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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

920 Journal of Intelligent Material Systems and Structures 23(8)



4/15

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

Vanli et al. 921



5/15

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.




6/15

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



7/15

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.




8/15

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.

Vanli et al. 925



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http://jim.sagepub.com/


10/15

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.

Vanli et al. 927



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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.




12/15

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.

Vanli et al. 929



13/15

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

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3030

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y

0 2 4 6 8 10 12

0

2

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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.




14/15

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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