COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Consideration of Sensor Placement Design for Damage Detection
Y. Q. Li 1, Z. H. Xiang 1*, M. S. Zhou 1, G. Swoboda 2, Z. Z. Cen 1
1
2
Department of Engineering Mechanics, Tsinghua University, Beijing, 100084 China
Department of Structural Engineering, University of Innsbruck, Innsbruck, A- 6020 Austria
Email: {liyq02, zms}@mails.tsinghua.edu.cn, {xiangzhihai, demczz}@tsinghua.edu.cn, [email protected]
Abstract A large portion of damage detection methods are kinds of parameter identification procedures. It is well
known that the result of parameter identification is sensitive to the sensor placement when measuring errors are
inevitable. Therefore, how to optimize the sensor placement has been an intriguing and challenging problem over
years. Up to now, most methods assume that the optimal sensor placement leads to an optimal value of certain property
of Fisher information matrix. However, these methods ignore the fact that this certain property need to be calculated
from true parameters, which are unknown in the design phase. In this paper, a new algorithm is proposed to solve this
problem. In this algorithm, the sensor placement design and the parameter identification procedure are conducted
alternatively until the discrepancy between identified parameters and guessed ‘true’ parameters vanishes. The validity
of this method is illustrated by an academic and a real engineering example.
Key words: sensor placement, damage detection, structural health monitoring
INTRODUCTION
Damage detection plays an important role to Structural Health Monitoring Systems (SHMS). Over the past years, it has
attracted extensive attention from engineering and academe [1]. However, although many proposed damage detection
methods have very good performances in laboratories, they fail in operation for practical engineering structures. This
is mainly because in the laboratory, simple specimens were easy to be modeled accurately and measuring errors were
well controlled. However, modeling errors are inevitable for real structures and on-site measurements are easily
contaminated by environmental changes. Therefore, a robust damage detection method is needed in engineering
practice.
The basic idea of damage detection is deducing the location and the severity of damages from on-site measurements.
To design a successful damage detection method, there are some questions should be answered: 1) what is the damage
of interest? 2) what is the proper model to describe the damage? 3) which kind of data should be measured? 4) how to
detect the damage based on the measurements? 5) where to measure? 6) how many measurements are needed?
The answers could be: 1) The damage of interest is surely the one, which is critical to the safety of structures. Usually,
stiffness is adopted as the damage parameter. 2) The adopted model should correctly describe the response of the
structure in working condition. Practically, this is not an easy job and modeling errors are inevitable. 3) When the
damage and the model are specified, one should measure the data sensitive to the damage. For example, displacements
are usually measured for damage detection because they are very sensitive to stiffness. 4) By taking damages as
structural parameters, most the damage detection methods are actually parameter identification procedures. Question 5)
and 6) are closely related with the sensor placement design, which is an important factor that affects the results of
damage detection, especially when measurements are contaminated with noises.
To design an optimal sensor placement, a criterion of the optimality is needed. Although many criteria have been
proposed, a widely recognized criterion does not existed so far. Most popular criteria are more or less related to certain
properties of Fisher Information Matrix (FIM) [2]. For example, the effective independence method [3] intends to
maximizing the items in the FIM so that the items in the covariance matrix between the measured and the calculated
data are minimized; the criterion of maximizing the system kinetic energy [4] could be regarded as a weighted form of
the FIM; the method of minimizing the information entropy [5] is almost equivalent to the “D optimality” method that
⎯ 491 ⎯
maximizes the determinant of the FIM [6, 7]; the criterion involving the eigenvalue of the covariance matrix [8] has
some relations with the condition number of the FIM; the FIM also gives a major contribution to the criterion proposed
in Ref. [9], which is based on the well-posedness analysis of parameter identification procedures.
As state above, no matter what criterion is adopted, the FIM should be calculated to design an optimal sensor
placement. However, this calculation should be based on true parameters, which are to be identified by a damage
detection method. To avoid this contradiction, many methods just using pre-assumed parameters to calculate the FIM.
This is certainly a rough approximation. In Ref. [9], a guess of true parameters was constructed according to the
engineering experience. This approach, however, is still too empirical to facilitate its implementation. In this paper, a
new iteration method is proposed to solve this problem. In this method, the sensor placement design and the parameter
identification procedure are conducted alternatively until the discrepancy between identified parameters and guessed
‘true’ parameters vanishes.
In the following text, a damage detection method based on static deformations is adopted to illustrate the general
theory. Firstly, the identification algorithm is briefly described. Then the criterion in Ref. [9] is implemented to this
algorithm and the proposed iteration method is presented in detail. Finally, the proposed method is verified by an
academic example and an instance of a real highway bridge.
THE DAMAGE DECTECTION METHOD
Generally, a damage could be regarded as the reduction of structural stiffness. Since structural deformations are very
sensitive to the change of stiffness, the damage detection method could be regarded as a parameter identification
problem based on structural deformations. Conventionally, it can be formulated as a nonlinear optimization problem
with least-squares objective:
Minimize :
f ( p) = RT R
Subjected to : K ( p ) u = F
p ∈ Dp
(1)
where K is the stiffness matrix; u is the vector of general displacements; F is the load vector; p is the parameter to be
identified; D p is the admissible set of p; and R is the residual vector, which is defined as:
R( p ) = u s* − Su
(2)
where us* are measured displacements and S is a Boolean matrix which specifies the number and location of
measurements.
Eq. (1) can be efficiently solved by using Gauss-Newton iteration method. According to Ref. [9], the iteration process
is:
(
)
(
)
p n = G p n −1 ≡ p n −1 − ⎡ J n −1 J n −1 ⎤
⎢⎣
⎥⎦
T
−1
(J ) R
n −1 T
n −1
n ≥1
(3)
where n is the iteration step; G is the mapping function; J is the Jacobean matrix; and J T J is the FIM. From Eq. (2),
it obtains:
J ( p) =
∂R
∂u
= −S
∂p
∂p
(4)
By deriving the governing equation K ( p ) u = F with respect to p, J can be calculated as:
J ( p ) = SK −1
∂K
∂K −1
u = SK −1
K F
∂p
∂p
(5)
Usually, Finite Element Method (FEM) can be conveniently used in the direct analysis in the parameter identification
procedure. The derivative of element stiffness matrix over p can be easily calculated and assembled together to
calculate J.
⎯ 492 ⎯
THE CRITERION FOR THE OPTIMAL SENSOR PLACEMENT
An ideal parameter identification method should be well-posed, i.e., valid solutions should exist by using this method
and the results are unique and stable. As an inverse problem, parameter identification methods are ill-posed in nature.
The well-posedness requirement can hardly be met in practice due to modeling and measuring errors. However, if the
sensor placement is carefully designed, the ill-posedness can be controlled to some extent. Accordingly, the criterion
for the optimal sensor placement should be based on the well-posedness analysis of parameter identification method.
As proved in Ref. [9], for a bounded convex set D p , the existence of the solution can be guaranteed by a continuous
mapping function G (see Eq. (3)), according to the Brouwer’s fixed-point theorem (see Ref. [10]) .
Supposing x and y are arbitrary two solutions, x, y ∈ D p ,
∂G (ζ )
(x − y)
∂p
G (x ) − G ( y) =
where ζ = y + α ( x − y ) ,
∂G ( p )
Ω ( p) ≡
= (J J )
T
∂p
−1
(6)
0 < α < 1 . According to Eq. (3), defining:
∂ (J TJ )
∂p
(J J )
T
−1
J T R − (J TJ )
−1
∂J T
R
∂p
Substituting Eq. (7) into Eq. (6) and applying the norm of infinity ⋅
G (x ) − G ( y)
∞
≤ Ω (ζ )
∞
x− y
∞
≤ LΩ x − y
∞
(7)
∞
, obtains:
(8)
where
LΩ ≡ Max Ω ( p )
(9)
∞
According to the contraction mapping theorem (See Ref. [10]), if LΩ < 1 the Gauss-Newton iteration process
convergences to a unique solution.
The stability analysis is closely related to the propagation of measuring errors during the parameter identification
procedure. Supposing there is a positive relative measuring error ε and the initial parameter p 0 is δ p , which is
positive, away from the true parameter p* :
us* = S (I + Iε ε )u*
(10)
p 0 = p* + I pδp
(11)
where I is the identity matrix; and I ε and I p are perturbation matrices with +1 or -1 on the diagonal positions. The
difference of p n to p 0 at iteration step n is:
h n = p n − p*
(12)
From Eqs. (2), (3), (7) and (10) to (12), it yields:
hn − hn −1 = p n − p n −1
( )
= [Ω ( pξ , ε ξ ) − I ]h − ⎡(J J )
⎢⎣
= Ω ( pξ , ε ξ )h + A( pξ )ε
= G p n −1 − p n −1
n −1
hn
ξ n−1
where p
(
n −1
n −1
n −1
n −1
(
n −1
T
−1
J T SI ε u*ε ⎤
⎥⎦ p = p ξn−1
(13)
n −1
)
= p* + β p n −1 − p* , 0 < β < 1 , ε ξ n−1 ∈ (0, ε ) and
A( p ) ≡ − J T J
)
−1
J T SIε u*
(14)
⎯ 493 ⎯
Define:
LA ≡ Max A( p )
(15)
∞
From Eqs. (9), (13) and (15), the following relation is valid:
hn
∞
≤ LΩ hn −1
∞
+ LAε ≤ L2Ω hn − 2
∞
+ (1 + LΩ )LAε L ≤ LnΩ δp
(
)
+ 1 + LΩ + L LnΩ−1 LAε
∞
(16)
If LΩ < 1 ,
lim hn
∞
n→∞
≤
1
LAε ≈ LAε (1 + LΩ )
1 − LΩ
(17)
The iteration process converges to a solution with bias LAε (1 + LΩ ) . This is coincident with the conclusion from the
analysis of uniqueness.
The above well-posedness analysis reveals that if LΩ < 1 , the parameter identification process will converge to a
unique solution with bias LAε (1 + LΩ ) . Since LΩ and LA contain S, the sensor placement should be optimized so
that LΩ < 1 and LAε (1 + LΩ ) reaches a minimum value. This is the criterion of optimal sensor placement proposed in
Ref. [9], in which further discussions on this criterion are available.
By applying the Taylor expansion around the true parameter p * , LΩ and LA can be evaluated as follows:
(
)
(
)(
(
) [( )
](
(
)
Ω ( p , ε ) ≈ Ω p * ,0 +
(
)
)
∂Ω p* ,0
∂Ω p* ,0
ε
p − p* +
∂ε
∂p
(18)
Denoting
*
−1
T
T ∂J
∂Ω p* ,0
= J* J* J*
∂p
∂p
( )
C p* ≡
[(
)
( )
]
*
−1 ⎧
T
∂Ω p* ,0
⎪∂ J
= − J* J* ⎨
D p ≡
∂ε
⎪⎩ ∂p
( )
*
)
(19)
T
[(
)
T
][(
](
−1
T
∂ J* J*
−
J* J* J*
∂p
)
⎫
) ⎪⎬ SI u
T
⎪⎭
*
ε
(20)
and observing
(
)
Ω p* ,0 = 0
(21)
it obtains:
Ω ( p, ε )
∞
( )
≤ C p*
∞
p − p*
Therefore,
( )
LΩ ≈ C p*
∞
( )
δp ∞ + D p*
Similarly,
( )
A( p ) ≈ A p* +
A( p )
∞
( )(
( )
( )
∞
( )
+ D p*
∞
( )
ε ≤ C p*
∞
( )
+ D p*
( )
∞
δp
( )
δp ∞ + D p*
∞
∞
ε
(22)
(23)
) ( ) ( )(
+ D p*
∞
∞ε
∂A p*
p − p* = A p* + D p* p − p*
∂p
≤ A p*
LA ≈ A p*
∞
p − p*
∞
( )
≤ A p*
)
∞
(24)
( )
+ D p*
∞
δp ∞
(25)
(26)
∞
THE DESIGN OF SENSOR PLACEMENT
The design of sensor placement is a combinatorial optimization problem, i.e., selecting a subset from all candidate
⎯ 494 ⎯
measurement locations to satisfy the criterion of optimal sensor placement. From Eqs. (18) to (26), it is noted that
during the optimization process, LΩ and LA have to be evaluated around p * . However, p * is unknown during the
design phase (maybe forever). Therefore, in Ref. [9], a guess of p * was used instead. From Eq. (11), p * could be
guessed as:
p *g = p 0 − I p δp
(18)
If there are m parameters, there will be 2 m + 1 guesses. Then the design algorithm proposed in Ref. [9] can be briefly
described as:
(1) According to the condition of structure and measurement, specify the candidate measurement set U c , the initial
parameter p 0 , the positive bias δp , the relative measurement error ε, the minimum measurement umin , the minimum
and maximum number of measurements ( N min and N max );
(2) For each p *g , remove the displacement less than umin from U c . The remaining j displacements are sorted in
ascending order with respect to their absolute values and are stored in U c1 . These displacements are also sorted in the
order that the diagonal items in the FIM are high sensitivities and they are stored in U c2 . Denote the number of
measurements as i. Set i = N min ;
(3) If the combination C ij =
j!
≥ 1× 106 , go to step (4).
i! ( j − i )!
Otherwise, enumerate all possible subset of U c1 to find the ones with LΩ < 1 . Go to step (5);
(4) Construct U cJM = U cJ + U cM , where U cJ contains the first i displacements in U c2 and U cM contains the first and
the last i displacements in U c1 . Denote the number of displacements in U cJM as N JM . Remove the last displacement
from U cJM ( N JM = N JM − 1 ), until C Ni JM < 1 × 106 . Enumerate all possible subset of U cJM and record the ones with
LΩ < 1 ;
(5) If i < N max , then i = i + 1 , go to (3). Otherwise, go to the next step;
(6) If all p *g guesses are examined, go to the next step. Otherwise, calculate the next guess and go to (2);
(7) Find out all common subsets from the recorded feasible ones for each p *g . The good subset is the one that has the
minimum LA (1 + LΩ )ε corresponding to the case of δp = 0 .
Because p *g is very empirical, the optimized sensor placement from the above algorithm may be far from optimal.
Consequently, there could be a large bias between the identified parameters to the true ones. To solve this problem, a
new algorithm is developed to implement the sensor placement design and the parameter identification alternatively as
follows:
(1) Empirically specify p 0 and δp 0 . Set k = 0 and S k −1 as an empty set;
(2) Based on p k and δp k , use the algorithm to find a good sensor placement S k ;
(3) If S k = S k −1 , stop. Otherwise, go to next step;
{
(4) Based on S k , identify the parameter p k +1 . Let δp k +1 = min δp 0 , p k +1 − p k
};
(5) Set k = k + 1 , go to (2).
Because modeling and numerical errors are always existed in practice, the stopping criterion in step (3) may hardly be
satisfied. However, as the iteration goes on, δp k is getting smaller and smaller and p k is approaching p * . Therefore,
it could reasonably use a soft criterion S k = S m , m < k instead.
⎯ 495 ⎯
EXAMPLES
1. Three-layered embankment As Fig. 1 shows, this plane strain model consists of three layers with different
materials and subjected to a linearly distributed load. To identify only three Young’s moduli, all displacements
calculated by FEM are taken as ‘measurements’. As what presented in Ref. [11], for an ideal case, let p 0 = p*
( δp = 0 ), ε = 0.001 , umin = 1 mm, N min = 3 and N max = 12 , the best measurement set contains Y displacements at
node 1, 3 and 4 (see Fig. 1). With these measurements, the estimated and calculated maximum bias is 5814Pa and
5787Pa, respectively.
E1 = 1 MPa
ν1 = 0.4
10 m
2
E2 = 3 MPa
4
4
ν2 = 0.3
3
1
9×10 Pa
10 m
E3 = 5 MPa
ν3 = 0.2
Y
X
30 m
Figure 1: A three-layered embankment
Practically, because p * is unknown, the design of measurement placement has to be based on empirical estimates. It is
(
reasonable to set p 0 = E10 , E20 , E30
)
= (0.95, 3.15, 4.75) MPa, and assumes δ p = (δ E10 , δ E20 , δ E30 )
T
T
T
= ( 0.1, 0.3, 0.5 ) MPa. The optimized measurements are Y displacements at node 1, 2 and 3, from which, the
calculated maximum bias is 6233Pa. Obviously, this is not optimal. However, if the sensor placement design and
parameter identification process are alternately conducted, it is possible to find a better placement. The whole process
is listed in Table 1.
T
Table 1: Sensor placement design and parameter identification process for three-layered embankment
Identified parameters (MPa)
E3
Estimate bias
LA (1 + LΩ )ε (Pa)
Calculated bias
(Pa)
3.004304
♣ 4.993767
6429
6233
1.000206
3.001640
♣ 4.994213
5806
5787
1.000206
3.001642
♣ 4.994213
5807
5787
Iteration
Measurements
1
1Y+, 2Y-, 3Y+
1.000897
2
1Y+, 3Y+, 4Y-
3
1Y+, 3Y+, 4Y-
E1
E2
+ with 1‰ error; - with −1‰ error; ♣ where the maximum bias locates
2. Highway bridge The proposed method was implemented to detect damages in Beixingtang bridge, which was
located on the highway connecting Shanghai and Nanjing, China. The main bridge consists three spans of reinforced
concrete continues box girder with a total length of 114 m. The main span is 50 m long and the other spans are 32 m
long (see Fig. 2).
1
2
32m
3
4
5
6
50m
7
8
9
32m
Figure 2: The main bridge and the measuring points
⎯ 496 ⎯
Because cracks, corrosion and water leakage were observed during the routine inspection, a careful examination was
conducted in September 2003, when both dynamic tests and static tests were carried out. It conceives that the damage
could be regarded as the reduction of bending stiffness for beam structures and deflections are very sensitive to the
change of bending stiffness. Therefore in this paper, only the static deflections, obtained from truck loading tests, were
used for the damage detection. In the tests, there were three measuring points evenly distributed over each span (see
Fig. 2). The loading process was divided into four steps. Six trucks of 30 Ton in weight were utilized in each loading
step. Because the damage would take effect only if the small cracks were sufficiently open, only the step with the
largest bending moment was considered in this example. The corresponding effective loading is depicted in Fig. 3.
360KN
18.4m
360KN 180KN 180KN 360KN
360KN
3.2m 1.3m 4.2m 1.3m 3.2m
18.4m
Figure 3: Truck loading on the main span
Because the cross-section of the box girder is almost in the same configuration over the three spans, the bending
stiffness should be almost the same over the whole main bridge in the intact state. If the bridge is subdivided into small
sections, the sudden drop of the bending stiffness in one section could imply the existence of a damage there. In this
example, the main bridge was discretized into 164 two-noded Euler-Bernoulli beam elements. Because there are only
nine measurements (see Fig. 2), these beam elements were divided into seven groups for damage detection. To make
sure the identified parameters can truly represent the damage, two subdivision schemes (see Fig. 4) were tested. In
Scheme 1, the main span is evenly subdivided; while in Scheme 2, section e and section g cover one fourth part each,
and section f covers half a span.
c
1
d
2
e
4
3
f
g
h
5
6
7
50m
32m
i
c
9
1
8
d
2
e
3
f
4
g
6
h
7
50m
32m
32m
5
Scheme 1
i
8
9
32m
Scheme 2
Figure 4: Two schemes of subdivision
Table 2: Parameter identification results of Beixingtang bridge
Identified bending stiffness （1011 Nm2）
2
3
4
5
6
Optimized
measurements
1
1
1, 3, 4, 5, 6, 7, 9
4.718
2.615
4.938
0.425
3.135
3.266
2.004
2
1, 3, 4, 5, 6, 7, 9
5.439
3.014
5.917
0.289
3.414
3.448
2.116
Deflection (mm)
Scheme
7
4
2
0
-2
-4
-6
-8
-10
-12
-14
10
20
30
40
50
60
70
Position (m)
80
90
100
110
Measured
Scheme 1
Scheme 2
-16
-18
Figure 5: Comparison between measured and calculated deflections of Beixingtang bridge
According to the original design, the inputs for measurement design and parameter identification are set as:
p 0 = 3.0 × 1011 Nm 2 , δp = 0.1 × 1011 Nm 2 , ε = 1% , umin = 0.01 mm . The optimized measurements and the
corresponding identified parameters are listed in Table 2. Because the discrepancy between the calculated and
⎯ 497 ⎯
measured deflections is very small (see Fig. 5), the validity of these identifications is thus confirmed. From Table 2, it
observes that the effective bending stiffness of section f is much smaller than those of other sections. This implies that
the middle part of the main span was seriously damaged. This is in good accordance with the observation of on-site
inspections, according to which, this bridge was demolished in May 2004.
CONCLUSIONS
In this paper, an additional iteration is proposed to modify a previously developed algorithm for the design of optimal
sensor placement in damage detection procedures. Because of this modification, the side effect of empirically setting
the initial parameters is remedied to some extent. This helps to obtain a more stable damage detection result. The
validity of this method is illustrated by an academic example and an engineering instance as well.
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