A hybrid method of optimal sensor placement for dynamic response monitoring of hydro-structures

Abstract

A hybrid method for optimal sensor placement is introduced to position detection sensors on hydro-structures to maximize the monitored structural dynamic information. In the hybrid method, modal assurance criterion matrices and effective independence vectors were applied as the optimization principle for adding and eliminating potential sensor locations in coordinate sets. The energy revision factor was implemented to measure the modal strain energy and ensure the arrangement of sensors at the large energy locations. QR decomposition of the modal matrix was employed to decrease the influence of the initial sensor positions. A computational simulation of an arch dam model, considering 20 sensors and the first six orders of modal vibrations, was used to demonstrate the feasibility of the method. The advantages and disadvantages of existing methods were demonstrated by comparison criteria including the modal assurance criterion, modal kinetic energy criterion, Fisher information matrix, and root-mean square error criterion. The results showed that the method proposed in this article provides linear independent and orthogonal modal vectors, minimizing the relative mean square error of the measured vibration modes, which fundamentally indicates that the identified vibration characteristics of the concrete arch dam are accurate and reasonable.

Introduction

A hydro-structure, such as a dam, is a large-scale concrete water retaining structure. A dam is built for comprehensive utilization of hydropower resources in a valley with heights of approximately 100–300 m.^1,2 Hidden dangers can be developed in these structures due to a combination of factors including high water pressure, alternating temperature, and seismic and other environmental erosion factors. Therefore, structural health monitoring is critical in these projects to discover and resolve the hidden safety problems.^3–5 One of the research focusing in hydro-structural health monitoring is to study how a limited number of sensors can be arranged in the appropriate positions ⁶ to gain more accurate structural dynamic response information and evaluate damage^7–9 under artificial ¹⁰ or environmental excitation measurements^11–13 such as earthquakes, flow pulsation, and hydroelectric unit vibration.

In recent years, optimization techniques such as the modal assurance criterion (MAC) method, effective independence (EI) method, and modal strain energy (MSE) method have been developed to optimize sensor placement for better identification of the dynamic response of structures. The MAC method^14–17 is commonly used to obtain the optimal sensor placement (OSP). This method is based on obtaining a large space angle for the measuring mode vector within the measurement selection process and preserving the characteristics of the original model. The principle of the EI method, proposed by Chang and Pakzad ¹⁸ and Kammer, ¹⁹ is to select a set of sensor locations from a larger candidate set according to their contribution to the linear independence of the target modal partitions. The final sensor placement maximizes the determinant of the corresponding Fisher information matrix. Other algorithms, including the EI driving-point residue (EFI-DPR) algorithm, ²⁰ energy coefficient-EI (EC-EI) algorithm ²¹ and orthogonal triangular decomposition (QR decomposition), ²² have been proposed to improve upon the EI algorithm. By weighting the average MSE value at the sensor elements, the MSE method^23–25 arranges the monitoring sensors at locations with high strain energy. However, the results of the above-mentioned methods are affected by the finite element model meshing. Without considering the symmetry of the measuring points, information redundancy will occur.

Given the disadvantage that some OSP methods only consider a single aspect in the fitness function, a class of comprehensive methods has been proposed. A hybrid optimization strategy ²⁶ is developed to use the MSE method to conduct the initial sensor placement, while an adaptive genetic algorithm is utilized to determine the optimal number of sensors and their location. The root-mean-square of the off-diagonal elements in the MAC matrix is the fitness function. Another algorithm ²⁷ integrates the advantages of the kinetic energy method, EI method, MAC, and many other optimal methods. This method is proposed to avoid information redundancy and maximize the signal-to-noise ratio. Some stochastic optimization algorithms, such as genetic algorithms, ²⁸ the random elimination (RE) algorithm, and the heuristic random elimination (HRE) ²⁹ algorithm, have been introduced to solve the OSP problem. Generally, these hybrid methods, usually integrated with optimization algorithms, are better for use in large-scale computation of huge hydro-structures because of their improved calculation efficiency and overall searching capability. However, disadvantages such as too many iterations and low search reliability are still present.

The existing single methods are implemented based on assumptions that are ineffective and inefficient for sensor location arrangement. Therefore, a hybrid method is proposed in this article to determine the OSP. In this method, the QR decomposition of a modal matrix was used to decrease the influence of the initial sensor positions, and the MAC matrix and EI vector were applied as the optimization principle for arranging sensor locations. The energy revision factor was utilized to maximize the MSE energy of the sensor placement. Significant research on sensor placement optimization has been performed in recent years. However, most of these studies have been demonstrated on simple structures, such as boards, beams, columns, simple trusses, and bridges. Insufficient research has been performed on dynamic response optimization of large and complex structures, such as high arch dams, which are often located in strong earthquake zones and are under high stress due to the surrounding land. Therefore, another goal of this article is to introduce and extend the sensor placement optimization theory to the monitoring of high dam dynamic response.

Basic theories

Optimal sensor placement for hydro-structures

According to vibration theory, the modal vectors of the theoretical modal matrix should be independent. Assuming that the value of the model degrees of freedom is N, the number of vibration modes needed to be identified is $m$ and the number of sensors to be optimized is $n$ , and based on the superposition theory of modal parameters, the structural response expression of the sensor placement is as follows

$$\mathbf{y}=\mathbf{\Phi }\mathbf{q}+\mathbf{\eta }=\sum _{i=1}^m{q}_i{\phi }_i+\eta$$

(1)

where $\mathbf{y}\in R^{n\times 1}$ is the output response of the structure, $\mathbf{\Phi }\in R^{n\times m}$ is the modal matrix, ${\phi }_i$ represents the $i\mathrm{th}$ vibration mode of the structure, $q_i\in R^{m\times 1}$ is the target mode coordinate, which represents the $i\mathrm{th}$ coordinate of the modal parameter, $\eta$ represents Gaussian white noise, and the effect of white noise on each sensor is assumed to be independent with variance ${\sigma }^2$ .

The purpose of OSP in hydraulic structure engineering is to obtain the best estimate of the target modal coordinates $q$ and identify the representational structural vibrational characteristics, such as inherent natural frequency ${\omega }_i$ , modal damping ratio ${\zeta }_i$ , and modal vibration mode ${\phi }_i$ . Therefore, for the selected frequency range ${\omega }_{\alpha }~{\omega }_{\beta }$ , $y(\omega )$ and $\tilde{y}(\omega )$ are the test value and the calculated value of the structural response, respectively. OSP for the vibration response of hydraulic structures can be expressed as searching for the optimal estimates of the target modal coordinate $q$ by minimizing the value of the objective function for the modal parameter identification as follows

$$f(x)=\min \sum _{i=1}^n{[\tilde{y}({\omega }_i,{\varsigma }_i,{\phi }_i)-y({\omega }_i,{\varsigma }_i,{\phi }_i)]}^2$$

(2)

where $f(x)$ is the objective function of OSP, $\tilde{y}({\omega }_i,{\varsigma }_i,{\phi }_i)$ is the identified modal parameter obtained by optimized sensors, $y({\omega }_i,{\varsigma }_i,{\phi }_i)$ is the actual modal information of the structure, and ${\omega }_i$ , ${\varsigma }_i$ , and ${\phi }_i$ are the inherent frequency, modal damping ratio, and modal vibration mode, respectively.

MAC matrix

According to the principle of structural dynamics, the natural frequencies and corresponding vibration modes that reflect the structure vibration characteristics are the eigenvalues and corresponding eigenvectors of the structural modal matrix, respectively. They can be formed with a set of orthogonal vectors by the values of the nodes of the natural structure vibration mode. For actual measurements, the number of measuring points is significantly less than the structural degrees of freedom of the numerical model. Additionally, the environmental noise and system measurement precision significantly affect the reliability of the measurements. Therefore, the structural modal vectors obtained commonly lack good orthogonality. Maximizing the obtained structural dynamic features and avoiding the loss of some important modes caused by a small modal vector space angle, the maximum space angle of the modal vectors is considered to be the objective function when optimizing sensor placement. Accordingly, the MAC matrix is introduced, and OSP is achieved by minimizing the values of the off-diagonal elements of the MAC matrix as follows

$$\mathrm{MA}{C}_{\mathrm{ij}}=\frac{{({\phi }_i^T{\phi }_j)}^2}{({\phi }_i^T{\phi }_j)({\phi }_i^T{\phi }_j)}$$

(3)

where ${\phi }_i$ and ${\phi }_j$ are the $i\mathrm{th}$ and $j\mathrm{th}$ modal vectors. Each MAC matrix element represents the cosine value of the intersection angle between two modal vectors. If two modal vectors are orthogonal, the two vectors are easily distinguished, and the corresponding MAC matrix off-diagonal element value is zero; if two vectors are parallel, the two vectors cannot be distinguished, and the corresponding MAC matrix off-diagonal element value is 1. Therefore, the OSP is achieved by minimizing the off-diagonal elements of the MAC matrix.

During the process of OSP based on the MAC matrix, a three-dimensional (3D) finite element model of the hydraulic structure is established, and the required structural modal information is computed by modal analysis with the finite element method (FEM) method; then, the MAC matrix of the initial measuring point group is calculated, and the remaining optional degrees of freedom of the structure are added to the initial point group successively as the MAC matrix is updated. Consequently, the measuring points with the minimum–maximum off-diagonal elements of the MAC matrix are selected and added to the initial point group; then, the iteration is repeated until the optimal dynamic sensor layout is obtained.

Diagonal element of the effective independence vector

To determine the best estimate of the modal coordinates $q$ , maximization of the matrix $\mathbf{Q}={\mathbf{\Phi }}^T\mathbf{\Phi }$ should be guaranteed. A matrix is defined to measure the contribution of alternative degrees of freedom to the linear independence of the objective modal matrix

$$\mathbf{E}=\mathbf{\Phi }\mathbf{ѱ}{\lambda }^{-1}(\mathbf{\Phi }\mathbf{ѱ}{)}^{-1}=\mathbf{\Phi }[{\mathbf{\Phi }}^T\mathbf{\Phi }{]}^{-1}{\mathbf{\Phi }}^T$$

(4)

where $\mathbf{E}$ is the defined matrix, $\lambda$ and $\mathbf{ѱ}$ are the eigenvalues and eigenvectors of the matrix $\mathbf{Q}$ , respectively, and $\mathbf{\Phi }\in R^{n\times m}$ is the modal matrix.

From equation (4), $\mathbf{E}$ is an idempotent matrix. The diagonal elements represent the contribution of the corresponding degrees of freedom to the linear independence of the target modal matrix; specifically, the $i\mathrm{th}$ diagonal element value indicates the contribution of the degrees of freedom to the linear independence of the matrix. The contribution of a matrix element to the linear independence of the matrix increases as the value of the corresponding diagonal element increases. This indicates that the corresponding matrix element is significant and should be reserved. Accordingly, points with lower diagonal element values should be removed. The diagonal elements of the matrix $\mathbf{E}$ can be expressed as a column vector

$${\mathbf{E}}_{\mathbf{diag}}=[{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n{]}^T$$

(5)

The elements of ${\mathbf{E}}_{\mathbf{diag}}$ should be sorted according to the size, and the sequence of the corresponding optional degrees of freedom is determined based on the diagonal element sequence. Then, the degrees of freedom with the lowest diagonal element values will be removed by an iterative method until the required control threshold is achieved.

From the derivation process of the methods detailed previously, if the obtained modal parameter vectors do not satisfy the largest space angle, orthogonality for all modal vectors cannot be ensured. In addition, similar to the MAC matrix method, considering the diagonal elements of only the EI vector may cause low-energy OSP selection and the loss of some important structural vibration information, without considering the inconsistent distribution of the structural energy.

Modification of the energy coefficient

To avoid selecting points with low energy as in the above methods, the energy modification coefficient ²⁶ is introduced to characterize the contribution of each candidate point to the structural MSE of each order.

By assuming the number of selected modal orders to be $m$ , the ECs of the structure candidate degrees of freedom can be defined as follows

$$\mathbf{H}=\mathrm{diag}(H_1,H_2,\dots ,H_m)$$

(6)

$$H_j=\frac{1}{m}\sum _{i=1}^m{\phi }_{\mathrm{ij}}^T{\mathit{k}}_j{\phi }_{\mathrm{ij}}$$

(7)

where $H_j(j=1,2,\dots ,m)$ are the ECs, $m$ is the number of selected modal orders, $\mathbf{H}$ is the diagonal matrix consisting of ECs, ${\phi }_{\mathrm{ij}}$ is the $i\mathrm{th}$ mode in the $j\mathrm{th}$ degree of freedom, and ${\mathit{k}}_j$ is the overall stiffness matrix in the $j\mathrm{th}$ degree of freedom.

The matrix $\mathbf{E}$ in equation (4) can be rewritten based on the EC matrix $\mathbf{H}$

$$\mathbf{E}=\mathbf{\Phi }[{\mathbf{\Phi }}^T\mathbf{\Phi }{]}^{-1}{\mathbf{\Phi }}^T\mathbf{H}$$

(8)

The corresponding candidate points with the minimum diagonal elements of $\mathbf{E}$ are removed successively by iterative calculation until the required number of measuring points is reached. The specific steps are as follows:

• Step 1. Establishing the finite element model, conducting the modal analysis, and determining the number of modals;

• Step 2. Calculating the effective independent distribution matrix by $\mathbf{E}=\mathbf{\Phi }[{\mathbf{\Phi }}^T\mathbf{\Phi }{]}^{-1}{\mathbf{\Phi }}^T$ ;

• Step 3. Calculating the contribution of each candidate point to the structural MSE of each order by $H_j=\frac{1}{m}\sum _{i=1}^m{\phi }_{\mathrm{ij}}^T{\mathit{k}}_j{\phi }_{\mathrm{ij}}$ ;

• Step 4. Modifying the effective independent distribution matrix $\mathbf{E}$ by $\mathbf{H}$ and computing the EI vector by $\mathbf{E}=\mathbf{\Phi }[{\mathbf{\Phi }}^T\mathbf{\Phi }{]}^{-1}{\mathbf{\Phi }}^T\mathbf{H}$ ;

• Step 5. Saving the modal parameters of the remaining points in the new modal matrix.

Steps 2–5 should be repeated until the required number of sensors is obtained.

Selection of the initial measuring points

To minimize the influence of the initial sensor layout scheme on sensor placement optimization and ensure the integrity of the obtained modal information, QR decomposition of the modal matrix was introduced to control the initial position of the sensor measuring points. Then, the measuring points with the minimum maximum off-diagonal MAC matrix element values were selected successively from the remaining degrees of freedom until reaching the threshold value and obtaining the OSP.

The measurable modal shape matrix of the established 3D finite element model was assumed to be $\mathbf{\Phi }(\mathbf{\Phi }\in R^{n\times m})$ . $m$ is usually less than $n$ , and $r(\mathbf{\Phi })=m$ ; specifically, $\mathbf{\Phi }$ is a full-rank matrix. According to matrix theory, a column vector subset of a matrix with larger norm can be obtained using QR decomposition, and the matrix of the decomposed subsets meets the requirements of a good matrix state. After QR decomposition of the column principal elements, a subset of column vectors are acquired. Therefore, the transpose matrix of the original modal matrix, specifically ${\mathbf{\Phi }}^T$ , was chosen to conduct QR decomposition. The decomposition expression is as follows

$$\begin{array}{c}{\mathbf{\Phi }}^T\mathbf{E}=\mathbf{QR}=\mathbf{Q}\left|\begin{array}{ccccc}{R}_{11}\hfill & ...\hfill & R_{1m}\hfill & ...\hfill & R_{1n}\hfill \\ \multicolumn{1}{c}{⋮}& \ddots \hfill & \hfill & \hfill & \hfill \\ \multicolumn{1}{c}{0}& ...\hfill & R_{\mathrm{mm}}\hfill & ...\hfill & R_{\mathrm{mn}}\hfill \end{array}\right|,\hfill \\ \mathbf{Q}\in {\mathbf{R}}^{m\times m},\mathbf{R}\in {\mathbf{R}}^{m\times n},\mathbf{E}\in {\mathbf{R}}^{n\times n}\hfill \\ R_{11}>R_{22}>\cdots >R_{\mathrm{mm}}\hfill \end{array}$$

(9)

where $\mathbf{E}$ is a permutation matrix, and according to the decomposition result, the corresponding row 1 of the decomposition matrix $\mathbf{R}$ from the original modal matrix $\mathbf{\Phi }$ is the subset with the greater row norm and good condition of the row vector group of matrix $\mathbf{\Phi }$ .

New analytical model

The implementation steps of the new hybrid method of sensor placement optimization for hydro-structures are as follows:

1. Establish a 3D finite element model of the dam structure and obtain a matrix of modal vectors through mode analysis to determine the focusing mode order value.

2. Use the MSE and EI methods to select all the degrees of freedom of the dam structure to determine the range of candidate sensors in the MAC method (assume n candidate sensors need to be deployed for measurements).

3. Perform QR decomposition for the modal matrix, acquire the initial sensor deployment, and calculate the corresponding MAC matrix and its maximum off-diagonal element value $\max$ .

4. Pick a point arbitrarily from n candidate points (assume the chosen point is measuring point k) and add it to the initial measurement points; then, calculate the new MAC matrix of the modal matrix and its maximum off-diagonal element value $\underset{k}{\max }$ .

5. Calculate the value of $f(k)=\underset{k}{\max }-\max$ .

6. Repeat steps 3–5 for each candidate measuring point or sensor and determine a set of values $f(i)i=1,2,\dots ,n$ ; then, add the measuring point to the initial sensor group corresponding to the minimum of $f(i)$ .

7. Replace the initial sensor group with the new group with the addition of one measurement point; repeat steps 3–6 until $f(i{)}_{\min }<0.25$ .

8. Visualize the sensor locations, remove the relatively nearby locations to avoid redundant information, and determine the sensor deployment arrangement. The quantitative standard of determining whether two sensors are nearby can be conducted by the following equation

$$\frac{\sum _{i=1}^n\underset{1\le j\le n}{\min }{\delta }_{\mathrm{ij}}}{(n-1)\underset{1\le j\le n,1\le j\le n}{\min }{\delta }_{\mathrm{ij}}}>3$$

(10)

where ${\delta }_{\mathrm{ij}}$ is the distance between the $i\mathrm{th}$ sensor and the $j\mathrm{th}$ sensor, $i$ and $j$ are the order of final sensors, $1\le j\le n$ , $1\le j\le n$ , $n$ is the number of sensors.

The calculation flow chart of the new MSE-EI-MAC hybrid method is shown in Figure 1.

Figure 1.

Flow chart of the new hybrid algorithm.

Numerical examples on an arch dam

Model information

A concrete arch dam, with a maximum dam height of 305 m, a crest width of 16 m, and a foundation thickness of 63 m, is used as an example. The dam is divided into 142,254 elements by HyperMesh software. The 3D FEM model of the arch dam is shown in Figure 2(a), and the model parameters of different materials are shown in Table 1.

Figure 2.

Mode shapes calculated using the finite element method (FEM): (a) 3D FEM model of an arch dam, (b) stationary state of the dam, (c) first-order modal shape, (d) second-order modal shape, (e) third-order modal shape, (f) fourth-order modal shape, (g) fifth-order modal shape, and (h) sixth-order modal shape.

Table 1.

Material parameters of the concrete arch dam for the FEM model.

No.	Material	Bulk density (kN/m3)	Elastic modulus (GPa)	Poisson’s ratio
1	Dam concrete	24.00	24.00	0.167
2	Rock II	28.00	25.00	0.250
3	Rock III 1	28.00	11.20	0.250
4	Rock III 2	28.00	6.50	0.300
5	Rock IV 1	27.50	1.40	0.350
6	Rock IV 2	27.50	0.40	0.350
7	Rock V 1	27.50	0.02	0.350
8	Pedestal concrete	24.00	2.00	0.167
9	Grouting concrete	27.50	0.40	0.350

FEM: finite element method.

The natural frequencies of a dam are influenced by the changing of environment loads, such as water level and temperature. Researches show that the impacts from the fluctuation of water level present in two aspects:^30,31 on one hand, a rising water level brings an increase in additional force upon dam, which results in reducing of natural frequencies of a structure; on the other hand, the joints and cracks in dam foundation may shut accounting for the increase in water pressure; in this case, the structural overall stiffness and corresponding natural frequencies will increase. Therefore, the changing relation between water level and hydro-structural natural frequencies is the integration of these two effects. For the environmental temperature factor, the natural frequencies will decrease when it increases, and the decrease in lower degree of frequencies is more obvious.^32,33 In order to minimize the influences of these factors, the numerical instance of OSP is conducted under the designed normal water level and long-time stable temperature field of the dam.

The structural modal analysis of the dam is performed using the ABAQUS finite element software. The natural frequencies of each order and corresponding structural mode shapes are shown in Figure 2 and Table 2.

Table 2.

Natural frequency.

Order	1	2	3	4	5	6
Frequency (Hz)	1.68	1.87	2.42	3.26	3.86	4.04

From the modal calculations, the participation factor of the arch dam in the first six orders of modal shapes is 0.96, and the structural vibrational energy in the downstream direction is greater than that in the cross-river direction and vertical direction. Therefore, the first six orders of modal shapes on the dam face of the downstream side are selected as the target modes for sensor placement optimization. Considering that the measuring point closer to the dam crest will have a larger vibration response and is more conducive to the identification of modal parameters, the degrees of freedom in the upper-half of the downstream dam face (758 total degrees of freedom) are selected as the candidate degrees of freedom for OSP.

Application of the new hybrid method

The new hybrid optimization algorithm is used to calculate the MAC matrices of the first six orders of modal displacement on the 758 candidate points. QR decomposition of the MAC matrices of the 758 measuring points is used to obtain six initial sensor positions and the corresponding MAC values of matrices. The results are shown in Figures 3 and 4. Considering the effectiveness of candidate points, from a total of 758 samples, 100 candidate points are selected based on the MSE-EI method. Then, the 100 candidate points are added sequentially to the initial measuring point group. It should be pointed out that the number of candidate points (namely, 100) is selected after comprehensive consideration of available degrees of freedom (758) and the size of a dynamical monitoring system in concrete arch dam (commonly 10–20 sensors). In the numerical example, an increase in sensor number brings a lower minimum off-diagonal value of MAC matrix. But a set of 100 candidate points is enough to get a satisfying sensor placement scheme.

Figure 3.

Six initial sensor locations.

Figure 4.

MAC values obtained by QR decomposition.

The iterative computation continues until the maximum off-diagonal element value of the MAC matrix exceeds the set threshold. The calculation of maximum value off-diagonal element value of the MAC matrix can be conducted by the method provided in “MAC matrix.” It is demonstrated that if the value is greater than 0.9, then the modal vectors are relevant and they are not easy to be resolved. However, if it is lower than 0.25, the two modal vectors can be considered as orthogonal to each other. ³⁴ Therefore, the threshold is picked as 0.25 in this numerical instance.

The changing curve of the maximum off-diagonal element of the MAC matrix is shown in Figure 5. The changing curve shows that the value begins to exceed the 0.25 threshold when the number of sensors reaches 20. Therefore, 20 locations are selected as the intermediate layout scheme of the sensors in the arch dam. Figure 6 is the preliminary sensor layout scheme.

Figure 5.

Maximum off-diagonal value of the MAC matrix.

Figure 6.

Preliminary sensor layout.

In the preliminary layout scheme, since some sensors near the dam crest are close to one another, the measured dynamic responses of these sensors are relatively similar. Therefore, the contribution of these close points to the variation of the MAC matrix maximum off-diagonal element value is calculated separately, and the sensor measurement location with a smaller contribution is removed. Meanwhile, to ensure that there are 20 total sensors, another sensor location from the candidate measuring point set is added to the set. The replaced sensor location should meet the two following requirements: minimizes the maximum off-diagonal element value of the ultimate MAC matrix and is less than the threshold value (0.25). After substituting these locations, the final sensor layout in the high arch dam is determined, as shown in Figure 7. The maximum off-diagonal element value of the MAC matrix is 0.249. The final MAC matrix of the optimal 20 sensors is shown in Figure 8.

Figure 7.

Final sensor layout after deployment optimization.

Figure 8.

Best MAC matrix after deployment optimization.

Result comparison

To investigate the effectiveness of the new hybrid method, the MAC, EI, and MSE-EI methods were also used with the same arch dam for result comparison. The final sensor layouts of the three methods are shown in Figures 9–11, and the corresponding MAC matrix is shown in Figures 12–14. In the MAC method, the maximum off-diagonal element of the MAC matrix is 0.279, and the method is feasible for OSP on the concrete arch dam. In the EI and MSE-EI methods, the minimum diagonal element value of the idempotent matrix E increases as points are excluded gradually, which means that the contribution from the remaining sensors to the modal vector linear independence increases gradually. Moreover, the minimum diagonal element of idempotent matrix E changes gradually before excluding the former 700 candidate locations for the EI method and 720 candidate locations for the MSE-EI method; then, a more rapid change is witnessed after excluding the measuring points.

Figure 9.

Sensor placement layout of the MAC optimization method.

Figure 10.

Sensor placement layout of the EI optimization method.

Figure 11.

Sensor placement layout of the MSE-EI optimization method.

Figure 12.

MAC matrix of the MAC method.

Figure 13.

MAC matrix of the EI method.

Figure 14.

MAC matrix of the MSE-EI method.

To compare the advantages and disadvantages of the layouts above, four types of commonly used evaluation criteria are selected. The evaluation criteria include the MAC, modal kinetic energy criterion, Fisher information matrix, and root-mean-square error criterion. ²⁶

MAC

According to the case studies of OSP by the hybrid, MAC, EI and MSE-EI methods, the maximum off-diagonal MAC matrix element value is 0.249, 0.279, 0.528, and 0.539, and the average off-diagonal MAC matrix element value is 0.060, 0.083, 0.123, and 0.146. The maximum values of the off-diagonal MAC matrix elements and the average diagonal MAC matrix element values for the hybrid method are less than the values obtained using other methods. Moreover, the sensor layout scheme determined by the hybrid and MAC methods would provide better orthogonality of the measured modal vectors.

Modal kinetic energy criterion

The average modal kinetic energy of each order for the four schemes is 5809, 3479, 4059, and 6204. The energy of the sensor optimization using the MSE-EI method is the highest, followed by the hybrid method, and those of the MAC method and EI method are the lowest. Compared with the EI and MAC methods, the former two methods have larger ratios of signal to noise, which provides higher precision in the modal parameters of dynamical response monitoring. In addition, this also avoids ignoring important modal information accounting for sensor placement in the low-energy position.

Fisher information matrix

Since it is difficult to determine the variance of noise in this case, the relative value of the Fisher information matrix is chosen. The Fisher information matrix values, which are generated by the modal matrix of the measuring sensors determined by the hybrid, MAC, EI, and MSE-EI optimization methods, are 25,673, 18,901, 19,420, and 29,678, respectively. This suggests that the hybrid and MSE-EI methods contain more modal information, which is beneficial for modal parameter identification of concrete arch dam structures.

Root-mean-square error criterion

Based on the sensor placement determined by the four methods, the dynamic response values at the dam crest, 275 m in elevation and 245 m in elevation, are calculated by FEM. The response values at other elevations are calculated with cubic spline interpolation. The relative root-mean-square error between the values calculated using FEM and the values obtained using the extended method is shown in Table 3. From Table 3, the relative root-mean-square error of the first six orders of modal vibration for the hybrid method is 3.92%, significantly less than the error of the three other methods; that is, the measured vibration modes of dynamic sensors placed by the hybrid method are closer to the theoretical modes, indicating that the arch dam modal vibration mode can be accurately identified.

Table 3.

Comparison of root-mean-square errors.

Method	Order
	1 (%)	2 (%)	3 (%)	4 (%)	5 (%)	6 (%)	Average (%)
Hybrid	3.0	3.5	4.9	3.7	4.1	4.3	3.92
MAC	6.5	7.7	6.8	8.5	7.9	11.0	8.05
EI	9.7	8.6	10.0	7.2	9.1	11.0	9.27
MSE-EI	5.1	6.3	6.9	6.1	5.6	4.6	5.77

MAC: modal assurance criterion; EI: effective independence; MSE: modal strain energy.

In summary, sensor placement based on the hybrid method presents satisfactory results in terms of accuracy of modal expansion, linear independence, and modal vector orthogonality. The relative mean square error of the measured vibration modes and theoretical modes is lowest using the hybrid method. Therefore, the hybrid method used for sensor deployment optimization in the arch dam has more advantages in the identification of dynamical modal parameters than other methods.

Conclusion

This article focuses on the optimal placement of dynamic response monitoring points of a high arch dam. A new hybrid algorithm has been proposed for OSP that guarantees good linear independence, orthogonality, signal-to-noise ratio, and modal information of modal vectors and the minimum relative root-mean-square error between the measured modes and theoretical modes. The hybrid algorithm accurately identifies the structural modal information and ensures modal expansion results that can be used for optimal placement of dynamic response monitoring points of a high arch dam.

In this article, only two aspects of OSP, including the number of sensors and their location, are studied using numerical simulation. However, the types of sensors selected and the arrangement of sensor angles are not considered. The designed normal storage upstream water level, the highest water level in normal operation of a reservoir, is considered in the numerical example of this article. However, the natural frequencies of a dam also depend on environmental variations, such as water level and temperature. Therefore, more effort should be made to solve these problems to decrease the gap between research and practical applications of OSP in hydro-structures.