CN112016253A - High-fidelity chaotic polynomial correction method suitable for CFD uncertainty quantification - Google Patents
High-fidelity chaotic polynomial correction method suitable for CFD uncertainty quantification Download PDFInfo
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Abstract
The invention discloses a high-fidelity chaotic polynomial correction method suitable for CFD uncertainty quantification, which is characterized by comprising the steps of obtaining a low-fidelity model by calculation on the level of the low-fidelity model, and establishing a rough chaotic polynomial expansion; a small amount of calculation is carried out on the high fidelity model level; and correcting the low-fidelity model to obtain a final high-fidelity correction model. The statistical information of the high fidelity model obtained by the invention has little difference with the original high fidelity model, the mean difference is less than three ten-thousandth and can be ignored, the error requirement of engineering application is completely met, the calculation time required by the high fidelity model established by the invention is about 40 percent of that of the original method, and the calculation cost is greatly saved.
Description
Technical Field
The invention relates to the field of computational fluid mechanics, in particular to a high-fidelity chaotic polynomial correction method suitable for CFD uncertainty quantification.
Background
CFD (Computational Fluid Dynamics) plays an increasingly important role in the fields of aerospace, land and water traffic, energy power, atmospheric oceans, and the like. However, there are a number of uncertainty parameters in CFD, such as turbulence model coefficients, thermophysical parameters, etc. This also results in significant uncertainty in the simulation results. Uncertainty factors can lead to product performance fluctuations and even functional failures. NASA has investigated 2500 on-track aircraft faults, of which about 52% are caused by uncertainty factors. Therefore, the influence of the uncertainty of the parameters on the numerical simulation needs to be quantified in the processes of aircraft optimization design, performance evaluation and the like.
The uncertainty parameters in CFD are numerous and present a significant high dimensional character. The chaotic polynomial method is the most commonly used parameter uncertainty quantification method. With the increase of the dimension of the uncertain parameters, the demands of the two methods on the calculation amount are increased sharply, so that the application of the two methods in engineering problems is limited, and the potential damage of the plurality of uncertain parameters on the product performance is difficult to evaluate by an industrial department. Therefore, a more efficient method is urgently needed to be developed, and a feasible solution is provided for the problem of parameter uncertainty quantification in engineering under the condition of limited computing resources.
Disclosure of Invention
The invention aims to provide a high-fidelity chaotic polynomial correction method, which can effectively reduce the calculation cost on the premise of not losing the accuracy of a calculation result.
In order to achieve the purpose, the invention adopts the following technical scheme:
and S1, obtaining a low-fidelity model through few calculations at the level of the low-fidelity model, and establishing a slightly rough chaotic polynomial expansion. The method comprises the following steps:
s101, sampling in a high-dimensional space formed by input parameters by using a Latin hypercube sampling method to obtain a sequence { ξ ] formed by N sample points1,ξ2,L,ξN}；
S102, transmitting the input parameters to a CFD solver to obtain the concerned output quantity of each sample point and obtain a sequence Y (Y) composed of N response quantities1,y2,L,yN}；
S103, selecting a proper orthogonal function sequence according to the probability density function of the input parameters;
the selection of the orthogonal basis function is determined by the probability density function of the random input variable, and satisfies the following conditions:
i.e. + -jJ ≧ 0} is an orthogonal polynomial sequence with a weight function of f (ξ), where f (ξ) is the probability density distribution satisfied by a random input variable ξ.
And S104, obtaining coefficients in the chaotic polynomial through a regression method.
Solving equations by least squares
Obtaining a vector consisting of the chaotic polynomial coefficient
Where Ψ is a measurement matrix, Ψij＝ψj(ξi)
And S105, calculating and sequencing the contribution degree of each item in the chaotic polynomial to the output variance.
The contribution of the jth term to the output variance in the expansion is determined by
And (4) calculating. The first bit in the ordering is fixed as a constant term in the expansion
At the high fidelity model level, a small number of deterministic calculations are performed S2. The method comprises the following steps:
s201, consistent with S101, obtaining a sequence { ξ'1,ξ′2,L,ξ'M}；
S202, in accordance with S102, a sequence Y ' = { Y ' consisting of M response amounts is obtained '1,y'2,L,y'M}
S3: and correcting the previously established low-fidelity model to obtain a final high-fidelity model.
S301, selecting the number of terms to be corrected as j (j is a positive integer between 1 and M), sorting and selecting corresponding correction terms according to the contribution sizes of the terms determined in S105, and evaluating the generalization error of the correction model by a cross-validation method.
When cross validation is carried out, all high-fidelity sample points are randomly divided into k parts, k-1 parts of the high-fidelity sample points are used as training samples every time, and a new coefficient of a correction term is obtained through a least square method. The remaining one was the test sample, passed R2The coefficients were determined to evaluate the accuracy of the model:
wherein y isiIs the system response calculated by the determined CFD,
is the average value of the response(s),
is a predicted value obtained from the corrected model, nsampleIs the number of verified sample points.
This results in k sets of training/test samples, which can be trained and tested k times. Calculate R on k test set2The mean and variance of the coefficients were determined and recorded.
S302, selecting j to be 1, 2, 3 to M respectively, and circulating the step 301 to obtain R corresponding to each item to be corrected2The mean and variance of the coefficients are determined. By R2The mean of the coefficients is determined to evaluate the accuracy of the modified model, and the variance is used to evaluate the stability of the modified model. And selecting the number of terms to be corrected with higher accuracy and smaller variance, and obtaining the final high-fidelity correction model on all high-fidelity sample points by a least square method again.
In summary, due to the adoption of the technical scheme, the invention has the beneficial effects that:
1. the statistical information of the high fidelity model obtained by the invention has little difference with the original high fidelity model, the mean difference is less than three ten-thousandths and can be ignored, and the error requirements of engineering application are completely met, and the method specifically refers to Table 1.
TABLE 1 statistical information of lift and drag coefficients obtained by the present invention, building a high fidelity model and an original high fidelity model
2. The calculation time required by the high fidelity model established by the invention is about 40 percent of that of the original method, so that the calculation cost is greatly saved, and the table 2 is referred.
TABLE 2 twoHeight ofDetermination by fidelity modelProperty of (2)Number of CFD calculations
| Number of calculations required for original mesh | Number of calculations required for coarse grid | Time (hours) | |
| The high fidelity model established by the invention | 20 | 110 | 21.8 |
| Original high fidelity model | 110 | 0 | 55 |
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The invention will now be described, by way of example, with reference to the accompanying drawings, in which:
FIG. 1 is a flow chart of the method of the present invention.
Detailed Description
All of the features disclosed in this specification, or all of the steps in any method or process so disclosed, may be combined in any combination, except combinations of features and/or steps that are mutually exclusive.
Any feature disclosed in this specification (including any accompanying claims, abstract and drawings), may be replaced by alternative features serving equivalent or similar purposes, unless expressly stated otherwise. That is, unless expressly stated otherwise, each feature is only an example of a generic series of equivalent or similar features.
As shown in fig. 1, the present embodiment includes the following steps:
s1, obtaining a low-fidelity model through calculation at the level of the low-fidelity model, and establishing a rough chaotic polynomial expansion;
s2, performing deterministic calculation on the high fidelity model level;
and S3, correcting the low-fidelity model established in S1 to obtain a final high-fidelity model.
Taking the analysis of the influence of the uncertainty of 9 coefficients of the SA model on the RAE2822 airfoil profile simulation as an example, the three steps are explained in detail:
(1) a low fidelity model is built by performing calculations on the coarse mesh.
a. Assuming that 9 coefficients of the SA model satisfy uniform distribution, sampling is performed in a 9-dimensional random space by using a Latin hypercube method to obtain 110 sample points.
b. And transmitting the 110 sample points to a CFD solver to obtain the wing profile lift and resistance coefficients corresponding to each sample point.
c. A legendre orthogonal function sequence is selected as the basis function for the unfolding.
d. And solving the least square problem to obtain the coefficient of the expansion.
e. And calculating the contribution of each term in the expansion to the variance of the lift coefficient and the resistance coefficient, and sorting the terms from large to small, wherein the first term is always a constant term in the expansion.
(2) And performing a small amount of calculation on the fine grid to obtain high-fidelity sample points.
a. Sampling is carried out in a 9-dimensional random space by using a Latin hypercube method to obtain 20 sample points.
b. And (4) transmitting the 20 sample points to a CFD solver to obtain the wing profile lift and resistance coefficients corresponding to each sample point.
(3) Modifying low fidelity models by high fidelity sample points
a. Assuming that the number of terms needing to be corrected is j, a correction term is determined according to the (1) e process, the coefficient of the correction term is set to be unknown, and the coefficients of other terms are kept unchanged. b. Using a k-fold cross-validation method (k 10 in this example), the model was modified and its generalization error was evaluated. The 20 sample points are evenly divided into 10 parts, 9 parts of the 20 sample points are taken each time to correct the model through a least square method, and the rest parts are used for calculating the prediction error of the corrected model. Repeat 10 times, take the average value of the prediction error.
c. Selecting j as 1, 2, 3 to 20, and repeating (3) a and (3) b to obtain the prediction error mean value corresponding to each j. J with the minimum prediction error is selected as the number of terms to be finally corrected. And correcting the model according to a least square method to obtain a final high-fidelity correction model.
The invention is not limited to the foregoing embodiments. The invention extends to any novel feature or any novel combination of features disclosed in this specification and any novel method or process steps or any novel combination of features disclosed.
Claims (6)
1. A high-fidelity chaotic polynomial correction method suitable for CFD uncertainty quantification is characterized by comprising the following steps:
s1, obtaining a low-fidelity model through calculation at the level of the low-fidelity model, and establishing a rough chaotic polynomial expansion;
s2, calculating on the high fidelity model level to obtain high fidelity sample points;
and S3, correcting the low-fidelity model established in the S1 by using the sample points obtained in the S2 to obtain a final high-fidelity correction model.
2. The high-fidelity chaotic polynomial modification method for CFD uncertainty quantization according to claim 1, wherein the S1 process comprises the following steps:
s101, sampling in a high-dimensional space formed by input parameters by using a Latin hypercube sampling method to obtain a sequence formed by a plurality of sample points;
s102, transmitting the input parameters to a CFD solver to obtain the concerned output quantity of each sample point and obtain a sequence formed by corresponding response quantities;
s103, selecting an orthogonal function sequence according to the probability density function of the input parameters;
s104, obtaining coefficients in the chaos polynomial expansion through a regression method;
and S105, calculating and sequencing the contribution degree of each item in the chaotic polynomial to the output variance.
3. The high-fidelity chaotic polynomial modification method for CFD uncertainty quantization according to claim 2, wherein the selection of the orthogonal basis functions in the orthogonal function sequence is determined by a probability density function of random input variables.
4. The high-fidelity chaotic polynomial modification method for CFD uncertainty quantization according to claim 3, wherein the probability density function satisfies the following requirements:
i.e. + -jJ ≧ 0} is an orthogonal polynomial sequence with a weight function of f (ξ), where f (ξ) is the probability density distribution satisfied by a random input variable ξ.
5. The high fidelity chaotic polynomial modification method for CFD uncertainty quantization according to claim 2, characterized in that the same steps as S101 and S102 are included in S2.
6. The high fidelity chaotic polynomial modification method for CFD uncertainty quantization according to claim 2, characterized in that in S3, the method comprises the following steps:
s301, determining items to be corrected according to the contribution size sequence determined in S105, correcting the low-fidelity model established in the S1 process by a least square method, and evaluating the generalization error of the corrected model by a cross validation method;
s302, according to the number of the items to be corrected, cycling S301 from small to large to obtain the generalized error estimation of the correction model corresponding to each number of the items to be corrected;
and S303, determining the number of terms to be corrected through the minimum generalization error, and correcting the low-fidelity model established in the S1 process through the least square method.
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Cited By (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113627098A (en) * | 2021-07-23 | 2021-11-09 | 北京理工大学 | CFD model confirmation method and product design method |
| CN114676656A (en) * | 2022-05-27 | 2022-06-28 | 中国空气动力研究与发展中心计算空气动力研究所 | Consistency measurement method, device, equipment and storage medium of multi-response CFD model |
| CN114692529A (en) * | 2022-06-02 | 2022-07-01 | 中国空气动力研究与发展中心计算空气动力研究所 | CFD high-dimensional response uncertainty quantification method and device, and computer equipment |
| CN114970396A (en) * | 2022-06-07 | 2022-08-30 | 北京理工大学 | CFD model correction method considering randomness and cognitive uncertainty |
| CN115618771A (en) * | 2022-12-16 | 2023-01-17 | 中国空气动力研究与发展中心计算空气动力研究所 | CFD software reliability quantitative evaluation method |
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2020
- 2020-09-09 CN CN202010937830.2A patent/CN112016253A/en active Pending
Cited By (8)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113627098A (en) * | 2021-07-23 | 2021-11-09 | 北京理工大学 | CFD model confirmation method and product design method |
| CN113627098B (en) * | 2021-07-23 | 2023-12-08 | 北京理工大学 | CFD model confirmation method and product design method |
| CN114676656A (en) * | 2022-05-27 | 2022-06-28 | 中国空气动力研究与发展中心计算空气动力研究所 | Consistency measurement method, device, equipment and storage medium of multi-response CFD model |
| CN114692529A (en) * | 2022-06-02 | 2022-07-01 | 中国空气动力研究与发展中心计算空气动力研究所 | CFD high-dimensional response uncertainty quantification method and device, and computer equipment |
| CN114692529B (en) * | 2022-06-02 | 2022-09-02 | 中国空气动力研究与发展中心计算空气动力研究所 | CFD high-dimensional response uncertainty quantification method and device, and computer equipment |
| CN114970396A (en) * | 2022-06-07 | 2022-08-30 | 北京理工大学 | CFD model correction method considering randomness and cognitive uncertainty |
| CN115618771A (en) * | 2022-12-16 | 2023-01-17 | 中国空气动力研究与发展中心计算空气动力研究所 | CFD software reliability quantitative evaluation method |
| CN115618771B (en) * | 2022-12-16 | 2023-03-10 | 中国空气动力研究与发展中心计算空气动力研究所 | CFD software reliability quantitative evaluation method |
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