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DOI: 10.1177/1045389X12440751
2012 23: 919 originally published online 8 May 2012Journal of Intelligent Material Systems and StructuresO Arda Vanli, Chuck Zhang, Annam Nguyen and Ben Wang
A minimax sensor placement approach for damage detection in composite structures
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A minimax sensor placement approachfor damage detection in composite
structures
O Arda Vanli1, Chuck Zhang1, Annam Nguyen2 and Ben Wang1
Abstract
This article proposes a new method for optimal placement of sensors for detecting damages in composite structures.The problem is formulated as a minimax optimization in which the goal is to find the coordinates of a given number ofsensors so that the worst (maximum) probability of nondetection of the sensor network is made as good as possible
(minimized). It is shown that a minimax approach can more efficiently place the sensors on complex geometries, com-pared to existing placement methods that consider average probability of detection. The method allows one to accountfor characteristics of sensors by assuming that the effectiveness of a sensor decreases with the distance from damage viaan experimentally determined sensor probability of detection function and sensor noise in sensor network optimization.The formulation also enables to account for nonuniform likelihood of damages on the structure, which often arises due
to irregular loading or boundary conditions, using a damage probability density. Numerical examples and an experimentalvalidation study involving a Lamb-wave sensing system are presented to show the effectiveness of the proposed method.
Keywords
damage detection, optimal sensor placement, Lamb-wave sensors
Introduction
Load-carrying composite structures operating under
tensile, fatigue, or impact loading or corrosive environ-
ments develop damages during service, including
matrix cracks, debonding, and delamination. These
damages are usually invisible to surface inspection, and
they do not immediately result in failure. Before these
damages reach critical size, the structure can continue
to safely operate. However, it is important to continu-
ously monitor the integrity of the structure in order to
detect these damages early and prevent them from
exceeding critical size and resulting in catastrophic fail-
ure. Commonly used nondestructive evaluation (NDE)
techniques, including x-ray and ultrasonics, require sig-
nificant labor and disassemble/reassemble time of the
components for inspection (Diamanti and Soutis,
2010). Structural health monitoring (SHM) system that
utilizes a set of built-in, distributed sensor network
embedded within composite structures has proven suc-
cessful as a cost-effective alternative to overcome the
shortcomings of NDE. It enables to more accurately
detect and locate damages. SHM can result in signifi-
cant cost reduction (by eliminating unnecessary mainte-
nance) and weight savings (by avoiding over safe
designs).
In this article, we focus on sensor placement aspectof health monitoring. While many aspects of SHM,
including damage detection and characterization, have
been studied extensively by many authors in the SHM
literature (see e.g. Worden and Manson, 2007), the sen-
sor placement problem received relatively small atten-
tion. Teo et al. (2009) proposed a sensor placement
approach using the scattering of stress waves as the
damage detection tool and optimizing the average prob-
ability of detection (POD) on the structure. Markmiller
and Chang (2010) optimized the locations of a set of
surface bonded piezoelectric sensors measuring strain
during impact using a finite-element analysis (FEA)model and genetic algorithms. Worden and Staszewski
(2000) used a neural network to locate and quantify the
1Department of Industrial Engineering, High Performance Materials
Institute, Florida A&M University, Florida State University, Tallahassee,
FL, USA2Department of Mechanical Engineering, Brown University, Providence,
RI, USA
Corresponding author:
O Arda Vanli, Department of Industrial Engineering, High Performance
Materials Institute, Florida A&M University, Florida State University, 2525
Pottsdamer Street, Tallahassee, FL 32310, USA.
Email: [email protected]
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extent of impacts from signals of a sensor network in
order to find the best sensor locations. Guo et al. (2004)
used genetic algorithms to search for optimal sensor
locations based on modal testing data. Hiramoto et al.
(2000) used the explicit solution of the algebraic Riccati
equation to solve for the optimal locations of actuators
and sensors in a vibration control system.In this study, the damage detection algorithm is
assumed to be given and we focus on the sensor place-
ment aspect. The detection problem has been studied
extensively by many authors. Worden and Manson
(2007) proposed a neural network and feature selection
approach for damage detection from vibration response
of aircraft structures. Sohn et al. (2001) considered time
series modeling and Mahalanobis distance outlier anal-
ysis for damage detection. In our article, we will follow
an outlier analysis for damage detection.
Due to the cost of installation of sensing elements
and wiring and reduced structural integrity concerns, it
is typically not desirable to very densely place a large
number of sensors on a structure; therefore, optimal
selection of sensor location is an important problem.
This article proposes a new minimax approach to find
the optimal number and location of sensors in health
monitoring of composite structures by minimizing the
maximum (worst) probability of nondetection (POND)
of a damage/impact anywhere on a two-dimensional
plane structure. In structural applications, it is crucial
for safety reasons that a damage or impact does not go
undetected. In minimax problems, we would like to
make the poorest response as good as possible; there-
fore, it is an appropriate measure for health monitoring.By contrast, in the commonly applied average probabil-
ity based approaches the overall response is made as
good as possible.
The proposed method assumes that the effectiveness
of a sensor decreases with the distance from damage.
The field of effectiveness of a sensor, referred to as the
sensor probability of detection function (SDF), is sta-
tistically estimated by fitting an exponential decay func-
tion to experimentally observed POD values. The
statistical model allows one to account for sensor noise
in the optimal solution through the use of confidence
intervals of mean response. The formulation alsoenables to account for nonuniform likelihood of dam-
ages on the structure using a damage probability distri-
bution (DPD) function.
Most structural damage detection and location
methods in the literature examine the changes in the
measured structural vibration response such as the
modal frequencies, mode shapes, and stiffness coeffi-
cients. The vibration-based damage detection can be
either active or passive. The passive methodologies con-
sider only the responses to operational vibrations, while
the active algorithms exert an auxiliary excitation by
means of an actuator to the system and examine the
system response (Doebling et al., 1996). In this article,
we will employ an active Lamb-wavebased actuator
sensor system for damage detection. More details on
the sensor system used are provided in section
Damage detection with Lamb waves and experiments
to quantify SDF and sensor noise.
Best location of sensors is a well-studied problem in
the operations research and optimization literature, aswell in application areas including placement of sentries
along a border to detect enemy penetration, facility
location, and detection of hazardous events (Cavalier et
al., 2007). Drezner and Wesolowsky (1997) formulated
the problem of locating identical sensors on a unit line
and a unit square as an optimization problem, called
the minimax problem, in which the objective is to mini-
mize the maximum POND. They considered exponen-
tial decay and power decay sensor detection probability
functions and proposed a special algorithm for the unit
line case that can achieve the necessary condition for
optimality. The minimax problem is a difficult non-
linear nonconvex problem even in the case of two sen-
sors. Cavalier et al. (2007) studied the minimax sensor
placement on a plane and proposed a heuristic based
on Voronoi polygons. It is shown that the proposed
heuristic can quickly generate high-quality solutions for
networks with large number of sensors.
The remainder of the article is organized as follows.
Section Proposed minimax sensor placement
approach presents the optimal sensor placement meth-
odology. Section Damage detection with Lamb waves
and experiments to quantify SDF and sensor noise
discusses the piezoceramic sensing system used in the
article and the experiments conducted to determine sen-sor characteristics. Section Examples illustrates the
application of the proposed method with numerical
examples. In section Experimental validation of the
approach, the proposed sensor placement method is
illustrated from data obtained from experiments. In the
experiments, composite panels were subjected to con-
trolled size of damages, and the damages are detected
with sensors. Different sensor placement configurations
were compared. Section Conclusion and future work
gives the concluding remarks.
Proposed minimax sensor placementapproach
Suppose that a damage can happen on the structure at
location given by a two-dimensional coordinate vector
z= (z1,z2) and the probability that an ith sensor located
at the coordinate vector xi = (xi,yi) detects this damage
is ps(di) where di =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz1 xi 2 + z2 yi 2
qis the
Euclidian distance between the sensor and the damage.
The function ps(:) will be referred to as the SDF and isused to model how much the effectiveness or sensitivity
of the sensor to a given size of damage decreases with
distance. SDF is represented using the traditional
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concept of POD defined as the proportion of damages
that are detected by an NDE system when applied to a
population of structural elements with damage of given
size (Berens and Hovey, 1983).
We will consider the exponential decay function sen-
sor probability function (Drezner and Wesolowsky,
1997)
ps d =b0eb1d 1where d is the distance of the damage from the sensor,
0\b0\1 is the POD at d= 0, and b1.0 is the exponent
that determines the rate of decay (if b1 is larger, the
decay rate is faster). As we will show later, the para-
meters of this model can easily be estimated from
experiments on the sensor using linear least squares.
It should be noted that other types of sensor func-
tions may be used depending on the application. For
example, a power decay function can be defined as
(Cavalier et al., 2007)
ps d = am+ dn
2
where a, m, and n are the parameters to be estimated.
The probability of detecting a damage at distance 0 is
given by the ratio a=m and exponent n gives the rate ofdecay.
The proposed methodology assumes that the likeli-
hood that damage occurs at different locations on the
structure may be nonuniform. This is incorporated in
the formulation using a DPD function, pD(z), which
gives the probability that a damage can occur at alocation z. Nonuniform probability distribution of
damages can arise due to irregular loading or bound-
ary conditions and can conveniently be obtained from
FEA.
The minimax criterion will place the sensors so that
the maximum POND anywhere on the structure is
made as small as possible. Suppose we want to place m
sensors and the coordinates of the sensors are given
with the vector X = (x1, . . . ,xm). Assume that the sen-
sors operate independently. Then, the probability of
not detecting (POND) a damage at location z is found
as the product of the probabilities that individual sen-sors do not detect the damage
POND zjX =pD z Ymi = 1
1 ps di 3
Thus, we want to find the set of sensor locations X
so that the maximum POND on the structure
PONDMAX X = maxz
POND zjX 4
is made as small as possible. Therefore, the minimax
problem is defined as
minX1, ...,Xm
PONDMAX X 5
= minX1, ...,Xm
= maxz
pD z Ymi = 1
1 ps d z,xi ( )
5a
and the solution of this problem x1
, . . . ,xm is the opti-
mal sensor locations that we are looking for.The DPD function pD z acts like a weight on the
objective function by specifying the probability that a
damage occurs at a spatial location z. In the case that
damage can occur anywhere on the structure with equal
probability, the DPD becomes a constant and can be
dropped without affecting the optimal solution. In this
case, the minimax problem can be written as
minX1, ...,Xm
maxz
Ymi = 1
1 ps d z,xi ( )
The effectiveness of the proposed minimax approachwill be compared to the existing sensor placement
method studied by Markmiller and Chang (2010),
which is based on maximizing the average POD by the
sensor network. To be equivalent with the proposed
method, we will formulate the Markmiller and Chang
method as the minimization of the average POND. An
advantage of average POND-based methods is that
they are simple to compute. However, they have vari-
ous shortcomings, including not being able to account
for complex, nonuniform loading conditions in the
optimal solution. The average POND for a sensor net-
work can be defined in the following. Suppose we want
to place m sensors attached on the plate and that dam-
age at a location z is detected by k(k\m) of these sen-
sors. Let an indicator variable be defined di(z) =1 if the
damage is detected by ith sensor i = 1, . . . , m anddi(z) =0 otherwise, then the average POND is
PONDAVG zjX = 1 1k
Xmi = 1
di z 6
The indicator variable is defined by specifying in the
SDF an appropriate threshold probability p0 below
which a damage cannot be detected by the sensor
di z = 1 if ps d z,xi ! p00 otherwise
&
The average POND for all damages on the plate is
found by considering a grid of n different damage loca-
tions zj(j= 1, 2, . . . , n)
PONDAVG X = 1n
Xnj= 1
PONDAVG zjjX
=1
nXn
j= 1
1 1kX
m
i = 1
di zj
" #
7
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The placement of the m sensors is found by minimiz-
ing PONDAVG X
minX1, ...,Xm
PONDAVG X 8
Damage detection with Lamb waves andexperiments to quantify SDF and sensor
noise
Due to environmental fluctuations and variations in
material properties, the sensor systems have variable
performance in detecting defects. One advantage of the
proposed minimax approach is the ease with such varia-
tions can be taken into account for sensor placement.
In this section, we present an experimental study we
conducted to quantify the variability in the SDF of a
Lamb-wave sensor. In section Examples, we will
show how to design an optimal sensor network whileaccounting for sensor noise using the SDF and its varia-
bility modeled in this section.
We will use a Lamb-wave sensor system for our
experiments (Qing et al., 2006). Lamb-wave
propagation-based piezoelectric sensor arrays are
becoming more popular in health monitoring of aero-
space and civil structures due to their low cost, good
performance, and ease of installation (Ihn and Chang,
2004; Kessler et al., 2002). For damage detection, a
piezoelectric actuator and a sensor are bonded on sur-
face or embedded between layers of multilayered car-
bon fiber-reinforced polymer composite laminate. For
health monitoring, diagnostic wave forms are generatedby the actuator, and the resulting structural response is
measured by the sensor. Cracks or defects that exist in
the material between the actuator and the sensor are
detected based on the difference in the shapes of the
transmitted and the received signals. The time of flight
of the wave packets from the actuator to the transducer
can further be used to locate the damage.
The experiment is conducted on a 10 3 26-in. three-
ply composite laminate. We used IM7GP 12K carbon
fibers from Cytec, and the thickness of the three-ply
laminates after resin infusion was 0.035 in. Polyester
resin and a fiber volume fraction of 40% were used in
the infusion process. Two piezoelectric sensors are
attached 24 in. apart as shown in Figure 1(a). Both sen-sors are set to work in a pulseecho mode; that is, they
work as both actuators and sensors. A three-peak, 20-V
amplitude and 400-kHz sine wave burst with a Hanning
window was used as the actuator signal. The amplitude,
frequency, number of cycles of the actuator signal
were determined in a preliminary experiment to minimize
the amount of dispersion in the actuator signal group
velocities. Figure 1(b) shows the actuator wave form
used in the tests.
Using the above actuator signal, we conducted a set
of damage experiments in which 3/8-in.-diameter holes
were created 6, 10, and 18 in. away from the actuatoron the left (sensor S1 in Figure 1(a)). Note that when
we set the sensor on the right (Sensor2 in Figure 1a) as
the actuator, we obtain another set of measurements
from the same damages. Therefore, a total of n = 5
sensor data from these damages were used (the second
replication at 6 in. was dropped due to hardware prob-
lems experienced during data collection).
Damages were created sequentially on the same
panel, and a given damage is detected by considering
the previous damage as the baseline. In order to mini-
mize the effect of interactions between sequential dam-
ages, the order of the damages is randomized. The
randomized order of the experiments was 6, 18, and 10in. Thus, for the damage at 18 in., the damage at 6 in.
was the baseline, and for the damage at 10 in., the dam-
age at 18 in. was the baseline.
In each test, the damage is detected by comparing
the sensor signal in the damaged state to the sensor sig-
nal from the baseline state. Figure 2(a) shows the sig-
nals from the baseline and damaged states obtained for
(a)
(b)
1 in1 in 6 in
26 in
Sensor 1
3/8 in Diameter 3/8 in Diameter
Sensor 2
4 in 8 in 6 in
5 in
5 in
0.5 1 1.5 2
105
800
600
400
200
0
200
400
Time (s)
Actuatoroutput
Actuator signal (400 kHz, three-peak sine wave with a Hanning window)
3/8 in
Diameter
Figure 1. (a) Damage experiment setup: 10 3 26-in. composite laminate, sensors S1 and S2, and the damages generated with 3/8-
in. drill bit. (b) Actuator signal for detecting damages: 400 kHz and three-cycle sine wave.
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the damage at 10 in. It can be seen that the amplitude is
attenuated due to the reflection of the wave from the
damage. The change in the time domain signal due to
the damage is measured as the reduction in the power
spectral energy in the frequency domain and is obtained
by taking the Fourier transformation
S v =
eivts t dt
where s(t) is the time domain sensor signal and v is fre-
quency. The fast Fourier transformation (FFT) plots of
the signals are shown in Figure 2(b). We defined a dam-age metric (DM) as the percent reduction in the area
under the power spectrum from the baseline to the
damaged states as
DM=AreaBaseline AreaDamaged
AreaBaseline3100 9
The area under the power spectral density S v j j2 ofthe signal, or the average power of the signal, is found
(Orfanidis, 2009: 713) as
Area =
1
2pp
pS v j j
2
dv
The range of the integration was set between 200 and
600 kHz since there were no other significant frequency
components outside this range.
It is important to point out that even though the
excitation frequency is 400 kHz, it is expected that the
response of the structure contains frequencies below
400 kHz (because of material damping of the vibra-
tions); however, no frequencies to be present in the
response above 400 kHz. In Figure 2(b), the few small
frequency peaks seen above 400 kHz are possibly due
to the spurious high-frequency components introduced
from the leakage effect in the Hanning window process
and also because the actual excitation frequency may
vary slightly from the setting of 400 kHz.
The DM values computed from the sensor signals of
the different damages are shown in Figure 3(a), which
shows a reduction with the distance from the actuator,
as expected. An exponential SDF (1) was fitted to these
observations using least squares. This can be written as
a linear regression after a logarithm transformation on
the response as
ln ps = ~b0 +b1d+ 2 10
where the new intercept is~b0 = ln (b0). The regression
analysis was conducted with statistical modeling software
Minitab (2006), and the output of the fitted model is
Denote the estimates of the linear model from theabove Minitab output as ~b0 = 0:9522 andb1 = 0:1637. Then, the estimated SDF is obtainedfrom the fitted model as ^ps = b0e
b1d = e0:9522 e0:164d
where b0 = e~b0 = e0:9522. Figure 3(b) shows the mean
and the upper and lower 95% confidence interval on
the mean from this fitted model. From the properties
of the least squares linear model, the probability distri-
butions of the coefficients are
~b0 0:952 0:559
;tnp andb1 0:164
0:042;tnp
11
2.1 2.2 2.3 2.4 2.5
104
15
10
5
0
5
10
15Sensor signals. (Damage 10 in away from S
1. Damage Size 3/8 in)
Time (s)
Sen
sor
Ou
tpu
t
Baseline
Damaged
0 2 4 6 8 10
x 105
0
0.1
0.2
0.3
0.4
0.5
Frequency (Hz)
Amp
litude,
|Y(f)|
1FFT of sensor signals. (Damage 10 in away from S . Damage Size 3/8 in)
Baseline
Damaged
(a) (b)
Figure 2. (a) Sensor signal in the baseline and damaged cases. Consider the sensor signal for time interval of 2025 ms. (b) FFTof
the signal. FFT plots of baseline and damaged signal. Take the integral between 200 and 600 kHz.FFT: fast Fourier transformation.
Vanli et al. 923
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where tnp is the Student t distribution with n pdegrees of freedom and n = 5 and p =2. Using these dis-
tributions, we generate random realizations of SDF by
first simulating realizations of ~b0 and b1 and calculat-
ing the corresponding ps(d) from equation (10). This
allows us to investigate the effect of sensor noise on
sensor placement. These distributions will be used in
finding the optimal placement of sensors by accounting
for the sensor noise, which we will discuss in section
Examples.
Examples
In this section, we illustrate the application of the pro-
posed method with two numerical examples.
Placement of two sensors on a line
Consider the case of placing two sensors on a one-
dimensional line of length 10 in. as shown in Figure 4.
This is a simple problem that can be solved with
exhaustive search without requiring any special optimi-
zation software; however, it will provide intuition for
more complex cases. The damage coordinate is z, andthe sensor coordinates we want to solve for are x1 and
x2. We assume that the probability of having a damage
anywhere on the line is equal (i.e. pD(z) is constant) and
that the SDF is the power decay function given by the
formula (2) and parameters a= 5, m= 5, and n = 1:05.We will compare the solutions of the proposed mini-
max POND problem (4) to the average POND prob-
lem (8) studied by Markmiller and Chang (2010). We
initially consider two arbitrary sensor placement con-
figurations: placement 1 at (3.33, 6.67) and placement 2
at (1.70, 8.30). The first placement has the sensors
equally spaced and second one as we will see later is the
optimal placement according to the minimax rule.
Figure 5(a) and (b) shows the POND(zjX) function ascalculated from formula (3) for two placements. As itcan be seen, optimal placement has a much smaller
maximum POND than that of equal spacing and is
more desirable. Figure 5(c) and (d) shows the average
POND, PONDAVG(zjX), calculated by formula (6) withthe two sensor placement configurations (a damage
detection threshold p0 = 0:70 was used). The PONDAVGmetric has a step behavior indicating that the sensor
network either detects it completely or misses. By con-
trast, the proposed POND definition as shown in
Figure 5(a) and (b) provides a more refined quantifica-
tion of sensor detection capability. We will show later
that this feature of the proposed method has the bene-fits over PONDAVG.
We will next illustrate how to find the optimal
solution X from the proposed PONDMAX(X) defini-tion (4) and the existing PONDAVG(X) definition (7).
Since we have only two variables, the minimizing
solution can be easily found by plotting the function
against x1 and x2. Figure 6 shows the contour plots
of PONDMAX and PONDAVG. As it can be seen, the
minimum PONDMAX is 0.18 and is achieved at two
alternative optimal solutions (x1,x2) = (1:7, 8:3) and(x1,x2) = (8:3, 1:7). The first optimal solution is whatwe considered in Figure 5. By contrast, the minimum
PONDAVG is 0.24, and instead of having optimum
0
0.1
0.2
0.3
0.4
0.5
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
D
M
Distance (d, in)
Damage metric values from experiments
0
0.1
0.2
0.3
0.4
0.5
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
DM
Distance (d, in)
Regression analysis on the DM data
Fied values
95% Confidence Bound
(a) (b)
Figure 3. Estimation of the SDF. (a) Experimental observations of DM. (b) Least squares model for SDF. Solid line is the mean SDF
values and dashed lines are the 95% confidence intervals.SDF: sensor probability of detection function; DM: damage detection metric.
Sensor 1: x1 Damage (z)
d1
Sensor 2: x2
10 in
Figure 4. Placement of two sensors on a line.
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solution at discrete points, there are continuous
regions of alternative optimum solutions that mini-
mize the function (the dark blue triangles in the upper
left given and the lower right). However, thePONDAVG definition does not give us the ability to dis-
tinguish between these solutions.
Placement of m sensors on a panel
The general case of the placement of m.2 sensors on a
two-dimensional plane is a nonlinear nonconvex opti-
mization problem, which requires the use of a numeri-
cal optimization algorithm. In this section, we will
illustrate the solution of the proposed minimax POND
problem using the MATLAB fminmax algorithm,
which is based on a sequential quadratic programming
method (MATLAB, 2010).
Consider a 12 3 12-in. square panel structure and
suppose we want to place four sensors on the panel. We
consider the two DPDs shown in Figure 7. Figure 7(a)
is a uniform DPD that assumes it is equally likely tohave damage anywhere on the panel while Figure 7(b)
is a nonuniform DPD that assumes that it is more likely
to have damages near the left edge (i.e. z1 = 12) than
near the right edge (i.e. z1 = 0) of the panel. If some
areas of the structure are more prone to damages, then
it may be desirable to have the sensors more densely
spaced near these areas, while placing them less densely
in other areas.
We solve the placement problem with the minimax
POND and average POND approaches. The optimum
sensor locations obtained for the uniform DPD are
shown in Figure 8(a). It can be seen that the minimax
POND approach places the sensors closer to the edges
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
z(damage location)
equally spaced sensors
max=0.24633mean=0.10184
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
z(damage location)
optimally placed sensors
max=0.17462mean=0.10457
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
z(damage location)
POND
ofsensorne
twork
equally spaced sensors
mean=0.27723
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
z(damage location)
optimally placed sensors
mean=0.24752
Proposed
P
OND(z|X)definion
AveragePOND(z|X)definion
(a) (b)
(c) (d)
Figure 5. (a and b) Proposed POND definition and (c and d) average POND definition from the literature for two sensors on a
line. (a) and (c): equally spaced sensors (x1,x2) = (3:33, 6:67) and (b) and (d): optimally spaced sensors (x1,x2) = (1:7, 8:3).POND: probability of nondetection.
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in Figure 10. As it can be seen, minimizing the maxi-
mum POND to select the sensor locations achieves a
lower maximum (worst) nondetection probability thanminimizing the average POND under both the uniform
and nonuniform damage probabilities. This is a desir-
able property of the minimax method and is achieved
because the objective in the minimax approach is to
find the sensor locations so that the worst performance
is made as good as possible. Moreover, as seen in
Figure 10, the improvement in the maximum POND
using the minimax over the average POND is larger
under the nonuniform DPD (from 0.555 to 0.355; a
36.0% reduction) than under the uniform DPD (from
0.645 to 0.577; a 4.4% reduction). Since in nonuniform
loading and damage probabilities will be more common
in engineering structures, the minimax approach is
expected to have more benefits in these cases for dam-
age detection.Next consider the design problem for the sensor net-
work in which we would like to find how many sensors
we should use and where we should place them so that
we can detect at least 75% of the damages on the panel.
We want to achieve this by accounting for our uncer-
tainty in estimating the sensor detection function. We
assume that Lamb-wave sensors are used and use the
SDF and the confidence intervals estimated in section
Damage detection with Lamb waves and experiments
to quantify SDF and sensor noise to account for sen-
sor noise. Consider the uniform DPD case only and
assume the sensors are identical. We solve the minimax
(a) (b)0 5 10
2
0
2
4
6
8
10
12
14sensor placement
minimax POND
0 5 102
0
2
4
6
8
10
12
14sensor placement
min-average POND
0.4
0.4
0.4
0.4
0.42
0.42
0.42
0.42
0.44
0.44
0.44
0.44
0.46
0.46
0.
46
0.
46
0.48
0.48
0.48
0.48
0.48
0.48
0.48
0.48
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
.52
.52
0.52
0.52
0.5
2
0.52
0.5
2
0.52
0.52
0.52
0.52
0.52
0.54
0.54
0.
540.
54
0.56
0.56
0.
56
0.5
6
0 5 100
2
4
6
8
10
12
0.35
0.3
5
0.
35
0.
35
0.4
0.4
0.4
0.4
0.4
0.4
5
0.45
0.45
0.4
5
0.45
0.45
0
.5
0.5
0.5
0.5
0.5
0.5
0.55
0.5
5
0.5
5
0.55
.6
0.6
0.6
0.6
0 5 100
2
4
6
8
10
12
Figure 8. Results for uniform DPD from minimax POND and min-average POND approaches: (a) optimal sensor placements and
(b) distribution of POND values with respect to x- and y-coordinates of the plate.DPD: damage probability distribution; POND: probability of nondetection.
(a) (b)
0 5 102
0
2
4
6
8
10
12
14sensor placement
minimax POND
0 5 102
0
2
4
6
8
10
12
14sensor placement
min-average POND
0.05 0.05
0.1 0.10.15 0.150.2 0.20.25 0.25
0.3
0.3
0.30.3
0.3
0.
0 5 100
2
4
6
8
10
12
0.05 0.05
0.1 0.1
0.15
0.15
0.2
0.2
0.25
0.250.3 0.3
0.35
0.35
0.4
0.40
.45
0.45
0.5
0.5
0 5 100
2
4
6
8
10
12
Figure 9. Results with nonuniform from minimax and min-average approaches: (a) optimal sensor placements and (b) distribution
of POND values on with respect to x- and y-coordinates of the plate.POND: probability of nondetection.
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problem for m = 2, 4, 5, and 10 sensors. Figure 11 shows
the optimal locations for the 4-, 5-, 6-, and 10-sensor
networks and the achieved PONDMAX values. As
expected, with increasing number of sensors, the
POND decreases.
In order to determine the number of sensors, we use
simulation to incorporate sensor noise in
PONDMAX(X) calculation of each network (note: maxi-
mum PONDs reported in Figure 11 are averages). To
account for the sensor noise, we place the sensors so
that we are 95% confident that at least 65% of the
damages are detected (or 95% confident that at most
35% of the damages are missed). This can be accom-
plished by having the 95% upper confidence bound of
PONDMAX(X) less than 0.35. The variability of each
sensor is simulated by the procedure described in sec-
tion Damage detection with Lamb waves and experi-
ments to quantify SDF and sensor noise, and this
variability is then propagated to the distribution of
PONDMAX X of the sensor network. We have simu-lated 100 SDF curves by assuming that the ~b0 and b1
coefficients of the curve are random variables
0 5 100
5
10
y(in
)
4 sensors.POND
MAX= 0.6143
0 5 100
5
10
5 sensors.POND
MAX= 0.54491
0 5 100
5
10
x(in.)
y(in
)
6 sensors. PONDMAX
= 0.48274
0 5 100
5
10
x(in.)
10 sensors. PONDMAX
= 0.29264
Figure 11. Optimal solutions for 4-, 5-, 6-, and 10-sensor networks.POND: probability of nondetection.
(a) (b)
min-averageminimax
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
POND
Uniform DPD: POND values fr om minimax and min-average approaches
Max = 0.577
Max = 0.645
min-averageminimax
0.6
0.5
0.4
0.3
0.2
0.1
0.0
POND
Nonuniform DPD: POND values from minimax and min-average approaches
Max = 0.3545
Max = 0.555
Figure 10. Distribution of the POND values and using the proposed minimax and average POND approaches under (a) uniform
DPD and (b) nonuniform DPD.DPD: damage probability distribution; POND: probability of nondetection.
928 Journal of Intelligent Material Systems and Structures 23(8)
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distributed according to the formulas (11). The simu-
lated SDF curves are shown in Figure 12(a), and the
sensor network PONDMAX X for the plate is computedby assuming this SDF for all sensors. For example,
Figure 12(b) gives the simulated PONDMAX values for
the optimally placed 10-sensor network.
Applying a one-sample t-test on these simulated val-ues, we construct a 95% confidence interval on the true
PONDMAX for the 10-sensor case as (0.265, 0.319).
Similarly, for the six-sensor case, the 95% interval can
be found as (0.440, 0.498). Figure 13 summarizes the
confidence intervals of all the cases. Since we want to
be confident that at most 35% of damages are missed,
we should use the optimally placed 10-sensor network.
Experimental validation of the approach
The experimental validation of the proposed sensor
placement method was performed on the piezoceramic
sensor system. We considered the case of placing four
sensors on three-ply, 12 3 12-in. carbon fiber laminates
(thickness = 0.035 in.) and compared the effectivenessof two different sensor placement arrangements: (a) an
arbitrary layout in which the sensors are placed at the
corners of the plate and (b) the optimal placement from
the minimax approach.
The two sensor layouts considered are shown in
Figure 14(a). Since there are four sensors, there are 12
paths for all pair of sensors: path between sensors 12,
21, 13, 31, 14, 41, 23, 32, 24, 42, 34, and 4
3. We calculate the DM for each of these paths. A
damage is detected to exist if the measured DM is signifi-
cantly different from a reference state in which there are
no damages. We conclude that there is a damage if DIijexceeds some prespecified threshold, which is selected in
order to achieve a desired false alarm rate. To establish
the threshold, we collected 10 reference data before creat-
ing any damages. The damage index (DI), used to decide
if there is damage on the path ij, is defined as
DIij =DMij mij
sij
where DMij is the DM found from sensor data (under
the damaged cases) using equation (9), and mij and sijare the mean and standard deviation of the reference
DM data for the path ij, respectively. The decision rule
used was to decide that there is a damage if the
observed damage is index is such that DIij .3 (i.e. out-
side 63 standard deviations from the mean). Note that
if DIij is normally distributed, then this threshold 3 for
DIij gives 0.0027 false alarm probability. We note that
this is an application of the Mahalanobis distance-
based outlier detection approach employed by Sohn et
al. (2001) for damage detection.
The effectiveness of the two layouts in detecting
damages was tested by generating a set of 3/8-in. holes
on the two plates at the locations shown in Figure
14(a). The experimental setup for the two sensor
PONDMAX
F
requency
0.60.50.40.30.20.10.0
20
15
10
5
0
Histogram ofPONDMAX for 12" square plate with 10 sensors
5 10 15 2000.2
0.4
0.6
0.8
Distance, d (in.)
Detectionprobability,
(d)
Figure 12. (a) Simulated SDF curves using the distributional properties of the regression model fitted in section Damage
detection with Lamb waves and experiments to quantify SDF and sensor noise. (b) Simulated PONDMAX values for the optimally
placed 10-sensor network.SDF: sensor probability of detection function; POND: probability of nondetection.
PONDMAX
10 sensors6 sensors5 sensors4 sensors2 sensors
0.8
0.7
0.6
0.5
0.4
0.3
0.2
95% CI for the Mean
Interval Plot ofPONDMAX for 2 sensors to 10 sensors
Figure 13. Confidence intervals of the POND
MAX(X) for the 12-
in. square panel with increasing number of sensors. Uniform DPD
function and the SDF estimated in the previous example were used.CI: confidence interval; POND: probability of nondetection; DPD: damage
probability distribution; SDF: sensor probability of detection function.
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layouts is shown in Figure 14(b) and (c). The damages
are created in a randomized order (as in the previous
example) so that the effect of the interactions between
the sequential damages on the damage detection mea-
sure is averaged. Similar to section Damage detection
with Lamb waves and experiments to quantify SDF
and sensor noise, the signal obtained for a damage
was used as the baseline for the damage generated
subsequently.One requirement of the Lamb-wave sensors is that
there should be a straight line between pairs of sensors
in order for the structural damages lying on the line
joining them can be detected. In order to cover all dam-
ages created inside the panel, we included a constraint
in the minimax problem so that the sensors are placed
on the edges of the plate not the interior of the plate.
The optimal sensor layout shown in Figure 14(a) is
obtained by solving this constrained optimization.
In comparison to the two sensor case studied in sec-
tion Damage detection with Lamb waves and experi-
ments to quantify SDF and sensor noise, where we
had only two sensor paths (S1 S2 and S2 S1) inthe four-sensor case, we have 12 paths. The existence of
damage on the plate was determined from defining a
network DM that combines the DI information from
all sensor paths
DInetwork = DI2
12+DI2
13+DI2
14+ +DI2
41
1=2
where DIij is the DM for path ij.Figure 15(a) compares the DInetwork values of the 17
damages considered under the optimal and arbitrary
sensor placement configurations. As it can be seen, it
is clear that the optimal placement provides signifi-
cantly larger DInetwork values than the arbitrary place-
ment, indicating that the optimal configuration will
be more effective in detecting damages. The Kruskal
Wallis test applied for the difference between the
median DI of these two layouts reveals that the med-
ian DI for the optimal configuration is significantly
larger (p value = 0.0103) than the median DI of the
suboptimal.
(a) (b) (c)
0 2 4 6 8 10 12
0
2
4
6
8
10
12
Optimal sensors
Suboptimal sensors
Figure 14. (a) Optimal and arbitrary sensor layouts and the damage locations. The damage locations are shown as asterisks, the
optimal sensor layout is shown as diamonds, and the arbitrary sensor layout is shown with squares. (b and c) The composite panels
with sensors installed according to the optimal (b) and arbitrary (c) configurations.
(a)
DInetwork, Optimal
(b)
DInetwork, Arbitrary
(c)10
15
20
25
30
35
40
45
50
yrartibrAlamitpO
DI for 17 damages under Optimal and Arbitrary placement
DIn
etwork
5
5
5
5
101
0
10
10
10
15
15
15
15
15
15
20
20 20
20
20
2020
20
20
20
25
25
25
25
25
25
25
25
30
30
30
20
20
30
3030
35
35
35
25
25
40
40
30
30
45
45
35
x
y
0 2 4 6 8 10 12
0
2
4
6
8
10
12
5
5
5
5
5
10
10
10
10
10
10
10
10
10
15
15
15
15
15
15
20
20
20
20
20
25
25
25
25
30
30
15
35
35
40
40
20
x
y
0 2 4 6 8 10 12
0
2
4
6
8
10
12
0
5
10
15
20
25
30
35
40
45
Figure 15. The overall DInetwork
from the two sensor configurations for the17 damages: (a) box plots and (b) contour plots of
damage metrics for the damages created in the experiments for optimal and suboptimal configurationsDI: damage index.
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If we use 3 as the threshold for each Dij, then the
threshold for DInetwork is 3ffiffiffiffiffi12
p= 10:4 (broken line in
Figure 15(a)), and we can find that the optimum layout
correctly detects 16 (94% correct detection) and the
arbitrary layout correctly detects 12 (71% correct detec-
tion) out of the 17 damages. Figure 15(b) and (c) is the
contour plots of the DIs of the two sensor layouts as afunction of the x- and y-coordinates of the damages on
the plate. As it can be seen with the optimal layout, the
bigger areas of the plate have larger DM values than
the suboptimal layout.
Conclusion and future work
This article presented a new minimax optimal sensor
placement approach for composite structures. The
effectiveness of the propose method was illustrated by
comparing it to existing sensor placement approaches
from the literature on two numerical examples and acase study that is based on a real composite panel struc-
ture and a Lamb-wave sensor network. It was shown
that the proposed method can provide sensor layouts
with significantly higher correct damage detection rates
than existing average POND-based methods.
The proposed method places a fixed number of sen-
sors on a structure with a specified geometry such that
the maximum or worst POND (PONDMAX) of damages
on the structure is minimized. In order to calculate this
probability, the sensor characteristics were modeled using
a sensor detectability function, which is the probability
that a sensor detects a damage at a given distance away.It was shown that the proposed optimal sensor placement
approach can effectively incorporate the physics of the
sensors of sensitivity range and sensor variability through
the use of SDF. By contrast, the existing average POND-
based methods do not consider sensor noise in the selec-
tion of number and locations of sensors.
Another advantage of the proposed method is the
ease with which the loading and boundary conditions
can be taken into account in sensor placement. Due to
irregular or complex loading or boundary conditions,
the likelihood that a damage can occur on the structure
may be highly nonuniform. It was illustrated how the
POND formulation can be easily adjusted to include a
DPD to account for such conditions. It is not clear how
to incorporate the DPD in the existing average POD-
based methods.
In experiments, we considered a fixed damage size
to illustrate the placement methodology. The sensitivity
of detection of the sensor network will is expected to
reduce with smaller damages. However, since the objec-
tive was sensor placement, the improvement of damage
detection performance was not considered in this arti-
cle. Possible extension of the placement methodology
would be to incorporate more advanced detection algo-
rithms with the placement optimization approach.
Other areas of future research include developing
placement algorithms robust to sensor noise and mate-
rial property variations and formulating methods to
incorporate FEA model results for determining DPD
functions based on the boundary and loading
conditions.
Funding
This research was supported by National Science Foundation
(grant number CMMI-0969413) and Florida State University,
High Performance Materials Institute.
Acknowledgment
The authors acknowledge the helpful comments from the
referees, which helped in improving the article significantly.
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932 Journal of Intelligent Material Systems and Structures 23(8)