A new neural-network-based method for structural damage identification in single-layer reticulated shells

Abstract

Single-layer reticulated shells (SLRSs) find widespread application in the roofs of crucial public structures, such as gymnasiums and exhibition center. In this paper, a new neural-network-based method for structural damage identification in SLRSs is proposed. First, a damage vector index, NDL, that is related only to the damage localization, is proposed for SLRSs, and a damage data set is constructed from NDL data. On the basis of visualization of the NDL damage data set, the structural damaged region locations are identified using convolutional neural networks (CNNs). By cross-dividing the damaged region locations and using parallel CNNs for each regional location, the damaged region locations can be quickly and efficiently identified and the undamaged region locations can be eliminated. Second, a damage vector index, DS, that is related to the damage location and damage degree, is proposed for SLRSs. Based on the damaged region identified previously, a fully connected neural network (FCNN) is constructed to identify the location and damage degree of members. The effectiveness and reliability of the proposed method are verified by considering a numerical case of a spherical SLRS. The calculation results showed that the proposed method can quickly eliminate candidate locations of potential damaged region locations and precisely determine the location and damage degree of members.

Graphical abstract

Keywords

single-layer reticulated shell / damage identification / neural network / convolutional neural network / cross-partitioning method

1 Introduction

In the field of civil engineering, structural damage can be caused by many factors, such as material aging, corrosion, load effect, and natural disasters [1], and their identification can be divided into two steps: determining the location of damage, and quantifying the damage [2–5].

When damage occurs in a civil engineering structure, changes in the structure’s physical characteristics, such as structural stiffness and damping, result in changes in the vibration response and modal parameters. The dynamic response of the structure can be monitored by installing various types of sensors on the structure to detect structural damage [6,7]. Owing to the development of damage identification theory, monitoring equipment, and modal analysis methods [8], the capability to identify damage has improved significantly [9]. Furthermore, in recent years, a structural health monitoring system has been developed to obtain the structural dynamic response [10,11]. From the structural dynamic response, structural damage can be detected through damage detection or modal analysis techniques [12].

As a universal function approximator, artificial neural networks (ANNs) have been successfully used to solve partial differential equation problems [13,14]. Similarly, structural damage identification can be accomplished by approximating the nonlinear mapping relationship between structural damage characteristics, damage locations, and damage degrees by using ANN [15]. However, shallow ANNs have difficulty in completing more complex nonlinear problems. To further research the problems in more complex engineering systems, application of deep neural network (DNN) has been introduced into the mathematical modeling of complex engineering problems [16,17]. DNNs achieve better approximation for complex engineering system behavior due to their stronger feature extraction and nonlinear mapping capabilities [18,19]. To solve the complex nonlinear damage identification problem, researchers have used DNNs, such as convolutional neural networks (CNNs) [20] and long short-term memory networks [21], and hybrid neural networks [22].

DNNs process the original data through multiple layers of nonlinear transformations and output higher dimensional and more abstract features. Recently, a deep CNN has been widely used for feature extraction; it can detect structural defects and identify structural damage [23] by directly extracting features from structural time history data [24] or frequency domain data [25]. Owing to the advantages of CNNs in feature extraction, it has been used for crack detection in steel structures [26], concrete members [27,28], and for damage identification in frame structures [29] and bridges [30]. However, there have been relatively few studies on the use of CNNs for damage identification in reticulated shells.

When neural networks are used for damage identification, the training data set is constructed by combining members with different damage patterns [31]. Researchers construct damage training data sets based on transmissibility data and they have accomplished structural damage identification using CNNs [32–34]. However, the transmissibility data are related to the damage location and degree simultaneously. For large-scale civil engineering structures containing many members, the direct construction of the training data set with transmissibility data or other indexes related to the damage location and degree simultaneously is prone to training data explosion, as the number of combinations of different member damage locations and degrees is extremely large. For such structures, a phased damage identification method has been developed [35]. The damage location is first determined, and the damage degree is then obtained. Such a stepwise damage identification method facilitates precise damage identification for framed structures [36] and spatial truss structures [37], and it can improve the identification efficiency.

The problems are yet to be overcome for damage identification in single-layer reticulated shells (SLRSs). First, a large number of training data sets should be constructed in advance because of the huge number of structural members. Secondly, some damage patterns are very similar because of the close spacing of modes [38]. Thirdly the presence of too many members in the damaged area reduces the accuracy of member damage identification when the traditional stepwise damage identification method is used. To solve these problems, this paper proposes: 1) a damage location index to determine the damaged region locations and damage degree index to determine the damage degree of members; 2) a CNN-based damaged region location method that can identify complex damage patterns and a cross-partitioning damaged region location method that can significantly narrow down the possible damaged region; 3) a damaged member identification method involving a fully connected neural network (FCNN) that should be applied to the narrowed-down damaged region.

This paper is organized as follows. First, the damage-sensitive feature of SLRSs is derived. Then, CNNs for damaged region location and a cross-partitioning damaged region localization method based on CNNs are developed. The CNN-based cross-partitioning damaged region localization method can help identify complex damage patterns and significantly narrow down the affected location, facilitating the achievement of a higher identification precision. Subsequently, FCNNs are constructed to identify the damaged member of the structure. Finally, the above methods are applied to a spherical SLRS to verify their effectiveness.

2 Damage sensitive features of single-layer reticulated shell

2.1 Damage localization index of single-layer reticulated shell

The vibration equation of an n-degree of freedom (n-DOF) undamped structure system can be written as

(1)\left(\mathit{K}-{\omega }_i^2\mathit{M}\right){\mathit{\Phi }}_i=0,

where K is the stiffness matrix of the structure, M is the mass matrix of the structure, {\omega }_i is the ith order circular frequency of the structure, and {\mathit{\Phi }}_i is the ith order vibration mode of the structure.

Assuming that only K of the structure changes and that M remains unchanged when the structure is damaged, we define the structural vibration equation for n-DOFs as

(2)\left[(\mathit{K}+\mathrm{\Delta }\mathit{K})-\left({\omega }_i^2+\mathrm{\Delta }{\omega }_i^2\right)\mathit{M}\right]\left({\mathit{\Phi }}_i+\mathrm{\Delta }{\mathit{\Phi }}_i\right)=0,

where \mathrm{\Delta }\mathit{K} is the stiffness variation of the structure, \mathrm{\Delta }{\omega }_i is the variation of the ith order circular frequency of the structure, and \mathrm{\Delta }{\mathit{\Phi }}_i is the variation of the ith order vibration mode of the structure.

Combing Eqs. (1) and (2) and ignoring the higher-order infinitesimal terms, we can write \mathrm{\Delta }{\omega }_i^2 as

(3)\mathrm{\Delta }{\omega }_i^2={\mathit{\Phi }}_i^{\mathrm{T}}\mathrm{\Delta }\mathit{K}{\mathit{\Phi }}_i/{\mathit{\Phi }}_i^{\mathrm{T}}\mathit{M}{\mathit{\Phi }}_i.

In a linear structure, a change in the structural stiffness is the linear superposition of changes in the stiffness of all elements. Hence, \mathrm{\Delta }\mathit{K} can be expressed as

(4)\mathrm{\Delta }\mathit{K}=\sum _{k=1}^L{\alpha }_k{\mathit{k}}_k\left(-1\le {\alpha }_k\le 0\right),

where {\alpha }_k is the damage coefficient of the kth element in the structure, {\mathit{k}}_k is the element-stiffness-contributed matrix of the kth element in the global stiffness matrix, and L is the total number of structural elements.

On the basis of the orthogonality of modal shapes, when r ≠ i, {\mathit{\Phi }}_r^{\mathrm{T}}\mathit{M}{\mathit{\Phi }}_i=0 and {\mathit{\Phi }}_r^{\mathrm{T}}\mathit{K}{\mathit{\Phi }}_i=0. By left-multiplying Eq. (2) with {\mathit{\Phi }}_r^{\mathrm{T}} and combining it with the equation {\mathit{\Phi }}_r^{\mathrm{T}}\left(\mathit{K}-{\omega }_r^2\mathit{M}\right)=0, we can obtain the following equation, in which the higher-order infinitesimal has been ignored:

(5)\left({\omega }_r^2-{\omega }_i^2\right){\mathit{\Phi }}_r^{\mathrm{T}}\mathit{M}\mathrm{\Delta }{\mathit{\Phi }}_i=-{\mathit{\Phi }}_r^{\mathrm{T}}\mathrm{\Delta }\mathit{K}{\mathit{\Phi }}_i.

Using the assumption of the linear model that the modal vibrations of the structure are independent of each other, we can express the variation of the ith order vibration mode as a linear combination of the modes:

(6)\mathrm{\Delta }{\mathit{\Phi }}_i=\sum _{i=1}^n{\epsilon }_i{\mathit{\Phi }}_i,

where {\epsilon }_i is the weight factor corresponding to {\mathit{\Phi }}_i.

Substituting Eq. (6) into Eq. (5) yields

(7)\left({\omega }_r^2-{\omega }_i^2\right){\mathit{\Phi }}_r^{\mathrm{T}}\mathit{M}\sum _{i=1}^n{\epsilon }_i{\mathit{\Phi }}_i=-{\mathit{\Phi }}_r^{\mathrm{T}}\mathrm{\Delta }\mathit{K}{\mathit{\Phi }}_i.

When r ≠ i, {\mathit{\Phi }}_r^{\mathrm{T}}\mathit{M}{\mathit{\Phi }}_i=0 and {\mathit{\Phi }}_r^{\mathrm{T}}\mathit{K}{\mathit{\Phi }}_i=0. Therefore, Eq. (7) can be simplified as

(8){\epsilon }_r{\mathit{\Phi }}_r^{\mathrm{T}}\left(\mathit{K}-{\omega }_i^2\mathit{M}\right){\mathit{\Phi }}_r=-{\mathit{\Phi }}_r^{\mathrm{T}}\mathbf{\Delta }\mathit{K}{\mathit{\Phi }}_i.

Herein we adopt the modal mass normalization method, i.e., {\mathit{\Phi }}_r^{\mathrm{T}}\mathit{M}{\mathit{\Phi }}_r=1. Then, Eq. (1) can be rewritten as {\mathit{\Phi }}_r^{\mathrm{T}}\mathit{K}{\mathit{\Phi }}_r={\omega }_r^2. The weight factor {\epsilon }_r corresponding to {\mathit{\Phi }}_r can be written as

(9)\begin{array}{c}{\epsilon }_r={\mathit{\Phi }}_r^{\mathrm{T}}\mathrm{\Delta }\mathit{K}{\mathit{\Phi }}_i/\left({\omega }_i^2-{\omega }_r^2\right),\left({\omega }_r\ne {\omega }_i\right),\\ {\epsilon }_r=0,\left({\omega }_r={\omega }_i\right).\end{array}

The parameter \mathrm{\Delta }{\mathit{\Phi }}_i can be rewritten as

(10)\mathrm{\Delta }{\mathit{\Phi }}_i=\sum _{\begin{array}{c}r=1\\ r\ne i\end{array}}^n\left({\mathit{\Phi }}_r^{\mathrm{T}}\mathrm{\Delta }\mathit{K}{\mathit{\Phi }}_i/\left({\omega }_i^2-{\omega }_r^2\right)\right){\mathit{\Phi }}_r.

Substituting Eq. (3) into Eq. (10) yields

(11)\mathrm{\Delta }{\mathit{\Phi }}_i=\sum _{k=1}^L{\alpha }_k\sum _{\begin{array}{c}r=1\\ r\ne i\end{array}}^n\left({\mathit{\Phi }}_r^{\mathrm{T}}{\mathit{k}}_k{\mathit{\Phi }}_i/\left({\omega }_i^2-{\omega }_r^2\right)\right){\mathit{\Phi }}_r.

Combining Eqs. (3) and (11), we obtain

(12)\frac{\mathrm{\Delta }{\mathit{\Phi }}_i}{\mathrm{\Delta }{\omega }_i^2}=\sum _{k=1}^L{\alpha }_k\sum _{\begin{array}{c}r=1\\ r\ne i\end{array}}^n\frac{{\mathit{\Phi }}_r^{\mathrm{T}}{\mathit{k}}_k{\mathit{\Phi }}_i}{{\omega }_i^2-{\omega }_r^2}{\mathit{\Phi }}_r/\sum _{k=1}^L{\alpha }_k{\mathit{\Phi }}_i^{\mathrm{T}}{\mathit{k}}_k{\mathit{\Phi }}_i.

If there is only one damaged element in the structure or all elements have similar damage degrees, the damage coefficient would be eliminated in Eq. (12). Subsequently, Eq. (13) below is derived from Eq. (12), and it is a damage-sensitive feature of the structure and is independent of the damage degree of the elements. Therefore, Eq. (13) is referred to as damage location feature.

(13)\frac{\mathrm{\Delta }{\mathit{\Phi }}_i}{\mathrm{\Delta }{\omega }_i^2}=\sum _{k=1}^L\sum _{\begin{array}{c}r=1\\ r\ne i\end{array}}^n\frac{{\mathit{\Phi }}_r^{\mathrm{T}}{\mathit{k}}_k{\mathit{\Phi }}_i}{{\omega }_i^2-{\omega }_r^2}{\mathit{\Phi }}_r/\sum _{k=1}^L{\mathit{\Phi }}_i^{\mathrm{T}}{\mathit{k}}_k{\mathit{\Phi }}_i.

Generally, the damage location feature is expressed using the damage location vector DL_i, which can be written as

(14){\mathit{D}\mathit{L}}_i=\frac{\mathrm{\Delta }{\mathit{\Phi }}_i}{\mathrm{\Delta }{\omega }_i^2}=\frac{{\mathit{\Phi }}_{\mathrm{u}i}-{\mathit{\Phi }}_{\mathrm{d}i}}{{\omega }_{\mathrm{u}i}^2-{\omega }_{\mathrm{d}i}^2},

where {\mathit{\Phi }}_{\mathrm{u}i} and {\mathit{\Phi }}_{\mathrm{d}i} are the ith-order modes of the undamaged structure and damaged structure, respectively, and {\omega }_{\mathrm{u}i}^{} and {\omega }_{\mathrm{d}i}^{} are the ith-order circular frequencies of the undamaged structure and damaged structure, respectively.

When the damage location vector DL_i is to be input into the neural network for machine learning, it should be normalized. The normalized damage location vector NDL_i can be written as

(15)\mathit{N}\mathit{D}{\mathit{L}}_i=\mathit{D}{\mathit{L}}_i/\sum _{i=1}^n\left|\mathit{D}{\mathit{L}}_i\right|,

where \left|\mathit{D}{\mathit{L}}_i\right| is the module of vector DL_i.

From Eqs. (14) and (15), a matrix related only to damage location can be constructed:

(16)\mathit{N}\mathit{D}\mathit{L}=\left[\mathit{N}\mathit{D}{\mathit{L}}_1,\mathit{N}\mathit{D}{\mathit{L}}_2,\dots ,\mathit{N}\mathit{D}{\mathit{L}}_i\right].

Using the damage location matrix NDL, we can perform the damage pattern identification of a damaged location by training a neural network with damage location vectors. In the damage identification of SLRS, the NDL based neural network is used to determine the potential damage areas of SLRS, thereby greatly reducing the range of candidates of damage patterns. Then, the location and degree of damaged members can be determined in the potential damage regions of SLRS by constructing corresponding neural network.

2.2 Damage degree index of single-layer reticulated shell

Equations (3) and (11) show that the vibration mode variation and the natural frequency variation of the structure are related to the damage location and degree. Hence, a vector related to both damage location and degree can be constructed:

(17)\mathit{D}\mathit{S}=\left[\mathrm{\Delta }{\mathit{\Phi }}_{N1},\mathrm{\Delta }{\mathit{\Phi }}_{N2},\dots ,\mathrm{\Delta }{\mathit{\Phi }}_{Nn},\mathrm{\Delta }{\omega }_1^2,\mathrm{\Delta }{\omega }_2^2,\dots ,\mathrm{\Delta }{\omega }_n^2\right],

where \mathrm{\Delta }{\mathit{\Phi }}_{Nn} is the variation of the normalized vibration mode vector of the nth order mode

On the basis of the vector DS, the neural network can learn the relevant features of damage location and degree to complete the damage location and degree identification of members in potential damage areas of SLRS.

3 Convolutional neural network-based cross-partitioning damaged region location method for single-layer reticulated shell

The number of members is large in an SLRS, and hence, there are numerous potential damage patterns. Therefore, it is challenging to directly determine the damaged member and its corresponding damage degree by using a neural network. However, the SLRS can be divided into substructures with similar or even identical geometric shapes, which implies that both the member distribution and damage patterns of the substructures have a certain regularity. A specific division method can used to divide the SLRS into several substructures. In the damage identification process, the damaged region locations can be determined first, which significantly reduces the number of unreasonable damage patterns. This section proposes a CNN-based cross-partitioning damaged location method for SLRS. The method can significantly narrow down the possible damaged location of the structure.

3.1 Construction of damage pattern

The nonlinear mapping relationship between the input and the damaged substructure can be determined by training the CNN with the common features of damage patterns in the substructure. The damage location vector can be processed and input into the CNN, which extracts the high-dimensional features in the damage location vector and performs damaged region location identification.

This section describes the transformation of the damage location matrix of the structure into images, and the training of the CNN with the features of the matrix to determine the mapping relationship between the damage pattern and the damaged region location. The trained CNN can be used to identify the damaged region location. The process of constructing the damage pattern that can be input into the CNN can be summarized as follows.

S1: Some members are randomly selected from the SLRS, and their stiffnesses are reduced to simulate structural damage. The damage location matrix NDL is calculated using Eqs. (11)–(16).

S2: In the NDL matrix, NDL_i is selected to construct a set of scatter points in the x, y, and z directions:

(18)\begin{array}{rl}& {\mathit{S}}_x=\left[{\mathit{S}}_{x1},{\mathit{S}}_{x2},\dots ,{\mathit{S}}_{xm}\right],\\ & {\mathit{S}}_y=\left[{\mathit{S}}_{y1},{\mathit{S}}_{y2},\dots ,{\mathit{S}}_{ym}\right],\\ & {\mathit{S}}_{\mathit{\text{z}}}=\left[{\mathit{S}}_{\mathit{\text{z}}1},{\mathit{S}}_{\mathit{\text{z}}2},\dots ,{\mathit{S}}_{\mathit{\text{z}}m}\right],\end{array}

where m is the number of joints in the SLRS and S_x, S_y, and S_z are the damage pattern scatter point sets for the x, y, and z directions, respectively. S_xm, S_ym, and S_zm are the three-dimensional coordinates of the mth point in S_x, S_y, and S_z, respectively, and they can be calculated as follows:

(19)\begin{array}{rl}& {\mathit{S}}_{xm}=\left({\mathit{N}\mathit{D}\mathit{L}}_{ixm},D_{ym},D_{\mathit{\text{z}}m}\right),\\ & {\mathit{S}}_{ym}=\left(D_{xm},{\mathit{N}\mathit{D}\mathit{L}}_{iym},D_{\mathit{\text{z}}m}\right),\\ & {\mathit{S}}_{\mathit{\text{z}}m}=\left(D_{xm},D_{ym},{\mathit{N}\mathit{D}\mathit{L}}_{i\mathit{\text{z}}m}\right),\end{array}

where D_xm, D_ym, and D_zm are the x, y, and z direction coordinates of the mth joint of the SLRS, respectively, and NDL_ixm, NDL_iym, and NDL_izm are the normalized damage location vectors at the mth node in the x, y, and z directions, respectively.

S3: The scattered points of Eq. (18) are fitted to a three-dimensional surface:

(20)h_x(x,y,\mathit{\text{z}})\to {\mathit{S}}_x,h_y(x,y,\mathit{\text{z}})\to {\mathit{S}}_y,h_{\mathit{\text{z}}}(x,y,\mathit{\text{z}})\to {\mathit{S}}_{\mathit{\text{z}}},

where h_x, h_y, and h_z are the fitting functions of S_x, S_y, and S_z, respectively. The fitting function form that has a better fitting result on scatter points is chosen among various fitting function forms.

S4: The three-dimensional surfaces formed by h_x, h_y, and h_z are projected onto the yz, xz, and xy planes to form projection images, respectively. The projection images on the yz, xz, and xy planes are denoted by H_x, H_y, and H_z, respectively. Here, H_x, H_y, and H_z are three-dimensional tensors for damage pattern identification.

3.2 Damaged region location method

The damage pattern construction method has been described in Subsection 3.1. The CNN-based damaged region location method is developed in this section. Herein, v damage patterns are constructed according to Subsection 3.1 to simulate the damage of the structure. For the vth damage pattern, the member damage is simulated by reducing the elastic moduli of k_v randomly selected members to a fixed value. The position vector of damaged members {\mathit{t}}_{k_v}^vand the set of damage patterns DP_C can be written as

(21){\mathit{t}}_{k_v}^v=\left[t_1^v,t_2^v,\dots ,t_{k_v}^v\right],

(22){\mathit{D}\mathit{P}}_{\mathrm{C}}=\left\{\mathit{H}\left({\mathit{t}}_{k_1}^1\right),\mathit{H}\left({\mathit{t}}_{k_2}^2\right),\dots ,\mathit{H}\left({\mathit{t}}_{k_v}^v\right)\right\},

(23)\mathit{H}\left({\mathit{t}}_{k_v}^v\right)=\left[{\mathit{H}}_{\mathrm{d}\mathrm{i}\mathrm{r}}^1\left({\mathit{t}}_{k_v}^v\right),{\mathit{H}}_{\mathrm{d}\mathrm{i}\mathrm{r}}^2\left({\mathit{t}}_{k_v}^v\right),\dots ,{\mathit{H}}_{\mathrm{d}\mathrm{i}\mathrm{r}}^{n_{\mathrm{f}}}\left({\mathit{t}}_{k_v}^v\right)\right],

where n_f is the number of selected vibration modes, dir represents the x, y, or z direction, \mathit{H}\left({\mathit{t}}_{k_v}^v\right) is the vth input three-dimensional tensor, and {\mathit{H}}_{\mathrm{d}\mathrm{i}\mathrm{r}}^{n_{\mathrm{f}}}\left({\mathit{t}}_{k_v}^v\right) is the n_fth three-dimensional tensor damage pattern in the x, y, or z direction for the position vector {\mathit{t}}_{k_v}^v of damaged members.

Assuming that the SLRS can be divided into n_a substructure regions under a certain partitioning scheme, and there exist combinations of damaged substructure regions S={\displaystyle \sum _{n_{\text{b}}=1}^{n_{\text{a}}}{C}_{n_{\text{a}}}^{n_{\text{b}}}} when the SLRS has n_a substructure regions. Based on the s types of damage substructure regions combinations, s types of subsets can be constructed in the damage pattern set DP_C:

(24){\mathit{D}\mathit{P}}_{\mathrm{C}}=\left\{\left\{{\mathit{H}}_{\mathrm{C}}^{(1)}\right\},\left\{{\mathit{H}}_{\mathrm{C}}^{(2)}\right\},\dots ,\left\{{\mathit{H}}_{\mathrm{C}}^{(\mathrm{s})}\right\}\right\},

where \left\{{\mathit{H}}_{\mathrm{C}}^{(\mathrm{s})}\right\} is the damage pattern subset under the sth damage substructure combination, which includes a series of damage patterns.

Based on the DP_C in Eq. (22), the corresponding damaged region location set OP_C can be written as

(25)\mathit{O}{\mathit{P}}_{\mathrm{C}}=\left\{\mathit{r}\left({\mathit{t}}_{k_1}^1\right),\mathit{r}\left({\mathit{t}}_{k_2}^2\right),\dots ,\mathit{r}\left({\mathit{t}}_{k_v}^v\right)\right\},

(26)\mathit{r}\left({\mathit{t}}_{k_v}^v\right)=\left[R{R}_1,R{R}_2,\dots ,R{R}_{n_{\mathrm{a}}}\right],

where \mathit{r}\left({\mathit{t}}_{k_v}^v\right) is the damaged substructure region location vector, R{R}_{n_{\mathrm{a}}} is the damage state of n_ath substructure region. It should be noted that R{R}_{n_{\mathrm{a}}} = 1 when the n_ath substructure region occurs damage, and R{R}_{n_{\mathrm{a}}} = 0 indicates that the n_ath substructure region is undamaged.

Similarly, s types of subsets can be constructed in the corresponding damaged region location set OP_C:

(27){\mathit{O}\mathit{P}}_{\mathrm{C}}=\left\{\left\{{\mathit{r}}_{\mathrm{C}}^{(1)}\right\},\left\{{\mathit{r}}_{\mathrm{C}}^{(2)}\right\},\dots ,\left\{{\mathit{r}}_{\mathrm{C}}^{(s)}\right\}\right\},

where \left\{{\mathit{r}}_{\mathrm{C}}^{(s)}\right\} is the corresponding output subset under the sth damage substructure combination, which includes a series of corresponding damaged substructure region location vectors.

According to Eqs. (24) and (27), the damage pattern data set and corresponding damaged region location data set can be divided into different categories, and the CNN can be used to complete the classification task. In the training process, the CNN is trained through DP_C and OP_C. Using the trained CNN, the damaged substructure regions can be identified.

3.3 Cross-partitioning damaged region location

The partitioning scheme of substructures is flexible, and SLRS can be divided into many different substructures by using different partition methods. A cross-partitioning method for determining the damaged region location of an SLRS is proposed in this section, which can significantly narrow down the range of possible damaged locations in the structure.

If it is assumed that the SLRS can be divided into substructures by using t types of partitioning schemes, and the SLRS can be divided into n_a,t substructure regions for the tth type of partitioning scheme, then the combination of damaged substructure regions s_t can be calculated as s_t={\displaystyle \sum _{n_{\text{b}}=1}^{n_{\text{a},t}}{C}_{n_{\text{a},t}}^{n_{\text{b}}}}. For the tth partitioning scheme, the damage pattern data set DP_C,t and corresponding damaged region location data set OP_C,t including s_t subsets can be constructed according to Eqs. (24) and (27). A series of damage pattern sets with different subsets can be constructed using similar methods; the damage pattern set can be expressed as DP_C,1,DP_C,2,…,DP_C,t, and the corresponding damaged location set can be expressed as OP_C,1,OP_C,2,…,OP_C,t.

For DP_C,1,DP_C,2,…,DP_C,t and corresponding OP_C,1,OP_C,2,…,OP_C,t, a total of t CNNs f₁,f₂,...,f_t are trained for classification, in which CNN f_t can classify the s_tth type of damage substructure regions for the tth type of partitioning scheme.

In the damaged region location determination stage, an observed damage pattern DP is input into the trained CNN classifier. The classification results of the CNN can be output as a probability vector {\mathit{P}}_t by the SoftMax layer:

(28){\mathit{P}}_1=f_1(\mathit{D}\mathit{P}),{\mathit{P}}_2=f_2(\mathit{D}\mathit{P}),\dots ,{\mathit{P}}_t=f_t(\mathit{D}\mathit{P}),

(29){\mathit{P}}_t=\left[p_1,p_2,\dots ,p_{s_t}\right],

where p_{s_t} is the probability of damage occurring in the s_tth substructure region combination under the tth type of partitioning scheme.

The damaged region locations determined by different CNNs usually have common regions. The final damaged region location is the intersection region of all identified damaged region locations. During the calculation of first-time cross-partitioning, the damaged region location probability matrix M₁ in each region can be written as {\mathit{M}}_1={\mathit{P}}_1^{\mathrm{T}}{\mathit{P}}_2, then the M₁ is changed into row vector {\mathit{V}}_1:{\mathit{M}}_1\to {\mathit{V}}_1. When the second-time cross-partitioning is calculated, the probability matrix of the second-time cross-partitioning damaged region location can be written as {\mathit{M}}_2={\mathit{V}}_1^{\mathrm{T}}{\mathit{P}}_3. The above cross-partitioning calculation is performed in order.

After cross-partitioning calculations are performed t times, the probability of damage in each region can be written as {\mathit{M}}_{t-1}={\mathit{V}}_{t-2}^{\mathrm{T}}{\mathit{P}}_t. In {\mathit{M}}_{t-1}, the regions corresponding to values close to 0 are considered undamaged, while other regions are considered damaged areas.

The cross-partitioning damaged region location method for an SLRS is described in Fig.1. This method involves the use of multiple CNNs connected in parallel, and the classification result vectors are multiplied t–1 times. A large region of potential damage as determined by a single CNN can be reduced to a smaller region when multiple CNNs are used in parallel. Since the number of potentially damaged members and the number of corresponding damage patterns can be reduced, it is easier to determine the more precise damage location and degree of members in the smaller region.

3.4 Convolutional neural network

The CNN is used to identify damage patterns to locate the damaged region locations, and it consists of convolution, pooling, fully connected (FC), and SoftMax layers. This section presents technical details of the CNN.

3.4.1 Convolutional layer

The convolution layer is used to compute the convolution of the input data and convolutional kernel. The convolution kernel slides over the input data with a fixed stride and convolves with the input data. The feature map matrix can be calculated after the convolution operation by using Eq. (30). Subsequently, the bias matrix is added to the feature map, as shown in Eq. (31).

The operations performed for the convolutional layer are as follows:

(30){\mathit{F}}_{\text{c}}\left(i_{\text{c}},j_{\text{c}}\right)={\mathit{I}}_{\text{c}}\otimes {\mathit{K}}_{\text{c}}=\sum _{m_{\text{c}}=1}^{M_{\text{c}}}\sum _{n_{\text{c}}=1}^{N_{\text{c}}}{\mathit{I}}_{\text{c}}\left(i_{\text{c}}+m_{\text{c}},j_{\text{c}}+n_{\text{c}}\right){\mathit{K}}_{\text{c}}\left(m_{\text{c}},n_{\text{c}}\right),

(31){\mathit{S}}_{\mathrm{c}}\left(i_{\mathrm{c}},j_{\mathrm{c}}\right)={\mathit{F}}_{\mathrm{c}}\left(i_{\mathrm{c}},j_{\mathrm{c}}\right)+{\mathit{b}}_{\mathrm{c}},

where {\mathit{F}}_{\mathrm{c}} is the feature map matrix; {\mathit{K}}_{\mathrm{c}} is the convolution kernel; {\mathit{I}}_{\mathrm{c}} is the input data matrix; \otimes represents the convolution operation; M_c and N_c are the horizontal and vertical dimensions of the two-dimensional convolution kernel, respectively; i_c and j_c represent the row number and column number of the elements in the matrix I_c, respectively; m_c and n_c represent the row number and column number of the elements in the matrix K_c, respectively; S_c is the biased characteristic mapping matrix; b_c is the bias matrix.

3.4.2 Pooling layer

The pooling layer can be divided into two layers: the maximum pooling layer and average pooling layer. If operations such as classification are performed directly after the convolutional layer, overfitting is likely to occur. The pooling layer is introduced to avoid this problem. The pooling layer resamples the feature map matrix extracted by the convolutional layer, and it compresses the feature map matrix to reduce the number of parameters in the CNN. The pooling layer can also enhance the anti-interference ability of CNN.

3.4.3 Local response normalization layer

The local response normalization (LRN) layer is added to some CNN models, and it is designed to increase the convergence speed and generalization ability. The calculation of the LRN layer can be expressed as

(32)b_{x,y}^i=a_{x,y}^i/{\left(k+a\sum _{j=min(0,i-n_k/2)}^{min(N_k-1,i+n_k/2)}{(a_{x,y}^i)}^2\right)}^{\beta },

where a_{x,y}^i is the activity of a neuron computed by applying convolutional kernel i at position (x,y) and then applying the ReLU nonlinearity, and b_{x,y}^iis the response-normalized activity. The summation is over n_k “adjacent” kernel maps at the same spatial position, and N_k is the total number of kernels in the layer. The constants k, n_k, α, and β are hyperparameters whose values are determined using a validation data set.

3.4.4 Architecture of convolutional neural network

On the basis of the aforementioned layers of CNNs, some CNN architectures have been proposed, such as AlexNet [39], VGG [40], GoogLeNet [41], and ResNet [42]. This section describes the use of the GoogLeNet architecture to identify the damaged region location.

The network structure of GoogLeNet is shown in Fig.2(a). Conv represents the convolutional layer, Maxpool represents the max pooling layer, LRN represents the LRN layer, Avepool represents the average pooling layer, FC represents the FC layer, and SoftMax represents the SoftMax layer. In addition, the inception layer is added in GoogLeNet (Fig.2(b)). After the convolution and pooling layers, the ReLU function is selected as the activation function.

Two auxiliary classifiers, shown in Fig.2(c), are added in the middle layer of the GoogLeNet model. The model can well differentiate the features in the lower layers when calculating the loss, and this capability accelerates the convergence of the entire network. At inference time, these auxiliary classifiers are discarded. The losses of the two auxiliary classifiers are multiplied by weight factors during training and added to the overall loss of the network before backward propagation. The identification of damaged region location is a classification task, so cross entropy loss function is selected in CNN. In the training process, the weights and biases are updated iteratively through the back propagation algorithm to reduce the loss function. The loss function of the network is given by

(33)LL=0.3L_1+0.3L_2+L_5,

where LL is the loss of the network during training; L₁ and L₂ are the losses of auxiliary classifiers 1 and 2, respectively; L₃ is the loss of the main network.

The hyperparameters of the CNN are introduced in Subsection 6.2.

4 Damaged member identification for single-layer reticulated shell by using a fully connected layer neural network

According to Eq. (17), the vector DS is related to the damage location and degree. Hence, part of the normalized vibration mode variation and circular frequency variation in DS can be selected to construct an input vector. The input in the input layer of the neural network is denoted as I = DS (n_f,n_ω), and it can be calculated as

(34)\begin{array}{rl}\mathit{I}=& \mathit{D}\mathit{S}\left(n_f,n_{\omega }\right)\\ =& \left[\mathrm{\Delta }{\mathit{\Phi }}_{\mathrm{N}1},\mathrm{\Delta }{\mathit{\Phi }}_{\mathrm{N}2},\dots ,\mathrm{\Delta }{\mathit{\Phi }}_{\mathrm{N}{n}_{\mathrm{f}}},\mathrm{\Delta }{\omega }_1^2,\mathrm{\Delta }{\omega }_2^2,\dots ,\mathrm{\Delta }{\omega }_{n_{\mathrm{e}}}^2\right],\end{array}

where I is a vector of length r, with r = m × n_f + n_ω, and n_ω is the number of natural frequencies selected in the calculation.

The output layer in the neural network is denoted as p, and it can be expressed as

(35)\mathit{p}=\left[0,\dots ,0,\dots ,D{D}_1,\dots ,D{D}_2,\dots ,D{D}_{n_{\mathrm{e}}},\dots ,0,\dots ,0\right],

where p is a vector with length n_m, n_m is the total number of members in the SLRS, n_e is the number of members in the damaged region location, and D{D}_{n_{\mathrm{e}}} is the damage degree of the n_eth member. The zero values in p represent the corresponding members are in undamaged states.

After the cross-partitioning damaged region location process, the size of the region that may be damaged is reduced. The FCNN is adopted for the damaged member identification. For the determined damaged region location, a total of w damage patterns are constructed to simulate the damages of the structure according to Eq. (34). The member damage is simulated by reducing the elastic moduli of k_w randomly selected members from the n_e members by n^w for the wth damage pattern. The position vector of damaged members {\mathit{t}}_{k_w}^w and the set of damage patterns DP_NN can be written as

(36){\mathit{t}}_{k_w}^w=\left[t_1^w,t_2^w,\dots ,t_{k_w}^w\right],

(37)\mathit{D}{\mathit{P}}_{\mathrm{N}\mathrm{N}}=\left\{\mathit{I}\left({\mathit{t}}_{k_1}^1,n^1\right),\mathit{I}\left({\mathit{t}}_{k_2}^2,n^2\right),\dots ,\mathit{I}\left({\mathit{t}}_{k_w}^w,n^w\right)\right\},

(38)\begin{array}{rl}\mathit{I}\left({\mathit{t}}_{k_w}^w,n^w\right)=& [\mathrm{\Delta }{\mathit{\Phi }}_{\mathrm{N}1}\left({\mathit{t}}_{k_w}^w,n^w\right);\mathrm{\Delta }{\mathit{\Phi }}_{\mathrm{N}2}\left({\mathit{t}}_{k_w}^w,n^w\right);\dots ;\mathrm{\Delta }{\mathit{\Phi }}_{\mathrm{N}{n}_f}\left({\mathit{t}}_{k_w}^w,n^w\right);\\ & \mathrm{\Delta }{\omega }_1{\left({\mathit{t}}_{k_w}^w,n^w\right)}^2;\mathrm{\Delta }{\omega }_2{\left({\mathit{t}}_{k_w}^w,n^w\right)}^2;\dots ;\mathrm{\Delta }{\omega }_{n_w}{\left({\mathit{t}}_{k_w}^w,n^w\right)}^2],\end{array}

where {\mathit{t}}_{k_w}^w is the k_wth damaged member for the wth damage pattern, \mathit{I}\left({\mathit{t}}_{k_w}^w,n^w\right) is the wth input vector of the FCNN, \mathrm{\Delta }{\mathit{\Phi }}_{\mathrm{N}{n}_{\mathrm{f}}}\left({\mathit{t}}_{k_w}^w,n^w\right) and \mathrm{\Delta }{\omega }_{n_w}\left({\mathit{t}}_{k_w}^w,n^w\right) are the normalized vibration mode variation and circular frequency variation for the position vector of damaged members {\mathit{t}}_{k_w}^w and the elastic modulus reduction n^w, respectively.

According to Eq. (35), the set of corresponding outputs OP_NN can be written as

(39)\mathit{O}{\mathit{P}}_{\mathrm{N}\mathrm{N}}=\left\{\mathit{p}\left({\mathit{t}}_{k_1}^1,n^1\right),\mathit{p}\left({\mathit{t}}_{k_2}^2,n^2\right),\dots ,\mathit{p}\left({\mathit{t}}_{k_w}^w,n^w\right)\right\},

(40)\mathit{p}\left({\mathit{t}}_{k_w}^w,n^w\right)=\left[0,\dots ,0,\dots ,D{D}_1,\dots ,D{D}_2,\dots ,D{D}_{n_{\mathrm{e}}},\dots ,0,\dots ,0\right],

where \mathit{p}\left({\mathit{t}}_{k_w}^w,n^w\right) is the wth output vector of the FCNN and D{D}_{n_{\mathrm{e}}} is the damage degree of the n_eth damaged member.

In the training process, the FCNN is trained through DP_NN and OP_NN. In the damaged member determination stage, an observed damage pattern I_ob is input into the trained FCNN, and the identification results p_id can be output by the FCNN, then the damaged member and damage degree can be determined by p_id.

The corresponding FCNN is then constructed to identify the damaged members and their damage degree in the smaller identified damaged region location. The architecture of the FCNN is shown in Fig.3, and mean squared error loss is used for the loss function in FCNN. Since the size of the damaged region location determined by the cross-partitioning damaged region location method is different in different damage patterns, the number of layers and neurons of the FCNN shown in Fig.3 may need to be different in order to achieve more precise identification results. We discuss in detail the hyperparameter of the FCNN used to identify damage elements in Subsection 6.3.

5 Damage identification process of single-layer reticulated shell

Combining the damaged region location method in Section 3 and the damaged member identification method in Section 4, the damage identification process of SLRS is shown in Fig.4.

In the damage identification process, the cross-partitioning damaged region location method is used to narrow down the range of possible damaged locations. After determining the damaged location, the damaged members can be identified and their damage degree can be determined.

6 Numerical example: Structural damage identification for spherical single-layer reticulated shell

6.1 Model parameters

A Kewitt-6 spherical SLRS model is used to verify the proposed damaged region location method and damaged member identification method. Model development and modal analysis of the structure are performed in ANSYS 18.0 [43], and the methods proposed in Sections 3 and 4 are used to identify the damage of the SLRS.

The Kewitt-6 SLRS has a span of 30 m, and there are 5 rings in the radial direction. The height-to-span ratio of the structure is 0.25, and the dead load is 1.5 kN/m². The structure is made of 6061-T6 aluminum alloy with a yield stress of f_0.2 = 240 MPa, an elastic modulus of E = 70 GPa, and a density of ρ = 2700 kg/m³. All the sectional dimensions of the member in SLRS are H250 mm × 150 mm × 8 mm × 10 mm. In the finite element model, pinned supports are positioned at the periphery, and element BEAM188 is used to simulate the structural members. Note that we ignore the contribution of the roof panel to the overall stiffness of the structure, yet the roof panel is simplified as concentrated mass at joints; the MASS21 element is used to simulate the mass. The finite element model of the SLRS is shown in Fig.5. The joint numbers and the member numbers are shown in Fig.6.

6.2 Cross-partitioning damaged region location

If it is assumed that damage only occurs at the members and the support always remains intact, 30 members at the outermost ring cannot affect the natural frequency and vibration mode of the structure. To simulate the random damage pattern of the structure, we randomly selected 1−10 members from the remaining 210 members to apply the damage. The elastic modulus of the damaged member is uniformly reduced by 20%. Vertical vibration modes dominate the structure because of the small height-to-span ratio. The first and second vibration modes in the z direction are selected to construct the damage pattern set. The calculation generates a total of 84000 damage patterns according to the process described by S1−S4 in Subsection 3.1. Combining Eqs. (21)–(23), we can write the input of the CNN as

(41)\mathit{D}{\mathit{P}}_{\mathrm{C}}=\left\{\mathit{H}\left({\mathit{t}}_1^1\right),\mathit{H}\left({\mathit{t}}_1^2\right),\dots ,\mathit{H}\left({\mathit{t}}_{k_v}^v\right),\dots ,\mathit{H}\left({\mathit{t}}_{10}^{84000}\right)\right\},

(42)\mathit{H}\left({\mathit{t}}_{k_v}^v\right)=\left[{\mathit{H}}_{\mathit{\text{z}}}^1\left({\mathit{t}}_{k_v}^v\right),{\mathit{H}}_{\mathit{\text{z}}}^2\left({\mathit{t}}_{k_v}^v\right)\right].

The spherical SLRS in Fig.5 is divided using two types of substructure partition schemes, which are named Sub1 and Sub2 in Fig.7.

In Fig.7, the SLRS with Sub1 is divided into six fan-shaped substructures of the same size, and the SLRS with Sub2 is divided into five ring-shaped substructures of similar size. The data set of damage patterns can be written with form of subsets according to Eq. (24):

(43)s_1=\sum _{n_{\mathrm{b}}=1}^6{C}_6^{n_{\mathrm{b}}}=63,\mathit{D}{\mathit{P}}_{\mathrm{C},1}=\left\{\left\{{\mathit{H}}_{\mathrm{C},1}^{(1)}\right\},\left\{{\mathit{H}}_{\mathrm{C},1}^{(2)}\right\},\dots ,\left\{{\mathit{H}}_{\mathrm{C},1}^{(63)}\right\}\right\},

(44)s_2=\sum _{n_{\mathrm{b}}=1}^5{C}_5^{n_{\mathrm{b}}}=31,\mathit{D}{\mathit{P}}_{\mathrm{C},2}=\left\{\left\{{\mathit{H}}_{\mathrm{C},2}^{(1)}\right\},\left\{{\mathit{H}}_{\mathrm{C},2}^{(2)}\right\},\dots ,\left\{{\mathit{H}}_{\mathrm{C},2}^{(31)}\right\}\right\}.

Two types of substructure division schemes are adopted for the spherical SLRS, and hence the corresponding data set of output OP_C is of two types:

(45)\mathit{O}{\mathit{P}}_{\mathrm{C},1}=\left\{{\mathit{r}}^{(1)}\left({\mathit{t}}_1^1\right),{\mathit{r}}^{(1)}\left({\mathit{t}}_1^2\right),\dots ,{\mathit{r}}^{(1)}\left({\mathit{t}}_{k_v}^v\right),\dots ,{\mathit{r}}^{(1)}\left({\mathit{t}}_{10}^{84000}\right)\right\},

(46){\mathit{r}}^{(1)}\left({\mathit{t}}_{k_v}^v\right)=\left[R{R}_1^{(1)},R{R}_2^{(1)},\dots ,R{R}_6^{(1)}\right],

(47)\mathit{O}{\mathit{P}}_{\mathrm{C},2}=\left\{{\mathit{r}}^{(2)}\left({\mathit{t}}_1^1\right),{\mathit{r}}^{(2)}\left({\mathit{t}}_1^2\right),\dots ,{\mathit{r}}^{(2)}\left({\mathit{t}}_{k_v}^v\right),\dots ,{\mathit{r}}^{(2)}\left({\mathit{t}}_{10}^{84000}\right)\right\},

(48){\mathit{r}}^{(2)}\left({\mathit{t}}_{k_v}^v\right)=\left[R{R}_1^{(2)},R{R}_2^{(2)},\dots ,R{R}_5^{(2)}\right],

where {\mathit{r}}^{(1)}\left({\mathit{t}}_{k_v}^v\right) and {\mathit{r}}^{(2)}\left({\mathit{t}}_{k_v}^v\right) are the damaged substructure region location vectors for the partitioning schemes Sub1 and Sub2, respectively.

Similarly, the corresponding damaged region location data sets can be written with form of subsets according to Eq. (27):

(49)\mathit{O}{\mathit{P}}_{\mathrm{C},1}=\left\{\left\{{\mathit{r}}_{\mathrm{C},1}^{(1)}\right\},\left\{{\mathit{r}}_{\mathrm{C},1}^{(2)}\right\},\dots ,\left\{{\mathit{r}}_{\mathrm{C},1}^{(63)}\right\}\right\},

(50)\mathit{O}{\mathit{P}}_{\mathrm{C},2}=\left\{\left\{{\mathit{r}}_{\mathrm{C},2}^{(1)}\right\},\left\{{\mathit{r}}_{\mathrm{C},2}^{(2)}\right\},\dots ,\left\{{\mathit{r}}_{\mathrm{C},2}^{(31)}\right\}\right\}.

For DP_C,1, DP_C,2, OP_C,1, and OP_C,2, CNN1 and CNN2 are trained for classification. The CNN1 can perform 63 types of damaged substructure region classifications for the fan-shaped substructure, and CNN2 can perform 31 types of classification of damaged substructure region for the ring-shaped substructure. The CNNs are trained in a Python environment using Keras [44], a high-level open-source deep-learning library built on top of Tensorflow [45]. The simulations described in this paper are performed on a PC with one Intel Core i5-9400F CPU and one NVIDIA RTX A4000 GPU card. In the training process, CNN1 is trained through DP_C,1 and OP_C,1, and CNN2 is trained through DP_C,2 and OP_C,2.

Based on Ref. [41] and the constructed damage data set, trial training is conducted for CNN1 and CNN2 to adjust hypermeters. The following hyperparameters of CNN with higher damage region classification accuracy are obtained, and the hyperparameters of Fig.2 are listed in Tab.1.

The hyperparameters of the CNN models used in this study are as follows.

1) The size of the input matrix is 224 × 224 × 3.

2) The dropout rates [46] in auxiliary classifiers 1 and 2 are set as 0.5. The dropout rate in the main network is set as 0.4.

3) On the basis of Eq. (32), The hyperparameters k, n_k, α, and β in the LRN layer are set to be 2, 5, 0.0001, and 0.75, respectively.

4) The adaptive momentum estimation (Adam) [47] is selected as the optimizer, and its learning rate and decay rate are 3 × 10⁻⁴ and 5 × 10⁻⁶, respectively.

5) Batch normalization is used during the training process, and the batch size is 32. The training epoch is set to be 300.

To avoid high variance and bias in the generated results, the K-fold cross validation is used to evaluate the CNN model more objectively. A 10-fold cross validation is used to randomly shuffle the data set and divide it into 10 mutually exclusive subsets of the same size. Each time, the current subset is used as the validation data set and all the remaining subsets are used as the training data set for training and evaluation of the model. The ratio of the training data set size to the validation data set size for each cross validation is 9:1, and the validation data set is not used for training the CNNs during each training process.

The accuracy curves and loss curves in validation data set of CNN1 and CNN2 in the first fold cross validation are shown in Fig.8. After 300 epochs of training, the CNN can successfully extract the damage characteristics of the spherical SLRS when different damage patterns occurred. When the accuracy and loss of each cross validation tend to stabilize, the average validation accuracy during stability of each cross validation is taken as the accuracy of each cross validation. The validation accuracy of each cross validation, N_K, in 10-fold cross validation and the average accuracy of the 10-fold cross validation are shown in Fig.9. According to Fig.9, the average accuracies of CNN1 (Sub1) and CNN2 (Sub2) reached 99.3% and 98.4%, respectively, for the validation data set. From the classification results for the validation data set, we can assume that the accuracy of damaged region location after one-time cross-partitioning for the validation data set is 99.3% × 98.4% = 97.7%. Then, the weights and hyperparameters of the trained CNN1 and CNN2 are used to test the accuracy of damaged region location on the testing data set.

For the trained CNN1 and CNN2, an additional 8400 testing damage patterns are constructed to input into CNN1 and CNN2, using the same process of constructing the training data set. Additionally, it should be noted that the 8400 damage patterns in the testing data set are not included in the damage pattern data set for training CNN1 and CNN2. The testing accuracies of CNN1 and CNN2 are 98.7% and 97.7%, respectively, indicating that CNN1 and CNN2 still maintain high accuracy in identifying unknown damage patterns. The one-time cross-partitioning for the testing data set is 98.7% × 97.7% = 96.4%. The testing results indicate that CNN1 and CNN2 have good generalization ability and can be used for location of damaged regions.

The six damage patterns shown in Tab.2 are selected in the testing data set as observed damage patterns, and the damage degree of selected members in Tab.2 are all set to 20%. The plane projection images corresponding to Tab.2 are shown in Fig.10. The cross-partitioning damaged region location method is used to determine the corresponding damaged region locations of the observed damage modes using trained CNN1 and CNN2. The plane projection image on the right side of Fig.10 is input into the trained CNN1 and CNN2. The CNN1 and CNN2 cross-partitioned the damaged region locations by using the cross-partitioning damaged region location method shown in Fig.1, and the damaged region location results are shown in Tab.2 and Fig.11. In Tab.2, result 1, result 2, and result 3 are the output identified damaged region locations of CNN1 and CNN2; F₁, R₃, and F₁R₃ represent the damaged region location is the first fan-shaped substructure, the third ring-shaped substructure, and the intersection region of the first fan-shaped substructure and the third ring-shaped substructure respectively. The cross-partitioning damaged region location results are shown in blue, yellow, and red colors in Fig.11, respectively.

The total number of members n_{\mathrm{e}}^{(1)}, n_{\mathrm{e}}^{(2)}, and n_{\mathrm{e}}^{(3)} in the damaged region location, obtained by CNN1, CNN2, and the cross-partitioning location method, are listed in Tab.3. With the cross-partitioning location method, the number of members in the located damaged region decreases to 18.3%−53.3% of the number of members obtained by the two CNNs. In particular, the accuracy loss of the cross-partitioning is acceptable. The identified damaged region location can be significantly reduced to a smaller region when the cross-partitioning damaged region location method is used. Hence, the cross-partitioning damaged region location method proposed in this paper can locate the damaged region of an SLRS more precisely.

6.3 Damaged member identification

Before identifying damaged members, it is necessary to determine the hyperparameters of FCNN with architecture shown in Fig.3. The following steps are adopted to determine the hyperparameters of the FCNN.

P1: The cross-partitioning region is constructed by randomly selecting fan-shaped and ring-shaped substructure in Subsection 6.2. The number of members in the cross-partitioning region is n_{\mathrm{e}}^{(3)}. The determination process of n_{\mathrm{e}}^{(3)} can refer to Fig.11 and Tab.3.

P2: For a determined n_{\mathrm{e}}^{(3)}, a total of 5000 damage patterns are constructed to simulate the damage of the spherical SLRS. Furthermore, 1−10 members are randomly selected from the n_{\mathrm{e}}^{(3)} members. Then, randomly select a reduction percentage for the elastic modulus of the selected members from the options of 5%, 10%, 15%, 20%, 25%, or 30%. The first and second vibration modes in the z direction and the first and second circular frequencies are selected to construct the damage pattern set. Equations (36)–(40), the set of damage patterns DP_NN and the corresponding set of outputs OP_NN can be written as

(51)\begin{array}{rl}\mathit{D}{\mathit{P}}_{\mathrm{N}\mathrm{N}}=& \{\mathit{I}\left({\mathit{t}}_1^1,0.05\right),\mathit{I}\left({\mathit{t}}_1^2,0.1\right),\dots ,\\ & \mathit{I}\left({\mathit{t}}_{k_w}^w,n^w\right)\dots ,\mathit{I}\left({\mathit{t}}_{10}^{5000},0.3\right)\},\end{array}

(52)\begin{array}{rl}\mathit{I}\left({\mathit{t}}_{k_w}^w,n^w\right)=& [\mathrm{\Delta }{\mathit{\Phi }}_{N1}\left({\mathit{t}}_{k_w}^w,n^w\right);\mathrm{\Delta }{\mathit{\Phi }}_{N2}\left({\mathit{t}}_{k_w}^w,n^w\right);\\ & \mathrm{\Delta }{\omega }_1{\left({\mathit{t}}_{k_w}^w,n^w\right)}^2;\mathrm{\Delta }{\omega }_2{\left({\mathit{t}}_{k_w}^w,n^w\right)}^2;\mathrm{\Delta }{\omega }_3{\left({\mathit{t}}_{k_w}^w,n^w\right)}^2]\end{array},

(53)\begin{array}{rl}\mathit{O}{\mathit{P}}_{\mathrm{N}\mathrm{N}}=& \{\mathit{p}\left({\mathit{t}}_1^1,0.05\right),\mathit{p}\left({\mathit{t}}_1^2,0.1\right),\dots ,\\ & \mathit{p}\left({\mathit{t}}_{k_w}^w,n^w\right),\dots ,\mathit{p}\left({\mathit{t}}_{10}^{5000},0.3\right)\},\end{array}

(54)\begin{array}{rl}\mathit{p}\left({\mathit{t}}_{k_w}^w,n_{r_w}^w\right)=& [0,\dots ,0,\dots ,D{D}_1,\dots ,D{D}_2,\dots ,\\ & D{D}_{n_e^{(3)}},\dots ,0,\dots ,0].\end{array}

P3: The number of FC layers is set to 2, 3, and 4, and 100, 200, 300 neurons per layer are selected to construct different FCNNs, to generate 9 different types of FCNN.

P4: The training and testing sets are randomly divided in an overall 8:2 ratio, to train the FCNNs. When the loss of FCNN tends to stabilize, the average value of the testing loss during the stable period is taken as the loss of FCNN.

P5: The steps of P1−P4 are repeated 20 times to obtain the loss value of FCNN under different damage regions and different network hyperparameters.

The other hyperparameters of the FCNN are set as follows.

1) The dropout rate in the network is set as 0.4.

2) The Adam is selected as the optimizer with a learning rate of 3 × 10⁻³ and a decay rate of 5 × 10⁻⁵.

3) The training epoch is set as 300.

For each selected cross-partitioning region, a surface is used to fit the relationship between testing loss, number of neurons, and number of layers. The surfaces determined by loss, number of neurons, and number of layers under 20 different cross-partitioning regions are drawn simultaneously in Fig.12, showing similar regular variation. In Fig.12, when using more than 3 FC layers and more than 200 neurons per layer, FCNN achieved lower loss for identifying damaged members in different cross-partitioning regions. Based on the above calculation results, the FCNN with 4 FC layers and 300 neurons per layer is finally determined for damaged member identification. The determined FCNN has lower loss values in different damage regions, which indicates that the selected FCNN has better performance in identifying damaged members. All the parameters of each layer in the FCNN are shown in Tab.4.

The damage patterns shown in Tab.2 are used to further identify the damage degree of members in determined damaged region locations. For the cross-partitioning damaged region location determined in Fig.11, the FCNN with same hyperparameters are used to further identify the damaged members and their damage degree. The total number of members in patterns 1−6 is n_{\mathrm{e}}^{(3)} (Tab.3). The input data set DP_NN and output data set OP_NN are constructed using the operations described in P2. The input and output data set in patterns 1−6 are denoted by, {\mathit{D}\mathit{P}}_{\mathrm{N}\mathrm{N}}^{(1)},…,{\mathit{D}\mathit{P}}_{\mathrm{N}\mathrm{N}}^{(6)} and {\mathit{O}\mathit{P}}_{\mathrm{N}\mathrm{N}}^{(1)},…,{\mathit{O}\mathit{P}}_{\mathrm{N}\mathrm{N}}^{(6)} for distinguishing between the inputs and outputs. Correspondingly, the trained FCNNs for identifying damaged members in patterns 1−6 are labeled FC1, FC2, …, FC6, respectively.

The 10-fold cross validation is also used to randomly shuffle the above data set and divide it into 10 mutually exclusive subsets of the same size. The accuracy curves and loss curves of FC1−FC6 in the 1st fold cross validation are shown in Fig.13. The average validation accuracy during the stability stage of each cross validation is also taken as the accuracy of each cross validation, and the validation accuracy of each cross validation in 10-fold cross validations are shown in Fig.14. According to Fig.14, the average validation accuracy of FC1−FC6 is in the range of 95.7%−96.8%. The weights and hyperparameters of the trained FC1−FC6 are then used to test the accuracy of damaged member identification on the testing data set.

For the trained FC1−FC6, a further 500 testing damage patterns are constructed using the same process of constructing a training data set for each FCNN. Each testing data set is not included in the training data set for FC1−FC6. The testing accuracies of FC1−FC6 are 94.4%, 96.0%, 95.8%, 95.0%, 95.4%, and 96.0%, respectively. The testing results indicate that FC1−FC6 have good generalization ability and can be used for damaged member identification in determined damaged region locations. For the observed damage patterns 1−6 in Tab.2, the vectors input to FCNNs can be constructed according to Eq. (52). Then the trained FC1−FC6 can output the damage degree of each member in the damaged region locations. The six damage patterns are also not included in the training data sets of FC1−FC6. The identification results are shown in Tab.5 and Fig.15. In Tab.5, results 1 represents the damaged member location results, and results 2 represents the damage degree identification results. The damaged members are marked with blue lines in Fig.15. For damage patterns 1−6, the location and damage degree of the damaged member are precisely identified. Herein, we have completed precise identification of damaged members of SLRS from the prior identification of damaged region locations.

7 Conclusions

A cross-partitioning damaged region locating method based on a CNN is proposed. The method can identify complex damage patterns and significantly narrow down the specific location within a region of possible damage. A FCNN is constructed to identify the damaged members and their degree of damage in the narrowed-down damaged region location. The effectiveness of the damaged region location method and damaged member identification method is verified using a spherical SLRS model. The conclusions of this study are as follows.

1) Damage localization and damage degree indices are proposed for an SLRS, and they can be used to determine the damage location and damage degree, respectively. These two indices can be used to construct damage patterns for training neural networks.

2) A CNN-based damaged region location method for SLRSs is proposed. Furthermore, a cross-partitioning damaged region locating method involving the use of parallel CNNs is proposed; the method can significantly narrow down the possible damaged region to a more precise location within the structure.

3) Owing to the significant reduction in the range of possibilities of damage patterns after narrowing down the location of the damaged region, the damage localization and damage degree identification of members can be performed by the FCNN.

4) The damaged region localization method and damaged member identification method are verified using a spherical SLRS. The cross-partitioning damaged region location method can successfully locate the damaged region under complex conditions. With the cross-partitioning location method, the number of members in the located damaged region decreased to 18.3%−53.3% of the value for the single CNN location results. After localizing the damaged region, the corresponding FCNN can further precisely identify the damage location and damage degree of members.

When using neural networks for damage identification, it is necessary to construct a large number of damage patterns; this relies on an accurate numerical model. Further research is still needed on how to modify the numerical model to be as close as possible to the behavior of the actual structure. In addition, the accuracy of SLRS modal identification directly affects the accuracy of damage identification during the damage identification process. The study of modal parameter identification methods applicable to SLRS is also a topic that needs to be studied in the future.

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