CN107480325B - The non-stationary non-gaussian earthquake motion time history analogy method of spatial variability - Google Patents
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Abstract
The present invention discloses a kind of non-stationary non-gaussian earthquake motion time history analogy method of spatial variability, this method passes through the iterative solution to non-stationary Gaussian power spectrum matrix, rather than it is simply assumed to and target non-stationary non-gaussian spectral power matrix, so that the non-stationary spectral power matrix of the non-stationary non-gaussian earthquake motion time history for the spatial variability simulated is consistent with target non-stationary power spectrum, the accuracy of simulation ensure that.After iteratively solving obtained potential non-stationary Gaussian power spectrum matrix, the non-stationary Gauss earthquake motion time history of spectral representation method simulation spatial variability can be used, then non-stationary non-gaussian earthquake motion time history is converted for the non-stationary Gauss earthquake motion time history of each point by the non-linear conversion between destination probability distribution function and Gaussian Profile again.The non-stationary non-gaussian earthquake motion time history precision of this method simulation is high, Iterations of Multi is strong, and can be used in combination with FFT, to ensure that simulation precision, is suitable for popularization and application.
Description
Technical Field
The invention relates to a method for simulating earthquake motion time course, in particular to a method for simulating nonstationary non-Gaussian earthquake motion time course with spatial variation, which belongs to the field of civil engineering anti-seismic design and can provide earthquake waves with input spatial variation for time course analysis of anti-seismic design of large-span structures such as bridges, tunnels and the like.
Background
For large-span structures, relevant anti-seismic specification requirements are met, and nonlinear dynamic time-course analysis needs to be carried out under the action of seismic waves with space variation in structural anti-seismic design. However, the relevant actual measurement seismic records show that the seismic motion time course shows certain non-gaussian property, so that a seismic motion method capable of simulating the non-gaussian non-stationary property reflecting the spatial variation of the real seismic motion property is needed to be provided.
At present, the research on the simulation aspect of the spatial variant non-stationary non-Gaussian seismic motion time interval mainly comprises the steps of establishing a non-stationary self-power spectrum model, a coherent function model and a non-Gaussian probability distribution model of actual measurement seismic motion through statistical distribution of the actual measurement seismic motion, generating the spatial variant non-stationary Gaussian seismic motion time interval by using a spectrum representation method, and obtaining the spatial variant non-stationary non-Gaussian seismic motion time interval according with given distribution characteristics by using a non-linear conversion model according to the non-Gaussian distribution characteristics of seismic motion of each point. The core steps are as follows:
(1) establishing a non-stationary self-power spectrum model, a coherent function model and a non-Gaussian probability distribution model of the measured seismic oscillation through the statistical distribution of the measured seismic oscillation;
(2) generating a potential spatial variation non-stationary Gaussian seismic motion time interval by using a certain non-stationary power spectrum matrix and adopting a spectral representation method;
(3) and converting the potential spatially variant non-stationary Gaussian seismic motion time interval into a spatially variant non-stationary non-Gaussian seismic motion time interval through nonlinear conversion.
Of the above steps, the most difficult is the determination of "some non-stationary power spectrum matrix" in step 2. No relevant research has been presented to reasonably determine a "certain non-stationary power spectrum matrix".
Disclosure of Invention
The purpose of the invention is as follows: the invention provides a method for simulating the time travel of the non-stationary non-Gaussian earthquake motion of spatial variation, which solves a non-stationary Gaussian power spectrum matrix through an iteration method, so that the non-stationary power spectrum matrix of the simulated non-stationary non-Gaussian earthquake motion time travel of the spatial variation is consistent with a target non-stationary power spectrum matrix, and the simulation accuracy is ensured.
The technical scheme is as follows: the invention relates to a method for simulating the time travel of a non-stationary non-Gaussian earthquake with spatial variation, which comprises the following steps:
1) giving target non-stationary non-Gaussian power spectrum matrixes of each point of spatial variation and seismic motion time-course distribution characteristics of each point;
2) an initial non-stationary Gaussian power spectrum matrix is assumed, and corresponding non-stationary Gaussian correlation function matrix elements are solved through the matrix elements;
3) respectively solving non-stationary non-Gaussian correlation function matrix elements corresponding to the non-stationary Gaussian correlation function matrix elements based on the distribution characteristics of the seismic motion time courses of all the points;
4) calculating non-stationary non-Gaussian power spectrum matrix elements from the non-stationary non-Gaussian correlation function matrix elements to obtain a non-stationary non-Gaussian power spectrum matrix corresponding to the non-stationary Gaussian power spectrum matrix, and comparing the error between the non-stationary non-Gaussian power spectrum matrix and a target non-stationary non-Gaussian power spectrum matrix;
5) decomposing a target non-stationary non-Gaussian power spectrum matrix, a non-stationary Gaussian power spectrum matrix and a non-stationary non-Gaussian power spectrum matrix, iterating decomposed matrix elements to obtain new non-stationary Gaussian power spectrum matrix elements, standardizing the new non-stationary Gaussian power spectrum matrix elements, and randomly sequencing all variables;
6) returning to the step 2), obtaining a new non-stationary non-Gaussian power spectrum matrix element, calculating an error between the new non-stationary non-Gaussian power spectrum matrix element and a target non-stationary non-Gaussian power spectrum matrix element, stopping iteration when the error meets the precision requirement, and otherwise, repeating the steps 2) to 5) until the requirement is met;
7) and simulating a spatially variant non-stationary Gaussian seismic motion time interval based on a spectral representation method by using a potential non-stationary Gaussian power spectrum matrix obtained by iteration, and converting the spatially variant non-stationary Gaussian seismic motion time interval into the required spatially variant non-stationary Gaussian seismic motion time interval through nonlinear transformation.
Preferably, in step 2), the target non-stationary non-gaussian power spectrum matrix is used as the initial non-stationary gaussian power spectrum matrix. And a target non-stationary non-Gaussian power spectrum matrix is used as an initial matrix, so that the convergence speed is high.
In the step 2), the non-stationary Gaussian power spectrum matrix element S is selected according to the following formula (6)Gjk(w, t) solving the corresponding non-stationary Gaussian correlation function matrix element RGjk(t,s)：
Wherein,respectively representing non-stationary gaussian power spectrum functions at time t and time s, and omega is frequency.
In step 3), the non-stationary non-Gaussian correlation function matrix element R is determined through the following formulas (7) to (8)NGjk(t,s)：
ρGjk(t,s)＝RGjk(t,s)/(σGj(t)σGk(s)) (8)；
In the above formula, FNGjAnd FNGkProbability distribution functions of the j-th point time interval and the k-th point time interval respectively, phi represents a two-dimensional standard normal distribution function, rhoGjkIs a normalized Gaussian correlation function of the time interval at the j point and the time interval at the k point.
In step 4), the non-stationary non-Gaussian correlation function matrix element R is obtained from the non-stationary non-Gaussian correlation function matrix by the following formula (9)NGjk(t, S) deducing the non-stationary non-Gaussian power spectrum matrix element SNGjk(w,t)：
In the above formula, the first and second carbon atoms are,is a non-stationary non-gaussian correlation function between the time interval at the j point at the time t and the time interval at the k point at the time t + tau,is the non-stationary non-Gaussian correlation function of the j-th time interval at the t moment and the k-th time interval at the t-tau moment.
Further, in step 4), comparing the error between the non-stationary non-gaussian power spectrum matrix element and the target non-stationary non-gaussian power spectrum matrix element according to the following formula (10):
in the above formula, the first and second carbon atoms are,is a target non-stationary non-Gaussian cross-power spectrum function of the j-th time interval and the k-th time interval,is a non-stationary non-Gaussian cross-power spectral function at the (i) th iteration of the j-th time interval and the k-th time interval.
In the step 5), the target non-stationary non-Gaussian power spectrum matrix is subjected toNon-stationary Gaussian power spectrum matrixAnd a non-stationary non-Gaussian power spectrum matrixDecomposing to obtain matrix of For decomposed non-stationary Gaussian power spectrum matrix element DGjk(w, t) iteratively updates according to equation (11) below:
specifically, in step 6), when the average error between the non-stationary non-gaussian power spectrum matrix element and the target non-stationary non-gaussian power spectrum matrix element is greater than the average error of the previous iteration, the iteration is stopped.
In the step 7), after the non-stationary gaussian seismic motion time interval of the spatial variation is simulated based on the spectral representation method, the non-stationary non-gaussian seismic motion time interval of the spatial variation to be simulated is obtained based on the nonlinear transformation relation between the gaussian distribution and the target distribution at each moment; wherein the nonlinear conversion formula is:
in the above formula, FGjAnd FNGjRespectively a gaussian probability distribution function and a target non-gaussian distribution function of the jth time interval.
In the step 3), when the formula (7) is used for solving, since the solution needs to be performed at every time, it takes much time. The distribution form of the earthquake motion obedience at each moment is generally considered to be the same, and the variances are different, so that the non-stationary non-Gaussian correlation function matrix elements when the variance is 1 under the target distribution form can be solved firstly, a table is established, the value when the variance is 1 is obtained by directly reading the data of the table, then the value is multiplied by the standard deviation to obtain the non-stationary non-Gaussian correlation function matrix elements under the corresponding variance, and the solving time is greatly saved.
Has the advantages that: compared with the prior art, the invention has the advantages that: (1) the simulation method has high precision, and the consistency of the final non-stationary non-Gaussian power spectrum matrix and a target power spectrum function can be ensured by carrying out iterative solution on the non-stationary Gaussian power spectrum matrix; moreover, after each iteration, all variables are randomly sequenced, so that the convergence speed consistency of all variables can be ensured; (2) the iteration method adopted by the invention has high efficiency, the iteration method directly carries out iteration solution through the theoretical relationship between the non-stationary Gaussian power spectrum matrix and the non-stationary Gaussian power spectrum matrix, and the matrix elements after decomposition are iterated in the solution process, so that the positive nature of the power spectrum matrix can be ensured, and the iterative convergence is ensured; (3) the simulation method of the invention uses a spectrum representation method when simulating the non-stationary Gaussian seismic motion time interval of the potential spatial variation, can overcome the difficulty of solving the non-stationary characteristic function by KL decomposition, and is combined with the FFT technology, thereby greatly improving the simulation efficiency.
Drawings
FIG. 1 is a non-stationary non-Gaussian self-power spectrum function of a predetermined target in an embodiment;
FIG. 2a is a potential non-stationary Gaussian self-power spectrum function of a first point obtained by iteration in the embodiment;
FIG. 2b is a potential non-stationary Gaussian cross-power spectral function of a first point and a second point obtained by iteration in the embodiment;
FIG. 2c is a potential non-stationary Gaussian cross-power spectral function of the first point and the third point obtained by iteration in the embodiment;
FIG. 2d is a diagram illustrating a potential non-stationary Gaussian self-power spectrum function of a second point obtained by iteration in the embodiment;
FIG. 2e is a potential non-stationary Gaussian cross-power spectral function of the second point and the third point obtained by iteration in the embodiment;
FIG. 2f is a diagram of a potential non-stationary Gaussian self-power spectrum function of a third point obtained by iteration in the embodiment;
FIG. 3a is a non-stationary Gaussian seismic motion time interval of a first point and a non-stationary non-Gaussian seismic motion time interval corresponding to the non-stationary Gaussian seismic motion time interval obtained by simulation in the embodiment; wherein 1 is a non-stationary Gaussian seismic motion time interval of a first point obtained by simulation; 2, obtaining a non-stationary non-Gaussian seismic motion time course of a first point by simulation;
FIG. 3b is a non-stationary Gaussian seismic motion time interval of a second point obtained by simulation in the embodiment and a non-stationary non-Gaussian seismic motion time interval corresponding to the non-stationary Gaussian seismic motion time interval; wherein 3 is a non-stationary Gaussian seismic motion time interval of the first point obtained by simulation; 4, obtaining the non-stationary non-Gaussian seismic motion time course of the first point by simulation;
FIG. 3c is a non-stationary Gaussian seismic motion time interval of a third point obtained by simulation in the embodiment and a non-stationary non-Gaussian seismic motion time interval corresponding to the non-stationary Gaussian seismic motion time interval; wherein 5 is a non-stationary Gaussian seismic motion time interval of a third point obtained by simulation; and 6, obtaining a non-stationary non-Gaussian seismic motion time course of a third point by simulation.
Detailed Description
The technical scheme of the invention is further explained by combining the attached drawings.
According to the non-stationary non-Gaussian seismic motion time-course simulation method for spatial variation, a non-stationary Gaussian power spectrum matrix required by non-stationary Gaussian seismic motion simulation for generating potential spatial variation is obtained through solving by an iterative algorithm, and the iterative algorithm can ensure the consistency of the final non-stationary non-Gaussian power spectrum matrix and a target power spectrum matrix.
The simulation method is explained by taking the non-stationary non-Gaussian seismic motion time course of three points simulating one space variation as an example; the simulated points are three points which are distributed on the ground surface at equal intervals, and the intervals are 100 m.
The invention relates to a method for simulating the time travel of a non-stationary non-Gaussian earthquake with spatial variation, which comprises the following steps:
1) firstly, assuming that the target non-stationary non-Gaussian self-power spectral function of each point is as follows:
wherein
In the formulas (1) to (3), σ is a standard deviation; omegagAnd ζgIs the characteristic frequency and damping of the graph, ωfAnd ζfAre filtering parameters, which can be set as: omegaf＝0.1ωg，ζf＝ζg(ii) a t is time, and lambda is a parameter representing seismic wave attenuation.
In this embodiment, the relevant parameter values are: sigma 110 (cm/s)3/2),ωg＝30-1.25t(rad/s),ζg＝0.5+0.005t,t1＝2s,t210s, λ 0.4; the obtained target non-stationary non-Gaussian self-power spectrum function is shown in figure 1.
Assume that the coherence function between points is:
ρjk(ω)＝Aexp[-2djk(1-A+αA)/αθ(ω)]+(1-A)exp[2djk(1-A+αA)/θ(ω)] (4)
wherein
θ(ω)＝K[1+(ω/2πf0)b]-1/2 (5)
In the formulae (4) to (5), A, α and K, f0And b is a model parameter whose values are a 0.63, α 0.0186, K31200, f01.51, b 2.95; the apparent wave velocity of seismic oscillation is 500 m/s; djkIs the distance between two points.
And constructing the target non-stationary non-Gaussian self-power spectrum matrix through the target non-stationary non-Gaussian self-power spectrum function and the coherent function.
Meanwhile, the seismic motion time course is assumed to be distributed according to Student's t, and the parameter c is 6.
2) And assuming the initial non-stationary Gaussian power spectrum matrix as a target non-stationary non-Gaussian self-power spectrum matrix.
3) Solving the non-stationary gaussian correlation function matrix elements by the non-stationary gaussian power spectrum matrix elements using the following formula (6):
wherein,respectively representing non-stationary gaussian power spectrum functions at time t and time s, and omega is frequency.
4) Solving the non-stationary non-Gaussian correlation function matrix elements by the non-stationary Gaussian correlation function matrix elements according to the following formulas (7) to (8):
ρGjk(t,s)＝RGjk(t,s)/(σGj(t)σGk(s)) (8)；
FNGjand FNGkProbability distribution functions of the j-th point time interval and the k-th point time interval respectively, phi represents a two-dimensional standard normal distribution function, rhoGjkIs a normalized Gaussian correlation function of the time interval at the j point and the time interval at the k point.
Generally, the distribution forms of the earthquake motion obeys at each moment are considered to be the same, only the variances are different, when the formula (7) is used for solving, firstly, the non-stable non-Gaussian correlation function matrix element when the variance is 1 under the target distribution form is solved, a table is established, and rho corresponding to the variance of 1 at each moment is recordedGjkAnd (t, s) directly reading the table data to obtain the value of the non-stationary non-Gaussian correlation function matrix element when the variance is 1, and then multiplying the value by the standard deviation to obtain the non-stationary non-Gaussian correlation function matrix element under the corresponding variance.
5) And (3) obtaining the non-stationary non-Gaussian power spectrum matrix element based on the non-stationary non-Gaussian correlation function matrix element by the following estimation formula (9):
in the above formula, the first and second carbon atoms are,is a non-stationary non-gaussian correlation function between the time interval at the j point at the time t and the time interval at the k point at the time t + tau,is the non-stationary non-Gaussian correlation function of the j-th time interval at the t moment and the k-th time interval at the t-tau moment.
And obtaining a non-stationary non-Gaussian power spectrum matrix corresponding to the non-stationary Gaussian power spectrum matrix according to the obtained non-stationary non-Gaussian power spectrum matrix elements.
6) Comparing the error between the non-stationary non-gaussian power spectrum matrix element and the target non-stationary non-gaussian power spectrum matrix element according to equation (10):
wherein,is a target non-stationary non-Gaussian cross-power spectrum function of the j-th time interval and the k-th time interval,is a non-stationary non-Gaussian cross-power spectral function at the (i) th iteration of the j-th time interval and the k-th time interval.
7) For target non-stationary power spectrum matrixNon-stationary Gaussian power spectrum matrixAnd a non-stationary non-Gaussian power spectrum matrixDecomposing to obtain matrix of
8) Updating the decomposed non-stationary Gaussian power spectrum matrix elements according to the following formula (11):
this results in new non-stationary gaussian power spectrum matrix elements.
Repeating the step 2) to the step 8), and performing iteration, wherein the final power spectrum error function obtained by 8 times of iteration is as follows:
the resulting potential non-stationary gaussian power spectrum matrix elements for each point are shown in fig. 2 a-2 f.
Then, a potential non-stationary Gaussian power spectrum matrix obtained through iteration is used, a spectrum representation method is used for simulating to obtain a potential non-stationary non-Gaussian seismic time interval of spatial variation, then a nonlinear conversion formula is shown as a formula (12) based on a nonlinear conversion relation between Gaussian distribution and target distribution, and finally the non-stationary non-Gaussian seismic time interval of the spatial variation to be simulated is obtained, and the time interval is shown in FIGS. 3 a-3 c.
The nonlinear conversion formula is:
in the above formula, FGjAnd FNGjRespectively a gaussian probability distribution function and a target non-gaussian distribution function of the jth time interval.
Claims (10)
1. A method for simulating the time travel of non-stationary non-Gaussian earthquake with spatial variation is characterized by comprising the following steps:
1) giving target non-stationary non-Gaussian power spectrum matrixes of each point of spatial variation and seismic motion time-course distribution characteristics of each point;
2) an initial non-stationary Gaussian power spectrum matrix is assumed, and corresponding non-stationary Gaussian correlation function matrix elements are solved through the matrix elements;
3) respectively solving non-stationary non-Gaussian correlation function matrix elements corresponding to the non-stationary Gaussian correlation function matrix elements based on the distribution characteristics of the seismic motion time courses of all the points;
4) calculating non-stationary non-Gaussian power spectrum matrix elements from the non-stationary non-Gaussian correlation function matrix elements, and comparing the non-stationary non-Gaussian power spectrum matrix elements with the target non-stationary non-Gaussian power spectrum matrix elements;
5) decomposing a target non-stationary non-Gaussian power spectrum matrix, a non-stationary Gaussian power spectrum matrix and a non-stationary non-Gaussian power spectrum matrix, iterating decomposed matrix elements to obtain new non-stationary Gaussian power spectrum matrix elements, standardizing the new non-stationary Gaussian power spectrum matrix elements, and randomly sequencing all variables;
6) returning to the step 2), obtaining a new non-stationary non-Gaussian power spectrum matrix element, calculating an error between the new non-stationary non-Gaussian power spectrum matrix element and a target non-stationary non-Gaussian power spectrum matrix element, stopping iteration when the error meets the precision requirement, and otherwise, repeating the steps 2) to 5) until the requirement is met;
7) and simulating a spatially variant non-stationary Gaussian seismic motion time interval based on a spectral representation method by using a potential non-stationary Gaussian power spectrum matrix obtained by iteration, and converting the spatially variant non-stationary Gaussian seismic motion time interval into the required spatially variant non-stationary Gaussian seismic motion time interval through nonlinear transformation.
2. The method as claimed in claim 1, wherein the target non-stationary non-gaussian power spectrum matrix is used as the initial non-stationary gaussian power spectrum matrix in step 2).
3. The method as claimed in claim 1, wherein the non-stationary Gaussian seismic motion time interval simulation method of spatial variation is characterized in that in step 2), the non-stationary Gaussian power spectrum matrix element S is obtained according to the following formula (6)Gjk(w, t) solving the corresponding non-stationary Gaussian correlation function matrix element RGjk(t,s)：
Wherein,andrespectively representing non-stationary gaussian power spectrum functions at time t and time s, and omega is frequency.
4. The method according to claim 3, wherein in step 3), the non-stationary non-Gaussian correlation function matrix element R is determined according to the following equations (7) to (8)NGjk(t,s)：
ρGjk(t,s)＝RGjk(t,s)/(σGj(t)σGk(s)) (8)；
In the above formula, FNGjAnd FNGkProbability distribution functions of the j-th point time interval and the k-th point time interval respectively, phi represents a two-dimensional standard normal distribution function, rhoGjkIs a normalized Gaussian correlation function of the time interval at the j point and the time interval at the k point.
5. The method as claimed in claim 4, wherein in step 4), the non-stationary non-Gaussian correlation function matrix element R is derived from the non-stationary non-Gaussian correlation function matrix by the following equation (9)NGjk(t, S) deducing the non-stationary non-Gaussian power spectrum function matrix element SNGjk(w,t)：
In the above formula, the first and second carbon atoms are,is a non-stationary non-gaussian correlation function between the time interval at the j point at the time t and the time interval at the k point at the time t + tau,is the non-stationary non-Gaussian correlation function of the j-th time interval at the t moment and the k-th time interval at the t-tau moment.
6. The method of claim 5, wherein in step 4), the error between the non-stationary non-Gaussian power spectrum matrix element and the target non-stationary non-Gaussian power spectrum matrix element is compared according to the following equation (10):
in the above formula, the first and second carbon atoms are,is a target non-stationary non-Gaussian cross-power spectrum function of the j-th time interval and the k-th time interval,is a non-stationary non-Gaussian cross-power spectral function at the (i) th iteration of the j-th time interval and the k-th time interval.
7. The method as claimed in claim 6, wherein the step 5) comprises applying a non-stationary non-Gaussian power spectrum matrix to the target non-stationary non-Gaussian seismic motion time intervalNon-stationary Gaussian power spectrum matrixAnd non-stationary non-Gaussian powerSpectral matrixDecomposing to obtain matrix of Andfor decomposed non-stationary Gaussian power spectrum matrix element DGjk(w, t) iteratively updates according to equation (11) below:
8. the method as claimed in claim 6, wherein in step 6), the iteration is stopped when the average error between the non-stationary non-Gaussian power spectrum matrix element and the target non-stationary non-Gaussian power spectrum matrix element is larger than the average error of the previous iteration.
9. The method for simulating the time interval of the spatially variant non-stationary non-gaussian seismic motion according to claim 1, wherein in step 7), after the time interval of the spatially variant non-stationary gaussian seismic motion is simulated based on the spectral representation, the time interval of the spatially variant non-stationary gaussian seismic motion to be simulated is obtained based on the nonlinear transformation relationship between the gaussian distribution and the target distribution at each moment; wherein the nonlinear conversion formula is:
in the above formula, FGjAnd FNGjRespectively a gaussian probability distribution function and a target non-gaussian distribution function of the jth time interval.
10. The method as claimed in claim 4, wherein in step 3), the distribution forms of the seismic motion obeys the same at each time and the variances are different, and when the solution is obtained by using the formula (7), the non-stationary non-gaussian correlation function matrix elements with the variance of 1 in the target distribution form are obtained first, a table is established, the value with the variance of 1 is obtained by directly reading the data of the table, and then the value is multiplied by the standard deviation to obtain the non-stationary non-gaussian correlation function matrix elements with the corresponding variances.
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| CN104573249A (en) * | 2015-01-16 | 2015-04-29 | 上海大学 | Time-variant ARMA model based non-stable wind speed simulation method |
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