Structural damage identification using piezoelectric

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Mthembu, L., Marwala, T., Friswell, M.I., and Adhikari, S., 2011, “Model selection in finite element model updating using the Bayesian evidence statistic,” ...
Structural Damage Identification Using Piezoelectric Impedance and Bayesian Inference Q. Shuai, K. Zhou and J. Tang Department of Mechanical Engineering University of Connecticut Storrs, CT06269, USA Phone: (860) 486-5911, Email: [email protected] ABSTRACT Structural damage identification is a challenging subject in the structural health monitoring research. The piezoelectric impedance-based damage identification, which usually utilizes the matrix inverse-based optimization, may in theory identify the damage location and damage severity. However, the sensitivity matrix is oftentimes ill-conditioned in practice, since the number of unknowns may far exceed the useful measurements/inputs. In this research, a new method based on intelligent inference framework for damage identification is presented. Bayesian inference is used to directly predict damage location and severity using impedance measurement through forward prediction and comparison. Gaussian process is employed to enrich the forward analysis result, thereby reducing computational cost. Case study is carried out to illustrate the identification performance. Key words: damage identification, impedance, Bayesian inference, Gaussian process

1. INTRODUCTION Structural health monitoring (SHM) is a process of implementing damage detection and damage identification strategy for engineering structures. In general, SHM includes two parts: damage detection and damage identification. The purpose of damage detection is to find out whether there is a damage occurrence. Based on different sensing mechanisms, there are different detection methods, such as vibration-based method (Prandey, et al, 1991; Salawu, 1997), lamb wav-based method (Rose, 1995; Giurgiutiu, 2005), impedance-based method (Giurgiutiu, 1999; Park et al, 2003; Wang and Tang, 2010) and some other methods. Among them, the piezoelectric impedance-based method has received attentions recently, since it has shown some promising aspects in including easy integration, high detection sensitivity, and large detection/monitoring range. Compared to damage detection, damage identification is a more challenging task. Its objective is to figure out where the damage is, how severe the damage is and even what kind of damage it is. For impedance-based method, damage identification can be divided into two major categories: data-oriented methods and model-based methods. In data-oriented methods, various damage indices are frequently used in the identification of damage location/severity/types (Park, et al, 1999; Zagrai and Giurgiutiu, 2001; Tseng and Naidu, 2002; Raju, 1998). These methods are mostly based on phenomenological characterizations of the impedance, and oftentimes cannot accurately relate the damage indices to the changes of local structural properties such as mass, stiffness, and damping ratio, etc. Alternatively, in model-based methods, damage identification can be achieved by combining the impedance signatures with mathematical models of the structure and the sensor. Structural damage can be modeled as change of a local structural property in finite element models or spectral element models (Yong and Hao, 2003; Wang and Tang, 2009). The changes of the impedance signature (e.g., resonant frequencies or response amplitudes) are then used as input to an inverse analysis process to predict the location and severity of the damage. Currently, most of such model-based methods use matrix inverse-based optimization which requires to solve the inverse of the sensitivity matrix. In most cases, however, the sensitivity matrix is ill-

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conditioned (Kim and Wang, 2014), because the useful impedance information is far less than that is needed to uniquely find the unknowns (i.e. damage location and severity). The objective of this research is to develop a robust algorithm to conduct damage identification based on piezoelectric impedance/admittance information. In particular, this proposed algorithm is built upon the Bayesian inference framework, which involves employing forward analysis-based instead of inversionbased identification procedures. Bayesian inference has been employed in model updating in the past. It can specify the model parameters with prior information in the form of probability density function (PDF), which may be viewed as imposing soft physical constraints to enable a unique and stable solution. Moreover, this approach allows the computation of any type of statistics of the model parameters to be identified (Antoni, et al, 2011). Several pioneering studies first employed Bayesian probabilistic framework to estimate the model parameters, and clearly demonstrated its applicability for simplified numerical models (Katafygiotis and Beck, 1998a; Katafygiotis and Beck, 1998b). Subsequent research has illustrated the effectiveness of this approach in the field of robust structural health assessment (Katafygiotis, et al, 1998). More recently, a wide range of research activities utilizing this approach have been explored, such as prognosis of fatigue crack growth, model selection, etc (Zarate, et al, 2012; Mthembu, et al, 2011). In this type of approach, the probability density function (PDF) of uncertain model parameters can be first assumed based on the prior knowledge and engineering judgment, and then can be updated using conditional PDF derived from sets of measured data based on Bayesian theorem. Under this framework, one may avoid the aforementioned matrix inversion. Nevertheless, there is a major challenge in Bayesian inference that is the high computational cost. Since the Bayesian inference is sampling-based algorithm, it requires repeated evaluation (e.g., finite element analysis (FEA)) under a large parameter space. Here in this research we propose to incorporate the Gaussian process to enrich the FEA results and then alleviate the computational burden. A Gaussian Process is a collection of random variables, any finite number of which have (consistent) joint Gaussian distributions (Rasmussen and Williams, 2006). Gaussian process has been used in many areas, such as statistics, signal process and machine learning (Lebarbier, 2005; Rasmussen and Williams, 2006). Recently Gaussian process is employed to emulate the frequency response in the field of structural dynamics analysis (DiazDelaO and Adhikari, 2010; Xia and Tang 2011). In this application, we use Gaussian Process for the purpose of regression. The rest of this paper is organized as follows. In Section 2, the piezoelectric impedance modeling, the general Bayesian inference framework and Gaussian process are outlined. In Section 3, a case study, damage identification of a clamped-clamped plate, is performed, to validate this proposed methodology. Section 4 summarizes the research progress.

2. APPROACH FORMULATION In this section, we first outline the modeling of a piezoelectric admittance-based damage detection scheme based on finite element analysis. We then formulate the Bayesian inference framework for damage identification, incorporated with Gaussian process. 2.1 Piezoelectric admittance modeling The following formulation is built upon the element basis. Without loss of generality, this formulation can be directly extended to the entire model with multiple elements. The constitutive relation for a piezoelectric transducer is given as σ = E p ε + hD (1) E = hT ε + β 33 D

(2)

where σ and ε are, respectively, are the stress and strain vectors ( 6 ×1 ). E p is the elastic matrix of piezoelectric transducer of dimension ( 6 × 6 ). h accounts for the electro-mechanical coupling property of the transducer, which is shown as [ − h31 , − h32 , 0, 0, 0, 0]T . D represents the electrical displacement and

E represents the electrical field. For a structural element attached with a piezoelectric transducer, following the general finite element modeling procedure, we can derive the governing equation as M ( e )δ&&( e ) + C( e )δ&( e ) + K ( e )δ ( e ) + k12( e ) Q = F( e )

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(3)

T k 22 Q + k12( e )δ ( e ) = U

(4)

where M ( e ) is the elemental mass matrix, K ( e ) is the elemental stiffness matrix, C( e ) is damping, δ ( e ) is the displacement vector, k22 is the inverse of the capacitance of the PZT transducer, Q is the charge accumulated, and k12( e ) is the coupling vector

∫ =

V

BT hdV

(5) A B is stress-strain transformation matrix which is related to the shape function, A is the surface area of the piezoelectric transducer, and h is the thickness of the transducer. The piezoelectric admittance-based damage detection can be modeled with finite element approach using Equations (3) and (4). k12( e )

2.2 Bayesian inference framework When applying Bayesian inference for structural damage identification, the hypothesis θ is interpreted as the vector of parameters that need to be identified, i.e. damage location and damage severity. D denotes the measured signature, which in this study is the electrical admittance of piezoelectric transducer. M denotes modeling assumptions, reflecting the experience and knowledge of the engineer. We then have p ( D | θ , M ) p (θ | M ) (6) p (θ | D, M ) = ∫ p( D | θ , M ) p(θ | M )dθ The prior distribution p(θ | M ) expresses the initial knowledge of concerned parameters, e.g., stiffness, mass, damage location. The choice of this distribution depends on how much information of the system is known. Since in most cases, there is no prior information on the damage scenario, it is reasonable to assume a multivariate uniform distribution. The posterior distribution p(θ | D, M ) indicates the updated knowledge of the parameters θ conditional on the prior knowledge and measured admittance information. The likelihood function p( D | θ , M ) is used to evaluate the agreement between the measurements and associated model output. Specifically, considering that the uncertainties exist in real measurement, here we define the likelihood function as a multivariate normal distribution to conduct the screen of model output over θ space.

p( D | θ ) =

1 (2π ) k | Σ |

e



( D − D (θ ))T Σ−1 ( D − D (θ ))) 2

(7)

where D is a measured admittance vector of length k , and D(θ ) is the model output parameterized by θ . Σ is the covariance matrix of D. Under this framework, the damage status of investigated structure can be eventually identified. It is worth mentioning that each individual model output is produced based on the finite element-based piezoelectric admittance analysis. As the number of repeated run for model output is extremely large, the computational cost is a significant challenge. Here we propose to incorporate Gaussian process to reduce the computational burden. 2.3 Gaussian process for regression The basic idea behind Gaussian process is to extend the discrete multivariate distribution on a finite dimensional space to a random continuous function defined on an infinite-dimensional space. For a system, we have m known input-output relations i.e., input X ( x1 , x2 , ⋅⋅⋅ xm ) and output Y ( y1 , y2 , ⋅⋅⋅ ym ) . We assume the distribution of the corresponding outputs Y to be a multivariate Gaussian, Y ~ N (m( X ), k ( X , X )) (8) The multivariate Gaussian distribution of a discrete subset of the range of the function can be extended to its entire continuous range. Thus, the multivariate Gaussian distribution for a finite-dimensional case is generalized to an infinite-dimensional case. So the predictions of Y * at target input points X * can be presented together with the observed data points in the standard form,

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⎡Y ⎤ ⎡ m( X ) ⎤ ⎡ k ( X , X ) k ( X , X * ) ⎤ , ⎢ * ⎥ ~ N (⎢ * ⎥ ⎢ * * * ⎥ ⎣Y ⎦ ⎣ m( X ) ⎦ ⎣⎢ k ( X , X ) k ( X , X ) ⎦⎥

(9)

It can be shown that the probability distribution of Y * at the target input points X * conditional on the data set ( X , Y ) is also a multivariate Gaussian (Rasmussen and Williams, 2006), p(Y * | X * , X , Y ) ~ N (m( X * ) + k ( X * , X )[k ( X , X ) + σ n2 I ]− (Y − m( X )), k ( X * , X * ) - k ( X * , X )[k ( X , X ) + σ n2 I ]− k ( X , X * ))

(10)

In the above expression, the expected value of Y * is m( X * ) + k ( X * , X )[ k ( X , X ) + σ n2 I ]− (Y − m( X )) and

the corresponding covariance matrix is k ( X * , X * ) - k ( X * , X )[ k ( X , X ) + σ n2 I ]− k ( X , X * ) . As we can see, Gaussian process can be fully specified by the mean function m( X ) and the covariance function k ( X , X ) . The mean function m( X ) is an assumed function that returns the expected value of a system. The covariance function k ( X , X ) is an assumed function to relate the observations at different input points. A wide body of conventionally used mean and covariance functions can be found in the literature (Rasmussen and Williams, 2006). In this research, we let each input point have two components (damage location L and severity S), and then the covariance function is chosen as k ( X , X ') = σ e 2 f

1 ( L − L ')2 ( S − S ')2 − ( + ) l1 l2 2

(11)

where (σ f , l1 , l2 ) is the hyper-parameters need to be determined.

3. CASE STUDY In this section, we proceed to carrying out the damage identification of a plate structure based on the proposed methodology. 3.1 Problem formulation The formulated problem is to identify the location and severity of small-size damage on a plate. The host structure is a clamped-clamped aluminum plate (size 0.610m × 0.508m × 0.004826m , density: 2780 Kg / m 3 , Young’s modulus: 73 Gpa) attached with piezoelectric transducer (Piezoelectric constant h31 and h32 : -140 ×10-12 m / V , β33 : 25 ×10−3 Vm / N ), as shown in Figure 1. In the corresponding finite element model, the plate is divided into 2688 ( 56 × 48 × 1 ) elements. It is worth noting that the piezoelectric transducer is a square one with edge length 0.004488m, which covers 25( 5 × 5 ) plate elements. A small resistance (100 ohm) is connected to the piezoelectric transducer to extract the current information.

For Bayesian inference, the hypothesis θ is selected as damage location L and damage severity S. Each element represents a possible damage location. The damage severity is modeled as density increase in each element. As no prior knowledge of possible damage location and damage severity exists, the prior distribution is chosen to be bivariate uniform distribution. The samples parameterized from such distribution are equal to the product of the possible damage severity candidates and damage location candidates. In real situation, the damage severity candidates are continuously sampled within certain range and damage can probably occur in each element of plate, which yields an extremely large sample number from prior distribution. Due to the essence of Bayesian inference, the repeated evaluation of the likelihood function corresponding to such samples using pure Monte Carlo simulation is computationally prohibitive, since each evaluation of likelihood function depends on the derivation of the associated admittance information by the harmonic forced response analysis over a range of significant frequencies, which is already costly. For the purpose of illustrating the effectiveness of proposed methodology with saved computational cost, we considered sib-region of plate where damage is likely to occur (including 165 elements as shown in Figure 2). Moreover, the damage severity is modeled as density increasing from 2 times to 10 times which has 9 levels in total.

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The admittance around the 31st natural frequency (1576.6 Hz) of the plate is taken into account (Figure 3). In each admittance curve, 50 frequency points are recorded. Taking uncertainties into account, the 0.1% admittance measurement standard deviation is introduced to establish the likelihood function. In this research the measured signature D is chosen according to the modal assurance criterion, which is used to indicate how close two admittance curves are,

D=

A1 ' A2 A2 A2

(12)

To facilitate the numerical analysis, we assume damage occurs in the single individual element (the 72th element) with a 5 times density increase (severity level 5), as shown in Figure 2. 3.2 Result demonstration For further alleviating the computational burden, we only conduct the finite element analysis to obtain the admittance information under three damage severity levels (2, 6, 10). We then employ Gaussian process to interpolate and predict the model assurance criterion under other damage severity candidates. The hyper-parameters used in this case is (1, 3.8, 3.8). Figure 4 shows the input data for Gaussian process. Figure 5 shows the corresponding outputs. From Figure 4 and Figure 5, we can observe that with Gaussian process, the surface becomes smoother and the data is enriched from 165 × 3 points to 165 × 9 points. In order to illustrate the accuracy of Gaussian process, the match of the input, Gaussian process output and actual data value from finite element analysis at one damage location is plotted in Figure 6. We can see that based on a small number of observations (at 3 damage severity level), the Gaussian process can properly predict the observations at other damage severity levels.

After all the model assurance criterion values under the full parameter space are acquired, we insert them into the Bayesian inference algorithm to evaluate the likelihood function and then obtain the posterior distribution. In order to have a clear observation, here we plot the derived probability distribution at different damage severity levels separately as shown in Figure 7. Each figure shows the probability of damage location candidates under the same damage severity. We can find that the Bayesian inference can predict possible damage locations. Table 1 gives the group of most probable damage parameters. We can observe that they have similar probabilities while the actual damage parameter has the 7th highest probability. It is worth mentioning that here in this set of results, only the admittance information around one natural frequency is taken into account. In order to improve the identification performance, we then incorporate another two sets of admittance information, i.e., one around the 24th natural frequency (1207.8 Hz) of the plate, the other around the 34th natural frequency (1804.4 Hz). Figure 8 shows the probability distribution, where we can find the probable damage area is narrowed down to the actual scenario. Clearly, with more information, this proposed approach can pinpoint the damage location/severity.

4. CONCLUDING REMARKS In this research, we present a Bayesian probabilistic approach to localize and identify the small-size damage of a plate structure based on the finite element piezoelectric admittance analysis. Gaussian process is employed to enrich the outputs of finite element analysis and then alleviate the computational burden. This proposed methodology can yield accurate and robust damage identification. More measurement information generally enhances the damage identification performance. The future work may concentrate on how to choose the admittance information to obtain the optimized identification performance and on experimental validation.

5. ACKNOWLEDGEMENT This research is supported by the AFOSR under grant FA9550-14-1-0384.

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REFERENCES Antoni, J., Fessel, P., and Zhang, E.L., 2011, “A comprehensive Bayesian approach for model updating and quantification of modeling errors,” Probabilistic Engineering Mechanics, V26, 550-560. DiazDelaO, F.A. and Adhikari, S., 2010, “Structural dynamics analysis using Gaussian process emulators,” Engineering Computations, 27, pp.580-605 Giurgiutiu, V., 1999, “Experimental investigation of E/M impedance health monitoring for spot-welded structural points,” Journal of Intelligent Material Systems and Structures, V10, 802-812. Giurgiutiu, V. 2005, "Tuned Lamb Wave Excitation and Detection with Piezoelectric Wafer Active Sensors for Structural Health Monitoring", Journal of Intelligent Material Systems and Structures, vol. 16, no. 4, pp. 291-305. Katafygiotis, L.S., and Beck, J.L., 1998a, “Updating models and their uncertainties. I: Bayesian statistical framework,” Journal of Engineering Mechanics, V124, 455-461. Katafygiotis, L.S., and Beck, J.L., 1998b, “Updating models and their uncertainties. II: Model identifiability,” Journal of Engineering Mechanics, V124, 455-461. Katafygiotis, L.S., Papadimitriou, C., and Lam, H.-F., 1998, “A probabilistic approach to structural model updating,” Soil Dynamics and Earthquake Engineering, V17, 495-507. Kim, J. and Wang, K.W., 2014, “Impedance-based damage identification enhancement via tunable piezoelectric circuitry”, Proc. SPIE 9061, Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems Lebarbier, E., 2005, “Detecting multiple change-points in the mean of Gaussian process by model selection,” Signal Processing, 85, pp.717-736 Mthembu, L., Marwala, T., Friswell, M.I., and Adhikari, S., 2011, “Model selection in finite element model updating using the Bayesian evidence statistic,” Mechanical System and Signal Processing, V25, 23992412. Pandey, A.K., Biswas, M. and Samman, M.M., 1991, “Damage detection from changes in curvature mode shapes,” Journal of Sound and Vibration, V145, 321-332 Park, G., Sohn, H., Farrar, C.R., and Inman, D. J., 2003, “Overview of piezoelectric impedance-based health monitoring and path forward,” The Shock and Vibration Digest, V35, 451-463. Park, G., Kabeya, K., Cudney, H. H. and Inman, D. J., 1999, “Impedance-based Structural Health Monitoring for Temperature Varying Applications”, JSME International Journal Series A Solid Mechanics and Material Engineering, vol. 42, no.2, pp.249–258 Raju, V., Park, G. and Cudney, H. H., 1998, “Impedance-based Health Monitoring Technique of Composite Reinforced Structures”, In: Proceedings of the Ninth International Conference on Adaptive Structures and Technologies, pp.448–457 Rasmussen, C.E. and Williams, C.K.I., 2006, Gaussian process for machine learning, MIT press Rose, J. L., 1995 “Recent Advances in Guided Wave NDE,” 1995 IEEE Ultrasonics Symposium, 761-770 Salawu, O.S., 1997, “Detection of structural damage through changes in frequency: a review,” Engineering Structures, V9, 718-723 Tseng, K. K-H. and Naidu, A. S. K. , 2002 “Non-parametric Damage Detection and Characterization Using Smart Piezoceramic Materials”, Smart Materials and Structures, vol. 11, no.3, pp.317–329 Wang, X. and Tang, J., 2009, “Damage Identification Using Piezoelectric Impedance Approach and Spectral Element Method”, Journal of Intelligent Material Systems and Structures, vol. 20, no.8, pp.907–921, Wang, X. and Tang, J., 2010, “Damage detection using piezoelectric admittance approach with inductive circuitry,” Journal of Intelligent Material Systems and Structures, V21, 667-676. Xia, Z. and Tang, J., 2011, “Uncertainty analysis of structural dynamics by using two-level Gaussian Processes,” Proceedings of ASME-IMECE, 891-898 Yong, X. and Hao, H., 2003, “Statistical damage identification of structures with frequency changes”, Journal of Sound and Vibration, vol. 263, no.4, pp.853–870 Zagrai, A.N. and Giurgiutiu, V., 2001, “Electro-Mechanical Impedance Method for crack detection in thin plates”, Journal of Intelligent Material Systems and Structures, vol. 12, no.10, pp.709–718 Zarate, B.A., Caicedo, J.M., Yu, J., and Ziehl, P., 2012, “Bayesian model updating and prognosis of fatigue crack growth,” Engineering Structures, V45, 2125-2134.

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PZT 117 mm 134 mm

Small resistance PZ

plate

Figure 1. Finite element model of the clamped-clamped plate

Damage

PZT

33

34

Possible damage locations

35

..

64

65

.. 3 4

3 5

3 6

2

6 ..

5

..

2

3

6 6

3

3 3

Figure 2. possible damage locations and corresponding element index

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Frequency (Hz)

Figure 3. Admittance around 31st natural frequency of plate 1.015 1.01 1.005 1 0.995 0.99 0.985 0.98 0.975 10

6

Damage severity

2

0

50

200

150

100

Damage location

Figure 4 Input data for Gaussian process

1.015 1.01 1.005 1 0.995 0.99 0.985 0.98 0.975 10

8

6

4

Damage severity

2

0

50

100

150

200

Damage location

Figure 5. Gaussian process outputs

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1.005 input data GP output FEA results

1

D

0.995 0.99 0.985 0.98 0.975 2

3

4

5

6 Severity

7

8

9

10

Figure 6. Gaussian process output at one damage location 4 3

Probability

2 1 0 6 4

Y coordinate

2 0

5

0

10

20

15

25

30

35

X coordinate

(a) Damage severity 5 4 3

Probability

2 1 0 5 4

Y coordinate

3 2 1

0

10

20

30

40

X coordinate

(b)Damage severity 10 Figure 7. Probability distribution under different damage severity

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10 8

Probability

6 4 2 0 5 4

Y coordinate

3 2 1

0

10

20

30

40

X coordinate

Figure 8. Improved probability distribution (severity 5)

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Table 1. Probability for the first couple of damage parameters Location Severity Probability 39

6

3.1551

143

5

3.1550

24

3

3.1550

49

2

3.1547

129

5

3.1546

58

2

3.1546

72

5

3.1541

Table 2. Improved probability for the first couple of damage parameters Location Severity Probability 72

5

10.18

36

2

10.12

159

2

9.99

15

3

9.93

105

5

9.88

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