CN116011337A - Efficient engineering design method for resisting numerical noise - Google Patents
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Abstract
The invention provides a high-efficiency engineering design method for resisting numerical noise, which comprises the following steps: optimizing the super parameters of the support vector regression model by adopting a super parameter optimization algorithm to obtain the optimal values of the super parameters; adding the optimal value of the super parameter to the diagonal of the correlation matrix during the training of the Kriging model to obtain the correlation matrix of the epsilon-Kriging model, and constructing to obtain the epsilon-Kriging model. Has the following advantages: when the invention is used for engineering design, the design period is greatly shortened, and the engineering practicability is stronger. The epsilon-Kriging proxy model creatively introduces the super parameter epsilon in the support vector regression model into the Kriging model training process, improves the traditional Kriging model into the regression model, effectively filters out the numerical noise influence in the data, and improves the noise filtering performance of the Kriging model.
Description
Technical Field
The invention belongs to the technical field of engineering design, and particularly relates to a high-efficiency engineering design method for resisting numerical noise.
Background
The design method in the current engineering design field is a deterministic design method capable of accurately calculating based on the response value, and the influence of numerical noise in the calculation process of the response value is not considered. For example, in engineering design problems based on numerical simulation, irregular fluctuation of the numerical simulation result, i.e. numerical noise, may occur due to discrete differences of grids and inaccurate setting of boundary conditions. Although slight numerical noise may not have a decisive impact on engineering design, strong numerical noise tends to have a serious impact on the design process. Therefore, how to design engineering problems under the condition of considering the influence of numerical noise and improve the accuracy of engineering design is a technical problem to be solved at present.
Currently, methods that can deal with numerical noise problems include polynomial fitting, deep neural networks, "Nugget-effect" Kriging models, support vector regression models (SVR), and the like. (1) polynomial fitting: polynomial fitting is a linear model whose goal is to construct an M-th order polynomial of the input x such that the polynomial can approximately represent the relationship of the input x and the output y. Studies have shown that the order of the polynomial model is more when the model is more complex. If numerical noise exists in the data, a higher order polynomial model needs to be constructed to fit the samples with the numerical noise, so that the calculation is very time-consuming and unacceptable in practical engineering problems. (2) deep neural network: deep neural networks exhibit good performance in such things as image recognition, object detection, semantic segmentation, and speech and natural language processing, but this approach does not effectively address the problem of numerical noise in the dataset. (3) support vector regression model: the support vector regression model can effectively weaken the influence of numerical noise on modeling due to the unique epsilon zone in the model, and the model considers that the sample points in the epsilon zone are correct sample points, so that the sample points are allowed to have small fluctuation in a certain range without influencing the model. However, the support vector regression model is built without theoretical assumption of a gaussian process, and uncertainty estimation cannot be performed on unknown points, so that the application of the support vector regression model in engineering design is limited. (4) Kriging model: the Kriging model is an interpolation model based on Gaussian process assumption, and is also the most widely applied machine learning method in the field of engineering design (such as pneumatic performance design). However, when the sample contains numerical noise, the Kriging model builds a non-smooth model with irregular fluctuations in the sample, resulting in modeling and design failures.
Therefore, for the engineering design problem that numerical noise exists in the response value, the existing anti-noise method has limitations, and a scheme meeting the design requirement cannot be obtained.
Disclosure of Invention
Aiming at the defects existing in the prior art, the invention provides the high-efficiency engineering design method for resisting the numerical noise, and the engineering design is influenced by the numerical noise.
The technical scheme adopted by the invention is as follows:
the invention provides a high-efficiency engineering design method for resisting numerical noise, which comprises the following steps:
Calculating the response value of each sample point, thereby obtaining n corresponding response values y (1) ,y (2) ,...,y (n) Form a response value matrix y s ＝[y (1) ,y (2) ,...,y (n) ] T ；
Sample point matrix S and response value matrix y s Form a sample set w= (S, y s )＝[(x (1) ,y (1) ),(x (2) ,y (2) ),...,(x (n) ,y (n) )] T ；
Step 2, establishing a support vector regression model SVR, taking a sample set W as a training sample, and optimizing the super parameters of the support vector regression model SVR by adopting a super parameter optimization algorithm to obtain an optimal value of the super parameters epsilon, wherein the optimal value is expressed as epsilon best ；
Step 3, according to the optimal value epsilon of the super parameter epsilon best Construction of improved Kriging model correlation matrix
The expression is:
wherein:
R(x (1) ,x (n) ) Representative sample point x (1) And sample point x (n) Correlation function values therebetween;
R(x (n) ,x (1) ) Representative sample point x (n) And sample point x (1) Correlation function values therebetween;
step 4, constructing an improved Kriging model correlation matrix according to the step 3
Taking a sample set W as a training sample, constructing an improved Kriging model, namely an epsilon-Kriging model, training the super-parameters of the epsilon-Kriging model by adopting a maximum likelihood estimation or cross verification method to obtain the super-parameter optimal value of the epsilon-Kriging model, wherein the super-parameter theta l The optimal value expressed as θ lbest ；
wherein:
β 0 is a constant, represents the global trend of the epsilon-Kriging model,
r (x) represents the unknown point x and n sample points x, respectively (1) ,x (2) ,...,x (n) Correlation vector formed by correlation function values between the correlation vectors, tableThe expression is:
wherein: r (x) (1) ,x),R(x (2) ,x),…,R(x (n) X) respectively represent an unknown point x and a sample point x (1) Correlation function value between unknown point x and sample point x (2) Correlation function value between, …, unknown point x and sample point x (n) Correlation function values therebetween;
t represents the transpose of the matrix;
f represents an n-dimensional identity matrix in the form of: f= [1 1 … 1] T ；
Step 6, inputting the value of any unknown point x, and outputting the predicted value of the response value of the unknown point x through the smooth pre-estimated expression in step 5
Filtering out the predicted value->
And guiding engineering design under the influence of medium-value noise.
Preferably, in step 1, the sampling method is a Latin hypercube sampling method or a uniform sampling method.
Preferably, in step 2, the adopted super-parameter optimization algorithm is a genetic algorithm, a covariance self-adaptive optimization method or a Bayesian optimization method.
Preferably, in step 3, the correlation function value between two sample points represents the correlation between the two sample points and is a function of the distance between the two sample points.
Preferably, for sample point x (1) And sample point x (n) The correlation function value is calculated using the following formula:
R(x (1) ,x (n) )＝exp(-θ lbest |x (1) -x (n) | p )
wherein:
p represents the anisotropic parameter of the correlation function.
The high-efficiency engineering design method for resisting numerical noise has the following advantages:
(1) When the invention is used for engineering design, the response value can be obtained by a low-credibility approximate calculation method while meeting the design requirement, and only the change trend of the response value along with the design variable is captured, so that the design period is greatly shortened, and the invention has stronger engineering practicability.
(2) The epsilon-Kriging proxy model creatively introduces the super parameter epsilon in the support vector regression model into the Kriging model training process, improves the traditional Kriging model into the regression model, effectively filters out the numerical noise influence in the data, and improves the noise filtering performance of the Kriging model.
Drawings
FIG. 1 is a graph of error contrast of modeling of three models based on different noise levels (R_square is R) 2 Mean value, representing modeling accuracy, the same applies below);
FIG. 2 is a graph of model accuracy versus three proxy models for a small sample provided by the present invention;
FIG. 3 is a graph of model accuracy versus three proxy models for a large sample provided by the present invention;
FIG. 4 is a flow chart of the method for efficient engineering design against numerical noise provided by the present invention;
FIG. 5 is a graph of the variation of RAE2822 airfoil drag coefficient with maximum relative thickness for numerical noise constructed during airfoil design in accordance with the present invention;
FIG. 6 is a comparison of the design airfoil and the reference airfoil of the present invention;
FIG. 7 is a graph of the RAE2822 airfoil modified maximum relative thickness with numerical noise figure fit and optimization results, where Samples represent sample points, min real Representing the minimum position, min, of the real function ε-Kriging Representing the minimum position of the real function, epsilon-Kriging represents the epsilon-Kriging model modeling curve.
Detailed Description
In order to make the technical problems, technical schemes and beneficial effects solved by the invention more clear, the invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the scope of the invention.
For engineering design problems of numerical noise in response values, the traditional method cannot effectively filter out the influence of the numerical noise, but the existing anti-noise method has limitations, and a scheme meeting design requirements cannot be obtained. Therefore, the invention provides a high-efficiency engineering design method for resisting numerical noise, which can be applied to the pneumatic design problem.
The invention is inspired by an epsilon band in a support vector regression model SVR, improves an original Kriging model, and provides a new proxy model, namely an epsilon-Kriging model, so that a numerical noise resistant efficient engineering design method is developed, and the main content of the invention is introduced by taking the problem of reducing cruising resistance in aerofoil aerodynamic performance design as an example. It should be noted that, the present invention is a general method applied to engineering design, and the embodiment uses only pneumatic performance design problem as an example, but the practical application scope of the present invention is not limited to this.
The invention provides a high-efficiency engineering design method for resisting numerical noise, which comprises the following steps of:
Calculating the response value of each sample point, thereby obtaining n corresponding response values y (1) ,y (2) ,...,y (n) Form a response value matrix y s ＝[y (1) ,y (2) ,...,y (n) ] T ；
Sample point matrix S and response value matrix y s Form a sample set w= (S, y s )＝[(x (1) ,y (1) ),(x (2) ,y (2) ),...,(x (n) ,y (n) )] T ；
When calculating the response value of each sample point, in order to improve the solution efficiency of engineering design, an approximate solution method may be generally used to calculate the response value of the sample point, for example, in the pneumatic design field, a low-reliability method is used to perform approximate CFD calculation on the sample point, although the calculation accuracy of the response value will be lost, resulting in irregular fluctuation phenomenon of the response value, but the trend feature of the design target along with the change of the design variable is not affected.
Step 2, establishing a support vector regression model SVR, taking a sample set W as a training sample, and optimizing the super parameters of the support vector regression model SVR by adopting a super parameter optimization algorithm to obtain an optimal value of the super parameters epsilon, wherein the optimal value is expressed as epsilon best ；
In the step, the adopted super-parameter optimization algorithm can be a genetic algorithm, a covariance self-adaptive optimization method or a Bayesian optimization method.
The merits of the super parameter epsilon of the support vector regression model SVR are evaluated by a generalization error, and the generalization error evaluation method comprises a cross-validation and a boundary-keeping method. The research result of the inventor shows that the super-parameter obtained by the super-parameter optimization method based on Bayesian optimization has good fitting effect, so that the super-parameter epsilon is optimized by adopting the Bayesian optimization method as a preferable mode.
Step 3, according to the optimal value epsilon of the super parameter epsilon best Construction of improved Kriging model correlation matrix R εbest The expression is:
wherein:
R(x (1) ,x (n) ) Representative sample point x (1) And sample point x (n) Correlation function values therebetween;
R(x (n) ,x (1) ) Representative sample point x (n) And sample point x (1) Correlation function values therebetween;
therefore, the invention leads the optimal value epsilon of the super parameter epsilon of the SVR of the support vector regression model obtained in the step 2 best Added to the diagonal of the correlation matrix of the Kriging model to obtain the correlation matrix of the epsilon-Kriging model
The correlation function value between two sample points, representing the correlation between the two sample points, is a function of the distance between the two sample points. The selection of the correlation function is required to satisfy the Gaussian assumption, and the correlation matrix is symmetrically and positively determined, and a Gaussian index model and a cubic spline function can be adopted.
As a specific embodiment, for sample point x (1) And sample point x (n) The correlation function value is calculated using the following formula:
R(x (1) ,x (n) )＝exp(-θ lbest |x (1) -x (n) | p )
wherein:
p represents the anisotropic parameter of the correlation function.
Step 4, constructing an improved Kriging model correlation matrix according to the step 3
The sample set W is taken as a training sample, an improved Kriging model is constructed, the improved Kriging model is called an epsilon-Kriging model, a maximum likelihood estimation or cross verification method is adopted to train the super-parameters of the epsilon-Kriging model, the mean value and the process variance of the epsilon-Kriging model in the training process can be obtained by analysis, the parameters in the correlation function can be obtained through a numerical optimization algorithm, and further the super-parameter optimal value of the epsilon-Kriging model is obtained, wherein the super-parameter theta l The optimal value expressed as θ lbest ；
wherein:
β 0 is a constant, represents the global trend of the epsilon-Kriging model,
r (x) represents the unknown point x and n sample points x, respectively (1) ,x (2) ,...,x (n) The expression of the correlation vector formed by the correlation function values is:
wherein: r (x) (1) ,x),R(x (2) ,x),…,R(x (n) X) respectively represent an unknown point x and a sample point x (1) Correlation function value between unknown point x and sample point x (2) Correlation function value between, …, unknown point x and sample point x (n) Correlation function values therebetween;
t represents the transpose of the matrix;
f represents an n-dimensional identity matrix in the form of: f= [1 1 … 1] T ；
Step 6, inputting the value of any unknown point x, and outputting the predicted value of the response value of the unknown point x through the smooth pre-estimated expression in step 5
Filtering out the predicted value->
Influence of medium value noiseAnd guiding engineering design to obtain an optimal design scheme.
The influence of numerical noise existing in the sample set on modeling can be filtered out through a smooth pre-estimated expression of the epsilon-Kriging model. The estimated value of the unknown point x can be obtained through the smooth estimated expression of the epsilon-Kriging model
And, while giving some unknown point x estimated value, can also get the mean square error estimation of the corresponding estimated value +.>
The engineering design method considering the influence of numerical noise provided by the invention can obtain an optimal design scheme and ensure higher design efficiency.
The main innovation point of the invention is based on the numerical noise phenomenon in the engineering design process and the problem of weak numerical noise processing capability of the traditional method, and provides an efficient engineering design method capable of resisting the influence of the numerical noise. In a specific engineering design problem, such as an airfoil design process, if a response value is affected by numerical noise, the conventional design method cannot handle the numerical noise, and the response value is affected by the numerical noise, but can generally correctly reflect the variation trend of a design target along with design variables. The epsilon-Kriging model provided by the invention is based on the sample points affected by numerical noise, and a smoother mathematical model is established, so that the noise filtering treatment of the sample points is realized. Therefore, the smooth pre-estimated expression of the epsilon-Kriging model can be solved by adopting a traditional method, so that the pre-estimated value of the design variable of the smooth pre-estimated expression of the epsilon-Kriging model is obtained.
One embodiment is described below:
the present embodiment is based on RAE2822 airfoil standard examples, developing airfoil designs that are subject to numerical noise. When CFD evaluation is generally performed on airfoil aerodynamic coefficients, numerical noise is caused by the merits of discrete grid quality, and sparse grids are divided for simulating numerical noise phenomena generated by grid quality in airfoil design. And adopting non-sticking attack angle calculation, and pneumatically designing by taking the maximum relative thickness of the airfoil as a design variable.
This embodiment uses the PARSEC airfoil parameterization method to describe the airfoil by 13 parameters. In the design state, the free-coming flow mach number Ma is 0.734, aoa=2.79 deg, and reynolds number re=6.5×106. The maximum relative thickness of the basic airfoil is known as 0.1207078, and the maximum relative thickness variation range is set as [0.07,0.19], namely: the design space X is [0.07,0.19].
And step 1, solving a non-stick equation when carrying out CFD evaluation on sample points in a design space X0.07,0.19. For the calculation grid, calculation was performed using an unstructured grid, and the far field boundary was 50 times the chord length from the object plane, and the grid cell number was 15000 (surface grid quantity 512).
In design space X [0.07,0.19]]Sampling uniformly, selecting 86 sample points in total to form a sample point matrix S= [ x ] (1) ,x (2) ,...,x (86) ] T The method comprises the steps of carrying out a first treatment on the surface of the Performing CFD evaluation on the sample points to obtain corresponding aerodynamic coefficient response values (resistance coefficients) so as to construct a response value matrix y s ＝[y (1) ,y (2) ,...,y (86) ] T ；
Sample point matrix S and response value matrix y s The sample set W is formed, and specifically, as shown in FIG. 5, the abscissa t/c is the maximum relative thickness and the ordinate Cd is the aerodynamic coefficient response value.
And 2, modeling the sample set W by adopting a support vector regression model SVR based on the sample set W in the step 1. Meanwhile, a Bayesian optimization algorithm is adopted to optimize the super-parameters of the support vector model SVR, and the optimal value epsilon of the super-parameters epsilon of the support vector regression model SVR is obtained best 0.008. In the process, the advantages and disadvantages of the super parameters are obtained based on the generalization error evaluation of the cross validation.
Step 3, extracting the optimal value epsilon of the super parameter epsilon of the SVR in the step 2 best Based on the sample set W in the step 1, in the proxy model training stageWill be optimal best Adding the correlation matrix into the diagonal of the Kriging correlation matrix to form a correlation matrix of epsilon-Kriging model
In addition, a Gaussian index model is selected as a correlation function for calculating a correlation function value between two sample points.
And 4, training the super-parameters of the epsilon-Kriging model according to the maximum likelihood estimation, wherein the mean value and the process variance of epsilon-Kriging in the training process can be obtained by analysis, and the parameters in the correlation function are obtained by a quasi-Newton method.
And 5, establishing a smooth pre-estimated expression of the epsilon-Kriging model, and filtering out numerical noise possibly existing in the sample set. The predicted value of the response value of the smooth pre-estimated expression of the epsilon-Kriging model to a certain sample point (the design variable of which is x) is
The smooth pre-estimated expression of epsilon-Kriging model can give out the pre-estimated value of response value of a certain sample point and at the same time can also obtain the mean square error estimate +.>
And 6, solving the smooth estimated expression of the epsilon-Kriging model established in the step 5 by adopting a genetic algorithm. Predictive value of response value obtained by prediction through smooth predictive expression of epsilon-Kriging model
The effect of numerical noise has been filtered out, so airfoils meeting design requirements can be obtained by genetic algorithms.
The airfoil design effect obtained by the method of the invention is as follows:
the reference airfoil baseline and the designed airfoil optimal profile pair are shown in fig. 6, the maximum relative thickness of the reference airfoil is 0.1218, and the maximum relative thickness of the designed airfoil is 0.1064, so that the maximum relative thickness of the designed airfoil is slightly smaller than the maximum relative thickness of the reference airfoil, and the designed airfoil lightens the shock intensity of the upper airfoil.
To verify the quality of the design results, performance of the airfoils before and after design was evaluated, and the aerodynamic coefficients of the reference airfoil and the designed airfoil are shown in table 1 for comparison.
TABLE 1RAE2822 design results of wing-type maximum relative thickness noise drag coefficient containing values
As can be seen from Table 1, the method provided by the invention has better drag reduction effect under the condition of meeting the design requirement, and can more accurately find the design point with the best performance, and the relative error with the actual optimal point position is controlled within 2%.
FIG. 7 shows the fitting result and the final design result of the epsilon-Kriging model provided by the invention, and as can be seen from FIG. 7, the epsilon-Kriging optimal solution is basically consistent with the true optimal solution in position. Therefore, the noise filtering effect of the noise filtering epsilon-Kriging model provided by the invention is good, and an optimal scheme can be accurately designed.
The invention also carries out a comparison test on the model precision of the epsilon-Kriging model, the traditional 'Nugget-effect' Kriging model and the traditional SVR model.
As shown in FIG. 1, error comparisons are modeled for three models based on different noise levels (R_square is R 2 Mean value, representing modeling accuracy, the same applies below); as shown in fig. 2, the model accuracy of the three proxy models in the case of a small sample is compared; as shown in fig. 3, the model accuracy of the three proxy models is compared in the case of a large sample.
By comparing the figures 1, 2 and 3, it can be seen that the epsilon-Kriging proxy model provided by the invention has higher modeling precision in the whole, small sample condition and large sample condition, and the modeling precision of the epsilon-Kriging proxy model is higher than that of the SVR model and the "Nugget-effect" Kriging model; under the condition of low-and-medium amplitude noise, the SVR modeling precision is higher than that of a Nugget-effect Kriging model, but under the condition of high-amplitude noise, the SVR modeling precision is higher than that of the SVR model.
The high-efficiency engineering design method for resisting numerical noise has the following advantages:
(1) When the invention is used for engineering design, the response value can be obtained by a low-credibility approximate calculation method while meeting the design requirement, and only the change trend of the response value along with the design variable is captured, so that the design period is greatly shortened, and the invention has stronger engineering practicability.
(2) The epsilon-Kriging proxy model creatively introduces the super parameter epsilon in the support vector regression model into the Kriging model training process, improves the traditional Kriging model into the regression model, effectively filters out the numerical noise influence in the data, and improves the noise filtering performance of the Kriging model.
The foregoing is merely a preferred embodiment of the present invention and it should be noted that modifications and adaptations to those skilled in the art may be made without departing from the principles of the present invention, which is also intended to be covered by the present invention.
Claims (5)
1. The high-efficiency engineering design method for resisting numerical noise is characterized by comprising the following steps of:
step 1, determining a design space X according to a specific engineering problem, and sampling n sample points X in the design space X (1) ,x (2) ,...,x (n) Form a sample point matrix s= [ x ] (1) ,x (2) ,...,x (n) ] T ；
Calculating the response value of each sample point, thereby obtaining n corresponding response values y (1) ,y (2) ,...,y (n) Form a response value matrix y s ＝[y (1) ,y (2) ,...,y (n) ] T ；
Sample point matrix S and response value matrix y s Form a sample set w= (S, y s )＝[(x (1) ,y (1) ),(x (2) ,y (2) ),...,(x (n) ,y (n) )] T ；
Step 2, establishing a support vector regression model SVR, taking a sample set W as a training sample, and optimizing the super parameters of the support vector regression model SVR by adopting a super parameter optimization algorithm to obtain an optimal value of the super parameters epsilon, wherein the optimal value is expressed as epsilon best ；
Step 3, according to the optimal value epsilon of the super parameter epsilon best Construction of improved Kriging model correlation matrix
The expression is:
wherein:
R(x (1) ,x (n) ) Representative sample point x (1) And sample point x (n) Correlation function values therebetween;
R(x (n) ,x (1) ) Representative sample point x (n) And sample point x (1) Correlation function values therebetween;
step 4, constructing an improved Kriging model correlation matrix according to the step 3
Taking a sample set W as a training sample, constructing an improved Kriging model, namely an epsilon-Kriging model, training the super-parameters of the epsilon-Kriging model by adopting a maximum likelihood estimation or cross verification method to obtain the super-parameter optimal value of the epsilon-Kriging model, wherein the super-parameter theta l The optimal value expressed as θ lbest ；
Step 5, obtaining a smooth estimated expression of the epsilon-Kriging model, wherein the smooth estimated expression is as follows:
wherein:
β 0 is a constant and represents the global trend of the epsilon-Kriging model, beta 0 ＝(F T R εbest -1 F) -1 F T R εbest -1 y s ；
r (x) represents the unknown point x and n sample points x, respectively (1) ,x (2) ,...,x (n) The expression of the correlation vector formed by the correlation function values is:
wherein: r (x) (1) ,x),R(x (2) ,x),…,R(x (n) X) respectively represent an unknown point x and a sample point x (1) Correlation function value between unknown point x and sample point x (2) Correlation function value between, …, unknown point x and sample point x (n) Correlation function values therebetween;
t represents the transpose of the matrix;
f represents an n-dimensional identity matrix in the form of: f= [1 1 … 1] T ；
Step 6, inputting the value of any unknown point x, and outputting the predicted value of the response value of the unknown point x through the smooth pre-estimated expression in step 5
Filtering out the predicted value->
And guiding engineering design under the influence of medium-value noise.
2. The method of claim 1, wherein in step 1, the sampling method is a latin hypercube sampling method or a uniform sampling method.
3. The efficient engineering design method for resisting numerical noise according to claim 1, wherein in the step 2, the adopted super-parameter optimization algorithm is a genetic algorithm, a covariance adaptive optimization method or a bayesian optimization method.
4. The method of claim 1, wherein in step 3, the correlation function value between two sample points represents a correlation between the two sample points as a function of a distance between the two sample points.
5. The method of claim 4, wherein for sample point x (1) And sample point x (n) The correlation function value is calculated using the following formula:
R(x (1) ,x (n) )＝exp(-θ lbest |x (1) -x (n) | p )
wherein:
p represents the anisotropic parameter of the correlation function.
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