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ARTIFICIAL NEURAL NETWORKS COMPARISON FOR A SHM PROCEDURE APPLIED TO COMPOSITE STRUCTURE CONFERENCE PAPER · SEPTEMBER 2012 DOI: 10.13140/2.1.1538.4006
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Italian Association of Aeronautics and Astronautics XXII Conference Napoli, 9-12 September 2013
ARTIFICIAL NEURAL NETWORKS COMPARISON FOR A SHM PROCEDURE APPLIED TO COMPOSITE STRUCTURE A. Alaimo1, A. Barracco1*, A. Milazzo2, C. Orlando1 1
Faculty of Engineering and Architecture, University of Enna “Kore”, Cittadella Universitaria, 94100, Enna, Italy 2
Dipartimento di Ingegneria Civile Ambientale e Aerospaziale dei Materiali, University of Palermo, Viale delle Scienze, 90128, Palermo, Italy *
[email protected]
ABSTRACT In this paper different architectures of Artificial Neural Networks (ANNs) for structural damage detection are studied. The main objective is to create an ANN able to detect and localize damage without any prior knowledge on its characteristics so as to serve as a realtime data processor for SHM systems. Two different architectures are studied: the standard feed-forward Multi Layer Perceptron (MLP) and the Radial Basis Function (RBF) ANNs. The training data are given, in terms of a Damage Index ℑD, properly defined using the piezoelectric sensor signal output to obtain suitable information on the damage position and dimensions. The electromechanical response of the assembled structure has been computed by means of a Dual Reciprocity Boundary Element code developed in the framework of piezoelectricity. On this basis, the neural networks are then used to recognize the location of the damage and its characteristics and the numerical results highlight the main differences on the the performances of the two different ANNs analyzed. Keywords: Structural Health Monitoring, Multilayer Perceptron, Radial Basis Function, Boundary Element Method
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INTRODUCTION The detection of damage in a structure is generally referred to as Structural Health Monitoring (SHM). The SHM is a very appealing technique that can allow to improve the safety and reliability of a structure and, particularly in aerospace industry, it can allow to reduce the downtime costs associated to the inspection procedure since automatic damage detection process can substitute the classical NDE test method used to monitor a component [1]. The problems related to damage detection represents a primary concern, particularly in the framework of composite structure. For this kind of structures, in fact, barely visible damage can occur. Moreover, one of the major in-service damage of composite aircraft strcutures is represented by disbonds between the stiffeners and the skin undergoing dynamic or post-buckling loads [2]. In order to address the problem of barely visible damage detection, different SHM systems, based on the use of smart materials, such as the piezoelectric materials, have been proposed [3-5]. Among this, piezoelectric patches have been widely used
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as strain sensing device that can effectively allow to detect damages in both metallic and composite structures [6-7]. The effective implementation of a SHM system relies on the synthesis of nondestructive technique (NDT), fracture mechanics, sensors technology, data manipulation and signal processing, and it can receive a great improvement through the use of an Artificial Neural Networks [8-10]. Artificial Neural Networks, usually called Neural Networks (NNs), are computational models which consist of an interconnected group of artificial neurons. These neurons process information using a connectionist approach to computation. The ANN technique generally provides an efficient tool to classify problems associated with nonlinearities, when they are well represented by input patterns, and to avoid the complexity introduced by conventional computational methods. Many researchers have studied ANN approaches for damage estimation of structures because of the versatility of the ANN model in dealing with various types of inputs and outputs. The debonding of a composite/aluminum beam structure with ANNs to identify the severity and presence of delamination is studied in [11], or the estimation of the delamination of composite beams using piezoelectric devices via modal analysis and ANNs [12]. Modal frequencies were used as multiple inputs, and the extent of delamination was successfully predicted. However, when the number of features was insufficient and the frequency ranges for measurements were varied, misclassification occurred and more features were needed for correct classification. Modal frequencies were used as multiple inputs, and the extent of delamination was successfully predicted. In [13] the back-propagation neural network technique with E/M impedance measurements is applied to identify damage and estimate its severity and location. In [14] NNs is used in conjuction with surface wave transmission functions to estimate crack depths in concrete. Through the blind tests using untrained data excluded during the training, it was found that ANNs are also an effective tool when the training of the neural networks is incomplete. In [15] is proposed a damage metric quantification based on the classification of impedance spectra achieved with a featuresbased probabilistic neural network. Feature vectors consisted of the resonance frequencies, resonance amplitudes, and damping factors. However, when the number of features was insufficient and the frequency ranges for measurements were varied, misclassification occurred and more features were needed for correct classification. On this basis, the present paper deals with the analyses of different Artificial Neural Networks (ANNs) architecture implemented in the framework of a SHM system. More particularly, two different architectures are studied: the standard feed-forward Multi Layer Perceptron (MLP) and the Radial Basis Function (RBF) ANNs. Training data, used as the inputs for the ANNs, are obtained in terms of a Damage Index ℑD which has been defined to manipulate the electrical response of the piezoelectric patch obtained by means of transient numerical Boundary Element analysis. This damage index has in fact revealed as an effective tool to identify the presence and the length of delamination on drop ply composite structures. The different ANNs architectures have been then analyzed and compared by a parametric study. For the first type, the MLP network, three training algorithms are considered: the Levenberg-Marquardt, the Conjugate Gradient backpropagation and the Gradient Descent with momentum. Moreover different complexity of the network are considered, by varying the number of hidden neurons. For the second type of ANNs instead, assumed that the number of hidden neurons is fixed due to the number of training data used to build and train all networks, a parametric analysis is made by changing the spread parameter.
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ARTIFICIAL NEURAL NETWORKS Neural Networks are biologically inspired artificial intelligence representations that mimic the functionality of nervous systems [16]. Biological neurons consist of synapses, dendrites, axons, and cell bodies: specific to the biological neuron is there ability to fire when activated. Artificial neurons are set on by the activation or transfer function. The activation function can be discontinuous (e.g. symmetrized Heaviside step function), or continuous (logistic sigmoid, or tangent sigmoid). Linear transfer function are also used. In an artificial neural network, each simulated neuron is viewed as a node connected to other nodes (neurons) through links to form a network. Each link is associated to a weight: the weight determines the nature and strength of one neuron's influence on another. Neural networks are organized in layers, input layer, output layer, and hidden layers (Figure 1).
Figure 1: Architecture of a Multi-Layer Artificial Neural Network: the 4-n-1 case.
ANNs can be implemented as software algorithms and/or hardware device. Currently, artificial neural network algorithms are widely available as software packages [17-19]. There are several types of ANNs: (a) feedforward; (b) autoassociative; (c) recurrent; (d) competitive; (e) probabilistic. As just pointed before in the present paper only the first and the latter architecture are studied.
4.1
Multi-Layer Feed Forward Perceptron
Feedforward neural networks compute the output directly from the input, in one pass. No feedback is involved. Feedforward networks can recognize regularity in data; they can serve as pattern identifiers. A generalized perceptron uses nonlinear transfer functions, φ j(·) (basis functions), which are placed between the input and the adder: y ( x )=g
(
N
∑ w i φi ( x ) i =0
)
(1)
where g (·) is the output transfer function, wi are the weights. Interconnected perceptrons form perceptron layers: several perceptron layers lead to a Multi-Layer Perceptron neural network. Multi-layer feedforward neural networks are trained through backpropagation, which is a generalized learning rule for neural networks with nonlinear differentiable transfer functions. Standard backpropagation uses the Gradient Descent algorithm; other used optimization techniques such as Newton-Raphson or Conjugate Gradients, etc. can also been used. The training through backpropagation is iterative, and can take several presentations of the training set. During training, the weights are optimized by minimization of a suitable error function, e.g., the sum-of-squared-errors
3
E=
4.2
2 1 y k ( x n ; wkj ) −t nk } { ∑ ∑ 2 n k
(2)
Radial basis neural network
Radial basis neural netowrks, a type of probabilistic neural networks, are hybrid multi-layer networks that use basis functions and competitive selection concepts. Probabilistic neural network achieve a Bayesian decision analysis with Gaussian kernel (Parzan window). A probabilistic neural network consists of a radial-basis layer, a feedforward layer, and a competitive layer. The radial basis functions depend on the distance between the input vectors and some prototype vectors, μj, representative of the various input classes. 2
M
y k ( x )= g
(∑
m=0
)
−
w km φm ( x ) , φm ( x )=e
∥x − μ m∥ 2σ
2 m
, φ0=1
(3)
Probabilistic neural networks identify class separation boundaries as hyper-spheres in the input space. They are trained in two stages [16]: the first stage consists of unsupervised learning: an adaptive design clusters the input vectors and determines prototype vectors that are representative of several input vectors. The radial spread parameters, σj, are adjusted to achieve optimal coverage of the input space. The second stage consists of supervised learning.
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MECHANICAL MODEL A drop-ply delaminated structure previously employed in [20] and [21], to represent a bidimensional simplification of the adasive joints between composite aircraft fuselage skins and stiffners is here analyzed to study the sensing capability of the SHM system proposed. The analyzed configuration is depicted in Figure 2 and consists of an host delaminated structure, made up by 0° and 90° graphite-epoxy (GE) plies and of a piezoelectric patch, employed to arrange the sensing device.
Figure 2: Drop-ply structure and piezoelectric sensor configuration.
The skin is arranged with unidirectional graphite-expoxy plies, whose material properties [22] are listed in Table 1, and is characterizied by a total tickness h1 = 5.08 mm and a length L1 = 50.8 mm. The geometry of the stringer foot is characterized by a length L2 = 35.6 mm, thickness of the 90° plies group h2 = 3.18 mm, thickness of the 0° plies group h3 = 1.9 mm. The drop-ply assembly deforms under plane strain conditions and it is clamped at the flange root, as shown in Figure 2. The structure undergoes several treansverse shear loads per unit length F acting on the free-edge of the skin. Moreover, the loads is modeled as a step load for
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dynamic analyses purpose with the aim of simulating the sudden raise of the shearing postbuckling action. Lastly a delamination of length a is assumed to occur at the skin-flange 0°/90° interfaces, starting from the flange free-edge. The piezoelectric patch is assumed to be bonded on the flange top surface. To realize a gridding electrodes configuration, both the top and bottom surfaces of the piezoelectric patch are supposed to be metallized with a pattern of evenly distributed electrode strips pairs Ei, as shown in Figure 2 by the bold black segments. C11 [109 Pa] C22 [109 Pa] C13 [109 Pa] C12 [109 Pa] C44 [109 Pa] ε 11 [10-9 F/m] ε 12 [10-9 F/m] e21 [C/m2] e22 [C/m2] e14 [C/m2] ρ [kg/m3]
PZT-4
PZT-5A
PECP
GE
138.499 114.745 77.371 73.643 25.600 1.306 1.115 -5.200 15.080 12.720 7600
99.201 86.856 54.016 50.778 21.100 1.530 1.500 -7.209 15.118 12.322 7750
79.700 66.800 35.800 35.800 17.200 15.920 15.920 10.500 15.200 -5.900 5730
137.699 13.699 6.166 6.166 5.520 1.530 1.530 5300
Table 1: Material properties.
Each couple of electrodes constitutes a probe of local strain and will be referred to as the ith sensor Ei, being the first the one closest to the flange free-edge. Electrodes length is LE = 4.5 mm and are placed at a distance de = 4 mm while the distances from the stringer foot and from the flange free-edge are d1 = 3.5 mm and d2 = 2 mm, respectively. For the transient analyses each electrodes pairs are considered to represent piezoelectric sources of time varying charge. The piezoelectric sensor Ei is modeled as a time varying charge source [23] Q(t) with a shunt capacitor Cp and a resistor Rp connected to acharge amplifier, Figure 3, in such a way the output voltage V(t) at the time instant t can be written as [24]: V ( t )=G ⋅ i ( t ) =
d Q (t ) dt
(4)
being G the charge amplifier constant gain that relates the output voltage to the electric current i(t) = dQ/dt, while the piezoelectric generated charge is computed as the integral on the electrodes surface of the electric displacement normal component Dn: Q ( t )=∫ Dn ( t ) dx
(5)
LE
The charge amplifies constant gain G can be appropriately selected to modify the piezoelectric frequency behavior [25] in such a way to let the output voltage V be proportional to the strain rate or to the strain of the piezoelectric itself.
Figure 3: Piezoelectric sensor circuit.
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TRAINING DATA GENERATION: BOUNDARY ELEMENT METHOD The boundary element model consists of 604 linear elements for a total of 2416 electromechnical dof. The mesh of the model is constructed in such a way the size of the boundary elements across the crack tip if ∆/a=0.05 and its effectiveness is assessed by comparing the fracture mechanics parameters computed via the BEM approach with the ones reported in [21] [26] obtained by means of FEM analysis. The dynamic response of the vertical displacements of the point where the load F acts is firstly computed and compared to that obtained through a FEA and good agreement is found. The time response of all the electrodes shows only the presence of the delamination by a change in the natural frequency of oscillation (as the crack induces an overall stiffness reduction) but no information on crack location are straightforwardly available. However, from the piezoelectric constitutive relationships it stems that the free charge Dn is proportional to the local strain. It is then expected that the crack causes a local strain change with respect to the undamaged case that originates from the stress field intensification near the crack tip or it is simply due to a local discontinuity in the beam deformation caused by the debonding. By observing the electric displacement Dn distribution along the piezoelectric patch top surface for two distinct delamination length in comparison with the undamaged configuration, the influences of strain variation induced by the delamination is found and, thus, it is inferred that analogue information can be extrapolated from the electrical current time response of each sensors. A Damage Index ℑD is then defined [27] in order to catch the delamination influence on the piezoelectric generated charge along the beam length: ℑ D=
∣qi∣− qi
(6)
qU
where the subscript i labels quantities pertaining to the i-th electrodes pair while the over bar is used to indicate the mean value of the variables. In Eq. 6 q i is a measure of the mean charge per unit of area accumulated at the electrodes and is defined as: 1 q i= Δt L E
t 0+ Δ t
(
t
∫ ∫ ii ( τ ) d τ i
t0
0
)
dt
(7)
while q U which is used to obtain a non-dimensional index, is the mean value of the mean surface charges qi read by the sensors when the structure is undamaged and, being NE the number of electrode pairs, is defined as: NE
1 qU= ∑q N E i=1 i , undamaged
(8)
In Eq. 7 Δt is the sampling time interval while t0 is the time delay between the dynamic response of the structures and the instant of interrogation of the SHM system. Thirty-two delamination length and twenty-one load cases are analyzed, namely a={1.02, 2.04, 3.06, ..., 31.62, 32.64} mm and F = {1.0, 2.0, 3.0, ..., 20.0, 21.0} kN/m. The damage index values computed for the four sensors are graphically reported in Figure 3 for a four delamination configuration, namely a = {5.1, 10.2, 15.3, 20.4} mm and F = 21,0 kN/m. The shaded areas are the ragions of the piezoelectric patch top surface (overtruned in site) where the electrodes have been placed while the grey bars represent the computed value of the Damage Index ℑD. The length of the flage-skin edge debond is also graphically given, as a red line stemming from x/L2 = 1, to help visualizing that the proposed index ℑD shows a peak only at the
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electrode closest to the delamination tip. Moreover it stems that electrodes which lie far from the crack tip respond with a zero Damage Index value.
Figure 4: Distribution of the Damage Index ℑD along the patch surface.
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NUMERICAL SIMULATIONS
5.1
Multilayer perceptron
Three types of ANNs are studied, a 4-n-1, a 7-n-1 and a 16-n-1 neural networks, where the first number is the number of neurons in the input layer (i.e. 4, 7 and 16 electrodes pairs), the n in the middle position is the variable number of neurons in the hidden layer, as explained in what follow, and the last number is the unique neuron in the output layer, i.e. the length of delamination. It is very difficult to know which training algorithm will be the fastest and precise for a give problem. It depends on many factors, including the complexity of the problerm, the number of data points in the training set, the number of weights and biases in the network, the error goal, and whether the network is being used for pattern recognition (discriminant analysis) or function approximation (regression). The training of the three MLP netowrks was made by use of a subset of the BEM generated datas, i.e. those corresonding to all delamination lengths but only for odd values of F: this choice is due to the possibility to test the extrapolation capacity of the networks since the simulation will be made on the even values of F. This section compare various training algorithms used to train the three types of MLP ANNs. Table 2 lists the tested algorithms and the acronyms used to identify them: Acronym LM GDX CGF
Algorithm Levenberg-Marquardt backprogation Gradient descent with momentum and adaptive learning rate back propagation Conjugate Gradient backpropagation with Fletcher-Reeves updates
Table 2: Training algorithm implemented.
In Figure 5 are plotted the training MSE vs the complexity of the first type of neural network (i.e. number of neurons in the hidden layer), for the three training algorithm in Table 2.
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Figure 5: MSE vs number of neurons, for the three algorithms, for the 4-n-1 network.
As it is clear the Levenberg-Marquardt algorithm has the best performances amongst the three. Moreover increasing the number of hidden neurons, say over 10 has a negligible effect.
Figure 6: MSE vs number of neurons, for the three algorithms, for the 7-n-1 network.
Figure 7: MSE vs number neurons, for the three algorithms, for the 16-n-1 network.
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A similar consideration can be done for the other two ANN types (Figure 6 and Figure 7), but with a slight difference: the minimum MSE is reached for a increasingly smaller hidden neurons number, respectively at 7-4-1 and 16-2-1 architectures. As aspected the 16-2-1 would be the best choice for accuracy, but, in order to contain the costs of an harware implementation of the systems, the 4-10-1 could be on optimum one. In Figure 8 the simulations results of the 4-10-1 network for the F = 1 kN/m and F = 21 kN/m case only are depicted: as it is clear the MLP neural network has a very good agreement with the expected values of the delaminations length a in both cases, with an improvement as the external load grow.
(a)
(b)
Figure 8: Prevision of the 4-10-1 MLP network with Levenberg-Marquardt training algorithm, for all delamination lengths and: (a) F = 1 kN/m, (b) F = 21 kN/m load.
5.2
Radial basis neural network
Radial basis networks can require more neurons than a standard feedforward backpropagation network, but often they can be designed in a fraction of the time it takes to train the latter. They work best when many training vectors are available [28]. It is a two-layer network. The first layer has radbas neurons, and calculates its weighted inputs using the euclidean distance and its net input with a product. The second layer has purelin neurons, and calculates its weighted input with dot product and its net inputs with sum. Both layers have biases. Initially the radbas layer has no neurons. The following steps are repeated until the network's mean squared error falls below goal: 1. the network is simulated; 2. the input vector with the greatest error is found; 3. a radbas neuron is added with weights equal to that vector; 4. the purelin layer weights are redesigned to minimize error. The main parameter of this type of ANN is the so-called Spread constant: this determines the width of an area in the input space to which each neuron responds. Spread should be large enough that neurons respond strongly to overlapping regions of the input space. A large spread means that a lot of neurons are required to fit a fast-changing function while a small value of spread means that many neurons are required to fit a smooth function, and the network might not generalize well. The standard RBF ANN is a network with zero error on training vectors. The only condition required is to make sure that spread is large enough that the active regions of the radial basis neurons overlap enough so that several radial basis neurons always have fairly large output at any given moment. This makes the network function smoother and results in better generalization for new input vectors occurring between input vectors used in the design.
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In order to compare the RBF network performaces, 4 different values of the spread constant are choosed, i.e. 1.0, 2.0, 5.0 and 10.0. After the 4 networks are created (4-320-1 with different spread constant), initialized and trained, a simulation was carried out, using the even subset of datas (Figure 9).
(a)
(b)
Figure 9: Radial Basis Function simulation as function of spread: (a) F = 1 kN/m, (b) F = 21 kN/m load.
As it is clear from the Figure 9, the best prevision is reached for a spread constant over 2.0. This reuslt is coherent with the observation that the larger spread is, the smoother the function approximation.
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CONCLUDING REMARKS This paper presents a study in order to assess the comparative performances of two important types of tneural networks, namely a Standard Multilayer Perceptron and a Radial Basis Fucntion neural networks for a structural damage detection system that uses a Damage Index ℑD distribution. Large number of simulations have been performed to generate data sets for training the networks and for subsequent testing and validation through a Dual Reciprocity Bourdary Element Method transient analysis of the host damaged structure. The BEM model allows to compute the electrical signals that are used to define the ℑD generated by an array of piezoelectric sensors bonded on a delaminated composite skin-stiffner configuration. It has been observed that the RBF tipically requires larger number of training patterns and also a larger network architecture to achieve the same lavel of desired accuracy as the MLP. On the other hand, an advantage of using RBF is that they require a training time inferior than the MLP. Thus, when the training data are not easily available or hard to generate, the MLP is more desiderable than the case when training data are abundantly available and adaptive control is preferential in which case the BRF is clearly superior. REFERENCES [1] W. J. Staszewski, C. Boller, G. Tomlison. Health Monitoring of Aerospace Structures – Smart sensor technologies and signal processing, John Wiley and Sons Ltd (2004). [2] S. Grondel, J. Assaad, C. Delebarre, E. Moulin. Health monitoring of composite wingbox structure, Ultrasonics, 42, pp. 819-824 (2004).
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[3] I. Chopra. Review of State of Art of Smart Structures and Integrated Systems, AIAA Journal, 40(10), pp. 2145-2187 (2002). [4] D. M. Pearis, G. Park, D. J. Inman. Improving accessibility of the impedance-based Structural Health Monitoring, Journal of Intelligent Material Systems and Structures, 15, pp. 129-139 (2004). [5] T. Liu, V. Martin, K. Sritawat. Modelling the input-output behaviour of piezoelectric structural heath monitoring systems for composite plates, Smart Materials and Structures, 12(5), pp. 836-844 (2003). [6] J. Beuth. Separation of crack extension modes in orthotropic delamination models, International Journal of Fracture, 77, pp. 305-321 (1996). [7] S. Narayan, J. Beuth. Designation of mode mix in orthotropic composite delamination problems, International Journal of Fracture, 90, pp. 383-400 (1998). [8] W. J. Staszewski. Intelling signal processing for damage detection in composite materials, Composite Science and Technology, 62, pp. 941-950 (2002). [9] S. Yuan, L. Wang, G. Peng. Neural network method based on a new damage signature for structural health monitoring, Thin-Walled Structures, 43, pp. 553–563 (2005). [10] G. R. Kirikera1, J. W. Lee1, M. J. Schulz, A. Ghoshal, M. J. Sundaresan, R. J. Allemang, V. N. Shanov, H. Westheider. Initial evaluation of an active/passive structural neural system for health monitoring of composite materials, Smart Materials and Structures, 15, pp. 1275–1286 (2006). [11] Z. Chaudhry, A. J. Ganino. Damage detection using neural networks: an initial experimental study on de-bonded beams, Journal of Intellingent Material Systems and Structures, 5, pp. 585–958 (1994). [12] A. C. Okafor, K. Chandrashekhara, Y. P. Jiang. Delamination prediction in composite beams with built-in piezoelectric devices using modal analysis and neural network, Smart Materials and Structures, 5, pp. 338–347 (1996). [13] Lopes V Jr, Park G, Cudney H H and Inman D J. Impedance-based structural health monitoring with artificial neural networks J. Intell. Mater. Syst. Struct., 11, pp. 206–214 (2000). [14] S. W. Shin, C. B. Yun, H. Furuta, J. S. Popovics. Nondestructive evaluation of crack depth in concrete using PCA-compressed wave transmission function and neural networks, Experimental Mechanics, 48, pp. 225–231 (2008) [15] V. Giurgiutiu. Comparative study of neural-network damage detection from a statistical set of electro-mechanical impedance spectra in ''Procedings of SPIE'', Smart Structures and Materials, 108, pp. 5047–15 (2002). [16] C. M. Bishop. Neural Networks for Pattern Recognition, Clarendon Press, ISBN 0-19853849-9 (1995). [17] R. G. Cowell, A. P. Dawid, S. L. Lauritzen,D. J. Spiegelhalter. Probabilistic Networks and Expert Systems, Springer-Verlag (1999). [18] H. Demuth, M. Beale. MATLAB Neural Network Tool Box, The Math Works, Inc.; Natick, MA (2000). [19] Hagan, M. T.; Demuth, H. B.; Beale, M. (1996) Neural Network Design, PWS Publishing Co.; Int. Thompson Publishing (2002). [20] 55J. Beuth. Separation of crack extension modes in orthtropic delamination models, International Journal of Fracture, 77, pp. 305-321 (1996). [21] 87S. Narayan, J. Beuth. Designation of mode mix in orthotropic composite delamination problems, International Journal of Fracture, 90, pp. 383-400 (1998).
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[22] A. Alaimo, A. Milazzo, C. Orlando. Numerical analysis of a piezoelectric structural health monitoring system for composite flange-skin delamination detection, Composite Structures, 100, pp. 343-355 (2013). [23] 127B. Lin, V. Giurgitiu. Modeling and testing of PZT and PVDF piezoelectric wafer active sensors, Smart Materials and Structures, 15, pp. 1085-1093 (2006). [24] 126 K. Kuang, S. Quek, W. Cantwell. Use of polymer sensors for monitoring the static and dynamic response of a cantilever composite beam, Journal of Material Science, 39, pp. 3839-3843 (2004). [25] 125J. Sirohi, I. Chopra. Fundamental understanding of piezoelectric strain sensors, Journal of Intelligent Material Systems and Structures, 11, pp. 246-257 (2000). [26] A. Alaimo, A. Milazzo, C. Orlando. Boundary elemets analysis of adhesively bonded piezoelectric active repair, Engineering Fracture Mechanics, 76, pp. 500-511 (2009). [27] A. Alaimo, A. Milazzo, C. Orlando. Numerical analysis of a piezoelectric structural health monitoring system for composite flange-skin delamination detection, Composite Structures, 100, pp. 343-355 (2013). [28] S. Chen, C. F. N. Cowan, P. M. Grant. Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks, IEEE Transactions on Neural Networks, 2, pp. 302209 (1991).
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