Multistrategy Cooperation Particle Swarm Optimization for FEM Model Update of the Continuous Warren Truss Steel Railway Bridge

It is very difcult to obtain an accurate fnite element method (FEM) model to further analyze structural mechanical properties. Terefore, as the main means of establishing accurate models, the model update has become a research hotspot in the dominion of bridge engineering. Particle swarm optimization (PSO) has the characteristics of being easy to implement, but it is easy to fall into the local optimum. Terefore, multistrategy cooperation particle swarm optimization (MCPSO) that balances exploration and exploitation of particle swarm is proposed. Tis algorithm achieves the goal of balancing exploration and exploitation by adopting diferent combinations of particle swarm velocity update strategies in diferent iteration stages. Te application efects of MCPSO in the FEM model update of the continuous Warren truss steel railway bridge are compared and analyzed, and the results show that the algorithm proposed in this paper outperforms the standard PSO (SPSO) algorithm. Tis paper provides a more efective algorithm for bridge model updates.


Introduction
Particle swarm optimization (PSO) is a stochastic population-based optimization method proposed by Shi and Eberhart [1].Since the algorithm was proposed, researchers have carried out long-term research and improvement work on it.It has the disadvantage that it is easy to fall into the local optimum [2][3][4].Tere are unimodal and multimodal problems in engineering practice.Unimodal problems have only one extreme point, while multimodal problems have multiple extreme points.For unimodal problems, this is advantageous, but when encountering multimodal problems, it is easy to obtain results that deviate from the global optimum [5][6][7].Te current solution to this problem is often to use some velocity update strategies to balance the exploration and exploitation of particle swarms in the solution domain [7][8][9].
Te article [10,11] indicates that multiswarm-combining dynamical topology is an efective strategy to improve PSO.
Li et al. [12] proposed four strategies to update the particles' positions called a self-learning particle swarm optimizer (SLPSO), in which each particle has four cooperation strategies implemented by an adaptive learning framework and can choose the optimal strategy according to its own local ftness landscape.Tang et al. [13] proposed multistrategy adaptive particle swarm optimization (MAPSO), which evaluates the population distribution, alternates strategy in real time, and has enhanced the research ability of PSO variants.Gülcü and Kodaz [14] proposed PSO variants, which set swarms as master and slave subswarms and make them work cooperatively and concurrently.Bonyadi and Michalewicz [15] conducted review research on PSO, and it is believed that the combination of multiple speed update strategies is one of the methods to improve the performance of PSO.Wang and Song [16] separated particles near the global best position and other particles and updated them in the population in diferent ways.It has good performance and high search precision than PSO and some other optimization algorithms.Xia et al. [17] presented dynamic multiswarm particle swarm optimization based on an elite learning strategy (DMS-PSO-EL), in which the whole computational process is divided into a former stage and a later stage.Tang et al. [18] presented dynamic multiswarm global particle swarm optimization (DMS-GPSO), which consists of two novel strategies balancing exploration and exploitation abilities.
However, benchmark functions are often used to test the pros and cons of most PSO variants [19][20][21][22], which are one sided.Te ultimate goal of researching algorithms is to apply them to engineering practice and solve practical engineering problems [23].In order to solve practical engineering problems, it is meaningful to design algorithms for this engineering problem.Tis paper proposes a PSO variant with combined strategies.Te algorithm evaluates the PSO results through roulette and selects diferent particle speed update methods according to the evaluation results, to achieve the purpose of improving the performance of the algorithm.
At present, researchers have developed some bridge structure model update methods [24][25][26][27], and the use of swarm intelligence algorithms to carry out bridge structure model update research is also one of the current research hotspots.Ho et al. presented a multiphase model update approach to system identifcation of a real railway bridge using vibration test results [28].Bayraktar et al. (2010) [29] obtained a railway bridge's dynamic characteristics experimentally, and according to them, the FEM model of the bridge was updated by changing some uncertain parameters, material properties and boundary conditions.Arisoy and Erol [30] compared an FEM model and experimental dynamic properties of a steel railway bridge and updated the FEM model by tuning material properties to match the real bridge.Tran-Ngoc et al. [31] updated a large-scale steel truss bridge using PSO and the genetic algorithm (GA) and found that PSO provides a better accuracy FE model and reduces the calculation cost compared to GA.
From the literature research, it is known that the research on the improvement of particle swarm optimization is one of the research hotspots of the current swarm evolutionary algorithm.However, no one PSO variant is superior to all algorithms.Terefore, for specifc problems, it is necessary to popularize the application of improved PSO algorithms and design targeted improved PSO algorithms.At the same time, the current standard for using benchmark functions to test algorithm performance is one sided.Attention should be paid to the comparative application efect of algorithms in practical engineering, and efectively improved algorithms should be recommended for solving practical engineering problems.
Tis study makes several contributions to the current literature.1.It is not a simple mathematical average that the article introduces roulette into particle swarm velocity updating strategy evaluation but considers the distribution of the particle updating process, which is more objective.2. Te improvement of the proposed algorithm is embodied in the comprehensive strategy to realize an optimal combination of the existing PSO algorithm.Te proposed algorithm is based on the evaluation result.Tis method can be used as a frame to replace the subswarm velocity update formula or introduce other optimization algorithms such as the ant colony algorithm, bee colony algorithm, wolf colony algorithm, and genetic algorithm to achieve the purpose of improving the competitiveness of the algorithm.
Te remainder of the article is organized as follows: Section 2 introduces the basic concepts of PSO and the related PSO variants.Section 3 elaborates the improvement strategies in proposed MCPSO.Section 4 applies the MCPSO to the structural update problem for the continuous Warren truss steel railway bridge, compares the results of MCPSO and SPSO, and evaluates the efectiveness of MCPSO.Finally, Section 5 provides the summary and conclusion.

PSO and Related PSO Variants
Canonical PSO is a swarm intelligence algorithm inspired by birds' natural behavior in search of food.It imitates the behavior they move, the positions they change, and trajectories to search their destinations.In mathematics, a single bird is regarded as a particle.Terefore, in the simulation process, each particle has two position-related characteristics, namely, the current position and moving velocity.In the PSO algorithm, each particle describes a feasible solution to a problem, a single particle is seen as a point in a D-dimensional space, and a population of N particles is used for an optimization problem generally.Te position of the ith particle is represented as x i = (x i1 , x i2 , . .., x ij , . .., x iD ), and its velocity is represented as Te best previous searched particle position with the best ftness value is saved and denoted as p i = (p i1 , p i2 , . .., p ij , . .., p iD ).Meanwhile, the particle position with the smallest objective function value among all the swarm particles is denoted as Te position and velocity of the ith particle at iteration t of the PSO algorithm are denoted as x i (t) and v i (t), respectively.Tus, the particle's position element is expressed as follows: From a variety of velocity update formulas, the classic particle velocity update formulas that can cover all kinds of velocity update strategies are selected as the subswarms' velocity update formulas of the proposed algorithm in this article.[1] introduced standard PSO, which added an inertia weight ω in PSO, to regulate the efect of the previous velocity on the updated velocity.Te particles' velocities at iteration t + 1 consist of three components.Te frst part is the velocity at iteration t, the second part is the individuality behaviors of the particles controlled by the random number r 1, ij uniformly distributed in [0, 1] and c 1 referred to as a cognitive parameter, and the last part is the sociality behavior of the particle considering the 2

SPSO. Shi and Eberhart
Advances in Civil Engineering random number r 2,ij uniformly distributed in [0, 1] and c 2 called a social parameter.Tus, the jth-dimensional velocities of the particle x i at t + 1 iteration are manipulated iteratively as follows: where ξ denotes the constriction factor, p n,ij (t) denotes the j element of the personal best position of the best neighborhood n of the particle x i (t), G ij (t + 1) is the global version velocity of the ith particle on the jth dimension, and L ij (t + 1) is the local version velocity.Te velocity update of UPSO is calculated as follows: where u denotes the unifcation factor controlling the global and local infuence on the velocity update, and 2.3.CLPSO.Comprehensive learning particle swarm optimization (CLPSO) [32] was proposed for better exploration.Te velocity updates are as follows: where f i determines which p ij is used to guide the particle x ij .Te decision depends on the learning probability p ci .Te learning probability p ci is expressed as follows: where a and b are the two parameters tuning the learning probability, a = 0 and b = 0.5 in this article.

Strategy for Proposed PSO
Te proposed strategies improve PSO as follows: 1. Tree algorithms are combined with diferent advantages, and the characteristics of diferent algorithms are integrated to work together to improve the performance of PSO variants.2. Te iterative process of the algorithm is divided into several calculation periods to prevent the accumulation of the limitations of a single algorithm strategy.3.After a calculation period, each particle update position is evaluated by objective function value change and the evaluation result guides the number of particles allocated to diferent particles' velocity update strategies in the next calculation period.Te swarm grouping strategy is shown in Figure 1.Te strategy with the best performance obtains the subswarm with the largest number of particles, and it is the green part in Figure 1.Te strategy with the second best performance obtains the subswarm with a larger number of particles and is shown as the blue part in Figure 1.Te strategy with the worst performance obtains the subswarm with the least number of particles, and it is the yellow part in Figure 1.Te pros and cons of the strategy are evaluated in a computing cycle, and the strategy with the largest change in the objective function in this cycle is the optimal strategy, and vice versa.Figure 2 shows the graphical fowchart of the MCPSO algorithm.
Tis research proposes a comprehensive strategy for the PSO algorithm that dynamically adjusts the number of subswarm particles.Te algorithm divides all particles into S subswarms (S = 1, 2, or 3 in this article) with strategies S and adjusts the particles in subswarms after each calculation period (C p ). Te subswarm with the strategy that signifcantly reduces the objective function value is allocated a larger number of particles.Te objective function value's average change is expressed as

Advances in Civil Engineering
where T s is calculation times for the subswarm S with the strategy S. Te best strategy is the strategy S resulting in maximum ∆f s (t), the earlier stage of the calculation period, or else it is that resulting in minimum ∆f s (t), the later stage of the calculation period.Te pseudocode of the MCPSO algorithm is shown in Table 1.N s (S ∈ {1, 2, 3}) is the proportion of subswarms S in particles, N is the population size, and C p is the calculation period.It outputs the optimal solution g, and it is also the particle position with the smallest objective function value among all the swarm particles.

Engineering Application
In this section, MCPSO proposed in this paper is applied to the FEM model update of the continuous Warren truss steel railway bridge.Te objective function for FEM model update is expressed as [33] where w ωj is the weight factor corresponding to the jth natural frequency, NF is the number of frequencies and mode shapes used in calculation, and ω C j and ω M j are the jth calculated and measured natural frequencies, respectively.

Introduction of the Bridge.
Te example bridge is a continuous Warren truss steel railway bridge, with a span of (48 + 3 × 64 + 48) m, and the span layout diagram is shown in Figure 3. Te main truss pattern is the Warren type, the height is 11 m, the internode length is 8 m, and the center distance of the main truss is 5.75 m.Te upper and lower chords of the main truss are welded H-shaped sections, some diagonal bars are box-shaped sections, the bridge deck structure is vertical and horizontal beams, and the upper and the lower horizontal longitudinal connections are welded Ishaped sections.Te section parameters are shown in Table 2. Te bridge bearing adopts a steel bearing, and the movable bearing is a roller bearing.Te bridge was built in 1996 and has been in service for more than 25 years.Currently, the main trains in operation are C70 trains, with an annual transportation volume of about 50 million tons.Due to the long service period of the bridge and the existence of microdamages, force analysis can provide suggestions for later maintenance, which is conducive to the safe operation of the bridge.
Compared with other simply supported bridges, continuous truss steel girder bridges have greater vertical and lateral stifness and their defection curves are relatively Te braking force of the continuous steel truss bridge is all here, so at the fxed support, the stress on the pier and foundation of the seat will be greater than that of other parts.

Conclusions
An MCPSO algorithm method is presented in this article, based on the PSO algorithm and the multiswarm strategy.Te experiment adopted a new technique for dividing the population into three subswarms and used strategies with diferent specialties to balance exploration and exploitation searches.Te MCPSO algorithm method is not easy to trap in the local optima, so it has a strong exploration and exploitation characteristic compared with that of SPSO.Te following conclusions can be drawn: (1) Tis paper proposes MCPSO, which divides the total particle swarm into three subswarms.Te number of subswarm particles is determined according to the evaluation of the objective function value change.
Te results show that the comprehensive evaluation strategy optimizes the computational performance of the PSO variant.(2) MCPSO algorithms achieved better performance than the SPSO algorithms in terms of the estimation accuracy and convergence velocity.A more accurate and efective structural FEM model considering damage possibility was obtained.(3) Te velocity update performance of a single strategy of the PSO variant algorithm with multistrategy combination is more critical to the algorithm.If the selected velocity update strategy is not suitable, the performance improvement of the PSO variant algorithm with the combined strategy will be not obvious compared to that of a single strategy particle.

4. 1 .
Objective Function.Natural frequencies and mode shapes are well known as the sensitive factors for the changes in structure properties.Te problem of FEM model update is transformed into the optimization problem of minimizing the objective function describing the gaps between the measured responses and the calculated ones.

3 .
Truss Steel Railway Bridge FEM Model.Te continuous Warren truss steel railway bridge model is established by using general fnite element software ANSYS, and the main truss, longitudinal and transverse beams, and connection systems are all simulated by three-dimensional Timoshenko beam element BEAM189, as shown in Figure4.Te connections between the main truss, beams, and connecting system are all rigid connections.In the design, to reduce the height of the bridge building, the longitudinal and transverse beams are set at unequal heights and rigid beam unit MPC184 is used to simulate the rigid connection between the longitudinal beams and transverse beams.Te sleepers and rails are simulated by threedimensional Euler beam unit BEAM4, with the sleeper interval being 0.4 m and the rails 60 kg/m.Te element sizes of the stringer, the rail, and the rest component are 0.2 m, 0.1 m, and 1.0 m, respectively.Te steel is 16 Mnq, the elastic modulus is 2.06 × 10 5 MPa, the density is 7850 kg/m 3 , and Poisson's ratio is 0.3.According to the quality of the gusset plate and high-strength bolts of the continuous steel truss bridge given by the design drawings, MASS21 mass elements are used, which are evenly distributed at each node, and other auxiliary masses are evenly distributed on the vertical and horizontal beams by using MASS21 mass elements.Te element COMBIN14 spring simulates the base plate between the beam and rail fastening or sleeper.Structural connections are achieved through shared nodes and rigid connections.Te longitudinal stifness of the bearing is R 1 , R 4 , R 7 , . .., R 28 , R 31 , R 34 , the vertical stifness of the bearing is R 2 , R 5 , R 8 , . .., R 29 , R 32 , R 35 , and the lateral stifness of the bearing is R 3 , R 6 , R 9 , . .., R 30 , R 33 , R 36 .

Figure 4 :
Figure 4: Finite element model of the bridge structure.
5 and ζ e = 0.5; and if ζ is c 2 , ζ b = 0.5 and ζ e = 2.5.t N is total number of iterations.Tis process of the velocity element v ij (t + 1), position element x ij (t + 1), individual best element p ij (t), and global best g(t) is iterated until a predetermined stopping condition is met.2.2.UPSO.Focusing on the balance abilities between global exploration and local exploitation, Parsopoulos and Varhatis [8] proposed unifed PSO (UPSO).It distinguishes the local and global velocity update of PSO.Te equations of global and local velocity updates are shown as follows:
4.4.Results.Vertical and transverse acceleration sensors are arranged in the frst, second, and third midspans to collect the response data when the train passes through.Te free attenuation data segment after the train passes through is used for spectrum analysis to obtain the natural frequency of the structure.Te natural vibration frequency measured of the bridge is the frequency corresponding to the frst-order transverse mode.Te range of parameters is determined according to empirical values.Te reasonable parameter range is [1 × 10 8 , 11 × 10 10 ] N/m.Te FEM update purpose of the bridge based on the test is to fnd an FEM model whose mechanical property is close to the reality.Te updated results are shown in Table3.Te objective function values of SPSO and MCPSO are average values with 30 analyses performed under the same conditions.Te tested frequency is 2.48, while the calculated frequencies after the FEM model update by SPSO and MCPSO are 2.463 and 2.464.Figure5shows the convergence of SPSO and MCPSO, and it shows that proposed MCPSO updated the FEM model and fnally obtained a more optimized FEM model.Te updated FEM model can be used to study the bridge response in a variant environment.

Table 3 :
Parameters of the range and update N/m.