Nonlinear Structural Discrete-Time Integration Method Based on the Update of Particle Swarm Optimization Model

Abstract

The present invention discloses a discrete-time integration method for nonlinear structures based on the update of a particle swarm optimization model. In the first step, the parameters of the particle swarm optimization integration algorithm are set. In the second step, according to the discrete-time domain formula of the structural dynamics equation, the initial parameters and initial conditions of the structure, the particle swarm optimization algorithm is used to obtain the optimal integration parameters. In the third step, the obtained optimal integration parameters are substituted into the iteration matrix, and the discrete-time domain motion equation is iteratively solved until the preset time condition is reached, and the change range of the structural characteristic parameters is output. N points with equidistant distribution are taken within the change range of the structural characteristic parameters, and according to the structural characteristic parameters corresponding to each point, particle swarm optimization is carried out to obtain the corresponding N groups of optimal integration parameters. In the fourth step, the motion equation is iteratively solved, and in each time step, the optimal integration parameters in the corresponding interval are selected according to the value of the structural characteristic parameters for calculation until the preset time limit is reached, and the structural response time history curve is output.

Description

Technical Field

[0001] The present invention belongs to the field of solving discrete-time structural dynamics equations in civil engineering, and particularly relates to a nonlinear structural discrete-time integration method based on particle swarm optimization model updating. Background Art

[0002] When analyzing the dynamic response of a structural system, generally two methods are adopted. The first is the mode superposition method, which can directly obtain the analytical solution of the structural dynamics problem. However, due to its iterative process, it requires a higher computational cost. Therefore, in the transient analysis of structural dynamics problems and various dynamic problems, the direct integration algorithm plays an important role. It can be used to analyze any set of nonlinear dynamic equations and coupled linear modal equations. This method is widely used in practice because it is easier to solve numerical nonlinear and linear systems without the need to solve the analytical solutions of most practical dynamic problems, thus improving computational efficiency. Therefore, the direct integration method is widely applied to various complex civil engineering problems such as structural dynamics analysis, finite element analysis, and real-time substructure testing of large complex structures.

[0003] The direct integration algorithm can be divided into two categories: explicit integration algorithm and implicit integration algorithm according to its characteristics. The implicit method can be designed to have the characteristics of unconditional stability and controllable numerical damping, but it requires matrix factorization. On the other hand, the explicit method is conditionally stable, but if the mass matrix is a diagonal matrix, matrix factorization is not required. For large nonlinear structural dynamics problems, the implicit method needs to construct and decompose the system matrix multiple times in each time step, and at the same time, it needs to conduct a convergence test for nonlinear iterative solution, occupying a large amount of computing resources. Therefore, for the problem of solving structural dynamics of complex nonlinear systems, the explicit method can better balance computational accuracy and computational efficiency.

[0004] However, when the system is in high-order modes, the explicit method must use a smaller time step to meet the stability condition, thus increasing the computing time. Since the efficiency of transient analysis and the accuracy of numerical solutions largely depend on the performance and characteristics of the selected time integration method, the development of improved direct time integration methods has always been a research hotspot in the field of structural dynamics. In recent years, many experts and scholars at home and abroad have achieved certain results in developing new explicit integration algorithms, among which the model-based integration algorithms have attracted much attention, such as the Chang algorithm and the CR algorithm. Although the model-based integration algorithm can ensure unconditional stability, it has no numerical damping in high-order modes and will exhibit overshoot phenomena, unable to guarantee its computational accuracy. Summary of the Invention

[0005] To solve the above problems, the present invention proposes a non - linear structural discrete - time integration method based on particle swarm optimization model updating. Based on the characteristic parameters of the structural model itself (mass and stiffness), the particle swarm optimization algorithm (Particle Swarm Optimization, PSO) is used to obtain the optimal integration parameters of the integration iteration gain matrix by considering the characteristics of the structure itself, so as to update the iteration gain matrix in the integration process. The algorithm has the characteristics of unconditional stability and controllable high - order numerical damping, and has the advantages of high precision, high computational efficiency and wide application range. The algorithm proposed by the present invention is of great significance for the structural dynamics analysis, finite element analysis and real - time sub - structure test of non - linear systems.

[0006] The technical solution of the present invention:

[0007] A non - linear structural discrete - time integration method based on particle swarm optimization model updating. The first step is to set the parameters of the particle swarm optimization integration algorithm; the second step is to use the particle swarm optimization algorithm to obtain the optimal integration parameters according to the discrete - time domain formula of the structural dynamics equation, the initial parameters and initial conditions of the structure; the third step is to substitute the obtained optimal integration parameters into the iteration matrix and substitute them into the discrete - time domain motion equation for iterative solution until the preset time condition is reached, and output the change range of the structural characteristic parameters (mass m and stiffness k); take N equally - spaced points within the change range of the structural characteristic parameters, and perform particle swarm optimization according to the structural characteristic parameters corresponding to each point to obtain the corresponding N groups of optimal integration parameters; the fourth step is to perform iterative solution of the motion equation, and select the optimal integration parameters in the corresponding interval according to the value of the structural characteristic parameters in each time step for calculation until the preset time limit is reached, and output the structural response time - history curve.

[0008] The specific steps are as follows:

[0009] Step 1: Set the parameters of the particle swarm optimization algorithm.

[0010] (1.1) Determine the optimization objective and optimization variables. From the structural dynamics equation and the discrete - time motion equation, we get:

[0011] kx + cv i+1 + ma i+1 =f i+1

[0012]

[0013] v i+1 =v i + β i Δta i

[0014] where k, c and m represent the stiffness, viscous damping and mass of the structure respectively; f i+1 =f(ti+1 ) represents the external force, x i+1 is the displacement at the (i + 1)-th step. Correspondingly, v i+1 is the velocity, a i+1 is the acceleration; Δt is the time interval; α 1,i , α 2,i and β i are the optimal integration parameters in the optimization process.

[0015] Define the optimization objective as the minimum error between the displacement, velocity, and acceleration under free vibration conditions and the expected values, as follows:

[0016]

[0017] Use a linear weighted model to linearly weight the three optimization objectives, set adjustable weight coefficients according to the emphasis on different optimization objectives, and obtain the comprehensive optimization objective function:

[0018]

[0019] where p i is the weight coefficient, p x + p v + p a = 1, and the specific values can be set according to the importance requirements of different optimization objectives.

[0020] (1.2) Selection of optimization process parameters. To balance the calculation efficiency and calculation accuracy, compare and set the population size and the number of iterations in the particle swarm optimization algorithm. Select appropriate population size N and the number of iterations Ger to ensure the convergence, calculation accuracy, and calculation efficiency of the algorithm.

[0021] Step 2: Use the particle swarm optimization algorithm to find the minimum value of the optimization objective under the constraint conditions.

[0022] (2) Use the particle swarm optimization method to perform a primary optimization to obtain the optimal integration parameters α 1 , α 2 and β corresponding to the initial state of the structure, and then obtain the corresponding discrete-time motion equation.

[0023] Step 3: Iteratively solve the discrete-time domain motion equation to obtain the structural response.

[0024] (3.1) Substitute the optimal integration parameters obtained in (2) into the matrix form of the discrete-time domain motion equation, solve for the structural response, and obtain the response of the next step.

[0025] (3.2) Define the structural response obtained in (3.1) as the new initial state of the structure and update it, then substitute it into the discrete-time domain motion equation for the next step of solution until the preset time condition is met, terminate the iteration, and output the change range of the structural characteristic parameters.

[0026] (3.3) Select N equally spaced points within the change range of the structural characteristic parameters, and repeat (2) according to the structural characteristic parameters at each point to obtain a corresponding set of optimal integration parameters.

[0027] Step Four: Conduct the calculation of the structural dynamic response.

[0028] (4) Conduct iterative solution of the discrete-time motion equation, and select the optimal integration parameters in the corresponding interval according to the value of the structural characteristic parameters at each time step for calculation until the preset time limit is reached, and output the structural response time history curve.

[0029] Based on the above entire process, the optimal integration parameters based on the model can be obtained, forming a non-linear structural discrete-time integration algorithm system based on the update of the particle swarm optimization model.

[0030] The beneficial effects of the present invention: This algorithm is a variable-display integration algorithm with variable integration parameters applicable to non-linear structures. When the structure is subjected to high-intensity seismic action, the non-linearity increases, and the structural stiffness and mass change accordingly; the real-time optimal change of the integration parameters in the discrete-time integration algorithm can ensure the accuracy of the calculation results, and at the same time, the limitation of the constraint conditions in the optimization process can ensure the unconditional stability of the algorithm; on the other hand, the determined integration parameters obtained by optimization in the algorithm can ensure higher calculation efficiency when this algorithm is applied to non-linear structures. In summary, a non-linear structural discrete-time integration algorithm based on the update of the particle swarm optimization model proposed by the present invention can demonstrate its advantages in the structural dynamics analysis, finite element analysis, and real-time substructure test of non-linear systems. Description of the Drawings

[0031] Figure 1 is the basic process of the method proposed by the present invention;

[0032] Figure 2 is the comparison chart of different parameters of the particle swarm optimization algorithm in Embodiment 1 of the present invention;

[0033] Figure 3(a), Figure 3(b) and Figure 3(c) are respectively the comparison of the displacement, velocity and acceleration time history curves obtained by using different integration algorithms for the stiffness softening system in Embodiment 1 of the present invention;

[0034] Figure 4(a), Figure 4(b) and Figure 4(c) are respectively the comparison of the displacement, velocity and acceleration time history curves obtained by using different integration algorithms for the stiffness hardening system in Embodiment 1 of the present invention;

[0035] Figure 5 The three - layer reinforced concrete frame shear model and information described in the second embodiment of the present invention;

[0036] Figures 6(a) and 6(b) are the third - layer displacement time - history curves of the simplified three - degree - of - freedom shear model obtained by using different algorithms in the second embodiment of the present invention; Detailed implementation manners

[0037] The following further illustrates the specific implementation manners of the algorithm proposed by the present invention in combination with the attached drawings and technical solutions.

[0038] The main steps of the present invention are as Figure 1 shown. In Embodiment 1, a single - degree - of - freedom undamped system is selected, and the parameters of the particle swarm optimization method are set. In Embodiment 2, a single - degree - of - freedom variable - stiffness system (stiffness - softening system and stiffness - strengthening system) is selected, and the accuracy of a non - linear structural discrete - time integration algorithm based on particle - swarm - optimization model updating and other typical discrete - time integration algorithms (explicit Newmark method with β = 0, γ = 0.5, Chang method, and CR method) is compared and analyzed. In Embodiment 3, taking a three - layer reinforced concrete frame as an example, the non - linear time - history analysis of the structure under earthquake action is carried out.

[0039] The implementation manner of setting the parameters of the particle swarm optimization algorithm is as follows:

[0040] (1) Select the weight coefficients. In this embodiment, only the minimum displacement error is selected as the optimization objective. Therefore, the weight parameters in the objective function are set as: a = 1, b = 0, c = 0.

[0041] (2) Set the analysis conditions for the population number and iteration times of the particle swarm optimization. The analysis conditions selected in this embodiment are as follows: N = 5000, Ger = 200; N = 8000, Ger = 100; N = 8000, Ger = 200; N = 10000, Ger = 200; N = 10000, Ger = 300; N = 30000, Ger = 200; N = 30000, Ger = 300; N = 300000, Ger = 200.

[0042] (3) Conduct particle swarm optimization analysis for the analysis conditions in (2), and the analysis results are shown in Figure (2).

[0043] (4) It can be obtained from Figure 2 that obviously, the increase of the population number (N) and the iteration times (Ger) is beneficial to the accuracy of the analysis results. When N = 8000, Ger = 200, the optimal solution remains below 2.5×10 -4 Hereinafter. When Ger is reduced to 200 and N reaches 5000, the optimal solution increases to 1×10 -3and 2.5×10 -4 Therefore, the population number and generation number of the PSO algorithm can be set according to the required accuracy of the optimization solution. However, to ensure the calculation accuracy, it is recommended to take N≥8000 and Ger≥200.

[0044] The implementation method of the single-degree-of-freedom variable stiffness system is as follows:

[0045] (1) Define the initial condition of the structure as x(0) = 1.25m, The initial structural parameters of the stiffness-softening system are m = 300kg, k 0 = 9000N / m; the initial structural parameters of the stiffness-hardening system are m = 300kg, k 0 = 6000N / m; the stiffness non-linearly changes in the following form:

[0046] k = k 0 [1 + α(u) 2

[0047] where u represents the relative displacement; take α = -0.3 for the stiffness-softening system; take α = 0.3 for the stiffness-hardening system.

[0048] (2) By changing the integration time step to select different Ω values, here take Ω = 0.0548, and use a non-linear structural discrete-time integration algorithm updated based on the particle swarm optimization model and three other typical explicit discrete-time integration algorithms respectively to obtain the structural response under free vibration. It should be noted that the result obtained by the explicit Newmark algorithm with Δt = 0.0001s is taken as the exact solution here.

[0049] (3) Compare the response results obtained by different methods with the exact solution, and then compare the accuracy of different algorithms. The response results of the stiffness-softening system are shown in Figures 3(a), 3(b) and 3(c), and the absolute errors from the exact solution are listed in Table 1; the response results of the stiffness-hardening system are shown in Figures 4(a), 4(b) and 4(c), and the errors from the exact solution are listed in Table 2.

[0050] (4) From the results of Figures 3(a), 3(b) and 3(c) and Figures 4(a), 4(b) and 4(c), it can be seen that for the variable stiffness system, the response time history curves obtained by the four algorithms are basically the same, but there are obvious period errors compared with the exact solution, and this phenomenon is particularly obvious in the acceleration comparison of the stiffness-hardening system (Figure 4(c)); however, the algorithm proposed in the present invention has smaller errors in both systems than the other three typical methods. These results provide strong evidence for the superiority of the algorithm proposed in the present invention in terms of accuracy.

[0051] The specific implementation method of the three-story reinforced concrete frame structure is as follows: ​

[0052] (1) Simplify the three - layer reinforced concrete frame structure into a three - degree - of - freedom shear model. The model schematic diagram and basic characteristic parameters are shown in Figure 5 .

[0053] (2) Use MATLAB R2010a software to establish a three - degree - of - freedom shear model and conduct a dynamic time - history analysis. Select the Northridge Beverly Hills - 14145 Mulhol ground motion and input the ground motion along the Figure 5 X - direction in the middle for the analysis. Computer performance for analysis: Intel Core I i5 - 6402P CPU, 24GB of memory.

[0054] (2) Consider the change of structural stiffness under load. Here, use the tri - linear stiffness degradation model. Use the algorithm proposed in the present invention and three other typical algorithms respectively to obtain the top - displacement response of this structure under the action, and at the same time compare the influence of different initial stiffnesses of the structure and ground - motion intensities on the results. The specific working conditions are listed in Table 3, and the corresponding response time - history curves and average time - consuming are shown in Figure 6(a), Figure 6(b) and Table 4.

[0055] (3) As can be seen from Figure 6(a) and Figure 6(b), under small ground - motion excitations, the displacement response curves obtained by various algorithms are basically the same. When the ground - motion intensity is large, the structure enters the non - linear stage. Overshoot phenomena occur in the Chang method and the CR method in the high - order modes, while the displacement response curves obtained by the method proposed in the present invention and the Newmark method are closer. On the other hand, as can be seen from Table 4, the time - consuming of the algorithm proposed in the present invention is significantly less than that of the other three typical algorithms.

[0056] From the comparative analysis, it can be seen that the algorithm proposed in the present invention can be applied to non - linear systems and has higher calculation accuracy and calculation efficiency compared with other typical algorithms. A non - linear structural discrete - time integration algorithm based on particle - swarm - optimization model updating proposed in the present invention can be a useful tool for structural dynamics analysis, finite - element analysis and real - time sub - structure testing of complex non - linear systems.

[0057] Table 1 is the comparison of absolute errors of the stiffness - softening system described in the first embodiment of the present invention;

[0058] Table 2 is the comparison of absolute errors of the stiffness - hardening system described in the first embodiment of the present invention;

[0059] Table 3 is the statistical table of analysis working - condition information of the three - layer reinforced concrete frame structure described in the second embodiment of the present invention.

[0060] Table 4 is the comparison of time - consuming of different analysis working - conditions of the three - layer reinforced concrete frame structure described in the second embodiment of the present invention.

[0061] Table 1 Comparison of absolute errors of stiffness softening systems

[0062]

[0063] Table 2 Comparison of absolute errors of stiffness strengthening systems

[0064]

[0065] Table 3 Statistical table of working conditions of three - layer reinforced concrete frame structures

[0066]

[0067] Table 4 Comparison of calculation time consumption of three - layer reinforced concrete frame structures using different algorithms

[0068]

Claims

1. A discrete-time integration method for nonlinear structures based on the update of the particle swarm optimization model, characterized in that, the steps are as follows: Step 1: Set the parameters of the particle swarm optimization algorithm; (1.1) Determine the optimization objective and optimization variables. From the structural dynamics equation and the discrete-time motion equation, we get: kx + cv i+1 + ma i+1 = f i+1 v i+1 = v i + β i Δta i where k, c, and m represent the stiffness, viscous damping, and mass of the structure, respectively; f i+1 = f(t i+1 ) represents the external force, x i+1 is the displacement at the (i + 1)-th step, and correspondingly, v i+1 is the velocity, a i+1 is the acceleration; Δt is the time interval; α 1,i , α 2,i and β i are the optimal integration parameters in the optimization process; Define the optimization objective as the errors between the displacement, velocity, and acceleration under free vibration conditions and the expected values, respectively, as follows: Use a linear weighted model to perform linear weighted processing on the three optimization objectives. Set adjustable weight coefficients according to the emphasis on different optimization objectives to obtain the comprehensive optimization objective function: where p x , p v , p a are weight coefficients, and p x + p v + p a = 1, and the specific values are specifically set according to the importance requirements of different optimization objectives; (1.2) Select the parameters of the optimization process. Compare and set the population size and the number of iterations in the particle swarm optimization algorithm; select the population size N and the number of iterations Ger; Step 2: Use the particle swarm optimization algorithm to find the minimum value of the optimization objective under the constraint conditions; (2) Use the particle swarm optimization method to perform a primary optimization to obtain the optimal integral parameters α 1,i , α 2,i and β i , and then obtain the corresponding discrete-time motion equation; Step 3: Perform iterative solution of the discrete-time domain motion equation to obtain the structural response; (3.1) Substitute the optimal integration parameters obtained in (2) into the matrix form of the discrete-time domain motion equation to solve for the structural response and obtain the next response; (3.2) Define the structural response obtained in (3.1) as the new initial state of the structure, update it, and substitute it into the discrete-time domain motion equation for the next solution until the preset time condition is met, terminate the iteration, and output the change range of the structural characteristic parameters; (3.3) Select N equally spaced points within the change range of the structural characteristic parameters, and repeat (2) according to the structural characteristic parameters at each point to obtain a corresponding set of optimal integration parameters; Step 4: Calculate the structural dynamic response; (4) Perform iterative solution of the discrete-time motion equation. Select the optimal integration parameters in the corresponding interval according to the value of the structural characteristic parameters at each time step for calculation until the preset time limit is reached, and output the structural response time history curve.

Images