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Advanced Materials Research Vol. 686 (2013) pp 266-272
© (2013) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/AMR.686.266
The Influence of Inertia Weight on Particle Swarm Optimization in
Boundary Element Inverse Analysis for Rebar Corrosion Detection
S. Fonna1,a, M. Ridha2,b, S. Huzni3,c and A. K. Ariffin4,d
1,4
Department of Mechanical and Materials Engineering
Universiti Kebangsaan Malaysia, Bangi 43600, Selangor DE, Malaysia
1,2,3
Department of Mechanical Engineering, Syiah Kuala University
Jl. Tgk. Syech Abdul Rauf 7 Banda Aceh, Indonesia
a
[email protected], b [email protected], c [email protected]
d
[email protected]
Keywords: PSO; BEM; inverse analysis; inertia weight; rebar corrosion
Abstract. Particle Swarm Optimization (PSO) has been applied as optimization tool in various
engineering problems. Inverse analysis is one of the potential application fields for PSO. In this
research, the behavior of PSO, related to its inertia weight, in boundary element inverse analysis for
detecting corrosion of rebar in concrete is studied. Boundary element inverse analysis was developed
by combining BEM and PSO. The inverse analysis is carried out by means of minimizing a cost
function. The cost function is a residual between the calculated and measured potentials on the
concrete surface. The calculated potentials are obtained by solving the Laplace’s equation using
BEM. PSO is used to minimize the cost function. Thus, the corrosion profile of concrete steel, such as
location and size, can be detected. Variation in its inertia weight was applied to analyze the behavior
of PSO for inverse analysis. The numerical simulation results show that PSO can be used for the
inverse analysis for detecting rebar corrosion by combining with BEM. Also, it shows different
behavior in minimizing cost function depending on inertia weight.
Introduction
Recently, there are many heuristic search algorithm based on nature that can be used to solve
engineering problems. One of them is the Particle Swarm Optimization (PSO). The use of PSO has
increased in popularity and such phenomena showed in Fig. 1 [1]. The important features that
proposed by PSO are include simple algorithm and easy to program, yet proven efficient [1].
Fig. 1: The Number of journal paper related PSO between 2000 and 2008 [1].
Those advantages have attracted many researchers to apply PSO to various engineering
problems. One of them is inverse analysis as in reference [2] and [3]. In this research, the utilizing of
PSO in rebar corrosion inverse analysis was studied. Important parameters especially inertia weight
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Advanced Materials Research Vol. 686
267
was studied in order to study PSO behavior in boundary element inverse analysis for detecting
corrosion of rebar in concrete.
Simulation Method
Modeling of Rebar Corrosion in Concrete. Concrete is referred to as homogeneous domain. There
is no accumulation or loss of ions in the bulk of the domain. Thus, the potential field in the concrete
domain ( ) can be modeled mathematically by the Laplace's equation [4-6]:
in
(1)
Eq. (1) shows the basic equation and boundary conditions used in the study. The density of
current across the boundaries, which will be denoted by , is given by:
(2)
Concrete
domain
( )
Concrete
domain
corroded
non-corroded
Fig. 2: Boundary condition.
The boundary conditions associated with Eq. 1, as given in Fig.2, are written as:
on
on
on
(3)
(4)
(5)
and
are the polarization curves
where is potential at any point, is current density, and
for corroded and non-corroded areas on the steel in concrete, respectively.
If the boundary conditions in Eq. 3 to 5 are known, Boundary Element Method (BEM) can be used
to solve the Laplace's equation in Eq. (1). Hence, the potential ( ) and current density ( ) on the whole
domain can be determined.
Inverse analysis for corrosion detection. Inverse analysis is used to construct the profile of
corroded area of rebar by comparing measured and calculated potential data (by BEM) on the top of
concrete [3]. In carrying inverse analysis, the cost function, Eq. (6) must be minimized.
(6)
where is the estimated corrosion profile (such as location, size and shape) and defined as particle in
PSO, is the number of potential data that are used in the inverse analysis. The and are the
calculated and measured potential, respectively.
is the highest potential value among
potential data.
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The Particle Swarm Optimization (PSO). To minimize the cost function in inverse equation, the
PSO was used. PSO is population-based search algorithm. It was inspired by flocking behavior of
birds [1]. Among the advantages of PSO are easy to implement and requires only few parameters.
Fig. 3 shows the flowchart of PSO in inverse analysis.
Initialization:
, , ,
and
Set constant
Randomly initialize particles position (corrosion
profile),
Randomly initialize particles velocity,
Set
-
Calculate the potential on concrete surface for each
corrosion profile using BEM
- Update each particle
velocity
- Update each particle
position
)
- Increment (
Evaluate cost function ( ) for each
corrosion profile:
If
, then
-
If
, then
no
,
,
measured
potential
data
or
yes
Stop
Fig. 3: The flowchart of inverse analysis using PSO.
The inverse analysis using PSO started with a population of particles that represent candidate
solutions. These particles will be subjected to cost function, and classified according to its fitness.
The particles then will move, using the local and global best particle as reference. Local best is the
best particle, in terms of its fitness, relative to one individual particle. Global best is the best particle,
in terms of its fitness, relative to whole population. The movement of the particles is described as
velocity by following equation:
(7)
The new profile of particles will be updated according the equation:
(8)
where
= next particle position,
= current particle position,
= next velocity,
velocity, = inertia weight, and = constants, and = random number (0-1),
best particle,
= global best particle and = iteration.
= current
= local
From the velocity formula Eq. (7), it can be seen that the inertia weight term ( ) will greatly
play role when the other two terms approaches zero. It will occur as the particles converge. Thus
inertia weight ( ) is an important factor that must be studied.
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Case study. A simple model of rebar corrosion in concrete domain was used as shown in Fig. 4. The
size of corroded area is assumed to be known.
Fig. 4: The simple model of RC corrosion for numerical simulation.
All the concrete surfaces assumed to have zero current density. The potential on the
non-corroded area of rebar is
mV, while the corroded area has
mV
potential on its surface [6-7].
By using BEM, the potential on the top of the concrete can be obtained. Then, this potential data
is randomly selected in specified number, and used as model of measured potential data in the
following inverse analysis.
For inverse analysis, the exact position of corroded area is not known. To locate the corroded
area, the inverse equation, Eq. (6), has to be minimized subjected to inverse analysis procedures as
shown in Fig. 3. Then, in order to study the behavior of PSO in minimizing cost function, several
values of inertia weight were selected that are 0.1, 0.5 and 1.
Numerical Simulation Results and Discussion
Fig. 5, 6 and 7 show results of using W value of 0.1, 0.5 and 1, respectively. Fig. 5 shows the particles
was spread randomly around the searching space for 1st iteration. By increasing iteration, the particle
look moved steadily toward the solution (the corroded area) and finally made cluster there. For this
condition, the solution was found.
22
Particle 1
Particle 2
Particle 3
Particle 4
Particle 5
20
18
16
Iteration
14
12
10
8
6
4
2
0
0
10
20
30
40
50
60
70
Position (cm)
Fig. 5: Particles behavior during searching corrosion location with
In Fig. 6, the particle also spread randomly for 1st iteration. It is clear that the particles movement
were unsteady toward the solution especially during 2nd to 16th iterations. However, the particles still
converged around the solution and thus, the solution was found.
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22
Particle 1
Particle 2
Particle 3
Particle 4
Particle 5
20
18
16
Iteration
14
12
10
8
6
4
2
0
0
10
20
30
40
50
60
70
Position (cm)
Fig. 6: Particles behavior during searching corrosion location with
Meanwhile, Fig.7 shows the behavior of particles in searching the solution for inertia weight
equal to 1. Same as before, the particle was initiated randomly for 1st iteration. The following
iterations show that the movement of particles became more unsteady during searching the corroded
area. Until 20th iteration, the corrosion location still did not be found. The iteration has been increased
till 100 to verify this behavior. It shows that the particles still not converged. Thus, the corrosion
location did not found for simulation with inertia weight equal to 1.
22
Particle 1
Particle 2
Particle 3
Particle 4
Particle 5
20
18
16
14
Iteration
12
10
8
6
4
2
0
0
10
20
30
40
50
60
70
Position (cm)
Fig. 7: Particles behavior during searching corrosion location with
Fig. 8, 9 and 10 show the cost function value of each particle in increasing iteration for selected
the inertia weight. Inline with the result in Fig 5, the cost function that given in Fig. 8 shows the
steady decreasing of cost function toward zero during increasing iteration. It is clear that for inertia
weight equal to 0.1, the simulation can find the expected solution effectively.
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0,09
0,08
0,07
Cost Function
Particle 1
0,06
Particle 2
Particle 3
0,05
Particle 4
0,04
Particle 5
0,03
0,02
0,01
0
0
2
4
6
8
10
12
14
16
18
20
Iteration
Fig. 8: Cost function for each iteration with
Fig. 9 shows the cost function behavior of each particle by increasing iteration for inertia weight
equal to 0.5. The cost functions also decreased approaching to zero as same as previous result. Thus,
the solution also can be found for such simulation with the selected inertia weight. Moreover, this
result is also agreeable with the result in Fig. 6.
0,09
0,08
0,07
Cost Function
Particle 1
0,06
Particle 2
Particle 3
0,05
Particle 4
0,04
Particle 5
0,03
0,02
0,01
0
0
2
4
6
8
10
12
14
16
18
20
Iteration
Fig. 9: Cost function for each iteration with
Meanwhile, the cost function for each iteration in PSO search with
is shown in Fig. 10. It
can be seen that for the first until eighth iteration, the cost function values decreased gradually toward
zero. However, for the following iteration, it did not move anywhere. The cost functions were
un-converged approaching zero value. Hence, for this simulation, the location of corrosion was not
found/detected.
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0,09
0,08
0,07
Cost Function
Particle 1
0,06
Particle 2
Particle 3
0,05
Particle 4
0,04
Particle 5
0,03
0,02
0,01
0
0
2
4
6
8
10
12
14
16
18
20
Iteration
Fig. 10: Cost function for each iteration with
Conclusions
The PSO has been applied for boundary element inverse analysis to predict the location of corrosion
in rebar. The PSO successfully minimize the inverse equation in order to determine the corroded area
of rebar by using some given potential. Different value of inertia weight will affect the movement and
path of the particles during optimization process. It seems that the appropriate value of inertia weight
for boundary element inverse analysis using PSO is less than 1 (one). For inertia weight less than one,
the behavior of PSO is tend to converge and so the expected solution can be achieved.
References
[1] Konstantinos E. Parsopoulos and Michael N. Vrahatis, 2010, Particle Swarm Optimization and
Intelligence: Advances and Applications, Information Science Reference, New York.
[2] Kyun Ho Lee, Seung Wook Baek, Ki Wan Kim, 2008, Inverse Radiation Analysis Using
Repulsive Particle Swarm Optimization Algorithm, International Journal of Heat and Mass
Transfer, 51: 2772-2783.
[3] Ridha, M., K. Amaya, and S. Aoki, 2001, Multistep Genetic Algorithm for Detecting Corrosion
of Reinforcing Steels in Concrete, Corrosion, 54(9):794-801.
[4] Aoki, S., Amaya, K. and Miyasaka, M., 1998, Boundary Element Analysis on Corrosion
Problems, Shokabo, Tokyo.
[5] Telles, J.C.F., Mansur, W.J., Wrobel, L.C. and Azevedo, J.P.S., 1985, Boundary Elements for
Cathodic Protection Problems, Proc. Int. Conf. Boundary Element VII. Editor: Brebbia, C.A.
and Maier, G., Springer-Verlag, pp. 73-83.
[6] Ridha, M., Amaya, K. & Aoki, S., 2005, Boundary Element Simulation for Identification of Steel
Corrosion in Concrete by Magnetic Field Measurement, Corrosion. 61 (8):784-791.
[7] Minagawa, K., Suga, K., Kikuchi, M. & Aoki, S., 2012, An Efficient Inverse Analysis
Considering Observation Error to Detect Corrosion in Concrete Structures Containing
Multilayered Rebar, International Journal of Mechanics and Materials in Design, 8(1): 81-87.