A smoothing factor order adjustment method for multi-order nonlinear weighted methods

Abstract

The invention discloses a smooth factor order adjustment method suitable for a multi-order nonlinear weighting method, and relates to the field of high-order format nonlinear weighting, comprising: selecting a non-equidistant template set based on precision K; respectively calculating and obtaining the smooth factors of three sub-templates in the non-equidistant template set; determining a smooth factor order adjuster on the longest sub-template; calculating and obtaining a corrected smooth factor on the longest sub-template according to the smooth factor order adjuster and the smooth factor; determining a smooth factor order adjuster on the shortest sub-template; calculating and obtaining a corrected smooth factor on the shortest sub-template according to the smooth factor order adjuster and the smooth factor; obtaining the final smooth factors of the three sub-templates; the invention can ensure the precision of the nonlinear weighted high-order format, and the nonlinear weight of the corrected smooth factor calculation is independent of the dimension of the control equation.

Description

Technical Field

[0001] The invention relates to the field of high-order format nonlinear weighting, and in particular to a smoothing factor order adjustment method suitable for a multi-order nonlinear weighting method. Background Art

[0002] Computational fluid dynamics (CFD) is one of the important means to study flow physics problems. As the requirements for numerical simulation accuracy in engineering applications become higher and higher, people are increasingly favoring the use of high-order numerical simulation methods with low dissipation, low dispersion and high resolution.

[0003] The core of high-order precision numerical simulation methods is high-order precision formats. After decades of continuous development and improvement, many high-order precision formats have been formed, including high-order precision finite differences, high-order precision finite volumes, and high-order precision finite elements. Among them, the high-order precision finite difference format based on structured grids is the most widely used due to its relatively small computational cost. High-order precision finite difference formats can be divided into high-order precision linear formats and high-order precision nonlinear formats. When the simulated flow problem contains discontinuous structures such as shock waves, high-order precision nonlinear formats need to be used.

[0004] The most commonly used high-order precision nonlinear format design method is to adopt a weighted combination design method: the overall template of high-order precision is divided into multiple candidate templates of lower-order precision, and the candidate templates of lower-order precision are combined to have high-order precision. Figure 2 A schematic diagram of the sub-template set of a 5-point template is given. The candidate template set for the fifth-order precision classical nonlinear interpolation on the 5-point template consists of three 3-point templates (sub-template 5, sub-template 6 and sub-template 7). A dynamic weight (nonlinear weight) is assigned to each candidate template according to its smoothness (represented by the smoothness factor). The smoother a candidate template is, the greater the weight assigned to it, and vice versa. If the overall template does not contain discontinuity structures such as shock waves, then each candidate template is smooth and the weight assigned to it is close to the ideal weight, and the result of the weighted combination is close to the high-order precision linear format. If the overall template contains discontinuity structures such as shock waves, then the candidate template containing the discontinuity is assigned a smaller weight, and other candidate templates that do not contain the discontinuity are assigned a larger weight, thereby avoiding interpolation across discontinuities. In this way, the high-order precision format of the weighted combination can capture discontinuity structures such as shock waves with basically no oscillation.

[0005] In summary, the core of the nonlinear weighted method includes the interpolation candidate template set, the smoothness factor and its functional relationship with the nonlinear weight. The research on nonlinear weighted formats has been going on for nearly 30 years. Among them, the research on the improvement of nonlinear weighted functions has proposed many types of nonlinear weighted functions with the goal of achieving optimal accuracy in the smooth area without non-physical fluctuations near the discontinuity.

[0006] In recent years, some researchers have adjusted the candidate template set, abandoning the classic optimal convex combination of equal-width templates and adopting non-equal-width candidate template sets, such as selecting a 5-point template (sub-template 0), a 3-point template (sub-template 6) and a 1-point template (sub-template 11). In order to ensure that nonlinear weighting can be performed on such a non-equidistant candidate template set (5-3-1) to obtain the expected format accuracy, a high-order smoothing factor is introduced. τ zq . For the specific implementation process, please refer to the following literature: Zhenming Wang, Jun Zhu, Linlin Tian, ​​Yuchen Yang, Ning Zhao. An efficient fifth-order finite difference multi-resolution WENO scheme for inviscid and viscous flow problems[J]. Computers&Fluids, 230(2021),105138.

[0007] In the above method, when a dimensional control equation is used for calculation, since the dimension of the high-order smoothing factor is inconsistent with the dimension of the sub-template smoothing factor, the nonlinear weight will be a value that changes with the dimension, and the final normalized nonlinear weight will change with the dimension, which may lead to incorrect calculation results. Summary of the invention

[0008] The present invention aims to solve the problem that the calculation result may be inaccurate due to the selection of the smoothing factor calculation method in the nonlinear weight function calculation using non-equidistant templates in high-order format construction.

[0009] To achieve the above-mentioned object of the invention, the present invention provides a smoothing factor order adjustment method applicable to a multi-order nonlinear weighting method, the method comprising:

[0010] Step 1: Select a non-isotropic template set based on the precision K S T , the non-equidistant template set includes 3 sub-templates, namely: the longest sub-template S long , Medium length sub-template S middle and the shortest sub-template S short ;

[0011] Step 2: Calculate and obtain the longest sub-template S long , Medium length sub-template Smiddle and the shortest sub-template S short The smoothness factor β long , β middle and β short ;

[0012] Step 3: Determine the longest sub-template S long Smooth factor order regulator on T long ;

[0013] Step 4: Scale the smoothing factor T long and smoothness factor β long , calculate the corrected longest sub-template S long Upper smoothing factor ;

[0014] Step 5: Determine the shortest sub-template S short Smooth factor order regulator on T short ;

[0015] Step 6: Scale the smoothing factor T short and smoothness factor β short , calculate the corrected shortest sub-template S short Upper smoothing factor ;

[0016] Step 7: The final smoothing factors of the three sub-templates are: , smoothness factor β middle and smoothness factor .

[0017] Among them, the present invention proposes a method for correcting the smoothing factor in a multi-order nonlinear weighted high-order format to obtain the desired smoothing factor order, so that the smoothing factor order relationship on templates of different widths satisfies the sufficient conditions for the optimal accuracy of nonlinear interpolation, and finally ensures the accuracy of the nonlinear weighted high-order format. The nonlinear weight calculated by the corrected smoothing factor proposed by the method is independent of the dimension of the control equation.

[0018] The present invention can obtain the desired smooth factor order, so that the smooth factor order relationship on templates of different widths satisfies the sufficient conditions for the optimal precision of nonlinear interpolation, and finally ensures the precision of nonlinear weighted high-order format.

[0019] Among them, the longest sub-template S long , Medium length sub-template S middle and the shortest sub-template S short The smoothness factor β long , β middle and β short The calculation method is:

[0020] ;

[0021] Among them, k=1, 2 and 3 correspond to β long , β middle and β short ;Longest sub-template S long , Medium length sub-template S middle and the shortest sub-template S short The number of grid points included are: r 1. r 2 and r 3. x j is the coordinate value of node j; h is the grid scale; p k ( x ) is the n Optimal precision interpolation polynomial of sub-template; The summation flag, ranging from 1 to r k ; x is the spatial coordinate; Express p k ( x ) Find the independent variable x of Derivative of order.

[0022] Among them, the smooth factor order regulator T long The calculation method is:

[0023] ;

[0024] in, T 2 to T m The grid scale h Some small quantities of order, L 2 to L m is the power exponent determined according to the order of magnitude, a 2. a 3. ... and a m The grid size is h The square, cubic, ... and m The coefficient of the power term, b 2. b 3. ... and b m The grid size is h The square, cubic, ... and m The coefficient of the power term, A and B is the smoothing factor of two sub-templates of the same width.

[0025] in, A and B The calculation method is:

[0026] ;

[0027] in, a 2. a 3. ... and a n The grid size is h The square, cubic, ... and n The coefficient of the power term, b 2. b 3. ... and b n The grid size is h The square, cubic, ... and n The coefficient of the power term, l The flag for summation, ranging from 2 to n.

[0028] Among them, the smoothness factor The calculation method is:

[0029] .

[0030] Among them, the smooth factor order regulator T short The calculation method is:

[0031] ;

[0032] in, C and D are the smoothing factors of the two two-point templates, L s is the quantity order, for The magnitude of h is the grid scale.

[0033] in, C and D The calculation method is:

[0034] ;

[0035] in, u is the physical quantity in the flow field, u j For Node j The corresponding physical quantity is u j+1 For Node j +1 corresponds to the physical quantity, u j-1 For Node j -1 corresponds to the physical quantity, h is the grid scale, x is the space coordinate.

[0036] Among them, the smoothness factor The calculation method is:

[0037] .

[0038] One or more technical solutions provided by the present invention have at least the following technical effects or advantages:

[0039] The smoothing factor obtained by the present invention can ensure that the nonlinear weighted interpolation reaches the designed target accuracy. At the same time, since the order regulators are all dimensionless quantities, the nonlinear weights calculated by using the smoothing factors adjusted by them are independent of the dimensions of the control equation, avoiding possible errors and ultimately ensuring the accuracy of the nonlinear weighted high-order format. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] The drawings described herein are used to provide a further understanding of the embodiments of the present invention, constitute a part of the present invention, and do not constitute a limitation on the embodiments of the present invention;

[0041] Figure 1 It is a flow chart of a smoothing factor order adjustment method applicable to a multi-order nonlinear weighting method;

[0042] Figure 2It is a schematic diagram of a sub-template set of a 5-point template;

[0043] Figure 3 Schematic diagram comparing the impact of the dimension of physical quantity on the weight function. DETAILED DESCRIPTION

[0044] In order to more clearly understand the above-mentioned purpose, features and advantages of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be noted that the embodiments of the present invention and the features in the embodiments can be combined with each other without conflict.

[0045] In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention may also be implemented in other ways different from those within the scope of this description. Therefore, the protection scope of the present invention is not limited to the specific embodiments disclosed below.

[0046] Embodiment 1;

[0047] Please refer to Figure 1 , Figure 1 The present invention is a flowchart of a smoothing factor order adjustment method applicable to a multi-order nonlinear weighting method. Embodiment 1 of the present invention provides a smoothing factor order adjustment method applicable to a multi-order nonlinear weighting method, including:

[0048] Step 1: Select a non-isotropic template set based on the desired accuracy of the format (denoted as K) S T The template set generally contains 3 sub-templates, denoted as S long , S middle and S short , respectively represent the longest sub-template, the medium-length sub-template, and the shortest sub-template. The number of grid points contained in the three sub-templates is recorded as r 1. r 2 and r 3.

[0049] Step 2: Calculate the three sub-templates according to the conventional smooth factor calculation method S long , S middle and S short Smooth factor on β long , β middle and β short . Smoothness factor β k(k=1, 2, 3 correspond to β long , β middle and β short ) is expressed as:

[0050] ;

[0051] in, x j is the coordinate value of node j, h is the grid scale, r k The number of grid points contained in the sub-template, p k ( x ) is the k The optimal precision interpolation polynomial of the sub-template, The summation flag, ranging from 1 to r k , x is the spatial coordinate, d α Express request α Derivatives, such as Represents the function p k ( x ) Find the independent variable x Derivative of order.

[0052] The conventional smoothing factor calculation method is referred to the following reference: Jiang, GS and Shu, CW (1996) Efficient Implementation of Weighted ENO Schemes. Journal of Computational Physics, 126, 202-228.

[0053] Step 3: Determine the longest sub-template S long Smooth factor order regulator on T long The relevant concepts, background knowledge and specific methods are as follows:

[0054] Nonlinear weighted interpolation formula:

[0055] ;

[0056] in, x j+1 / 2 Half Node j The coordinates at +1 / 2, u j Representation Nodej The physical quantity at u The true value of u It can be flow field variables such as density and velocity. The physical quantities at the nodes are known quantities, and the physical quantities at the half nodes are is the unknown quantity we hope to obtain by interpolating the known quantity. ω n is the nonlinear weight of the nth sub-template, p n ( x j+1 / 2 ) is the n Sub-template in half node j Best-efficiency interpolation polynomial at +1 / 2.

[0057] The expression of nonlinear weighted interpolation truncation error is:

[0058] ;

[0059] in, is the truncation error of the n-th sub-template interpolation polynomial.

[0060] According to the target accuracy (K) in step 1, there is the following order relationship:

[0061] ;

[0062] in, L k for ω k The magnitude order of r k is the order of the truncation error (the same as the number of grid points contained in the sub-template), β k1 and β k2 are the smoothing factors on sub-template k1 and sub-template k2 respectively, L k2 and L k1 They are ω k2 and ω k1 The above formula shows that the sum of the order of the nonlinear weight function and the order of the interpolation polynomial truncation error is greater than the target accuracy. The order of the ratio of the smoothing factors of different sub-templates is equal to the order of the reciprocal of the ratio of the corresponding nonlinear weight functions. O() represents the order of the physical quantity and is the abbreviation of Order.

[0063] According to the above nonlinear interpolation accuracy requirements, in the smooth area, the smoothing factor on the sub-template in the multi-order nonlinear weighted method is the same as the grid scale. hThe order of the related small quantities and the number of grid points they contain must satisfy:

[0064] ;

[0065] in, m 1 and m 2 is the order of the smoothing factor on sub-template k1 and sub-template k2, r1 and r2 are the order of the truncation error on sub-template k1 and sub-template k2. The above formula is a sufficient condition for nonlinear weighted interpolation to meet the design target accuracy.

[0066] According to the three sub-templates given by the conventional method S long , S middle and S short The smoothing factors are β long , β middle and β short In the smooth region without extreme value points, the above three smooth factors are of the same magnitude. O ( h 2 ), regardless of the template width, there must be m 1- m 2=0, thus it cannot meet the above requirements of the multi-order nonlinear weighting method on the order of the smoothing factor on the sub-template.

[0067] The design idea of ​​the smooth factor regulator is as follows:

[0068] set up A and B is the smoothing factor of two sub-templates of the same width, which can be expressed in the smooth area as:

[0069] ;

[0070] Among them, a2, a3, ...a n The grid size is h The coefficients of the square, cubic, ...nth power terms, b2, b3, ...b n The grid size is h The coefficients of the square, cube, ...nth power terms of Indicates that the subscripts from 2 to l The items are summed.

[0071] Define the function:

[0072] ;

[0073] Among them, a m 、b m They are A and B The grid size in the expression h The m-th power term ( h m ), a n = b n =0, n =2,…,m-1, a m ≠0, b m ≠0. Then in the smooth region, T m = O ( h L ), L is a positive integer. It can be seen that T m The grid scale h The L-order small quantity can be used T m The order of the smoothing factor on the sub-template is adjusted so that the adjusted smoothing factor satisfies the above sufficient conditions.

[0074] In the specific implementation, the present invention designs the following smooth factor order regulator: T :

[0075] ;

[0076] Among them, the power index L 2 to L m It is determined according to the quantity order requirement (expressed by the above sufficient condition). For specific methods, please refer to the specific implementation examples.

[0077] Step 4: According to the smooth factor adjuster on the longest sub-template determined in step 3 and the conventional method on the longest sub-template obtained in step 2, the smooth factor on the longest sub-template after adjustment (correction) is obtained. The details are as follows:

[0078] ;

[0079] Step 5: Determine the shortest sub-template S short Smooth factor order regulator on T short Usually the shortest sub-template contains one dot, mainly to ensure that at least one template does not contain a discontinuity when multiple discontinuities occur.

[0080] For the shortest one-point template, its smoothing factor regulator is based on the smoothing factors of the two two-point templates. C and D To build, C and D The expression is:

[0081] ;

[0082] in, u It represents the physical quantity in the flow field, which can be flow field variables such as density and velocity.

[0083] It is easy to prove that , which is independent of the order of the extreme point, so the order regulator degenerates into:

[0084] ;

[0085] Among them, the power index L s It is determined according to the quantity order requirement (expressed by the above sufficient condition). For specific methods, please refer to the specific implementation examples.

[0086] Step 6: According to the smooth factor adjuster on the shortest template determined in step 5 and the conventional method on the shortest template obtained in step 2, the smooth factor on the shortest sub-template after adjustment (correction) is obtained. The details are as follows:

[0087] ;

[0088] At this point, the final smoothing factors on the three sub-templates are obtained , β middle , , the expressions of these three smoothing factors can ensure that the nonlinear weighted interpolation achieves the designed target accuracy. At the same time, due to the order regulator T long , T short They are all dimensionless quantities, so the nonlinear weights calculated using the smoothing factors adjusted by them are independent of the dimensions of the control equations, avoiding possible errors.

[0089] Among them, after obtaining the final smooth factor, the method can perform corresponding numerical simulations, such as verifying the aerodynamic performance of an aircraft or calculating the flow field. Based on the smooth factor in the present invention, a three-dimensional flow field can be constructed, and then the aerodynamic performance of the aircraft can be verified based on the three-dimensional flow field and the aircraft to be tested. The specific verification method and method can refer to CN118683749A, and the present invention will not be repeated accordingly. The flow field calculation method is to obtain a template based on the smooth factor, and perform flow field calculation based on the template. The specific calculation method can be adjusted according to the aircraft parameters and process parameters, and the embodiment of the present invention does not make specific limitations.

[0090] Please refer to Figure 2-Figure 3 , the following 5-point template is given ( j -1, j , j +1, j +2, j +3) A specific implementation example of fifth-order nonlinear interpolation on the 5-3-1 sub-template.

[0091] Select three sub-templates, namely sub-template 1, sub-template 7, and sub-template 11, corresponding to S long , S middle and S short .

[0092] Calculated according to the conventional method S long Smoothness factor on β long The details are as follows:

[0093] S long The two three-point templates included ( j -1, j , j +1) (corresponding to Figure 2 Neutron template 6) and ( j +1, j +2, j +3) (corresponding Figure 2 The smoothing factors on the neutron template 8) are:

[0094] ;

[0095] Will u j-1 , u j+1 , u j+2 and u j+3 Respectively for uj Performing Taylor expansion, we can get:

[0096] ;

[0097] in,( f 1 )、( f 2 )、( f 3 )、( f 4 )、( f 5 )、( f 6 )、( f 7 )and( f 8 ) are j 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, and 8th derivatives of a quantity at a point.

[0098] S long Smooth factor on β long for:

[0099] ;

[0100] Calculated according to the conventional method S middle Smooth factor on β middle The details are as follows:

[0101] ;

[0102] Will u j+1 and u j+2 For u j Performing Taylor expansion, we can get:

[0103] ;

[0104] Calculated according to the conventional method S short Smooth factor on β short The details are as follows:

[0105] ;

[0106] Will u j-1 and uj+1 Respectively for u j Performing Taylor expansion, we can get:

[0107] ;

[0108] ;

[0109] Calculate 5-point template S long The order regulator is used to adjust the order relationship between the 5-point template smoothing factor and the 3-point template smoothing factor.

[0110] Taking the case where the extreme value point does not exceed the 4th order as an example, the method for determining the coefficients in the order regulator is explained.

[0111] ;

[0112] ;

[0113] So we have:

[0114] ;

[0115] At the third-order extreme point, ( f 1 )=0, ( f 2 )=0, ( f 3 )=0, ( f 4 )≠0;

[0116] ;

[0117] at this time, r 1=5, r 2=3, m 1=8, m 2=8, r 1- r 2=2, m 1- m 2=0, not satisfied r 1- r 2< m 1- m 2 conditions need to be adjusted m 1, so that m 1>10;

[0118] ;

[0119] So the smoothing factor after adjustment , whose magnitude is , L 8The minimum value is 3 to satisfy m 1=11>10;

[0120] So far, confirm L 8=3;

[0121] At the second-order extreme point, ( f 1 )=0, ( f 2 )=0, ( f 3 )≠0;

[0122] ;

[0123] at this time, r 1=5, r 2=3, m 1=6, m 2=6, r 1- r 2=2, m 1- m 2=0, not satisfied r 1- r 2< m 1- m 2 conditions need to be adjusted m 1, so that m 1>8;

[0124] ;

[0125] So the smoothing factor after adjustment , whose magnitude is , L 6The minimum value is 3 to satisfy m 1=9>8;

[0126] So far, confirm L 6=3;

[0127] When the first-order extreme point is f 1 )=0, ( f 2 )≠0;

[0128] ;

[0129] at this time, r 1=5, r 2=3, m 1=5, m 2=4, r 1- r2=2, m 1- m 2=1, not satisfied r 1- r 2< m 1- m 2 conditions need to be adjusted m 1, so that m 1>6;

[0130] ;

[0131] So the smoothing factor after adjustment , whose magnitude is , L 2The minimum value is 1. L 4When the minimum value is 1, it can satisfy m 1=7>6;

[0132] So far, confirm L 4=1;

[0133] When the first-order extreme point is f 1 )≠0;

[0134] ;

[0135] at this time, r 1=5, r 2=3, m 1=4, m 2=2, r 1- r 2=2, m 1- m 2=2, not satisfied r 1- r 2< m 1- m 2 conditions need to be adjusted m 1, so that m 1>4;

[0136] ;

[0137] So the smoothing factor after adjustment , whose magnitude is , L When 2 takes the minimum value of 1, it can satisfy m 1=8>4;

[0138] So far, confirm L 2=1;

[0139] Therefore, without exceeding the 4th order extreme point, the order adjuster:

[0140] ;

[0141] 5 Ideas Template S long The smoothing factor after the upper adjustment is; ;

[0142] Calculate 1 point template S short The order adjuster is used to adjust the order relationship between the one-point template and the three-point template;

[0143] The smoothing factor adjuster is constructed based on the following two-point template smoothing factors:

[0144] ;

[0145] It is easy to prove that , which is independent of the order of the extreme points. Smooth factor regulator ;

[0146] At the third-order extreme point, ( f 1 )=0, ( f 2 )=0, ( f 3 )=0, ( f 4 )≠0;

[0147] ;

[0148] at this time, r 1=3, r 2=1, m 1=8, m 2=8, r 1- r 2=2, m 1- m 2=0, not satisfied r 1- r 2< m 1- m 2 conditions need to be adjusted m 2. Make m 2<6;

[0149] Adjusted smoothing factor , whose magnitude is , L s When the value is -3, it can satisfy m 2=5<6;

[0150] A similar method can be used to determine the second-order extreme point, the first-order extreme point, and the zero-order extreme point. L s A value of -3 satisfies the sufficient condition.

[0151] Therefore, the step regulator ;

[0152] A little template S short The smoothing factor after adjustment is ;

[0153] Obviously, for the 5-3-1 candidate template set, in the absence of discontinuity, the smoothness factor of the fixed 3-point template is β middle , two-step regulator ( T long and T short ) can be used together to ensure 1 point template S short With 3-point template S middle and 5-point template S long The order relationship of the upper smoothing factor meets the requirements of the optimal accuracy of the format. Therefore, for the 5-3-1 template set, two order adjusters are used to adjust the smoothing factor between templates of different widths to achieve the requirements of optimal accuracy of nonlinear interpolation.

[0154] The smoothing factor order adjuster is a dimensionless quantity of a combination of smoothing factors. Its existence is to meet the requirements of the multi-order nonlinear interpolation method on the order of the smoothing factor, with the purpose of achieving the optimal accuracy of nonlinear weighted interpolation.

[0155] Figure 3 A comparison of the influence of the dimension of the physical quantity on the weight function is given. The weight function calculated by the method proposed in the present invention is independent of the grid dimension, which can avoid the occurrence of incorrect flow field calculation results that may be caused by different calculation results of the weight function under different dimensions. Figure 3 The literature method used in the paper is: Zhun J. and snu,cw (2018), A newtype of multi-resolution WENO schemes with increasingly higher order of accuracy Journal of Computational Physics, 375, 659-683. Figure 3 middle ω 1 and ω 2 represents the linear weight of two of the sub-templates.

[0156] In the solution of the nonlinear weighted function of the high-order format, the present invention uses an order adjuster to adjust the order of the smooth factor obtained by the conventional method, so that the order of the smooth factor on different sub-templates after adjustment meets the sufficient condition of the optimal accuracy of nonlinear interpolation, and finally ensures the accuracy of the nonlinear weighted high-order format. At the same time, since the order adjusters are all dimensionless quantities, the nonlinear weights calculated by using the smooth factors adjusted by them are independent of the dimension of the control equation, avoiding possible errors.

[0157] Although the preferred embodiments of the present invention have been described, those skilled in the art may make other changes and modifications to these embodiments once they have learned the basic creative concept. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments and all changes and modifications that fall within the scope of the present invention.

[0158] Obviously, those skilled in the art can make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if these modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include these modifications and variations.

Claims

1. A smoothing factor order adjustment method applicable to a multi-order nonlinear weighting method, characterized in that: The method comprises: Step 1: Select a non-isotropic template set S based on the accuracy K T , the non-equidistant template set includes 3 sub-templates, namely: the longest sub-template S long , medium length sub-template S middle and the shortest sub-template S short ; Step 2: Calculate and obtain the longest sub-template S long , medium length sub-template S middle and the shortest sub-template S short The smoothing factor β long , β middle and β short ; Step 3: Determine the longest sub-template S long Smooth factor order regulator T long ; Step 4: Adjust the smoothing factor T long and smoothing factor β long , calculate the modified longest sub-template S long Upper smoothing factor Step 5: Determine the shortest sub-template S short Smooth factor order regulator T short ; Step 6: Adjust the smoothing factor T short and smoothing factor β short , calculate the corrected shortest sub-template S short Upper smoothing factor Step 7: The final smoothing factors of the three sub-templates are: Smoothing factor β middle and smoothness factor A three-dimensional flow field is constructed based on the final smoothing factors of the three sub-templates, and the aerodynamic performance of the aircraft is verified based on the constructed three-dimensional flow field and the aircraft to be tested; Smoothing factor order regulator T long The calculation method is: Among them, T2 to T m is a small quantity of several orders of magnitude of the grid size h, L2 to L m are the power exponents determined according to the order of magnitude, a2, a3, ... and a m are the coefficients of the square, cubic, ... and mth powers of the grid size h, b2, b3, ... and b m are the coefficients of the square, cubic, ... and mth power of the grid size h, respectively. A and B are the smoothing factors of two sub-templates with the same width. A and B are calculated as: Among them, a2, a3, ... and a n are the coefficients of the square, cubic, ... and nth powers of the grid size h, b2, b3, ... and b n are the coefficients of the square, cubic, ... and nth power of the grid size h, respectively. l is the symbol for summation, with values ​​ranging from 2 to n. Smooth Factor The calculation method is: Smoothing factor order regulator T short The calculation method is: Among them, C and D are the smoothing factors of the two-point templates, L s is the quantity order, for The order of , h is the grid scale; The calculation method of C and D is: Among them, u is the physical quantity in the flow field, u j is the physical quantity corresponding to node j, u j+1 is the physical quantity corresponding to node j+1, u j-1 is the physical quantity corresponding to node j-1, h is the grid scale, and x is the spatial coordinate; Smooth Factor The calculation method is:

2. The smoothing factor order adjustment method applicable to the multi-order nonlinear weighting method according to claim 1 is characterized in that: The longest sub-template S long , medium length sub-template S middle and the shortest sub-template S short The smoothing factor β long , β middle and β short The calculation method is: Among them, k = 1, 2 and 3 correspond to β long , β middle and β short ; The longest sub-template S long , medium length sub-template S middle and the shortest sub-template S short The number of grid points included are: r1, r2 and r3, x j is the coordinate value at node j; h is the grid scale; p k (x) is the optimal precision interpolation polynomial of the k-th sub-template; α is the summation mark, ranging from 1 to r k ; x is the spatial coordinate; Indicates that p k (x) Find the α-order derivative of the independent variable x.

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