P Blanloeuil
email: p.blanloeuil@uq.edu.au

A Meziane

A N Norris

M Renier

M Veidt

NUMERICAL COMPUTATION OF THE NONLINEAR FAR FIELD OF ULTRASONIC WAVES SCATTERED BY CLOSED CRACKS OF VARIOUS ORIENTATIONS

Keywords: Nonlinear acoustics, Scattering, Higher Harmonics, Finite Elements

The directivity patterns of the higher harmonics generated by the interaction between a bulk wave and a closed crack are obtained in the far field. A Finite Element (FE) model gives the near field solution which is then analytically propagated with expressions based on Hankel functions. The crack is modeled by an interface of unilateral contact with Coulomb's friction which takes into account a compression pre-stress σ 0 that closes the crack. The mode conversion occurring at the crack investigated. Moreover, the diffracted fields of the generated higher harmonics are compared to those of the fundamental harmonic.

INTRODUCTION

The evaluation of damage at an early stage of fracture is relevant in many technologies. Ultrasonic methods based on linear wave scattering are efficient for detecting defects and characterizing material elasticity, but are less sensitive to micro-cracks or closed cracks. However, if the excitation amplitude is sufficient, the latter can behave nonlinearly because of contact dynamics occuring between the faces of the crack. This effect, called Contact Acoustic Nonlinearity (CAN) [START_REF] Solodov | CAN: an example of nonclassical acoustic nonlinearity in solids[END_REF], has potential for characterization of closed cracks and is of great interest for the application of Structural Health Monitoring (SHM) using nonlinear ultrasonics.

Like classical nonlinearities [START_REF] Zarembo | Nonlinear phenomena in the propagation of elastic waves in solids[END_REF], CAN generates higher harmonics [START_REF] Buck | Acoustic harmonic generation at unbonded interfaces and fatigue cracks[END_REF]. This spectral enrichment has been largely studied on contacting interfaces [START_REF] Richardson | Harmonic generation at an unbonded interface -I Planar interface between semiinfinite elastic media[END_REF][START_REF] Biwa | Quantitative evaluation of harmonic generation at contacting interface[END_REF] and it has been shown that the nonlinear response is dependent on the normal stress applied on the interface. Higher harmonics generation has also been applied to closed cracks [START_REF] Lee | Experimental investigation of nonlinear acoustic effect at crack[END_REF], which demonstrated its potential to reveal this kind of defect.

To set up a SHM method based on nonlinear acoustics, a good understanding of the nonlinear effects is necessary [START_REF] Blanloeuil | Numerical study of nonlinear interaction between a crack and elastic waves under an oblique incidence[END_REF] as well as the propagation of the higher harmonics in the surrounding media. Moreover, if the crack is tilted (e.g. oblique incidence), mode conversions will occur, and each wave may carry the nonlinear signature of the crack. Therefore, it is interesting to obtain the directivity patterns corresponding to the higher harmonics and different wave modes in order to know in which directions these nonlinear components are preferentially generated.

The purpose of this paper is to examine the frequency content of waves scattered from a closed crack of different orientations. A two step procedure is used to plot the directivity pattern of the scattered field. First, a Finite Element (FE) model is used to solve the nonlinear problem of a wave interacting with a closed crack. The crack is pre-stressed and modeled by a unilateral contact with Coulomb's friction. Limited by the number of unknowns, the near field solution given by the FE model is extended in the outside domain using an analytical method. This method uses Green's functions and is similar to the one used by Hunt et al. to compute the radiated field from elastic structures in a fluid domain [START_REF] Hunt | Finite element approach to acoustic radiation from elastic structures[END_REF]. The results provide valuable information about the higher harmonic propagation, especially regarding mode conversions.

FINITE ELEMENT MODEL

A Finite Element model using the software Plast2 [START_REF] Baillet | Mixed finite element methods for the Signorini problem with friction[END_REF] is built to treat the interaction between an acoustic wave and a crack of finite extent. An isotropic and homogeneous solid is considered, its mechanical properties are those of aluminum. The Young's modulus is E = 69 GPa, the Poisson's coefficient is ν = 0.33 and the density is ρ = 2700 kg.m -3 . This solid contains a crack tilted by an angle θ . Perfectly matched layers (PML) are set on both sides of the solid to model an infinite solid in the direction x x x. Finally, a plane wave of five cycles is generated on the top face of the solid. The central frequency of this pulse is 0.5 MHz for the shear wave and 1 MHz for the longitudinal wave. The wavelength of the incident waves is thus λ ≃ 6 mm. Finally, the amplitude of the incident wave is of 10 nm. Fig. 1 shows the characteristics of the model. The spatial discretization is essential in the FE method. In order to have an accurate solution up to the third harmonic, the corresponding wavelength has to be sufficiently discretized. The maximal size of the elements is a = 0.2 mm which means that the wavelength of the third harmonic is divided in ten elements (λ 3 f /a ≃ 10). The mesh is refined at the vicinity of the crack tips because of the stress singularities (σ ∼ 1/ √ r where r is the radial distance from the crack tip). The software Plast2 uses a time explicit scheme and therefore the time step is subject to the Courant-Friedrichs-Lewy stability condition. The time step is ∆t = 3 ns. The convergence has been validated in time and space. The displacements are recorded at each node belonging to a circle of radius R = 15 mm, as shown in Fig. 1. The length of the crack is d = 10 mm.

The crack is modeled by an interface of unilateral contact with Coulomb's friction. Therefore, three states can be observed simultaneously at different places of the interface: separation, slipping contact and sticking contact. The crack can be either closed by the weight of the structure or an external load or a residual stress due to plasticity. Therefore, it is assumed that a static normal stress σ 0 < 0 closes the crack. This pre-stress σ 0 is directly considered in the contact laws.

The crack is made of two faces which are denoted by i = 1, 2 respectively for the top and bottom face. Let be u u u i the displacement and n n n i the outward normal vector of the face i of the crack, the normal jump of displacements is defined by:

[u n ] = u u u 1 • n n n 1 + u u u 2 • n n n 2 = (u u u 1 -u u u 2 ) • n n n 1 (1)
The incident wave creates stresses represented by the Cauchy stress tensor σ σ σ (u u u i ). Denoting σ n (u u u i ) its normal component and τ(u u u i ) its tangential component, the unilateral contact law taking into account σ 0 is given by the following equation:

     σ n (u u u i ) + σ 0 ≤ 0 [u n ] ≤ 0 (σ n (u u u i ) + σ 0 ).[u n ] = 0 (2)
The first equation states that only a compression can be transmitted through the interface. The normal stress σ n is allowed to be positive although the crack is closed as long as σ n ≤| σ 0 |. When σ n = -σ 0 the interface opens. The second line corresponds to the nonpenetration condition. Finally, the third line, or complementary condition, indicates that the contact interface is either open or closed. Denoting µ the friction coefficient and [u t ] the tangential jump of displacements, the Coulomb's law is used for the tangential behavior. The pre-stress σ 0 is also introduced in the classical law:

           | τ(u u u i ) |≤ µ | σ n (u u u i ) + σ 0 | If | τ(u u u i ) |< µ | σ n (u u u i ) + σ 0 | ⇒ sticking: [u t ] = 0 If | τ(u u u i ) |= µ | σ n (u u u i ) + σ 0 | ⇒ { sliding: ∃α ≥ 0; [u t ] = -ατ(u u u i ) τ(u u u i ) = ±µ | σ n (u u u i ) + σ 0 | (3)
The shear stress τ has to be equal to µ | σ n + σ 0 | to generate sliding. When sliding occurs, the value of the shear stress depends on the global normal stress σ n + σ 0 . The switches between the different contact states introduce the nonlinearity in the model. These boundary conditions can be referred as the non-smooth contact dynamics [START_REF] Jean | The non-smooth contact dynamics method[END_REF].

ANALYTICAL SOLUTION FOR THE RADIATED FIELD

The FE model gives the near field solution which is limited in space due to computational cost. The scattered field is defined as the difference between the total field and the incident field (obtained without the crack). Therefore, the crack is considered like a secondary source. The knowledge of the scattered field on the circle of radius R = 15 mm is used to obtain the solution in the far field, assuming that the material has a linear behavior. The cylindrical displacements obtained at r = R are converted is the frequency domain using the Fourier transformation. Then, it is possible to obtain the radiated field for r ≥ R from an integral equation at r = R. This kind of approach has already been used in fluid mechanics [START_REF] Hunt | Finite element approach to acoustic radiation from elastic structures[END_REF].

The solution is assumed time harmonic, with the time convention e -iωt . The displacements are written in the cylindrical coordinate system, in the frequency domain. û u u(r, θ ) = ûr e e e r + ûθ e e e θ = ∇ϕ + ∇ ∧ ψe e e z , r ≥ R, [START_REF] Richardson | Harmonic generation at an unbonded interface -I Planar interface between semiinfinite elastic media[END_REF] where ϕ and ψ are respectively the velocity potential for the longitudinal wave and the shear wave. Then,

û u u(r, θ ) = ( ∂ ϕ ∂ r + 1 r ∂ ψ ∂ θ ) e e e r + ( 1 r ∂ ϕ ∂ θ - ∂ ψ ∂ r
)

e e e θ .

(

) 5
The velocity potentials have to satisfy the Helmholtz equation. Since we consider the waves scattered by a source located at the origin of the coordinate system, the potentials can be written with the Hankel's functions :

       ϕ = ∑ n A n k L H (1) n (k L r)e inθ + ∑ n C n k L H (2) n (k L r)e inθ ψ = ∑ n B n k T H (1) n (k T r)e inθ + ∑ n D n k T H (2)
n (k T r)e inθ [START_REF] Lee | Experimental investigation of nonlinear acoustic effect at crack[END_REF] where the notation

∑ n = +∞ ∑ n=-∞
is used. The terms k L = ω/c L and k T = ω/c T are respectively the wave numbers for the longitudinal wave and the shear wave. The scattered waves have to verify the Sommerfeld condition, which states that the radiated energy has to be zero at infinity. With the time dependence e -iωt and for a two dimensional space, this condition is given by:

lim r→∞ √ r ( ∂ ∂ r -ik ) û u u(r, θ ) = 0. ( 7
)
The Sommerfeld condition is obtained when C n = 0, D n = 0, ∀n. Replacing the potential expressions in Eq. 5, the radial and the tangential displacements are given by:

       ûr (r, θ ) = ∑ n A n H (1) ′ n (k L r)e inθ + in r ∑ n B n k T H (1) n (k T r)e inθ ûθ (r, θ ) = in r ∑ n A n k L H (1) n (k L r)e inθ -∑ n B n H (1) ′ n (k T r)e inθ (8)
The coefficient A n and B n have now to be computed from an integral at r = R where the displacements are known. To do that, each term of Eq. 8 is multiplied by e -imθ and integrated over the circular domain. Using the orthogonality of the functions e inθ for the scalar product ⟨ f , g⟩ =

∫ 2π 0 f (θ )g(θ )dθ , we obtain:        ∫ 2π 0 ûr (R, θ )e -imθ dθ = 2π ( A m H (1) ′ m (k L R) + in k t R B m H (1) m (k T R) ) ∫ 2π 0 ûθ (R, θ )e -imθ dθ = 2π ( im k L R A m H (1) m (k L R) -B m H (1) ′ m (k T R) ), ∀m (9)
Defining the matrix

M M M n (r) =    H (1) ′ n (k L r) in k T r H (1) n (k T r) in k L r H (1) n (k L r) -H (1) ′ n (k T r)    , (10)
the coefficients A n and B n are the solution of a linear system, and are given by:

( A n B n ) = M M M -1 n (R)     1 2π 2π ∫ 0 ûr (R, θ )e -inθ dθ 1 2π 2π ∫ 0 ûθ (R, θ )e -inθ dθ     . ( 11
)
Finally, the displacement field is obtained for any point of the outside domain from Eq. 8 and 11:

û u u(r, θ ) = ∑ n M M M n (r)M M M -1 n (R) e inθ 2π 2π ∫ 0 û u u(R, θ ′ )e -inθ ′ dθ ′ , r ≥ R, ∀θ (12)
This solution is valid for a given frequency. The directivity patterns are obtained by plotting the amplitude of the displacements over θ ∈ [0; 2π] for a radial distance r = 100 mm.

DIRECTIVITY PATTERNS

In this part, the directivity patterns are plotted for the fundamental frequency and the two first higher harmonics. Radial displacements u * r correspond to longitudinal waves and tangential displacements u * θ to shear waves. They are normalized by the maximal displacement measured at r = R in the case of a linear simulation, i.e with free boundary conditions applied on the crack faces.

In-plane longitudinal wave

A longitudinal wave is considered and the directivity patterns are plotted in Fig. 2 for three orientations of the crack, θ = 0 • , 25 • , 45 • . The directions of propagation are marked by the dashed lines. Regarding the fundamental harmonic, the scattered displacement is mostly radial in the case of a normal incidence and the wave propagates in the same direction as the incident wave. There is no mode conversion. Under oblique incidence, the reflected wave propagates in the direction predicted by the Snell-Descartes law for simple reflection. Mode conversions occur and the reflected shear waves also propagate along the directions given by the Snell-Descartes law. The second and the third harmonic are generated by the clapping occurring at the interface. These harmonics propagate in the same directions as the coresponding fundamental harmonic. However, the directivity seems enhanced in the case of the higher harmonics because of the shorter wavelength. As a consequence, the alignment of the transducers during a nonlinear experiment is a parameter that should be checked precisely, as demonstrated in [START_REF] Liu | Effects of experimental variables on the nonlinear harmonic generation technique[END_REF]. The amplitude of the higher harmonics contained in the shear waves (Fig. 2.(e) and (f)) increases with the angle of incidence whereas it decreases for the longitudinal waves. The mode conversion is stronger for high incidence angle and more energy is transfered to the shear wave components.

U θ * u * r u * θ A 1 A 2 A 3 (a) (b) (c) (d) (e) (f) Incident wave θ = 0 • θ = 25 • θ = 45 •

In-plane shear wave

The directivity patterns are plotted in Fig. 3 for an incident shear wave. The directions of propagation are marked by the dashed lines. Three crack orientations are considered, θ = 0 • , 25 • , 45 • , and the directivity patterns are obtained for the fundamental and the two first higher harmonics. The contact parameters are σ 0 = -0.25 MPa and µ = 0.3. In this case there is no activation of the clapping and only sliding is activated at the interface whatever the angle of incidence [START_REF] Blanloeuil | Numerical study of nonlinear interaction between a crack and elastic waves under an oblique incidence[END_REF].

U θ * u * r u * θ A 1 A 2 A 3 (a) (b) (c) (d) (e) (f)
Incident wave Similarly to the longitudinal wave, the higher harmonics, when they are generated, have directivity patterns of the same form as those of the fundamental harmonic. One can see that under normal incidence, the third harmonic is preferentially generated for the tangential displacements. Under oblique incidence, things get more intricate. For θ = 25 • mode conversions occur and there is an increase of the second harmonic, both for the tangential and the radial displacements. For θ = 45 • the values of the harmonics are insignificant and there are no scattered field. The reason can be understood under the consideration of contact stresses. An in-plane shear wave generates a shear stress σ xy (t) in the solid which maximal amplitude is 0.23 MPa. The normal and tangential stress generated by the incident wave at the interface are given by:

θ = 0 • θ = 25 • θ = 45 •
{ σ inc n (t) = (σ σ σ (t).n n n) .n n n = -σ xy (t) sin 2θ τ inc (t) = (σ σ σ (t).n n n) .t t t = σ xy (t) cos 2θ (13)
where σ σ σ is the stress tensor and n n n and t t t are respectively the normal and tangent vector at the interface.

For θ = 45, the shear stress τ inc (t) becomes zero and consequently it can not trigger sliding at the interface. The normal stress being to small to open the crack, there is no nonlinear effect and the wave propagation is not affected by the crack.

Directivity patterns of the scattered waves are closely linked to the orientation of the crack. It is well established that the nonlinear interaction between a wave and a closed crack is related to the state of stress at the interface. Other properties of the crack may also modify the scattering of higher harmonics. Here, the effect of the coefficient of friction µ is studied for a shear wave propagating with normal incidence. Since odd harmonics are mainly generated in this case, the directivity patterns of the third harmonics are plotted in As expected, the third hamonic is mainly contained in the waves propagating along the direction normal to the interface. The coefficient of friction has only an amplitude effect on the third harmonic. Its amplitude is plotted in Fig. 4.(b) for the direction of propagation 270 • as a function of the normal-ized parameter ξ defined as :

ξ = µ|σ 0 + σ inc n | |τ inc | (14)
The amplitude shows an optimal value which is obtained for ξ ≃ 0.4 (µ = 0.4). This result agrees with previous work [START_REF] Meziane | Nonlinear shear wave interaction at a frictional interface: Energy dissipation and generation of harmonics[END_REF]. Under oblique incidence, the third harmonic still has an optimal value when the coefficient of friction changes but is obtained for a different value of ξ . This is due to mode conversion and energy transfer between the different waves.

CONCLUSION

In this paper, a Finite Element model is used to treat the nonlinear interaction between a plane wave and a closed crack, using an unilateral contact law with Coulomb's friction. The numerical solution computed at the near field is propagated semi-analytically to obtain the scattered field of the frequency components corresponding to the fundamental harmonic and the two first higher harmonics. The study of the directivity patterns provides valuable information about the higher harmonics propagation, especially regarding mode conversions. A major conclusion is that the direction of propagation of the higher harmonics agrees with those of the fundamental harmonic. The scattering behavior is also related to the mechanical properties of the crack, such as the state of stress or the coefficient of friction. This method enables computation of the scattered field at any position in the solid. It could be used to predict signals measured by transducers, and then help to position them to extract more information about closed cracks. A study of the influence of frequency should also be interesting for a SHM application.

Figure 1 :

 1 Figure 1 : Modeling of a wave interacting with a crack orientated by the angle θ . The length of the crack is d = 10 mm. The crack is represented by the dashed line and a pre-stress σ 0 is applied on the interface. The numerical output is defined on a circle of radius R = 15 mm.

Figure 2 :

 2 Figure 2 : Directivity patterns for an incident longitudinal wave, for θ = 0 • , 25 • , 45 • . (a), (b) and (c) give the radial displacements u u u * r for the fundamental and the two first higher harmonics. (d), (e) and (f) give the tangential displacements u u u * θ for the same frequencies. σ 0 = -0.15 MPa and µ = 0.8.

Figure 3 :

 3 Figure 3 : Directivity patterns for an incident shear wave, for θ = 0 • , 25 • , 45 • . (a), (b) and (c) give the radial displacements u u u * r for the fundamental and the two fisrt higher harmonics. (d), (e) and (f) give the tangential displacements u u u * θ for the same frequencies. σ 0 = -0.25 MPa and µ = 0.3.

Fig. 4 .

 4 (a) for the tangential component and for different values of µ.

Figure 4 :

 4 Figure 4 : The crack is horizontal. (a) The third harmonic directivity patterns of the tangential displacements u u u * θ are plotted for different friction coefficients, respectively µ = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1 for the curves 1 to 8. The pre-stress is σ 0 = -0.25 MPa. (b) The amplitude of the third harmonic contained in the wave propagating in the direction 270 • is plotted as a function of the normalized load ξ .

EWSHM 2014 -Nantes, France


João Santos
email: josantos@lnec.pt

Luís Calado
email: 2calado@ist.utl.pt

Paulo Silveira
email: 3paulo.silveira@lnec.pt

Christian Crémona
email: 4christian.cremona@cerema.fr

BASELINE-BASED DETECTION OF STRUCTURAL CHANGES: DRAWBACKS AND AN ALTERNATIVE APPROACH

Keywords: Structural Changes, Bridges, Neural Networks, Auto-Correlation

The present paper addresses the detection of structural changes using statistical learning methods with the objective of exposing the major limitations of defining baselines in which structural systems must be assumed unchanged. For this purpose, the data acquired during approximately two years in a cable-stayed bridge is used and was chosen since it comprises unchanged structural responses as well as measurements comprising changes imposed by creep and shrinkage. In addition to the study of baseline-based detection, the present paper proposes a simple yet effective baseline-free approach relying auto-correlation functions. The effectiveness of the proposed strategy in clearly distinguishing between changed and unchanged structural responses was demonstrated using the same data set.

INTRODUCTION

Structural Health Monitoring (SHM) is generally defined as the development and implementation of strategies to detect structural changes [START_REF] Santos | Baseline-free real-time assessment of structural changes[END_REF]. A large number of these strategies is nowadays proposed in the literature and used in practical applications relying on statistical learning algorithms to obtain accurate and non-parametric descriptions of the monitored structural responses [START_REF] Santos | Multivariate Statistical Analysis for Early Damage Detection[END_REF], [START_REF] Cury | Application of symbolic data analysis for structural modification assessment[END_REF].

Regardless of the algorithms underpinning their definition, these strategies rely almost exclusively on the assumption that the structural systems are healthy and remain unchanged during a certain period of time, which is named baseline or training period [START_REF] Santos | Baseline-free real-time assessment of structural changes[END_REF], [START_REF] Santos | Baseline-Free Real-Time Novelty Detection Using Vibration-Based Symbolic Features[END_REF]. The responses acquired during this period are in-putted into the statistical learning algorithms, which are trained and output a set of coefficients that are posteriorly used for predicting structural responses [START_REF] Posenato | Methodologies for model-free data interpretation of civil engineering structures[END_REF], [START_REF] Sohn | Structural Health Monitoring Using Statistical Process Control[END_REF].

Changes are then identified by comparing the errors associated with these predictions for two distinct periods: the baselines and posterior testing periods, which are associated with unknown structural conditions [START_REF] Santos | Baseline-free real-time assessment of structural changes[END_REF], [START_REF] Posenato | Methodologies for model-free data interpretation of civil engineering structures[END_REF]. These errors are generally named residuals and the comparison of their distributions, obtained from the baselines and testing subsets, is generally based on hypothesis testing [START_REF] Santos | Identificação de Comportamento Anómalo sem Conhecimento Prévio da Condição Estrutural[END_REF], [START_REF] Zhou | Structural damage alarming using auto-associative neural network technique: Exploration of environment-tolerant capacity and setup of alarming threshold[END_REF].

When the baselines are acquired from healthy and unchanged structural responses, the statistical learning algorithms learn the "regular" structural behaviour and can therefore outline future structural changes. However, this assumption may not always be met, especially for SHM applications comprising aged structural systems [START_REF] Santos | Multivariate Statistical Analysis for Early Damage Detection[END_REF], [START_REF] Guo | A vibration-based framework for structural health monitoring of railway bridges[END_REF]. If a baseline is defined during the occurrence of structural changes, chances are that the statistical learning algorithms will learn these changes as "regular" and thus will not be capable of identifying future ones.

The present paper aims at addressing this problem by conducting baseline-based and baseline-free detection of structural changes using the two statistical learning algorithms most reported in SHM literature: the multivariate linear regression (LR) [START_REF] Mata | Constructing statistical models for arch dam deformation[END_REF], [START_REF] Posenato | Model-Free Data Interpretation for Continuous Monitoring of Complex Structures[END_REF] and the multi-layer perceptron neural networks (MLP) [START_REF] Ni | Generalization Capability of Neural Network Models for Temperature-Frequency Correlation Using Monitoring Data[END_REF], [START_REF] Mata | Interpretation of concrete dam behaviour with artificial neural network and multiple linear regression models[END_REF].

In the present paper, section 1 provides a brief overview of the statistical methods used and section 2 aims at presenting the case study used herein. Section 3 aims at exposing the main limitations and hazards of using baseline-based detection while section 4 presents an alternative approach, which does not require the definition of baseline, as well as the evaluation and comparison of the two methods' performances. Afterwards, the main conclusions are drawn from the work described are presented.

STATISTICAL LEARNING ALGORITHMS

Multivariate linear regression (LR) is the most straightforward regression method reported in the SHM literature. Its formulation is based on the assumption that the structural response, denoted Y and referred to as "output", is linearly or near-linearly dependent on a set of action-related measured quantities, X, denoted as "input". Considering a vector of structural response data, Y, and a set of d action-related quantities (operational or environmental) acquired at the same n time-instants, X; a generic multivariate linear regression model, capable of explaining the changes observed in Y using the ones obtained from X, is generically represented in Figure 1a and can be defined as follows [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF] (1) where U is the vector of regression weights (or coefficients) and U (0) is the "bias" vector, which allows for any fixed offsets to be considered in the regression model. The detection of structural changes based on LR consists in finding the optimal set of coefficients U by minimizing the prediction's squared error and in evaluating the residuals' (RE) distributions.

The Multi-Layer Perceptron (MLP) comprising a single layer of hidden units (Figure 1b) and trained with the back-propagation algorithm is the most used method among those classified as "artificial neural networks" [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF]. The mathematical model which describes the output variables, Y, in terms of the inputs X, can be defined as [START_REF] Santos | Multivariate Statistical Analysis for Early Damage Detection[END_REF] where U (iii) and U (ii) are the regression coefficients of the output and hidden layers, respectively. The function h allows the MLP to find non-linear relations among the input and output variables and is taken herein as in the majority of applications found in the literature as the hyperbolic tangent.

Unlike the LR, the problem of finding the optimal set of coefficients does not exhibit a unique and linear solution. Hence, this solution if found through the iterative back-propagation optimization technique based on gradient descent. Over-fitted solutions are obviated herein using ( ) EWSHM 2014 -Nantes, France the early-stopping technique, while under-fitted solutions are avoided by defining a number of hidden units equal to the number of input variables. The effects of ill initialization are minimized by performing multiple repetitions of each training. As with the LR, the detection of structural changes based on MLP consists of finding the optimal set of coefficients, U (ii) and U (iii) , and in evaluating the residuals' (RE) distributions. Additional information regarding the MLP can be found in [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF], [START_REF] Hastie | The Elements of Statistical Learning, Data Mining, Inference and Prediction[END_REF]

n.p (III) m.p (II) q.m n.q n.p (III) m.p n.m n.p RE U U X h RE U Z Y + = + = ˆn.p d.p n.d (0) n.p n.p RE U X U Y + + =

CASE STUDY -THE SALGUEIRO MAIA BRIDGE

The cable-stayed Salgueiro Maia Bridge is located in the Portuguese highroad A13, over the river Tagus, 70km upstream of Lisbon. The structural system is 570m long and partially suspended over a length of 489m by a single plane of seventy two stay-cables, eighteen in each of four semi-fans. Due to the high seismic risk of the region where the bridge was built, the superstructure is structurally independent from the infrastructure, which supports the first by means of fifty six high damping rubber bearings (HDRB), which provide a high level of seismic isolation.

The superstructure is symmetric and composed of a 570m long box girder deck monolithically connected to two pylons, which are 50m high (Figure 2a). The central suspended span is 246m long while the lateral counterweight spans are 78m and 42m long. The box girder section is 2.62m high and 27.7m wide, composed of a rectangular shape plus two cantilevers punctually supported by prefabricated concrete struts. The pylons are divided into a bottom part composed of a rectangular reinforced concrete section and a steel-concrete composite part comprising two concrete rectangles connected by steel elements which anchor the stay-cables. The seventy two stay cables are 31m to 131m long and exhibit vertical inclinations varying from 35º to 69º. Their cross sections vary from 55 to 73 strands and their anchoring devices are spaced by 6m on the deck and 1.2m on the pylons. Besides individual mono strand sheathing, the stay cables are protected by a global sheath of high density polyethylene.

During and immediately after the bridge's construction (from 1998 to 2000), an automatic monitoring system comprising more than two hundred sensors was deployed in situ with the objective of studying its behaviour and of controlling its safety in mid and long term. The sensorial system comprises acoustic (vibrating wire) strain gages and embedded resistance thermometers (Figure 2b,c), a hydrostatic levelling system and stay-cables' load cells (4 per semi-fan).

The data set used in the present paper was obtained from the thermometers and strain gages of sections S2, S4 and S10 as well as from the load-cells identified in Figure 4a. The location of the thermometers and strain gages in the three concrete cross-sections can be observed in Figure 4b,c. For each of these sections, the data used was fused without loss of structural relevant information into average strains: C2m, C4m and C10m (time-series shown in Figure 5b); as well as in average and differential temperatures: T2m, T10m, T2d and T10d (time-series shown in Figure 5a). The stay-cables' forces were used as obtained in situ and their time-series are shown in Figure 5c.

As it can be observed in Figure 5a,c, the temperatures and forces reveal important reversible variations which are directly correlated with the time of the year (time index's null value defined at 0:00 of the 29 th December 2004). Nonetheless, their overall trend appears to be horizontal, thus suggesting that the structural system is not changing over time. Conversely, the strain measurements exhibit, in addition to the variations imposed by temperature, an irreversible downward trend generated by the structural changes imposed by creep and shrinkage.

BASELINE-BASED DETECTION OF STRUCTURAL CHANGES

The availability of one set of structural responses influenced only by the "regular" action of temperature (the four stay-cables' forces, SC24, SC29, SC33 and SC36 -see Figure 2a) and another influenced also by irreversible structural changes (average strain measurements, C2m, C4m and C10m) motivated the evaluation of the two learning methods concerning their ability to: (i) learn the "regular" behaviour (first set of responses) and to (ii) outline structural changes (second set of responses). When using baselines to train the regression methods, the detection is based on comparing the distributions of the residuals obtained from the baselines and from the testing subsets. In the present paper, choice was made to carry out this comparison using a single-valued quantity, D, defined from the residuals interquartile intervals

where i is the index identifying each of the seven structural responses and Q (i,25,B) as well as Q (i,75,B) are the 25% and 75% quantiles obtained from the residuals of the baseline subset of sensor i. Similarly, Q (i,25,T) and Q (i,75,T) are the corresponding quantiles obtained for the testing subset. Since this indicator is based on interquartile intervals, its values must reflect changes in the residuals' distributions (not only on the expected value, but also in its variance, symmetry and flatness). Large values of D(B,T) indicate very dissimilar distributions between a baseline and the testing set, and thus the presence of structural changes, while small values suggest similarity between them, and thus unchanged behaviour. Hence, bearing in mind the effects influencing the stay-cables forces and average strains, an effective baseline-based detection algorithm must be capable of:

• generating small D(B,T) values for the stay-cables' forces (SC24, SC29, SC33 and SC36);

• generating high D(B,T) values for the three average strains (C2m, C4m and C10m). To study the effectiveness of baseline-based detection, four distinct lengths of baseline data (shown in green colour in Figure 4) were used for training the LR and the MLP: (i) 300 days, (ii) 350 days, (iii) 400 days, (iv) 450 days. The time-series shown in Figure 4 illustrate the analysis conducted herein for the average strain on the deck, C10m, using the MLP neural network. From this figure it can be observed that the residuals' distributions, are highly influenced by the length of the chosen baseline.

The detection procedure exemplified in Figure 4 was repeated using the MLP method for the seven structural responses analysed. Its outcome is summarized in Figure 5, where it can be readily observed that the Dsc(B,T) values (obtained from the stay-cables) are low and not dependent on the baseline length. Conversely, the values of Dc(B,T) (obtained from the average strains) are much higher for small baseline lengths, thus leading to the conclusion that if the learning methods are trained with baselines comprising structural changes, they become ineffective in outlying future variations of identical nature.

The same analysis was conducted also using the multivariate linear regression and the results of this analysis along with those presented in Figure 6 are summarized in Figure 7 using Dc(B,T) and Dsc(B,T), which consist of the average D(B,T) obtained for the three average strains and for the four stay-cables' forces. From this figure it can be once again observed that if the learning methods are trained with unchanged structural responses (Figure 7b), their effectiveness is not dependent on the chosen baselines. However, if these methods are trained with structural responses comprising changes (Figure 7a), their effectiveness can be highly variable. These results expose the main limitation of baseline-based SHM strategies: if no structural changes occur during the baseline period, the outcome of the detection can be relied on; however, if structural changes occur during this period, the estimation methods "learn" them as normal behaviour and are incapable of pointing them out in future data subsets. In this case, the premise in which baseline-based detection relies is not valid and SHM methods will suggest the non-existence of novelties when they might, in fact, exist.

( ) ( ) ( ) ( ) ( ) { } ( ) ( ) p 1,..., i ; Q - Q Q - Q ; Q - Q max T B, D B i,25, B i,75, T i,75, B i,75, T i,25, B i,25, i = =

BASELINE-FREE ALTERNATIVE PROCEDURE

To avoid the hazards of having SHM strategies which might be unable to detect changes, a simple yet effective strategy is proposed herein. It consists of not drawing assumptions about the structural condition during an analysed monitoring period by using its entire length to train the statistical learning algorithms. Since no separation of the data is performed, no comparison between residuals' distributions can be carried out, and information obtained from one single time-series must be used for detecting structural changes. Herein, choice was made to use the concept of auto-correlation function along with the knowledge that random-like time-series exhibit null (or nearly null) auto-correlation [START_REF] Shumway | Time Series Analysis and its Applications[END_REF]. Ideally, the residuals of the LR or of the MLP must exhibit this type of distributions since they are independent from the "regular" action of temperature. However, if any structural changes occur, the residuals' auto-correlations must be high, thus signalling the existence of structural changes.

The auto-correlation of a residuals' time-series is simply defined as [START_REF] Shumway | Time Series Analysis and its Applications[END_REF] (4)

where RE (t) and RE (t+h) are subsets of RE, each comprising n-h elements and starting, respectively, as instant t+h and t. The quantity σ 2 is the variance and the auto γ -covariance, defined as in [START_REF] Shumway | Time Series Analysis and its Applications[END_REF] The auto-correlation function, ACF, can then be simply defined by varying the lag values, from 1 to a specified chosen value. To assess the effectiveness of the proposed baseline-free SHM strategy, the ACF were obtained for each of the 14 residuals' series (obtained from 7 sensors and 2 methods) and for h=1 to 100. The resulting ACF distributions are shown in the box-and-whiskers plots in Figure 7. As it can be observed, when applied under the proposed baseline-free strategy, both methods appear to be highly effective in distinguishing between changed and unchanged structural responses. To obtain a more accurate comparison, the ratio of the average ACF value computed for the strains and for the forces was obtained and resulted in 5.1 for the MLP and 4.6 for the LR. The higher this ratio, the more effective the method in distinguishing between changed and unchanged conditions. Hence, it can be concluded that the iterative nature of the MLP allow it to achieve higher performance in detecting structural changes under the proposed baseline-free approach.

CONCLUSIONS

In the present paper, the multivariate linear regression (LR) and the multi-layer perceptron neural network (MLP) were applied to a data set acquired from the Salgueiro Maia Bridge. This set comprises structural responses describing changed and unchanged behaviour, which were analysed under baseline-based and baseline-free approaches. The application conducted using baselines exposed an important limitation of this type of SHM strategies: if no structural changes occur during the baseline period, the outcome of the detection can be relied on; however, if structural changes occur during this period, the estimation methods "learn" them as normal behaviour and are incapable of pointing them out in future data subsets. In this case, the premise in which baseline-based detection relies is not valid and SHM methods will suggest the non-existence of novelties when they might, in fact, exist. To overcome this disadvantage, a baseline-free approach was proposed and consists in analysing whether the statistical learning methods (LR and MLP) remove the auto-correlation from the time-series of structural responses acquired on site. This univariate statistical feature (auto-correlation) is not generally used to evaluate the outcome of SHM normalization. Nonetheless, it proved effective in distinguishing time-series influenced by structural changes from those influenced only by temperature. In addition, it was also observed that even if the relations between structural responses and temperatures are generally linear, the MLP outperformed the LR when applied under the proposed baseline-free methodology.

( ) ( ) ( ) (t) n 2 h) (t h - n (t) h - n 2 n h RE σ RE ; RE γ RE AC + =

Figure 1 :

 1 Figure 1: Graph representations of: (a) linear regression, (b) multi-layer perceptron.

Figure 2 :

 2 Figure 2: Salgueiro Maia Bridge: (a) overall view, (b) sensors on the deck and (c) sensors on the pylon.

Figure 3 :

 3 Figure 3: Time-series of: (a) average and differential temperatures, (b) average strain and (c) stay-cables' forces. Time index's null value defined at 0:00 of the 29 th December 2004.

Figure 4 :

 4 Figure 4: Time-series of residual errors obtained for C10m, considering four different baseline lengths: (a) 300 days, (b) 350 days, (c) 400 days, (d) 450 days.

Figure 5 :

 5 Figure 5: Bar-plots of the D(B,T) obtained using MLP-based normalization and four distinct baseline subsets lengths: (a) 300 days, (b) 350 days, (c) 400 days and (d) 450 days.

Figure 6 :

 6 Figure 6: Average D(B,T) values for the four baselines: (a) Dc(B,T), (b) Dsc(B,T).

Figure 7 :

 7 Figure 7: Box-and-whiskers plots of the ACF obtained from the residuals of: (a) LR and (b) MLP.

EWSHM 2014 -Nantes, France


João Santos
email: josantos@lnec.pt

Luís Calado
email: 2calado@ist.utl.pt

André D Orcesi
email: 3orcesi@lnec.pt

Christian Crémona
email: 4christian.cremona@cerema.fr

Adaptive detection of structural changes based on unsupervised learning and moving time-windows

Keywords: cluster analysis, symbolic data, suspended bridge, adaptive assessment

The present paper addresses data-driven structural health monitoring to propose a real-time strategy for adaptive structural assessment. The adaptive character is achieved using unsupervised discrimination machine-learning methods, widely known as clustering algorithms. Real-time capability is based on the definition of symbolic data, which allow describing large amounts of information without loss of related information. The efficiency of the proposed methodology is illustrated using an experimental case study in which structural changes were imposed to a suspended bridge during an extensive rehabilitation program.

INTRODUCTION

Structural health monitoring (SHM) is generally defined as the process of implementing a damage identification strategy for any engineering structure, infrastructure or system [START_REF] Farrar | An introduction to structural health monitoring[END_REF]. Damage occurrences, which can be defined as changes affecting present or future structural performance may have significant human and social impact. In this context, an effective SHM strategy should be able to provide real-time structural feedback and detect any abnormal structural behaviour before damage occurrences take place and affect structural performance.

Structural health monitoring techniques may follow inverse or forward approaches [START_REF] Glaser | Sense of Sensing: From Data to Informed Decisions for the Built Environment[END_REF] for identifying structural changes. Inverse approaches are frequently named model updating and consist of fitting analytical or numerical responses to experimental data with the objective of inferring structural quantities that cannot be directly measured [START_REF] Glaser | Sense of Sensing: From Data to Informed Decisions for the Built Environment[END_REF]. Forward SHM approaches do not require the development of models [START_REF] Glaser | Sense of Sensing: From Data to Informed Decisions for the Built Environment[END_REF]. Instead, they rely on data mining techniques for extracting sensitive information from time-series acquired on site [START_REF] Sohn | A Review of Structural Health Monitoring Literature: 1996 -2001[END_REF]. Their computational simplicity makes them the most suitable candidates for carrying out real-time SHM in large-scale structures [START_REF] Posenato | Model-Free Data Interpretation for Continuous Monitoring of Complex Structures[END_REF].

Feature extraction and discrimination are common steps to all data-driven SHM strategies [START_REF] Farrar | An introduction to structural health monitoring[END_REF], [START_REF] Sohn | A Review of Structural Health Monitoring Literature: 1996 -2001[END_REF]. A feature can be defined as a type of information, extracted from time-series acquired on site, which may highlight structural changes by being correlated with them [START_REF] Glaser | Sense of Sensing: From Data to Informed Decisions for the Built Environment[END_REF]. Its extraction is generally necessary since raw data is not informative about the occurrence of structural changes. Widely used examples of SHM features are modal quantities [START_REF] Alvandi | Assessment of vibration-based damage identification techniques[END_REF], autoregressive models [START_REF] Lautour | Damage classification and estimation in experimental structures using time series analysis and pattern recognition[END_REF], principal components [START_REF] Santos | Multivariate Statistical Analysis for Early Damage Detection[END_REF] or symbolic data [START_REF] Santos | Multivariate Statistical Analysis for Early Damage Detection[END_REF], [START_REF] Cury | Application of symbolic data analysis for structural modification assessment[END_REF]. The latter was only recently applied to SHM and proved particularly effective in fusing multi-sensor data without loss of damage-related information.

Feature discrimination consists of classifying the extracted features as related to monitored structural behaviours (known or unknown). This step is generally based on training machine-learning algorithms with baseline references of extracted features, where the structure is assumed to be undamaged and unchanged. New features are then analysed, by the pre-trained machine-learning algorithms, to assess if the structure's behaviour remains unchanged. Statistical process control [START_REF] Posenato | Methodologies for model-free data interpretation of civil engineering structures[END_REF], multi-layer perceptron neural networks [START_REF] Lautour | Damage classification and estimation in experimental structures using time series analysis and pattern recognition[END_REF] or support vector machines [START_REF] Kim | Wavelet-based AR-SVM for health monitoring of smart structures[END_REF]) are among the most common algorithms for discrimination strategies. Cluster analysis can be seen as an alternative approach since it does not require a prior baseline to perform feature discrimination, thus being more flexible and adaptable. Even though this type of unsupervised analysis has been reported as effective [START_REF] Sohn | Reference-Free Damage Classification Based on Cluster Analysis[END_REF], its computational complexity has discouraged its use in SHM of civil structures. To circumvent these disadvantages, symbolic data has been coupled to cluster analysis, providing baseline-free classification with sufficient computational efficiency [START_REF] Santos | Multivariate Statistical Analysis for Early Damage Detection[END_REF], [START_REF] Cury | Application of symbolic data analysis for structural modification assessment[END_REF].

SHM discrimination methodologies have essentially been reported, in the literature, as off-line (posterior) analyses. Instead of classifying structural behaviour upon detection of the first discrepant feature, these strategies have been mostly applied to data sets composed of large amounts of data, acquired before and after the occurrence of structural changes. The identification of structural changes is then performed by analysing different densities of discrepant features, such as outliers, in similar monitoring periods.

The objective of this paper is to define an adaptive strategy for detection of structural changes upon their occurrence. To test and validate the proposed concepts, data acquired from the suspended Samora Machel Bridge during retrofitting works, in which structural changes were imposed to the structure, is used. This study is divided into three main sections. After this introduction, section 1 describes the suspended bridge, the monitoring program, the acquired data and a brief preliminary analysis. Section 2 describes the data-driven techniques and shows their usefulness when analysing acquired data. In section 3, the real-time strategy is presented and tested in the Samora Machel Bridge data. Afterwards, the main conclusions from this analysis are drawn.

CASE STUDY -THE SAMORA MACHEL BRIDGE

The five-span suspended Samora Machel Bridge is located in the province of Tete, Mozambique. The bridge has four concrete towers, a deck 720m long, two main cables and 144 diagonal hangers (see Figure 1a). The cables have a nominal diameter of 170mm and the hangers are composed of 63 steel wires, with 5mm diameter each. Suspended by each pair of hangers are 72 precast transverse beams with variable cross-sections (Figure 1b,c). Simply supported on each pair of transverse beams are 71 post-tensioned deck modules shown in Figure 1b,c. Each module is composed of nine precast concrete longitudinal beams of variable depth. An extensive refurbishment program was conducted on the Samora Machel Bridge from 2009 to 2012. Some of the most important refurbishment tasks included the generalized repairing / replacement of concrete, replacement of suspension cables, hangers and supports as well as the installation of stabilization systems in each of the seventy-two transverse beams. The latter consisted of the greatest change to the original structural system of the bridge and was designed to reduce the range of rotations generated on the transverse beams along their longitudinal axes, when heavy trucks and medium-weight off-road vehicles cross the bridge. These rotations were observed to be the main cause for damage, not only in transverse beams but also in the adjacent deck modules, supports and anchoring devices. To stabilize each transverse beam, two 140mm diameter steel tubes (Figure 2a,b) were installed to connect each transverse beam to the adjacent deck panel. Both connections are pinned and anchored to concrete with metal plates (Figure 2d). The Samora Machel Bridge SHM was monitored with the objective of assessing the effectiveness of the transverse beams stabilization systems. For this purpose, a transverse beam considered representative of the structural degradation was monitored before and after the stabilization procedure described in the previous section. Data was obtained from two precision tilt-meters fixed to one transverse beam near the tubes' anchorages. The tilt-meters fixed near the upstream and the downstream plates are named herein TM1 and TM2. To achieve these objectives, monitoring of the structural response was conducted at 10 Hz under operational loading, or regular traffic, in three phases:

1. before retrofitting of any beam (phase 1); 2. after stabilization of the instrumented beam (phase 2); 3. after stabilization of the two beams adjacent to the one monitored (phase 3). The data obtained from the monitoring campaign consisted of the rotations' ranges, registered at 10-second intervals since only rotations generated by heavy trucks loads and medium-weight off-road vehicles were of interest to analyse the efficiency of the stabilizing system. The time-series of Figure 3a,b reveal rotations ranges higher in phase 1 than in phases 2 and 3. In the first phase, the measured ranges were reflecting the rotations exhibited by the unrestrained transverse beams. After the installation of the stabilization system, the transverse beams became rigidly connected to the adjacent deck panels and their rotation ranges are much smaller, reflecting only the global response (deflection) of the entire superstructure.

FORWARD STRUCTURAL HEALTH MONITORING

Feature extraction

Feature extraction is addressed in this paper using symbolic data objects, which are less voluminous and more generic types of information when compared to classical data such as SHM time-series measurements [START_REF] Cury | Application of symbolic data analysis for structural modification assessment[END_REF]. While classical data mining focuses on finding groups or patterns in individual measurements, symbolic data deals with concepts, such as time-intervals (Santos, Crémona, Orcesi & Silveira, 2013), structural elements or structural systems, described by statistical quantities such as interquartile intervals or histograms [START_REF] Cury | Application of symbolic data analysis for structural modification assessment[END_REF]. In this paper, interquartile intervals are chosen over histograms since they lead to a greater degree of data fusion [12] while exhibiting similar ability to describe SHM data [START_REF] Cury | Application of symbolic data analysis for structural modification assessment[END_REF]. A symbolic data object, T, described by interquartile intervals of multiple sensors, can be defined as

( ) ( ) ( ) [ ] (p) sup (p) inf (i) sup (i) inf (1) sup (1) inf T ; T T ; T T ; T L L = T (1)
where p is the number of sensors describing the symbolic object. In the present work, p=2 and thus symbolic objects exhibit a rectangular shape in a bi-dimensional space, as shown in Figure 4a-c, where the limits of interquartile intervals define the four sides of the rectangles. For p=3, the symbolic objects can be represented by parallelepiped solid shapes, and for p>3 by hyper-volumes of dimension p.

The usefulness of symbolic data objects in the realm of SHM heavily relies on the definition of techniques that allow the analysis of this type of data using conventional machine-learning algorithms [12]. Among these techniques, symbolic distances have assumed particular importance since they enable cluster analysis to be applied to large data sets, such as those generated in the framework of SHM. Symbolic distances can take a variety of forms and some applications might require specific ones. In the present work, the normalized Euclidean Ichino-Yaguchi distance [12] is used since it exhibits a high sensitivity to structural changes, when compared to other measures found in literature [START_REF] Santos | Multivariate Statistical Analysis for Early Damage Detection[END_REF]. To present its theoretical background, two symbolic objects T i and T j , obtained from a data set of N , ... , 1 s = objects are defined. The normalized Euclidean Ichino-Yaguchi distance [12] between these objects is defined as

( ) [ ] 2 1 2 p 1 r j i r r ij T , T φ Y 1 p 1 d         = ∑ = (2)
where ϕ r is given by

( ) ( ) (r) j (r) i (r) j (r) i (r) j (r) i (r) j (r)
i j i r T T T T 2 γ T T T T T , T φ - - ⊗ + ⊗ - ⊕ = (3)
and γ a pre-specified constant ranging from 0 to 0.5. The Cartesian join ⊕ and meet ⊗, as well as the norm |[…]| and the normalizing term |Y r |, are defined by

( ) ( ) [ ] (r) sup j, (r) sup i, (r) inf j, (r) inf i, (r) j (r) i T , T max , T , T min T T = ⊕ (4) ( ) ( ) [ ] (r) sup j, (r) sup i, (r) inf j, (r) inf i, (r) j (r) i T , T min , T , T max T T = ⊗ (5) [ ] inf sup sup inf T T T , T T - = = (6) ( ) ( ) (r) inf s (r) sup s r T min T max Y - = (7)
For the data acquired from the Samora Machel Bridge SHM, a symmetric Ichino-Yaguchi distance matrix of size 131x131, obtained according to Eq.(3) was defined using symbolic objects of 5 minutes. This matrix is presented in Figure 4d, where brighter colours stand for closer data objects while darker colours represent larger distances. This figure suggests the existence of two distinct groups in the data set, corresponding to two distinct structural behaviours. The first one being associated with data acquired during phase 1 and the second one comprising data acquired during phases 2 and 3.

Feature discrimination

Symbolic objects and distance measures can help perform data analysis and suggest the existence of different structural behaviours within a data set, as shown in the previous section. However, the development of real-time SHM strategies requires the use of machine-learning algorithms which can analyse these features and help decide whether one or more distinct structural behaviours is being observed. Hence, feature discrimination is addressed herein using cluster analysis, which appears to be the family of most promising machine-learning methods for conducting adaptive SHM. The goal of a clustering process is to divide a data set into groups, which must be as compact and separate as possible [START_REF] Santos | Multivariate Statistical Analysis for Early Damage Detection[END_REF]. This aim can be mathematically posed as the attempt to minimize the overall within-cluster distance W(P k ) which, consequently, maximizes the overall between-cluster distance [START_REF] Cury | Application of symbolic data analysis for structural modification assessment[END_REF]. Considering a given partition containing K clusters, the overall within-cluster distance W(P k ) can generally be defined as [START_REF] Theodoridis | Pattern Recognition[END_REF] ( ) ∑ ∑ ∑

= = = = K 1 k k (i) C k (j) C ij k d 2 1 P W (8)
where C(i) is an allocation rule which assigns element i to cluster k, based on the distance measure. The total distance of data can be defined as in Eq. [START_REF] Kim | Wavelet-based AR-SVM for health monitoring of smart structures[END_REF], where N is the number of symbolic objects defined for the analysed data set. The overall between-cluster distance B(P k ) is simply obtained by subtracting the other two distances (Eq.( 11)).

∑∑

= = = N 1 i N 1 j ij d 2 1 T (9)
After partitioning the data set into K clusters, the data allocated to each of them can be simply described by a prototype, which generally consists of an object of the same type as those being clustered. The location and features of the prototype are sufficient to represent the structure of clustered data. The location of each prototype is obtained herein by computing the centroid of each cluster's members.

Several families of clustering methods can be found in the literature, from which the most well-known and used are the combinatorial methods. This type of cluster analysis is iterative in nature and requires the input of an initial set of K<N cluster prototypes and their centroids, generally randomly defined [START_REF] Theodoridis | Pattern Recognition[END_REF]. Afterwards, each iteration starts by allocating data objects to the closest cluster prototype. After the allocation, the new set of prototypes is defined by computing the centroids of the new clusters (defined through the allocation). This procedure is then repeated until a general criterion, which is usually a function of the overall within-cluster distance, is met. The most used clustering method is based on greedy descent and is named k-means. This algorithm uses as loss function the squared overall within-cluster distance. A generalization of this method, named dynamic cloud, was proposed by Diday for application to symbolic data sets [START_REF] Cury | Application of symbolic data analysis for structural modification assessment[END_REF] and is used herein.

The structural engineering problem posed for the presented case study consists in determining which of the two monitored structural changes, generated by the installation of the retrofitting systems (in the monitored beam and in the adjacent ones), were significant in changing the monitored structural behavior. Since the number of significant structural changes is not known in advance, this type of problems must be tackled by allocating data objects to all possible cluster partitions. For the case study used herein, the analysis of cluster partitions comprising 2 and 3 structural behaviors, respectively, suffices since only two structural changes were imposed to the structural system.

Cluster analysis performed on the data set acquired from the Samora Machel SHM was based on the dynamic cloud algorithm applied to the Ichino-Yaguchi distance matrix, shown in Figure 4d and obtained from 5-minute symbolic objects. Two and three clusters were, respectively, defined in the initialization phases of these analyses and their outputs, presented as object allocations, are shown in Figure 5. When studying the possibility of one significant structural change, by assuming two clusters (Figure 5a), it was observed that most data objects allocated to cluster 1 were acquired during phase 1 whereas objects assigned to cluster 2 were mostly obtained during phases 2 and 3. These results strongly suggest that the most important structural change observed in the Samora Machel Bridge took place between phases 1 and 2, thus generating two distinct structural behaviors: the first one observed during phase 1 and the second one during phases 2 and 3. If two significant structural changes have occurred in the Samora Machel bridge, a cluster analysis comprising three clusters should allocate the objects belonging to each experimental phase into each distinct cluster. However, the allocations shown in Figure 5b indicate otherwise. Even though cluster 3 is almost completely composed of objects acquired during phase 1, cluster 2 is assigned with objects from phases 2 and 3 and cluster 3 is scattered among the three phases. Finally, the possibility of having observed unchanged structural behavior throughout the SHM program of the Samora Machel Bridge must also be considered. Provided that cluster analysis cannot analyse single-cluster partitions, this possibility must be considered by analyzing the remaining cluster partitions. A data set obtained during an unchanged structural behavior would generate random allocation of data objects among any number of considered clusters. The results discussed in the two previous paragraphs clearly show that this is not the case for the present case study, thus leading to the conclusion that structural changes were observed on the Samora Machel bridge.

Clustering algorithms are able to allocate data objects to any amount of predefined structural behaviours (each represented by one cluster), irrespective of whether this number of distinct behaviours has really been observed in the structural system. Even though an analysis of object allocations, such as the one performed in the previous paragraph, is able to suggest the number of structural behaviours (and structural changes), its conclusions might be subjective.

To obtain a quantitative and objective estimation of the most truthful number of observed behaviours (clusters) existent in a data set, an evaluation known as cluster validity [START_REF] Santos | Multivariate Statistical Analysis for Early Damage Detection[END_REF] must be conducted on the allocations provided by cluster analyses. Cluster validity consists of computing indices for all possible cluster partitions [START_REF] Theodoridis | Pattern Recognition[END_REF]. Each validity index provides a quantitative measure of cluster compactness and separation, regardless of the number of clusters present in the evaluated partition. The most truthful number of clusters (or structural behaviours) is then assessed by comparing the indices obtained from each of the evaluated partitions. In the present work the global Silhoutte index was used since it achieved best performance in previous works [START_REF] Santos | Multivariate Statistical Analysis for Early Damage Detection[END_REF], and allowed concluding that two clusters (and distinct structural behaviours) are present in the analysed data set.

ADAPTIVE REAL-TIME ASSESSMENT

An adaptive strategy capable of conducting automatic structural assessment in real-time is proposed in this section. Its real-time capabilities are based on the definition of fixed-length moving time-windows, each of which is defined upon the acquisition of new data. Its ability to perform baseline-free assessment is achieved through the use of cluster analysis (explained in the previous section), followed by an outlier analysis.

Conventional methods proposed in SHM-related literature cannot generally perform real-time assessment due to their difficulty in providing reliable conclusions upon detection of the first discrepant value (outlier). This is due to the high probability of obtaining false detections when using features that normally describe single-sensor data or exhibit small generalization capacity. Herein, the false detections are avoided by using a single-valued quantity based on cluster analysis and which describes the entire data set analysed within each time-window: the distance between clusters, B(Pk). At each time-window defined over time, this index is computed along with its confidence interval defined in the previous window. This procedure results in an on-going process with an adaptive confidence boundary capable of outlining structural changes in real-time. This strategy was applied to the Samora Machel Bridge data set using time-windows of one hour and 10-minute symbolic objects. Each time-window was defined 10 minutes after the previous one, thus simulating a real-time assessment procedure with a time resolution of 10 minutes. The output obtained from this procedure consists of the B(P k ) values along with their adaptive confidence boundary, as show in Figure 6. In this plot, the B(P k ) exceeds the adaptive boundary in the first two time-windows containing data from phases 1 and 2 (B(P k ) (values outlined with a red circle), thus detecting the structural change 10 minutes after its occurrence. This time resolution evidences the effectiveness of the proposed strategy.

CONCLUSIONS

This paper presents a data-driven SHM methodology for conducting adaptive real-time structural assessment based on cluster analysis and symbolic data. Its ability to detect structural changes in real-time without resorting to baselines were tested using data acquired from the Samora Machel Bridge. It was observed that symbolic objects based on interquartile intervals are able to accurately generalize the information present in data, while performing significant data fusion. Furthermore, it was observed that the Ichino-Yaguchi distance is sensitive to structural changes concealed in symbolic data sets. Concerning cluster analysis, it was observed that the dynamic cloud method is effective in allocating data objects. The moving time-windows strategy proposed herein proved capable of describing multi-sensor data and point out structural changes imposed in the experimental study only 10 minutes after their occurrence. In addition, it was concluded that the variations exhibited by the proposed single-valued quantity, B(P k ), over time reflect structural changes measured across multiple sensors and that the proposed discrimination method based on cluster analysis and adaptive confidence boundaries allows detecting structural changes upon their occurrence, while avoiding false detections.

Figure 1 -

 1 Figure 1 -The Samora Machel Bridge: (a), perspective view (b) longitudinal section, (c) transverse section.

Figure 2 -

 2 Figure 2 -Stabilization of the transverse beams: (a) longitudinal section, (b) overall view, (c) placement of new concrete, (d) anchoring plates and concrete encasement.

Figure 3 -

 3 Figure 3 -Time series of rotation ranges acquired from the tilt-meter: (a) TM1, (b) TM2.

Figure 4 -

 4 Figure 4 -Symbolic data objects defined for phases (c) 1, (d) 2 and (e) 3; symbolic distance matrix.

Figure 5 -

 5 Figure 5 -Data objects allocation to partitions comprising (a) 2 clusters, (b) 3 clusters.

Figure 6 .

 6 Figure 6. Real-time structural assessment output using a 10-minute time-resolution.

EWSHM 2014 -Nantes, France


Fabio Casciati

Sara Casciati

Zhicong Chen

Lucia Faravelli

Michele Vece

DEVELOPMENT OF A RELIABLE WIRELESS GNSS SENSOR NETWORK

Keywords: Critical Asset Tracking; GNSS sensor, Wireless sensor network, ZigBee

To localize, track and trace critical assets and first responders for remote emergency management are primary needs after a catastrophic event. In disaster areas, however, it is likely that the existing terrestrial communication network and the power grid will be cut off. Therefore, a reliable asset tracking system should be independent of those terrestrial networks, especially in terms of communication. This paper explores the development of a reliable low power local wireless communication network for portable GNSS-based positioning sensors and a centralized recording unit, as part of the independent communication network. The wireless sensors network is based on the ZigBee standard protocol and the IEEE802.15.4 compliant transceiver CC2530 from Texas Instrument. The network features low cost, low power, mesh topology, high reliability and high security.

INTRODUCTION

An ongoing research effort aims at improving the state of the art related to the exploitation of the potential offered by the satellite network. Based on precise positioning and timing system, coordination and operation in emergency scenarios would find a significant help in the transfer of information coming from a networks of units reading the Global Navigation Satellite System (GNSS). It is well known [START_REF] Casciati | Engineering Vibration Monitoring by GPS: Long Duration Records[END_REF][START_REF] Casciati | Monitoring a steel building using GPS sensors[END_REF][START_REF] Wu | Local positioning systems versus structural monitoring: a review[END_REF] that for structures requiring the accuracy of centimetres, global positioning systems (GPS) offer a valid solution because the technology covers a large area and it is suitable for the monitoring of constructions characterized by a large own period. One feature of GPS units to be placed across the area under inspection is to rely on internet for transferring the local information to a recording nucleus. However, in any disaster areas, it is quite likely that the existing terrestrial communication network and power grid will be cut off after the catastrophic event. Therefore, a reliable asset tracking system should be independent of those terrestrial networks, especially in terms of communication. This paper explores the development of a reliable low power local wireless communication network for portable GNSS based positioning sensors and a centralized recording unit, as part of the independent communication network.

Actually, the research activity reported in this paper is devoted to the implementation of a basic component within the European Union FP7 project named "Satellite Based Asset Tracking for Supporting Emergency Management in Crisis Operations (Spartacus)": a local low power wireless communication network. The Spartacus project aims to develop Galileo-ready satellite-based applications to address three main societal needs, i.e., (1) Track, trace, and localize critical transport assets especially in times of crisis and in case of major failure of existing networks (communications and power); (2) Track the flow of relief support goods from the sending side to the receiving side; (3) Support coordination and ensure the safety of first responders in disaster management operations. One major component of the project is to develop a portal, mobile and rapid-deployment communication infrastructure on site by integrating terrestrial wireless access technologies with satellite backhauling technologies, in which the low power local communication network is required to connect GNSS based tracking units and a centralized collecting unit. In terms of position tracking of critical assets, in addition to the basic position of the whole transportation vehicles, the high-accuracy displacement monitoring of their key mechanical components is important for the health monitoring to ensure the integrity and thus safety. Since many GNSS based tracking units are expected to be distributed in the transportation vehicle, the conventional cabling method suffers problems, such as inflexible installation and negative impact on the vehicle structure, which suggests the adaption of a local wireless communication network [START_REF] Casciati | Wireless links for global positioning system receivers[END_REF][START_REF] Casciati | Energy harvesting and power management of wireless sensors for structural control applications in civil engineering[END_REF][START_REF] Casciati | A multi-channel wireless connection system for structural health monitoring applications[END_REF]. Furthermore, in terms of position tracking of first responders and other onsite users, a wireless solution is also required. The local wireless communication network should operate long enough to support the first 24 hours after an emergency, without connection to the grid. Therefore, it should features low power. Furthermore, the network should have encryption function to ensure a high security. These requirement can be well met by the ZigBee standard network, which is a low-cost, low-power, wireless mesh network standard. ZigBee operates in the industrial, scientific and medical (ISM) license-free radio bands: 868 MHz in Europe, 915 MHz in the USA and Australia and 2.4 GHz in most jurisdictions worldwide. Data transmission rates vary from 20 kilobits/second in the 868 MHz frequency band to 250 kilobits/second in the 2.4 GHz frequency band. ZigBee networks are secured by 128 bit symmetric encryption keys. Its transmission distances range from 10 to 100 meters lineof-sight, depending on power output and environmental characteristics. In order to have a longer range, an external radio power amplifier can be adopted with ZigBee compliant transceiver. There are many ZigBee compliant transceivers and protocol stacks offered by different companies, which can implement the ZigBee network. In this paper, the CC2530 and the Z-Stack are adopted. As shown in Figure 1, the local wireless GNSS sensors network consists of several GNSS sensors integrated with a ZigBee node through RS232 serial port, and a collecting unit integrated with a ZigBee base station also through RS232 serial port. The GNSS sensors acquire their positions from the GNSS satellites signal periodically and send the position data to the centralized collecting unit through the multi-hop ZigBee mesh network. In order to pass the collected data to the remote centre, the collecting unit is connected to the internet through a satellite backhauling device. In this way, a local existing terrestrial communication infrastructure can be avoided.

SYSTEM ARCHITECTURE

Satellite Backhauling

GNSS Sensor

ZigBee Node

ZigBee Base Station

Collecting Unit

GNSS Antenna

GNSS Receiver

ZigBee Node RS232

DEVELOPMENT ENVIRONMENT OF THE TI CC2530 BASED ZIGBEE NETWORK

The CC2530 is a true system-on-chip (SoC) solution for IEEE 802.15.4, Zigbee and RF4CE applications, from Texas Instrument. It enables robust network nodes to be built with very low total bill-of-material costs. The CC2530 combines the excellent performance of a leading RF transceiver with an industry-standard enhanced 8051 MCU, in-system programmable flash memory, 8-KB RAM, and many other powerful features. In order to develop the CC2530-based ZigBee device, there are many software tools involved, as shown in Fig. 2 In addition to the software tools mentioned above, some hardware tools are needed to construct the whole development environment for the CC2530-based ZigBee-Serial Port adapter. The relevant hardware tools include debugger, development board, transceiver module and its adapting board, as shown in Figure 2. The debugger is used to download and debug the firmware for the CC2530 8051 microcontroller core. The development board is used to facility the access of the resource of the CC2530 transceiver module. The transceiver module integrates a CC2530 chip and its peripheral passive components pin out its GPIO (general purpose input and output) and power supply ports on a small PCB board.

HARDWARE OF THE ZIGBEE NODE AND BASE STATION

As shown in Figure 3, the ZigBee node is a ZigBee to RS232 adapter, simply consisting of a CC2530 module, an antenna, an UART to RS232 converter , a DB9 serial port connector, a 5V power supply connecter and a JTAG interface for downloading and debugging the firmware. As shown in Figure 4, the ZigBee base station is a ZigBee to USB (Virtual RS232) adapter, simply consisting of a CC2530 module, an antenna, an UART to USB bridge, an USB connector and a JTAG interface for downloading and debugging the firmware. The firmware application (named SerialApp, as shown in Figure 5) of the ZigBee nodes and the base station is built on the ZigBee protocol stack Z-Stack (Version ZStack-CC2530-2.5.1a) and implements a transparent data transfer between the serial port and the ZigBee stack. When a data packet is received from the air (ZigBee stack), the SerialApp will pass it to the serial port. When a data stream is sent to the serial port, the SerialApp will pass it to the air. Since the data over the air is based on packet and the packet length is limited, the data stream from serial port may need to be fragmented. The SerialApp adopts software flow control to frame the data steam from serial port into packets: the received serial data will be processed when the data number or the idle time exceeds a predefined threshold. For the reliability of the wireless data transfer, in addition to the single-hop acknowledgement, the Z-stack offer end to end acknowledgement function to enable the sending device get confirmation that a data packet has been delivered to its destination device. Since many GNSS sensors send their data to the centralized collecting unit simultaneously and the data of one GNSS sample is long and should be divided into more than one packet, the integrity of GNSS sample data many be damaged.

To avoid this problem, Z-Stack's message fragmentation function can be used, which is a process where a large message is broken down and transmitted as smaller fragments. The fragments of the larger message are then reassembled by the receiving device. Through the about discussed techniques, the SerialApp can achieve a reliable transparent data transfer between GNSS sensors and the collecting unit.

CONCLUSIONS

This paper presents a preliminary solution for the ZigBee-based low power wireless communication network of the European FP7 project Spartacus, adopting the CC2530 SoC transceiver and Z-Stack. The preliminary solution includes the hardware and basic firmware implementation of the standard ZigBee wireless sensors node, which was validated in laboratory environment.

Figure 1 :

 1 Figure 1: Wireless GNSS sensors network

  . Firstly, an integrated development environment (IDE) is needed to program and compile the firmware code for the 8051 microcontroller core integrated in the CC2530 transceiver. The adopted and compatible IDE software is IAR Embedded Workbench 8051. Secondly, a Zigbee protocol stack is needed, which is provided for free by Texas Instrument and named Z-Stack™. It is an open source stack. The latest version is ZStack-CC2530-2.5.1a. The developer just needs to design the application program based on the stack. Lastly, Texas Instrument also provides two other software tools for free to facilitate the debug and testing of ZigBee transceiver and wireless network, including SmartRF Studio 7 and Packet Sniffer.

( 5 )Figure 2 :

 52 Figure 2: Development Environment of the CC2530 based ZigBee Network

Figure 3 :Figure 4 :

 34 Figure 3: The ZigBee node hardware -Top vie w (left), bottom view (right), and block diagram

Figure 5 :

 5 Figure 5: Block diagram of the firmware application of the ZigBee node and base station

ACKNOWLEDGMENTS

The authors acknowledge the support of the European Union FP7 via the project SPARTACUS (313002) coordinated by Dr. A. Ferrari and Dr. C. Fuggini of D'Appolonia s.p.a..