An efficient two-stage approach for structural damage detection using meta-heuristic algorithms and group method of data handling surrogate model

Abstract

In this study, the performance of an efficient two-stage methodology which is applied in a damage detection system using a surrogate model of the structure has been investigated. In the first stage, in order to locate the damage accurately, the performance of the modal strain energy based index for using different numbers of natural mode shapes has been evaluated using the confusion matrix. In the second stage, to estimate the damage extent, the sensitivity of most used modal properties due to damage, such as natural frequency and flexibility matrix is compared with the mean normalized modal strain energy (MNMSE) of suspected damaged elements. Moreover, a modal property change vector is evaluated using the group method of data handling (GMDH) network as a surrogate model during damage extent estimation by optimization algorithm; in this part of methodology, the performance of the three popular optimization algorithms including particle swarm optimization (PSO), bat algorithm (BA), and colliding bodies optimization (CBO) is examined and in this regard, root mean square deviation (RMSD) based on the modal property change vector has been proposed as an objective function. Furthermore, the effect of noise in the measurement of structural responses by the sensors has also been studied. Finally, in order to achieve the most generalized neural network as a surrogate model, GMDH performance is compared with a properly trained cascade feed-forward neural network (CFNN) with log-sigmoid hidden layer transfer function. The results indicate that the accuracy of damage extent estimation is acceptable in the case of integration of PSO and MNMSE. Moreover, the GMDH model is also more efficient and mimics the behavior of the structure slightly better than CFNN model.

Keywords

two-stage method / modal strain energy / surrogate model / GMDH / optimization damage detection

Introduction

Health monitoring of structures, involves monitoring and analyzing the static and dynamic behavior of structures, has become a topic of great importance in recent years. Aging and damage in different types of structures, such as dams, bridges, and buildings are becoming an important issue nowadays [¹–³]. Long-term monitoring of these structures, particularly large-scale ones, requires dealing with high dimensional data sets, captured from different types of sensors [⁴]. These raw data sets must be effectively processed and interpreted. There are two distinct approaches for an interpretation of SHM data, so-called model-based and model-free [⁵]. In the model-based approaches, development of the large-scale structural model is time-consuming; on the other hand, using this approach is useful in the cases that prediction of structural behavior is of the main concern. Model-free approaches are applicable for damage detection and localization but not for prediction [⁶].

In the model-based approach, the most prevalent scheme for global assessment of the structural condition is the idea of measuring the changes in structural dynamic characteristic data [⁷,⁸]. Modal properties are a set of structural dynamic characteristics, that are calculated based on the results of structural modal analysis during free vibration. Modal properties, namely, natural frequency, mode shape, modal flexibility, or modal strain energy (MSE) are changing due to damage disrupting the structural performance [⁹]. The earliest modal indices are obtained based on natural frequencies. The natural frequencies are easily measurable with high accuracy, but their sensitivity to damage is low [¹⁰,¹¹]. It was concluded that mode shapes are more sensitive to damage than natural frequencies and are able to directly provide damage location information [¹²]. The MSE is quite sensitive to damages in structures and can provide detailed information with respect to the extent and location of the structural damage [¹³]. In this article, the sensitivity of these modal properties due to damage has been compared and investigated.

In the model-based approach, in order to reduce the search domain, two-stage methods are used [¹⁴–¹⁶]. At first, damage localization is done by damage indices. In the second stage, in order to estimate the damage extent in structural systems, methods based on the structural model updating are applied. As one of the model updating methods, the damage extent estimation is considered as a solution of an optimization problem in which the damage extent of each element is considered as a design variable. The objective function is defined to minimize the differences between the measured properties of actual structure captured from sensors and properties of finite element (FE) model of the structure. Over the past two decades, optimization algorithms have been extensively applied in various domains of civil engineering including design optimization [¹⁷–²⁴], prediction of material properties [²⁵,²⁶], damage identification/detection [²⁷–³¹], system identification [³²,³³], etc. Accordingly, in this study, the performance of three popular optimization algorithms which have been used separately in this stage of structural damage detection [³⁴–³⁶], is investigated simultaneously; a comparison of reliability and efficiency of optimization algorithms is a notable subject in damage quantification. These algorithms include particle swarm optimization (PSO), bat algorithm (BA), and colliding bodies optimization (CBO). Moreover, the purpose of the application of three different algorithms is to examine the performance of the proposed damage parameter based on MSE for various optimization algorithms.

Nevertheless, the process of updating the FE model for structures with a large number of degrees of freedom is computationally expensive [³⁵]. Xia et al. [³⁷] used a FE model of continuous three-span bridge which has 907 elements and 949 nodes to be updated by an optimization algorithm. The optimization algorithm required 155 iterations and took about 420 h to estimate the damage extents. Thus, using the surrogate model can be considered as an appropriate way to reduce the cost of computing and the amount of computer memory usage in model updating methods.

Using suitable artificial neural network (ANN) as a surrogate of the FE model of the structure has been computationally efficient for the model updating [³⁴,³⁸–⁴¹]. In this study, a feed-forward neural network has been employed based on heuristic self-organization learning approach. Feed-forward neural networks have been widely used to solve the inverse and optimization problems [⁴²–⁴⁴]. This approach is named group method of data handling (GMDH). This type of ANN can automatically select the optimum neural network architecture by using heuristic self-organization method [⁴⁵]. According to Anastasakis and Mort [⁴⁶], perceptron type structure of GMDH neural network needs a fewer number of samples and consequently training and test is faster in GMDH compared to other types of neural networks.

As the modern world is industrialized and developed, there is a larger demand for the use of structures with large spans [⁴⁷]. A Space trusses are structures with a large number of degrees of freedom. The most common type of the space trusses are double-layer grids in which two top and bottom chords are interconnected by inclined or vertical web members. Space trusses are subjected to multiple forms of damage, such as impact damage, structure assembling errors, faulty materials or element connections, and others [⁴⁸]. Because of the interest in these types of structures in recent years and also noting that the great number of their elements causing computationally expensive model-based damage detection methods [¹⁶], features such as an automatic health monitoring or a damage detection system based on optimal computational cost are highly desirable. In this study, to implement a computationally efficient damage detection system for these large-scale structures, an efficient surrogate model is used.

Hence, this article deals with using modal properties to detect damage and proposes an efficient two-stage methodology to implement a damage detection system using the surrogate model of the structure. In the first stage, in order to locate damage, performance of modal strain energy based index (MSEBI) for using different numbers of first natural mode shapes has been evaluated using the confusion matrix. Confusion matrix is a table that illustrates the performance of a classifier [⁴⁹]. In the second stage, in order to estimate the damage extent, the most used modal properties, such as natural frequency, flexibility matrix (as a combination of natural frequencies and mode shapes) and mean normalized modal strain energy (MNMSE) of suspected damaged elements as a proposed damage index are used in a modal property change vector as the objective function for optimization algorithm. In addition, the performance of three optimization algorithms for damage extent estimation is also investigated. After the selection of the suitable modal property and optimization algorithm, modal property change vector of the structure is evaluated using the GMDH network as a surrogate model. The measurement noise effect has also been studied.

The structure of paper is described in the following. The introduction given in this section is followed by the presentation of the MSEBI and its performance evaluation based on the confusion matrix concept given in Section 2. Section 3 then describes the procedure of structural damage extent estimation and proposed damage detection procedure. Two examples are studied in Sections 4 and 5 includes a summary of the study and presents the conclusions.

Structural damage localization

The concept of MSE has been applied vastly for damage identification [⁴⁸,⁵⁰]. Results show that the sensitivity of MSE changes in the structural elements before and after the damage is very significant. Accordingly, one of the most useful damage indices is the MSEBI which is introduced by Seyedpoor [¹⁴]. In this article, the confusion matrix is used to determine the optimal number of mode shapes, needed to achieve the best MSEBI values.

The MSE of each structural element is:

(1)ms{e}_i^e={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\phi }_i^e{\mathit{K}}^e{\phi }_i^e,\hspace{0.17em}\hspace{0.17em}i=\mathrm{1},...,ndf,\hspace{0.17em}\hspace{0.17em}e=\mathrm{1},...,nte,

where {\mathit{K}}^e is the stiffness matrix of each element and {\phi }_i^e is the vector of mode shapes corresponding to each element’s nodal displacements. The total strain energy of the structure in i\mathrm{th} mode is calculated by the sum of the MSE of each structural element nte:

(2)ms{e}_i={\displaystyle \sum _{e=\mathrm{1}}^{nte}ms{e}_i^e},\hspace{0.17em}i=\mathrm{1},...,ndf.

Then the MSE of each element is normalized to the total MSE of the structure:

(3)nms{e}_i^e={\displaystyle \frac{ms{e}_i^e}{ms{e}_i}},

where nms{e}_i^e is the normalized MSE of each element for each mode. The obtained values of nms{e}_i^e which are equal to the number of first used modes are then averaged:

(4)mnms{e}^e={\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^{nm}nms{e}_i^e}}{nm}},\hspace{0.17em}e=\mathrm{1},...,nte.

Since the damage in the structural element is modeled by decreasing the parameters of element stiffness, then the damage causes an increase in the MSE and the MNMSE. Therefore, the MNMSE parameter is defined for each structural element for two undamaged and damaged status and an index is determined based on MNMSE as follows for detecting the damage in each element:

MSEB{I}^e=\max \left[\mathrm{0},\hspace{0.17em}{\displaystyle \frac{{\left(mnms{e}^e\right)}^d-{\left(mnms{e}^e\right)}^h}{{\left(mnms{e}^e\right)}^h}}\right],

(5)e=\mathrm{1},...,nte.

It should be noted that, since there is no information about the damaged structure, thus the stiffness matrix of the undamaged structure is used to calculate the MSE of each element in both undamaged and damaged status. Based on the Eq. (5), the value of the MSEBI for an undamaged element is zero and for the damaged one it will be greater than zero [¹⁴]. In this study, in order to evaluate the MSEBI performance, the confusion matrix has been used.

The confusion matrix is a table which contains four metrics for the performance of a classifier. These metrics are the numbers of false positives (FP), false negatives (FN), true positives (TP), and true negatives (TN). This table provides more detailed metrics than the mere proportion of correct classifications which is called accuracy [⁴⁹]. In this article, the classifier is MSEBI whose values determine the state of damage for each structural element. The confusion matrix of MSEBI is formed for use of the first mode shape up to first 10 mode shapes of the structure. The terminology of outputs derived from a confusion matrix can be described as follows.

1)‚TP are defined in such a way that the elements are detected as damaged ones; and they are actually damaged structural elements.

2)‚TN are defined in such a way that the elements are detected as healthy or undamaged ones; and they are actually healthy or undamaged structural elements.

3)‚FP are defined in such a way that the elements are detected as damaged ones; but they are not actually damaged elements.

4)‚FN are defined in such a way that the elements are detected as healthy ones; but they are not actually healthy elements.

5)‚TP rate is the result of dividing the number of truly predicted damaged elements into the number of actual damaged elements. (TP/No. of actual damaged elements).

6)‚FP rate is the result of dividing the number of wrongly predicted damaged elements into the number of actual healthy elements. (FP/No. of actual healthy elements).

The graph presenting TP rate and FP rate is called a receiver operating characteristic (ROC) curve. In this study, ROC curve is plotted for illustrating the diagnostic ability of the MSEBI calculated using up to first ten mode shapes. The procedure of MSEBI calculation and its performance evaluation as described above is illustrated in Fig. 1.

Structural damage extent estimation

In this section, the damage extents of suspected damaged elements which are identified in Section 2 are estimated by using optimization algorithms. Accordingly, for solving this problem, the objective function is defined to minimize the difference between the measured and analytical modal properties such as natural frequencies, diagonal elements of the modal flexibility matrix and the MNMSE of suspected elements (Eq. (4)). At first, the efficiency of modal properties which are listed above is studied comparatively by integration of three examined optimization algorithms and FE model. Detailed description of these optimization algorithms can be found in the Appendix. Then, the performance of the GMDH surrogate model integrated with selective optimization algorithm is evaluated. Details are presented in this section. The flowchart of damage extent estimation procedure is illustrated in Fig. 2.

Modal flexibility matrix

One of the modal properties including the influence of both the natural mode shapes and frequencies is the flexibility matrix, which can be written as [⁵¹]:

(6)\mathit{F=\Phi }{\mathit{\Lambda }}^{-\mathrm{1}}{\mathit{\Phi }}^{\mathrm{T}}={\displaystyle \sum _{j=\mathrm{1}}^{nd}{\displaystyle \frac{\mathrm{1}}{{\omega }_j^{\mathrm{2}}}}{\phi }_j}{\phi }_j^{\mathrm{T}},

where \mathit{F} is defined the modal flexibility matrix, \mathit{\Phi } contains the mode shape vectors which are normalized to the mass, and {\mathit{\Lambda }}^{-\mathrm{1}} is a matrix that includes the inverse of the squared circular frequencies of the structure. {\omega }_j and {\phi }_j are j\mathrm{th} circular frequency and mode shape of the structure, respectively. nd is the number of degrees of freedom. As the frequency increases, the values of the flexibility matrix elements decreases, therefore, using only a few first vibrating modes of the structure, a proper estimation of the flexibility matrix can be acquired. Moreover, employing the mode shape vectors which are normalized to the mass in Eq. (6) leads to a diagonal flexibility matrix and hence the diagonal elements’ sensitivity to damage is more considerable. Hence, in this regard, the first diagonal elements of flexibility matrix have been used as a parameter vector in damage extent estimation process.

Definition of objective function

To estimate the extent of induced damage in suspected elements, two parameter vectors are defined. A parameter vector consists of modal property changes (\mathrm{\Delta }\mathit{P}) due to structural damage. In this study, natural frequencies, diagonal elements of modal flexibility matrix and the MNMSE of suspected elements (Eq. (4)), have been used as the modal property in same formula, i.e.,

(7)\mathrm{\Delta }\mathit{P=}{\displaystyle \frac{{\mathit{P}}_{\mathrm{h}}-{\mathit{P}}_{\mathrm{d}}}{{\mathit{P}}_{\mathrm{h}}}}.

When using natural frequency

(8){\mathit{P}}_{\mathrm{h}}^{\mathrm{T}}=\left\{f_{\mathrm{h}\mathrm{1}},\hspace{0.17em}{f}_{\mathrm{h}\mathrm{2}},...,\hspace{0.17em}{f}_{\mathrm{h}n}\right\},\hspace{0.17em}{\mathit{P}}_{\mathrm{d}}^{\mathrm{T}}=\left\{f_{\mathrm{d}\mathrm{1}},\hspace{0.17em}{f}_{\mathrm{d}\mathrm{2}},...,\hspace{0.17em}{f}_{\mathrm{d}n}\right\},

where n is the number of first modes used, f_{\mathrm{h}} is natural frequency of healthy structure in Hertz, f_{\mathrm{d}} is natural frequency of damaged structure in Hertz which is assumed to have been extracted from the sensor.

When using diagonal element of flexibility matrix:

{\mathit{P}}_{\mathrm{h}}^{\mathrm{T}}=\left\{F_{\mathrm{h}\mathrm{11}},\hspace{0.17em}{F}_{\mathrm{h}2\mathrm{2}},...,\hspace{0.17em}{F}_{\mathrm{h}nn}\right\},

(9){\mathit{P}}_{\mathrm{d}}^{\mathrm{T}}=\left\{F_{\mathrm{d}\mathrm{11}},\hspace{0.17em}{F}_{\mathrm{d}\mathrm{22}},...,\hspace{0.17em}{F}_{\mathrm{d}nn}\right\},

where n is the number of used first modes, F_{\mathrm{h}ii} is the i\mathrm{th} diagonal element of modal flexibility matrix for healthy structure, F_{\mathrm{d}ii} is the i\mathrm{th} diagonal element of modal flexibility matrix for damaged structure.

When using MNMSE:

(10)\begin{array}{l}{\mathit{P}}_{\mathrm{h}}^{\mathrm{T}}=\left\{{\left(mnms{e}^{\mathrm{1}}\right)}^{\mathrm{h}},\hspace{0.17em}{\left(mnms{e}^{\mathrm{2}}\right)}^{\mathrm{h}},...,\hspace{0.17em}{\left(mnms{e}^e\right)}^{\mathrm{h}}\right\},\hfill \\ {\mathit{P}}_{\mathrm{d}}^{\mathrm{T}}=\left\{{\left(mnms{e}^{\mathrm{1}}\right)}^{\mathrm{d}},\hspace{0.17em}{\left(mnms{e}^{\mathrm{2}}\right)}^{\mathrm{d}},...,\hspace{0.17em}{\left(mnms{e}^e\right)}^{\mathrm{d}}\right\},\hfill \end{array}

where e is the number of suspected damaged elements which has been identified in the first stage. mnmse is MNMSE calculated based on Eq. (4).

Another parameter vector is determined as:

(11)\delta \mathit{P}\left(\mathit{X}\right)={\displaystyle \frac{{\mathit{P}}_{\mathrm{h}}-\mathit{P}\left(\mathit{X}\right)}{{\mathit{P}}_{\mathrm{h}}}},

where \mathit{P}\left(\mathit{X}\right) is a selected parameter (natural frequency, diagonal element of flexibility matrix or suspected element’ MNMSE) that can be predicted from an updated analytical model. {\mathit{X}}^{\mathrm{T}}=\left\{x_{\mathrm{1}},...,x_i,...,x_n\right\} is a vector of damage extents \left(x_i,\hspace{0.17em}i=\mathrm{1},...,n\right) of all n suspected damaged elements.

The objective function minimized using optimization algorithm to estimate the damage extents of suspected damaged elements is termed the root mean square deviation (RMSD) which is defined based on the difference between \mathrm{\Delta }\mathit{P} and \delta \mathit{P}\left(\mathit{X}\right) vectors:

(12)RMSD=\sqrt{{\displaystyle \sum _{i=\mathrm{1}}^{dim}{\left(\left|\mathrm{\Delta }{\mathit{P}}_i\right|-\left|\delta \mathit{P}{\left(\mathit{X}\right)}_i\right|\right)}^{\mathrm{2}}}},

where dim is the length of \mathrm{\Delta }\mathit{P} and \delta \mathit{P}\left(\mathit{X}\right) vectors.

GMDH model

Recently, various ANNs have been employed in implementation of different components of structural health monitoring to perform pattern classification, function approximation, and regression [⁵²,⁵³]. Among them, multilayer feedforward networks are the most popular. The optimum weight values of these networks which have a fixed topology are determined by using standard backpropagation algorithm and gradient descent method [⁵⁴]. GMDH is a neural network that automatically selects its optimum architecture. This feature results in a more accurate or unbiased model.

In GMDH, all possible combinations of a pair \left(x_i,\hspace{0.17em}{x}_j\right) of input variables (all possible number of neurons) are considered. Then by using one of the available minimizing methods such as the principle of least-squares error or singular value decomposition (with training data) and a polynomial, the combinations (neurons) that have the least error (for test data) are selected. Subsequently, the next layer will be generated, and this process continues up to the last layer [⁵⁵].

For a given input vector \mathit{X=}\left(x_{\mathrm{1}},\hspace{0.17em}{x}_{\mathrm{2}},x_{\mathrm{3}},...,\hspace{0.17em}{x}_N\right), the output ({\widehat{y}}_i) should be predicted as close as possible to (y_i). With P samples:

(13)y_i=f\left(x_{i\mathrm{1}},\hspace{0.17em}{x}_{i\mathrm{2}},\hspace{0.17em}{x}_{i\mathrm{3}},...,\hspace{0.17em}{x}_{in}\right),\hspace{0.17em}i=\mathrm{1},\mathrm{2},\mathrm{3},...,P.

This training data set (85% of all data) is applied for training the GMDH network to predict {\widehat{y}}_i with a given input vector \mathit{X=}\left(x_{\mathrm{1}},\hspace{0.17em}{x}_{\mathrm{2}},x_{\mathrm{3}},...,\hspace{0.17em}{x}_N\right), therefore:

(14){\widehat{y}}_i=\widehat{f}\left(x_{i\mathrm{1}},\hspace{0.17em}{x}_{i\mathrm{2}},\hspace{0.17em}{x}_{i\mathrm{3}},...,\hspace{0.17em}{x}_{in}\right),\hspace{0.17em}i=\mathrm{1},\mathrm{2},\mathrm{3},...,P.

The parameters of GMDH neural network are calculated based on minimizing the squared differences between the actual and predicted outputs.

(15)E={\displaystyle \sum _{i=\mathrm{1}}^P{\left(\widehat{f}\left(x_{i\mathrm{1}},\hspace{0.17em}{x}_{i\mathrm{2}},\hspace{0.17em}{x}_{i\mathrm{3}},...,\hspace{0.17em}{x}_{in}\right)-y_i\right)}^{\mathrm{2}}}\to \min .

By using discrete analog of Volterra functional series named Kolmogorov-Gabor polynomial [⁴⁶]:

\widehat{y}={\beta }_{\mathrm{0}}+{\displaystyle \sum _{i=\mathrm{1}}^P{\beta }_i{x}_i}+{\displaystyle \sum _{i=\mathrm{1}}^P{\displaystyle \sum _{j=\mathrm{1}}^P{\beta }_{ij}{x}_i{x}_j}}

(16)+{\displaystyle \sum _{i=\mathrm{1}}^P{\displaystyle \sum _{j=\mathrm{1}}^P{\displaystyle \sum _{k=\mathrm{1}}^P{x}_j{\beta }_{ijk}{x}_i{x}_j{x}_k}}}+\cdots .

Ivakhnenko represented the Kolmogorov-Gabor polynomial (Eq. (16)) by using low order polynomials for every pair of the input variables. Using a quadratic polynomial:

(17)\widehat{y}=G\left(x_i,x_j\right)={\beta }_{\mathrm{0}}+{\beta }_{\mathrm{1}}{x}_i+{\beta }_{\mathrm{2}}{x}_j+{\beta }_{\mathrm{3}}{x}_i^{\mathrm{2}}+{\beta }_{\mathrm{4}}{x}_j^{\mathrm{2}}+{\beta }_{\mathrm{5}}{x}_i{x}_j.

The principle of least-squares method which is used to determine the coefficient {\beta }_i in Eq. (17) is:

(18)E={\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^O{\left({\widehat{y}}_i-y_i\right)}^{\mathrm{2}}}}{P}}\hspace{0.17em}\to \min .

In the GMDH network, to form the regression polynomials (Eq. (16)), all possible scenarios are used to select two independent variables among n input variables. Therefore, \left(\begin{array}{l}n\hfill \\ \mathrm{2}\hfill \end{array}\right)={\displaystyle \frac{n\left(n-\mathrm{1}\right)}{\mathrm{2}}} neurons is built in the first hidden layer from the samples \{\left(y_i,\hspace{0.17em}{x}_{ir},\hspace{0.17em}{x}_{iq}\right)\mathrm{;}\hspace{0.17em}(i=\mathrm{1},\mathrm{2},\mathrm{3},...,P)\} for different \left(r,\hspace{0.17em}q\in \left\{\mathrm{1},\mathrm{2},\mathrm{3},...,n\right\}\right).

(19)\mathit{Y}=\mathit{A}\beta ,

where \mathit{Y=}\left\{y_{\mathrm{1}},\hspace{0.17em}{y}_{\mathrm{2}},\hspace{0.17em}{y}_{\mathrm{3}},...,\hspace{0.17em}{y}_P\right\}is the actual output vector, \mathit{A=}\left\{a_{\mathrm{0}},\hspace{0.17em}{a}_{\mathrm{1}},\hspace{0.17em}{a}_{\mathrm{2}},\hspace{0.17em}{a}_{\mathrm{3}},\hspace{0.17em}{a}_{\mathrm{4}},\hspace{0.17em}{a}_{\mathrm{5}}\right\} is the vector of quadratic polynomial coefficients calculated as:

(20)\left[\begin{array}{llllll}\mathrm{1}\hfill & x_{\mathrm{1}r}\hfill & x_{\mathrm{1}q}\hfill & x_{\mathrm{1}r}{x}_{\mathrm{1}q}\hfill & x_{\mathrm{1}r}^{\mathrm{2}}\hfill & x_{\mathrm{1}q}^{\mathrm{2}}\hfill \\ \mathrm{1}\hfill & x_{\mathrm{2}r}\hfill & x_{\mathrm{2}q}\hfill & x_{\mathrm{2}r}{x}_{\mathrm{2}q}\hfill & x_{\mathrm{2}r}^{\mathrm{2}}\hfill & x_{\mathrm{2}q}^{\mathrm{2}}\hfill \\ \mathrm{1}\hfill & x_{\mathrm{3}r}\hfill & x_{\mathrm{3}q}\hfill & x_{\mathrm{3}r}{x}_{\mathrm{3}q}\hfill & x_{\mathrm{3}r}^{\mathrm{2}}\hfill & x_{\mathrm{1}q}^{\mathrm{2}}\hfill \\ \cdots \hfill & \hfill & \hfill & \cdots \hfill & \hfill & \cdots \hfill \\ \mathrm{1}\hfill & x_{Pr}\hfill & x_{Pq}\hfill & x_{Pr}{x}_{Pq}\hfill & x_{Pr}^{\mathrm{2}}\hfill & x_{Pq}^{\mathrm{2}}\hfill \end{array}\right].

Finally the coefficient \beta is determined as follows:

(21)\beta ={\left({\mathit{A}}^{\mathrm{T}}\mathit{A}\right)}^{-\mathrm{1}}{\mathit{A}}^{\mathrm{T}}\mathit{Y}.

For each neuron in all hidden layers the above procedure is done.

For minimizing the difference of the actual output (y) and predicted output (\widehat{y}) for each pair \left(x_i,\hspace{0.17em}{x}_j\right) of independent variables (n), (Eq. (18)) in each layer, a parameter called alpha is defined in the range of zero and one, whose task is similar to the task of selection pressure parameter in the genetic algorithm.

For this purpose, at first based on Eq. (18) the error of the actual output (y) and predicted output (\widehat{y}) for each pair \left(x_i,\hspace{0.17em}{x}_j\right) of input variables are calculated:

(22)\mathit{E}=\left[E_{\mathrm{1}},E_{\mathrm{2}},...,E_f\right],\hspace{0.17em}f=\left(\begin{array}{l}n\hfill \\ \mathrm{2}\hfill \end{array}\right)={\displaystyle \frac{n\left(n-\mathrm{1}\right)}{\mathrm{2}}},

where f is all possible scenarios used to select two independent variables among n input variables for each layer. After that, the members of vector \mathit{E} are sorted in ascending order and finalerror is calculated:

(23)

Finally, a pair \left(x_i,\hspace{0.17em}{x}_j\right) of input variables is selected for the current layer that has the least error (). Subsequently, to compose second layer, number of independent variables (n) for second layer is equal to the number of first layer neurons \left(\begin{array}{l}n\hfill \\ \mathrm{2}\hfill \end{array}\right)={\displaystyle \frac{n\left(n-\mathrm{1}\right)}{\mathrm{2}}}.

In our employment of GMDH network as a surrogate model, in order to optimize the performance of the network, the optimal value of alpha has been determined. For each value of alpha between zero and one at the pace of 0.05, the network is trained and tested via a test data set. Finally, the alpha value is selected for which the network has the minimum standard deviation of test data set error:

(24)\begin{array}{l}TestTargets=\left[\mathrm{\Delta }{p}_{\mathrm{1}},\hspace{0.17em}\mathrm{\Delta }{p}_2,...,\hspace{0.17em}\mathrm{\Delta }{p}_n\right],\hfill \\ \hfill \\ \mathrm{\Delta }{p}_i={\displaystyle \frac{p_{\mathrm{h}i}-p_{\mathrm{d}i}}{p_{\mathrm{h}i}}},\hspace{0.17em}i=\mathrm{1},\mathrm{2},...,n,\hfill \\ \hfill \\ Error=TestTargets-TestOutputs,\hfill \end{array}

(25)ErrorStd=\sqrt{{\displaystyle \frac{\mathrm{1}}{n-\mathrm{1}}}{\displaystyle \sum _{i=\mathrm{1}}^n{\left|Erro{r}_i-\mu \right|}^{\mathrm{2}}}},\hspace{0.17em}\mu ={\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \sum _{i=\mathrm{1}}^{n}Erro{r}_i},

where P_{\mathrm{h}} and P_{\mathrm{d}} denote the natural frequency or diagonal element of flexibility matrix or suspected elements’ MNMSE vectors of a healthy and damaged structure, respectively. n is equal to the number of first modes used when using natural frequency or diagonal elements of flexibility matrix or n is equal to the number of suspected damaged elements when using MNMSE.

Numerical examples

Two numerical examples are discussed in this section. A 52-bar space truss is used as the first example. Damage location has been identified using MSEBI. The effect of a number of used modes on the MSEBI performance has been demonstrated using ROC and confusion matrix concept; damage extent estimation is performed using the RMSD objective function. In the second example which has more elements and therefore has more degrees of freedom, a 200-bar double layer grid, the performance of the GMDH surrogate model used in the stage of damage extent estimation is evaluated. In these examples, in the stage of damage extent estimation, the sensitivity of different modal properties due to damage and the performance of PSO, BA, and CBO as three commonly used optimization methods are compared. In addition, the effect of noise, caused by the measurement of structural response, on the accuracy of selected procedure is also studied.

52-bar space truss

The first example is the 52-bar space truss shown in Fig. 3. Non-structural masses of 50 kg are applied to all free nodes [⁵⁶]. Material properties of this structure are presented in Table 1. Three damage scenarios are considered as reported in Table 2.

Evaluating the performance of MSEBI in identifying suspected damaged elements

At first, MSEBI values of elements using first mode shape to first 10 mode shapes of 52-bar space truss are calculated. Then, the confusion matrix with the use of first mode shape up to first 10 mode shapes is formed to calculate the TP and FP rate. Table 3 is presented as an example and represents the confusion matrix for the cases of using the first 7, 3, and 1st mode shapes to compute MSEBI for scenario 1.

Results of Table 3 indicate that using first 7 mode shapes, the actual damaged elements are exactly identified (TP=3) for scenario 1; therefore its TP rate is equal to one and its FP rate is equal to zero (Fig. 4). Using the MSEBI and first 3 mode shapes of the structure, 8 elements are identified as damaged elements; meaning that 5 false damaged elements have been identified for scenario 1 (FP=5). Using the MSEBI and 1st mode shape of the structure has the worst result; 24 elements are predicted as damaged elements; meaning that 21 false damaged elements have been identified (FP=21). Finally, ROC curve is created for the case of using first mode shape up to first 10 mode shapes of 52-bar space truss for each damage scenario.

As illustrated in Fig. 4, using the first 7 mode shapes the MSEBI provides the best performance in the detection of damaged elements.

Results of Table 4 indicate that using first 7 mode shapes, 5 elements are identified as damaged elements; meaning that one false damaged element has been identified for scenario 2 (FP=1). Therefore, its TP rate is equal to one and its FP rate is equal to 0.02 (Fig. 5). Using the MSEBI and first 3 mode shapes of the structure, six elements are identified as damaged elements; meaning that two false damaged elements have been identified for scenario 2 (FP=2).

As illustrated in Fig. 5, the damage is located properly using first 7 mode shapes of space truss but with one false damaged element identification (Table 4).

Results of Table 5 indicate that using first 7 and 3 mode shapes, all the actual damaged elements are exactly identified (TP=5) for scenario 3. Therefore, there TP rate is equal to one and its FP rate is equal to zero (Fig. 6). Furthermore, the MSEBI has the worst performance for using 1st mode shape of the structure. As can be seen in Table 5, for the case of using 1st mode shape only 3 elements among 5 actual damaged elements (FN=2), are identified as damaged elements and 13 healthy elements have been wrongly diagnosed as damaged ones (FP=13).

According to the ROC plots for 52-bar space truss (Figs. 4, 5, and 6) using first 3 or first 7 mode shapes can be considered as good thresholds for the selection of the number of considered modes.

Estimation of the damage extent

In this phase, damage detection problem has become easier to solve as the number of suspected elements is much less than 52, the number of all elements. First, the specifications of the PSO, BA, and CBO algorithms are specified in Table 6. Population size and the maximum number of iterations for each scenario are considered the same for all three algorithms. Therefore, the number of function evaluations (i.e., number of analyses) is the same and the performance of the three algorithms can be fairly evaluated and compared.

Performance of MSEBI in locating the damaged elements for each damage scenario is illustrated in Figs. 4, 5, and 6. Accordingly, using the first three mode shapes has an acceptable performance in damage localization stage. Hence at the second stage, in order to estimate the damage extent, the performance of modal properties including natural frequencies and diagonal elements of modal flexibility matrix are investigated by using the first three modal responses.

To evaluate the above-mentioned optimization algorithms, the index termed as root mean square error (RMSE) is defined as below:

(26)RMSE=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^n{\left(estimated\hspace{0.17em}exten{t}_i-actual\hspace{0.17em}exten{t}_i\right)}^{\mathrm{2}}}}{n}}},

where n is the number of suspected damaged elements identified in the previous stage. To specify the best modal property and the best optimization algorithm for damage extent estimation, the best RMSEs of estimated extents over 10 independent runs for scenario 1 are presented in Table 7. The performance of damage extent estimation using MNMSE of suspected damaged elements is compared with the use of two previous modal properties (natural frequency and flexibility matrix). MNMSE is calculated with the first 2 mode shapes. As described in Section 3.2, using the MNMSE of suspected damaged elements means that a parameter vector \mathrm{\Delta }\mathit{P} (Eq. (7)) is composed of MNMSE of suspected damaged structural members.

Table 7 reports two important points. First, using the MNMSE of suspected damaged elements, compared to two other types of modal properties (natural frequency and flexibility matrix), can more effectively estimate damage extents in all three optimization algorithms. Moreover, damage extents are estimated by MNMSE using only first two mode shapes related to suspected elements.

Second, for this scenario, PSO has the best performance by using MNMSE of suspected damaged elements among three algorithms.

To have a better evaluation, the details of the 10 independent runs to estimate the damage extents of suspected damaged elements, performed by the three algorithms with using the MNMSE are shown in Table 8.

As can be seen in Table 8, PSO with MNMSE estimates the damage extents of suspected elements more accurately compared with BA and CBO (its best and average RMSE over 10 independent runs are the least among three algorithms). Moreover, PSO is more reliable algorithm compared with BA and CBO (it has the least RMSE standard deviation during 10 independent runs).

Similar to scenario 1, the best RMSEs of estimated extents over 10 independent runs for scenario 2 are reported in Table 9.

For the scenario 2, by increasing the number of suspected elements, PSO algorithm with any type of modal property has the best performance among all examined algorithms. Hence, similar to scenario 1, the best accuracy belongs to PSO with using MNMSE. Then, in order to optimally represent the results, Table 10 only reports the average and standard deviation of RMSEs of estimated damage extents over 10 independent runs by three examined algorithms with using MNMSE. As shown in Table 10, PSO with MNMSE estimates the damage extents of suspected elements more accurately compared with BA and CBO.

The best RMSEs of estimated extents over 10 independent runs by three examined algorithms are reported in Table 11 for scenario 3.

The results for scenario 3 are the same as the scenario 2. It can be concluded that using MNMSE as a modal property for the use of any of three optimization algorithms has better accuracy than two other modal properties. Like previous scenario the average and standard deviation of RMSEs of estimated damage extents over 10 independent runs by three examined algorithms with using MNMSE are presented in Table 12. As shown in Table 12, PSO is the most reliable among all three algorithms.

In Fig. 7 the convergence history of RMSD (objective function) which is defined based on the difference of modal property change vector of MNMSE (proposed damage extent indicator) for use of three examined optimization algorithms (PSO, BA, and CBO) is illustrated for comparison.

The CPU time of damage extent estimation is 15 s (using a Core^TM i5 2.67 GHz CPU) when using natural frequency and 17 s when using diagonal elements of flexibility matrix, respectively.

Increasing the number of structural elements and, consequently, the number of suspected damaged elements leads to an increase in computational cost of structural analysis and dimensions of the search space to estimate the damage extents, respectively. Using ANN-based surrogate models to reduce the time of structural analysis and, consequently, reducing the PSO implementation time is helpful. In computing the time to estimate damage extents with using ANN-based surrogate models, data generation time, training and testing time and PSO implementation time are all considered together. For this example, the sum of the times listed above is almost equal to PSO implementation time when using the FE model. However, as will be seen in the next example, for larger-scale structures, the use of the ANN-based surrogate model has noticeable effect in total CPU time.

It can be concluded from the results reported in this example, that using the MNMSE of suspected damaged elements, compared to two other types of modal properties (natural frequency and flexibility matrix), damage extents can be effectively estimated for all damage scenarios.

200-bar double layer grid

The second example is a 200-bar with two 25 m × 25 m grid shown in Fig. 8. This structure has a Young’s modulus of E=2×10¹¹N·m⁻² and mass density of \rho =\mathrm{2770}kg·m⁻³. The diagonal, bottom and top layers elements’ cross sections are A_d=10cm², A_b=12cm², and A_t=18cm², respectively. Two different damage scenarios are considered as depicted in Table 13.

Evaluating the performance of MSEBI in identifying suspected damaged elements

First, MSEBI values of elements using first mode shape to first 10 mode shapes of 200-bar double layer grid are calculated. Secondly, a confusion matrix with the use of first mode shape up to first 10 mode shapes is formed to calculate the TP and FP rate. Table 14 is depicted as example and represents the confusion matrix for the cases of using the first 8, 4 and 1st mode shapes in computation of MSEBI.

Results of Table 14 indicate that using first 8 mode shapes, 10 elements are identified as damaged elements; meaning that three false damaged elements have been identified for scenario 1 (FP=3). Therefore, its TP rate is equal to one and its FP rate is equal to 0.0155 (Fig. 9). Using first 4 mode shapes of the structure, 25 elements are identified as damaged elements; 19 damaged elements have been falsely identified for scenario 1 (FP=19); in this case only 6 elements among 7 actual damaged elements (FN=1) are identified as damaged elements.

ROC curve is created with using first mode shape up to first 10 mode shapes of 200-bar double layer grid for each damage scenario (Fig. 9).

As illustrated in Fig. 9, using first 8 mode shapes the MSEBI provides the best performance in the detection of damaged elements. After that, using first 7 mode shapes of the structure has the best performance.

Effect of measurement noise

Modal response of the existing structure which is assumed to have been extracted from the sensors ({\mathit{P}}_{\mathrm{d}} in Eq. (7)) is usually contaminated with noise during measurement. Hence, in this paper, the effect of measurement noise is studied for scenario 1 of 200-bar double layer grid. Measurement noise is considered for both frequency and mode shape as referred in Refs. [¹⁵,⁵⁷,⁵⁸]:

(27)f_i^{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}}=f_i^{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}-\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}\left(\mathrm{1}+{\eta }_{\mathrm{f}}\left(rand-\mathrm{1}\right)\right),

(28){\phi }_{ij}^{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}}={\phi }_{ij}^{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}-\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}\left(\mathrm{1}+{\eta }_{\phi }\gamma {\max }_i\left\{\left|{\phi }_{ij}^{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}-\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}}\right|\right\}\right),

where f_i^{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}} and f_i^{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}-\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}} are the i\mathrm{t}\mathrm{h} natural frequency in noisy and noise-free status, respectively. {\eta }_{\mathrm{f}} is the noise caused by the measurement of natural frequency; randis random value in [0,1]; {\phi }_i^{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}}and {\phi }_i^{\mathrm{n}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{e}-\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e}} are the i\mathrm{th} mode shape vector in noisy and noise-free status, respectively; {\eta }_{\phi } is the noise caused by the measurement of structural mode shape; \mathit{\gamma } is random number with zero mean and unit variance. Noise levels considered in this paper are: ({\eta }_{\mathrm{f}}=\mathrm{1}\mathrm{\%} & {\eta }_{\phi }=3\mathrm{\%}).

MSEBI was calculated 100 times independently for each case of using first mode shape to first 10 mode shapes of 200-bar double layer grid, the mean values of MSEBI vector for each case (first mode shape to first 10 mode shapes) are considered. Table 15 represents the confusion matrix for the cases of using the first 8, 4 and 1st mode shapes in computation of MSEBI for scenario 1 under the effect of noise.

As can be seen in Table 15, using first 8 mode shapes, 11 elements have nonzero MSEBI value and are identified as damaged elements; one more identified damaged element than the case of without noise effect; meaning that four false damaged elements have been identified for scenario 1 (FP=4). Therefore, its TP rate is equal to one and its FP rate is equal to 0.0207 (Fig. 10). Using first 4 mode shapes of the structure, similar to the case of without noise effect, 25 elements are identified as damaged elements; 19 damaged elements have been falsely identified for scenario 1 (FP=19); in this case only 6 elements among 7 actual damaged elements (FN=1) are identified as damaged elements.

ROC curve is created using first mode shape up to first 10 mode shapes of 200-bar double layer grid for scenario 1 with the consideration of noise effect (Fig. 10).

As illustrated in Fig. 10, MSEBI identification performance is almost similar to the previous case where noise effects were not considered (Fig. 9).

Results of Table 16 indicate that using first 7 mode shapes, 11 elements are identified as damaged elements; meaning that one false damaged element has been identified for scenario 2 (FP=1). Therefore, its TP rate is equal to one and its FP rate is equal to 0.0053 (Fig. 11). Using first 4 mode shapes of the structure, 14 elements are identified as damaged elements; meaning that 4 false damaged elements have been identified for scenario 2 which is not satisfactory (FP=4).

As illustrated in Fig. 11, the damage is located properly using first 7 mode shapes of space truss but with one false damaged element identification (Table 16).

According to ROC plots (Figs. 9, 10, and 11), for 200-bar double layer grid, using first 7 mode shapes can be considered as a good threshold for selection of the number of considered mode shapes.

Estimation of the damage extent

In this section, selected optimization algorithm with an ANN model (GMDH) is applied for estimation of damage extent. Accordingly, for the first scenario, performance of PSO, BA, and CBO is investigated for the presence and absence of measurement noise (scenario 1); then, the optimization algorithm with the best performance and the most suitable modal property is applied to estimate damage extents using GMDH surrogate model. Regarding the PSO specifications applied in this example, the number of particles is 60; the maximum number of iterations for scenario 1 is 80 and for scenario 2 is 100, and other specifications are the same as the ones used in the previous example (Table 6).

Estimation of the damage extent using selective optimization algorithm engaged by FE model

Identification performance of MSEBI for each damage scenario is illustrated in Figs. 9, 10, and 11. Accordingly, for the case of using the first eight mode shapes for scenario 1 and the first seven ones for scenario 2, the MSEBI performance is the best during damage localization. Therefore, in the second stage, for this structure which has more elements than the previous example, the use of the first 7 mode shapes to investigate the performance of modal properties including natural frequencies and diagonal elements of modal flexibility matrix, can be a reasonable choice.

To specify the best modal property and the best optimization algorithm for damage extent estimation, the best RMSEs of estimated extents for scenario 1 over 10 independent runs are reported in Table 17 for two cases of with and without the measurement noise effect.

As can be seen in Table 17, for both cases of with and without measurement noise effect, damage extent estimation is more accurate when MNMSE of suspected elements is used as modal response which is calculated using only the first two mode shapes. Overall, the results are not satisfactory when using first seven natural frequencies or even first seven diagonal elements of flexibility matrix. Also, as can be observed, PSO has a better performance than two other algorithms for both cases of with and without measurement noise effect. Then, in order to observe more precisely, Table 18 shows the average and standard deviation of RMSEs of estimated damage extents over 10 independent runs by three examined algorithms with using MNMSE. As shown in Table 18, PSO with MNMSE estimates the damage extents of suspected elements more accurately compared with BA and CBO even with noise effect. It is worth noting, that if more modes for calculation of MNMSE are considered, the RMSE will also be further reduced.

A plot showing the relation between the average RMSE of damage extent estimation and noise percentage in the case of using PSO with MNMSE is presented in Fig. 12. As can be seen in Fig. 12, by increasing the noise percentage from 1 to 3 percent, RMSE is also increased.

In Fig. 13 the convergence history of RMSD (objective function) which is defined based on the difference of modal property change vector of MNMSE (proposed damage extent indicator) for use of three examined optimization algorithms (PSO, BA, and CBO) is illustrated for comparison in the cases of with and without measurement noise effect.

Damage extent estimation using MNMSE

The best estimated damage extents over 10 independent runs of PSO with using MNMSE are presented in Tables 19, 22, and 23. Estimated damage extents with using only first 2 mode shapes to compute the suspected damaged elements’ MNMSE and RMSD objective function are best converged in comparison to other modal properties and optimization algorithms (compared to Table 17).

Damage extent estimation procedure using surrogate model

In this section, a new procedure has been proposed for estimation of damage extent. During this procedure, the PSO (selective optimization algorithm) along with an optimized GMDH model as shown in Fig. 2 is applied. Accordingly, the performance of the GMDH model is compared with the cascade feed-forward neural network (CFNN) model with a log-sigmoid transfer function for the hidden layer [³⁴].

According to the results of the previous section and also previous example, the suspected damaged elements’ MNMSE, which is obtained from the first two mode shapes, is used as a modal property and RMSD is used as an objective function. For scenario 1, the number of input features (N) is equal to the number of suspected damaged elements (10). The number of output features (neurons) is equal to the number of suspected damaged elements (10) when using the suspected damaged elements’ MNMSE modal property as outlined in Section 3.2, Eq. (10) and Fig. 2. The number of hidden neurons is 20 for CFNN. For the GMDH neural network, the maximum number of layers is 6 and the maximum number of neurons in a layer is 25. In all of the solution procedures, PSO specifications are the same.

All possible cases per ten marked suspected damaged elements (scenario 1) for damage extents vary from 0 to 0.5 at the pace of 0.05, is equal to 10¹⁰ (Eq. (29)).

(29)All\hspace{0.17em}possible\hspace{0.17em}cases=S^d,

where S is the number of existing damage extents vary from 0 to 0.5 at the pace of 0.05 and d is the number of suspected damaged elements.

Determining the sufficient number of training and test data set is necessary. By increasing the number of training and test data sets for the neural network, the data generation time increases due to the increase in the number of FE analysis which in turn increases the computational cost of training and testing of the network. Also, the use of more efficient sampling methods for data sampling, like modified LHS method, has led to an enhanced optimized sampling with the least possible number of training and test data sets. LHS samples are iteratively generated to find the best one according to the criterion based on reducing the correlation between produced points. Accordingly, it is iterated up to 10 times in an attempt to improve the design space spanning. To determine the effect of a number of points in data sets on the final prediction of GMDH, the root mean square error of test data sets for a different number of training and test data sets has been calculated and illustrated in Fig. 14.

As can be seen in Fig. 14, the test data set error when using 1100 training and test data sets is higher than the corresponding error of using 200 data sets. This means that the model generalization is decreased [⁵⁹]. Therefore 400 damage cases as training and test data sets are generated using latin hypercube sampling (LHS) method. Then, a suitable network is selected based on the root mean square error (RMSE) of test data sets.

For both damage scenarios, results of two kinds of ANN as surrogate models in terms of CPU time and accuracy are reported in Tables 20, 22, and 24. It should be noted that in computing the time to solve the problem of damage extent estimation with using ANN-based surrogate models, data generation time, training and testing time and, PSO implementation time are considered together (using a Core^TM i5 2.67 GHz CPU).

First, results of Tables 19 and 20 indicate that applying a surrogate of FE model, decreases the CPU time of damage extent estimation. As the numbers indicate, the CPU time of performing the procedure is reduced by half while maintaining an acceptable accuracy. Consequently, using ANN model requires only 400 FE structural analyses for generation of training and test data sets, whereas using FE model leads to 4800 FE structural analyses (the number of objective function evaluations by PSO).

Secondly, the use of GMDH model has almost the same RMSE of estimated damage extent as that of CFNN model, but the time to solve the problem of damage extent estimation, when using GMDH model, is far less than that of using CFNN model. According to these results, GMDH model is a more efficient surrogate model for damage extent estimation.

Similar to the case of without noise effect, the best results acquired in ten independent implementations of PSO along with surrogate model for the case of noise effect ({\eta }_{\phi }=3\mathrm{\%}) are presented for scenario 1 in Tables 21 and 22.

For damage scenario 2, the maximum number of iterations for this scenario is 100. Results are reported in Tables 23 and 24.

According to the results obtained in this example, it is observed that the estimated damage extents have an acceptable accuracy when using PSO along with GMDH as a surrogate model and RMSD as an objective function. Moreover, using the MNMSE of suspected damaged elements is a more appropriate modal property to estimate the extent of damage even with noise effect.

Conclusions

In this paper, after locating the suspected damaged elements using MSEBI and evaluating the MSEBI performance based on a ROC curve concept, an efficient surrogate model has been used for damage extent estimation. To reduce the computational time of model updating, the optimization algorithm is used along with an ANN model as a surrogate of the FE model. Accordingly, to evaluate the performance of this surrogate model, two damage scenarios were studied, one of which was studied under effect of measurement noise. Moreover, the GMDH network performance has been compared with the performance of the CFNN network. In addition, the performance of three popular optimization algorithms (PSO, BA, and CBO) and the performance of the most used modal properties are investigated simultaneously. Based on the numerical results, the following conclusions can result.

1)‚In both examples, for all damage scenarios and for both cases of the presence and absence of measurement noise, the results of damage extent estimation obtained by PSO are more accurate than those obtained by other two examined optimization algorithms (BA and CBO).

2)‚For the 200-bar double layer grid, the use of natural frequencies and diagonal elements of flexibility matrix to estimate the extent of damage, even for the case of using the first 7 modal properties did not produce satisfactory results.

3)‚Using the MNMSE of suspected damaged elements can effectively be used to estimate damage extent using only first two mode shapes even with the presence of measurement noise.

4)‚The time of damage extent estimation using PSO along with GMDH model is significantly reduced compared to using CFNN model due to shorter training and testing time of GMDH network than CFNN network (reduced less than 50€%). In addition, both GMDH and CFNN provide the same level of accuracy in the estimation of damage extent even with the presence of measurement noise.

5)‚In this employment of GMDH network, in order to optimize the performance of the network as a surrogate model, the optimal value of alpha (selection pressure) has been determined. For each value of alpha between zero and one at the pace of 0.05, the network is trained and tested via test data sets. Finally, the alpha value is selected for which the network has the least standard deviation of the test data set error. Consequently, the selection of the best value of alpha enhances the quality of GMDH in damage extent estimation.

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