CN107169188A - A kind of multidimensional multivariable non-gaussian spatial random field analogy method - Google Patents
A kind of multidimensional multivariable non-gaussian spatial random field analogy method Download PDFInfo
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Abstract
The invention discloses a kind of multidimensional multivariable non-gaussian spatial random field analogy method, comprise the following steps：A target multidimensional non-gaussian spectral power matrix and the distribution character of parameter are given first, and assumes an initial multidimensional Gaussian power spectrum matrix, are solved multidimensional non-gaussian spectral power matrix, are compared its error with target power spectrum matrix；Secondly target multidimensional spectral power matrix, multidimensional Gaussian power spectrum matrix and multidimensional non-gaussian spectral power matrix are decomposed, and the element of matrix after decomposition is iterated；Solve new multidimensional Gaussian power spectrum matrix；Randomly ordered, iteration is carried out to each variable, precision is met requirement；The last Gaussian power Spectral matrix simulation Gaussian spatial random field obtained by iteration, further obtains multidimensional multivariable non-gaussian spatial random field.Multidimensional multivariable non-gaussian spatial random field precision that the present invention is simulated is high, Iterations of Multi is strong, and can be used in combination with FFT, so that simulation precision is ensure that, suitable for popularization and application.
Description
Technical field
The present invention relates to civil engineering reliability analyzing method, more particularly to a kind of random field stimulation of multidimensional multivariate space
Method.
Background technology
The spatial random field simulation of material parameter is one of important process of the research of civil engineering reliability and design field,
Because the defect of civil engineering material manufacturing process in the fabrication process often results in the physico-mechanical properties of material in material space
Inside there is certain Spatial Variability, rather than a constant；Equally, the engineering properties of natural rock-filled due to being deposited, afterwards deposition,
Chemical weathering and effect and the influence of different loading history such as carry and certain Spatial Variability is presented.Therefore, to soil
Wood engineering works carries out being contemplated that material, the uncertainty of Rock And Soil physical and mechanical parameter during security evaluation, should use
Civil Engineering Design theory based on Reliability Theory, and the spatial random field for providing material parameter is RELIABILITY DESIGN and research
Element task.
Research in terms of current multidimensional multivariable non-gaussian spatial random field analogy method, mainly first passes through given table
The correlation distance of parameter space variation characteristic is levied, using existing coherency function model, the multidimensional Power Spectrum Model of parameter is set up, then
Multidimensional multivariate Gaussian spatial random field is generated with spectral representation method, then according to the non-gaussian distribution characteristic of material parameter, fortune
With nonlinear transformation model, obtain meeting the multidimensional multivariable non-gaussian spatial field of the material parameter of given distribution character.Its core
Heart step is：By experimental data, the correlation distance of parameter is tried to achieve；The target multidimensional non-gaussian power spectrum set up between each parameter
Matrix；Using some multidimensional spectral power matrix, potential multidimensional multivariate Gaussian spatial random field is generated using spectral representation method；It is logical
Non-linear conversion is crossed, it is random that potential multidimensional multivariable non-gaussian spatial random field is converted into multidimensional multivariable non-gaussian space
.
Wherein, the most it is difficult to the determination of " some multidimensional spectral power matrix ".Traditional method, which is mainly, directly to be used
Target multidimensional non-gaussian spectral power matrix, non-linear conversion is employed due to being transformed into non-gaussian spatial field from Gaussian spatial,
The random spectral power matrix in the non-gaussian space finally given is so caused to differ with target power spectrum matrix so that analog result
There is error.
The content of the invention
Goal of the invention：Potential multidimensional Gaussian power spectral moment is solved using iterative program it is an object of the invention to provide one kind
Battle array, so that the power spectrum for the multidimensional multivariable non-gaussian spatial random field the simulated multidimensional consistent with target power spectrum is more
Variable space random field analogy method.
Technical scheme：A kind of multidimensional multivariable non-gaussian spatial random field analogy method, comprises the following steps：
(A) by the correlation distance of parameter, the target multidimensional non-gaussian spectral power matrix set up between parameter gives one
The distribution character of target multidimensional non-gaussian spectral power matrix and parameter；
(B) it is target multidimensional non-gaussian spectral power matrix to assume an initial multidimensional Gaussian power spectrum matrix；
(C) by above-mentioned multidimensional gaussian spectrum Matrix Solving multidimensional non-gaussian spectral power matrix；
(D) error between multidimensional non-gaussian spectral power matrix and target multidimensional non-gaussian spectral power matrix is compared；
(E) to target multidimensional non-gaussian spectral power matrix, multidimensional Gaussian power spectrum matrix and multidimensional non-gaussian power spectrum
Matrix is decomposed；
(F) element of matrix after decomposition is iterated using exponential type iterative formula；
(G) by solving new multidimensional Gaussian power spectrum matrix to the element obtained after iteration, and matrix element is carried out
Standardization；
(H) each variable is carried out randomly ordered；
Repeat step (C) is to step (H), until mean error of the mean error more than preceding an iteration of error matrix,
Obtain potential multidimensional Gaussian power spectrum matrix；
(I) the potential multidimensional Gaussian power Spectral matrix simulation multidimensional multivariate Gaussian spatial random field obtained using iteration,
Again by nonlinear transformation, the multidimensional multivariable non-gaussian spatial random field to be simulated is obtained.
In step (C), formula is changed using multidimensional Fourier, Gauss is obtained by the multidimensional gaussian spectrum Matrix Solving
Cross-correlation functionNon-gaussian cross-correlation function is obtained further according to following equations：
Wherein, φ [gj1,gk2；ρjk(ξ)] it is j-th of element and k-th element in correlation coefficient ρjkJoint under (ξ)
Normal distyribution function,Represent the inverse function of j-th of function probability distribution function；
Multidimensional non-gaussian spectral power matrix is obtained by the non-gaussian cross-correlation function solution.
Step (I) is specially：The potential multidimensional Gaussian power Spectral matrix simulation for obtaining iteration first by spectral representation method
Multidimensional multivariate Gaussian spatial random field, secondly by based on the non-linear conversion relation between Gaussian Profile and target distribution,
Obtain the multidimensional multivariable non-gaussian spatial random field to be simulated.
In step (F), following exponential type iterative formula is employed：
Wherein,WithFor target multidimensional non-gaussian spectral power matrixMultidimensional gaussian spectrum
MatrixAnd multidimensional non-gaussian spectral power matrixThe element of matrix after decomposition, parameter beta is used for control convergence speed；
Parameter beta is chosen for 0.5-2, preferably 0.8-1.2.
Operation principle：The present invention proposes a kind of multidimensional Gaussian power spectral moment determined used in potential Gaussian spatial field stimulation
The iterative method of battle array, obtains enabling to final multidimensional non-gaussian spectral power matrix and target multidimensional non-gaussian power spectrum
The consistent multidimensional Gaussian power spectrum matrix of matrix；After the potential multidimensional Gaussian power spectrum matrix obtained by iterative, use
Spectral representation method simulates multidimensional multivariate Gaussian spatial random field, then again by between destination probability distribution function and Gaussian Profile
Non-linear conversion, be translated into multidimensional variable non-gaussian spatial random field.
Beneficial effect：The present invention compared with prior art, has the following advantages that：1st, the precision of analogy method has been obtained significantly
Raising：The present invention to multidimensional Gaussian power spectrum matrix by being iterated solution, rather than to be simply assumed to target more
Tie up power spectrum, it is ensured that the uniformity of final non-gaussian spatial field spectral power matrix and target power spectrum matrix；2nd, alternative manner
Convergence is strong：This method directly passes through the theory relation between multidimensional Gaussian power spectrum matrix and multidimensional non-gaussian spectral power matrix
Solution is iterated, and spectral power matrix is so ensure that by being iterated to matrix element after decomposition in solution procedure
Orthotropicity, so as to ensure the convergence of iteration.
Brief description of the drawings
Fig. 1 is the iterative algorithm flow chart of the present invention；
Fig. 2 is target multidimensional auto-power spectrum schematic diagram；
Fig. 3 is the destination probability distribution function function and gauss of distribution function of each variable；
Fig. 4 (a) -4 (c) is respectively the non-gaussian spatial random field signal of variable 1, variable 2 and variable 3 that simulation is obtained
Figure；
Fig. 5 (a) -5 (c) is respectively the probability-distribution function and destination probability of variable 1, variable 2 and variable 3 that simulation is obtained
The comparison schematic diagram of distribution function.
Embodiment
Technical scheme is described in further detail with reference to embodiment and accompanying drawing.
As shown in figure 1, the present embodiment is illustrated exemplified by simulating a two-dimentional ternary non-gaussian spatial random field, wrap
Include following steps：
(A) a target multidimensional non-gaussian spectral power matrix and the distribution character of each parameter are given.It is assumed that each parameter is more
Tieing up auto-power spectrum is：
Wherein, parameter a1=a2=2 and θ1=10m, θ2=2m；According to above-mentioned parameter, obtained multidimensional auto-power spectrum is as schemed
Shown in 2.
Multidimensional crosspower spectrum is：
Wherein parameter ρ13=ρ31=0.4, ρ12=ρ21=0.3, ρ23=ρ32=0.6.
As shown in Figure 3, it is assumed that the probability density function of Gaussian Profile is variable 1, it obeys logarithm normal distribution；Variable 1
Logarithm normal distribution probability density function be variable 2, its obey distribution of mean value；Variable 2 is uniformly distributed probability density function
For variable 3, it obeys beta distribution.
(B) initial multidimensional Gaussian power spectrum matrix is assumedFor target multidimensional non-gaussian spectral power matrix
(C) by multidimensional Gaussian power spectrum matrix, formula is changed according to multidimensional Fourier, solution obtains Gauss cross-correlation letter
NumberNon-gaussian cross-correlation function is obtained further according to following equations：
Wherein, φ [gj1,gk2；ρjk(ξ)] it is j-th of element and k-th element in correlation coefficient ρjkJoint under (ξ)
Normal distyribution function,Represent the inverse function of j-th of function probability distribution function；
The element of non-gaussian spectral power matrix is obtained by above-mentioned non-gaussian cross-correlation function solution
(D) according to following formula comparison multidimensional non-gaussian spectral power matrixsWith target multidimensional non-gaussian spectral power matrixBetween error：
Initial error matrix is：
(E) to target multidimensional non-gaussian spectral power matrixMultidimensional Gaussian power spectrum matrixAnd multidimensional not high
This spectral power matrixDecomposed, decomposing obtained matrix is respectivelyWith
(F) renewal is iterated according to the following formula to the element of matrix after decomposition：
Wherein,WithFor target multidimensional non-gaussian spectral power matrixMultidimensional gaussian spectrum
MatrixAnd multidimensional non-gaussian spectral power matrixThe element of matrix after decomposition, introducing parameter beta is used for control convergence speed
Rate, the recommended value of parameter beta is 0.8-1.2, when convergence effect is bad, can reset its value, then be iterated again, one
As be chosen in the range of 0.5-2 value just can be so that iteration has preferable convergence；In the present embodiment, parameter beta is chosen for 1.
(G) decomposed according to inverse of a matrix and try to achieve new multidimensional Gaussian power spectrum matrixAnd to the matrix of new production
It is standardized；
(H) randomly ordered, the thought randomly ordered to the progress of each variable after each iteration is carried out to each variable order,
The uniformity of the convergence rate of each variable can be ensured.
Repeat step (C) is to step (H), by 6 iteration, obtains potential multidimensional Gaussian power spectrum matrix；Final mistake
Poor matrix is：
Each element in above-mentioned matrix, is the mean error of each element of spectral power matrix.
(I) the potential multidimensional Gaussian power spectrum matrix obtained using iterationObtain potential using spectral representation method simulation
Multidimensional multivariate Gaussian spatial random field, then again by based between Gaussian Profile and target distribution non-linear conversion close
System, obtains the multidimensional multivariable non-gaussian spatial random field to be simulated, and such as Fig. 4 (a), Fig. 4 (b) and Fig. 4 (c) are shown respectively
To simulate the non-gaussian spatial random field schematic diagram of obtained variable 1, variable 2 and variable 3.
Fig. 5 (a), Fig. 5 (b), Fig. 5 (c) are respectively the probability-distribution function of variable 1, variable 2 and variable 3 that simulation is obtained
With the comparison of destination probability distribution function；Wherein, the destination probability distribution function of 5 pairs of dependent variables 1；The simulated space of 6 correspondences
The probability-distribution function of variable 1 that field analysis is obtained；The destination probability distribution function of 7 pairs of dependent variables 2；The simulated space of 8 correspondences
The probability-distribution function of variable 2 that field analysis is obtained；The destination probability distribution function of 9 pairs of dependent variables 3；The simulated space of 10 correspondences
The probability-distribution function of variable 3 that field analysis is obtained；Analyzed by the spatial random field to simulation, it can be found that variable 1, change
The probability-distribution function of amount 2 and variable 3 is consistent with destination probability distribution function.
Claims (5)
1. a kind of multidimensional multivariable non-gaussian spatial random field analogy method, it is characterised in that comprise the following steps：
(A) by the correlation distance of parameter, the target multidimensional non-gaussian spectral power matrix set up between parameter gives a target
The distribution character of multidimensional non-gaussian spectral power matrix and parameter；
(B) it is target multidimensional non-gaussian spectral power matrix to assume an initial multidimensional Gaussian power spectrum matrix；
(C) by above-mentioned multidimensional gaussian spectrum Matrix Solving multidimensional non-gaussian spectral power matrix；
(D) error between multidimensional non-gaussian spectral power matrix and target multidimensional non-gaussian spectral power matrix is compared；
(E) to target multidimensional non-gaussian spectral power matrix, multidimensional Gaussian power spectrum matrix and multidimensional non-gaussian spectral power matrix
Decomposed；
(F) element of matrix after decomposition is iterated using exponential type iterative formula；
(G) by solving new multidimensional Gaussian power spectrum matrix to the element obtained after iteration, and standard is carried out to matrix element
Change；
(H) each variable is carried out randomly ordered；
Repeat step (C) until the mean error of error matrix is more than the mean error of preceding an iteration, is obtained to step (H)
Potential multidimensional Gaussian power spectrum matrix；
(I) the potential multidimensional Gaussian power Spectral matrix simulation multidimensional multivariate Gaussian spatial random field obtained using iteration, then lead to
Nonlinear transformation is crossed, the multidimensional multivariable non-gaussian spatial random field to be simulated is obtained.
2. multidimensional multivariable non-gaussian spatial random field analogy method according to claim 1, it is characterised in that：Step
(C) in, formula is changed using multidimensional Fourier, Gauss cross-correlation function is obtained by the multidimensional gaussian spectrum Matrix SolvingNon-gaussian cross-correlation function is obtained further according to following equations：
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Wherein, φ [gj1,gk2；ρjk(ξ)] it is j-th of element and k-th element in correlation coefficient ρjkJoint normal state under (ξ)
Distribution function,Represent the inverse function of j-th of function probability distribution function；
Multidimensional non-gaussian spectral power matrix is obtained by the non-gaussian cross-correlation function solution.
3. multidimensional multivariable non-gaussian spatial random field analogy method according to claim 1, it is characterised in that step
(I) it is specially：The potential multidimensional Gaussian power Spectral matrix simulation multidimensional multivariable for obtaining iteration first by spectral representation method is high
This spatial random field, secondly by based on the non-linear conversion relation between Gaussian Profile and target distribution, obtains being simulated
Multidimensional multivariable non-gaussian spatial random field.
4. multidimensional multivariable non-gaussian spatial random field analogy method according to claim 1, it is characterised in that：Step
(F) in, following exponential type iterative formula is employed：
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Wherein,WithFor target multidimensional non-gaussian spectral power matrixMultidimensional Gaussian power spectrum matrixAnd multidimensional non-gaussian spectral power matrixThe element of matrix after decomposition, the value of parameter beta is 0.5-2.
5. multidimensional multivariable non-gaussian spatial random field analogy method according to claim 4, it is characterised in that：The ginseng
Number β value is 0.8-1.2.
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| CN107657127A (en) * | 2017-10-11 | 2018-02-02 | 河海大学 | One-dimensional multi-variate random process efficient analogy method based on energy interpolations such as frequency domains |
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| CN112284518A (en) * | 2020-09-30 | 2021-01-29 | 南京航空航天大学 | non-Gaussian multi-dimensional random vibration waveform generation method |
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| CN107657127A (en) * | 2017-10-11 | 2018-02-02 | 河海大学 | One-dimensional multi-variate random process efficient analogy method based on energy interpolations such as frequency domains |
| CN110069851A (en) * | 2019-04-18 | 2019-07-30 | 中铁二院工程集团有限责任公司 | A kind of Geotechnical Parameter random field inversion method |
| CN110196960A (en) * | 2019-04-24 | 2019-09-03 | 重庆大学 | A kind of ergodic wind field efficient analogy method based on Two-dimensional FFT |
| CN110196960B (en) * | 2019-04-24 | 2022-12-27 | 重庆大学 | Two-dimensional FFT-based efficient simulation method for wind field of each experience |
| CN110826179A (en) * | 2019-09-29 | 2020-02-21 | 贵州电网有限责任公司 | Intelligent substation relay protection real-time reliability prediction method |
| CN112284518A (en) * | 2020-09-30 | 2021-01-29 | 南京航空航天大学 | non-Gaussian multi-dimensional random vibration waveform generation method |
| CN113505470A (en) * | 2021-06-11 | 2021-10-15 | 中国海洋大学 | Novel non-Gaussian random wave simulation method |
| CN113505470B (en) * | 2021-06-11 | 2023-12-01 | 中国海洋大学 | Novel non-Gaussian random wave simulation method |
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Application publication date: 20170915 |