Optimized damage identification in CFRP plates by

Engineering Structures 181 (2019) 111–123

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Optimized damage identification in CFRP plates by reduced mode shapes and GA-ANN methods

T

⁎

Guilherme Ferreira Gomesa, , Fabricio Alves de Almeidab, Diego Morais Junqueiraa, Sebastiao Simões da Cunha Jr.a, Antonio Carlos Ancelotti Jr.a a b

Mechanical Engineering Institute, Federal University of Itajubá, Brazil Institute of Industrial Engineering and Management, Federal University of Itajubá, Brazil

A R T I C LE I N FO

A B S T R A C T

Keywords: Damage identification Sensor placement optimization Structural health monitoring Inverse problem Artificial neural networks Composite plates

Delamination is one of the most common failure mode in laminated composites that leads the separation along the interfaces of the layers. The structural performance can be significantly affected by this degradation. Such damages are not always visible on the surface and could potentially lead to catastrophic structural failures. The existence of delamination alters the vibration characteristics of the laminated structures, so if they are detected and measured previously, they can be used as indicator for quantifying health and the potential risk of catastrophic failures. To ensure structural performance and integrity, accurate Structural Health Monitoring (SHM) is crucial. In this study, an optimized methodology for delamination identification on laminated composite plates involving the use of reduced mode shapes and computational tools, i.e., Genetic Algorithm (GA) and Artificial Neural Networks (ANN) is performed. In a first step, the sensor distribution on the surface of the structure was optimized using Fisher Information Matrix (FIM) criteria. After, GA and ANN were applied in order to identify and predict delamination location. A feed-forward based neural network is used in order to detect damage on the laminated plate using data obtained from Finite Element Analysis (FEA). The present methodology identifies damage localization in structures and also quantifies damage severity. The applicability of the technique is demonstrated on laminated plates and results are compared with numerical algorithms. This paper shows the effectiveness of GA and ANN as tools for delamination damage identification problem. The algorithms in their inverse formulations are capable of predicting accurately delamination position in plates-like structures.

1. Introduction The application of advanced materials such as composites in components and structures have envolved in last decades due to the need for improvements in terms of weight/strength ratio performance. In addition, good fatigue strength and high structural performance, have also been significant contributions to the rapid increase of the application of those materials. However, according to [29], composite materials are subjected to high structural load demands due to their good mechanical performance. Due to high loads, damage may occur internally to the material. Knowing the damage and Being able to detect, locate and identify the type of damage in a structure is crucial for engineers. However, the development of techniques capable of detecting, locating, identifying and characterizing composite damages are still considered a challenge. Actually, most of the non-destructive inspection techniques applied in composite structures require high levels of operator experience since the inspection procedure and the interpretation

⁎

of the results are very complex steps. A reliable and effective non-destructive damage identification method is crucial to maintain the safety and integrity of mechanical structures (aircrafts, ships, buildings, etc.). The most common non-destructive damage identification techniques include visual inspection and conventional nondestructive testing (NDT), such as tap coin, ultrasonic inspection, penetrating liquids and thermography. However, the visual inspection techniques are unable to detect damage which is embedded in a structure or invisible to human eyes while the conventional NDT requires that the vicinity of damage be known a priori and readily accessible for testing [12]. Vibration-based damage identification method has been widely used as NDT [23,24]. Many approaches based on this method have been proven to be effective in addressing problems in both basic and complex structures [20]. This approach explains that damage can affect both the physical and dynamic characteristics of the structural properties. Physical characteristics include the mass, stiffness and damping, while

Corresponding author. E-mail address: [email protected] (G.F. Gomes).

https://doi.org/10.1016/j.engstruct.2018.11.081 Received 29 March 2018; Received in revised form 30 October 2018; Accepted 30 November 2018 0141-0296/ © 2018 Published by Elsevier Ltd.

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Nomenclature

[A] [B] [C ] [K ] [M ] FIM α β λ ν ω Φ ϕ ρ

E G J Ke Ne u, v, w x , y, z CFRP FEA [Q] FRF GA NDT SHM SPO

coordinates matrix shape function matrix elastic matrix stiffness matrix mass matrix Fisher information matrix stiffness multiplier stiffness reduction factor eigenvalue Poisson’s ratio natural frequency mode shape eigenvector density

Young’s modulus shear modulus objective function damaged element damaged element number Cartesian displacements Cartesian coordinates carbon fiber reinforced polymer finite element analysis Fisher information matrix frequency response function genetic algorithm non destructive testing structural Health Monitoring sensor placement optimization

identification method, damage scenario and sensing technology. The characteristics of these components are closely interlinked and, together, define the performance of the proposed strategy. Ideally, a good strategy combines a high level of detection with a low number of false positives. The success of a damage identification strategy depends, however, on the structure and damage scenario that is considered. The selection of the most appropriate approach is therefore far from trivial and direct [41]. As already pointed out, SHM technology is a field that requires a deep understanding of materials, sensors, and the ability to perform sophisticated numerical and analytical modeling and signal processing. Each of these topics is a matter in its own right, and developing a work incorporating all of the above is really a difficult task [27]. Modeling is an important component in SHM technology. Simulated data are used to support the development of new algorithms for damage detection/ identification, or for a better understanding of the effects of structure response damage. One of the specific objectives of this research is therefore to present some computational tools that can be used for structural damage identification. Approaches based on mechanical vibrations are emphasized in this study because the presence of a damage in the structural components results in the modification of the stiffness matrices thereof, which implies in the modification of the structural dynamic response. The use of dynamic methods is justified in the first place because they do not require the structure to be easily accessible and because low frequency methods provide relatively easy data to be interpreted. The ability to identify damaged components in aerospace, mechanical and civil systems is becoming increasingly important [9,12,8]. Although considerable efforts have been devoted to the diagnostic and prognostic community to develop effective methods of diagnosing and identifying damages over the past decades, the identification of structural damage is still considered a practical challenge for the safety assurance of engineering structures [28]. Although some studies have been reported on the structural damage identification based on vibration signals, very few have been focused on the complete analysis using sensor optimization and GA-ANN. The work presented here assesses the potential of computational techniques (global optimization and ANN). Studies investigating the use of ANN based in reduced mode shapes are scarce and need be more developed. The reduced mode shape is then obtained by FIM optimization. In other words, the objective and main focus of this paper is based on the solution of the inverse problem and the recognition of patterns of damage identification in composites plates, making use mainly of computational intelligence using structural modal parameters in order to identify the possible location of a structural damage. This study will contribute to the scientific community and engineers for the development of tools and in the diagnosis of damage identification in composite materials.

dynamic characteristics include the frequency response functions (FRFs), natural frequencies, damping ratio and mode shapes [42,16]. Finding the best solution for a particular problem is an important area of research and its application can be found in many engineering fields. A variety of different optimization problems can be identified in Structural Health Monitoring (SHM) community, where a number of different possible mathematical tools are offered to find a better solution [51]. GA’s have been of considerable interest since they provide a robust solution to complex problems. Because of the way in which the genetic algorithm exploits the region of interest, it avoids getting stuck to a local minimum point, i.e., a non-optimal solution of the problem in question. ANN’s have evolved as one of the promising artificial intelligence concepts used in real world applications. In like manner, ANNs have been extensively used in structural engineering applications in the area of failure prediction, delamination identification, crack identification, etc. ANN is essentially a system that contains many simple and highly interconnected neurons which processes information based on an architecture inspired by the structure of the cerebral cortex of the brain. ANN is used to derive a relationship between a set of input parameters and their output responses [30,22]. Some researchers have proposed damage assessment based on analysis of measured dynamic responses of the structure before and after damage [25,4,13]. More recent works in the structural health monitoring are tending to utilize artificial intelligence tools as well ANN to assess damages in mechanical structures [11,42,5,17]. The most significant advantage of using ANN is that it gives excellent pattern recognition, auto-association, self-organization, self learning, and nonlinear modeling capability. ANN is capable of extracting and obtaining precise and reliable information from imprecise, unreliable, inconsistent, uncertain, and noise-polluted data. As a result, ANN is fault tolerant which makes the fault diagnosis procedure automatic, once the network is correctly trained [1]. Some studies in the literature present a wealth of detail of several specific contributions of damage identification from modal data [54,39,49,52]. However, there is still a certain space to be explored with regard to predicting damage through artificial neural networks. The main novelty of this study is the introduction of reduced mode shapes in the ANN input. The reduced mode shapes are obtained by Sensor Placement Optimization (SPO) using Fisher Information Matrix (FIM) criteria. This strategy allows a reduction of cost and weight in aircraft structural health monitoring systems, since the technique allows reducing the number of sensors needed to obtain a good level of damage identification. The development of a strategy for SHM involves relevant research challenges. The multidisciplinary framework associated with this challenge includes four main components: structure, damage 112

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This manuscript is organized as follows: Section 2 the methodological procedure is presented. Damage identification modeling is shown, including SPO and some evaluation metrics. Section 3 presents the main results and discussion. Finally, Section 4 draws the conclusions.

Table 1 Mechanical properties of the CFRP applied in FEM direct problem modeling.

2. Damage identification problem modeling 2.1. Direct problem: finite element modeling Proper mathematical models are required for the post-processing of the response to predict the location of damage. Among the most used mathematical models is the finite element method (FEM). FEM is a powerful numerical technique for solving problems governed by differential equations in complex domains. It is usually adopted to solve direct problems in structures, that is, for a given load (input), it can be determined the stresses and deformations that the structures undergo (output) [32]. However, SHM requires an estimate of the state of the structure from the measured output (strain, stress, natural frequencies, mode shapes, etc.) to a given predefined input. Therefore, SHM falls under the domain of the system identification problem. These problems are also called inverse problems [26]. In this Section, the main purpose is not to address the theoretical foundation and procedure of the FEM as such, since many classic texts and with a wealth of detail are already available in this area. The use of FEM in this paper is justified by the application of the finite element updating model, where the numerical model in finite elements will be coupled with intelligent signal processing techniques and both will work together in an attempt to locate structural damage. As mentioned by [3] and emphasized by [14], there is no need for a SHM system to locate damage with millimetric precision. The cost and effort involved in predicting damage to a high level of accuracy can be prohibitive. Furthermore, due to measurement inaccuracies in the model and signal processing, an SHM system that claims to predict damage with great accuracy is likely to give false alarms. A better idea, according to [3] is to locate the damage in about a meter of precision using the SHM methodology and then use traditional inspection methods for a more detailed analysis of the damaged area. Therefore, the modeling of the damage as a reduction of local stiffness in this study justifies the choice in the interaction with all the proposed methodology. Even more, the method is used to calculate modal parameters such as natural frequency, mode shapes, and strain energy distribution of each mode for a laminated composite plate with or without delamination. By introducing a Cartesian global coordinate system x , y, z into a rectangular plate of uniform thickness, the displacements of a point ( x , y, z ) on the plate along the axes x , y and z are u, v and w, respectively. The finite element used in this paper is the eight-node rectangular shell element. For each node, there are six degrees of freedom, that is, translations and rotation in the axes. For a finite element of eight nodes with six degrees of freedom per node, the stiffness matrix of the element can be written as [53]:

[K e] =

∫V [B]T [A][C ][A]−1 [B] dV e

Value

E1 E2 G12 ν12 ρ

83.02 GPa 5.13 GPa 8.37 GPa 0.32 1408.10 kg/m3

About the model used in this study, two-dimensional studies on plate-like structures are considered. All the direct problem modeling was made in ANSYS®mechanical APDL software. The quality of a good mesh is essential in obtaining reliable results. It was decided to use a structured mesh, since the structure in question is uniform, two-dimensional and it is added the fact that a computational economy is obtained by opting for such a configuration. The amount of elements was also chosen in such a way as to obtain a sufficient convergence in the evaluated responses. Section 3 will provide more information about the decision to use a discrete structure in 10 × 10 elements. The plate was clamped in all four edges. Regarding geometry, the direct problem was modeled as a square side plate equal to 30 cm. The structure studied consists of a composite laminated plate made of unidirectional carbon/epoxy layers assembled in a stacking sequence [0/90]3S . It should be emphasized that this paper is exclusively aimed at the study of the method of detecting damages in laminated composite materials, not emphasizing the geometric parameters and specific characteristics of the material in question. Table 1 displays the set of main properties that were employed in modeling the problem via FEM. In this study, the modal results calculated for different structural conditions (undamaged and damaged states) are used in the corresponding damage identification algorithms to locate simulated damages. 2.1.1. Damage: local stiffness reduction It has been previously argued that there is an increase in the reduction of stiffness due to the increase in the size of the delamination [36]. Still, several works [55,48,56,35,40] have studied, addressed and made evident the reduction of stiffness due to delamination damages. For this, the modeling of delamination by means of a numerical approach of the percentage reduction of local stiffness is valid. According to [10], for a discrete, undamaged structure, the eigenvalue equation can be written as:

[K ] qi = λi [M ] qi for i = 1, …, n

(3)

where [K ] and [M ] are the stiffness and mass matrices, respectively. λi the i-th eigenvalue, qi the i-th eigenvector, and n is the number of modes evaluated or available for the structure in question. If the structure is then subjected to some kind of damage, its stiffness is then altered (as seen in Section 1, this research field treats structural damage as a local change in structural stiffness), so Eq. (3) can be rewritten as:

(1)

where [C ] is the elastic constant matrix of the material, [A] the coordinate transformation matrix and [B] represents the matrix related to the shape function of the element. Assuming that the composite plate has a harmonic motion with an angular frequency ω , and using the Lagrange principle, the motion equation for the free vibration of the composite plate is reduced to a standard problem of eigenvalue as follows:

([K ] − ω2 [M ]){δ } = 0

Property

∼ [K ] qj ̃ = λj ̃ [M ] qj ̃ for j = 1, …, m

(4)

∼ where [K ] is the mass matrix, λj ̃ the j-th eigenvalue, qj ̃ the j-th eigenvector and m the number available or calculated modes of the damaged structure. Here, the mass matrix [M ] of the structure is maintained constant even after the insertion of the structural damage, which translates into a mathematical character the behavior of a delamination, which is a common and dangerous damage in composite materials, where there is detachment without loss of material (mass). The matrices [K ] and [M ] are assumed to be definite positive

(2)

where [M ] is the global mass matrix. Modal plate parameters, such as mode shapes, natural frequency and modal deformation, can be obtained by solving Eq. (2). 113

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where the sensors are located in the structure. Cost and practicality issues prevent the instrumentation of all points of interest in the structure and lead to the selection of a smaller set of measurement sites [2,19]. Traditionally, a successful sensor distribution has been heavily dependent on the knowledge and experience of those conducting experimental tests. Practical methods, for example, by choosing sites close to anti-knots of low-frequency mode shapes, are combined to create coherent sensor distributions [2]. However, for a lot of the times, a single mode shape does not have enough information on a damaged structural state, being necessary the use of a set of modes [21]. Therefore, a distribution in the anti-nodes would not be feasible for a pre-defined number of sensors. According to the theory discussed in [2], the objective of sensor placement can be stated as the need to select a subset of measurement locations from a large finite set of locations, so as to represent the system with the highest possible accuracy using a limited number of degrees of freedom accessible. In general, on the first aspect, the minimum requirement for the system to be observable is that the number of sensors required cannot be less than the number of modes to be identified. For a limited number of available sensors, the problem is the development of a suitable sensor placement performance measurement to be optimized and the selection of an appropriate method. Some approaches require a single calculation to be performed, some are iterative, and many others take the form of an objective function to which an optimization technique must be applied. The third and last aspect includes several possibilities for evaluating the performance of chosen sensor sets. In this study, preferably the positioning item will be approached, where a pre-defined number of sensors will be fixed. In addition, the sensor placement issue attracts a lot of attention from academia and industry, especially due to the growing number of large instrumented monitoring structures over the last decade. This is due in part to economic reasons, to the high cost of data acquisition systems (sensors and their supporting instruments), partly because of the limitations of structural accessibility [44]. Some performance indices have been developed for the sensor distribution problem, but it is only comparatively recent that the problem was considered in the SHM perspective, according to [43]. In this study, a Fisher Information Matrix (FIM) criteria is used to obtain the best sensor layout.

symmetries and thus the eigenvalues are positive and the eigenvectors ∼ can be taken as [K ]-ortogonal. Similar conditions apply to [K ], λj ̃ and qj .̃ Considering the mass normalization of the modes, orthogonality conditions are defined by:

δij = 0 for i ≠ j ∼ qiT̃ [K ] qj ̃ = δij λi ̃ with ⎧ = = ⎨ ⎩ δij 1 for i j

(5)

Since the stiffness matrix of the damaged structure is given by ∼ [K ] = [K ] − δ [K ], the perturbation in the matrix corresponding to the element e is given by δK e = αK e where α ∈ [0, 1] is a local stiffness multiplier. The α parameter translates the severity of the structural damage, it can writed as β the local stiffness reduction percentage as β = (1 − α ) × 100 . As mentioned earlier, the plate under study was discretized into 100 finite elements, the reference system for such elements is shown in Fig. 1, in addition, an element will be predefined as a damaged element where computational methodologies here shall be applied. It is important to note that there is a difference in the detection, location and identification of damages. In this study, the damage is induced to the structure so that the ability to locate it in an already known position can be verified. Therefore, the focus of this study is restricted to the identification (location) of the damage, starting from the hypothesis that the damage is already present in the structure. This study is dedicated to the identification of structural damages in laminated composite structures, especially regarding the identification of delamination. Damage was modeled as a local loss of structural stiffness, since this is a the main failure modes of composite materials. Simulated damage as a reduction of stiffness faithfully models the mechanically structural behavior of delamination, where there is in fact a very strong relationship between the size of delamination and reduction of stiffness [31]. Equally important, It is true that there is an increase in the reduction of stiffness due to the increase in the size of the delamination [36]. However, several other works [55,48,56,35,40] have studied, addressed and made evident the reduction of stiffness due to delamination damages. The delamination, bearing, an important failure mode in composite materials that may not be visible on the structural surface, but is capable of affecting strength and stiffness [15]. 2.2. Sensor placement optimization

2.3. Fisher information matrix

The basic problem of damage detection/identification is to deduce the existence of damage to a structure from measurements made on distributed sensors. It is known that the quality of these measurements, that is, the quality of the structural monitoring, is largely dependent on

As shown by [33], the array of sensors can be given in the form of a estimation problem with a corresponding Fisher Information Matrix (FIM) given by

Fig. 1. Numerical structural damaged modeling by stiffness reduction. 114

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[Q] = ϕsT [W ] ϕs

modified due to damage, the stiffness matrix and mass are also modified in a certain way. For the sake of simplicity, assume that only the stiffness is reduced, keeping the mass constant in this particular case of damage, the problem of the eigenvalue of the damaged structure is expressed as:

(6)

where [W ] is a weighting matrix. The modal response is estimated based on the data measured by the sensors. The maximization of [Q] results in the minimization of the corresponding error covariance matrix, which results in the best estimate. The sensors must be placed in such a way that [Q] is maximized in an appropriate matrix norm. The maximization of the FIM determinant is a commonly used criterion for the estimation of optimal parameters. Maximizing the determinant of the FIM will maximize a combination of the spatial independence of the target modal partitions and their signal strength in the sensor data [34]. It has been shown that the FIM can be decomposed into the contributions of each candidate sensor location in the form [33]: nc

[Q] =

∑ i=1

[([K ] + [ΔK ]) − (λi + Δλi )[M ]]([Φi] + [ΔΦi]) = 0

where Δλi and ΔΦi the i-th eigenvalue and variation of the mode shape with respect to the reduction of stiffness ΔK , respectively. Eq. (9) can be simplified by neglecting the terms of the second order:

([K ] − λi [M ])ΔΦi = Δλi [M ]Φi − [ΔK ]Φi

=

∑

Qi

i=1

(10)

The variation in the mode shape ΔΦi can be described as a linear combination of the so-called ”pristine” modes, [57], such as:

nc

ϕsiT ϕsi

(9)

P

(7)

ΔΦi =

being ϕsi a i-th line of the mode partition array associated with the i-th candidate sensor location, nc is the number of candidate sensors. Then, the sensors must be placed so as to provide the best estimate of the target modal response. The maximization of the determinant of the information matrix is chosen as the criterion of positioning of the sensor, since it results in the maximization of the signal intensity and the independence of the main directions [45].

∑

dik Φk

k=1

(11)

being dik a scalar factor corresponding to the i-modes of the original system and P the number of total modes. In order to determine dik , Eq. (11) is replaced in Eq. (10), where both sides are multiplied by ΦTr (r ≠ i) . The coefficient dir can then be computed as:

dir = −

2.4. Modeling the inverse problem of sensor optimization

ΦTr ΔK Φi λr − λi

(12)

Here, r = i and no variation is present in the system mass matrix, dii = 0 , using the orthogonality principle (ΦTi M Φi = 1). Substituting Eqs. (11) and (12) into Eq. (10), the variation in the i-th mode shape due to the damage can be expressed as:

The objective in this part of the study is to apply the evolutionary method of optimization of positioning of sensors using GA, for the precise modal identification in mechanical structures. A discrete-type optimization problem using genetic algorithm is formulated by defining the positions of the sensor according to the criteria quoted in the previous paragraphs. The modal parameters (modes shapes and natural frequencies) of the real structure are obtained numerically using the finite element model. Thus, to identify the optimum location of n sensor candidates, global optimization algorithms will be used for this purpose. For example, if one were to work with 9 sensors distributed on the surface of the structure, taking into account a search space composed of 341 possible nodes (plate meshed in 100 elements with 8 nodes per element), this would result in a combination of 1.55 × 1017 locations. The time of evaluation in all these possible combinations justifies the use of advanced optimization methods. The mathematical formulation of the inverse problem can be summed up in maximizing de FIM determinant J = det ([Q]) . Subject to the constraints imposed of lower and upper bounds in relation to the maximum number of nodes and type x i − x i − 1 ⩾ 1, where x i is the position of the candidate node. This restriction is necessary to not get multiple sensors in the same location. The search space is displayed in Fig. 2 where the possible 341 sensor positions are considered.

p

ΔΦi =

∑ r = 1, r ≠ i

−ΦTr ΔK Φi Φr λr − λi

(13)

It is notoriously difficult to model a general type of damage in its details. In this research, the damage is modeled as a loss of local stiffness, that is, simulated as a scalar multiplier in the global stiffness matrix: N

ΔK =

∑ k=1

αk K k

(14)

being Kk the stiffness of k-element and αk the coefficient corresponding to the reduction of stiffness of this same element and L the total number

2.5. Criteria and metrics for damage identification As reported in the previous paragraphs, it is known that the local change in stiffness induces a change in the dynamic response. The response is dependent on the mass and stiffness matrices. To prove that in fact the modal response, more precisely the modal deformation is altered as a result of the presence of a local damage, the difference of the modes of vibration is introduced. The already widely known eigenvalue problem of a dynamic system can be defined as [58] apud [46]:

([K ] − λi [M ])Φi = 0

(8)

where K and M are the stiffness and mass matrices, Φi and λi the eigenvector and eigenvalue associated with the i-th mode shape. Since the structural properties (either material or geometry) are

Fig. 2. Configuration of 341 possible sensor positions (nodes) on the 10 × 10 elements meshed plate. 115

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iteration by the optimization algorithm referring to the mode i and ϕireal ,s are the known displacements of the structure which possibly possesses some structural damage, obtained at the optimal nodal points s. Another important aspect refers to the parameters used in the optimizer. In this case, genetic algorithms were used because of their proven high efficiency in similar cases. The configuration of the GA’s main parameters was established after performing some test cases until a satisfactory set of values were obtained. The main advantage of GAs is in their robustness, being able to find the global optimum without being stagnant in local points in multimodal functions. In fact, no optimal criterion of the algorithm parameters was employed in this study. A trial and error was employed until an acceptable minimum and a repeatability were obtained in the different runs of the optimizer. In addition, there is no mathematical proof that in practical, complex cases the GA converges to the global optimum. This can be proven only by tests: trying out your GA on several appropriate example problems. GA can only approach the global optimum with appropriate preciseness (just like the nature does). This is why GA is especially appropriate for use in this inverse damage identification problem, where no absolute preciseness is needed. Table 2 displays the genetic operators used in this part of the search. Population size was set at 10 × Nvar , where Nvar is total number of design variables in the problem, that is, in the case where the aim is to identify the position of the damage and its local severity, there are two x = [Ne , α ], so a population of 20 inproject variables involved → dividuals was sufficient to achieve convergence in the solution of the problem. The criteria for the issue is defined by the maximum number of generations of the GA individuals, that is, 100. This value was chosen because the convergence of the algorithm for several cases occurred in much smaller generations. Therefore, 100 generations would guarantee a convergence to another similar case. For the problem of optimal positioning of the sensors, where modeling the inverse problem has been discussed in previous sections of this chapter, if needed, to expand the number of individuals present in the population to 10 × Nvar × Nmodes , where Nmodes is the number of modes considered in the process of optimal positioning of the sensors. The stopping criterion was also established as the maximum number of generations, but increased to 1000per generation to meet sufficient convergence criteria.

of elements in the structure. It should be noted that the 0 ⩽ αk < 1. This model is suitable for most types of damage in real structures. It works even for a damage whose change is not proportional to elemental stiffness, because this assumption gives only a small error to a large structure and does not change the essence of the requirements at the location of damage [47]. By replacing Eqs. (14) and (13), the change of the i-th mode can be represented as a sum of the contribution of each damage to the modal form in the structure. Thus (13) becomes: L

n

∑

ΔΦi =

∑

αk

k=1

r = 1, r ≠ 1

−ΦTr Kk Φi Φr = F (K ) δA λr − λi

(15)

where n

−ΦTr K1 Φi ⎛ Φr F (K ) = ⎜ ∑ λr − λi ⎝ r = 1, r ≠ 1 n

∑

−ΦTr KL Φi λr − λi

r = 1, r ≠ 1

n

∑ r = 1, r ≠ 1

⎞ Φr ⎟ ⎠

−ΦTr K2 Φi Φr … λr − λi

(16)

With:

δA = {α1 α2 … αL }T

(17)

where F (K ) is the vector of sensitivity coefficients of the i-th modal form in relation to the damage vector δA . If several modes are used, ΔΦi would become the shape change matrix of the measured mode and F (K ) would become the sensitivity matrix for the selected modes. 2.6. Inverse problem modeling: optimization and artificial neural networks Structural damage assessment can be performed by comparing measured data (actual data) with simulated data. To provide the simulated data, a numerical code is necessary in which a problem model is used by an inverse problem algorithm. For the direct problem, a model is needed to obtain information about the response of interest of the structure with damage, for this, the boundary conditions and the positioning of the damage are considered. For the inverse problem, a model is required for the procedure to locate the damage in the structure, providing some (partial) information about the amount of interest in some specific locations [37]. Then, for the inverse problem modeling, optimization algorithm and artificial neural network is used. The algorithms were created in MATLAB®environment. The parameters of each algorithm will be described in the next sections.

2.6.2. Identification by artificial neural networks Artificial neural networks have shown promise for fault diagnosis. In the most basic approach to supervised learning, the network is presented with data vectors, the input being the vector of system measurements and the output being the fault classification desired. In each presentation of the data, the internal structure of the network is modified to bring the actual network outputs in correspondence with the desired outputs. This iterative procedure is terminated when the network outputs have the required properties over the entire training set [50]. In this study, the use of multiple inputs was chosen because it is identified that artificial neural networks trained with multiple modal responses generate more accurate damage identification results compared to the precision of trained artificial neural networks with individual response functions [1]. By attacking the proposed problem, an

2.6.1. Identification by optimization algorithm It was discussed previously the wide applicability of evolutionary optimization algorithms and their great importance in engineering problems, where finding the best solution for a given problem is not always trivial [7]. The idea of using genetic algorithms in this part of the work is due to the reality that a damage identification problem is in fact an inverse problem. It is known that a damage, whether it be translated as a change in the geometric or mechanical properties, promotes a change in the responses structures that differ from those taken as the base response of an initial structure without presence of damages [18]. In order for the algorithm to be able to solve the identification problem, it is necessary to introduce objective functions that, when properly constructed and minimized, are capable of providing the expected global solution, which in our case translates to the position of the damage. Thus, for the proposed damage identification process, the function JΦ is constructed to be: n

JΦ =

∑ i=1

where

2

ϕicalculated ⎛ ⎞ ,s ⎜1 − ϕ real ⎟ i, s ⎝ ⎠

ϕicalculated ,s

Table 2 Genetic operators considered for the inverse problem of parameter (damage) identification.

(18)

is the calculated nodal displacements obtained at each 116

Genetic operator

Damage identification Value

Sensor placement Value

Population Crossover Mutation Elitism Generation

10 × Nvar 60% 1% 1 100

10 × Nvar × Nmodes 60% 1% 1 1000

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(s) so that its response is identical or as close as possible to the real structure response. Regarding the second method (ANN), in a first instant, only the numerical model was necessary. A base of damage must be raised by acquiring the ”optimized” structural response in several damage scenarios, i.e., modal responses due to damages at different positions and intensities (dimension/intensity). Based on this data base, artificial neural network training is performed, and after this procedure, the network is expected to be able to identify and quantify the damage from unknown damage (which were not used in its training phase). It is important to highlight that the method proposed in this paper is developed with potential application to larger mechanical structures, for example in civil structures, in the naval, aerospace and other industries. In general way, it is expected to find by the techniques presented here an approximate location of the damage, causing a high saving of time and costs with inspection. Since it is possible to know the possible location and/or vicinity of a probable region that is subject to failure. There is no clear restriction on the minimum limit on the size of the damage detected by the proposed inverse method methodology. As long as the structural response of the actual structure is changed from a pre-established threshold, the algorithm is able to provide a possible location of the damage.

ANN was developed for the same problem by evaluating two distinct inputs. In a first attempt, it was decided as input of the neural network data referring to the first 6 natural frequencies, of the damaged structure with known damage in only a few elements. Then the network was created from a series of trial and error until a set of optimal parameters for neural architecture was obtained. The evaluated and desired response at the network exit is the location of the damage (element) and its severity. Following the identification through ANN, a second neural network architecture was developed, aiming to identify the location of the structural damage, now modified by its input. The evaluated input for this second network corresponds to the mode shape, that is, the local amplitudes were evaluated at an optimal number of points (calculated from the optimal sensor distribution) and from there the network is expected to be able to treat this data well enough to predict damage from untrained data. In summary, Fig. 3 displays the flowchart of the work methodology adopted in this paper in a logical sequence. Firstly, it is necessary to know the structure in which it is desired to apply and develop the structural monitoring. Then the numerical modeling of a given structure becomes necessary, where the optimal configuration of the sensors is applied and obtained. It can be said that ”optimized responses” was obtained, that is, responses with sufficient information and no redundancy for the application of the damage identification tools (specific algorithms). From this point the work takes two different paths: one by means of optimization algorithms (GA) and another by means of ANNs. The optimization method requires that frame data at the optimal points be acquired and then a communication with the numerical model in finite elements is performed, where the optimizer, from its specific operators, is able to minimize a written function of the evaluated structural response until the numerical model presents damage

3. Numerical results and discussion In this section, the main results are presented and discussed in chronological order, namely: (i) preliminary results regarding numerical modeling and direct effect of the damage in the structural response; (ii) results obtained by the optimal positioning of the sensors; (iii) results of damage identification by optimization heuristics and (iv) results of damage location by artificial neural networks. The direct problem (step 1 - Fig. 3) was modeled on a square plate

Fig. 3. Flowchart of the inverse structural damage identification problem modeling. 117

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which could lead to some small structural variations. Table 3 displays the optimization results for 4, 6 and 9 sensors. It can be seen that the optimal distribution of sensors occurred near of points with great modal deformation. By expanding the number of sensors on the plate to 6, more sensor set options are available. Table 3 displays the graphical results obtained by GA. Similarly, the results obtained by the matrix criteria generated optimum values in the vicinity of greater deformation/modal amplitude of the plate for most of the methods and modes studied. In addition, the figures in Table 3 exhibit the same phenomenon in behavior for 9 distributed sensors. In a general way, all methods obtained similar performance, generating a distribution of sensors in antiknots of the mode shapes. A particular case is given by the method of effective Independence, which has some discrepancy in relation to the other criteria. A possible explanation for this phenomenon, as discussed by [2], is that this criterion can select sensor locations that exhibit low signal strength, which can be considered as a disadvantage. Fig. 6 displays the final optimization results of the n = 6 first mode shape of the plate in question. It can be verified that the final result was still made up of sensor points, in general, points at the ends of the plate. The limitation of 4, 6 and 9 sensors may still be low enough to not obtain somewhat more distributed points still added by the fact that the structure studied, if it is a symmetrical square plate.

with equal sides a = b = 30 cm. The structure consists of a symmetrical laminate of composite material consisting of 12 layers of different orientations. The layer thicknesses are all 0.18 mm symmetrically oriented [0/90]3S . 3.1. Structural damage direct effect In this study, the damage was adopted as a local reduction of stiffness (step 2 - Fig. 3). As shown in Section 1, it is usual to adopt this model in the study of structural damage. The damage is then considered by the multiplication of the Ne element by a loss of stiffness α . The plaque is then composed of 100 elements, numbered in ascending order from the origin. Fig. 4 shows the numbering and location of the damages considered in this study, that is, the elements of numbers 19, 45 and 89. The numbering of these elements was purposely chosen in order to study a damage in (element 45), and two asymmetric damages in relation to one of the reference axes, in this case the axis y(elements 19 and 89). As a consequence, the insertion of a structural damage modifies the dynamic response and from this order of variation, it is possible to treat it by means of intelligent processing until the location of the expected damage is obtained. The purpose of a damage identification system is the solution of an inverse problem, that is, the identification of a system that describes the relationship between an unknown input and a known output. This means that the purpose of damage identification is to describe an existing structural model, based on data obtained experimentally or simulated. Given the above, it is necessary to investigate if the insertion of a localized damage, the response evaluated in this study (mode shapes) are really affected significantly. As main damage index (DI), mode shape difference is used. For this, it was used data from a pristine (healthy) structure and mode shape data after insertion of the damage into a known location. Fig. 5 shows, the results obtained by the absolute difference of the mode shapes for the third mode shape of the plate under study, that is, Fig. 5 is the result of the surface generated by ∣Φundamaged − Φedamaged ∣, where e is the damaged element. It can be concluded that in fact the difference of mode shape results in vibration amplitude peaks at points very close to the damage (damaged element), as seen by the image. However, there are adjacent regions with smaller amplitudes known as the ”neighborhood effect” that may lead to problems of identification of false positives in the identification of damage. As seen by the results of this topic, in fact structural damage produces a certain discontinuity in modal forms and when properly treated (mode shape difference) may indicate the possible location of certain faults. However, to obtain a complete mode shape of certain mechanical structure can be an onerous task (mainly for time and signal processing) due to its complexity, by using usual methods. In an attempt to overcome all the problems arising from a complete modal acquisition and sensing, optimal sensor placement strategies is used, in which, from a specific number of points, optimally distributed, a relevant amount of information structural response without loss or redundancy of information is obtained. Next, the results obtained by the sensor placement methods will be exposed, and from these results, the capacity of the damage identification methods will be addressed.

3.3. Damage identification using GA The first computational methodology to detect damage is related to heuristic optimization techniques (step 5 - Fig. 3). In this paper, genetic algorithms was adopted due to its advantages mentioned in the previous section of this paper. As the problem here deals with the use of implicit functions (there is no function that models the modal response as a function of the damage), the use of such evolutionary techniques is justified. Taking into account the problem of damage detection/identification and the importance of evolutionary algorithms in the solution of the inverse problem, a optimization heuristic (GA) is applied in the engineering SHM problem. Since the genetic algorithm is a zero-order method, that is, independent of function derivatives, it is necessary to obtain an average from some simulations, in order to obtain an overall mean of the result. Thus, the study was performed considering an average for 4 simulations. This value was obtained considering a complete factorial where if a number of experiments of the order of 2n [38] is necessary, being n the number of variables. As in the evaluated problem n = 2 , since the damage is modeled as a local stiffness reduction in the Ne element with a given severity α , it results in a total of 22 = 4 random searches for that a certain level of reliability is obtained in the results obtained by the optimization. Table 4 displays the results of the damage search at the known (target) positions. Three possible damage scenarios were

3.2. Optimum positioning of sensors The optimal layout of the sensor distribution was obtained using the genetic algorithm optimization method. The common basic understanding of these methods is to obtain optimal sensor positions where the greatest amount of modal information is obtained. Only the position of a predefined number of sensors at 4, 6 and 9 were adopted with optimization criteria. The sensors were then distributed in the nodal positions of the plate. It should also be remembered that the type of sensor adopted is an ideal sensor, where its physical-mechanical characteristics, such as weight and stiffness, are not taken into account,

Fig. 4. Location of the three damage position scenarios considered in this studied on elements 19, 45 and 89. 118

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Fig. 5. Absolute mode shape difference in the three considered damage scenarios (bi-dimensional and three-dimensional view) showing peak amplitude close to the damage site.

considered in this study, considering damages in different positions of the plate. The damage was then obtained by the inverse problem considering the minimization of the objective function JΦ :

n

JΦ =

∑ i=1

119

2

ϕicalculated ⎛ ⎞ ,s 1 − ⎜ ⎟ ϕireal s , ⎝ ⎠

(19)

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Table 3 Optimal sensor placement result considering FIM criteria for a clamped CFRP plate. 4 sensors

6 sensors

Table 4 Structural damage identification results by GA inverse problem considering a reduced number of sensors.

9 sensors

Scenario I

Mode 1

Mode 2

Mode 3

Mode 4

Scenario II

Scenario III

Ne

α

Ne

α

Ne

α

Objective

19

20%

45

20%

89

20%

Search Search Search Search

1 2 3 4

19 19 19 19

20.0046 19.9998 20.0002 19.9999

45 45 45 45

20.0016 20.0079 20.0000 19.9287

89 89 89 89

20.0110 20.0030 19.9966 20.0116

Average

19

20.0011

45

19.9846

89

20.0056

St. Deviation

0

0.0023

0

0.0374

0

0.0071

Table 5 Graphical structural damage identification results by GA inverse problem considering a reduced number of sensors. Scenario I

Mode 5

Scenario II

Scenario III

Objective

Mode 6 Result

The results obtained from Table 4 show the good performance of the algorithm in identifying the damage, particularly at the position of the damage. There were small errors in relation to the identification of the severity of the damage, however, the target that is most desired in this study refers to the location of the damage, being the severity a step forward in the SHM methodology, referring to the prognosis of failures, which is not the objective of this paper. Table 5 shows the graphical results of the damage identification, displaying the perfect identification at the exact position of the target (damage inserted with known location, or simply data from a real damaged structure - step 3 of Fig. 3).

Table 6 ANN parameters considered for the inverse problem of parameter (damage) prediction. Variable

Value

Trainning function Activation function Error

Levenberg–Marquardt Bayesian regulation

Neuron Number Learning Rate Iterations

10−3 100 0.02 10000

3.4. Damage identification using ANN possible structural state, that is, in an attempt to find damages. By evolving the analyzes to a more local condition, the mode shapes are used in the training of the neural network. For this, it was considered the reduced modes shapes, or rather, the vibrations in a limited number

As seen in the previous chapters, according to the entire SHM philosophy, natural frequencies possess information concerning the overall character of damages. However, when you properly combine, considering a set n, you can aggregate interesting information about the

Fig. 6. Overall sensor placement results considering the six first modes for the clamped plate (optimum location highlighted in red). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 120

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Fig. 7. ANN architecture considering reduced mode shapes (limited sensors) in damage location prediction.

training data set. When a neural network is well trained, it has a high capacity for generalization. The quality of your predictions relies heavily on the network architecture as well as the richness of the training data set. Thus, changes in the network architecture and in the training data set were made and throughout the training process and tested several times before reaching an ideal architecture, which not only leads to a well-trained network, but also ensures subsequent reasonably accurate prediction when exposed to the unknown data set [6]. Once the network was trained to the desired level of accuracy, it was tested for unknown data set and only after a satisfactory performance in predicting the outputs of the unknown data-set was obtained, network learning was stopped. Table 7 displays the validation results of the neural network. Fig. 8 shows the graphical results. It can be seen that the network was able to predict with some efficiency the location of the damage. This method does not objectively evaluate millimeter accuracy in the damage identification process. The results generated can be considered as satisfactory, because the results obtained as global results in the process of the location of defects in structures are used. There are numerous specific tools that are able to detect damage on milli and micrometric scale and are not designed to replace them by means of this. Although neural network-based approaches have been successful because of their generalization capabilities, they have been shown to require relatively large numbers of training cases. In this step, 70% of data was used in the training phase. The ability to predict the location of damage can be improved by considering a greater amount of data in the training phase of the network. Although the results obtained by ANN do not result in exactly the actual ideal location of the damage, the network gives results very close to the existing damage, which in fact is objective of this study through modal metrics, being used as the initial global criterion. From obtaining a possible damage region, an specific conventional inspection technique (visual, ultrasonic, etc.) can be applied. In general, it can be concluded that all the methodology developed for the identification of delamination was successfully performed. The optimization algorithms can accurately identify even the severity of certain structural damage. Artificial neural networks in addition were able to make a satisfactory prediction of the possible location of damage from unknown validation data.

Table 7 Structural damage identification prediction results by ANN inverse problem considering a reduced number of sensors. Index

Ne Target

Ne ANN

Index

Ne Target

Ne ANN

a b c d e f g h i j k l m n o

11 12 14 17 18 29 30 34 36 38 43 48 52 56 58

15 3 17 6 19 26 44 39 59 43 46 57 58 61 59

p q r s t u v w x y z aa ab ac ad

61 63 66 67 70 70 72 74 77 79 80 83 84 87 91

57 83 90 91 59 62 77 94 65 84 77 97 82 99 80

of points, these points being obtained by the optimal distribution of the sensors. Thus, ANN is applied in order to predict damage location in composites plates (step 6 - Fig. 3). After many ANN tries, an ideal configuration was reached that could predict with better efficiency untrained data. The architecture of the neural network was conceived considering n entries, being n the number of sensors distributed in the plate. A minimum number of neurons, Nn = 10, was defined. Thus avoiding an additional unnecessary confounding in the neural network. Only one hidden layer was sufficient and the desired outputs were modeled only as the possible location of the damage. Once again, the term of α severity of the damage has been omitted since it is understood that this term adds some degree of disorientation in predicting the location of the damage, and in this study, such a term is somewhat superfluous. The ANN parameters are shown in Table 6. The input E (Eq. (20)) of the network has information regarding the nodal displacements ϕsi in the sensors si . It was used 70% were used for training, that is, 70 of the 100 structural elements of the plaque were taken in the training phase on a random basis. The input E still takes into account the first 6 modes shapes of the plate, in order to obtain as much information as possible from the structural state. The architecture of the neural network is shown in Fig. 7. 1

s12 s1e ⎤ s22 s2e ⎥ ⎥ s32 s3e ⎥ ⋮ ⋮⎥ ⎥ sn2 sne ⎥ ⎦N

sne

ϕ1n, e .

⎡ s1 ⎢s 1 ⎢ 2 E = ⎢ s31 ⎢⋮ ⎢ 1 ⎢ ⎣ sn

sensors × Nelements

4. Conclusion In this paper a vibration-based structural delamination identification and prediction was conducted as an inverse optimization problem using GA and ANN, respectively. Based on present computational results, the following conclusions can be made:

• Delamination

(20)

modeling as a local variation of stiffness in fact faithfully simulates the behavior of this type of defect. From the results obtained by FEM, it can be observed that a change in

where = The trained network is tested with data that is not present in the 121

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Fig. 8. Structural damage identification prediction (ANN output) results for an untrained data set considering the reduced mode shapes (legend: prediction).

GA Computational cost Time of inspection Structural testing preparation Instrumentation cost Operator’s skill Struxtural accessibility Knowledge of a neighborhood of the damage Possible automatic damage prediction

• • •

•• •∘ •∘ •∘ •∘ •• •∘

• ∘ ∘ ∘ ∘ ∘ ∘

•∘ ∘∘ ∘∘ ∘∘ ∘∘ ∘∘ ∘∘

••∘∘∘

Conventional NDT

In general, Table 8 shows in summary a general conclusion of the advantages and disadvantages of the proposed method over conventional inspection techniques. Conventional inspection techniques are understood to be those already practiced in the literature (for example: ultrasonic techniques, acoustic emission, magnetic particles, visual inspection and others). Here, each specific particularity and/or innovation of each conventional method in a specific character is not thoroughly addressed.

ANN •• •∘ •∘ •∘ •∘ •• •∘

•∘∘ ∘∘∘ ∘∘∘ ∘∘∘ ∘∘∘ ∘∘∘ ∘∘∘

••••∘

ANN

By removing the term of severity and aiming to detect the possible location of delamination, it obtained excellent accuracy results, even with only 70% of the data used for training.

Table 8 General conclusion about the proposed method compared to conventional inspection techniques. Present Study

Target,

•∘ •• •• •• •• •• ••

∘ • • • • • •

∘∘ •∘ •∘ •∘ •• •• ••

•∘∘∘∘

Acknowledgment The authors would like to acknowledge the financial support from the Brazilian agency CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico, CAPES Coordenação de Aperfeiçoamento de Pessoal de Nível Superior and FAPEMIG - Fundação de Amparo à Pesquisa do Estado de Minas Gerais (APQ-00385-18).

stiffness, or simply a delamination, promotes a change in the modal characteristics, especially on the mode shapes. The optimization of the sensors using fisher criterion generated results in points of sensory where it is possible to guarantee the highest level of modal quality. These points were collected to construct reduced mode shapes, which in turn will be used in inverse methods using GA and ANN. In relation to the identification of damage itself, the genetic algorithms presented excellent results, being able, besides identifying the location of the delamination, also the degree of severity (α ) of this damage. Damage prediction was addressed using training data containing the reduced mode shapes at optimal points generated by the SPO. At first, the insertion of two output data (Ne and α ) generated a certain level of confusion in the network, generating results not so precise.

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