CN107480325A - The non-stationary non-gaussian earthquake motion time history analogy method of spatial variability - Google Patents
The non-stationary non-gaussian earthquake motion time history analogy method of spatial variability Download PDFInfo
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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 to non-stationary Gaussian power spectrum matrix, rather than simply it is 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 the potential non-stationary Gaussian power spectrum matrix obtained by iterative, the non-stationary Gauss earthquake motion time history of spectral representation method simulation spatial variability can be used, then the non-stationary Gauss earthquake motion time history of each point is converted into by non-stationary non-gaussian earthquake motion time history by the non-linear conversion between destination probability distribution function and Gaussian Profile again.Non-stationary non-gaussian earthquake motion time history precision height, the Iterations of Multi of this method simulation are strong, and can be used in combination with FFT, so as to ensure that simulation precision, suitable for popularization and application.
Description
Technical field
The present invention relates to one kind to belong to earthquake motion time history analogy method, more particularly to a kind of non-height of the non-stationary of spatial variability
This earthquake motion time history analogy method, belong to civil engineering Aseismic Design field, can be longspan structure, such as bridge, tunnel
Time-history analysis etc. Aseismic Design provides the seismic wave of input space variation.
Background technology
For longspan structure thing, related earthquake resistant code requirement, need to carry out spatial variability in seismic design of structures
Nonlinear dynamical damage under ground seismic wave function.But related actual measurement earthquake record shows that earthquake motion time history shows one
Fixed non-Gaussian system, it is therefore necessary to propose that a kind of non-gaussian for the spatial variability that can be simulated and reflect true ground motion characteristic is non-
The earthquake motion method of smooth performance.
Research in terms of the non-stationary non-gaussian earthquake motion time history simulation of spatial variability at present, is mainly first passed through to reality
The statistical distribution of geodetic vibrations, establishes its non-stationary auto-power spectrum model, coherency function model and non-gaussian probability distribution mould
Type, then the non-stationary Gauss earthquake motion time history with spectral representation method generation spatial variability, then according to the non-height of each point earthquake motion
This distribution character, with nonlinear transformation model, the non-stationary non-gaussian that obtains meeting the spatial variability of given distribution character
Shake time-histories.Its core procedure is as follows：
(1) by survey earthquake motion statistical distribution, establish its non-stationary auto-power spectrum model, coherency function model with
And non-gaussian probability Distribution Model；
(2) some non-stationary spectral power matrix is used, the non-stationary that potential spatial variability is generated using spectral representation method is high
This earthquake motion time history；
(3) by non-linear conversion, the non-stationary Gauss earthquake motion time history of potential spatial variability is converted into space and become
Different non-stationary non-gaussian earthquake motion time history.
In above-mentioned steps, the most it is difficult in the 2nd step " some non-stationary spectral power matrix " determination.At present there has been no
Correlative study provides how a kind of method reasonably determines " some non-stationary spectral power matrix ".
The content of the invention
Goal of the invention：The present invention is directed to problems of the prior art, there is provided a kind of non-height of the non-stationary of spatial variability
This earthquake motion time history analogy method, the analogy method solve non-stationary Gaussian power spectrum matrix by alternative manner, so that
The non-stationary spectral power matrix of the non-stationary non-gaussian earthquake motion time history for the spatial variability simulated and target non-stationary power spectrum
Matrix is consistent, ensure that the accuracy of simulation.
Technical scheme：A kind of non-stationary non-gaussian earthquake motion time history analogy method of spatial variability of the present invention, bag
Include following steps：
1) target non-stationary non-gaussian spectral power matrix and the distribution of each point earthquake motion time history for giving spatial variability each point are special
Property；
2) an initial non-stationary Gaussian power spectrum matrix is assumed, it is high by non-stationary corresponding to the solution of its matrix element
This correlation function matrix element；
3) distribution character based on each earthquake motion time history, non-stationary Gauss correlation function matrix element pair is tried to achieve respectively
The non-stationary non-gaussian correlation function matrix element answered；
4) inquire into non-stationary non-gaussian spectral power matrix element from non-stationary non-gaussian correlation function matrix element, and then obtain
To the non-stationary non-gaussian spectral power matrix corresponding with non-stationary Gaussian power spectrum matrix, compare itself and the non-height of target non-stationary
Error between this spectral power matrix；
5) to target non-stationary non-gaussian spectral power matrix, non-stationary Gaussian power spectrum matrix and non-stationary non-gaussian work(
Rate spectrum matrix is decomposed, and the matrix element after decomposition is iterated, and obtains new non-stationary gaussian spectrum matrix element,
It is standardized, and each variable carried out randomly ordered；
6) step 2) is returned to, new non-stationary non-gaussian spectral power matrix element is tried to achieve, it is non-with target non-stationary to calculate it
Error between gaussian spectrum matrix element, when error meets required precision, stop iteration, otherwise repeat step 2)~
5), until meeting to require；
7) the potential non-stationary Gaussian power spectrum matrix obtained using iteration, based on the non-of spectral representation method simulation spatial variability
Stable Gaussian earthquake motion time history, then pass through nonlinear transformation, the non-stationary non-gaussian of the spatial variability required for being converted into again
Earthquake motion time history.
Preferably, in step 2), initial non-stationary Gaussian power is used as using target non-stationary non-gaussian spectral power matrix
Spectrum matrix.It is very fast using target non-stationary non-gaussian spectral power matrix as initial matrix, convergence rate.
Above-mentioned steps 2) in, according to following formula (6), by non-stationary gaussian spectrum matrix element SGjkCorresponding to (w, t) is solved
Non-stationary Gauss correlation function matrix element RGjk(t,s)：
Wherein,T and the non-stationary Gaussian power spectral function at s moment are represented respectively, and ω is frequency
Rate.
In step 3), non-stationary non-gaussian correlation function matrix element R is determined by following formula (7)~(8)NGjk(t,s)：
ρGjk(t, s)=RGjk(t,s)/(σGj(t)σGk(s)) (8)；
In above formula, FNGjAnd FNGkRespectively jth point time-histories, the probability-distribution function of kth point time-histories, φ represent two-dimentional standard
Normal distyribution function, ρGjkFor the standardization Gauss correlation function of jth point time-histories and kth point time-histories.
In step 4), by following formula (9) from the non-stationary non-gaussian correlation function matrix element RNGjk(t, s) inquires into non-
Steady non-gaussian spectral power matrix element SNGjk(w,t)：
In above formula,For jth point time-histories t and kth point time-histories the t+ τ moment non-stationary non-gaussian
Correlation function,For jth point time-histories t and kth point time-histories the t- τ moment the related letter of non-stationary non-gaussian
Number.
Further, in step 4), according to following formula (10) the non-stationary non-gaussian spectral power matrix element and mesh
Mark the error between non-stationary non-gaussian spectral power matrix element：
In above formula,For the target non-stationary non-gaussian crosspower spectrum function of jth point time-histories and kth point time-histories,For jth point time-histories and kth point time-histories (i) secondary iteration when non-stationary non-gaussian crosspower spectrum function.
Above-mentioned steps 5) in, to target non-stationary non-gaussian spectral power matrixNon-stationary Gaussian power spectrum matrixAnd non-stationary non-gaussian spectral power matrixDecomposed, the matrix obtained after decomposition is To the non-stationary gaussian spectrum matrix element D after decompositionGjk(w, t) (11) are changed according to the following formula
Generation renewal：
Specifically, in step 6), when non-stationary non-gaussian spectral power matrix element and target non-stationary non-gaussian power spectrum
When the mean error of matrix element is more than the mean error of last iteration, stop iteration.
Above-mentioned steps 7) in, after the non-stationary Gauss earthquake motion time history based on spectral representation method simulation spatial variability, based on spectrum
Representation simulation spatial variability non-stationary Gauss earthquake motion time history after, based on Gaussian Profile with it is each when inscribe target distribution it
Between nonlinear transformation relation, obtain needed for simulation spatial variability non-stationary non-gaussian earthquake motion time history；Wherein, this is non-thread
Property conversion formula is：
In above formula, FGjAnd FNGjThe gaussian probability distribution function of respectively j-th time-histories and target non-gaussian distribution function.
,, will be quite time-consuming due to need to be solved under each time when being solved using formula (7) in step 3).Due to
It is generally acknowledged that the distributional pattern that the earthquake motion at each moment is obeyed is identical, simply variance is different, therefore can first solve target point
Non-stationary non-gaussian correlation function matrix element when variance is 1 under cloth form, and form is established, by directly reading form number
According to come try to achieve variance for 1 when value, be then multiplied by standard deviation, obtain the non-stationary non-gaussian correlation function matrix under corresponding variance
Element, the time is solved so as to greatly save.
Beneficial effect：Compared with existing skill, the advantage of the invention is that：(1) precision of analogy method of the invention is high, should
Analogy method thereby may be ensured that final non-stationary non-gaussian work(by being iterated solution to non-stationary Gaussian power spectrum matrix
Rate spectrum matrix and the uniformity of target power spectral function；Moreover, after each iteration, it is randomly ordered to the progress of each variable, can
Ensure the uniformity of the convergence rate of each variable；(2) the alternative manner efficiency high used in the present invention, the alternative manner are direct
Solution is iterated by the theory relation between non-stationary Gaussian power spectrum matrix and non-stationary non-gaussian spectral power matrix, and
By being iterated to matrix element after decomposition in solution procedure, it can so ensure the orthotropicity of spectral power matrix, so as to protect
Demonstrate,prove the convergence of iteration；(3) analogy method of the invention is when simulating the non-stationary Gauss earthquake motion time history of latent space variation,
Spectral representation method is used, this method can overcome KL to decompose the difficulty for solving non-stationary characteristic function, and mutually be tied with FFT technique
Close, greatly improve simulation precision.
Brief description of the drawings
Fig. 1 is target non-stationary non-gaussian auto-power spectrum function previously given in embodiment；
Fig. 2 a are the potential non-stationary Gauss auto-power spectrum function of first point that iteration obtains in embodiment；
Fig. 2 b are the potential non-stationary Gauss crosspower spectrum letter of first point that iteration obtains in embodiment and second point
Number；
Fig. 2 c are the potential non-stationary Gauss crosspower spectrum letter of first point that iteration obtains in embodiment and the 3rd point
Number；
Fig. 2 d are the potential non-stationary Gauss auto-power spectrum function of second point that iteration obtains in embodiment；
Fig. 2 e are the potential non-stationary Gauss crosspower spectrum letter of second point that iteration obtains in embodiment and the 3rd point
Number；
Fig. 2 f are the potential non-stationary Gauss auto-power spectrum function of the 3rd point that iteration obtains in embodiment；
Fig. 3 a are that the obtained non-stationary Gauss earthquake motion time history of first point and corresponding is simulated in embodiment
Non-stationary non-gaussian earthquake motion time history；Wherein, the non-stationary Gauss earthquake motion time history of 1 first point obtained for simulation；2 be mould
Intend the obtained non-stationary non-gaussian earthquake motion time history of first point；
Fig. 3 b are that the obtained non-stationary Gauss earthquake motion time history of second point and corresponding is simulated in embodiment
Non-stationary non-gaussian earthquake motion time history；Wherein, the non-stationary Gauss earthquake motion time history of 3 first point obtained for simulation；4 be mould
Intend the obtained non-stationary non-gaussian earthquake motion time history of first point；
Fig. 3 c are that the obtained non-stationary Gauss earthquake motion time history of the 3rd point and corresponding is simulated in embodiment
Non-stationary non-gaussian earthquake motion time history；Wherein, the non-stationary Gauss earthquake motion time history of 5 the 3rd points obtained for simulation；6 be mould
Intend the obtained non-stationary non-gaussian earthquake motion time history of the 3rd point.
Embodiment
Technical scheme is described further below in conjunction with the accompanying drawings.
The non-stationary non-gaussian earthquake motion time history analogy method of the spatial variability of the present invention, solves to obtain by iterative algorithm
The non-stationary Gaussian power spectrum matrix needed for the non-stationary Gauss earthquake synthesis of latent space variation is generated, iterative algorithm can be with
Ensure the uniformity of final non-stationary non-gaussian spectral power matrix and target power spectrum matrix.
Exemplified by simulating the non-stationary non-gaussian earthquake motion time history of three points of a spatial variability, to simulation side of the invention
Method illustrates；The point of simulation be earth's surface equidistantly distributed three points, spacing 100m.
The non-stationary non-gaussian earthquake motion time history analogy method of the spatial variability of the present invention, comprises the following steps：
1) the target non-stationary non-gaussian auto-power spectrum function for assuming first that each point is：
Wherein
In formula (1)~(3), σ is standard deviation；ωgAnd ζgIt is characteristic frequency and the damping of figure, ωfAnd ζfIt is filtering parameter,
They can be set as：ωf=0.1 ωg, ζf=ζg；T is the time, and λ is the parameter for embodying the attenuation of seismic wave.
In the present embodiment, related parameter values are：σ=110 (cm/s3/2),ωg=30-1.25t (rad/s), ζg=0.5+
0.005t,t1=2s, t2=10s, λ=0.4；Gained target non-stationary non-gaussian auto-power spectrum function such as Fig. 1.
It is assumed that the coherent function between each point 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+ (the π f of ω/20)b]-1/2 (5)
In formula (4)~(5), A, α, K, f0It is model parameter with b, their value is A=0.63, α=0.0186, K=
31200, f0=1.51, b=2.95；Earthquake motion regards velocity of wave as 500m/s；djkFor the distance between 2 points.
Build to obtain the non-height of target non-stationary by above-mentioned target non-stationary non-gaussian auto-power spectrum function and coherent function
This auto-power spectrum matrix.
Assume that earthquake motion time history obeys Student ' s t distributions simultaneously, parameter c is 6.
2) assume that initial non-stationary Gaussian power spectrum matrix is target non-stationary non-gaussian auto-power spectrum matrix.
3) by non-stationary gaussian spectrum matrix element, non-stationary Gauss correlation function matrix is solved using following formula (6)
Element：
Wherein,Represent that t and the non-stationary Gaussian power spectral function at s moment, ω are respectively
Frequency.
4) non-stationary non-gaussian correlation is solved by non-stationary Gauss correlation function matrix element according to following formula (7)~(8)
Jacobian matrix element：
ρGjk(t, s)=RGjk(t,s)/(σGj(t)σGk(s)) (8)；
FNGjAnd FNGkRespectively jth point time-histories, the probability-distribution function of kth point time-histories, φ represent two-dimentional standard normal point
Cloth function, ρGjkFor the standardization Gauss correlation function of jth point time-histories and kth point time-histories.
It is generally acknowledged that the distributional pattern that the earthquake motion at each moment is obeyed is identical, simply variance is different, is asked using formula (7)
Xie Shi, non-stationary non-gaussian correlation function matrix element when variance is 1 under target distribution form is first solved, and establishes form,
Record corresponding ρ when variance is 1 under each momentGjkThe value of (t, s), by directly read list data try to achieve variance for 1 when
The value of non-stationary non-gaussian correlation function matrix element, is then multiplied by standard deviation, obtains the non-stationary non-gaussian under corresponding variance
Correlation function matrix element.
5) non-stationary non-gaussian correlation function matrix element is based on, by following estimation equations (9), tries to achieve the non-height of non-stationary
This spectral power matrix element：
In above formula,For jth point time-histories t and kth point time-histories the t+ τ moment non-stationary non-gaussian
Correlation function,For jth point time-histories t and kth point time-histories the t- τ moment the related letter of non-stationary non-gaussian
Number.
It is available relative with non-stationary Gaussian power spectrum matrix according to the non-stationary non-gaussian spectral power matrix element tried to achieve
The non-stationary non-gaussian spectral power matrix answered.
6) non-stationary non-gaussian spectral power matrix element and target non-stationary non-gaussian power spectrum are compared according to following formula (10)
Error between matrix element：
Wherein,For the target non-stationary non-gaussian crosspower spectrum function of jth point time-histories and kth point time-histories,For jth point time-histories and kth point time-histories (i) secondary iteration when non-stationary non-gaussian crosspower spectrum function.
7) to target non-stationary spectral power matrixNon-stationary Gaussian power spectrum matrixAnd non-stationary
Non-gaussian spectral power matrixDecomposed, the matrix obtained after decomposition is
8) to the non-stationary gaussian spectrum matrix element after decomposition, (11) are updated according to the following formula：
Thus obtain new non-stationary gaussian spectrum matrix element.
Repeat step 2) to step 8), it is iterated, by 8 iteration, obtained final power spectrum error function is：
Potential non-stationary gaussian spectrum matrix element such as Fig. 2 a~2f of each point finally given.
Then the potential non-stationary Gaussian power spectrum matrix obtained using iteration, simulate to obtain potentially using spectral representation method
The non-stationary non-gaussian earthquake motion time history of spatial variability, then again by based on non-linear between Gaussian Profile and target distribution
Transformational relation, non-linear conversion formula such as formula (12), finally give the non-stationary non-gaussian earthquake of the spatial variability to be simulated
Shi Shicheng, such as Fig. 3 a~3c.
Non-linear conversion formula is：
In above formula, FGjAnd FNGjThe gaussian probability distribution function of respectively j-th time-histories and target non-gaussian distribution function.
Claims (10)
1. the non-stationary non-gaussian earthquake motion time history analogy method of a kind of spatial variability, it is characterised in that comprise the following steps：
1) the target non-stationary non-gaussian spectral power matrix and each point earthquake motion time history distribution character of spatial variability each point are given；
2) assume an initial non-stationary Gaussian power spectrum matrix, pass through non-stationary Gauss phase corresponding to the solution of its matrix element
Close Jacobian matrix element；
3) distribution character based on each earthquake motion time history, try to achieve respectively corresponding to non-stationary Gauss correlation function matrix element
Non-stationary non-gaussian correlation function matrix element；
4) inquire into non-stationary non-gaussian spectral power matrix element from non-stationary non-gaussian correlation function matrix element, compare itself and mesh
Mark the error between non-stationary non-gaussian spectral power matrix element；
5) to target non-stationary non-gaussian spectral power matrix, non-stationary Gaussian power spectrum matrix and non-stationary non-gaussian power spectrum
Matrix is decomposed, and the matrix element after decomposition is iterated, and new non-stationary gaussian spectrum matrix element is obtained, to it
It is standardized, and each variable is carried out randomly ordered；
6) step 2) is returned to, new non-stationary non-gaussian spectral power matrix element is tried to achieve, calculates itself and target non-stationary non-gaussian
Error between spectral power matrix element, when error meets required precision, stop iteration, otherwise repeat step 2)~5), directly
Required to satisfaction；
7) the potential non-stationary Gaussian power spectrum matrix obtained using iteration, the non-stationary based on spectral representation method simulation spatial variability
Gauss earthquake motion time history, then pass through nonlinear transformation, the non-stationary non-gaussian earthquake of the spatial variability required for being converted into again
Dynamic time-histories.
2. the non-stationary non-gaussian earthquake motion time history analogy method of spatial variability according to claim 1, it is characterised in that
In step 2), initial non-stationary Gaussian power spectrum matrix is used as using the target non-stationary non-gaussian spectral power matrix.
3. the non-stationary non-gaussian earthquake motion time history analogy method of spatial variability according to claim 1, it is characterised in that
In step 2), according to following formula (6), by the non-stationary gaussian spectrum matrix element SGjkNon-stationary corresponding to (w, t) solution is high
This correlation function matrix element RGjk(t,s)：
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Wherein,WithT and the non-stationary Gaussian power spectral function at s moment are represented respectively, and ω is frequency.
4. the non-stationary non-gaussian earthquake motion time history analogy method of spatial variability according to claim 3, it is characterised in that
In step 3), the non-stationary non-gaussian correlation function matrix element R is determined by following formula (7)~(8)NGjk(t,s)：
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</mrow>
ρGjk(t, s)=RGjk(t,s)/(σGj(t)σGk(s)) (8)；
In above formula, FNGjAnd FNGkRespectively jth point time-histories, the probability-distribution function of kth point time-histories, φ represent two-dimentional standard normal
Distribution function, ρGjkFor the standardization Gauss correlation function of jth point time-histories and kth point time-histories.
5. the non-stationary non-gaussian earthquake motion time history analogy method of spatial variability according to claim 4, it is characterised in that
In step 4), by following formula (9) from the non-stationary non-gaussian correlation function matrix element RNGjk(t, s) inquires into the non-height of non-stationary
This power spectrum function matrix element SNGjk(w,t)：
<mrow>
<msubsup>
<mi>S</mi>
<mrow>
<mi>N</mi>
<mi>G</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
</msubsup>
<mrow>
<mo>(</mo>
<mi>&omega;</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>&ap;</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mi>&pi;</mi>
</mrow>
</mfrac>
<msubsup>
<mo>&Integral;</mo>
<mn>0</mn>
<mrow>
<mo>+</mo>
<mi>&infin;</mi>
</mrow>
</msubsup>
<mo>&lsqb;</mo>
<msubsup>
<mi>R</mi>
<mrow>
<mi>N</mi>
<mi>G</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
</msubsup>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>t</mi>
<mo>+</mo>
<mi>&tau;</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<msubsup>
<mi>R</mi>
<mrow>
<mi>N</mi>
<mi>G</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
</msubsup>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>,</mo>
<mi>t</mi>
<mo>-</mo>
<mi>&tau;</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mi>&omega;</mi>
<mi>&tau;</mi>
<mo>)</mo>
</mrow>
<mi>d</mi>
<mi>&tau;</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
<mo>;</mo>
</mrow>
In above formula,It is related in the non-stationary non-gaussian at t+ τ moment to kth point time-histories in t for jth point time-histories
Function,For jth point time-histories t and kth point time-histories the t- τ moment non-stationary non-gaussian correlation function.
6. the non-stationary non-gaussian earthquake motion time history analogy method of spatial variability according to claim 5, it is characterised in that
In step 4), according to following formula (10) the non-stationary non-gaussian spectral power matrix element and target non-stationary non-gaussian power
Error between spectrum matrix element：
<mrow>
<msubsup>
<mi>&epsiv;</mi>
<mrow>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
</msubsup>
<mo>=</mo>
<msqrt>
<mfrac>
<mrow>
<msubsup>
<mo>&Integral;</mo>
<mn>0</mn>
<mrow>
<mo>+</mo>
<mi>&infin;</mi>
</mrow>
</msubsup>
<msubsup>
<mo>&Integral;</mo>
<mn>0</mn>
<mi>&infin;</mi>
</msubsup>
<msup>
<mrow>
<mo>&lsqb;</mo>
<mo>|</mo>
<msubsup>
<mi>S</mi>
<mrow>
<mi>N</mi>
<mi>G</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mrow>
<mi>T</mi>
<mi>a</mi>
<mi>r</mi>
</mrow>
</msubsup>
<mrow>
<mo>(</mo>
<mi>&omega;</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>|</mo>
<mo>-</mo>
<msubsup>
<mi>S</mi>
<mrow>
<mi>N</mi>
<mi>G</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
</msubsup>
<mrow>
<mo>(</mo>
<mi>&omega;</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mn>2</mn>
</msup>
<mi>d</mi>
<mi>&omega;</mi>
<mi>d</mi>
<mi>t</mi>
</mrow>
<mrow>
<msubsup>
<mo>&Integral;</mo>
<mn>0</mn>
<mrow>
<mo>+</mo>
<mi>&infin;</mi>
</mrow>
</msubsup>
<msubsup>
<mo>&Integral;</mo>
<mn>0</mn>
<mi>&infin;</mi>
</msubsup>
<msup>
<mrow>
<mo>&lsqb;</mo>
<mo>|</mo>
<msubsup>
<mi>S</mi>
<mrow>
<mi>N</mi>
<mi>G</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mrow>
<mi>T</mi>
<mi>a</mi>
<mi>r</mi>
</mrow>
</msubsup>
<mrow>
<mo>(</mo>
<mi>&omega;</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>|</mo>
<mo>&rsqb;</mo>
</mrow>
<mn>2</mn>
</msup>
<mi>d</mi>
<mi>&omega;</mi>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</msqrt>
<mo>&times;</mo>
<mn>100</mn>
<mi>%</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>10</mn>
<mo>)</mo>
</mrow>
<mo>;</mo>
</mrow>
In above formula,For the target non-stationary non-gaussian crosspower spectrum function of jth point time-histories and kth point time-histories,For jth point time-histories and kth point time-histories (i) secondary iteration when non-stationary non-gaussian crosspower spectrum function.
7. the non-stationary non-gaussian earthquake motion time history analogy method of spatial variability according to claim 6, it is characterised in that
In step 5), to target non-stationary non-gaussian spectral power matrixNon-stationary Gaussian power spectrum matrixIt is and non-
Steady non-gaussian spectral power matrixDecomposed, the matrix obtained after decomposition is With
To the non-stationary gaussian spectrum matrix element D after decompositionGjk(w, t) (11) are iterated renewal according to the following formula：
<mrow>
<msubsup>
<mi>D</mi>
<mrow>
<mi>G</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</msubsup>
<mrow>
<mo>(</mo>
<mi>&omega;</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msup>
<mrow>
<mo>&lsqb;</mo>
<mfrac>
<mrow>
<msubsup>
<mi>D</mi>
<mrow>
<mi>G</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mrow>
<mi>T</mi>
<mi>a</mi>
<mi>r</mi>
</mrow>
</msubsup>
<mrow>
<mo>(</mo>
<mi>&omega;</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msubsup>
<mi>D</mi>
<mrow>
<mi>N</mi>
<mi>G</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
</msubsup>
<mrow>
<mo>(</mo>
<mi>&omega;</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>&rsqb;</mo>
</mrow>
<mi>&beta;</mi>
</msup>
<msubsup>
<mi>D</mi>
<mrow>
<mi>G</mi>
<mi>j</mi>
<mi>k</mi>
</mrow>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mo>)</mo>
</mrow>
</msubsup>
<mrow>
<mo>(</mo>
<mi>&omega;</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mo>)</mo>
</mrow>
<mo>.</mo>
</mrow>
8. the non-stationary non-gaussian earthquake motion time history analogy method of spatial variability according to claim 6, it is characterised in that
It is flat when the non-stationary non-gaussian spectral power matrix element and target non-stationary non-gaussian spectral power matrix element in step 6)
When equal error is more than the mean error of last iteration, stop iteration.
9. the non-stationary non-gaussian earthquake motion time history analogy method of spatial variability according to claim 1, it is characterised in that
In step 7), based on spectral representation method simulation spatial variability non-stationary Gauss earthquake motion time history after, based on Gaussian Profile with it is each
When inscribe nonlinear transformation relation between target distribution, obtain needed for simulation spatial variability non-stationary non-gaussian earthquake motion
Time-histories；Wherein, the non-linear conversion formula is：
<mrow>
<msub>
<mi>y</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msubsup>
<mi>F</mi>
<mrow>
<mi>N</mi>
<mi>G</mi>
<mi>j</mi>
</mrow>
<mrow>
<mo>-</mo>
<mn>1</mn>
</mrow>
</msubsup>
<mo>{</mo>
<msub>
<mi>F</mi>
<mrow>
<mi>G</mi>
<mi>j</mi>
</mrow>
</msub>
<mo>&lsqb;</mo>
<msub>
<mi>x</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mi>t</mi>
<mo>&rsqb;</mo>
<mo>}</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>12</mn>
<mo>)</mo>
</mrow>
<mo>;</mo>
</mrow>
In above formula, FGjAnd FNGjThe gaussian probability distribution function of respectively j-th time-histories and target non-gaussian distribution function.
10. the non-stationary non-gaussian earthquake motion time history analogy method of spatial variability according to claim 1, its feature exist
In in step 3), it is considered that the distributional pattern that the earthquake motion at each moment is obeyed is identical, and simply variance is different, uses formula
(7) when solving, non-stationary non-gaussian correlation function matrix element when variance is 1 under target distribution form is first solved, and establish
Form, by the value for directly reading list data to try to achieve when variance is 1, standard deviation is then multiplied by, is obtained under corresponding variance
Non-stationary non-gaussian correlation function matrix element.
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| CN113468784A (en) * | 2021-06-25 | 2021-10-01 | 深圳大学 | Bridge non-stationary non-Gaussian random wind field simulation method, device, equipment and medium |
| CN114139363A (en) * | 2021-11-25 | 2022-03-04 | 四川农业大学 | Method for simulating space variation non-stationary earthquake motion time course |
| CN114239348A (en) * | 2021-11-30 | 2022-03-25 | 中铁二院工程集团有限责任公司 | Bridge seismic reliability analysis method |
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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 |
| US20160109591A1 (en) * | 2014-10-15 | 2016-04-21 | Westerngeco L.L.C. | Noise model estimation in multimeasurement data |
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| CN103364826A (en) * | 2012-03-30 | 2013-10-23 | 中国石油化工股份有限公司 | An earthquake blind source deconvolution method based on independent component analysis |
| CN103714252A (en) * | 2013-12-29 | 2014-04-09 | 中国地震局工程力学研究所 | Method for recognizing modal parameters of structural vibration table model based on earthquake response record end data |
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| CN113468784A (en) * | 2021-06-25 | 2021-10-01 | 深圳大学 | Bridge non-stationary non-Gaussian random wind field simulation method, device, equipment and medium |
| CN113468784B (en) * | 2021-06-25 | 2023-06-20 | 深圳大学 | Bridge non-stationary non-Gaussian random wind field simulation method, device, equipment and medium |
| CN114139363A (en) * | 2021-11-25 | 2022-03-04 | 四川农业大学 | Method for simulating space variation non-stationary earthquake motion time course |
| CN114139363B (en) * | 2021-11-25 | 2022-07-01 | 四川农业大学 | Method for simulating space variation non-stationary earthquake motion time course |
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