Computational Framework for DenseSensor Network Evaluation Based onModel Assisted Probability of DetectionJin Yan1, Simon Laflamme2, and Leifur Leifsson3

1,2Dept. of Civil, Constr., and Env. Engineering, Iowa State University, Ames IA, USA2Center for Nondestructive Evaluation, Iowa State University, Ames, IA, USA3Dept. of Aerospace Engineering, Iowa State University, Ames, IA, USA

ABSTRACT

Recent sensing advances empower the deployment of dense sensor networks (DSNs) that can be usedin automating the condition assessment process of large-scale structural and mechanical systems. Tofully enable DSN technologies, it is critical to develop and implement co-design techniques that allow forthe evaluation of their performance to attain the condition assessment targets. In this paper, we proposea methodology to validate the design of DSNs. The methodology consists of constructing a physicalsurrogate of the DSN-equipped component based on the sensor configuration, updating the physicalsurrogate using sliding mode theory based on generated or collected sensor data, and quantifyingthe performance of the DSN based on model assisted probability of detection theory. The proposedmethodology is numerically verified and validated on a cantilever beam subjected to damage at its rootand equipped with a network of soft elastomeric capacitors measuring strain. Various uncertainties areconsidered in the simulated system. The verification consists of confirming the capability of the adaptiveprocess for the surrogate model at reaching an accurate representation of the full system. The validationconsists of ranking the performance of various DSN configuration and benchmarking results againstthose obtained from the full finite element model. Results show that the proposed methodology canbe used to evaluate the performance of DSN configurations, but that the damage thresholds used indetermining damage need calibration for successful field applications.

Keywords: nondestructive evaluation, structural health monitoring, dense sensor network, damagedetection, probability of detection, physical surrogate

1 INTRODUCTIONStructural health monitoring (SHM) is the automation of the structural integrity assessment task conductedusing sets of measurements from permanently mounted sensors (Lynch et al., 2016). Of interest to thispaper are SHM systems based on dense sensor networks (DSNs), which have recently been empoweredfrom advances in smart materials and signal processing. The promise of DSNs lies in their high spatialresolution that yields rich spatio-temporal information about the monitored component (Ubertini et al.,2017). If properly harnessed through strategically tailored signal processing algorithms, such informationcould dramatically improve structural condition assessment capabilities, directly enabling condition-basedmaintenance decisions.

Similar to other SHM solutions, linking sensor measurements to decision is a difficult task. This taskis ever more complex with DSNs because of the large amount of temporal data acquired over a highspatial resolution. In addition, several complexities in the monitored system may affect the performanceof the damage detection capabilities, including weak and/or noisy sensor signal, parametric uncertaintiesin the modeled system, and nonstationarity of the dynamics. To address these challenges, data-drivensignal processing methods (Downey et al., 2018b; Liao et al., 2019) have been proposed to detect damagewithout relying on physical knowledge, but results often fail to provide a clear link to physical parameters.The injection of physical knowledge in data-based techniques gives rise to hybrid approaches (Downeyet al., 2019) that have shown promise at assessing the structural condition in a more useful manner.

In this paper, hybrid procedures are leveraged to provide an integrated approach to evaluating DSNs,therefore empowering condition-based maintenance through improved efficiency of the SHM system.

The concept is based on probability of detection (POD) theory used in the field of nondestructiveevaluation (NDE). POD is generally used to evaluate the capability of an NDE technique at detectingparticular damage under sources of uncertainties. POD curves in NDE applications were initially based onprobabilistic methods and required the generation of multiple sets of repeated measurements on realisticspecimens, is a very costly procedure (Forsyth, 2016). Model-Assisted POD (MAPOD) has been proposedto complement limited empirical experiments with physical information (Memmolo et al., 2016; Du et al.,2018a). Some studies (Aldrin et al., 2011; Moriot et al., 2018) have extended to the concept of MAPODfor SHM applications, where a particular challenge is to quantify the performance of multiple sensorsmonitoring multiple types of damage.

A key difficulty in applying MAPOD is in the creation of a physical representation. A popular methodin SHM is to create a high-fidelity physical representation and update it based on measurements. Manyapplications of such model updating methods are found in vibration-based SHM where the physicalrepresentation is updated to match modal parameters extracted from measurements (Cancelli et al., 2019;Li et al., 2018; Cui and Lanza di Scalea, 2019). However, this methodology is difficult to apply becauseit requires complex optimization and thus requires large computational time, and can be ill-conditionedwhen there are more unknown parameters than available data (Benner et al., 2017; Peherstorfer et al.,2018). In MAPOD applications, it was discussed that these high-fidelity models remain computationallyintensive and usually lack robustness with respect to parameter variations (Amsallem and Farhat, 2008).A solution is to leverage iterative procedures adopting simplified model representation for near real-timeapplications, yet at the cost of lower accuracy (Benner et al., 2015).

In this paper, a MAPOD-based technique for the evaluation and design of DSNs is presented, usingthe application example of a strain-based DSN developed by the authors (Laflamme et al., 2015). Thetechnique consists of creating a physical surrogate model of the monitored system, and using the surrogateto develop a number of scenarios based on uncertainties and damage of interest. Instead of laboratory-produced data like it is common in MAPOD-based techniques, this research uses synthetic signalsproduced through numerical simulations. This is due to the early development stage of studied sensingtechnology, which limits laboratory testing capabilities at this stage. In the numerical simulations, differentuncertainty scenarios are considered and results assembled in the form of POD curves to quantify andrank the performance of a given DSN configuration. The presented work is an extension to preliminaryfindings published in conference proceedings (Yan et al., 2018; Du et al., 2018b; Yan et al., 2019), wherewe now validate the proposed technique on a realistic model and formalize the study by using MAPOD tovalidate the performance of various DSN configurations under different uncertainty scenarios.

The remainder of the paper is organized as follows. Section 2 provides the background on thebenchmark model, including a description of its dynamic representation and the DSN under investigation.Section 3 introduces the proposed MAPOD-based methodology, including the construction of the physicalsurrogate and its adaptation procedure. Section 4 presents and discusses the results from numericalsimulations. Section 5 concludes the paper.

2 BACKGROUNDThis section provides a background on the SEC sensors, followed by an overview of model adaptationand the model-assisted probability of detection techniques.

2.1 SEC-Based Sensing SkinIn this paper, the DSN of interest consists of a densely deployed flexible electronics termed soft elastomericcapacitors (SECs), previously proposed by the authors. The SEC is a low-cost, flexible sensor constitutedfrom a block copolymer matrix (SEBS) mixed with titania to form the dielectric, and with carbon black toform the electrodes. Details on the fabrication process and derivation of the electromechanical modelcan be found in Reference (Laflamme et al., 2013, 2015). This sensing skin has the capabilities scale todifferent sizes and covers large areas which is suitable for strain sensing over large-scale surfaces. Figure1(a) shows a picture of an SEC measuring 76 mm × 76 mm. The sensing principle is based on a changein the sensor’s geometry, provoking a measurable change in capacitance. The capacitance (C) of an SECcan be estimated from the equation for a parallel plate capacitor:

C = ε0εrAh

(1)

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where ε0 is the vacuum permittivity, εr is the relative permittivity, A = d · l is the sensing area with widthd and length l, and h is the thickness of the dielectric, as illustrated in Figure 1(b). Assuming small strains,a relative change in capacitance can be obtained by differentiating Equation 1:

∆CC0

=∆dd0

+∆ll0− ∆h

h0= εx + εy− εz =

11−ν

(εx + εy) = λ (εx + εy) (2)

where ν is the Poisson’s ratio, λ ≈ 2 is the gauge factor, and εx, εy, and εz are the strains along the x, y,and z planes, respectively. It follows from Equation 2 that the SEC measures the average additive in-planestrain of the material located underneath the bottom electrode.

(a) (b)

dielectric

conductiveplatecontacts

εy

εz

εx

h

d

l

x

yz

Figure 1. (a) Picture of a soft elastomeric capacitor (SEC); and (b) schematic of an SEC with keycomponents annotated.

2.2 Benchmark ModelThe example system used in this study is known as DROPBEAR (Dynamic Reproduction of Projectiles inBallistic Environments for Advanced Research) (Joyce et al., 2018a). It is selected due to the availabilityof experimental data that can be used to construct an accurate numerical model for simulation purposes.It features a large, rectangular steel cantilever beam of length lx = 505 mm, width ly = 51 mm, andthickness lh = 6.3 mm, illustrated in Figure 2. Its materials properties are taken as follows: Young’smodulus E = 190 GPa, density ρ = 7970kg/m3, and Poisson’s ratio ν = 0.3 The study starts with thebeam virtually equipped with five SECs with the root of the beam elastically restrained. The boundaryelement has a width lbc = 5 mm and length ly = 51 mm (see Figure 3). The equation of motion governingthe system can be written:

Mq(t)+Cq(t)+Kq(t) = B f f(t) (3)

where t denotes time, the dot a time derivative, q(t) ∈ Rn×1 the displacement vector, M ∈ Rn×n, C ∈ Rn×n

and K ∈ Rn×n the mass, proportional damping, and stiffness matrices, respectively, B f ∈ Rn×n the forceapplication vector, and f(t) ∈ Rn×1 the vector of external forces for an n degrees-of-freedom (nDOF)representation. To create a physical surrogate, a reduced-order finite element model is discretized as afunction of the number of sensors and their sizes, plus an additional element linking to the fixity. Eachelement of cross section area A, Young’s Modulus E and material density ρ comprises two DOFs at eachend, one translational and one rotational, as illustrated in Figure 2.A state space vector Xe = [qe, qe]

T is defined for each element e = 1,2, . . . ,6:

Xe =[qe1,e1 qe1,e2 qe2,e1 qe2,e2 qe1,e1 qe1,e2 qe2,e1 qe2,e2

]T (4)

and the state space formulation adopted:

X = AX+BfY = HX (5)

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1 2

q 1

le

ρAee,e

q 1e,e1

q 2e,e2

q 2e,e1f(t)

lh

(a) (b)

SEC e,E ,

partially retrainedboundary element

x2

Figure 2. Degrees of freedom notation for one element.

where X = [X1,X2, . . . ,X6]T is the assembled state vectors and has a total of 24 states under five SECs, Y

is the measurement output, and H is the observation matrix, with

A =

[0 I

−M−1K −M−1D

],B =

[0

−M−1

],H =

0

He. . .

He

(6)

and

He =lh2[0 −1 0 1

]T (7)

for constant thickness lh. The system’s matrices were constructed by assembling the element mass Me,stiffness Ke, and damping Ce matrices:

Me =ρAele420

156 22le 54 −13le22le 4l2

e 13l −3l2e

54 13le 156 −22le−13le −3l2 −22le 4l2

e

,Ke =EeIe

l3e

12 6le −12 6le6le 4l2

e −6le 2l2e

−12 −6le 12 −6le6le 2l2

e −6le 4l2e

,Ce = βKe

(8)

where for each element, Ae is its cross-sectional area, le the length, I the moment of inertia, and β = 3e−05the stiffness proportionality term. The displacement and velocity feedback can be obtained from themeasured strain by fitting and integrating spatial strain measurements using an kth order polynomialfunction ε = ξ0 +∑

kj=1 ξ jx j. The fitting coefficients ξ j are obtained through a least squares estimator:

ξ = (WTW)−1WTε (9)

where

W =

1 x1 x2

1 . . . x3k

1 x2 x22 . . . x3

k...

......

...1 x5 x2

5 . . . x3k

(10)

Integrating Equation 10 and assigning the proper boundary conditions yield an expression for q

q =− 2lh

x112 x2

1 . . . 1k+1 xk+1

112 x1

16 x2

1 . . . 1(k+1)(k+2)xk+1

212 x2

16 x2

2 . . . 1(k+1)xk+1

212 x2

16 x2

2 . . . 1(k+1)(k+2)xk+1

2...

......

...12 x5

16 x2

5 . . . 1(k+1)(k+2)xk+1

5

ξ0ξ0ξ1ξ1...

ξk

(11)

3 METHODOLOGYThis section presents the model adaptation technique used to parametrize the physical surrogate andameliorate the quality of the representation. After, the adapted model is used to conduct MAPOD.

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3.1 Model AdaptationThe parametrization of DROPBEAR’s dynamic model is relatively simple given known material geome-tries and dimensions, except at the root where the beam may not be perfectly fixed but instead partiallyrestrained. Here, the model is adapted by altering the bending rigidity at the first element (i.e., at theroot) using experimental data. This adaptation is conducted by sequentially updating the model using thesliding mode theory. Note that, while this could easily be conducted heuristically given the simplicityof the model, the methodology presented herein extends to higher-dimensional problems. Note that theperformance of the model adaptation algorithm is expected to reduce with the increasing number ofsensors used in the DSN. While this is out-of-the-scope of the paper, it is anticipated that the incorporationof an event localization algorithm prior to model adaptation would limit the search space of the slidingmode-based algorithm, thus augmenting its stability.

Assume that the system’s stiffness K is the only adjustable parameters, and take the estimated systemas:

˙X = AX+BfY = CX

(12)

with

A =

[0 I

−M−1K −M−1C

]=

[0 I

−M−1K −βM−1K

](13)

where the hat denotes an estimate. The error between both systems can be written A = A− A, where thetilde denotes an estimation error. The estimation error matrix A can be written A = θQ, where θ is thevector of adjustable parameters that pre-multiplies non-adjustable parameters Q. In this example where apartially restrained boundary condition is of concern, the bending rigidity EI is pre-multiplied by θ . Theestimation error matrix becomes:

A =−θ

M−1

1 K1 0 . . . 0 βM−11 K1 0 . . . 0

0 0 . . . 0 0 0 . . . 0...

......

......

......

...0 0 . . . 0 0 0 . . . 0

(14)

Consider the sliding surface s

s =(

ddt

+ c)

e = Pe (15)

where e = X− X is the state error, P = [1,c] is a user-defined vector, and c is a strictly positive constant,and take the following Lyapunov function

V =12(s2 +Γ

−1θ

θ2) (16)

where Γθ is a diagonal matrix of strictly positive elements representing learning parameters. In thispaper, we assigned Γθ = 1e−13 and c = 1.5e5. The function V is positive definite and contains all of thetime-varying parameters. Taking its time derivative and substituting Equations (9), (12) and (13) yields:

V = sTPe+ θΓ−1θ

˙θ

T

= sTP[AX− AX

]+ θΓ

−1θ

˙θ

T

= sTP[AX− (A− A)X

]+ θΓ

−1θ

˙θ

T

= sTP[Ae+ AX

]+ θΓ

−1θ

˙θ

T

= eTPTPAe+ sTPAX+ θΓ−1θ

˙θ

T

(17)

The first term in Equation 17 is negative semi-definite, and the adaptation rule is selected such that

sTPAX+ θΓ−1θ

˙θ

T < 0 (18)

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Substituting Equation 18 into Equation 17, and noting that ˙θ = θ − ˙

θ =− ˙θ yields

V = eTPTPAe+ sTPAX− θΓ−1θ(Γθ sTPQX)

= eTPTPAe+ sTPAX− sTPAX

= eTPTPAe

(19)

showing the stability of the adaptation rule (Equation 18), with expected convergence to the correct modelunder persistent excitation (Joyce et al., 2018b). In the discrete time form, Equation 12 becomes

θk+1 = θk−Γθ ∆tsTPQXk (20)

3.2 Model-Assisted Probability of Detection-Based EvaluationMAPOD is leveraged to quantify the performance of a given DSN by building the POD curves as afunction of the damage of interest under the considered uncertainties. In this study, the damage of interestis loosening of the fixity. The process starts by defining the uncertain model parameters as randomvariables with specific probability distributions. For the system considered in this study, the uncertaintiesare arbitrary selected as arising from the magnitude of the applied load and SEC measurement noise.These uncertainties are introduced in the updated reference model, and several realizations are generatedat different damage levels, including the baseline (i.e., “healthy” structure). After, damage indicatorsare computed by extracting user-defined damage-sensitive features. These features are temporarilycompared using the baseline strain measurements S0 and new measurements S, which are respectivelycomposed of measurements Si,k

0 and Si,k, with i corresponding to the sensor location and k to the discretetime step. Here, three damage-sensitive features are taken from literature (Lu and Michaels, 2009) andtheir performance compared as an exercise to demonstrate the promise of the proposed MAPOD-basedmethodology:

Mean squared error J1: a measure of how close the shape of the measured signal Si,k is to the baselinesignal Si,k

0:

J1 = ∑i,k[Si,k−S0

i,k]2 (21)

Loss of correlation J2: a measure of the overall match between two signals’ waveform shape

J2 = 1−∑i,k(Si,k−µS)(S0

i,k−µS0)

σSσS0(22)

where µS and µS0are the means of Si,k and Si,k0, respectively,σS and σS0 their standard deviations.

Differential curve length (DCL) J3: a measure of the signal complexity:

J3 = ∑i,k|dk−dk−1| (23)

where dk = Si,k−S0i,k.

The J−a plots, where a denotes the damage severity (a = 1−θ ), as constructed from the realizationsand data fitted using a linear regression (Gratiet et al., 2016)

a = β0 +β1a+ ε (24)

where coefficients β0 and β1 are determined through a least squares estimator, and error ε has a normaldistribution with zero mean and standard deviation σε . Damage is detected when the damage indicator Jbecomes larger than the threshold values a. For a given a, POD is computed as follows

POD(a) = P(a > a) = 1−Φ

(a−β0−β1a

σε

)(25)

where P(a > a) is the probability that the degree of damage is higher than the threshold, and Φ is thestandard normal distribution function. The POD function becomes a cumulative normal distributionfunction with mean (a−β0)/β1 and standard deviation σε/β1.

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4 RESULTS4.1 Numerical ValidationSynthetic strain data from DROPBEAR was produced in the commercial finite element software ABAQUS.The ABAQUS model, shown in Figure 3, consists of 1010 S4R shell elements and is taken as the real ortrue system. SEC data measurements were simulated by averaging the strain of all the elements locatedunder a given SEC. The damage location used in the study is at the root of the beam which is processedby reducing the bending rigidities to θ = 1−a of the elements.

xl

lbc SEC #1 SEC #2 SEC #3 SEC #4 SEC #5

ly

x

y

f(t)

Figure 3. ABAQUS model of DROPBEAR.

To verify the model adaptation stage, a damage case of a = 0.2 was generated and the model adapted.An implicit dynamic analysis was utilized with a 100 N magnitude white noise excitation of 100 Hzbandwidth applied at the tip of the beam. A non-simultaneous harmonic excitation of 20 N at 20 rad/swas also simulated at the tip of the beam to verify the adapted model.

After verifying model adaptation, the MAPOD methodology was assessed. This was done bysubjecting the cantilever to a harmonic excitation with an amplitude of 100 N at 5 rad/s. Two sources ofuncertainties were introduced in the system: one on the force, where its variation is assumed to follow auniform distribution (Bartel, 2005), and one on SEC measurement noise, where its variation is assumed tofollow a Gaussian distribution as established in a previous study (Downey et al., 2018a). Three uncertaintycases were considered, each constructed by combining different uncertainty levels on the input force andsensor measurements: case #1: ±0.5N variation on the load and 10% on measurement noise; case #2:±1N variation on the load and 10% on measurement noise; and case #3: ±1N variation on the load and5% on measurement noise. Damage cases were generated by reducing the bending stiffness randomlybetween a = 0 and a = 0.6. The analysis was conducted using the adapted simplified physical surrogatemodel using 1000 realizations of synthetic data sets produced by the Latin hypercube sampling technique(Haddad et al., 2013). This technique was used as an improvement of the traditional Monte Carlo methodto produce more even independent samples from the uncertainty distributions.

4.2 Model AdaptationFigure 4 plots the results for the model adaptation verification task. Figure 4(a) shows the evolution of theestimated parameter θ (red dashed line) versus its real value (black solid line), where θ represents thebending rigidity. The offset between the estimated and real bending rigidities is attributed to the level ofnoise in the sensor, observable measurement error time histories (Figures 4(b) and (c)), quality of the leastsquares estimate used to convert strain into displacements and velocities, and level of simplification of thephysical surrogate model. Figure 4(b) plots the measurement error between the physical surrogate andmeasured (i.e., from ABAQUS) strains before the adaptation of the surrogate, while the same measurementerror is plotted in Figure 4(c) after the adaptation of the surrogate. The estimation error is significantlyminimized post model adaptation.

Figure 5 plots the first five seconds of results from the harmonic load to further verify the modeladaptation technique, before and after the adaptation of θ . The updated model reproduces strain outputssimilar to that from ABAQUS from the error plotted in Figure 5(b) with the maximum strain error ofSEC #1 reducing from 32.7 µε to 16.4 µε at the steady state, and the maximum strain error of SEC #5reducing from 1.3 µε to 0.8 µε at the steady state. The algorithm was capable of tracking changes inbending rigidity, which can be used as a measure of damage.

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(a)

(b)

(c)

0 5 10 15 20 25 30 35 40 45 50time (s)

0.7

0.8

1

estimation real SEC #1 SEC #2 SEC #3 SEC #4 SEC #5

0 5 10 15 20 25 30 35 40 45 50time (s)

-1

0

1104

0 5 10 15 20 25 30 35 40 45 50time (s)

-1

0

1104

0.9

Figure 4. Verification of model adaptation using white noise excitation: (a) adaptation of parameter θ ;(b) measurement error before model adaptation; and (c) measurement error after model adaptation.

SEC #1 SEC #2 SEC #3 SEC #4 SEC #5

time (s)(a) (b)

0 1 2 3 4 5time (s)

-30

-20

-10

0

10

20

30

0 1 2 3 4 5

-30

-20

-10

0

10

20

30

Figure 5. Verification of model adaptation using harmonic load: (a) measurement error before modeladaptation; and (b) measurement error after model adaptation.

4.3 Model Assisted Probability of Detection-Based EvaluationThe performance of damage detection algorithms for a given DSN configuration was assessed using theproposed methodology using the harmonic load applied at the tip of the beam. Threshold values a, listed inTable 1, are taken as the intercept of the 95% upper confidence interval at a = 0 for each damage detectionalgorithm, therefore minimizing false positives. Figures 6(a-c) are the J−a plots showing the simulatedvalues, linear regressions, and damage detection thresholds under the first uncertainty combination casefor the three damage indicators, respectively. Figure6(d) is the resulting POD curves, with the dashedcurve representing the upper 95% confidence intervals of the linear regression. The smoother slope inFigure 6(d) is from damage indicator J3, which implies a lower performance of the damage indicatorunder uncertainty combination #1. Conversely, damage indicator J2 exhibits the best performance. Twometrics a50/95 (bottom dash dotted line) and a90/95 (top dash dotted line) are specified for assessing theparticular damage detection algorithms. These two metrics represents there is a 50% probability with 95%

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confidence detecting the change of bending rigidity higher than a50/95 and 90% probability with 95%confidence of detecting the change of bending rigidity higher than a90/95, respectively. Metrics a50/95and a90/95, listed in Table 1, are used for quantifying the performance of a given DSN configuration andits associated damage detection algorithm under different uncertainty cases. Damage indicator J2 hasa 90% probability with 95% confidence of detecting damage higher than 43% under uncertainty case#1. However, damage indicator J3 exhibits better performance under uncertainty cases #2 and #3 with a90% probability with 95% confidence of detecting damage higher than 89% and 85%, respectively. Thus,indices J2 and J3 are selected to perform the evaluation of DSN configurations under uncertainty case #1and cases #2-3, respectively, in what follows.

0 0.2 0.4 0.6degree of damage, a

0.02

0.025

0.03

0.035

0.04

0.045

resp

onse

, J3

data pointslinear regression95% confidence intervaldetection threshold

(a) (b)

(c) (d)

a

0 0.6

90/95a50/95

0.2 0.4

Figure 6. Uncertainty case #1: (a) J1−a plot; (b) J2−a plot; (c) J3−a plot; and (d) POD curves andthe upper (conservative) 95% confidence interval (CI) of the linear regression for J1, J2 and J3.

Table 1. Probability of detection results under each uncertainty case

Uncertainty case Damage indicator Detection threshold a a50/95 a90/95

Case #1J1 1.18e-06 0.24 0.44J2 8.38e-04 0.23 0.43J3 0.03 0.25 0.46

Case #2J1 1.60e-06 0.49 0.92J2 1.5e-3 0.47 0.89J3 0.04 0.47 0.89

Case #3J1 8.66e-7 0.48 0.90J2 1.2e-3 0.47 0.89J3 0.03 0.45 0.85

4.3.1 Model-Assisted Probability of Detection-Based Evaluation for DSN DesignHere, the proposed MAPOD-based technique is used to evaluate the performance of DSN configurations.The performance of each configuration is evaluated and ranked using the selected damage indicators(J2 for uncertainty case #1 and J3 for uncertainty cases #2-3). Figures 8 show 10 arbitrarily selected

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configurations, which include various numbers, sizes, and locations of sensors in both symmetric orasymmetric configurations. These configurations were selected to cover different part of the beam,some providing a higher resolution for strain close to damage (e.g., configurations 1 and 7), higherresolution much farther to damage (e.g., configuration 10), higher resolution over a longer lengths (e.g.,configurations 5, 6), high resolution equally spaced sensor measurements (e.g., configuration 3), mixtureof high resolution measurement locations (e.g., configurations 2, 4), and lower resolution obtained withlarger sensors (e.g., configurations 8, 9). SEC measurements from each surrogate model are taken as theaveraged strain over the reparametrized elements (shown as dashed lines).

The performance of each configuration is investigated under uncertainty case #1. Configurationsillustrated in the Figure 7(a) are ranked by performance using the a90/95 metric established by the surrogatemodel. Performance results obtained from the surrogate are benchmarked against those obtained usingABAQUS. In ABAQUS, damage cases at the root of the beam were generated at levels ranging from 0to 0.6 at 0.05 steps and conducting 100 realizations under each damage case. The POD curves obtainedfor each DSN configuration from the surrogate model and ABAQUS are plotted in Figures 7(b) and (c),respectively. While the ranking of configurations produced by the surrogate model and ABACUS areidentical, values for a90/95 differ (see Figure 8). This is investigated further in what follows.

(b) (c)

configuration 1 configuration 2

configuration 6

configuration 3

configuration 7 configuration 9configuration 8

configuration 4 configuration 5

configuration 10(a)

Figure 7. Uncertainty case #1: (a) 10 DSN configurations with sensors of different numbers and sizes;(b) 90%/95% POD surface plots obtained from the surrogate model; and (c) 90%/95% POD surface plotsobtained from ABAQUS.

The capability of the proposed method at ranking DSN configurations is investigated under differentuncertainty cases. Figure 8 plots the results for the a50/95 and a90/95 metrics obtained using the surrogatemodel and ABAQUS. Results show that the produced ranking of configurations is consistent between boththe surrogate model and ABAQUS. Remark that the ranking of results provides insights on the efficiencyof each sensor configurations. For instance, configurations 1-4 consistently perform best under the threeuncertainty cases attributable to a general concentration of measurements closer to damage, configuration5’s performance degrades over higher load noise levels attributable to the availability of measurementscloser to that load, and configuration 10 consistently provides the worst performance attributable to thelack of available measurements close to damage. While these results are intuitive for a cantilever beam,they could teach more important causalities for more complex geometries. This is left to future work.One can also observe that the two metrics a50/95 and a90/95 are different between the surrogate model andABAQUS, but yet appear to be mappable through a linear or first degree nonlinear function. This meansthat the damage threshold identified by the surrogate model cannot be used for field application without acalibration.

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(a) (b) (c)configuration # configuration # configuration #

Figure 8. Metrics a50/95 and a90/95 under uncertainty: (a) case #1; (b) case #2; and (c) case #3.

5 CONCLUSIONIn this paper, a methodology for the validation of sensor networks used for structural health monitoringwas presented. The methodology consisted of constructing a physical surrogate model based on agiven DSN configuration, adapting the surrogate model using field data and sliding mode theory, andusing model-assisted probability of detection (MAPOD) to study the performance of various networkconfiguration and damage detection algorithm under various sources of uncertainty.

Numerical simulations were conducted to verify and validate the proposed methodology on a straight-forward example consisting of an elastically restrained beam equipped with a dense sensor network(DSN) measuring strain. The verification of the model adaptation technique showed that sliding modetheory could be used to update the physical surrogate using measurements, where measurements weresynthetically produced using a high-resolution finite element model constructed in ABAQUS. The updatedphysics surrogate was then used to evaluate the performance of various damage detection algorithms.After, the surrogate was used to investigate the performance of various sensor configurations at detectingdamage at the root of the cantilever. Results showed that the proposed method successfully rankedthe performance of various DSN configurations. Such ranking could be critical is designing efficientDSNs. Results also showed that the damage detection threshold would need to be calibrated to enablesurrogate-based damage detection and quantification in the field.

ACKNOWLEDGMENTSThe work presented in this paper is partially funded by a fellowship award from the American Societyfor Nondestructive Testing (ASNT) and by the Air Force Office of Scientific Research (AFOSR) underaward number FA9550-17-1-0131. Their support is gratefully acknowledged. Any opinions, findings, andconclusions or recommendations expressed in this material are those of the authors and do not necessarilyreflect the views of the sponsors.

REFERENCESAldrin, J. C., Medina, E. A., Lindgren, E. A., Buynak, C. F., Knopp, J. S., Thompson, D. O., and Chimenti,

D. E. (2011). Case Studies for Model-assisted Probabilistic Reliability Assessment for Structural HealthMonitoring. volume 1335, pages 1589–1596. American Institute of Physics Conference Proceedings.

Amsallem, D. and Farhat, C. (2008). Interpolation method for adapting reduced-order models andapplication to aeroelasticity. AIAA Journal, 46(7):1803–1813.

Bartel, T. (2005). Uncertainty in NIST force measurements. Journal of Research of the National Instituteof Standards and Technology, 110(6):589.

Benner, P., Gugercin, S., and Willcox, K. (2015). A survey of projection-based model reduction methodsfor parametric dynamical systems. SIAM Review, 57(4):483–531.

Benner, P., Ohlberger, M., Cohen, A., and Wilcox, K. E. (2017). Model Reduction and Approximation:Theory and Algorithms. SIAM-Society for Industrial & Applied Mathematics.

Cancelli, A., Laflamme, S., Alipour, A., Sritharan, S., and Ubertini, F. (2019). Vibration-based damagelocalization and quantification in a pretensioned concrete girder using stochastic subspace identificationand particle swarm model updating. Structural Health Monitoring, page 147592171882001.

11/13

Cui, R. and Lanza di Scalea, F. (2019). On the identification of the elastic properties of composites byultrasonic guided waves and optimization algorithm. Composite Structures, 223:110969.

Downey, A., Lui, Y. H., Hu, C., Laflamme, S., and Hu, S. (2019). Physics-based prognostics of lithium-ionbattery using non-linear least squares with dynamic bounds. Reliability Engineering and System Safety,182:1–12.

Downey, A., Sadoughi, M., Laflamme, S., and Hu, C. (2018a). Fusion of Sensor Geometry into AdditiveStrain Fields Measured with Sensing Skin. Smart Materials and Structures, 27(7):75033.

Downey, A., Sadoughi, M., Laflamme, S., and Hu, C. (2018b). Incipient Damage Detection for LargeArea Structures Monitored With a Network of Soft Elastomeric Capacitors Using Relative Entropy.IEEE Sensors Journal, 18(21):8827–8834.

Du, X., Gurrala, P., Leifsson, L., Song, J., Meeker, W., Roberts, R., Koziel, S., and Tesfahunegn, Y.(2018a). Stochastic-expansions-based model-assisted probability of detection analysis of the spherically-void-defect benchmark problem. In Lecture Notes in Computer Science, pages 593–603. SpringerInternational Publishing.

Du, X., Yan, J., Laflamme, S., Leifsson, L., Tesfahunegn, Y., and Koziel, S. (2018b). Model-AssistedProbability of Detection for Structural Health Monitoring of Flat Plates. In Lecture Notes in ComputerScience, volume 10861 LNCS, pages 618–628. Springer International Publishing.

Forsyth, D. S. (2016). Structural Health Monitoring and Probability of Detection Estimation. AIPPublishing LLC.

Gratiet, L. L., Iooss, B., Blatman, G., Browne, T., Cordeiro, S., and Goursaud, B. (2016). Modelassisted probability of detection curves: New statistical tools and progressive methodology. Journal ofNondestructive Evaluation, 36(1).

Haddad, R. E., Fakhereddine, R., Lecot, C., and Venkiteswaran, G. (2013). Extended Latin HypercubeSampling for Integration and Simulation. In Monte Carlo and Quasi-Monte Carlo Methods 2012,pages 317–330. Springer Berlin Heidelberg.

Joyce, B., Dodson, J., Laflamme, S., and Hong, J. (2018a). An experimental test bed for developinghigh-rate structural health monitoring methods. Shock and Vibration, 2018:1–10.

Joyce, B. S., Hong, J., Dodson, J. C., Wolfson, J. C., and Laflamme, S. (2018b). Adaptive Observers forStructural Health Monitoring of High-Rate, Time-Varying Dynamic Systems. In Structural HealthMonitoring, Photogrammetry & {DIC}, Volume 6, pages 109–119. Springer International Publishing.

Laflamme, S., Saleem, H. S., Vasan, B. K., Geiger, R. L., Chen, D., Kessler, M. R., and Rajan, K.(2013). Soft Elastomeric Capacitor Network For Strain Sensing Over Large Surfaces. {IEEE}/{ASME}Transactions On Mechatronics, 18(6):1647–1654.

Laflamme, S., Ubertini, F., Saleem, H., D’Alessandro, A., Downey, A., Ceylan, H., and Materazzi, A. L.(2015). Dynamic Characterization Of A Soft Elastomeric Capacitor For Structural Health Monitoring.Journal Of Structural Engineering, 141(8):4014186.

Li, D., Dong, X., and Wang, Y. (2018). Model updating using sum of squares (SOS) optimization tomminimize modal dynamic residuals. Structural Control and Health Monitoring, 25(12):e2263.

Liao, Y., Kiremidjian, A. S., Rajagopal, R., and Loh, C.-H. (2019). Structural Damage Detection andLocalization with Unknown Postdamage Feature Distribution Using Sequential Change-Point DetectionMethod. Journal of Aerospace Engineering, 32(2):4018149.

Lu, Y. and Michaels, J. (2009). Feature extraction and sensor fusion for ultrasonic structural healthmonitoring under changing environmental conditions. IEEE Sensors Journal, 9(11):1462–1471.

Lynch, J. P., Farrar, C. R., and Michaels, J. E. (2016). Structural health monitoring: Technologicaladvances to practical implementations [scanning the issue]. Proceedings of the IEEE, 104(8):1508–1512.

Memmolo, V., Ricci, F., Maio, L., Boffa, N. D., and Monaco, E. (2016). Model assisted probabilityof detection for a guided waves based SHM technique. In Kundu, T., editor, Health Monitoring ofStructural and Biological Systems 2016. SPIE.

Moriot, J., Quaegebeur, N., Le Duff, A., and Masson, P. (2018). A Model-Based Approach for StatisticalAssessment of Detection and Localization Performance of Guided Wave–Based Imaging Techniques.Structural Health Monitoring, 17(6):1460–1472.

Peherstorfer, B., Willcox, K., and Gunzburger, M. (2018). Survey of multifidelity methods in uncertaintypropagation, inference, and optimization. SIAM Review, 60(3):550–591.

Ubertini, F., Laflamme, S., Chatzi, E., Glisic, B., and Magalhaes, F. (2017). Dense Sensor Networks for

12/13

Mesoscale {SHM}: Innovations in Sensing Technologies and Signal Processing. Measurement Scienceand Technology, 28(4):40103.

Yan, J., Du, X., Downey, A., Cancelli, A., Laflamme, S., Leifsson, L., Chen, A. A., Ubertini, F., Du, X.,Laflamme, S., and Leifsson, L. (2018). Surrogate Model for Condition Assessment of Structures UsingA Dense Sensor Network. In Sohn, H., editor, Sensors and Smart Structures Technologies for Civil,Mechanical, and Aerospace Systems 2018, volume 10598. SPIE.

Yan, J., Du, X., Laflamme, S., Leifsson, L., Hu, C., and Chen, A. (2019). Model-Assisted Validationof A Strain-Based Dense Sensor Network. In Wang, K.-W., Sohn, H., Huang, H., and Lynch, J. P.,editors, Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems2019, page 80. SPIE.

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