Journal of Sensors

Volume 2016 (2016), Article ID 2567305, 10 pages

http://dx.doi.org/10.1155/2016/2567305

## Determining the Optimal Placement of Sensors on a Concrete Arch Dam Using a Quantum Genetic Algorithm

^{1}State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China^{2}National Engineering Research Center of Water Resources Efficient Utilization and Engineering Safety, Hohai University, Nanjing 210098, China^{3}College of Water-Conservancy and Hydropower, Hohai University, Nanjing 210098, China^{4}Zhejiang Institute of Hydraulics and Estuary, Hangzhou 310020, China^{5}Engineering Safety and Disaster Prevention Department, Changjiang River Scientific Research Institute, Wuhan 430010, China

Received 18 September 2015; Accepted 1 December 2015

Academic Editor: Kourosh Kalantar-Zadeh

Copyright © 2016 Kai Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Structural modal identification has become increasingly important in health monitoring, fault diagnosis, vibration control, and dynamic analysis of engineering structures in recent years. Based on an analysis of traditional optimization algorithms, this paper proposes a novel sensor optimization criterion that combines the effective independence (EFI) method with the modal strain energy (MSE) method. Considering the complex structure and enormous degrees of freedom (DOFs) of modern concrete arch dam, a quantum genetic algorithm (QGA) is used to optimize the corresponding sensor network on the upstream surface of a dam. Finally, this study uses a specific concrete arch dam as an example and determines the optimal sensor placement using the proposed method. By comparing the results with the traditional optimization methods, the proposed method is shown to maximize the spatial intersection angle among the modal vectors of sensor network and can effectively resist ambient perturbations, which will make the identified modal parameters more precise.

#### 1. Introduction

Under the effects of outburst accidents (such as earthquakes and wind) and operating loads, engineering structures can accumulate damage, which may lead to destructive accidents. In fact, internal damage of engineering structures inevitably leads to changes in structural dynamic parameters, such as the natural frequency, damping, and mode shapes. Therefore, structural modal identification has become the core technology of modern dynamic testing and on-line monitoring of complex engineering structures.

Modal identification and damage diagnosis based on structural vibrations have been widely adopted in many fields of civil engineering, but the application of this technology to hydraulic structures is still in the initial stage. In the 1960s, the construction bureau of California firstly developed the mechanical vibration machine and conducted an operational dynamic test on the Montst arch dam. The first four natural frequencies of the dam were successfully measured. In the 1990s, Houqun et al. [1] conducted prototype dynamic tests on the Dongjiang arch dam in Hunan Province and on the Longyangxia arch dam in Qinghai Province using a blasting method and extracted the corresponding modal parameters from the test results. Yifeng and Ming [2] established a weighted rubber model of a double-curvature arch dam. Jijian et al. [3] built a large-scale hydroelastic model of the Laxiwa arch dam to conduct the experimental modal analysis in order to identify the structural modal parameters. Mridha and Maity [4] investigated the nonlinear response of a concrete gravity dam-reservoir system using laboratory experiments on a small-scale model of the Koyna dam. Altunişik et al. [5] studied the variations of the modal parameters of a damaged arch dam before and after retrofitting using laboratory model experiments. Darbre et al. [6] researched the relationship between the dam natural frequency and reservoir water level using a plaster model of an arch dam. Loh and Wu [7] identified the modal parameters of the Fei-Tsui arch dam from monitoring data during a strong earthquake and studied the influence of the reservoir water level on the structural modal parameters and nonuniform input on the dynamic response of the arch dam. Mau and Wang [8] performed a system identification of an arch dam using vibrational test data. Sevim et al. [9] identified the modal parameters of the Berke arch dam using the frequency domain method based on the environmental excitations and then calibrated a three-dimensional FEM model with the results.

During modal experiments, determining the optimal number of sensors and corresponding configuration on structures has received increasing attention because inappropriate sensor placement can reduce the accuracy of the identified modal parameters. Meanwhile, modern concrete arch dams with complex structures and giant volumes may contain considerable degrees of freedom (DOFs); thus, the arrangement of sensors would particularly influence the measurement accuracy of the modal parameters [10, 11]. During traditional dynamic tests of dams, the sensors are arranged based on engineering experience, which does not guarantee optimal sensor placement [12–14]. Therefore, research of the optimal sensor placement on concrete arch dams has become an important subject in recent years.

Many researchers have studied optimization criteria of sensor placement for structural modal identification in the past few years. In 1990, Kammer [15] proposed the effective independence (EFI) method to obtain the greatest spatial resolution of the targeted modes. The Fisher information matrix (FIM) is established to guarantee linear independence of interesting modal vectors and reserve the DOFs that contribute most to the independence of the targeted modes. Liu and Tasker [16] proposed the Multiple-Reference Ibrahim Time Domain method for sensor placement and developed a relationship between the sensor locations and variance of identification. By combining the Eigensystem Realization algorithm, Lim [17] arranged sensors to minimize the condition number of the Hankel matrix to maintain the independence of the targeted modes. Rafajłowicz [18] revealed a relationship between the information matrix and density of the input spectrum and measurement positions and studied the problem of sensor placement for parameter identification in the frequency domain. Xing and Bainum [19] researched the problem of optimal sensor placement based on the degree of controllability and observability of the discrete system. Reynier and Abou-Kandil [20] placed sensors by maximizing the minimum eigenvalue of the Gramian matrix. Shih et al. [21] defined the degree of controllability and observability as a second-order ordinary differential equation and deduced a relationship between the equation and frequency-response function. The sensors are arranged based on the contribution of each DOF to the index. Salama et al. [22] claimed that the sensors should be placed on the locations with great energy because it is good for modal identification and optimized the sensor locations by maximizing the modal strain energy (MSE). Baruh and Choe [23] used spline interpolation of the response of the measured points to obtain information about the unmeasured points. The sensors were optimized by minimizing the fitting error of interpolation. Breitfeld [24] arranged sensors by minimizing the off-diagonal elements of the Modal Assurance Criterion (MAC) matrix. Cruz et al. [25] established the fitness function by maximizing the natural frequency identification effectiveness and the mode shape independence and adopted the custom genetic algorithm to optimize the sensor configuration. Debnath et al. [26] evaluated the modal participation at individual degree of freedom (DOF) for the target modes and proposed the modal contribution in output energy (MCOE) as the optimization criterion. Papadimitriou [27] adopted the theory of information entropy to measure the uncertainty in the system parameters and proposed the nominal structural model to optimize the sensor configuration on the truss structure. In recent years, various optimization methods have been used in the process of sensor placement, such as the serial method [28, 29], particle swarm optimization [30], simulated annealing algorithm [31], and genetic algorithm [32].

This paper studied two traditional methods for maximizing the modal information, the effective independence method based on the maximal determinant of the FIM and the Kinetic Energy method based on the maximum modal strain energy. One drawback of the EFI method is that the locations with low modal stress energy may be selected, which would result in the loss of modal information. One drawback of the Kinetic Energy method is that the optimization process is highly dependent on the partitioning of the finite element mesh. Based on the above analysis, a novel optimization method combining the EFI method with the MSE method is proposed that could effectively resolve the drawbacks of the two methods. Considering the enormous number of DOFs of the hydraulic structure, a quantum genetic algorithm (QGA) is adopted to increase the computational efficiency and accuracy in this paper. Finally, a specific concrete arch dam is used as an example, and the sensors on the upstream surface are optimized with the proposed method. A comparison of the traditional and proposed methods shows that the proposed method has a higher convergence speed and better optimization accuracy, which has theoretical and practical application values.

#### 2. The Basic Principle of the Optimal Sensor Placement

##### 2.1. The Effective Independence Method

In 1991, Kammer presented the effective independence method and introduced the Fisher information matrix, which is based on the displacement modal matrix. To make the concerned modal vectors linearly independent, the measured points are ranked based on an effective independence value of the modal matrix and the measurement point with the minimum contribution is deleted in succession. For modal experiments that only identify the structural mode shapes, the generalized coordinates of the mode shapes can represent the identified parameters. The output value of the sensor can be expressed as follows:where represents the output value of the sensor; is the modal matrix; and is the vector of the modal coordinates.

The least square estimation of can be written as follows:

By introducing measurement noise, (3) is defined aswhere is the Gaussian white noise of a uniform distribution with the variance of .

By minimizing the covariance matrix of the modal coordinates, the effective unbiased estimate of can be obtained as follows:where is the Fisher information matrix (FIM).

Therefore, an optimal estimation of the modal coordinates can be obtained when the determinant of the FIM is maximized. Then, the covariance matrix is minimized and the targeted modal vectors are linearly independent, which is good for structural modal identification.

Then, the corresponding fitness value of the EFI method can be represented as follows:

##### 2.2. The Modal Strain Energy Method

Modal strain energy (MSE) is an index that is sensitive to the variations of the structural parameters and the ratio between the element MSE and total structural kinetic energy and is one of the system eigenvalues. Therefore, the MSE index is usually applied in structural damage identification.

The element MSE is defined as follows:where is the th normalized mode shape and is the element stiffness matrix of the th element.

Similarly, the th structural modal strain energy can be expressed as follows:where is the global stiffness matrix of the structure.

This paper defines the fitness value of the optimal sensor placement based on the MSE as follows:where represents the stiffness coefficient between the th DOF and th DOF; is the deformation of the th element in the th mode; is the deformation of the th element in the th mode; and and are the numbers of modes and measurement points, respectively.

##### 2.3. The Combined Optimization Algorithm

Although the EFI method can make the selected modal vectors approximately linearly independent, the measured MSE of the selected points may be low, which would result in the loss of modal information. Because of this effect, this paper combines the EFI method with the MSE method and defines the fitness value as follows:where is the adjustment parameter that scales the fitness value into the appropriate range.

#### 3. The Basic Theory of the Quantum Genetic Algorithm

The quantum genetic algorithm (QGA) is a recently developed probability evolutionary algorithm that combines quantum computing with a genetic algorithm [33, 34]. Based on quantum theory, the quantum probability vector is used to encode chromosomes, and the population is updated and optimized by adopting quantum-rotating doors to search for a globally optimal solution.

The smallest unit of information in the QGA is called a quantum bit and the state of a quantum bit can be expressed as follows:where represent the probability amplitudes of the quantum bit and meet the following normalized condition:

Thus, the state of a quantum bit can also be written as follows:

In the quantum genetic algorithm, quantum information is encoded by pairs of complex numbers. The quantum chromosome composed of sets of quantum bits can be written as follows:where .

This method can represent the random linear superposition of quantum states. For example, a chromosome with 3 quantum bits can be expressed as follows:

An update of a quantum gate can be represented aswhere is the quantum-rotating gate, among which the variables are denoted as follows:where is the adaptive variable; is the evolutionary population; and is a constant that depends on the complexity of the optimization problem.

The search strategy of is shown in Table 1. In Table 1, and are the probability amplitudes of the global optimal solutions, and . are the probability amplitude of the current solutions, and . If both and are greater than 0, the current solutions and global optimal solutions will be in the first or third quadrant. When , the current solution should be rotated counterclockwise, ; otherwise, . The three other rotational criteria can be determined using the same method.