A non-iterative canonical response spectrum matching method for reduced dimension modeling of random ground motions

Abstract

The application discloses a non-iterative high-matching normative response spectrum random seismic motion dimension reduction modeling method, and belongs to the technical field of engineering structure anti-seismic, which comprises the following steps: S1, a stationary seismic motion power spectrum model is established; S2, a non-stationary seismic motion evolution power spectrum model is established; S3, a dimension reduction method based on spectral representation-random orthogonal function is adopted to express the non-stationary seismic motion process as a function form of a single random variable; S4, based on a particle swarm optimization algorithm, the undetermined parameters of the evolution power spectrum model are determined with the normative response spectrum as a target; and S5, a representative sample set of seismic motion acceleration is generated, the average response spectrum is calculated and compared with the normative response spectrum. The average response spectrum of the representative sample set of random seismic motion generated by the method can be directly and high-precisely fitted with the normative response spectrum without iteration, and the representative sample set generated by the application is a complete probability set, which lays a foundation for realizing the anti-seismic reliability analysis of complex engineering structures in the full probability sense.

Description

Technical Field

[0001] This invention belongs to the field of seismic resistance technology for engineering structures. Specifically, it relates to a method for dimensionality reduction modeling of stochastic ground motions using non-iterative standard response spectrum matching. Background Technology

[0002] In the field of seismic analysis of engineering structures, achieving a high-precision match between the response spectra of representative simulated seismic ground motion samples and the code response spectra is crucial for accurate seismic reliability analysis results. The code response spectrum reflects the spectral characteristics of ground motion under specific site conditions and serves as the legal basis for seismic design of structures. Therefore, generating a highly matched seismic ground motion time history is essential for conducting structural dynamic response and seismic reliability analysis.

[0003] Currently, mainstream methods generally rely on iterative correction of the original power spectrum to gradually approximate the target gauge response spectrum. Although such methods can achieve high accuracy, their iterative process involves a large amount of computation, resulting in low computational efficiency, and the generated samples differ significantly from actual ground motion.

[0004] To overcome the aforementioned technical bottlenecks, there is an urgent need to develop a non-iterative, high-precision, and fully probabilistic method for simulating stochastic ground motion. Summary of the Invention

[0005] The purpose of this invention is to provide a non-iterative canonical response spectrum matching method for stochastic ground motion dimensionality reduction modeling, overcoming the shortcomings of traditional ground motion simulation methods that rely on multiple iterations to correct the power spectrum and suffer from low computational efficiency. A non-iterative, fully non-stationary evolving power spectrum model is constructed, and combined with the PSO optimization algorithm and dimensionality reduction method, it can directly generate ground motion samples that accurately match various canonical response spectra in a single step, while retaining complete probabilistic information, thus laying a solid foundation for refined analysis of the seismic reliability of engineering structures.

[0006] To achieve the above objectives, this invention provides a method for dimensionality reduction modeling of stochastic ground motion using non-iterative canonical response spectrum matching, specifically including the following steps:

[0007] S1. Establish a steady ground motion power spectrum model, derive the steady ground motion power spectrum density function based on the dual-filter model, and determine the site soil layer characteristic parameters of the power spectrum model according to the correspondence between the design earthquake group and the site category in the target seismic design code.

[0008] S2. Establish a non-stationary ground motion evolution power spectrum model, and multiply the stationary ground motion power spectrum density function with a time-frequency modulation function to obtain the evolution power spectrum density function.

[0009] S3. Using a dimensionality reduction method based on spectral representation and random orthogonal functions, the non-stationary ground motion process is expressed as a function of a single random variable, generating representative samples of ground motion acceleration.

[0010] S4. Based on the particle swarm optimization algorithm, with the goal of minimizing the error between the average response spectrum of the generated representative sample set and the target standard response spectrum, optimize the undetermined parameters in the evolution power spectral density function.

[0011] S5. Repeat step S3 using the optimized parameters to generate the final representative sample set of seismic motion as the output of the method, and determine that the error between the calculated average response spectrum and the target specification response spectrum is within the allowable range.

[0012] Preferably, in step S1, the expression for the power spectral density function of steady-state ground motion is:

[0013] ;

[0014] in, and These are the characteristic frequency and damping ratio of the site soil, respectively. and These are the characteristic frequencies and damping ratios of the bedrock, respectively. It is the angular frequency (unit: rad / s). The spectral intensity factor is calculated as follows:

[0015] ;

[0016] in, The peak factor is determined by the particle swarm optimization algorithm. This represents the peak ground acceleration.

[0017] Preferably, in step S2, the time-frequency modulation function is: The evolution power spectral density function expression is:

[0018] ;

[0019] Among them, the time-frequency modulation function The expression is: ;

[0020] in, For the duration of the earthquake, Indicates the natural index; , It is a constant.

[0021] Preferably, step S3 specifically includes:

[0022] S31. Based on the source spectrum representation of non-stationary random processes, the seismic acceleration process is expressed... It can be represented in the following form:

[0023] ;

[0024] in, , For frequency intervals, It is a set of orthogonal random variables;

[0025] S32. Constructing a set of orthogonal random variables by introducing a random orthogonal function: Define a set of orthogonal random variables in the interval... Basic random variables with uniform distribution And construct a set of intermediate orthogonal random variables. :

[0026] ;

[0027] S33. Determine the target's set of orthogonal random variables through one-to-one mapping: [This involves] determining the intermediate set of orthogonal random variables... Through a deterministic, one-to-one mapping relationship, it is transformed into a set of orthogonal random variables of the target. ;

[0028] S34. Generate representative seismic motion samples: The mapped samples... Substituting into the formula in step S31, we obtain the seismic acceleration process expressed by a single basic random variable. Furthermore, a representative set of points is selected for a single basic random variable, each representative point has an assigned probability, and a corresponding representative sample with the same assigned probability is generated.

[0029] Preferably, in step S33, the one-to-one mapping relationship is achieved in the following way:

[0030] Using MATLAB's built-in random permutation function, the intermediate orthogonal random variable set is transformed. index Rearrange and determine the set of random variables orthogonal to the target standard. index A one-to-one correspondence between them.

[0031] Preferably, in step S4, the error (including average relative error) between the average response spectrum of the generated representative sample set of seismic ground acceleration and the target specification response spectrum is used. and maximum relative error ), which serves as the fitness function for the particle swarm optimization algorithm;

[0032] Mean relative error The calculation formula is:

[0033] ;

[0034] Maximum relative error The calculation formula is:

[0035] ;

[0036] in, To generate the average response spectrum of a representative set of seismic motion samples in the first... The average response spectrum at each control point is obtained through the following steps: calculate the response spectrum of each representative sample of ground motion acceleration in the representative sample set of ground motion, and then take the average value of all response spectra; The target specification response spectrum is represented in the first... The values ​​at each control point The first period of the natural vibration of the structure is represented by the first period of the natural vibration. discrete point values The structural damping ratio, This represents the total number of discrete points representing the natural vibration period of the structure within a preset period range. The average tolerance is... This represents the maximum permissible error.

[0037] Preferably, in step S4, the execution of the particle swarm optimization algorithm includes the following steps:

[0038] S41. Initialize the particle population: For the first... Each particle, and its initial position. With initial velocity Randomly generated within a preset range:

[0039] ;

[0040] in, and These are the preset minimum and maximum values ​​of the undetermined parameters for the evolution power spectrum, respectively. In order to be in A random number that is uniformly distributed within the range; and These are the preset minimum and maximum particle movement speeds, respectively.

[0041] S42, Iteratively update particle state:

[0042] In each iteration, based on the individual's historical best position and the global optimal position of the group Update the velocity and position of each particle:

[0043] ;

[0044] in, and These are the updated particle velocities and particle positions. Indicates the number of iterations; and It is a learning factor. and is a random number uniformly distributed in the interval [0,1].

[0045] S43. Calculate adaptive random weights:

[0046] Weights in the speed update formula The adaptive random weights are calculated using the following formula:

[0047] ;

[0048] in, It is a constant. Represents a random number that follows a standard normal distribution; , and It is a constant.

[0049] Preferred, average tolerance Maximum permissible error .

[0050] Therefore, this invention employs the aforementioned non-iterative canonical response spectrum matching method for stochastic ground motion dimensionality reduction modeling. It constructs a non-iterative evolving power spectrum model and, combined with the PSO optimization algorithm and dimensionality reduction method, can directly generate fully non-stationary ground motion samples consistent with multiple types of canonical response spectra. This significantly improves computational efficiency while maintaining fitting accuracy and preserving complete probabilistic information, laying a solid foundation for the seismic reliability analysis of complex engineering structures.

[0051] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0052] Figure 1 This is a flowchart of a method according to an embodiment of the present invention;

[0053] Figure 2 This is the acceleration time history diagram of the second representative sample in the embodiments of the present invention;

[0054] Figure 3 This is the acceleration time history diagram of the 123rd representative sample in this embodiment of the invention;

[0055] Figure 4 This is the acceleration time history diagram of the 233rd representative sample in this embodiment of the invention;

[0056] Figure 5 This is an evolutionary power spectral density diagram of an embodiment of the present invention;

[0057] Figure 6 This is a comparison chart of the average response spectrum of the embodiments of the present invention and the standard response spectrum specified in the "Code for Seismic Design of Buildings" (GB 50011-2010);

[0058] Figure 7 This is a comparison chart of the average response spectrum of the embodiments of the present invention and the standard response spectrum specified in the "Code for Seismic Design of Highway Bridges" (JTG / T 2231-01-2020);

[0059] Figure 8 This is a comparison chart of the average response spectrum of the present invention and the standard response spectrum specified in the "Seismic Design Standard for Hydraulic Structures" (GB 51247-2018). Detailed Implementation

[0060] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0061] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning understood by one of ordinary skill in the art to which this invention pertains. The terms "first," "second," and similar terms used in this invention do not indicate any order, quantity, or importance, but are merely used to distinguish different components. Terms such as "comprising" or "including" mean that the element or object preceding the word encompasses the elements or objects listed following the word and their equivalents, without excluding other elements or objects. Terms such as "connected" or "linked" are not limited to physical or mechanical connections, but can include electrical connections, whether direct or indirect. Terms such as "upper," "lower," "left," and "right" are used only to indicate relative positional relationships; when the absolute position of the described object changes, the relative positional relationship may also change accordingly.

[0062] Example

[0063] like Figure 1 As shown, a non-iterative canonical response spectrum matching method for dimensionality reduction modeling of stochastic ground motion includes the following steps:

[0064] S1. Establish a steady ground motion power spectrum model, derive the steady ground motion power spectrum density function based on the dual-filter model, and determine the site soil layer characteristic parameters of the model according to the correspondence between the design earthquake group and the site category in the target seismic design code.

[0065] The expression for the power spectral density function of steady-state ground motion is:

[0066] ;

[0067] in, and These represent the characteristic frequency and damping ratio of the site soil, respectively. For ease of application, based on the characteristic period values ​​given in the "Code for Seismic Design of Buildings" (GB50011-2010) and existing research results, suggested parameter values ​​for the steady-state ground motion power spectrum model are directly provided, as shown in Table 1. and These are the characteristic frequencies and damping ratios of the bedrock, respectively. It is the angular frequency (unit: rad / s). The spectral intensity factor is calculated as follows:

[0068] ;

[0069] in, The peak factor is determined by the particle swarm optimization algorithm. This represents the peak ground acceleration.

[0070] Table 1. Parameter values ​​for the power spectrum model of steady ground motion.

[0071]

[0072] S2. Establish a non-stationary ground motion evolution power spectrum model, in which the stationary ground motion power spectrum density function is multiplied by a time-frequency modulation function to obtain the evolution power spectrum density function.

[0073] The time-frequency modulation function is The evolution power spectral density function expression is:

[0074] ;

[0075] Among them, the time-frequency modulation function The expression is:

[0076] ;

[0077] in, For the duration of the earthquake, Represents the natural index. , It is a constant.

[0078] S3. Using a dimensionality reduction method based on spectral representation-random orthogonal functions, the non-stationary ground motion process is represented as a function of a single random variable, generating the ground motion acceleration time history;

[0079] S31. Based on the spectral representation theory of non-stationary random processes, the seismic acceleration process... It can be represented in the following form:

[0080] ;

[0081] in, , For frequency intervals, It is a set of orthogonal random variables;

[0082] S32. Constructing a set of orthogonal random variables by introducing a random orthogonal function: Define a set of orthogonal random variables in the interval... Basic random variables with uniform distribution And construct a set of intermediate orthogonal random variables. :

[0083] ;

[0084] S33. Determine the target orthogonal random variable set through one-to-one mapping: [This involves] mapping the intermediate orthogonal random variable set... A deterministic, one-to-one mapping relationship is established (which can be achieved using rand('stae',0) and randperm(2N) in the Matlab program). Mapped one by one to Transform it into a target orthogonal random variable set. ;

[0085] S34. Generate representative seismic motion samples: The mapped samples... Substituting into the formula in step S31, we obtain the seismic acceleration process expressed by a single basic random variable. Furthermore, a representative set of points is selected for a single basic random variable. Each representative point has an assigned probability, and a corresponding representative sample with the same assigned probability is generated. Based on this, a representative sample set of seismic motion is generated for subsequent seismic design analysis.

[0086] S4. Based on the particle swarm optimization algorithm, optimize the undetermined parameters in the evolution power spectrum with the goal of minimizing the error between the average response spectrum of the generated time history and the target standard response spectrum;

[0087] The error between the average response spectrum generated by the acceleration time history and the target specification response spectrum (including the average relative error) With the maximum relative error ), which serves as the fitness function for the particle swarm optimization algorithm;

[0088] Mean relative error The calculation formula is:

[0089] ;

[0090] Maximum relative error The calculation formula is:

[0091] ;

[0092] in, To generate the average response spectrum of a representative set of seismic motion samples in the first... The average response spectrum at each control point is obtained through the following steps: calculate the response spectrum of each representative sample of ground motion acceleration in the representative sample set of ground motion, and then take the average value of all response spectra; The target specification response spectrum is represented in the first... The values ​​at each control point The first period of the natural vibration of the structure is represented by the first period of the natural vibration. discrete point values The structural damping ratio, This represents the total number of discrete points representing the natural vibration period of the structure within a preset period range. The average tolerance is... This represents the maximum permissible error.

[0093] The execution of the particle swarm optimization algorithm includes the following steps:

[0094] S41. Initialize the particle population: For the first... Each particle, and its initial position. With initial velocity Randomly generated within a preset range:

[0095] ;

[0096] in, and These are the preset minimum and maximum values ​​of the undetermined parameters for the evolution power spectrum, respectively. In order to be in A random number that is uniformly distributed within the range; and These are the preset minimum and maximum particle movement speeds, respectively.

[0097] S42, Iteratively update particle state:

[0098] In each iteration, based on the individual's historical best position and the global optimal position of the group Update the velocity and position of each particle:

[0099] ;

[0100] in, and These are the updated particle velocities and particle positions. Indicates the number of iterations; and It is a learning factor. and is a random number uniformly distributed in the interval [0,1].

[0101] S43. Calculate adaptive random weights:

[0102] Weights in the speed update formula The adaptive random weights are calculated using the following formula:

[0103] ;

[0104] in, It is a constant. Represents a random number that follows a standard normal distribution; , and It is a constant.

[0105] S5. Repeat step S3 using the optimized parameters to generate the final representative sample set of seismic motions, and calculate the relative error between the average response spectrum and the gauge response spectrum.

[0106] Example 1

[0107] The specific implementation process is as follows: Figure 1 As shown.

[0108] Step 1: Run the particle swarm optimization algorithm to determine the parameters to be determined in the evolution spectrum.

[0109] (1) Algorithm initialization:

[0110] Set PSO algorithm parameters: Population size Solution vector dimension The number of iterations is 40.

[0111] Parameter boundary definition:

[0112] Search range for each parameter:

[0113]

[0114]

[0115] Initialize particle position and velocity.

[0116] (2) Parallel computing environment configuration:

[0117] Detect and start the MATLAB parallel computing pool to evaluate particle fitness in a parallel manner, significantly improving computational efficiency.

[0118] (3) Phased parameter configuration and population management:

[0119] A phased strategy is adopted to manage key parameters: During the PSO optimization process, the population size, learning factor, and inertia weight are not fixed, but are systematically adjusted according to the iteration stage.

[0120] Phase 1 (Generations 1-5): Focus on overall exploration.

[0121] Phase 2 (Generations 6-10):

[0122] Balance exploration and development

[0123] Phase 3 (Generations 11-40):

[0124] Focus on local search

[0125] Implement a phased population reduction strategy: retain 350 optimal particles in the 6th generation and 250 optimal particles in the 11th generation.

[0126] (4) Fitness assessment:

[0127] The fitness function is defined as the weighted sum of the average relative error and the maximum relative error, taking into account both overall and local fitting accuracy. The convergence criteria are set as follows: average relative error < 5% and maximum relative error < 15%. During optimization, all feasible solutions that meet these criteria are recorded for subsequent analysis and result selection.

[0128] (5) Convergence control:

[0129] When premature convergence is detected, a perturbation strategy is used to reset the particle position.

[0130] (6) Output of results:

[0131] Output the optimal parameter combination and its corresponding fitness value; save all feasible solutions to a text file.

[0132] This embodiment integrates staged parameter configuration, population simplification strategy, and perturbation mechanism to construct a highly efficient and robust PSO optimization method. By pre-setting search strategies and population optimization strategies for different stages, this method effectively balances the algorithm's global exploration and local search capabilities, achieving higher accuracy in optimizing evolutionary power spectrum parameters.

[0133] Step 2: Apply dimensionality reduction methods to generate 233 representative samples of ground motion.

[0134] Based on the above method, representative samples of time-frequency fully non-stationary random ground motions are generated, with parameters... , , , , , The parameters were obtained from the particle swarm optimization algorithm in step one, with the remaining parameters taken according to the standard values. All parameters used in this numerical example are detailed in Table 2, and representative samples obtained from the simulation are shown in [Table 2]. Figures 2-4 .

[0135] Table 2 Parameter values ​​for the embodiments

[0136]

[0137] Step 3: Calculate the average reaction spectrum and compare it with the standard reaction spectrum.

[0138] The 233 representative acceleration samples generated in step two above are used as input, and the following methods are applied: The reaction spectrum was calculated and averaged, and finally compared with the standard reaction spectrum. Figure 6 .

[0139] It should be noted that, Figures 2-4 Representative sample diagram of acceleration, Figure 5 The evolution of power spectral density plot and Figure 6 All results are based on the "Code for Seismic Design of Buildings" (GB50011-2010). The "Code for Seismic Design of Highway Bridges" (JTG / T2231-01-2020) and the "Standard for Seismic Design of Hydraulic Structures" (GB 51247-2018) are presented only as comparisons of the final response spectra. Figure 7 and Figure 8 .

[0140] Therefore, the present invention employs the above-mentioned non-iterative canonical response spectrum matching method for dimensionality reduction modeling of stochastic ground motion, which has the following advantages:

[0141] (1) The method proposed in this invention can directly generate fully non-stationary ground motions that are consistent with various standard response spectra without the need for traditional iterative processes, thereby significantly improving computational efficiency.

[0142] (2) Compared with the traditional Monte Carlo method, the ground motion generated by the dimensionality reduction method has higher computational efficiency and can better balance accuracy, demonstrating good engineering applicability. In addition, the random ground motion generated by the dimensionality reduction method retains complete probabilistic information, laying a solid foundation for the analysis of the stochastic dynamic response and reliability of engineering structures under seismic loading.

[0143] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for dimensionality reduction modeling of stochastic ground motions using non-iterative canonical response spectrum matching, characterized in that, Includes the following steps: S1. Establish a steady ground motion power spectrum model, derive the steady ground motion power spectrum density function based on the dual-filter model, and determine the soil layer characteristic parameters in the power spectrum model according to the correspondence between the design earthquake group and the site category in the target seismic design code. S2. Establish a non-stationary ground motion evolution power spectrum model, and multiply the stationary ground motion power spectrum density function with a time-frequency modulation function to obtain the evolution power spectrum density function. S3. Using a dimensionality reduction method based on spectral representation and random orthogonal functions, the non-stationary ground motion process is expressed as a function of a single random variable, generating representative samples of ground motion acceleration. Specifically, it includes: S31. Based on the spectral representation theory of non-stationary random processes, the seismic acceleration process... It can be represented in the following form: ； in, , For frequency intervals, Given a set of orthogonal random variables, The power spectral density function is the evolution function. S32. Constructing a set of orthogonal random variables by introducing a random orthogonal function: Define a set of orthogonal random variables in the interval... Basic random variables with uniform distribution And construct a set of intermediate orthogonal random variables. : ； S33. Determine the target orthogonal random variable set through one-to-one mapping: [This involves] mapping the intermediate orthogonal random variable set... Through a deterministic, one-to-one mapping relationship, it is transformed into a target orthogonal random variable set. ; S34. Generate representative samples of random ground motion: (The mapped samples are then...) Substituting into the formula in step S31, we obtain the seismic acceleration process expressed by a single basic random variable; a representative set of points is selected for the single basic random variable, each representative point has an assigned probability, and a corresponding representative sample with the same assigned probability is generated. S4. Based on the particle swarm optimization algorithm, with the objective of minimizing the error between the average response spectrum of the representative sample set generated in step S3 and the target specification response spectrum, optimize the undetermined parameters in the evolution power spectral density function; the undetermined parameters include the characteristic frequencies of the site bedrock. Damping ratio of the bedrock of the site Peak factor and the constants in the time-frequency modulation function. , , ; S5. Repeat step S3 using the optimized parameters to generate the final representative sample set of random ground motions as the output of the method, and determine that the error between the calculated average response spectrum and the target specification response spectrum is within the allowable range.

2. The method for dimensionality reduction modeling of stochastic ground motions based on non-iterative canonical response spectrum matching according to claim 1, characterized in that, In step S1, the expression for the power spectral density function of steady-state ground motion is: ； in, and These are the characteristic frequency and damping ratio of the site soil, respectively. and These are the characteristic frequencies and damping ratios of the bedrock at the site. It is the angular frequency. The spectral intensity factor is calculated as follows: ； in, The peak factor is determined by the particle swarm optimization algorithm; This represents the peak ground acceleration.

3. The method for dimensionality reduction modeling of stochastic ground motion based on non-iterative canonical response spectrum matching according to claim 2, characterized in that, In step S2, the time-frequency modulation function is: The evolution power spectral density function expression is: ； Among them, the time-frequency modulation function The expression is: ； in, For the duration of the earthquake, This represents the natural exponential function. , It is a constant.

4. The method for dimensionality reduction modeling of stochastic ground motion based on non-iterative canonical response spectrum matching according to claim 1, characterized in that, In step S33, the one-to-one mapping relationship is achieved in the following way: Using MATLAB's built-in random permutation function, the intermediate orthogonal random variable set is transformed. index Random rearrangement to determine a set of random variables orthogonal to the target standard. index The correspondence between them.

5. The method for dimensionality reduction modeling of stochastic ground motion based on non-iterative canonical response spectrum matching according to claim 1, characterized in that, In step S4, the error between the average response spectrum of the representative sample set and the target standard response spectrum is calculated, including the average relative error. With the maximum relative error , which serves as the fitness function for the particle swarm optimization algorithm; Mean relative error The calculation formula is: ； Maximum relative error The calculation formula is: ； in, To generate the average response spectrum of a representative set of seismic motion samples in the first... The average response spectrum at each control point is obtained through the following steps: calculate the response spectrum of each representative sample of ground motion acceleration in the representative sample set of ground motion, and then take the average value of all response spectra; The target specification response spectrum is represented in the first... The values ​​at each control point The first period of the natural vibration of the structure is represented by the first period of the natural vibration. discrete point values The structural damping ratio, This represents the total number of discrete points representing the natural vibration period of the structure within a preset period range. The average tolerance is... This represents the maximum permissible error.

6. The method for dimensionality reduction modeling of stochastic ground motion based on non-iterative canonical response spectrum matching according to claim 5, characterized in that, In step S4, the execution of the particle swarm optimization algorithm includes the following steps: S41. Initialize the particle population: For the first... Each particle, and its initial position. With initial velocity Randomly generated within a preset range: ； in, and These are the preset minimum and maximum values ​​of the undetermined parameters for the evolution power spectrum, respectively. In order to be in A random number that is uniformly distributed within the range; and These are the preset minimum and maximum particle movement speeds, respectively. S42, Iteratively update particle state: In each iteration, based on the individual's historical best position and the global optimal position of the group Update the velocity and position of each particle: ； in, and These are the updated particle velocities and particle positions. Indicates the number of iterations; and It is a learning factor. and is a random number uniformly distributed in the interval [0,1]. S43. Calculate adaptive random weights: Weights in the speed update formula The adaptive random weights are calculated using the following formula: ； in, It is a constant. Represents a random number that follows a standard normal distribution; , and It is a constant.

7. The method for dimensionality reduction modeling of stochastic ground motion based on non-iterative canonical response spectrum matching according to claim 5, characterized in that, Average tolerance Maximum permissible error .

Images