1. Introduction
As bridges move towards higher heights, larger spans and more complex structural forms, structural health monitoring (SHM) systems using modal- and vibration-based methods have been widely investigated [
1,
2]. SHM, as a control system, monitors the state of a large civil engineering structure by measuring the structural response. Monitoring and assessing the health status of structures throughout their life cycle is essential to reduce maintenance expenses and labor costs and ensure the safe operation of structures [
3,
4]. As an important method of SHM, the main purpose of dynamic detection technology is to identify and determine structural dynamic parameters [
5]. Through a gradual change in parameters, faults can be predicted in advance and the occurrence of major safety accidents can be prevented. These parameters are mainly obtained using experimental modal analysis technology; that is, structural response data are obtained through sensors and data acquisition equipment, and then modal parameters are obtained through parameter identification. Experimental modal analysis has always been one of the research hotspots in the field of health monitoring. Modal tests of large and complex bridge structures to capture more accurate and complete structural responses from the structure and obtain structural dynamic characteristics for condition assessments and safety warnings have become a hot issue of interest, bridging the academic and engineering communities [
6,
7].
The first step of modal testing is to determine the number and location of the measurement points of the bridge structure, that is, to install sensors at key parts of the structure to obtain the dynamic characteristics of the bridge and other structural data. Theoretically, the more sensors deployed, the more accurate the response data available. This is feasible for structures such as small bridges but difficult for large and complex bridges [
8]. Considering the engineering environment, cost and risk assessments and other factors, the needs of SHM should be met with the lowest number of sensor measurement points. At this time, the result of the modal test depends heavily on the position and number of sensor measurement points, and the optimal number is determined and configured in the best position so as to enhance the accuracy of information acquisition in the modal test. Therefore, research on the optimal arrangement of sensor measurement points is extremely important and has important engineering value and social significance.
Scholars have done extensive research on sensor measurement point optimization. Kammer [
9] proposed a famous efficient independent method in 1991 for optimizing sensor placement in large spatial structures. Liu et al. [
10] used modal strain energy (MSE) and the modal assurance criterion (MAC) as fitness functions to study the optimization problem of spatial structure sensors using an improved genetic algorithm. Yin et al. [
11] proposed a novel relaxation sequence algorithm. A truss structure and a rigid-frame arch bridge were used as examples to verify the effectiveness of the new algorithm in optimizing sensor placement. Chai et al. [
12] proposed an improved optimal sensor placement method based on the sensitively effective independence method, which further optimized sensor placement by using the sensitivity coefficient reflecting structural damage. These studies have achieved good results in the optimized arrangement of sensor measurement points and have provided some reference for further development in this field. However, they did not carry out actual tests to verify whether the accuracy of the test results obtained based on this distribution method met actual engineering requirements after the optimization arrangement was completed.
In the optimization of measurement points, a finite element model of the structure needs to be established by finite element analysis software, and then the node data is extracted as the training data of the optimization algorithm; hence, how to effectively extract a large number of node data is vital for measurement point optimization. Liu et al. [
13] studied the optimal arrangement of bridge sensors based on the single parent genetic algorithm. Taking Guiliuhe Arch Bridge as an example, they used the APDL module of ANSYS to divide the bridge into 1440 units and 1276 nodes and then extracted node data to train the optimization algorithm. Zhang et al. [
14] proposed a nested stacked genetic algorithm (NSGA) to solve the sensor position optimization problem in an existing bridge health monitoring system and established the finite element fishbone model of the Wuhai Yellow River cable-stayed bridge with ANSYS (R15.0). The main beam was divided into 259 nodes and 258 elements; then, the stiffness and mass matrices were extracted from the model as data for the subsequent optimization algorithm. He et al. [
15] proposed a hybrid optimization strategy named MSE-AGA in view of some defects in the existing bridge sensor arrangement methods, established a finite element model of a three-span suspension bridge with ANSYS, and optimized the sensor layout with the generated 1276 nodes as data. The sensor measurement point placement methods based on the optimization algorithm proposed by the above scholars can improve the identification ability of the actual dynamic characteristic parameters of the structure. However, in the selection of optimal measurement point training data, only the finite element model is used to select the target node data, resulting in the insufficient extraction of training data because the upper limit of selected data in this method is only about 100. Insufficient training data will have a great impact on the optimization results. Therefore, there is an urgent need for a method that can extract enough node data, effectively solving the problem of insufficient training data, is easy to operate, with fast extraction efficiency and high accuracy, and provides important basic support for the optimal arrangement of sensor measurement points. This paper requires a lot of node data in order to make sure that the sensors are arranged in the most critical positions of the structure and that the most accurate dynamic response of the structure is collected. Therefore, the tetrahedral mesh type is used and the structure is meshed with a smaller mesh size to get enough nodes.
In recent years, many scholars have been interested in using swarm intelligence algorithms to solve the problem of sensor measurement point arrangement optimization in complex civil engineering fields. Inspired by the group foraging behavior of birds, fish and other creatures in nature, Eberhart and Kennedy [
16] first proposed a new intelligent algorithm called the particle swarm optimization (PSO) algorithm in 1995. The algorithm has the advantages of few parameters, fast convergence, a simple program and a large search range, so it is widely used in neural network training, signal processing, the sensor arrangement of complex civil structures and other engineering fields. Fakhouri et al. [
17] studied multi-vector particle swarm optimization (MVPSO) for solving single-objective optimization problems. Yu et al. [
18] proposed an advanced particle swarm optimization algorithm (APSO) to solve the flutter problem of long-span bridges and compared the flutter prediction results of the proposed method with the actual test results and obtained a good agreement. Huang et al. [
19] proposed a hybrid optimization algorithm combining particle swarm optimization and cuckoo search. The hybrid optimization algorithm was used to test the damage identification performance of a real steel–concrete composite bridge under temperature variation and verified its good robustness and engineering applicability. Tran et al. [
20] proposed a new method for the model updating of large railway bridges using orthogonal diagonalization (OD) combined with improved particle swarm optimization (IPSO).
Scholars in various fields have solved many problems by using the advantages of the PSO algorithm; however, the standard PSO algorithm has the defects of easily falling into local optimums and premature convergence. To this end, some scholars have also proposed many improvement strategies. Jiang et al. [
21] proposed an improved particle swarm optimization algorithm and simulated three benchmark functions. The results showed that IPSO has a better global optimization ability than the basic PSO algorithm. Liu et al. [
22] proposed a new state assessment method for reinforced concrete (RC) bridge superstructures based on fuzzy C-mean clustering using a particle swarm optimization (FCM-PSO) algorithm. It has the advantage of the PSO algorithm in global optimization. Nguyen et al. [
23] proposed a new method to identify damage in structures using particle swarm optimization (PSO) combined with artificial neural networks (ANNs), which avoids the possibility that ANNs will get trapped in local minima while seeking the best solution.
When solving specific engineering problems, especially when solving combinatorial optimization problems such as the optimal placement of sensor measurement points, the two random weights of social learning and individual cognition in the PSO algorithm will directly affect the search speed and search accuracy of the optimal solution of the sensor measurement point arrangement scheme. Considering that the search performance of the PSO algorithm depends on the balance between its global search and local improvement ability, this paper will improve the inertia weight in the algorithm to make up for the defect of the linear decline in so that the particles take into account both local and global searches in the optimization process and improve the efficiency of the algorithm. In the beginning, it is anticipated that the particles can search the whole space quickly at a fast speed to find the general position of the optimal solution and then conduct a fine search of the general position at a slow speed, finally converging to the optimal solution.
The rest of this paper is organized as follows. In
Section 2, aiming at the problem that the standard particle swarm optimization algorithm is prone to falling into local optimal and premature convergence due to the influence of weight, the PSO is improved, and an improved particle swarm optimization algorithm based on random weight (IPSOARW) is proposed. In
Section 3, on the basis of setting the fitness function, the optimization strategy of the number and location of measurement points is proposed for the large-span steel beam structure. In
Section 4, the optimization strategy of the number and location of measurement points in
Section 3 is applied to the deck of the steel tube truss girder bridge (STTGB). The number and position of measurement points in the modal test of the STTGB bridge deck are optimized by taking the vibration data of the deck as the optimization data. In
Section 5, based on the optimized number and position of bridge deck measurement points obtained in
Section 4, the bridge deck measurement points are arranged according to the uniform distribution method and the IPSOARW distribution method and the modal test is completed. The test results based on the two methods are compared and analyzed, and the feasibility and superiority of the measurement point arrangement strategy based on IPSOARW are verified. The purpose of this paper is to provide a reference for the dynamic test of long-span steel beam structures, save test costs and improve test efficiencies on the basis of ensuring the accuracy of the test results.
3. Measurement Point Optimization Strategy of Long-Span Steel Beam Based on the IPSOARW
Based on the simulation verification results in
Section 2.3, the IPSOARW can be used to study the optimal arrangement strategy of measurement points in the modal test of long-span beams. The modal confidence criterion was used to optimize the number of measurement points of the long-span beam structure. The IPSOARW program is realized by MATLAB to optimize the location of the measurement points. After obtaining the modal data of the beam structure model, the optimization algorithm was applied to the optimization problem of the measurement point positions.
3.1. Modal Assurance Criterion (MAC)
As a multi-degrees of freedom linear vibration system, the dynamic equation of the long-span steel beam is
where
is the damping matrix;
is the mass matrix;
is the stiffness matrix;
is the disarrangement vector;
is the excitation force.
Many experimental studies show that the damping of steel structures is generally small, resulting in a small and negligible effect of damping on the mode; the solution to this equation will be obtained by a linear combination of harmonic functions, and then an eigenvalue problem will be obtained. Its characteristic equation is
where
is its corresponding eigenvector and
is the eigenvalue. When the quality matrix is a diagonal matrix, its corresponding eigenvector is an orthogonal vector. On this basis, the
is used to differentiate the eigenvalues. The independence between the modal vectors is tested by the
. The
matrix is calculated by the following equation:
where
is the element of the
row and
column of the matrix,
is the
order vibration vector,
is the
order vibration vector, and
and
are transpose of vectors.
The value of the matrix elements reflects the similarity of the two vibration patterns, so the maximum value of the non-diagonal element of the matrix is used to determine whether the results of the measurement point arrangement are reasonable. The value of Equation (8) indicates the correlation between the order vector and the order vector, and the value of it takes the range . When the equation value is 1, it means that the two vectors are completely correlated. When the equation value is 0, it means that the two vectors are completely independent. Therefore, in modal tests, the smaller the value of , the more accurate the test results.
3.2. Establishment of the Fitness Function
The fitness function in the optimization procedure is also a key factor affecting the optimization arrangement of the measurement points. Therefore, it is necessary to debug and program the measurement point optimization program of the IPSOARW. The fitness function is designed according to the and finally applied to the optimization of the measurement point arrangement of the modal test of the long-span steel beam.
It is important to simplify the fitness function as much as possible in the design of the optimization algorithm to reduce the calculation time. A fitness function based on the
is shown in Equation (10).
where
is the maximum value and
is the absolute value. The smaller the value of
, the better the optimization effect when optimizing the measurement point arrangement.
3.3. Acquisition of Training Data for Bridge Decks Based on IPSOARW
In order to optimize the modal test points of long-span steel beam structures, the modal data of the structure is needed, no matter the number optimization or position optimization. Therefore, the finite element model of the long-span steel beam structure was established in ANSYS (R19.2), and then modal analysis was carried out to obtain the frequencies and modes of the structure in the frequency band of interest. In the follow-up work, the IPSOARW will take the modal shape data of the structure as the training data so as to optimize the position of the structure’s modal test measuring points.
3.4. Strategy for Optimizing the Number of Measurement Points for the Modal Test of Large-Span Steel Beams
The number of measurement points is optimized by:
(1) constructing a fitness function with the minimum value of the non-diagonal elements of the matrix; (2) increasing the number of randomly selected measurement points and observing the change in the value of the fitness function. (3) The number of measurement points is determined by the relation curve between the number of measurement points and the fitness function value.
The steps for optimizing the number of measurement points are as follows.
Construct the
matrix formula.
Set the maximum value of the non-diagonal elements of the matrix to be the function .
Select one node at random from all the nodes as a measurement point and calculate the relationship between the number of measurement points and , select one node at random from the remaining nodes to add a measurement point, calculate the number of measurement points at this moment as a function of and repeat the superposition of measurement points.
Using the above iterative method, the value of the function can be derived under different numbers of measurement points.
On this basis, the most reasonable number of measurement points is determined by plotting the value of the function against the number of measurement points.
3.5. Optimization Strategy for the Location of Measurement Points on Large-Span Steel Beams Based on the IPSOARW
Based on the known data of the structural mode and the optimized number of measuring points, the fitness function is taken as the optimal number of measuring points in the particle dimension. The structural mode data are used as the training data of the optimization algorithm.
The steps of measurement point position optimization are as follows:
Construct the fitness function with the maximum value of the off-diagonal elements of the matrix.
Set IPSOARW parameters, such as population size, particle dimension, maximum number of iterations, learning factor, particle speed, etc.
Calculate and obtain the modal shape data of the structure.
Use the IPSOARW to optimize the position of the measurement point, calculate the current fitness value of each particle and determine the group extreme value and the individual extreme value of the particle. The loop iterates. When the maximum number of iterations is reached, the loop ends and the optimal value is output. The value is the node number of the optimal location of the measurement points.