A deep feed-forward neural network for damage detection in functionally graded carbon nanotube-reinforced composite plates using modal kinetic energy

Abstract

This paper proposes a new Deep Feed-forward Neural Network (DFNN) approach for damage detection in functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates. In the proposed approach, the DFNN model is developed based on a data set containing 20 000 samples of damage scenarios, obtained via finite element (FE) simulation, of the FG-CNTRC plates. The elemental modal kinetic energy (MKE) values, calculated from natural frequencies and translational nodal displacements of the structures, are utilized as input of the DFNN model while the damage locations and corresponding severities are considered as output. The state-of-the art Exponential Linear Units (ELU) activation function and the Adamax algorithm are employed to train the DFNN model. Additionally, in order to enhance the performance of the DFNN model, the mini-batch and early-stopping techniques are applied to the training process. A trial-and-error procedure is implemented to determine suitable parameters of the network such as the number of hidden layers and the number of neurons in each layer. The accuracy and capability of the proposed DFNN model are illustrated through two distinct configurations of the CNT-fibers constituting the FG-CNTRC plates including uniform distribution (UD) and functionally graded-V distribution (FG-VD). Furthermore, the performance and stability of the DFNN model with the consideration of noise effects on the input data are also investigated. Obtained results indicate that the proposed DFNN model is able to give sufficiently accurate damage detection outcomes for the FG-CNTRC plates for both cases of noise-free and noise-influenced data.

Graphical abstract

Keywords

damage detection / deep feed-forward neural networks / functionally graded carbon nanotube-reinforced composite plates / modal kinetic energy

1 Introduction

Advanced materials have become the subject of intense interest due to the necessity of high-performance for technical products. Thanks to outstanding mechanical, thermal and chemical properties such as high strength, high stiffness, high aspect ratio with low density, temperature resistance and isolation, composite materials have widely been applied to various engineering disciplines such as aerospace, defense, energy, automobile, medicine, and construction. Some glances at the application of composite materials to airplane products can be found in the presentation of Pora [1]. Additionally, composite materials have been applied to various civil engineering constructions, several of which can be seen in the paper of Biswal and Swain [2]. For the past few decades, functionally graded materials (FGMs) have emerged as an outstandingly inventive advance in the field of composite materials, and are created from multi phases of constituent materials with certain distributions in order to achieve specific engineering aims. An extensive account of FGMs was thoroughly presented in the book of Miyamoto et al. [3]. Moreover, fiber reinforced composites have attracted significant attention from researchers and engineering manufacturers due to their high performance in practical applications. Highlights and applications of the CNT can be found in several previous articles [4–7]. Motivated by the specialized functions that the FGMs effectively offer, the idea of grading the CNT-reinforced fibers into a matrix material has been applied to develop a new pattern of composite material—functionally graded carbon nanotube-reinforced composite (FG-CNTRC). Details related to mechanical properties and analyses of the CNT and FG-CNTRC materials can be found in the review articles of Liew et al. [ 8] and Imani Yengejeh et al. [9]. Due to the great potential in practical and in-depth applications of the FG-CNTRC materials, the engineering tasks related to “structural health monitoring” for (engineering) products constituted from these materials demand to be studied.

In general, the term “structural health monitoring (SHM)” is related to the processes of providing the diagnosis, notification, and information which show the current condition of structures during their lives. General reviews of the SHM can be referred to several previous publications [10–12]. In recent considerations, SHM usually refers to the process of monitoring the behaviors of the structures such as aircraft and infrastructural buildings, via the signal collected from a sensor network. The growing concern about the safety insurance of important structures, such as bridges, highways, and high-rise buildings, using smart composite materials, together with the rapid development in computational technologies have created demand for the development of more effective SHM methods, which motivates the authors to conduct this study.

In the literature of the SHM discipline, vibration-based damage detection methods have been broadly applied to the composite structures [13–17]. In the general view of those methods the changes in vibration properties, caused by the occurrences of damage, are extracted and used as information for the detecting processes. For instance, structural natural frequency [18,19], nodal displacements [20], curvature of mode shape [21,22], modal strain energy [23–25], modal kinetic energy (MKEn) [26], structural flexibility characteristics [27–32], structural stiffness characteristics [33], and Lamb wave-based damage identification approaches [34] have been utilized to develop the damage detection methods for composite structures. Across other research interests, apart from the above-mentioned studies conducted in terms of composite materials, various vibration-based damage detection methodologies have been proposed for different types of structure in the past ten years. Liu et al. [35] proposed a scheme of using frequency response function (FRF) shapes to localize damage occurrences in a cantilever beam. Jafarkhani and Masri [36] investigated the performance of an evolutionary strategy using the finite element model updating approach for damage detection in a quarter-scale two-span reinforced concrete bridge system. Hadjian Shahri and Ghorbani-Tanha [37] proposed a damage detection method based on the damage sensitive feature parameter named modal kinetic change ratio (MKECR) to detect the damage of beam-like structures. Li et al. [38] investigated the efficiency of MSEn indices for damage detection, and then proposed an index of total modal energy, which is the summation of MSEn and MKEn. The study, conducted in numerical and laboratory simulations, targeted detecting the damage of offshore wind turbine structures. Nguyen-Thoi et al. [39] used vibration properties of a 72-bar space truss and a one-span steel portal frame to develop a two-stage damage detection method, which employed the damage locating vector (DLV) method and the differential evolution (DE) algorithm, and hence to detect location and extent of damage in space truss and steel portal frame structures. Further references of the vibration-based damage detection methods can be found in the review articles of Das et al. [40] and Gomes et al. [41]. Methods proposed in previous studies show abilities in detecting the damage in various structures; however, there have still been some challenging aspects of these methods. For example, some damage detection indices and the objective functions built from structural properties are not suitable to represent the physics-based nature of the current state of structures, which decreases the accuracy of the algorithms. Or, the degrees of freedom of structures which are used to compute the value of MSEn [42–45] or MKEn [26,37,46] are generally difficult to obtain and measure by acceleration sensor systems in practice, especially for plate/shell structures.

In recent years, applying the Deep Neural Networks (DNN) to various engineering problems has become an accelerating trend thanks to the effective assistance of computational technologies including algorithms and hardware platforms. In brief introduction, DNN is a representation of a broader category, namely Deep Learning (DL) which is indeed a sub-brand of Machine Learning (ML) within the Artificial Intelligence (AI) context. Thorough fundamentals about DNN can be found in the book by Goodfellow et al. [47]. Thanks to vigorous productivity, ML-based algorithms have been widely applied in computational mechanics and shown high effectiveness. Samaniego et al. [48] utilized DNN as an option of approximation for an energy-based approach to the solution of partial differential equations in mechanical problems. Anitescu et al. [49] presented a method for solving the partial differential equations using artificial neural networks and adaptive collocation strategy, which was numerically performed on benchmark scalar-valued partial differential equations problems of Poisson and Helmholtz equations, and an inverse acoustics problem. Zhuang et al. [50] applied a specific type of feedforward DNN called Deep Autoencoding, combined with an energy method to analyze the bending, vibration, and buckling characteristics of Kirchhoff plates. Guo et al. [51] analyzed the bending behaviors of the same type of plate as the formerly mentioned study by employing the Deep Collocation method with the combination of optimizers in the backpropagation process during training the model. Recently, there are growing interests in utilizing DNN for damage detection problems thanks to the ability to solve more complex and highly nonlinear problems [52], which traditional shallow neural networks could not address with high accuracy. Since the emergence of this trend, DNN-based SHM approaches have primarily been developed to correlate information, extracted from image (vision) data, to automatic recognition of structures. Xu et al. [53] proposed a modified faster region-based convolutional neural network (faster R-CNN) to develop a multitype seismic damage identification and localization method, from images, for reinforced concrete columns. Wang et al. [ 54] also used the faster R-CNN to develop an automatic damage detection technique for the efflorescence and spalling of historic masonry structures. Liu and Zhang [55] generated the image data by capturing the analytical results obtained from finite element simulation and used a deep convolutional neural network (DCNN) model to assess the damage condition of steel structures. Vision-based SHM methods through DL can also be found in many studies such as crack-deflection coupling [56], corrosion [57,58], concrete spalling [59–61], crack detection [62–66], fatigue detection [67], and surface and subsurface damages of steel structures [68,69]. However, detecting damage using image data is usually costly and generally difficult to apply to large-scale structures. Additionally, image data cannot illustrate imperceptible forms of damage such as the physical defects occurring inside the structural elements. Therefore, during the progression of the concept, the vibration properties of structures which show high effectiveness and accuracy have attracted great interest in utilizing as the input data of DNN models for predicting damage occurrences. For example, Teng et al. [70] used the structural MSEn, obtained through finite element simulation, to construct the training data for the proposed convolutional neural network (CNN) model for damage detection of steel frame structures. Khodabandehlou et al. [71] proposed a vibration-based structural condition assessment method using acceleration response histories and a two-dimensional DCNN. The proposed method was validated through vibration data which was recorded during the extensive shake-table testing of a highway bridge model at the University of Nevada, Reno. Li et al. [72] established a scale-down model of cable-stayed bridge by an experimental platform which was used to develop a deflection monitoring approach for continuous bridges. Truong et al. [52] proposed a damage detection method using the deep feed-forward neural network (DFNN) for truss structures including a 31-bar planar truss structure and a 52-bar dome-like space truss structure. Their study focused on dealing with the noisy data and noisy-incomplete data of nodal displacements which are the input of the training process. Further, extensive coverage in the data-driven structural health monitoring and damage detection through Deep Learning can be found in a recent review article published by Azimi et al. [73].

From the above literature review, it can be seen that the study of developing a deep learning model based on the Deep Feed-forward Neural Network (DFNN) accompanied with MKEn for damage detection in FG-CNTRC plates has not yet been performed. Therefore, this study is conducted to fill this research gap. In the proposed DFNN model, the elemental MKEn values calculated from the natural frequencies and translational displacements are used to construct the input data set, while the locations and corresponding severities of the damage occurrences are designed as the output data set. A trial-and-error procedure is applied to determine an optimal architecture of the DFNN model. In the training process, the exponential linear units (ELU) activation function (AF) and the Adamax optimizer are adopted to build up the DFNN model. Additionally, the mini-batch and early-stopping techniques are employed to increase the training rate and prevent the over-fitting issue, respectively. The accuracy, stability and applicability of the proposed method are demonstrated through its application to the FG-CNTRC plates, constituted from two distinct configurations of the CNT-fibers including uniform distribution (UD) and functionally graded-V distribution (FG-VD), and with different boundary conditions. Moreover, the performance and stability of the proposed DFNN model with the consideration of noise effects on the input data are also investigated.

This paper is presented as following: Section 2 presents the deep feed-forward neural network (DFNN)-based structural damage detection; Section 3 shows the numerical results and gives discussions on the results; and finally, Section 4 points out the conclusions of the study.

2 Deep feed-forward neural network (DFNN)-based structural damage detection

2.1 Deep feed-forward neural network

2.1.1 Architecture

Basically, the architecture of a typical DFNN model includes three main parts, namely, input layer, hidden layers, and output layer, as shown in Fig. 1(a). The first layer is called the input layer where the input vector of a sample is stored for each iteration. The number of nodes in this layer, therefore, matches the length of an input vector. The number of hidden layers and number of nodes per each hidden layer are manually designed such that the model can reach its optimum performance for a particular data set. The output layer is the final stage of the DFNN model by which the predicted results are obtained, so the number of nodes in this layer is set according to the dimension of the output data. As for the implementation in the model, each hidden layer gets the output of the previous layer as the input, and then the input is transformed to be the (next) output which will be used as the input of the next hidden layer. Figure 1(b) illustrates the structure and the operations at a node of a hidden layer. Particularly, each node of a hidden layer (or of the output layer) has a particular bias coefficient and is connected with all nodes of the previous layer via a particular weight factor \left(w_{ij}\right) at each node (of the previous layer). Additionally, there are two operators at the node which are linear summation (Eq. (1)) and transformation by AF (Eq. (2)). Assuming that the model consists of NoL hidden layers, and there are l⁽ⁱ⁾ nodes at the ith hidden layer, the transformation of signal at the jth node of the hidden layer is performed through the following mathematical operations:

(1)z_j^{(i)}=\sum _{k=1}^{l^{(i-1)}}{a}_k^{(i-1)}\times w_{kj}^{(i)}+b_j^{(i)},

(2)a_j^{(i)}=\sigma \left(z_j^{(i)}\right),

where a_k^{(i-1)} is the output signal at kth node of \left(i-1\right)\mathrm{t}\mathrm{h} hidden layer, a_j^{(i)} is the output signal at jth node of ith hidden layer, and is \sigma \left(z_j^{(i)}\right) the AF.

2.1.2 Internal parameters

First, for a typical DFNN model the raw data is transformed into the output signal by a function, AF, and takes place at each node of the neural network during the feed-forward procedure of the training process (Fig. 1, Eqs. (1) and (2)). Functions of sigmoid and hyperbolic tangent were formulated and set for applications to neural network models in the initial period of the progression of this discipline, and showed the high applicability for various problems. However, growths in complexity, volume, and nonlinearity level of problems promote the inventions of new AF for enhancing the prediction efficiency of the neural network model. Literature related to AF for DNN models can be found in the publication of Nwankpa et al. [74]. In-depth considerations including discovering new AF or the performance of the AF in DNN models could be referred to the articles of Agostinelli et al. [75] and Ramachandran et al. [76]. Next, the performance of the DFNN model is evaluated by a loss function which measures the difference between the actual (output) data and the predicted outcome. In this study, the performance of the model is appraised via the mean squared error (mse) value whose mathematical expression is shown by:

(3)mse=\frac{1}{N}\sum _{n=1}^N{\left(y_n^{\text{pred}}-y_n\right)}^2,

where y^pred and y are the predicting and actual output values, respectively; N is the number of samples of the data set. For a DFNN model designed with M nodes at the output layer, the predicted result is a vector with the size of (1 × M) accordingly. Therefore, after an epoch, the output results are stored in a (N × M) matrix. As a consequence, Eq. (3) can be rewritten more specifically as:

(4)mse=\frac{1}{N\times M}\sum _{n=1}^N\sum _{m=1}^M{\left(y_n^{\text{pred}}-y_n\right)}^2.

Finally, the DFNN model is trained by a back-propagation algorithm, or optimizer, by which the internal weight factors of the model are updated such that the loss function (or loss value) tends to be minimized. There are a number of algorithms developed to undertake the task such as Adaptive Gradient Algorithm–Adagrad [77], Adaptive Learning Rate method–Adadelta [78], RMSProp [79], FTRL–Proximal–Furl [80], Stochastic Gradient Descent–SGD [81], Adaptive Moment Estimation–Adam and Adamax (a variant of Adam) [82], and Nesterov-accelerated Adaptive Moment Estimation–Nadam [83]. In practice, each algorithm carries characteristic properties, which makes the suitability of its applications vary depending on particular attributes of the considered problem and the available data. The performance of the optimizer should be understood as the mutual dependencies between the optimizer and the chosen AF, which will be studied in the next section.

2.2 Formulations of modal kinetic energy

Within the field of SHM, it is accepted that the damage of an element could be considered as a reduction in the elemental stiffness. In previous studies [42–45], the MSEn was utilized and estimated according to the intact elemental stiffness value, with the assumption that the mass matrix is unchanged (or insignificantly changed). Recent studies [26,37,46] illustrated that the MKEn offers high efficiency in developing methods of structural damage detection. This is because the MKEn value is formulated with the participation of the elemental mass matrix. Therefore, the estimation of the MKEn value does not require additional assumptions, resulting in higher accuracy of the developed methods.

By assuming that the vibration behavior of the FG-CNTRC plates is harmonic, the free vibration properties of a linearly undamped system are obtained by solving the eigenvalue problem in the form of:

(5)\left(\mathit{K}-{\omega }^2\mathit{M}\right)\mathit{d}=0,

where {\omega }^2 and d are the eigenvalue and the eigenvector, respectively; K and M are the global stiffness matrix and global mass matrix of the FG-CNTRC plates, respectively, which are obtained through the assembling procedure from the elemental stiffness matrix {\mathit{K}}^e and elemental mass matrix {\mathit{M}}^e,

(6)\mathit{K}=\underset{e=1}{\stackrel{N_e}{\cup }}{\mathit{K}}^e;\mathit{M}=\underset{e=1}{\stackrel{N_e}{\cup }}{\mathit{M}}^e.

The MKEn [37] of jth element of ith mode is formulated as

(7)MK{E}_{ij}=\frac{1}{2}{\omega }_i^2{\left({\mathit{\phi }}_i\right)}^{\text{T}}{\mathit{M}}_j{\mathit{\phi }}_i,

where {\omega }_i, {\mathit{\phi }}_i, and {\mathit{M}}_j are the natural frequency, vector of nodal displacements of the ith mode, and elemental mass matrix of the jth element, respectively.

The total MKEn of the ith mode is calculated as:

(8)MK{E}_i=\sum _{j=1}^{noe}MK{E}_{ij},

where noe is the total number of elements of the plates.

The MKEn value of each element of a mode is then normalized in terms of the total MKEn value by:

(9){\overline{MKE}}_{ij}=\frac{MK{E}_{ij}}{MK{E}_i}.

3 Results and discussions

3.1 Free vibration analysis of FG-CNTRC plates

In this study, the first order shear deformation theory (FSDT) [84] is employed for the free vibration analysis of the plates, in which the constitutive equations for FG-CNTRC plates are constructed in terms of orthotropic material. In the FE simulation, the 9-node iso-parametric element [25,85] is selected with the mesh option of 10 by 10 along the x and y directions, respectively, which corresponds to the geometrical accuracy level of 100 rectangular detecting areas (elements) for the damage assessment.

For the constitutive materials of the FG-CNTRC plates, the matrix is chosen to be the poly (methylmethacrylate) (PMMA) and poly {(m-phenylenevinylene)-co-[(2,5-dioctoxy-p-phenylene) vinylene]} (PmPV) whose parameters are estimated by means of molecular dynamics simulation [86]. The parameters are based on the study by Zhu et al. [87], with {\nu }^{\text{m}}=0.34, {\rho }^{\text{m}}=1.15\mathrm{g}/\text{c}{\text{m}}^3, and E^{\mathrm{m}}=2.1\text{GPa} (at room temperature, T = 300K). The armchair (10,10) Single-Walled Carbon-Nanotube (SWCNT) ( L=9.26\text{nm}, R=0.68\text{nm}, h=0.067\text{nm}, {\nu }_{12}^{\text{CNT}}=0.175) is chosen as the reinforcement. According to the results in the study of Shen [88], the parameters at room temperature of the (10, 10) SWCNT are listed as E_{11}^{\text{CNT}}=5.6466\text{TPa}, E_{22}^{\text{CNT}}=7.0800\text{TPa}, G_{12}^{\text{CNT}}=1.9445\text{TPa}, and V_{\text{CNT}}^{\ast }=0.11. From these material parameters, the rule of mixture [89,90] is employed to determine the effective material properties of the FG-CNTRC plates whose formulations can be referred to the former publications of Zhu et al. [87] and Shen [88]. Note that the orientation angle of the reinforced fibers is chosen to be zero for this study. The graphical illustrations of the UD CNTRC, FG-VD CNTRC plates, and the orientation angle of the fibers are shown in Fig. 2.

The free vibration analysis of the FG-CNTRC plates is verified by comparing the analyzed results of this study to those of previously published papers [87,91,92]. The non-dimensional frequency of the FG-CNTRC plates is calculated by \overline{\omega }=\omega \left(a^2/h\right)\sqrt{{\rho }^{\text{m}}/E^{\text{m}}}. The comparison of the first six non-dimensional natural frequencies of the FG-CNTRC plates is shown in Table 1. The results show that the present free vibration analysis of FG-CNTRC plates has acceptable accuracy and reliability.

3.2 Data generation

In a computational context, the damage of an element can be considered as a degradation of its stiffness [42]. Consequently, the damage occurring in the jth element can be simulated by using a stiffness reduction parameter of {\alpha }_j. This parameter varies from 0 to 1 and the reduction of stiffness is indicated as:

(10){\mathit{K}}_j^{\mathrm{d}}=\left(1-{\alpha }_j\right){\mathit{K}}_j,

where {\mathit{K}}_j^{\mathrm{d}} and {\mathit{K}}_j are the elemental stiffness matrix of the jth element in the damaged and intact conditions, respectively. Thus, a damage case of the plates is then designated via a vector of \mathit{\alpha }={\left\{{\alpha }_j\right\}}_{j=1:noe}, in which noe is the total number of elements of the plates discretized.

The output data set is constructed by generating a collection of damage cases. In each case, the positions and corresponding severities of the damage occurrences are randomly created. In this study, the possible damage occurrences are assumed to happen at maximum of three positions. The output data set is of the following form

(11)[Y]={\left[\begin{array}{c}{\left\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{\text{noe}}\right\}}^1\\ {\left\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{\text{noe}}\right\}}^2\\ \dots \\ {\left\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_{\text{noe}}\right\}}^N\end{array}\right]}_{N\times (noe)},

where N and noe denote the total number of samples and number of elements discretized, respectively.

In a supervised DL-based method, the input data is mapped to output results thanks to the DL model trained. According to the data set of damage cases presented in Eq. (11), the corresponding vibration properties of the FG-CNTRC plates including natural frequencies and nodal displacements are obtained by FE simulation. In this work, only a small and easy-to-measure number of degrees of freedom of the mode shapes are utilized as the input data. We only extract the translational degrees of freedom, without using the rotational degrees of freedom, to estimate the values of MKE_ij in Eq. (7). Additionally, the nodal displacements are only gathered at the four corners of the element. As a consequence, the dimension of the elemental mass matrix is diminished according to the obtained nodal displacements. The first six vibration modes are adopted to determine the elemental MKEn values (Eq. (9)) which are then assembled to establish the input data set [X], presented as:

(12)\begin{array}{rl}& [X]=\\ & {\left[\begin{array}{rl}& {\left\{{\overline{MKE}}_{1j},{\overline{MKE}}_{2j},{\overline{MKE}}_{3j},{\overline{MKE}}_{4j},{\overline{MKE}}_{5j},{\overline{MKE}}_{6j}\right\}}^1\\ & {\left\{{\overline{MKE}}_{1j},{\overline{MKE}}_{2j},{\overline{MKE}}_{3j},{\overline{MKE}}_{4j},{\overline{MKE}}_{5j},{\overline{MKE}}_{6j}\right\}}^2\\ & \dots \\ & {\left\{{\overline{MKE}}_{1j},{\overline{MKE}}_{2j},{\overline{MKE}}_{3j},{\overline{MKE}}_{4j},{\overline{MKE}}_{5j},{\overline{MKE}}_{6j}\right\}}^N\end{array}\right]}_{N\times (6\times noe)}.\end{array}

In this study, two configurations of the CNTRC plates including the uniform distribution (UD) and the functionally graded-V distribution (FG-VD) are investigated. Herein, the plates are geometrically discretized into 100 elements for the damage assessment. The longitudinal measurements in x- and y- directions and the thickness of the plates are chosen with the ratio being a=b=10h. The sketch of the plates is shown in Fig. 3.

Three boundary conditions including simply supported (SSSS), clamped-simply supported (CSSS), clamped (CCCC), and mixed clamped-free (CFCF) edges are considered. The restraints corresponding to these boundary conditions are indicated in Table 2.

The effects of noise on collecting modal properties are taken into investigation by adding a random variance to each value of the natural frequencies and the nodal displacements. The noise inclusion is implemented by the following equation:

(13)R_{\text{noise}}=R+\gamma (2\text{rand}[0,1]-1)R,

where R is an individual value of the natural frequencies or the nodal displacements, and the subscript of “noise” denotes data with noise; \gamma is the noise level and \text{rand}\left[0,1\right] is a uniformly distributed random number ranging from 0 to 1.

3.3 Construction of DFNN model

In this study, the DFNN model is built by the programming language of Python and the deep learning libraries provided by Keras [93]. The internal parameters of the DFNN model are selected through the following efficacy investigations.

3.3.1 Optimizer and activation function

Selecting the AF and optimizer (Op) is made by evaluating the mse value in Eq. (4) for each pair of AF/Op. The overfitting level is assessed via the ratio of the validation loss value to the training loss value. Details of those are presented in Table 3 and illustrated in Figs. 4 and 5. Initially, the number of hidden layers is set to be three layers each of which consists of 200 nodes (denoted as 3-[200]). The training process is implemented through 3000 epochs with the batch size of 1% of the training data set (160 samples). The data case chosen for the process is the UD with simply supported plate (denoted by UD-SSSS) in which 80% of the data set is used for the training while the remaining 20% is used for the testing.

Figure 5 shows that combining the AF of ReLU with the optimizer of Adamax gives the smallest loss values for the training and testing process, but results in an extreme overfitting. Therefore, in terms of the ReLU AF, the selection should be ReLU/RMSprop which gives a relatively equal result of validation loss value without leading to large overfitting. Next, among the conjunctions of Leaky ReLU with the optimizers, the choice should be Leaky ReLU/Adam due to the lowest loss and validation loss values it produces, and the resulting overfitting status is not much larger than other cases. Finally, the results of the ELU AF share relative efficiencies in terms of the training procedure and the training-testing loss values gap. In the further assessment, the stability of these six pairs of AF/Op is observed via the training history whose results are presented in Fig. 6.

It can be seen from Fig. 6 that the combination of ELU and Adamax provides outstanding stability when compared with other combinations. Therefore, in this study the AF/Op of ELU/Adamax is selected for training the DFNN model. The formulation of the ELU AF is shown as follows:

(14)\sigma (\chi ,\mathit{\text{z}})=\{\begin{array}{ll}\mathit{\text{z}},& \mathit{\text{z}}>0,\\ \chi \left({\text{e}}^{\mathit{\text{z}}}-1\right),& \mathit{\text{z}}⩽0,\end{array}\in (\chi ,\mathrm{\infty }),

where \chi is the designed parameter of the function which is set to be 0.5 in this study.

Note that in terms of computational time, the ReLU AF is certainly better than its ELU counterpart. This is because the ReLU function possesses absolute linear characteristics; meanwhile, the ELU function is constituted of both linear part and nonlinear (exponential) part which consumes more time for the gradient computation during the backpropagation process. However, due to the zero-part of the ReLU, the function possibly produces vanishing gradient during the backpropagation of the training process. As a result, the feature and information carried in the data may be missed in the optimizing phase, which may make the ReLU a poor choice for this highly nonlinear problem with a large database.

3.3.2 Number of hidden Layers (NoL) and Number of Nodes per layer (NoN)

The learning capability of a DFNN model depends on its scale which is shaped by the coupling of the number of hidden layers and the number of nodes per layer (denoted by [NoL-NoN]). Generally, the model could provide more accurate outcomes if its size is enlarged; however, the calculation costs will consequently increase. Furthermore, at a particular depth level of a DFNN model (i.e., a particular number of hidden layers), the learning ability tends to reach the utmost at a certain number of nodes contained in the hidden layers. Therefore, investigation of the coupling of [NoL-NoN] needs to be conducted. In this study, the architecture of the DFNN model is designed with the same number of nodes among the hidden layers. Based on the result of the previous section, the AF of ELU and the optimizer of Adamax are selected for training the model. The same data case and the batch size are applied for the training process, i.e., UD-SSSS and batch size of 160 samples. The training operation is stopped if the validation loss value cannot be minimized further, and the optimal weight factors of the DFNN model are then retrieved and recorded for future predictions. Each case of the [NoL-NoN] is executed three times, and then the mean value and the standard deviation are used for the evaluation, as presented inTable 4.

Numerical results from Table 4 illustrate that the association of five hidden layers each of which consists of 600 nodes gives the best performance. Hence, this solution of five hidden layers and 600 nodes per each layer is adopted for the DFNN model. The training history of this architecture is additionally presented in Fig. 7.

3.3.3 Batch size

In practice, training the model by using the whole data set for each updating of the set of weight factors may push up the calculation costs as well as limit the effectiveness of the training process. To improve the situation, the mini-batch technique [94] is introduced, by which the training data set is split into groups, each containing fewer samples. This number of samples is called the batch size. Generally, the model tends to learn more specifically if the batch size is small, which means that the overfitting may appear with a high level. Meanwhile, if this parameter is set too large, the model may be trained superficially; hence, the model may consequently provide low predictive performance. The search for the optimal solution in terms of the batch size is carried out with the architecture obtained from the previous investigations, i.e., five hidden layers, 600 nodes per hidden layer, the AF of ELU, and the optimizer of Adamax. The training result for each case of the batch size is recorded for three separate times (of training operation), and the evaluation is then carried through the mean and standard deviation values among different settings of the batch size, which are summarized in Table 5.

The results in Table 5 show that the batch size of one percent (i.e., 160 samples) provides the best outcome, with the respective mse value and accuracy of \left(3.8327\pm 0.0124\right)\times {10}^{-4} and \left(96.36\pm 0.20\right)\mathrm{\%}. Therefore, in this study the value 160 is used as the parameter of batch size for the training processes.

3.3.4 Architecture and parameters of the DFNN model

According to the investigations conducted in the previous sections, the architecture and the parameters of the DFNN model which show the outstanding performance are finally determined and listed as follows:

• AF: Exponential Linear Unit–ELU \left(\chi =0.5\right);

• optimizer: Adamax;

• number of hidden layers: 5;

• number of nodes per layer: 600;

• batch size: 160.

The training results including the loss value and accuracy value for each case of the material configurations and the boundary conditions are summarized in Table 6.

3.4 Performance evaluation

The performance of the proposed method for damage detection is evaluated via two damage scenarios which are given randomly with different severities and dissimilar locations of the damage occurrences. For each scenario, the damage prediction is implemented in turns for each case of the CNTRC plates including the UD and FG-VD and the boundary conditions of SSSS, CSSS, CCCC, and CFCF. Two damage scenarios of the FG-CNTRC plate discretized into 100 elements are assumed as in Table 7.

3.4.1 Scenario 1st

First, the first scenario with only one damage element is graphically illustrated in Fig. 8, in which the 80th element located near the edge of the plates shows a damage of 19.84%. Figures 9 and 10 indicate the detected results of the UD and FG-VD plates in the boundary conditions of SSSS, CSSS, CCCC, and CFCF, respectively. It is observed that for both UD and FG-VD plates, the model can give accurate detected results for the case of noise-free data with the largest mse values of only 7.080×10⁻⁷ (UD-CFCF) and 1.533×10⁻⁶ (VD-SSSS). For the data with 5% of noise, the model can identify the damage occurrences for all boundary conditions, but the errors tend to be larger, with the largest error of 1.055×10⁻⁴ (UD-CSSS) and 8.090×10⁻⁵ (VD-CSSS).

Next, the second case of the first scenario is graphically presented in Fig. 11, in which the 55th and 65th element located around the middle of the plates show damage occurrences of 36.56% and 24.91%, respectively. Figures 12 and 13 illustrate detected results of the UD and FG-VD plates in the boundary conditions of SSSS, CSSS, CCCC, and CFCF, respectively. It can be seen that the DFNN model is able to give explicit identifications of the damaged elements of 55th and 65th elements for all cases of the surveyed boundary conditions, both for noise-free and free noise-influenced data. The model tends to give large errors when detecting low damage occurrences, which can be seen in the detections of the 65th element. However, this damaged position of the plates is still identified via the exceptionally high values compared to the near-zero values of the other intact elements.

Finally, the final case of the first scenario is depicted in Fig. 14, in which the damage locations are assumed at 8th, 59th, and 98th elements with the corresponding severities of 67.46%, 59.86%, and 21.38%. Figures 15 and 16 illustrate the corresponding results of the damage detection for the UD and FG-VD plates in the boundary conditions of SSSS, CSSS, CCCC, and CFCF, respectively. The results show that the given damaged state is detected for all cases of the boundary conditions whether the data is influenced by noise or not. Particularly, the DFNN model can exactly recognize the damaged elements of 8th and 59th elements; meanwhile, similarly to the previous case, the element with a low level of damage (98th) is generally recognized but with larger errors than the two remaining elements. Additionally, the damaged elements in the plate with mixed boundary of CFCF are detected with relatively lower accuracies compared to the other boundary conditions whose errors for noise-free and noise-existed data are 2.183×10⁻⁴, 1.345×10⁻³ (UD), and 3.176×10⁻⁴, 1.117×10⁻³ (FG-VD), respectively.

3.4.2 Scenario 2nd

First, the second scenario with only one damage element is shown in Fig. 17, in which a damage of 44.13% is assumed to occur at the 72nd element of the plates. Figures 18 and 19 illustrate detected results of the UD and FG-VD plates in the boundary conditions of SSSS, CSSS, CCCC, and CFCF, respectively. It can be seen that the model detects the given damage state with high accuracy, both for data with and without noise of 5%, in which the largest MSE values are only 5.932×10⁻⁵ (UD-SSSS with 5% noise) and 9.554×10⁻⁵ (VD-SSSS with 5% noise).

Next, Fig. 20 graphically indicates the second case of the second scenario in which two damage occurrences of 30.96% and 48.44% are assumed to occur at the 21st and 85th elements, respectively. Figures 21 and 22 illustrate detected results of the UD and FG-VD plates in the boundary conditions of SSSS, CSSS, CCCC, and CFCF, respectively. The figures show that two damage elements of the given scenario are distinctly detected by the DFNN model for all cases of the boundary conditions regardless of the effects of noise on the data, in which all the stiffness degradation factors \left(\alpha \right) of the healthy elements are very close to zero. Generally, in this scenario although relatively large errors arise in the detections with 5% noise-affected data of the 85th element (for FG-VD plate with CCCC boundary condition) and 21st element (for UD plate with CFCF boundary condition), the predictions still show justifiable accuracies.

Finally, the final case of the second scenario is graphically shown in Fig. 23, in which three damage positions are assumed to occur at the 37th, 75th, and 79th elements with the corresponding percentages of 38.61%, 40.50%, and 36.75%. Figures 24 and 25 show detected results of the UD and FG-VD plates in the boundary conditions of SSSS, CSSS, CCCC, and CFCF, respectively, in which all the damaged elements are identified by the proposed DFNN model for both cases of noise-free and 5% noise-influenced data. It can be further seen that there are some over-quantity recognitions for the UD plates subjected to SSSS and CSSS boundary conditions with 5% noise data, and the detections made with noisy data for the UD plate subjected to CFCF boundary condition and the FG-VD plate subjected to SSSS boundary condition produce relatively large MSE values of 5.185×10⁻⁴ and 5.316×10⁻⁴, respectively. However, the proposed DFNN model can generally provide acceptable accuracy for damage detection of the case.

Overall, from the above-presented numerical results, it can be seen that the proposed method can detect the damage occurrences in the CNTRC plates in terms of all given scenarios (Table 7), in which the DFNN model shares relatively equal performances on noise-free data among different boundary conditions. The appearance of noise in the data may escalate the errors in the detected results for the considered boundary conditions, in which the mixed-boundary condition of CFCF tends to be influenced to a greater degree, but the damaged elements are still justifiably identified. Additionally, the proposed method possibly recognizes the damage areas located near the edges of the plates, for which previous methods usually give poor detected results. Furthermore, the proposed DFNN model can detect damage occurrences with equivalent accuracies between the two configurations of the plates including UD and FG-VD. This demonstrates the generalization characteristics of the proposed method in terms of different material configurations of the composite plates. However, for damage occurrences with low severities, the model tends to yield larger errors compared to occurrences with great severities. This is because low extent of damage results in minor change in the vibration responses, hence the DFNN model may experience difficulty in extracting the features of such data during the training process. Especially for noise-impacted data, the overlap between the changes in the vibration responses caused by minor damage occurrence and the effect of noise may blur the perception of the model on the correlation of the input signal to the (output) detected result. Therefore, the model tends to produce more errors in detecting minor damage with noisy data. This aspect will be further studied in future work in order to improve the performance of the proposed method for damage detection.

4 Conclusions

A deep feed-forward neural network model is developed and applied to the damage detection of FG-CNTRC rectangular plates in this work. The numerical examples of two configurations of the CNT-fibers including uniform distribution (UD) and functionally graded-V distribution (FG-VD) with different boundary conditions are performed to validate the effectiveness and applicability of the proposed method. In the construction of the input data set, the MKEn value is incompletely formulated from only translational nodal displacements. The DFNN model can sufficiently detect various cases of damage occurrences including positions along or near the edges of the plates which may not be identifiable with high accuracy by previous methods. Additionally, apart from the high performance for noise-free input data, the proposed approach is able to detect the damage occurrences with justifiable accuracy for noise-existed input data, with the noise level of 5% in the natural frequencies and nodal displacements. Based on the results and discussions presented in the above section, the salient points of the research work are summarized as follows.

1) The successful application of the DFNN model to the prediction of damage occurrences in the FG-CNTRC plates demonstrates the feasibility of DL-based methods in this engineering problem of damage detection. This affords newly promising directions for leveraging vast data sources to deal with engineering tasks which may be difficult to be solved by conventional theoretical approaches.

2) The MKEn value is formulated only from translational nodal displacements with the corresponding reduction in the dimension of the elemental mass matrix, but this still offers adequate training outcomes for the DFNN prediction model. This demonstrates that the DL-based method provides a high flexibility in the design of input data for the detection which traditional damage indicators may not be able to undertake.

3) The proposed DFNN model is trained through samples contained in the input data set. Therefore, this concept shows a possibility for the detection of more than three damage occurrences, which may be achieved thanks to the training data set whose samples indicate various damage states.

4) However, the model tends to make more mistakes in the damage identification if the severity of the damage appears to be small. In future work, this aspect will be further improved; the internal framework of the DL-based model will be amended so that the model is expected to be able to provide increased accuracy in the detection of minor damage occurrences. Additionally, techniques for eliminating the influences of noise, which suitably associates with the DL-based damage detection method, will also be considered and studied.

5) Furthermore, the geometry of the FG-CNTRC plate will be further studied and the concept of damage detection will be developed and expanded to plates with arbitrary shapes.

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