An engine sensitivity analysis method based on the stochastic collocation method

Abstract

The present invention provides a method for engine sensitivity analysis based on the stochastic collocation method, belonging to the field of engine analysis. First, determine the random variables affecting the response of the engine model and the mean and standard deviation corresponding to the random variables, and establish the input-output relationship between the random variable input and the engine model response output. Secondly, calculate the contribution of each random variable to the variance of the engine model response based on the stochastic collocation method. Finally, calculate the sensitivity index of each random variable to the engine model response. The present invention solves the problem of engine sensitivity analysis through the stochastic collocation method, filling the research gap in engine sensitivity analysis; constructs a sample point set based on the idea of the stochastic collocation method, with higher accuracy and reduced computational effort required for sensitivity analysis; does not depend on a specific engine model construction method, is applicable to engines of different models and types, and can achieve the purpose of efficient engine sensitivity analysis.

Description

Technical Field

[0001] The present invention belongs to the field of engine analysis and relates to an engine sensitivity analysis method based on the stochastic perturbation collocation method. Background Technique

[0002] Performing sensitivity analysis on an engine is an important means to study and measure the influence of different random variables on the engine, and it can provide an important reference basis for the subsequent design optimization of the engine. The engine has a complex structure, numerous parts, and complicated processes. Different processing methods will cause a large number of uncertain variables in the part sizes and assemblies, ultimately resulting in different performances of the engine products. Therefore, in order to ensure a high yield rate in engine mass production, performing sensitivity analysis on each uncertain variable affecting the engine and screening out the most significant factors affecting the engine are essential steps in engine production and manufacturing.

[0003] To perform engine sensitivity analysis, two key steps need to be completed. First is to construct an accurate input-output model between the random variables and the engine response, and second is to sample the random variables to construct the sample set required for subsequent global sensitivity analysis. Due to the complex structure of the engine and the large number of uncertain variables involved, performing sensitivity analysis on the engine is a computationally intensive task. To improve the computational efficiency and reduce the computational effort required for engine sensitivity analysis, there have been a large number of studies. For example, in the Chinese invention patent "A global sensitivity analysis method for an aeroengine disk based on Copula" (Application No.: CN202410126072.4), a global sensitivity analysis method for an aeroengine disk based on space partitioning and combined with the Kriging surrogate model is developed. By using the surrogate model technology, the computational effort required for each single sample calculation is reduced. In the Chinese invention patent "A method for predicting the unsteady characteristics of shock trains in an isolator based on an artificial neural network" (Application No.: N202410658945.6), the artificial neural network technology is applied to the global sensitivity analysis of a dual-mode scramjet engine, and a simplified network model between the random variable input and output is constructed. The above patents all start from simplifying the first key step, that is, simplifying the input-output model between the random variables and the engine response, and reducing the computational effort for each single sample calculation to reduce the overall computational effort of sensitivity analysis. However, regarding the second key step, since these methods still remain at random sampling, a large number of samples are still required to ensure the computational accuracy during global sensitivity analysis.

[0004] In view of the deficiencies of the prior art, the present invention proposes an engine sensitivity analysis method based on the stochastic perturbation collocation method. Starting from improving the sampling method of sample points, the number of sample points required for calculation is reduced while ensuring the calculation accuracy, and further reducing the computational effort required for engine sensitivity analysis. The stochastic perturbation collocation method is a numerical method proposed by scholar Wu in solving the problem of uncertainty quantification of random variables [Wu F, Gao Q, Xu X M, et al. A modified computational scheme for the stochastic perturbation finite element method[J]. Latin American Journal of Solids and Structures, 2015, 12: 2480-2505.], [Wu F, Zhong W X. A modified stochastic perturbation method for stochastic hyperbolic heat conduction problems[J]. Computer Methods in Applied Mechanics and Engineering, 2016, 305: 739-758]. At present, it has been used in the fields of civil engineering, aerospace, nuclear engineering, etc. Based on the idea of the stochastic perturbation collocation method, the present invention constructs a set of sample points required in the engine sensitivity analysis process, and solves the difficulty that a large number of sample points are required to ensure the calculation accuracy in the existing methods. Since the present invention starts from improving the sampling method, it is applicable to existing different engine input-output models, can be combined with existing different methods, has extremely strong versatility, and can further reduce the overall computational effort required for engine sensitivity analysis. Summary of the Invention

[0005] The present invention mainly solves the problems of large computational effort and low computational efficiency of traditional engine sensitivity analysis methods, and proposes an engine sensitivity analysis method based on the stochastic perturbation collocation method. Based on the stochastic perturbation collocation method, the present invention can complete the engine sensitivity analysis with multiple random variables using fewer sample points, saving costs and time.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is as follows:

[0007] An engine sensitivity analysis method based on the stochastic collocation method. First, determine the random variables that affect the engine model response and the corresponding mean values and standard deviations of the random variables, and establish the input-output relationship between the random variable input and the engine model response output. Second, calculate the contribution of each random variable to the variance of the engine model response based on the stochastic collocation method. Finally, calculate the sensitivity index of each random variable to the engine model response. It includes the following steps:

[0008] In the first step, determine the random variables that affect the engine model response and the corresponding mean values and standard deviations of the random variables, and establish the input-output relationship between the random variables and the engine model response. Specifically:

[0009] Step 1.1, it is known that there are a total of random variables that affect the engine model response. Represent these random variables as a vector , the corresponding mean vector is , and the corresponding standard deviation vector is . Among them, represents the -th component of the vector , corresponding to the -th random variable; represents the mean value of the -th random variable; represents the standard deviation of the -th random variable; .

[0010] Step 1.2, convert the random variable vector to a zero-mean random variable vector , where , and the standard deviation of is . The zero-mean random variable and the original random variable are in one-to-one correspondence.

[0011] Step 1.3, construct a numerical model of the engine. The input of the numerical model is the zero-mean random variable vector obtained in Step 1.2, and the output of the numerical model is the engine model response .

[0012] After Steps 1.1 to 1.3, we obtain the output-output relationship between the zero-mean random variable vector corresponding to the random variable vector and the engine model response .

[0013] Further, in step 1.3, a numerical model of the engine is constructed using the finite element method or the surrogate model method.

[0014] Second, based on the stochastic perturbation collocation method, calculate the contribution of each random variable to the response variance of the engine model. Specifically:

[0015] Step 2.1: Construct two different zero-mean random variable vector sample input sets based on the stochastic perturbation collocation method, and calculate the corresponding two different engine model response output sets using the input-output relationship obtained in the first step. The calculation methods for the two different zero-mean random variable vector sample input sets and the corresponding two engine model response output sets are as follows:

[0016] Zero-mean random variable vector sample input set 1:

[0017] ,

[0018] where, , represents the -th component of the zero-mean random variable vector sample input ; , represents the expectation of the fourth power of the zero-mean random variable and the ratio of the fourth power of the standard deviation of the zero-mean random variable ; represents the calculation of the mathematical expectation; represents the -th component of the zero-mean random variable vector , which is a zero-mean random variable; represents the standard deviation of the zero-mean random variable , . When the -th component of the zero-mean random variable has a uniform distribution, ; when the -th component of the zero-mean random variable has a Gaussian distribution, . Substitute the zero-mean random variable vector sample input set 1 into the engine model input-output relationship obtained in the first step to obtain the corresponding engine model response output set 1: ;

[0019] Random variable vector sample input set 2:

[0020] , , .

[0021] Among them, and are calculated in the same way as in the random variable sample input set 1. Substitute the random variable vector sample input set 2 into the engine model input-output relationship obtained in the first step, and calculate the corresponding engine model response output set 2: .

[0022] Step 2.2, based on the engine model response output set 1 calculated in Step 2.1, calculate the absolute contribution of the zero-mean random variable to the variance of the engine model response, as shown in the following formula:

[0023] ,

[0024] Among them, represents the absolute contribution of the zero-mean random variable to the variance of the engine model response; , which is an intermediate variable used in the calculation; represents the engine model response output when the random variable vector input is ; represents the engine model response output when the random variable vector input is ; , which is an intermediate variable used in the calculation; represents the engine model response output when the random variable vector input is ;

[0025] Step 2.3, based on the engine model response output set 2 calculated in Step 2.1, calculate the absolute contribution of the coupling of the zero-mean random variables and to the variance of the engine model response, as shown in the following formula:

[0026] ,

[0027] Among them, represents the direct contribution of the coupling of the zero-mean random variables and to the variance of the engine model response; represents the ratio of the expectation of the fourth power of the zero-mean random variable to the fourth power of the standard deviation of the zero-mean random variable , and its calculation method is the same as , only with different subscripts; ; ; , , They are all intermediate variables used in the calculation. The calculation method is as follows. Specifically:

[0028] .

[0029] Among them, ; Substitute the random variable vector sample input set 2 into the input-output relationship of the engine model obtained in the first step, and calculate the zero-mean random variable and The absolute contribution of the coupling to the response variance of the engine model.

[0030] Step 2.4, calculate the variance of the engine model response , .

[0031] After the above steps 2.1 to 2.4, we have calculated the variance of the engine model response , the zero-mean random variable The absolute contribution to the response variance of the engine model , the zero-mean random variable and The absolute contribution of the coupling to the response variance of the engine model .

[0032] The third step is to calculate the sensitivity index corresponding to each random variable and the sensitivity index corresponding to the coupling relationship between the random variables and based on the calculation results of the second step. Specifically:

[0033] Step 3.1, according to the calculation results of steps 2.2 and 2.4, calculate , It reflects the relative contribution of the zero-mean random variable to the variance . Since the zero-mean random variable corresponds one-to-one with the original random variable , is also The corresponding sensitivity index.

[0034] Step 3.2, according to the calculation results of steps 2.3 and 2.4, calculate , where , It reflects the relative contribution of the coupling effect of the random variables and to the variance . Since the zero-mean random variable corresponds one-to-one with the original random variable , Also and The sensitivity index corresponding to the coupling effect

[0035] Based on the above steps 3.1 to 3.2, we calculated the sensitivity index of the original random variable acting alone on the variance of the engine response ; We also calculated the sensitivity index of the original random variable ; We also calculated the sensitivity index of the original random variable and the coupling effect on the variance of the engine response The sensitivity index of the impact .

[0036] Compared with the prior art, the present invention has the following beneficial effects:

[0037] (1) The present invention solves the problem of engine sensitivity analysis by the stochastic perturbation collocation method, filling the research gap in engine sensitivity analysis.

[0038] (2) Based on the idea of the stochastic perturbation collocation method, the present invention constructs a sample point set. Compared with the traditional sensitivity analysis sampling method, it requires fewer sample points, higher accuracy, and reduces the computational amount required for sensitivity analysis.

[0039] (3) The present invention does not depend on a specific engine model construction method, is applicable to engines of different models and types, and achieves the purpose of efficient engine sensitivity analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] Figure 1 is a flow chart of the engine sensitivity analysis method based on the stochastic perturbation collocation method. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0041] The present invention will be described in detail below with reference to the drawings and specific embodiments. This embodiment is implemented on the premise of the technical solution of the present invention, and gives detailed implementation manners and specific operation processes, but the protection scope of the present invention is not limited to the following embodiments.

[0042] Refer to Figure 1 the flow chart of the engine sensitivity analysis method based on the stochastic perturbation collocation method, and perform sensitivity analysis on the pressure at key points in the engine compartment of a certain model four-stroke engine. This embodiment includes the following steps:

[0043] The first step is to establish the input-output relationship between the random variable and the pressure at the key points in the engine compartment. Specifically:

[0044] Step 1.1, it is known that there are a total of random variables that affect the pressure at the key points in the engine model compartment There are 5, namely ambient pressure, initial cylinder temperature, initial cylinder pressure, engine speed, and air-fuel ratio. These 5 random variables all follow a uniform distribution. Represent these 5 random variables as a vector , and the corresponding mean vector is , and the corresponding standard deviation vector is . Among them, represents the th component of the vector , corresponding to the th random variable; represents the mean of the th random variable; represents the standard deviation of the th random variable; .

[0045] Step 1.2, convert the random variable vector to a zero-mean random variable vector , where , and the standard deviation of is . The zero-mean random variable and the original random variable are in one-to-one correspondence.

[0046] Step 1.3, construct a numerical model of the engine, and the construction method uses the finite element method. The input of the numerical model is the zero-mean random variable vector obtained in Step 1.2, and the output of the numerical model is the pressure at the key points in the engine compartment.

[0047] After the above three steps, we obtain the output-output relationship between the zero-mean random variable vector corresponding to the random variable vector and the pressure at the key points in the engine compartment.

[0048] Second step, calculate the contribution of each random variable to the variance of the pressure at the key points in the engine compartment based on the stochastic perturbation collocation method. Specifically:[[]]

[0049] Step 2.1, construct 2 different zero-mean random variable vector sample input sets based on the stochastic perturbation collocation method, and calculate the corresponding 2 different pressure output sets at the key points in the engine compartment by using the input-output relationship obtained in the first step. The calculation methods of the 2 different zero-mean random variable vector sample input sets and the corresponding 2 pressure output sets at the key points in the engine compartment are as follows:[[]]

[0050] Zero-mean random variable vector sample input set 1:[[]]

[0051] ,

[0052] where represents the \(i\)-th component of the zero-mean random variable vector sample input the -th; represents the ratio of the expectation of the fourth power of the zero-mean random variable to the fourth power of the standard deviation of the zero-mean random variable ; represents the calculation of the mathematical expectation; represents the \(j\)-th component of the zero-mean random variable vector the -th, which is a zero-mean random variable; represents the standard deviation of the zero-mean random variable ; . Since the distributions of the zero-mean random variables are all uniform distributions, . Substitute the zero-mean random variable vector sample input set 1 into the engine model input-output relationship obtained in the first step to obtain the corresponding pressure output set 1 of the key points in the engine compartment: ;

[0053] Random variable vector sample input set 2:

[0054] , , .

[0055] where and are calculated in the same way as in the random variable sample input set 1. Substitute the random variable vector sample input set 2 into the engine model input-output relationship obtained in the first step to calculate the corresponding pressure output set 2 of the key points in the engine compartment: .

[0056] Step 2.2, based on the pressure output set 1 of the key points in the engine compartment calculated in Step 2.1, calculate the absolute contribution of the zero-mean random variable to the pressure variance of the key points in the engine compartment, as shown in the following formula:

[0057] ,

[0058] where represents the absolute contribution of the zero-mean random variable to the pressure variance of the key points in the engine compartment; is an intermediate variable used in the calculation; denotes the pressure output at the key points in the engine compartment when the random variable vector is input ; denotes the pressure output at the key points in the engine compartment when the random variable vector is input ; , which is an intermediate variable used in the calculation; denotes the pressure output at the key points in the engine compartment when the random variable vector is input ;

[0059] Step 2.3, based on the pressure output set 2 of the key points in the engine compartment calculated in Step 2.1, calculate the absolute contributions of the zero-mean random variables and coupled to the pressure variance at the key points in the engine compartment, as shown in the following formula:

[0060] ,

[0061] where denotes the direct contribution of the zero-mean random variables and coupled to the pressure variance at the key points in the engine compartment; denotes the ratio of the expectation of the fourth power of the zero-mean random variable to the fourth power of the standard deviation of the zero-mean random variable , and its calculation method is the same as that of , only with different subscripts; ; ; , , are all intermediate variables used in the calculation, and the calculation methods are as follows. Specifically:

[0062] .

[0063] where ; . Substitute the random variable vector sample input set 2 into the input-output relationship of the engine model obtained in the first step to calculate the absolute contributions of the zero-mean random variables and coupled to the pressure variance at the key points in the engine compartment.

[0064] Step 2.4, calculate the variance of the pressure at the key points in the engine compartment , .

[0065] After the above 4 steps, we have calculated the variance of the pressure at the key points in the engine compartment and the zero-mean random variables Absolute contribution to the variance of the pressure at key points in the engine compartment , zero-mean random variable and Coupling's absolute contribution to the pressure at key points in the engine compartment .

[0066] Step 3: Calculate the sensitivity indices corresponding to each random variable and the sensitivity index corresponding to the coupling relationship between the random variable and based on the calculation results of Step 2. Specifically:

[0067] Step 3.1: Calculate , which reflects the relative contribution of the zero-mean random variable to the variance . Since the zero-mean random variable corresponds one-to-one with the original random variable , is also the corresponding sensitivity index.

[0068] Step 3.2: Calculate , which reflects the relative contribution of the coupling effect between the random variables and to the variance . Since the zero-mean random variable corresponds one-to-one with the original random variable , is also and the sensitivity index corresponding to the coupling effect.

[0069] Based on the above two steps, we have calculated the sensitivity index of the influence of the original random variable acting alone on the variance of the pressure at key points in the engine compartment (subscripts 1 - 5 correspond to ambient pressure, initial cylinder temperature, initial cylinder pressure, engine speed, and air-fuel ratio respectively); we have also calculated the sensitivity index of the influence of the coupling effect between the original random variables and on the engine response variance . The final calculation results of the sensitivity indices are shown in the following table: .

[0070] ​

[0071] According to the data analysis in the table, the pressure at the key points in the engine compartment is most affected by the initial cylinder temperature and the air-fuel mixture ratio. The ambient pressure, the initial cylinder pressure, and the engine speed have little effect on the pressure at the key points in the engine compartment.

[0072] The above-described embodiments only represent the implementation modes of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that for those skilled in the art, without departing from the concept of the present invention, several modifications and improvements can still be made, and these all belong to the protection scope of the present invention.

Claims

1. An engine sensitivity analysis method based on random perturbation collocation method, characterized in that: The engine sensitivity analysis method comprises the following steps: The first step is to determine the random variables that affect the engine model response and the mean and standard deviation corresponding to the random variables, and establish an input-output relationship between the random variables and the engine model response; including the following steps: Step 1.1, assuming that there are q random variables that affect the response of the engine model, these q random variables are represented as a vector X = [X1, X2, ..., X q ] T , the corresponding mean vector is The corresponding standard deviation vector is Among them, X i represents the i-th component of vector X, corresponding to the i-th random variable; represents the mean of the i-th random variable; represents the standard deviation of the i-th random variable; i=1,2,…,q; Step 1.2, convert the random variable vector X into a zero-mean random variable vector ε = [ε1, ε2, …, ε q ] T ,in ε i The standard deviation of Zero mean random variable ε i With the original random variable X i There is a one-to-one correspondence; Step 1.3, construct a numerical model of the engine; the input of the numerical model is the zero-mean random variable vector ε = [ε1, ε2, …, ε q ] T , the output of the numerical model is the engine model response u(ε); After steps 1.1 to 1.3, the output-output relationship between the zero-mean random variable vector ε corresponding to the random variable vector X and the engine model response u(ε) is obtained; The second step is to calculate the contribution of each random variable to the engine model response variance based on the random perturbation collocation method; the steps include: Step 2.1, based on the random perturbation collocation method, two different zero-mean random variable vector sample input sets are constructed, and the input-output relationship obtained in the first step is used to calculate the corresponding two different engine model response output sets, including zero-mean random variable vector sample input set 1 and random variable vector sample input set 2; specifically: In step 2.1, the calculation method of two different zero-mean random variable vector sample input sets and the corresponding two engine model response output sets is as follows: Zero mean random variable vector sample input set 1: in, Represents a zero-mean random variable vector sample input b i The i-th component of ; represents a zero-mean random variable ε i The expectation of the fourth power and the zero-mean random variable ε i The ratio of the fourth power of the standard deviation of i represents the i-th component of the zero-mean random variable vector ε, which is a zero-mean random variable; σ i represents a zero-mean random variable ε i , i = 1, 2, ..., q; Substitute the zero-mean random variable vector sample input set 1 into the engine model input-output relationship obtained in the first step, and obtain the engine model response output set 1 corresponding to one: {u(b ±i )}; Random variable vector sample input set 2: Among them, b i and b j The calculation method is the same as that of random variable sample input set 1 b i The calculation method is the same as that of ; the random variable vector sample input set 2 is brought into the engine model input-output relationship obtained in the first step, and the engine model response output set 2 corresponding to one is calculated: {u(b ±i,±j )}; Step 2.2, based on the engine model response output set 1 calculated in step 2.1, calculate the zero-mean random variable ε i absolute contribution to the variance of the engine model response; Step 2.3, based on the engine model response output set 2 calculated in step 2.1, calculate the zero-mean random variable ε i and ε j The absolute contribution of the coupling to the variance of the engine model response; Step 2.4, calculate the variance D(u) of the engine model response; The third step is to calculate the sensitivity index of each random variable to the engine model response based on the calculation results of the second step.

2. The engine sensitivity analysis method based on random perturbation collocation method according to claim 1 is characterized in that: In the step 1.3, the numerical model of the engine is constructed by using the finite element method or the proxy model method.

3. The engine sensitivity analysis method based on random perturbation collocation method according to claim 1 is characterized in that: Calculate the zero-mean random variable vector sample input set 1, when the i-th component ε of the zero-mean random variable ε i When the distribution of is uniform, When the i-th component ε of the zero-mean random variable ε i When the distribution of is Gaussian, ρ iiii =3.

4. The engine sensitivity analysis method based on random perturbation collocation method according to claim 1, characterized in that: The steps 2.2 to 2.4 are specifically as follows: Step 2.2, calculate the zero-mean random variable ε i The formula for the absolute contribution to the variance of the engine model response is: Among them, D u,i represents a zero-mean random variable ε i The absolute contribution to the variance of the engine model response; w u,i =u(b i )-u(b -i ), is the intermediate variable used in the calculation; u(b i ) means that when the random variable vector input ε = b i The engine model response output when u(b -i ) means that when the random variable vector input ε = b -i The engine model response output when z u,i =u(b i )+u(b -i )-2u(0), is the intermediate variable used in the calculation; u(0) represents the engine model response output when the random variable vector input ε=0; Step 2.3, calculate the zero-mean random variable ε i and ε j The formula for the absolute contribution of the coupling to the variance of the engine model response is: Among them, D u,ij represents a zero-mean random variable ε i and ε j Direct contribution of coupling to the variance of engine model response; represents a zero-mean random variable ε j The expectation of the fourth power and the zero-mean random variable ε j The ratio of the fourth power of the standard deviation of iiii Same, just different subscript; w u,i =u(b i )-u(b -i );w u,j =u(b j )-u(b -j );o u,ij , p u,ij ,q u,ij are all intermediate variables; the random variable vector sample input set 2 is brought into the engine model input-output relationship obtained in the first step, and the zero-mean random variable ε is calculated. i and ε j The absolute contribution of the coupling to the variance of the engine model response; Step 2.4, the formula for calculating the variance of the engine model response is:

5. The engine sensitivity analysis method based on random perturbation collocation method according to claim 4 is characterized in that: The intermediate variable o u,ij , p u,ij ,q u,ij The calculation method is as follows: Among them, i=1,2,…,q; j=i+1,i+2,…,q.

6. The engine sensitivity analysis method based on random perturbation collocation method according to claim 4 is characterized in that: The third step comprises the following steps: Step 3.1, based on the calculation results of steps 2.2 and 2.4, calculate S i (u) reflects the zero-mean random variable ε i Relative contribution to the variance D(u); due to the zero-mean random variable ε i With the original random variable X i One-to-one correspondence, S i (u) is also X i Corresponding sensitivity index; Step 3.2, calculate S based on the calculation results of steps 2.3 and 2.4 ij (u), where S i (u) reflects the random variable ε i and ε j The relative contribution of the coupling effect to the variance D(u); due to the zero-mean random variable ε i ,ε j With the original random variable X i ,X j One-to-one correspondence, S ij (u) is also X i and X j The sensitivity index corresponding to the coupling effect.

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