COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Optimization of Observation Condition on Inverse Analysis
for Identifying Corrosion of Steel in Concrete
Kazuhiro Suga 1*, M. Ridha2, Shigeru Aoki 3
1
Center for Computational Mechanics Research , Toyo University, Toyo, Japan
Department of Mechanical Engineering, Syiah Kuala University, Banda Aceh, Indonesia
3
Department of Computational Science and Engineering, Toyo University, Japan
2
Email: [email protected]
Abstract The purpose of this study is to optimize the observation condition on the inverse problem for
estimating the real and imaginary parts of the concrete conductivity and the impedance of the steel-concrete
interface. The optimization is achieved by minimizing the average of eigen values of a posteriori estimate
error covariance matrix. We performed a numerical identification to demonstrate the effectiveness of the
optimized observation condition. The estimation is carried out by using the Kalman Filter algorithm. The
simulation result shows that the real part of the impedance can be estimated with a high accuracy while the
others cannot be well estimated. To overcome the above difficulty, a priori information and other kinds of
observation conditions are considered.
Key words: Inverse Problem, Corrosion Identification, Measurement Optimization, Reinforce Concrete,
Kalman Filter Algorithm
INTRODUCTION
Corrosion of steels in concrete is a major cause of premature deterioration and failure of the reinforced
concrete structure. It is a worldwide problem and has attracted more attentions of engineers and practitioners
due to its serious implications on the infrastructure [1]. The technique for diagnosis and monitoring the
corrosion of the concrete steels in concrete is widely acknowledged to maintain a long life of the structures
and to reduce the const of maintenance.
Detection of reinforced concrete corrosion can be performed by identifying the parameters such as
conductivity of the concrete and the impedance of the steel-concrete interface [2,3].
Due to impossibility of direct observation of the steel in concrete and the low conductivity of the concrete,
detection of the steel corrosion in the concrete is difficult.
It has been reported in the previous study that these parameters were obtained by the boundary element
inverse analysis. The inverse problem is carried out using some potential data which are measured on the
concrete surface while the alternating electrical current was injected to the concrete [4].
It is well known that many inverse problems are ill-posed [5-7]. In some cases, the ill-posedness could be
reduced by using many observation data. However, it takes more cost and consumes more time. Therefore,
optimization of the observation condition is important to obtain the maximum likelihood solution with less
effort [8,9].
In this paper, we introduce the method to optimize the optimization condition of the inverse problem for
detection of reinforced concrete corrosion. In addition, to demonstrate the effectiveness of the proposed
method, the analysis was performed using numerical simulation data of the potential on the concrete surface.
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OPTIMIZATION METHOD
Generally, an inverse problem can be represented with the discrete form Eq. (1),
yM = hM (x) + wM ,
(1)
where yM is m-dimensional observation(measurement) vector, x is n-dimensional state vector, hM is
nonlinear m-dimensional vector function which represents a system, the index
M
means that these values
are depended on a observation condition. wM represents m-dimensional observation error vector. wM is
stochastic vector and follows the probably density function given by Eq. (2)
p(w) = N (wM ,WM ) ,
(2)
where N represents Gaussian distribution, wM is the average vector, WM is a covariance matrix of wM .
Pre-estimator means estimator before observation, while post-estimator means estimator after observation.
The covariance matrix of the post-estimator’s error, which is represented by PM , can be calculated by Eq. (3).
PM ≡ (QM−1 + H Mt WM−1H M ) −1
(3)
QM is the covariance matrix of the pre-estimator’s error and this matrix is known, H M is the Jacobian
matrix of hM .
Figure 1 shows the relationship between likelihood of estimation and the eigen values of PM in case that the
estimator is 2-dimensions, where p( x | yM ) is a conditional probability density function of x when the yM
was observed. The eigen values coincide with long radius or short radius of the ellipsoid which given by
p( x | yM ) = const. .
Figure 1: Eigen values of probability density function
p( x | yM ) of well estimated x has sharp peak, i.e. the eigen values are small. Therefore the observation
optimization problem can be reduced to the combinatorial optimization problem defined by Eq. (4).
min(λavg (PM )) (M ⊂ U )
(4)
arg=M
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Where λavg (PM ) represents the average value of the eigen values of PM , U represents whole set of the
candidates of measurement condition.
IDENTIFICATION OF CONDUCTIVITY OF CONCRETE AND IMPEDANCE OF STEELCONCRETE INTERFACE
1. Inverse Problem Setup Fig. 2(a) shows the numerical model of a reinforced concrete. An alternating
current is injected at the surface of the concrete. The injected current intensity i0 is 0.1 mA, the angular
frequency ω is 2π rad. The observations of this inverse problem are the real part of the electrical potential
φRe on the upper surface of the concrete specimen shown in Fig. 2(a).
This inverse problem was used to identify the conductivity of the concrete κ (≡ κ Re + jκ Im ) and the
impedance of the steel-concrete interface Z (≡ Z Re + jZIm ) , where j , Re, Im are the imaginary unit, the
value of real component and the value of imaginary component, respectively.
Analysis conditions are shown in the Eqs.(5-8) [1].
∇2φ = 0
∂φ
i ≡κ = 0
∂n
i = i0e jωt
− φ = Zi
(Concrete)
(5)
(Concrete surface)
(6)
(Impressed current point)
(7)
(Rebar surface)
(8)
Where, φ (≡ φRe + φIm ) represents electrical potential, i represents the current density,
outward normal derivative.
Figure 2: Numerical model and ranking
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∂
represents the
∂n
2.Optimized Observation Condition This optimization was performed to select the efficient observation
points for identifying the conductivity of the concrete κ and the impedance of the steel-concrete
interface Z .
We set up the observation error of electrical potential to be 1mV and the number of feasible obseravtion
points on the upper surface was 25. The range of each parameter was set as the following equations (9).
0.001≤ κ Re ≤ 0.1
⎫
− 0.01 ≤ κ Im ≤ −0.0001 ⎪⎪
⎬
0.33 ≤ Z Re ≤ 103
⎪
− 21.03 ≤ Z Im ≤ −0.020 ⎪⎭
(9)
The above parameters have differently range values. For making possible comparison of the covariance of
the estimated parameters, we normalized all of the parameters in range between 0 and 1.
We assumed that useful a priori information can not be obtained in this problem. Therefore, the
pre-estimator and the covariance matrix of the estimator were given as in the following equations (10) and
(11).
( κ Re κ Im Z Re Z Im ) = ( 0.5 0.5 0.5 0.5 )
(10)
⎛1.0
⎜
⎜ 0.0
QM = ⎜
0.0
⎜
⎜ 0.0
⎝
(11)
0.0
1.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0⎞
⎟
0.0⎟
0.0⎟
⎟
1.0 ⎟⎠
We performed the observation condition optimization under the above settings. Fig. 1(b) shows the result of
the optimization. The number in the figure denotes the ranking of efficient observation points.
3.Numerical Simulation We performed the numerical simulation to solve the inverse problem to identify κ
and Z by employing the Kalman filter algorithm [10].
A preliminary numerical simulation was carried out using a model of reinforced concrete specimen as given
in Fig. 2(a). To obtain the electrical potentials at four locations on the concrete surface, i.e. point 1 - 4 in
Fig. 2(b), the boundary element method [11] was used to solve the Laplace equation in Eq.(1). These
potential values were rounded off into three digits in order to simulated modeling and observation errors and
used as the observed potential data.
Only one initial value for estimation can not be selected before estimation since we assumed that useful a
priori information can not be obtained. Therefore, our estimation starts with some initial values. The values
consist of the combination of each parameter which was varied from 0.2 to 0.8 with interval of 0.1. The total
number of initial values was 2041( 7×7×7×7).
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2.Result and Discussion The result of estimation was shown in Fig. 3. In this histogram, the horizontal axis
shows converged values, the vertical axis shows the frequency of converged values and the marked bars
show the areas included the correct answer of each parameter.
Fig.2 Result of estimation
It is can be seen in the Fig. 3 that Z Re was estimated with high accuracy, while the other parameters, ZIm ,
κ Re , κIm , cannot be well estimated. It is because the low-sensitivity of the electrical potential on the
concrete surface for each parameter except of ZRe .
For this example, the optimized observation condition was carried out at only single frequency, i.e. 1Hz.
Since the electrical potential is a frequency dependent, the observations at more than one frequency might
improve the estimation accuracy of each parameter.
CONCLUSIONS
The observation condition was optimized on the inverse problem for identifying of reinforced concrete
corrosion.
We performed the numerical simulation to solve the inverse problem under the optimized observation
condition to obtain the conductivity of the concrete and the impedance of steel-concrete interface. The real
part of the impedance can be estimated with high accuracy but the other parameters cannot be
well-identified.
Further study is necessary for better identify of the all parameters, for instance, by taking into account a
priori information and/or introducing the other types of observation.
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REFERENCES
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Cost of Corrosion. http://www.corrosioncost.com/home.html, 2006.
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Broomfield JP et al. Corrosion of Steel in Concrete; Understanding Investigation and Repair, E&FN
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L. Lemoine et al. Study of the Corrosion of Concrete Reinforcement by Electrochemical Impedance
Measurement”, ASTM STP 1065, Ed. byN. S. Berke, V. Chaker and D. Whiting, American Society for
Testing and Materials, 1990, p.118-133
4.
M Ridha et al., Improvement of AC Impedance Method for Monitoring Corrosion of Rebar Structure by
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S Kubo, Gyakumondai, Baifukan , 1998 (in Japanese)
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series, 67, 2001, p.16-21 (in Japanese)
SPON, London, UK, 1997.
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http://www.cs.unc.edu/~welch/media/pdf/Kalman_intro.pdf, 2006
nd
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Mechanics Publications, 1992
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edition, Computational