Keywords

1 Introduction

Structural Health Monitoring (SHM) is a significant factor in research over the past two decades because assessing the location and extent of damaged elements in the structure will ensure effective working of the structure and timely repair. The inverse method combined with model updating is a reliable method to predict damaged elements in SHM. This approach has been used widely and successfully in recent studies [1,2,3]. Friswell [4] introduced a brief overview of applying the model updating technique and inverse methods for SHM from vibration data obtained from measurements. The main features of this method include three steps: (i) establish the objective function, (ii) the model updating technique is used to calculate the objective function, and (iii) finding a reliable algorithm to minimize the objective function. Thus, the selection of a suitable optimization algorithm takes a crucial role in the successful method. The robust development of metaheuristic algorithms with different inspirations has brought many choices to solve optimization problems. There are many algorithms such as Atom search algorithm (ASO) [5], Cuckoo search (CS) [6], Gravitational Search Algorithm (GSA) [7], Black Hole [8], Grey Wolf optimizer (GWO) [9], The Whale Optimization Algorithm (WOA) [10], and so on. These algorithms have proven effective in their application to SHM. Minh et al. [11] presented a new model updating technique combined with an improved PSO named (EHVPSO) for damage detection in a real 3D structure. In this study, a real 3D transmission tower was used to prove the effectiveness of the method. The results obtained in this study prove the reliability and high level. Cuong-Le et al. [12] utilize PSO to enhance the ability of support vector machine in damage detection. Alkayem [13] used a finite element (FE) model built in MATLAB and some algorithms, including Particle Swarm Optimization (PSO), and Differential Evolution (DE) and Genetic Algorithm (GA), to detect damaged elements in 3D frame structures. Chen [14] proposed a new method for calculating the objective function, then the new objective function collaborated with Whale optimization algorithm (WOA) to predict the damaged elements in a simply-supported beam and 31-bar truss structures. Nozari et al. [15] used a FE model updating for damage detection of 10-story building using ambient vibration measurements, etc. Almost all previous public studies in this field concerned with simple structures including 2D or 3D frames.

In this paper, to assess generally this method, a real structure named Guangzhou New TV Tower located China is used to detect the damaged elements. First, For simplicity and to reduce the number of degrees of freedom of the structure, a reduced-order FE model generated from the full-order model is conducted by MATLAB. Then, a recent optimization algorithm named Atom search optimization (ASO) is employed as a reliable algorithm to detect damaged elements in this structure. The damaged detection process will be secured by the exchange data between the FE model and the ASO algorithm. This process will stop if the objective function achieves a suitable convergence rate and at the same time, these variants in the objective function will determine the extent of damage in the structure.

2 Structural Modeling of the Guangzhou New TV Tower

In this section, a real 3D tower named the Guangzhou New TV Tower is selected to validate the proposed method. The system of this structure is tube-in-tube with 600 height. The tower includes two main parts; the first part is the main tower with 454 m height, and the second part is the antenna mast with 146 m height, according to Chen et al. [16] as shown in Fig. 1a. Because of the complicated geometry of the structure, using the full-scale FE model will have difficulties in this study. To simplify the model, a reduced-order FE model is simulated in a simple way. Thus, the simple FE model includes 37 elements; each element is simulated as a linear elastic beam element. Consequently, the FE model is modelled as a cantilever beam including 37 elements. Each element registers two nodes at start and end, respectively. The total number of nodes is 38, the label of nodes will in a gradual increase from 1 to 38, the vertical coordinates of each node are given in Table 1. In the reduced model, the mass of each story is lumped at a node that is located at the same high level of the story mentioned, then these masses are lumped to the adjacent reducing center, as shown in Fig. 1b. The vertical displacement following the Z direction is ignored in the FE reduced model. Thus, each node registers 5 DOFs, including two horizontal translational DOFs and three rotational DOFs, as shown in Fig. 1b. Thus, each element has 10 DOFs and will register stiffness matrix with dimensions 10 × 10. The FE model in this study is referred to the model presented by Ni, Xia [21] and Chen, Lu [20]. Thus, each element stiffness matrix is given in Eq. (1).

$$K_{i}^{e} = \left[ {\begin{array}{*{20}c} {K_{11}^{e} } & {K_{12}^{e} } & {K_{13}^{e} } & {K_{14}^{e} } & {K_{15}^{e} } & {K_{16}^{e} } & {K_{17}^{e} } & {K_{18}^{e} } & {K_{19}^{e} } & {K_{110}^{e} } \\ {} & {K_{22}^{e} } & {K_{23}^{e} } & {K_{24}^{e} } & {K_{25}^{e} } & {K_{26}^{e} } & {K_{27}^{e} } & {K_{28}^{e} } & {K_{29}^{e} } & {K_{210}^{e} } \\ {} & {} & {K_{33}^{e} } & {K_{34}^{e} } & {K_{35}^{e} } & {K_{36}^{e} } & {K_{37}^{e} } & {K_{38}^{e} } & {K_{39}^{e} } & {K_{310}^{e} } \\ {} & {} & {} & {K_{44}^{e} } & {K_{54}^{e} } & {K_{46}^{e} } & {K_{47}^{e} } & {K_{48}^{e} } & {K_{49}^{e} } & {K_{410}^{e} } \\ {} & {} & {} & {} & {K_{55}^{e} } & {K_{56}^{e} } & {K_{57}^{e} } & {K_{58}^{e} } & {K_{59}^{e} } & {K_{510}^{e} } \\ {} & {} & {} & {} & {} & {K_{66}^{e} } & {K_{67}^{e} } & {K_{68}^{e} } & {K_{69}^{e} } & {K_{610}^{e} } \\ {} & {} & {} & {} & {} & {} & {K_{77}^{e} } & {K_{78}^{e} } & {K_{79}^{e} } & {K_{710}^{e} } \\ {} & {} & {} & {Sym} & {} & {} & {} & {K_{88}^{e} } & {K_{89}^{e} } & {K_{810}^{e} } \\ {} & {} & {} & {} & {} & {} & {} & {} & {K_{99}^{e} } & {K_{910}^{e} } \\ {} & {} & {} & {} & {} & {} & {} & {} & {} & {K_{1010}^{e} } \\ \end{array} } \right]$$
(1)
Fig. 1
figure 1

a The Guangzhou New TV Tower according to Chen et al., b a reduced-order FE model

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Table 1 The nodal coordinate (z) of the reduced model
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3 The Objective Functions

The objective function is selected from the correlation of natural frequencies obtained from the FE model with cases of damage and data measured from testing of healthy structure. By solving Eq. (2) using the FE model, we can get the natural frequencies.

$$\left[ {K - \omega_{i}^{2} M} \right]\left\{ {\phi_{i} } \right\} = \left\{ 0 \right\}$$
(2)

where K and M are the structural stiffness matrix and mass matrix, respectively. \(\omega_{i}\) is natural period and \(\phi_{i}\) is mode shape vector.

From \(\omega_{i}\), we can calculate the frequencies as follows Eq. (3):

$$T_{i} = \frac{2\pi }{{\omega_{i} }},\;f_{i}^{FE} = \frac{1}{{T_{i} }}$$
(3)

The objective function is selected by using Root-Mean-Square-Error (RMSE) as shown in Eq. (4).

$$f_{ofun} = \frac{{\sqrt {\sum\limits_{i = 1}^{n} {\left( {f_{i}^{Measured} - f_{i}^{FEM} } \right)^{2} } } }}{n}$$
(4)

where n is the number of frequencies obtained using the FE model.

4 Atom Search Algorithm (ASO)

ASO [5] is inspired by basic molecular dynamics. From this perspective, each position \(X_{i} \;(i = 1,2,...,N)\) is considered as a candidate solution in an unknown search space dimension.

4.1 Interaction Force in ASO

The interaction force obtained from the Lennard–Jones (L-J) potential is a simple mathematical revised to obtain more positive attraction and less negative repulsion as iterations increase. The interaction force impact on the ith Atom from jth Atom at tth iteration is shown in Eq. (5)

$$\begin{gathered} F_{ij}^{d} = - \eta (t)\left[ {2\left( {h_{ij} (t)} \right)^{13} - \left( {h_{ij} (t)} \right)^{7} } \right] \hfill \\ \eta (t) = \alpha \left( {1 - \frac{t - 1}{{T_{\max } }}} \right)^{3} e^{{ - \frac{20t}{{T_{\max } }}}} ,\quad \alpha = \left[ {10,\;20,\;30,...,100} \right] \hfill \\ \end{gathered}$$
(5)

where \(\alpha\) is the depth weight, t is the current of iteration, and Tmax is the maximum of iterations.

\(h_{ij} (t)\) is the ratio of the distance between two ith and jth atoms to the length scale \(\sigma (t)\) as given in Eq. (6).

$$h_{ij} (t) = \left\{ \begin{gathered} h_{\min } \quad \quad \frac{{\left\| {r_{ij} (t)} \right\|_{2} }}{\sigma (t)} < h_{\min } \hfill \\ \frac{{\left| {r_{ij} (t)} \right|}}{\sigma (t)}\quad h_{\min } \le \frac{{\left\| {r_{ij} (t)} \right\|_{2} }}{\sigma (t)} \le h_{\max } \quad \hfill \\ h_{\max } \quad \quad \frac{{\left\| {r_{ij} (t)} \right\|_{2} }}{\sigma (t)} > h_{\max } \quad \quad \hfill \\ \end{gathered} \right.$$
(6)

where \(h_{\max }\) and \(h_{\min }\) are lower and upper boundaries, respectively and defined as Eq. (7).

$$\left\{ \begin{gathered} h_{\min } = g_{0} + g(t) \hfill \\ h_{\max } = u \hfill \\ \end{gathered} \right. = \left\{ \begin{gathered} h_{\min } = g_{0} + 0.1\sin \left( {\frac{\pi t}{{2T_{\max } }}} \right) \hfill \\ h_{\max } = u \hfill \\ \end{gathered} \right.$$
(7)

\(\sigma (t)\) is denoted the length scale, and it can be expressed by Eq. (8)

$$\begin{gathered} \sigma (t) = \left\| {X_{ij} (t),X_{Kbest} (t)} \right\|_{2} \hfill \\ X_{Kbest} (t) = \frac{{\sum\limits_{{j \in K_{best} }} {X_{ij} (t)} }}{K(t)} \hfill \\ \end{gathered}$$
(8)

where \(K_{best}\) can be defined as a high-reliability search space for next iterations, and it is described by Eq. (9).

$$K_{best} (t) = N - (N - 2)\sqrt {\frac{t}{{T_{\max } }}}$$
(9)

4.2 Mathematical Representation of Geometric Constraint and the Mass of Atom

To increase the convergence rate, each Atom is linked with the best Atom. In other words, each Atom is affected by the best Atom through the force called geometric constraint force shown in Eq. (10).

$$\begin{gathered} G_{i}^{d} (t) = \lambda (t)\left[ {X_{best}^{d} (t) - X_{i}^{d} (t)} \right] \hfill \\ \lambda (t) = \beta e^{{ - \frac{20t}{{T_{\max } }}}} \hfill \\ \beta = \left[ {0.1;\;0.2;\;0.3;\;0.4;\;0.5;\;0.6;\;0.7;\;0.8;\;0.9;\;1} \right] \hfill \\ \end{gathered}$$
(10)

where \(X_{best}^{d} (t)\) is the best solution at tth iteration, \(\lambda (t)\) is the Lagrangian multiplier.

In ASO, each Atom \(X_{i}^{d} (t)\) registers a changeable mass and can be expressed in Eq. (11).

$$\begin{gathered} M_{i} (t) = e^{{ - \frac{{\left( {Fit_{i} (t) - Fit_{best} } \right)}}{{Fit_{worst} - Fit_{best} }}}} \hfill \\ m_{i} (t) = \frac{{M_{i} (t)}}{{\sum\limits_{j = 1}^{N} {M_{j} (t)} }} \hfill \\ \end{gathered}$$
(11)

where \(Fit_{best} (t)\) and \(Fit_{worst} (t)\) are the maximum and minimum values of the objective function at the tth iteration, respectively. N is the number of Atom, \(Fit_{i} (t)\) is the value of the objective function of Atom ith at tth iteration. \(Fit_{best} (t)\) and \(Fit_{worst} (t)\) are expressed as Eq. (12).

$$\begin{gathered} Fit_{best} (t) = \mathop {\min }\limits_{{\left( {i = 1,2,...,N} \right)}} \left[ {Fit_{i} (t)} \right] \hfill \\ Fit_{worst} (t) = \mathop {\max }\limits_{{\left( {i = 1,2,...,N} \right)}} \left[ {Fit_{i} (t)} \right] \hfill \\ \end{gathered}$$
(12)

4.3 Atomic Motion

The process of acceleration updating is calculated by Eq. (13).

$$\begin{gathered} a_{i}^{d} (t) = \frac{{F_{i}^{d} (t)}}{{m_{i}^{d} (t)}} + \frac{{G_{i}^{d} (t)}}{{m_{i}^{d} (t)}} \hfill \\ = \alpha \left( {1 - \frac{t - 1}{{T_{\max } }}} \right)^{3} e^{{ - \frac{20t}{{T_{\max } }}}} \sum\limits_{{j \in K_{best} }} {\frac{{rand_{j} \left[ {2\left( {h_{ij} (t)} \right)^{13} - \left( {h_{ij} (t)} \right)^{7} } \right]}}{{m_{i} (t)}}\frac{{\mathop {r_{ij} }\limits^{ \to } }}{{\left\| {r_{ij} } \right\|_{2} }}} \hfill \\ + \beta e^{{ - \frac{20t}{{T_{\max } }}}} \frac{{\left( {X_{best}^{d} (t) - X_{i}^{d} (t)} \right)}}{{m_{i} (t)}} \hfill \\ \end{gathered}$$
(13)

where \(\mathop {r_{ij} (t)}\limits^{ \to }\) and \(\left\| {r_{ij} (t)} \right\|_{2}\) are the position difference vector and Euclidean distance between the ith and the jth Atoms, respectively and can be expressed by Eq. (14).

$$\mathop {r_{ij} (t)}\limits^{ \to } = \left( {X_{j}^{d} (t) - X_{i}^{d} (t)} \right),\quad \left\| {r_{ij} (t)} \right\|_{2} = \sqrt {\sum\limits_{k = 1}^{D} {(x_{jk} - x_{ik} )^{2} } }$$
(14)

The position updating of each Atom is updated through the velocity updating process and can be written as Eqs. (15 and 16).

$$V_{i}^{d} (t + 1) = rand_{i}^{d} V_{i}^{d} (t) + a_{i}^{d} (t)$$
(15)
$$X_{i}^{d} (t + 1) = X_{i}^{d} (t) + V_{i}^{d} (t + 1)$$
(16)

5 Application to Structural Health Monitoring

5.1 Structural Damage Identification Approach

The damaged identification can be illustrated by a scalar vector \(X_{i} = \left( {x_{1,} x_{2} ,...,x_{n} } \right)\) with each individual \(x_{i} \;(i = 1,2,..n)\) bounded in the ranged \(\left[ {0,\;1} \right]\). Thus, the global stiffness matrix will reduce at damaged stage and given in Eq. (17).

$$\left[ K \right] = \sum\limits_{i = 1}^{n} {(1 - x_{i} )} k_{i} ;\quad 0 \le x_{i} \le 1$$
(17)

where \(k_{i}\) is the stiffness matrix of element ith at the healthy stage.

The goal of structural damage identification is to determine the scalar vector \(X_{i} = \left( {x_{1,} x_{2} ,...,x_{n} } \right)\), which agrees with the objective function with acceptable error. The process of detecting \(X_{i} = \left( {x_{1,} x_{2} ,...,x_{n} } \right)\) is secured by an optimization algorithm. In this paper, ASO is employed to do this. And the process of damage identification using FE model updating and ASO is illustrated in Fig. 2.

Fig. 2
figure 2

The process of detecting damaged structures using a reduced-order FE model and Atom search algorithm

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5.2 Application to Guangzhou New TV Tower

To demonstrate the reliability of the proposed method, two damaged cases with different reductions of stiffness of each story are considered as given in Table 2. The value of frequencies at two stages damaged and undamaged are shown in Table 3.

Table 2 Reduction in stiffness in stories of Guangzhou New TV Tower for different damage cases
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Table 3 The values of the 40 first frequencies using the reduced-FE model at undamaged stage and two cases of damaged stage
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The results of the damaged indicator process will be presented according to the following:

  • The convergence trend of objective function: These curves show the trend of the value of the objective function over the course of iterations.

  • The historical trend of damaged elements: These curves show the changes in the values of the damaged variables, which are defined as a reduction of stiffness. They also illustrate the fast or slow convergence rate of ASO algorithm.

  • The damage identification bar chart: The final values are shown in the bar chart. The chart shows the correlation between the values of two damaged cases and damaged indicators using FE model updating combined with ASO.

The results are shown from Figs. 3, 4, 5, 6, 7, and 8 and the statistical table in true value and indicator value is given in Table 4.

Fig. 3
figure 3

The convergence trend of objective function in Case 1

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Fig. 4
figure 4

The historical trend of damaged elements in Case 1

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Fig. 5
figure 5

The damage identification bar chart in Case 1

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Fig. 6
figure 6

The convergence trend of objective function in Case 2

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Fig. 7
figure 7

The historical trend of damaged elements in Case 2

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Fig. 8
figure 8

The damage identification bar chart in Case 2

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Table 4 The results of ASO algorithms for predicting damage elements
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The results in Table 4 show that ASO can predict accurately the damaged stories in Case 1 and acceptable errors in Case 2. There is a general acceptance that the ability to explore ASO during the first few iterations does not appreciate. However, this ability is improved more clearly during the last iterations, and ASO achieves stability in convergence rate and accuracy level because of reducing the Lennard-Jones (L-J) potential force during the last iterations.

6 Conclusion

The paper introduced an effective method to predict damaged structures for a real complex structure named Guangzhou New TV Tower. In the paper, a complex model structure can be simplified by a simple model structure in which each floor is lumped at the same level as the real floor. The stiffness of each story is converted to the stiffness of the frame whose registers ten degrees of freedom. The process of damaged identification is performed using the FE model updating and inverse method. The results obtained in this paper prove the effectiveness of the method and the reliability of the ASO algorithm. However, this method still has limitations because it cannot predict the level of damage severity of the individual elements on each floor. This method can be used as a reference method to assess the level of damage story quickly. For a more detailed assessment, we need to apply a new model updating technique with a full-scale model, which can be simulated by finite element software.