A distance coefficient-multi objective information fusion algorithm for optimal sensor placement in structural health monitoring

Abstract

Optimal sensor placement (OSP) plays a key role in the construction and implementation of an effective structural health monitoring system (SHM). In this study, a novel and effective method named the distance coefficient-multi objective information fusion algorithm (D-MOIF), which is different from the conventional method and easier to be implemented, is developed to select the best sensor location for large-scale structures. An integrated information matrix including mode independence, damage sensitivity and modal strain energy is deduced from the structural motion equation to meet multiple needs of SHM. A European distance derived from the analytic geometry is proposed to overcome the information redundancy between sensors. Based on the principle of information entropy, an optimized objective function is constructed, which could balance the sensitivity and robustness of the algorithm. A computational case of a high arch dam is implemented to demonstrate the effectiveness of the modified algorithm, and three classical evaluation criteria are used to estimate the comparison between the D-MOIF algorithm and four traditional OSP methods. Finally, the optimization of the number of sensors based on different algorithms is discussed in detail. Results indicate that the proposed D-MOIF algorithm could generate more applicable sensor configurations for large-scale structures.

Keywords

optimal sensor placement information fusion optimized objective function distance coefficient large-scale structures

Introduction

In recent years, in order to enhance safety measures in large-scale infrastructure, research activities focus on the significant technological advances in sensing and communication technology. The concept of the SHM system proposed by Housner et al. (1997) opened a breakthrough to solve this problem. The integration and operation of the SHM system are similar to human organs and circulation, as shown in Figure 1. The structure is like the muscles and bones of the human body. The health of these muscles and bones are what people care about most; sensors placed on structures are like the nervous system all over the human body. Just as the nervous system could perceive and feed back pain on the body, sensors in health monitoring systems could measure changes in the characteristics of structures in real-time; the information control system in the SHM system is like the human brain. The structural information collected by the sensors is aggregated and stored in the “brain” to classify and analyze the information, and the results evaluated by the “brain” are used as the basis for structural health assessment. Therefore, OSP is one of the core technologies of the SHM system, which has a crucial impact on the effectiveness of data acquisition and the accuracy of structural parameter identification. For structures with a simple form and few degrees of freedom (DOF), the optimal arrangement of sensors can be directly determined according to the experience of experts (Feng et al., 2018; Xu et al., 2020; Zhao et al., 2020). However, for complex large-scale structures, whose DOFs may be tens of thousands, it is necessary to study the arrangement of sensors in theory (Yi et al., 2011).

Figure 1.

Analogy diagram of the SHM system.

In the early days of this study, many scholars and engineers have carried out widely and deeply theoretical research (Hemez and Farhat, 1994; Li et al., 2008; Papadopoulos and Garcia, 1998; Udwadia, 1994). The most significant OSP algorithm is the effective independence (EfI) method proposed by Kammer (1991), which attempted to exclude the test points which have little contribution to the target modal partitions from all possible test points, and then use as few sensors as possible to achieve the best estimation of modal parameters. Based on the theory of the EfI, Kammer and Tinker (2004) presented a new technique named EfI3, which could make the three directions of the selected measuring point simultaneously satisfy the maximum linear independence of the mode. Schedlinski and Link (1996) choose the best placement of the sensors by QR decomposition method. Salama et al. (1987) proposed the modal kinetic energy (MKE) method. The basic principle of this method is to arrange the measuring points according to the order of modal kinetic energy on each degree of freedom (DOF) of the structure. Kim and Stubbs (1995) proposed modal strain energy (MSE) algorithm, which use modal strain energy as the principle of determining optimal pick-up locations. Based on the theory of MKE, Chung and Moore (1993) proposed a method to rank the importance of candidate sensor positions by using the average kinetic energy and weighted average kinetic energy (WAKE). Shi et al. (2000) put forward a method of OSP based on damage sensitivity. The sensor placement scheme obtained by this method is more sensitive to damage. Meo and Zumpano (2005) developed the variance method (VM) based on the most informative subset (MIS) with the aim to reduce the data amount. Li et al. (2009) developed a fast computational algorithm for the computation of the EI method which does not need to form explicitly the information matrix. The idea of these classical methods continues to this day, but these criteria only meet a certain demand of the SHM system. In 1997, Wolpert and Macready (1997) published an article entitled “No Free Lunch Theorem for Optimization” in the IEEE Transaction on Evolutionary Computation magazine. The main idea is that for any algorithm, any elevated performance over one class of problems is offset by performance over another class. Therefore, with the complexity of structural form, single-objective information matrix hardly meets multiple needs of the SHM system, multi- objective information integration is the inevitable development trend of OSP.

In recent years, the OSP method considering multi-object has also made great progress. Imamovic (1998) proposed effective independence driving-point residue (EfI-DPR) method. This technology could ensure that the selected nodes are distributed as much as possible in the position with larger modal kinetic energy, so as to reduce the loss of effective information in the field test with large environmental noise. Based on the theory of EfI-DPR method, Lian et al. (2012) developed the triaxial effective independence driving-point residue (EfI3-DPR3) algorithm, which could take into account both the maximum triaxial modal kinetic energy and the linear independence of the triaxial target mode. Dai and Ji (2015) put forward the effective independence-average modal kinetic/strained energy coefficient methods and introduced a weight coefficient reflecting the contribution proportion of a higher-order model. Gomes et al. (2019) adopted multi-objective genetic algorithm to search the position of sensors, and proposed a multi-objective optimization function combining Fisher information matrix (FIM) and mode interpolation information. Lin et al. (2019) developed a multi-objective OSP method based on response covariance. According to the determined optimal sensor number and location, the multi-sensors were installed on the frame structure and the advanced nature of the method was verified by the damage identification results based on the monitored data. Pei et al. (2019) proposed an OSP method based on conditional information entropy, and studied the influence of environmental noise and model error on sensor deployment. Sahu et al. (2020) proposed a method of OSP utilizing the majorization-minimization algorithm and the duality principle. This method could give the precise direction of sensors and could easily deal with the related noise. Yang et al. (2019) investigated the combined objective function based on the effective independence method and the three-dimensional redundancy elimination model to achieve a balance between the reliability of sensor placement and redundant information. These methods are mainly carried out in the field of spacecraft, rocket, launch pad, machinery and bridge, but they hardly be directly used in large-scale hydraulic structures with dense grids because similar FIM may be generated by two degrees of freedom close to each other. In the field OSP of large-scale hydraulic structures, the nearest neighbor index and distance coefficient (He et al., 2014; Lian et al., 2013) were induced to avoid information redundancy between the selected measuring points. In addition, it should be noted that in recent years, lots of research findings on the OSP lies in how to improve the global search capability through improved evolutionary algorithms. Feng and Jia (2018) proposed a microhabitat frog-leaping algorithm based on shuffled frog-leaping algorithm and effective independence method, which could more effectively optimize the configuration of the three-axis acceleration sensor in the vibration test of SHM. Zhou et al. (2019) proposed the automatic-learning firefly algorithm by combining the original firefly algorithm with the Lévy flight and the automatic learning mechanism, and combined the algorithm with traditional optimization criteria, demonstrating that the algorithm has a good global search capability. Similar studies include (Banik and Das, 2019; Houssein et al., 2020; Kaveh and Eslamlou, 2019). These studies have played a vital role in the development of OSP. But exploring the objective function that contains more monitoring indicators is as important as the development of evolutionary algorithms. Therefore, in this article, in order to highlight the improvement of the optimization function, there is no effort to choose which evolutionary algorithm, but the traditional serial method for OSP. So far, no technology could satisfy the following three conditions at the same time: including multi-objective information matrix, avoiding information redundancy and ensuring the stability of the algorithm.

In this paper, a novel and effective method named the distance coefficient-multi objective information fusion algorithm (D-MOIF), which can meet the meet multiple needs of SHM for large-scale structures, is developed and carried out on the Baihetan arch dam, the largest hydropower station under construction in China. The method proposed in this paper features several new innovations. An integrated information matrix including mode independence, damage sensitivity and modal strain energy is deduced from the structural motion equation to meet multiple needs of SHM. A European distance derived from the analytic geometry is proposed to overcome the redundancy of information between the sensors. Based on the principle of information entropy, an optimized objective function is constructed, which could balance the sensitivity and robustness of the algorithm at the same time.

The remainder of this paper is arranged as follows. In the section “Theory,” the mathematical model and optimization method of the D-MOIF algorithm are derived in detail, and the implementation logic and flow of the algorithm are given. In the section “Application,” a computational case is carried out to verify the effectiveness of the algorithm. Besides, three classical evaluation criteria are used to compare the D-MOIF algorithm with four traditional sensor optimization methods, and the optimal number of sensors is discussed detailed. Finally, some concluding remarks of this study are summarized.

Theory

Mathematical model

The free vibration equation of a dynamic system with n degrees of freedom can be defined as (Chopra, 2007):

(K - λ_{i} M) ϕ_{i} = 0

(1)

where K and M are the stiffness matrix and mass matrix of the system, respectively; $λ_{i}$ is the ith eigenvalue (modal frequency) of the system, $ϕ_{i}$ is the corresponding eigenvector (target modal shape).

Using the perturbation finite element method, assuming that the damage only causes structural stiffness to disturb, ignoring changes in mass and damping, then the vibration equation of the system can be described as:

[(K - Δ K) - (λ_{i} - Δ λ_{i}) M] (ϕ_{i} - Δ ϕ_{i}) = 0

(2)

where $Δ K, Δ λ, Δ ϕ$ are the variation values of system stiffness, eigenvalue, and mode shape. If the higher-order term is ignored, equation (2) then implies that:

(K - λ_{i} M) Δ ϕ_{i} = Δ λ_{i} M ϕ_{i} - Δ K ϕ_{i}

(3)

Since the target mode shape partitions are linearly independent, the variation of the ith mode shape ( $Δ ϕ_{i}$ ) can be expressed as a linear combination of the mode shapes of the structure ( $ϕ_{r}, r = 1, 2, 3, \dots, n$ ) in the initial state:

Δ ϕ_{i} = \sum_{r = 1}^{n} k_{ir} ϕ_{r}

(4)

where $k_{ir}$ represents the mode variation participation factor of the rth mode. The calculation process of $k_{ir}$ can be divided into the following two cases:

(1) if $r \neq i$ , premultiplying both sides of equation (3) by $ϕ_{r}^{T}$ produces:

\sum_{r = 1}^{n} k_{ir} ϕ_{r}^{T} (K - λ_{i} M) ϕ_{r} = Δ λ_{i} ϕ_{r}^{T} M ϕ_{i} - ϕ_{r}^{T} Δ K ϕ_{i}

(5)

According to the orthogonalization principle of the vibration mode, it is obvious that: $ϕ_{r}^{T} K ϕ_{i} = 0, ϕ_{r}^{T} M ϕ_{i} = 0, ϕ_{r}^{T} K ϕ_{r} = λ_{r}, ϕ_{r}^{T} M ϕ_{r} = I .$ The form of equation (5) implies that:

k_{ir} = \frac{ϕ_{r}^{T} Δ K ϕ_{i}}{λ_{i} - λ_{r}}

(6)

Combining equations (6) and (4), implying:

Δ ϕ_{i} = \sum_{\binom{r = 1}{\begin{matrix} r \neq i \end{matrix}}}^{n} \frac{ϕ_{r}^{T} Δ K ϕ_{i}}{λ_{i} - λ_{r}} ϕ_{r}

(7)

(2) if $r = i$ , then $k_{ir} = 0$ . The perturbation of the stiffness of each element can be expressed as:

Δ K = \sum_{k = 1}^{Q} K_{k} d_{k}

(8)

where Q is the number of elements; $K_{k}$ and $d_{k}$ are stiffness matrix and stiffness damage coefficient of the kth element, respectively. Combining equations (7) and (8), implying:

Δ ϕ_{i} = \sum_{k = 1}^{L} (d_{k} \sum_{\binom{r = 1}{\begin{matrix} r \neq i \end{matrix}}}^{n} \frac{ϕ_{r}^{T} K_{k} ϕ_{i}}{λ_{i} - λ_{r}} ϕ_{r}) = S_{i} Ω

(9)

By definition, $S_{i}$ means the sensitivity of the ith mode to damage, $Ω$ represents the damage vector of each element. The quantities $S_{i}$ and $Ω$ can be written as follows:

S_{i} = (\sum_{\binom{r = 1}{\begin{matrix} r \neq i \end{matrix}}}^{n} \frac{ϕ_{r}^{T} K_{1} ϕ_{i}}{λ_{i} - λ_{r}} ϕ_{r}, \sum_{\binom{r = 1}{\begin{matrix} r \neq i \end{matrix}}}^{n} \frac{ϕ_{r}^{T} K_{2} ϕ_{i}}{λ_{i} - λ_{r}} ϕ_{r}, \dots, \sum_{\binom{r = 1}{\begin{matrix} r \neq i \end{matrix}}}^{n} \frac{ϕ_{r}^{T} K_{n} ϕ_{i}}{λ_{i} - λ_{r}} ϕ_{r})

(10)

Ω = {[\begin{matrix} d_{1} & d_{2} & \dots & d_{Q} \end{matrix}]}^{T}

(11)

According to mode superposition method, the dynamic response of the damaged structure can be expressed as:

Y_{D} = (Φ + Δ Φ) q = (Φ S) (q α q)^{T}

(12)

where $α = diag {Ω, Ω, \dots, Ω}$ is a diagonal matrix and q is the coefficient response vector.

By definition, the sensitivity matrix is $Γ = (Φ S), θ = {(q α q)}^{T}$ . The form of $Y_{D}$ reduced to:

Y_{D} = Γ θ

(13)

Under the influence of measurement noise, equation (13) implies that:

Y_{d} = Γ θ + γ

(14)

where $γ$ denotes the vector of measurement noise, which is assumed to be zero-mean Gaussian noise with a variance of $σ^{2}$ . When the number of actual sensors is less than the number of candidate measurement points, the best state estimation is needed to keep as much linearly independent information as possible. Assuming that the process is unbiased estimation, the covariance matrix of the estimated error is given by:

J = E [(θ - \bar{θ}) {(θ - \bar{θ})}^{T}] = {[\frac{1}{σ^{2}} Γ^{T} Γ]}^{- 1} = \frac{1}{σ^{2}} Q^{- 1}

(15)

in which E denotes the expected value.

Q_{Γ} = Γ^{T} Γ

(16)

where $Q_{Γ}$ is the Fisher information matrix (FIM).

Maximizing $Q_{Γ}$ could contribute to the minimization of the covariance matrix and the best state estimate $\hat{θ}$ . FIM not only reflects the contribution of each DOF to the linear independence of the target mode but contains the sensitivity information of the structural damage.

The vibration energy of each test freedom is another important factor to be considered in the optimal arrangement of sensors.

If the sensor is placed in the position where the modal kinetic energy is small, the signal-to-noise ratio of the measurement point will be a low level, which is not conducive to the accurate identification of parameters. The effective independence (EfI) method does not consider the energy factor, while the modal kinetic energy (MKE) method and the modal strain energy (MSE) method aim to arrange the sensors at the places with larger kinetic energy or strain energy. These three methods may ignore some measuring points with small energy, resulting in the lack of information of some important measuring points, making the characteristics of the structure incomplete. The element energy method considers the influence of the structural stiffness on the energy of the measuring points, so the energy of the element stiffness is used to weight the information matrix, so as to overcome the situation that the energy information of the measuring points is less. Therefore, the element energy method is used to further modify the FIM.

According to the principle of the element energy method, the contribution of each DOF to the MUE_i can be expressed in the form:

\begin{matrix} MU E_{i} = ϕ_{i}^{T} \frac{K}{M} ϕ_{i} = ϕ_{i}^{T} [\frac{k_{1}}{m_{1}}, \frac{k_{2}}{m_{2}}, \dots, \frac{k_{n}}{m_{n}}] ϕ_{i} \\ = \sum_{j = 1}^{n} ϕ_{i}^{T} \frac{k_{j}}{m_{j}} ϕ_{ij} \end{matrix}

(17)

where MUE_i is the modal kinetic energy of the unit stiffness of the ith mode, K is the stiffness matrix of the structure, $ϕ_{i}$ is the ith modal shape, n is the number of structural degrees of freedom, and $ϕ_{ij}$ is the vibration mode value of the jth DOF in the ith mode. The contribution of the jth DOF to the modal kinetic energy of the unit stiffness of the ith mode can then be obtained as:

MU E_{ij} = ϕ_{i}^{T} \frac{k_{j}}{m_{j}} ϕ_{ij}

(18)

By definition, the energy coefficient of each candidate measurement point can be given as:

C_{E C_{j}} = \frac{1}{N} \sum_{i = 1}^{N} ϕ_{i}^{T} \frac{k_{j}}{m_{j}} ϕ_{ij}

(19)

C_{EC} = diag (C_{EC 1}, C_{E C_{2}}, \dots, C_{E C_{n}})

(20)

where N is the total number of modes selected. The sensitivity matrix $Γ$ is corrected by equation (20):

\bar{Γ} = Γ C_{EC}

(21)

Then form of $Q_{Γ}$ is corrected to:

{\bar{Q}}_{Γ} = {\bar{Γ}}^{T} \bar{Γ}

(22)

The modified $\bar{Q_{Γ}}$ and $\bar{Γ}$ not only reflect the contribution of each DOF to the independence of mode shape partitions, but include the magnitude of modal kinetic energy of the unit stiffness of each DOF and the sensitivity information of structural damage.

In the calculation process of large spatial structure, there are high requirements for fine mesh division of finite element model. The number of nodes or degrees of freedom of the finite element model can reach tens of thousands when the grid is fined meshed. In this case, two relatively close spatial degrees of freedom may contribute a lot to the target matrix, but the information provided by both is often repeated. By using the above algorithm to gradually select candidate nodes or degrees of freedom that contribute to the target matrix, it is inevitable that the information provided by the selected points is redundant. The solution to this problem is increase the size of the grid. However, rough grid is likely to make the sensor layout lose its best position. Therefore, this problem is solved by introducing a distance coefficient into the algorithm.

The information matrix specifying the kth candidate point is:

A^{k} = {\bar{Γ}}_{k}^{T} {\bar{Γ}}_{k}

(23)

where $\bar{Γ_{k}}$ is the sensitivity matrix corresponding to the kth measuring point, $A_{k}$ is a symmetric matrix. The closer the information collected from two measuring points is, the more similar the information matrix is. In this paper, the information independence of two measuring points or degrees of freedom is evaluated by introducing European distance, defining (He et al., 2013):

d_{kl} = \sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{m} {| A_{ij}^{k} - A_{ij}^{l} |}^{2}}

(24)

where d_kl represents the spatial difference of information matrix corresponding to measuring point k and l; m represents the number of modal shapes to be identified. In order to introduce the weight coefficient more intuitively, it is necessary to standardize the coefficient. Assuming that the maximum Euclidean distance in a set of candidate measuring points is d_max, the standardized Euclidean distance D_kl can be expressed as follows:

D_{kl} = \frac{d_{kl}}{d_{max}} = \frac{\sqrt{\sum_{i = 1}^{m} \sum_{j = 1}^{m} {| A_{ij}^{k} - A_{ij}^{l} |}^{2}}}{d_{max}}

(25)

For any two points in the layout scheme, the following condition is met:

0 \leq D_{kl} \leq 1, \forall k, l

(26)

When the information matrix corresponding to two measuring points is identical, D_kl takes the minimum value of 0; when the information matrix corresponding to two measuring points is completely independent, D_kl takes the maximum value of 1. For any measuring point k, the weight coefficient of the information matrix is defined as:

R_{k} = min (D_{ks}), \forall s

(27)

where s represents all selected sensor measuring points in the scheme.

Premultiplying left side of equation (22) by R_k produces:

{\bar{Q}}_{Γ}' = R_{k} {\bar{Γ}}_{k}^{T} {\bar{Γ}}_{k}

(28)

It can be seen that the sensor arrangement scheme which maximizes the modified information matrix $\bar{Q_{Γ}}'$ is the best scheme corresponding to this algorithm. It should be pointed out that the information matrix in different directions cannot be compared in the calculation process, because even if the information matrix corresponding to two measuring points in different directions is very close, there is still no information redundancy between them. Therefore, if the sensor layout is to be optimized in multiple directions, the candidate measurement points or degrees of freedom in each direction should be optimized separately.

Optimization method

As mentioned above, in order to obtain the best estimate of the parameter θ, the 2-norm of the information matrix $\bar{Q_{Γ}}'$ should be maximized. In order to ensure the stability of the solution of the equations, the 2-condition of the sensitivity matrix $\bar{Γ}$ should be minimized. The 2-norm of information matrix $\bar{Q_{Γ}}'$ and the 2-condition of sensitivity matrix $\bar{Γ}$ can be expressed by the following formula respectively:

‖ {\bar{Q}}_{Γ}' ‖_{2} = ‖ R_{k} {\bar{Γ}}^{T} \bar{Γ} ‖_{2} = R_{k} ‖ \bar{Γ} ‖_{2}^{2} = R_{k} λ_{max} ({\bar{Γ}}^{T} \bar{Γ})

(29)

con d_{2} (\bar{Γ}) = ‖ \bar{Γ} ‖_{2} ‖ {\bar{Γ}}^{- 1} ‖_{2} = \sqrt{λ_{max} ({\bar{Γ}}^{T} \bar{Γ}) / λ_{min} ({\bar{Γ}}^{T} \bar{Γ})}

(30)

The relationship between 2-norm of $\bar{Q_{Γ}}'$ and 2-condition of $\bar{Γ}$ can be obtained by singular value decomposition (SVD) of matrix. The SVD of matrix can be written as:

\bar{Γ} = U [\begin{matrix} Σ_{1} & 0 \\ 0 & Σ_{2} \end{matrix}] V^{T}

(31)

where $\sum_{1} = diag (σ_{1}, σ_{2}, \dots, σ_{r})$ , $\sum_{2} = 0$ , $σ_{i} = \sqrt{λ_{i} {\bar{Γ}}^{T} \bar{Γ}}$ is the nonzero eigenvalue of matrix ( $σ_{1} > σ_{2} > \dots > σ_{r}$ ).

Equation (29) then implies that:

‖ {\bar{Q}}_{Γ}' ‖_{2} = R_{k} λ_{max} ({\bar{Γ}}^{T} \bar{Γ}) = R_{k} σ_{1}^{2}

(32)

Equation (30) then implies that:

{cond}_{2} (\bar{Γ}) = σ_{1} / σ_{r}

(33)

It can be seen from equations (32) and (33) that when the 2-norm takes the maximum value, the 2-condition may not be the smallest. To maximize the 2-norm of information matrix $\bar{Q_{Γ}}'$ and minimize the 2-condition of sensitivity matrix $\bar{Γ}$ , the following objective function is defined to coordinate the two optimization objectives:

f = \max {w ‖ {\bar{Q}}_{Γ} ‖_{2} + (1 - w) / {cond}_{2} (\bar{Γ})}

(34)

where $ω$ is the weight coefficient. It can be considered that the 2-norm of information matrix $\bar{Q_{Γ}}'$ is equally important to the 2-condition of sensitivity matrix $\bar{Γ}$ , so that $ω$ can be taken as 0.5.

The target function f takes the maximum value only when $‖ \bar{Q_{Γ}}' ‖_{2}$ and $con d_{2} (\bar{Γ})$ take the maximum value at the same time. In order to prevent the information inundation caused by the large difference between the two values, the following methods are adopted to normalize:

ζ_{ij} = \frac{τ_{ij} - τ_{j}^{min}}{τ_{j}^{max} - τ_{j}^{min}} (i = 1, 2, \dots, n; j = 1, 2)

(35)

where $j = 1$ means the objective function is $‖ \bar{Q_{Γ}}' ‖_{2}$ , $j = 2$ means the objective function is $con d_{2} (\bar{Γ})$ . $τ_{ij}$ is the value of $‖ \bar{Q_{Γ}}' ‖_{2}$ and $con d_{2} (\bar{Γ})$ after selecting the ith DOF. $τ_{j}^{\max}$ and $τ_{j}^{\min}$ are the maximum and minimum values corresponding to $τ_{ij}$ , respectively.

Combining equations (34) and (35), implying:

\begin{matrix} f = max {ζ_{i 1} ({‖ {\bar{Q}}_{r}' ‖}_{2}) + ζ_{i 2} (1 / {cond}_{2} (\bar{Γ}))} \\ = max {ζ_{i 1} (R_{k} λ_{max} ({\bar{Γ}}^{T} \bar{Γ})) + ζ_{i 2} (1 / {cond}_{2} (\bar{Γ})} \end{matrix}

(36)

Algorithm implementation logic

Traditional calculation methods include inference algorithm, random algorithm, and serial method. The serial method involves generalized genetic algorithm and generalized eliminative algorithm. Considering that the series method is relatively mature and the number of measuring points of the large-scale structure is far less than the total free degree of the structure, the method of generalized genetic algorithm is more effective than generalized eliminative algorithm. Suppose there are N candidate degrees of freedom, m target modes to be identified, and n sensors to be arranged. The D-MOIF algorithm is illustrated in Figure 2, and the key steps are as follows:

Step 1: Calculating the stiffness matrix and mode matrix of the structure;

Step 2: Calculating the damage sensitivity matrix $S$ , the energy coefficient matrix $C_{EC}$ , the sensitivity matrix $\bar{Γ}$ , and the modified information matrix $\bar{Q_{Γ}}'$ by equations (10), (20), (21), and (28), respectively;

Step 3: Doing loop calculations in all DOFs and sort by the value of the objective function f. Selecting the DOF corresponding to f to obtain the maximum value as the position of the first sensor;

Step 4: Calculating the Euclidean distance $D_{kl}$ between the remaining (N−1) candidate degrees of freedom and the first measuring point according to formula (25), where $d_{\max}$ is the maximum of the Euclidean distance between the (N−1) candidate degrees of freedom and the first measuring point. Since only one measuring point is selected, the distance coefficient $R_{k} = D_{k 1}$ ;

Step 5: Calculating the value of f corresponding to the first measuring point and each remaining candidate DOF by the equation:

\begin{matrix} T_{k}^{(2)} = f_{1} + max \\ {ζ_{i 1} (R_{k} λ_{max} ({\bar{Γ}}_{k}^{T} {\bar{Γ}}_{k})) + ζ_{i 2} (1 / {cond}_{2} ({\bar{Γ}}_{k}))} \end{matrix}

(37)

Figure 2.

Flowchart for OSP based on D-MOIF.

The candidate DOF that maximizes $T_{k}^{(2)}$ is used as the second measurement point of the sensor;

Step 6: Calculating the Euclidean distance $D_{kl}$ between the remaining (N−2) candidate degrees of freedom and the second measuring point according to formula (25), where $d_{\max}$ is the maximum of the Euclidean distance between the (N−2) candidate degrees of freedom and the second measuring point. If $D_{k 1} < R_{k}$ , then update $R_{k} = D_{k 2}$ . Otherwise, $R_{k}$ remains unchanged;

Step 7: Calculating the value of f corresponding to the first two measuring points and each remaining candidate DOF by the equation:

\begin{matrix} T_{k}^{(3)} = f_{1} + ζ_{i 1} (R_{2} λ_{max} ({\bar{Γ}}_{2}^{T} {\bar{Γ}}_{2})) + ζ_{i 2} (1 / {cond}_{2} ({\bar{Γ}}_{2})) \\ + max {ζ_{i 1} (R_{2} λ_{max} ({\bar{Γ}}_{k}^{T} {\bar{Γ}}_{k})) + ζ_{i 2} (1 / {cond}_{2} ({\bar{Γ}}_{k}))} \end{matrix}

(38)

The candidate DOF that maximizes $T_{k}^{(3)}$ is used as the third measurement point of the sensor;

Step 8: When s measurement points have been selected by the algorithm, (N−s) candidate degrees of freedom are still left. Calculating the Euclidean distance $D_{kl}$ between the remaining (N−s) candidate degrees of freedom and the sth measuring point according to formula (25), where $d_{\max}$ is the maximum of the Euclidean distance between the (N−s) candidate degrees of freedom and the sth measuring point. If $D_{ks} < R_{k}$ , then update $R_{k} = D_{ks}$ . Otherwise, $R_{k}$ remains unchanged;

Step 9: Calculating the value of f corresponding to the first s measuring points and each remaining candidate DOF by the equation:

\begin{matrix} T_{k}^{(s + 1)} = \sum_{i = 1}^{s} ζ_{i 1} (R_{i} λ_{max} ({\bar{Γ}}_{i}^{T} {\bar{Γ}}_{i})) + ζ_{i 2} (1 / {cond}_{2} ({\bar{Γ}}_{i})) \\ + ζ_{i 1} (R_{k} λ_{max} ({\bar{Γ}}_{k}^{T} {\bar{Γ}}_{k})) + ζ_{i 2} (1 / {cond}_{2} ({\bar{Γ}}_{k})) \end{matrix}

(39)

Step 10: Repeating steps 8 to 9 until N sensors are selected. So far, the sensor placement scheme based on the D-MOIF algorithm is obtained.

Application

Target modes and candidate DOFs

Baihetan arch dam is a large-span double curvature arch dam with advanced technology. The dam height reaches 289 m, which is the largest hydropower station in the world. Therefore, Baihetan arch dam is selected as the research object. In order to test the accuracy of the objective function and guide the improvement of the next step, numerical calculations are selected as analytical tools. The full-scale three-dimensional finite element model of arch dam is shown in Figure 3. The dam model consists of 25,776 eight–node solid elements and 39,410 nodes, and the mesh size is approximately 2 m in the vertical direction. In this paper, the main mechanical properties of concrete are defined as follows: (1) for the dam concrete, average mass density ρ = 2400 kg/m³, the Poisson’s ratio γ = 0.167, the static elastic modulus E = 24 GPa. The standard value of dynamic elastic modulus can be increased by 50% compared with the standard static value; (2) for the foundation rock, average mass density ρ = 2800 kg/m³, the initial modulus E = 26 GPa, and Poisson’s ratio γ = 0.24. The damping of the dam material is assumed to be Rayleigh type and the damping ratio is 5%.

Figure 3.

The finite element model of the Baihetan arch dam.

Because the vibration of arch dam is mainly along the river flow direction (Y direction), without loss of generality, this paper focuses on the OSP along the Y direction, and 1226 DOFs of the downstream face of the arch dam are selected as the candidate DOFs. The low-order modes of large-scale hydraulic structures have large mode participation coefficients, which can usually describe the dynamic characteristics of structural systems. In addition, in practical engineering, the higher frequency modes of large structures is difficult to measure. Therefore, the first 6 modes of the high arch dam are selected as the target modes. In order to compare the effects of the proposed algorithm and several traditional algorithms, it is assumed that only 30 sensors are available. The first six modes of the structure are shown in Figure 4.

Figure 4.

First six frequencies and mode shapes: (a) first mode shape (1.52 Hz), (b) second mode shape (1.66 Hz), (c) third mode shape (2.16 Hz), (d) fourth mode shape (2.72 Hz), (e) fifth mode shape (2.97 Hz), and (f) sixth mode shape (3.08 Hz).

Contrast and validation

In this section, the algorithm D-MOIF proposed in this paper and four mature algorithms EfI, DPR, DS, D-EfI are used to optimize the layout of sensors for the arch dam. The first six modes corresponding to the candidate degrees of freedom are regularized and the layout results are shown in Figure 5. It can be seen that, according to the algorithm EfI, DPR, and DS, the positions of sensors are relatively concentrated, and the problem of redundant information appears to some extent. Another way to think of this is that selecting these sensors is nearly equivalent to selecting some of them. By comparison, the two algorithms D-EfI and D-MOIF that introduce the distance coefficient effectively avoids this problem, indicating that the validity of introducing distance coefficient.

Figure 5.

Sensor placement scheme obtained by the five algorithms: (a) EfI, (b) MUE, (c) DS, (d) D-EfI, and (e) D-MOIF.

Table 1 shows sensor location and the corresponding distribution coefficient of each DOF when using the five algorithms to optimize the sensor layout on the downstream face of the arch dam. It can be found that although the optimal order of the positions of the test points obtained by different algorithms is different, the DOFs in the top of the order are preserved.

Table 1.

Comparisons of sensor placement results with different criterion.

ID	DOF	EfI distribution coefficient	DOF	MUE distribution coefficient	DOF	DS distribution coefficient	DOF	D-EfI distribution coefficient	DOF	D-MOIF distribution coefficient
1	4938y	0.4726	5103y	0.3885	4131y	0.5469	5435y	0.5292	5435y	0.6209
2	4015y	0.4697	4015y	0.3801	3916y	0.5295	4938y	0.5265	4938y	0.6133
3	4016y	0.4366	4016y	0.3767	3917y	0.5237	4015y	0.5109	4015y	0.6028
4	4131y	0.4292	4131y	0.3748	3918y	0.4992	4131y	0.494	4131y	0.5980
5	3917y	0.4214	3916y	0.3693	4189y	0.4904	3918y	0.4923	3918y	0.5894
6	3918y	0.4186	3917y	0.3551	477y	0.4862	389y	0.4874	478y	0.5636
7	4189y	0.4068	3918y	0.3524	478y	0.4775	709y	0.4832	554y	0.5622
8	477y	0.4059	4189y	0.3477	553y	0.4610	834y	0.4818	833y	0.542
9	478y	0.4030	477y	0.3317	554y	0.4512	1489y	0.4728	1424y	0.5265
10	389y	0.3991	478y	0.3307	709y	0.4366	2385y	0.456	1770y	0.5263
11	553y	0.3926	389y	0.3153	710y	0.4273	2947y	0.4437	2385y	0.5095
12	554y	0.3903	553y	0.2919	833y	0.4188	4150y	0.4383	2787y	0.4955
13	709y	0.3744	554y	0.2833	834y	0.4167	573y	0.4164	4892y	0.4858
14	710y	0.3517	709y	0.2784	1423y	0.4160	1379y	0.4154	4087y	0.478
15	833y	0.3495	710y	0.2733	4167y	0.3995	2121y	0.4113	3845y	0.4681
16	834y	0.3379	833y	0.2726	523y	0.3942	4899y	0.4049	500y	0.468
17	1423y	0.3362	834y	0.2610	499y	0.3920	4081y	0.4043	856y	0.4589
18	1490y	0.3212	1424y	0.2518	385y	0.3625	3851y	0.398	1849y	0.4395
19	2385y	0.3105	1489y	0.2257	575y	0.3552	4173y	0.3864	2934y	0.4309
20	4668y	0.3044	1490y	0.2176	4169y	0.3417	494y	0.3832	4150y	0.4211
21	1400y	0.2939	2385y	0.2171	522y	0.3322	751y	0.3666	573y	0.3988
22	3846y	0.2937	862y	0.1991	498y	0.3145	1406y	0.3636	2121y	0.3986
23	496y	0.2801	4151y	0.1911	384y	0.3135	1856y	0.3585	4173y	0.3911
24	382y	0.2704	3873y	0.1904	574y	0.3108	3879y	0.3456	751y	0.3893
25	4173y	0.2641	4170y	0.1902	598y	0.3045	375y	0.3427	566y	0.3487
26	518y	0.2527	521y	0.1893	738y	0.2916	511y	0.3115	4075y	0.3446
27	494y	0.2470	497y	0.1872	3874y	0.2900	724y	0.3099	1412y	0.3283
28	380y	0.2337	383y	0.1740	4171y	0.2894	369y	0.2802	511y	0.3274
29	570y	0.2306	573y	0.1606	382y	0.2832	872y	0.2759	724y	0.3159
30	734y	0.2217	597y	0.1576	596y	0.2715	3000y	0.2549	369y	0.3013

Evaluation

In this paper, the above five sensor layout schemes are further evaluated by the following three evaluation criteria.

(1) Modal assurance criterion (MAC)

The modal vector of structure is orthogonal in theory. However, due to the limited number of measurement points, the limitation of the accuracy of measurement point identification, and the influence of noise and other factors, the modal vector obtained can not satisfy the orthogonality. Under some uncertain conditions, the small intersection angle between the modal vectors may lead to the loss of some modes information in the test. MAC is a good criterion for evaluating the space angle, which is defined as:

MA C_{ij} = \frac{{(ϕ_{i}^{T} ϕ_{j})}^{2}}{(ϕ_{i}^{T} ϕ_{i}) (ϕ_{j}^{T} ϕ_{j})}

(40)

where $ϕ_{i}$ and $ϕ_{j}$ are the ith and jth mode shapes. Mac_ij = 0 means the ith modal shape and the jth modal shape are orthogonal, and the two modal shapes are easy to distinguish; Mac_ij = 1 means the angle between the ith modal vector and the jth modal vector is equal to zero, which is difficult to distinguish. The closer the value of the non diagonal element of MAC matrix is to 0, the less the correlation between the two mode shapes is, and the better the measuring points can reflect the dynamic characteristics. Therefore, the requirement of measuring point arrangement is to minimize the maximum non diagonal element of MAC matrix. Figure 6 shows that the optimal values of the MAC obtained by the five algorithms.

Figure 6.

The optimal values of the MAC obtained by the five algorithms: (a) EfI, (b) MUE, (c) DS, (d) D-EfI, and (e) D-MOIF.

In order to describe the modal independence of different sensor layout schemes more clearly, the average value of the non diagonal elements of the MAC matrix (MAC-AVE) and maximum value of the non diagonal elements of the MAC matrix (MAC-MAX) are selected as the evaluation indexes. The smaller the two values, the better the sensor layout scheme is. Figure 7 shows two evaluation indexes of five layout schemes. Obviously, the MAC-AVE and MAC-MAX obtained by D-EfI algorithm and D-MOIF algorithm which considering the distance coefficient are far less than the other three algorithms, which once again illustrates that considering the distance coefficient for large-scale grid intensive structures is necessary. However, the MAC-AVE and MAC-MAX obtained by using the D-MOIF algorithm proposed in this paper are slightly larger than that of the D-EfI algorithm, indicating that the proposed algorithm takes into account more factors, resulting in a lower modal independence than the D-EfI algorithm which only considers mode shape. Therefore, this method still has some room for performance improvement.

(2) FIM evaluation criterion

Figure 7.

Two evaluation indexes of five layout schemes: (a) MAC-AVE, and (b) MAC-MAX.

FIM evaluation criteria can be defined as the ratio of the determinant value of the FIM obtained by part of the freedom of the structure to the determinant value of the FIM obtained by all the freedom of the structure, which can be expressed as:

η = \frac{\det ({\bar{Q}}^{'})}{\det ({\bar{Q}}^{'}_{0})} * 100 %

(41)

where ${\bar{Q}}_{0}'$ and $\bar{Q}'$ indicate the FIM obtained by part of the freedom and all the freedom of the structure. The larger $η$ is, the more information the measuring point contains, the better the sensor arrangement. Figure 8 presents the $η$ of five layout schemes. It can be found that the a obtained by the D-MOIF algorithm proposed in this paper is significantly larger than the others. This discovery further confirms that the D-MOIF algorithm can be used to find the placement position of sensors from multiple dimensions, so that the obtained monitoring results contain more information. Meanwhile, this layout scheme can provide more help for damage monitoring.

(3) Singular value ratio evaluation criteria

Figure 8.

FIM evaluation criterion of five layout schemes.

The singular value decomposition of eigenvector matrix based on the DOFs can be used to determine the applicability of the selected measurement locations. In this method, the ratio of the maximum singular value to the minimum singular value of the eigenvector matrix is simply obtained. The closer the ratio is to 1, the better the positions of the measuring points are. The larger the ratio, the worse the positions of the measuring points. The singular value ratio evaluation criteria can be described as:

S = \frac{σ_{max}}{σ_{min}}

(42)

where $σ_{\max}$ and $σ_{mim}$ are the largest and smallest singular value of the eigenvector matrix.

It can be seen from the Figure 9 that the maximum singular value ratio of the proposed sensor optimization algorithm D-MOIF is the smallest of the five algorithms, which implies that the algorithm is the most stable for the calculation of information matrix, and also shows that the optimization method introduced in this paper plays a positive role in the stability of sensor layout.

Figure 9.

Singular value ratio of five layout schemes.

Optimal number of sensors

The reasonable number of sensors in the research of OSP is a crucial problem. If the number of sensors is too large, the dynamic time-varying information of the structure may be collected more comprehensively. However, the monitoring cost of the whole project will be greatly pressured by equipment layout, data transmission, data processing and other factors. In addition, the redundant sensors will also cause information redundancy and more invalid acquisition, resulting in low identification accuracy and unstable identification results. On the contrary, if the number of sensors is too small, the dynamic information of the structure may not be truly reflected by the whole sensor network, and the monitoring accuracy will be affected accordingly. Combining with the specific mathematical expression of the objective function in this paper, the mode of gradually increasing the number of sensor layouts and repeatedly calculating are used to find the optimal number of sensor layouts. Set the number of sensors from 1 to 50 and the increment step is 1, and record the change of objective function after each trial calculation. As can be seen from Figure 10, with the increase of the number of sensors, the objective function value increases steadily. Especially in the initial stage of the trial calculation, the objective function increases roughly with a higher slope, but when the number of sensors increases to 26, the growth slope of the objective function decreases, and keeps decreasing. The reason is that with the increase of the number of sensors, the global effectiveness of the signal collected by the additional sensors to the overall dynamic characteristics of the structure decreases. Mathematically speaking, the linear correlation between the modal information collected by the added sensor and the modal information collected before is large. Therefore, the number of sensors corresponding to the growth turning point of the objective function is the optimal number of sensors in this paper. Besides, Figure 11 shows the ratio of Fisher determinants of the five OSP algorithms as the number of sensors increases. Obviously, no matter how many sensors are deployed, compared with the other four algorithms, the D-MOIF algorithm can reflect the state information of the structure to a greater extent, and the optimal number of sensors corresponding to D-MOIF is the least of the five optimization algorithms.

Figure 10.

FIM evaluation criterion of five layout schemes.

Figure 11.

The relationship of objective function value and the number of sensors.

To sum up, the D-MOIF proposed in this paper plays an improvement over single-objective OSP methods, and the solution determined by the D-MOIF method could provide reliable technical support for OSP problem for large-scale structures.

Conclusion

The single-objective OSP algorithm scarcely meets the needs of the SHM system sensor arrays, and the multi-objective OSP algorithm is the inevitable trend of future development. In this study, a novel and effective method named the distance coefficient-multi objective information fusion algorithm (D-MOIF), which is different from the conventional method and is easier to be implemented, is developed to select the best sensor location for large-scale structures. There are some innovations in this algorithm, such as an integrated information matrix, the Euclidean distance and a new objective function. The effectiveness of the algorithm is verified in detail by an arch dam example, and the results of the algorithm and four traditional algorithms are compared by three evaluation criteria. In addition, the optimal number of sensors corresponding to each algorithm is studied as well. Some main conclusions and recommendations of the study are summarized as follows:

Although previous algorithms (EfI, DPR, DS) could achieve a good estimate estimation of modal parameters, it ignores that the similar FIM may be generated by two degrees of freedom close to each other. The proposed European distance operator derived from the analytic geometry is effective to overcome the redundancy of information between the selected sensors.

D-EfI algorithm can effectively avoid the problem of information redundancy caused by the DOFs close to the space, but the sensor placement scheme based on this algorithm hardly meet the multiple monitoring needs of the SHM. In order to make the sensor layout scheme in the SHM system meet more requirements, an integrated information matrix is deduced from the structural motion equation, which includes information on mode independence, damage sensitivity and modal strain energy. Based on an example analysis, it is proved that the sensor layout scheme considering more monitoring requirements can be obtained by this method.

In order to balance the sensitivity and robustness of the algorithm at the same time, a new objective function with the criterion of maximum norm of the information matrix and minimum condition number of the sensitivity matrix is set up, which is certified effective to obtained a more stable OSP scheme.

Numerical studies on an arch dam is carried out to validate and demonstrate the efficacy of the proposed D-MOIF algorithm. Three classical evaluation criteria are used to compare the D-MOIF algorithm with four traditional sensor optimization methods. The results highlight that the new optimized objective function could balance the sensitivity and robustness of the algorithm at the same time and could generate superior sensor configurations for large-scale structures with a refined mesh compared to the conventional algorithm.

D-MOIF algorithm can get more comprehensive structure state information when the same number of sensors are deployed. Based on the quantity optimization analysis of the sensors, it is concluded that the optimal number of sensors corresponding to D-MOIF is the least of the five optimization algorithms.

To monitor the dynamic characteristics of the system in real time during the operation of the structure, the OSP has become a crucial task in the SHM system. The D-MOIF algorithm developed in this paper improves the optimal placement of sensors from multiple dimensions. It is worth noting that this method can also be used effectively for small space structures. This study could provide reliable guidance for engineers in the process of sensor deployment.

In this paper, a numerical simulation model verified by experiments conducted by the authors (Fan et al., 2009) is used to verify the numerical simulation of the sensor layout. Considering the algorithm could be verified more effectively through the experimental examples, an experimental study will be the further research work by the authors.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are indebted to the editor, and the referee, whose careful reading and constructive comments helped improving an earlier version of the paper. The authors are also grateful for the joint support of the National Key R&D Program of China (Grant No. 2017YFC0404900), National Natural Science Foundation of China (Grant No. 52079025) and the National Natural Science Foundation of China (Grant No. 51679030).

ORCID iD

Xiangyu Cao

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