148
3.0476
14
A
14
1.0000
1.0000
3.3276
1.0034
0.5947
15
y
3
1.2086
1.1998
0.9637
1.0010
0.8756
16
y
5
1.5788
1.6553
1.3978
1.3909
1.2546
17
y
7
1.6719
1.9652
1.5929
1.5893
1.4446
18
y
9
1.7703
2.0737
1.8812
1.7507
1.5889
19
y
11
1.8502
2.3050
2.0856
1.8336
1.6480
Natural frequencies (Hz)
f
1
20.0850
20.0013
20.0001
20.056
20.020
f
2
42.0743
40.0305
40.0003
40.035
40.022
f
3
62.9383
60.0000
60.0001
60.030
60.233
f
4
74.4539
73.0444
73.0440
74.387
72.137
f
5
90.0576
89.8244
89.8240
85.929
84.065
Best weight (kg)
366.50
368.84
377.20
364.04
362.8812
Number of structural analyses
—
—
20,000
32,000
8,000


## 4.4. Planar 47-Bar Truss

The original geometry of planar 47-bar truss is shown in Figure5(a). Four nodes (ID: 15, 16, 17, and 22) are totally fixed and the y-coordinates of other two nodes (ID: 1 and 2) are fixed as well. The loading conditions are listed in Table 9.Table 9
Loading conditions acting on the planar 47-bar truss.
Case
Node
F
x (kips)
F
y (kips)
1
17 and 22
6.0
−14.0
2
17
6.0
−14.0
3
22
6.0
−14.0Layout optimization of the planar 47-bar truss.
(a)
Geometry and element definitions of the planar 47-bar truss
(b)
Best solution of the planar 47-bar trussTo ensure the structural symmetric about they-axis, all design variables are classified into 44 groups:sizing variables:A
1
=
A
3; A
2
=
A
4; A
5
=
A
6; A
7; A
8
=
A
9; A
10; A
11
=
A
12; A
13
=
A
14; A
15
=
A
16; A
17
=
A
18; A
19
=
A
20; A
21
=
A
22; A
23
=
A
24; A
25
=
A
26; A
27; A
28; A
29
=
A
30; A
31
=
A
32; A
33; A
34
=
A
35; A
36
=
A
37; A
38; A
39
=
A
40; A
41
=
A
42; A
43; A
44
=
A
45; A
46
=
A
47;
shape variables:x
1
=
-
x
2; x
3
=
-
x
4; y
3
=
y
4; x
5
=
-
x
6; y
5
=
y
6; x
7
=
-
x
8; y
7
=
y
8; x
9
=
-
x
10; y
9
=
y
10; x
11
=
-
x
12; y
11
=
y
12; x
13
=
-
x
14; y
13
=
y
14; x
19
=
-
x
20; y
19
=
y
20; x
18
=
-
x
21; y
18
=
y
21.Material parameters and design constraints are listed in Table10.Table 10
Material parameters, design constraints, and search range of the planar 47-bar truss optimization problem.
Category
Values
Material parameters
Density
0.3 lb/in3
Modulus of elasticity
3
×
1
0
4 ksi
Constraints
Stress
The allowable elements stress interval:[−15 ksi, 20 ksi]
Local buckling
|
(
σ
c
)
i
|
≤
β
E
A
i
/
l
i
2
,
i
=
1,2
,
…
,
47
,
β
=
3.96
Search range
Shape variables
0
in.
≤
x
i
≤
120
in.
,
(
i
=
2,4
,
6,8
)
;
-
30
in.
≤
x
j
≤
90
in.
(
j
=
10,12,14,20
)
;
30
in.
≤
x
21
≤
150
in.;
60
in.
≤
y
4
≤
180
in.
;
180
in.
≤
y
6
≤
300
in.
;
300
in.
≤
y
8
≤
420
in.
;
360
in.
≤
y
10
≤
480
in.
;
420
in.
≤
y
12
≤
540
in.
;
480
in.
≤
y
14
≤
600
in.
;
540
in.
≤
y
20
≤
660
in.
;
540
in.
≤
y
21
≤
660
in.
Sizing variables
S
=
{
0.1,0.2,0.3
,
…
,
4.8,4.9,5.0
}
in
.
2
A
i
∈
S
,
i
=
1,2
,
…
,
47Figure5(b) shows the optimum design identified by this work. Table 11 compares the best design of this work with those reported in the literature. It can be seen that by using the proposed algorithm it is possible to achieve better results at a lower computational cost.Table 11
Comparison of optimized designs found for the planar 47-bar truss.
No.
Variable
GA [21]
FSD-ES [22]
PSO [17]
CPSO [17]
SCPSO [17]
FEPSO-AP
1
A
3
2.50
2.70
2.80
2.60
2.50
3.20
2
A
4
2.20
2.50
2.70
2.50
2.50
2.80
3
A
5
0.70
0.70
0.80
0.70
0.80
0.70
4
A
7
0.10
0.10
1.10
0.30
0.10
0.10
5
A
8
1.30
0.90
0.80
1.20
0.70
0.60
6
A
10
1.30
1.10
1.30
1.10
1.40
1.60
7
A
12
1.80
1.80
1.80
1.60
1.70
1.90
8
A
14
0.50
0.70
0.90
0.80
0.80
0.90
9
A
15
0.80
0.90
1.20
1.10
0.90
1.00
10
A
18
1.20
1.30
1.40
1.30
1.30
2.00
11
A
20
0.40
0.30
0.30
0.30
0.30
0.10
12
A
22
1.20
1.10
1.40
0.80
0.90
0.40
13
A
24
0.90
1.00
1.10
1.00
1.00
1.40
14
A
26
1.00
0.90
1.20
1.00
1.10
1.50
15
A
27
3.60
0.80
1.60
0.90
5.00
1.20
16
A
28
0.10
0.10
1.00
0.10
0.10
0.70
17
A
30
2.40
2.70
2.80
2.70
2.50
5.00
18
A
31
1.10
0.80
0.80
0.90
1.00
1.00
19
A
33
0.10
0.10
0.10
0.10
0.10
0.10
20
A
35
2.70
3.00
3.00
3.00
2.80
3.00
21
A
36
0.80
0.90
0.90
1.00
0.90
0.50
22
A
38
0.10
0.00
0.10
0.20
0.10
0.10
23
A
40
2.80
3.20
3.30
3.30
3.00
3.30
24
A
41
1.30
1.00
0.90
0.90
1.00
0.30
25
A
43
0.20
0.10
0.10
0.10
0.10
0.10
26
A
45
3.00
3.30
3.30
3.30
3.20
3.50
27
A
46
1.20
1.10
1.20
1.10
1.20
0.40
28
x
2
114.0000
100.9724
98.8628
99.3630
101.3393
87.7275
29
x
4
97.0000
80.4772
78.6595
83.4439
85.9111
72.4352
30
y
4
125.0000
136.8699
146.7331
126.3845
135.9645
162.6451
31
x
6
76.0000
64.3908
66.5231
69.5148
74.7969
67.2113
32
y
6
261.0000
247.0491
239.0901
218.2013
237.7447
218.2041
33
x
8
69.0000
55.2589
55.6936
58.0004
64.3115
50.6507
34
y
8
316.0000
338.4534
327.7882
322.2272
321.3416
375.4549
35
x
10
56.0000
48.7333
48.8641
51.4015
53.3345
36.6525
36
y
10
414.0000
409.7380
398.6775
401.5626
414.3025
408.7230
37
x
12
50.0000
43.4742
43.1400
46.8605
46.0277
36.9960
38
y
12
463.0000
472.1479
464.7831
458.3021
489.9216
483.4295
39
x
14
54.0000
44.8349
37.8993
46.8885
41.8353
37.9558
40
y
14
524.0000
512.1901
511.0450
527.8575
522.4161
535.7644
41
x
20
1.0000
3.8414
18.2341
16.2354
1.0005
4.6875
42
y
20
587.0000
591.1449
594.0710
610.8496
598.3905
599.7416
43
x
21
99.0000
84.5040
90.9369
98.3239
97.8696
101.4535
44
y
21
631.0000
630.3472
621.3943
624.9580
624.0552
605.4302
Best weight (lb)
1925.7897
1842.6609
1975.8393
1908.8301
1864.0985
1799.7037
Maximum stress (ksi)
19.9528
20.0000
19.0636
19.3351
19.4735
19.9808
Minimum stress (ksi)
−14.9973
−15.0000
−14.9999
−14.9986
−15.0000
−14.9986
Number of buckling elements
0
0
0
0
0
0
Number of structural analyses
100,000
55,802
25,000
25,000
25,000
20,000


## 5. Conclusion

In this work, a new hybrid PSO algorithm is proposed to solve a quite challenging task in truss optimization area: truss layout optimization with multiple constraints.Two computational techniques are adopted to further enhance the performance of PSO algorithm. In the first fitness estimation strategy, the evaluation of particles is partly substituted by the estimation of similar particles, with the purpose to reduce the computational cost of real world optimization problem. In the second adaptive penalty function approach, the iteration information is merged into the penalty function to find a good balance between the exploration and exploitation of the constrained design domain. The resulted algorithm is termed as FEPSO-AP.Four benchmark truss layout optimization problems, subject to nodal displacement constraints, element stress constraints, natural frequency constraints, and local buckling constraints, are used to verify the performance of FEPSO-AP. Numerical results demonstrate that three out of four benchmarks, to which the FEPSO-AP based optimization is applied, delivered the best feasible designs to the author’s knowledge. Moreover, the convergence rate of the FEPSO-AP algorithm is quite competitive comparing to other state-of-the-art hybrid algorithms published in the former literatures.



---

*Source: 671872-2014-09-29.xml*