A multi-order sub-template nonlinear weighted interpolation method with adaptive smooth scale separation and sixth-order accuracy

Abstract

The present invention discloses a multi-order sub-template nonlinear weighted interpolation method for adaptive smooth scale separation of sixth-order precision, which relates to the field of nonlinear weighting of high-order precision finite difference format, including: obtaining four multi-order sub-templates, the four multi-order sub-templates including three third-order precision sub-templates and one sixth-order precision sub-template; respectively calculating and obtaining smooth metric factors of the four multi-order sub-templates; constructing and obtaining a smooth scale separator based on the smooth metric factors; obtaining the first-order derivative, second-order derivative, third-order derivative and fourth-order derivative of the physical quantity at the half-node based on discrete calculation of six template points; and obtaining the first-order derivative, second-order derivative, third-order derivative and fourth-order derivative of the physical quantity at the half-node based on the first-order derivative of the physical quantity at the half-node. An adaptive scale identifier is constructed by using the first-order derivative, second-order derivative, third-order derivative and fourth-order derivative; the adaptive scale identifier is used as the judgment condition of the smooth scale separator to calculate the weighted coefficients of four multi-order sub-templates respectively; based on the weighted coefficients of the four multi-order sub-templates, the four multi-order sub-templates are weightedly calculated to obtain the physical quantity values ​​of the solution interface; this method can solve the problem of failing to achieve high-order precision interpolation calculation in the smooth area when using multi-order sub-templates in nonlinear weighted interpolation in high-order format construction, resulting in inaccurate calculation results, thereby ensuring the accuracy of turbulent numerical simulation.

Description

Technical Field

[0001] The present invention relates to the field of nonlinear weighting of high-order precision finite difference formats, and in particular to a multi-order sub-template nonlinear weighted interpolation method with sixth-order precision adaptive smooth scale separation. Background Art

[0002] With the rapid development of computational theory and computers, computational fluid dynamics (CFD) has grown rapidly as an emerging discipline and has begun to play an important role in engineering turbulence problems. By discretizing the flow field in time and space and converting the nonlinear NS equations into large linear equations, CFD can solve and analyze complex flows under various complex boundary conditions. It is widely used in various industries such as aerospace, energy and power, saving a lot of manpower, material resources and time costs.

[0003] Mathematically, for a k-order precision numerical format, its numerical error e is proportional to the k-th power of the discrete grid size h, that is, e~h k Therefore, under the same numerical error, improving the scheme's accuracy k can significantly increase the discrete grid size h, reducing the overall computational effort. In complex multiscale turbulence numerical simulations of compressible turbulence, it is necessary to simultaneously consider the scheme's ability to stably capture shock wave discontinuities and the high resolution of multiscale turbulent structures. Improving the scheme's accuracy can effectively reduce the scheme's numerical dissipation and increase its ability to resolve turbulence. However, the stable capture of shock waves requires sufficient numerical dissipation error to suppress non-physical numerical oscillations. This contradiction makes the development of high-order precision numerical methods extremely challenging.

[0004] Among the high-order precision finite difference methods, the nonlinear shock wave capture scheme is one of the most successful theoretical methods. The multiple template points involved in the high-order precision interpolation calculation of the variables are divided into nonlinear weighted sub-templates of different low-order precision. Figure 2 A schematic diagram of the sub-template set of a 6-point template is given. The candidate template set for sixth-order precision classical nonlinear interpolation on the 6-point template consists of three 3-point templates (sub-template K1, sub-template K2, and sub-template K3) and one 6-point template (sub-template K4). The smoothness factor of the sub-template is constructed to detect discontinuities caused by phenomena such as shock waves in the flow field, and the weighting coefficient of the sub-template is calculated based on the detected smoothness factor. If the sub-template does not contain discontinuities such as shock waves, the weighting coefficient of each sub-template is close to the ideal weight, and the result of the weighted combination calculation is close to the high-order precision linear format. Conversely, when the sub-template contains discontinuities such as shock waves, the corresponding sub-template weighting coefficient is assigned a smaller weight, and the other sub-templates that do not contain discontinuities still use the ideal weight. By designing this nonlinear weighting method, the cross-discontinuity template is avoided from participating in the high-order precision interpolation calculation, and a shock wave capture capability with almost no oscillation can be achieved.

[0005] In summary, with the goal of achieving optimal accuracy in the smooth region without unphysical oscillations near the discontinuity, a better nonlinear weighted interpolation method can be proposed from three perspectives: interpolation candidate sub-templates, smoothness measurement factors, and their relationship with nonlinear weighting methods.

[0006] In order to reduce the numerical dissipation of high-precision numerical formats and improve the resolution of turbulence, scholars proposed a multi-order sub-template nonlinear weighted method. The specific implementation process is referred to the following literature:

[0007] Yankai Ma, Meiliang Mao, Zhen-Guo Yan, et al.Amulti-order candidatesweighting framework for discontinuity capturing of hyperbolic conservationlaws[J].Journal of Computational Physics,2024,510,113064.

[0008] Although the above method ensures the match between the smoothness factor and the sub-template order, and improves the resolution of the numerical format at discontinuities, the nonlinear weighted interpolation is designed for calculations in the entire flow field area and cannot guarantee high-order precision interpolation calculations in the smooth area, which in turn affects the accuracy of turbulent numerical simulations. Summary of the Invention

[0009] The purpose of the present invention is to solve the problem that high-order precision interpolation calculation cannot be achieved in the smooth area when using multi-order sub-templates in nonlinear weighted interpolation in high-order format construction, resulting in inaccurate calculation results, which in turn affects the accuracy of turbulence numerical simulation.

[0010] To achieve the above-mentioned object, the present invention provides a multi-order sub-template nonlinear weighted interpolation method with sixth-order precision adaptive smooth scale separation, the method comprising:

[0011] Step 1: Divide the six adjacent template points into four multi-order sub-templates, which are three third-order precision sub-templates K1, K2 and K3 and one sixth-order precision sub-template K4;

[0012] Step 2: Calculate the smoothness metrics β1, β2, β3 and β4 of the four multi-order sub-templates respectively;

[0013] Step 3: construct a smooth scale separator based on smooth metric factors β1, β2, β3 and β4;

[0014] Step 4: Obtain the first-order derivative, second-order derivative, third-order derivative, and fourth-order derivative of the physical quantity at the half-node based on the discrete calculation of the six template points;

[0015] Step 5: construct an adaptive scale identifier based on the first-order derivative, second-order derivative, third-order derivative and fourth-order derivative of the physical quantity at the half-node;

[0016] Step 6: Using the adaptive scale identifier as the decision condition of the smooth scale separator, respectively calculate and obtain the weighted coefficients of the four multi-order sub-templates;

[0017] Step 7: Based on the weighting coefficients of the four multi-order sub-templates, the four multi-order sub-templates are weighted and calculated to obtain the physical value of the solution interface;

[0018] Step 8: Perform turbulence numerical simulation based on the physical values ​​of the interface.

[0019] Among them, the present invention proposes a multi-order sub-template nonlinear weighted interpolation method with sixth-order precision adaptive smooth scale separation. When calculating the physical value of the interface, the template composed of six solution points near the interface is split into weighted interpolation of four multi-order sub-templates. Through adaptive smooth scale separation, it is determined whether the four multi-order sub-templates are smooth or discontinuous, and then adaptive weighted interpolation is realized, avoiding the discontinuous sub-template from participating in the weighted interpolation calculation, thereby achieving high-order precision in the smooth area and oscillation-free capture of the discontinuity.

[0020] The present invention enables the multi-order sub-template weighted calculation of the smoothing factor order relationship on templates of different widths to meet the optimal accuracy of nonlinear interpolation while ensuring that the smooth area of ​​the flow field achieves the expected high-order accuracy weighted interpolation and discontinuous oscillation-free capture, ultimately ensuring the accuracy of the nonlinear weighted high-order format.

[0021] Specifically, the three third-order precision sub-templates K1, K2 and K3 correspond to three template points approximated as and The six template points corresponding to the sixth-order precision sub-template K4 are approximately The calculation method is:

[0022]

[0023] Where U is the physical quantity in the flow field, subscripts j+n and jn represent the solution points j+n and jn respectively, U j+n and U j-n Represent the physical quantities at the solution points j+n and jn, n is an integer, U j+12 Represents the physical quantity at half node j+1 / 2, O(h k ) represents the first small quantity greater than order k, h is the grid scale, and j is the number of the computing node.

[0024] Specifically, the smoothness metrics β1, β2, β3, and β4 of the four multi-order sub-templates are calculated as follows:

[0025]

[0026] Where U is the physical quantity in the flow field, subscripts j+n and jn represent the solution points j+n and jn respectively, U j+n and U j-n They represent the physical quantities at the solution points j+n and jn respectively, n is an integer, and j is the number of the calculation node.

[0027] Specifically, the smooth scale separator is constructed based on the smooth metric factors β1, β2, β3 and β4 as follows:

[0028]

[0029] Among them, δ k is the smooth scale separator corresponding to the four multi-order sub-templates, k = 1, 2, 3, 4, h is the grid scale, χ k is the intermediate process variable, C T is the adaptive scale identifier, γ k is the smoothness index, ε is the second smallest quantity, ε=1.0×10 -40 .

[0030] Among them, the tanh function is the hyperbolic tangent function, which is to calculate the value of the hyperbolic tangent function according to the parameter χ. k and C T The relative relationship between the values ​​is used to achieve a fast and continuous switching of the scale separator between 0 and 1. The size of the scale separator can be calculated based on the functional relationship between the calculated smoothness metric factor and the adaptive scale identifier. For the multi-order sub-template in the smooth area, the smoothness metric factor is small, and the corresponding scale separator δ k The value is 1, and when the sub-template is discontinuous, the smoothness factor is large, and the corresponding scale separator δ k Close to 0.

[0031] Specifically, the calculation method for obtaining the first-order derivative, second-order derivative, third-order derivative, and fourth-order derivative of the physical quantity at the half-node based on the discrete calculation of six template points is as follows:

[0032]

[0033] Among them, ΔU 1,j+1 / 2 is the first-order derivative of the physical quantity at the half node, ΔU 2,j+1 / 2 is the second-order derivative of the physical quantity at the half node, ΔU 3,j+1 / 2 is the third-order derivative of the physical quantity at the half node, ΔU 4,j+1 / 2is the fourth-order derivative of the physical quantity at the half node, U is the physical quantity in the flow field, the subscripts j+n and jn represent the solution points j+n and jn respectively, U j+n and U j-n They represent the physical quantities at the solution points j+n and jn respectively, n is an integer, and j is the number of the calculation node.

[0034] Specifically, the adaptive scale identifier C T The calculation method is:

[0035]

[0036] Where h is the grid size, g(k ESW ) is the wave number switching function, k ESW is the effective wave number, Expressed as Gaussian notation, is the magnitude parameter, a1 and a2 are constants, ε1 is the third smallest quantity, ε1=1.0×10 -12 , ΔU 1,j+12 is the first-order derivative of the physical quantity at the half node, ΔU 2,j+12 is the second-order derivative of the physical quantity at the half node, ΔU 3,j+12 is the third-order derivative of the physical quantity at the half node, ΔU 4,j+12 is the fourth-order derivative of the physical quantity at the half node, and j is the number of the calculation node.

[0037] Specifically, the weighting coefficients of the four multi-order sub-templates are calculated as follows:

[0038] ω k =d k δ k / ∑(d k δ k );

[0039] Among them, ω k are the weighted coefficients of the four multi-order sub-templates, k = 1, 2, 3, 4, d4=100,δ k It is the smooth scale separator corresponding to the four multi-order sub-templates.

[0040] Specifically, the calculation method of the physical quantity at the half node is:

[0041]

[0042] in, is the approximate value of the physical quantity at half node j+1 / 2, k=1,2,3,4,ω k are the weighted coefficients of the four multi-order sub-templates, and There are three third-order precision sub-templates K1, K2 and K3 corresponding to the approximate values ​​of three template points respectively. It is the approximate value of six template points corresponding to the sixth-order precision sub-template K4.

[0043] This method performs turbulence numerical simulation based on the physical values ​​of the interface. Accurate physical values ​​of the interface can be obtained through the above method, and then accurate turbulence numerical simulation can be performed based on the accurate physical values ​​of the interface to obtain accurate simulation results.

[0044] One or more technical solutions provided by the present invention have at least the following technical effects or advantages:

[0045] The present invention uses a smooth scale separator to ensure that the sixth-order precision of the multi-order sub-template nonlinear weighted interpolation can achieve the designed sixth-order precision in the smooth region while maintaining the ability to capture shock waves. Furthermore, an adaptive scale identifier is designed based on a six-point template to ensure that the smooth scale separation can adaptively identify smooth regions in the local flow field. Ultimately, a high-order precision nonlinear weighted interpolation format is achieved, guaranteeing the accuracy of the interpolation and subsequent turbulence numerical simulation. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] 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 of the embodiments of the present invention;

[0047] Figure 1 The figure is a flowchart of a multi-order sub-template nonlinear weighted interpolation method with sixth-order precision adaptive smooth scale separation;

[0048] Figure 2 Schematic diagram of the template set in 6th order precision format. DETAILED DESCRIPTION

[0049] In order to more clearly understand the above-mentioned objects, 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 therein can be combined with each other without conflict.

[0050] 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 described herein. Therefore, the scope of protection of the present invention is not limited to the specific embodiments disclosed below.

[0051] Embodiment 1;

[0052] Please refer to Figure 1 , Figure 1The present invention provides a nonlinear weighted interpolation method for multi-order sub-templates with adaptive smooth scale separation and sixth-order precision, and the method includes:

[0053] Step 1: Divide the six adjacent template points into four multi-order sub-templates, which are three third-order precision sub-templates K1, K2 and K3 and one sixth-order precision sub-template K4;

[0054] Step 2: Calculate the smoothness metrics β1, β2, β3 and β4 of the four multi-order sub-templates respectively;

[0055] Step 3: construct a smooth scale separator based on smooth metric factors β1, β2, β3 and β4;

[0056] Step 4: Obtain the first-order derivative, second-order derivative, third-order derivative, and fourth-order derivative of the physical quantity at the half-node based on the discrete calculation of the six template points;

[0057] Step 5: construct an adaptive scale identifier based on the first-order derivative, second-order derivative, third-order derivative and fourth-order derivative of the physical quantity at the half-node;

[0058] Step 6: Using the adaptive scale identifier as the decision condition of the smooth scale separator, respectively calculate and obtain the weighted coefficients of the four multi-order sub-templates;

[0059] Step 7: Based on the weighting coefficients of the four multi-order sub-templates, the four multi-order sub-templates are weighted and calculated to obtain the physical value of the solution interface;

[0060] Step 8: Perform turbulence numerical simulation based on the physical values ​​of the interface.

[0061] Among them, through the above method, accurate physical values ​​of the interface can be obtained, and then based on the accurate physical values ​​of the interface, accurate turbulence numerical simulation can be performed to obtain accurate simulation results.

[0062] Fluxes and flux derivatives are calculated based on the physical values ​​of the interface, thereby completing the spatial discretization of the CFD control equations. This is combined with available time discretization methods to obtain discrete numerical solutions to the control method, thereby achieving turbulent numerical simulation. The turbulent numerical simulation can refer to existing turbulent numerical simulation methods or approaches, such as aircraft model turbulence numerical simulation, and is not specifically limited in this embodiment of the present invention.

[0063] The specific steps are:

[0064] Step 1: The sixth-order precision nonlinear weighted interpolation method uses four multi-order sub-templates containing six adjacent template points to calculate the variable value at the interface, such as Figure 2 As shown, Figure 2 x in j 、x j-1 、x j-2 、x j+1 、x j+2 、x j+3 The four multi-order sub-templates are three third-order precision sub-templates K1, K2, K3 and one sixth-order precision sub-template K4, where the three third-order precision sub-templates correspond to the three template point approximations. The calculation methods are:

[0065]

[0066] A sixth-order precision sub-template corresponds to six template point approximations The calculation method is:

[0067]

[0068] Step 2: After obtaining the specific expressions of the four multi-order sub-templates, it is necessary to calculate the weighting coefficients of the four templates. First, the smoothness metric factors β1, β2, β3, and β4 of the four multi-order sub-templates are calculated separately:

[0069]

[0070] Step 3: Construct a smooth scale separator using the smooth factor. For k = 1, 2, 3, 4, the smooth scale separators of the four multi-order sub-templates are defined as:

[0071]

[0072] Step 4: Discretely calculate the first-order derivative, second-order derivative, third-order derivative, and fourth-order derivative of the variable value at the half point through six template points:

[0073]

[0074] Step 5: Construct the adaptive scale identifier C by solving the four derivatives of the variable value at the interface through step 4 T :

[0075]

[0076] Step 6: Use the adaptive scale identifier in step 5 as the judgment condition of the smooth scale separator in step 3, and calculate the weight coefficients ω of the four multi-order sub-templates respectively. k :

[0077] ω k =d k δ k / ∑(d k δ k );

[0078] in, d4=100.

[0079] Step 7: According to the weighting coefficient ω of the four multi-order sub-templates k The variable values ​​of the solution interface are obtained by weighted calculation of the four multi-order sub-templates in step 1.

[0080]

[0081] In solving the nonlinear weighted function of the high-order format, the present invention defines a scale separator according to the smoothness metric factor and constructs an adaptive scale identifier using multi-order derivatives, thereby adaptively separating the scales of the multi-order sub-templates involved in the weighted calculation. When the template is in the smooth area, the scale separator takes the value of 1, realizing the designed sixth-order precision numerical format. When the template is discontinuous, the scale separator will take the value of 0, and the template will not participate in the weighted summation, thereby realizing discontinuous oscillation-free capture.

[0082] Although the preferred embodiments of the present invention have been described, those skilled in the art may make additional 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.

[0083] Obviously, those skilled in the art may make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if such changes and modifications fall within the scope of the claims and their equivalents, the present invention is intended to include such changes and modifications.

Claims

1. A multi-order sub-template nonlinear weighted interpolation method with sixth-order precision adaptive smooth scale separation, characterized by: The method comprises: Step 1: Divide the six adjacent template points into four multi-order sub-templates, which are three third-order precision sub-templates K1, K2 and K3 and one sixth-order precision sub-template K4; Step 2: Calculate the smoothness metrics β1, β2, β3 and β4 of the four multi-order sub-templates respectively; Step 3: construct a smooth scale separator based on smooth metric factors β1, β2, β3 and β4; Step 4: Obtain the first-order derivative, second-order derivative, third-order derivative, and fourth-order derivative of the physical quantity at the half-node based on the discrete calculation of the six template points; Step 5: construct an adaptive scale identifier based on the first-order derivative, second-order derivative, third-order derivative and fourth-order derivative of the physical quantity at the half-node; Step 6: Using the adaptive scale identifier as the decision condition of the smooth scale separator, respectively calculate and obtain the weighted coefficients of the four multi-order sub-templates; Step 7: Based on the weighting coefficients of the four multi-order sub-templates, the four multi-order sub-templates are weighted and calculated to obtain the physical value of the solution interface; Step 8: Perform numerical simulation of turbulence of the aircraft model based on the physical values ​​of the interface.

2. The method of claim 1 for the multi-order sub-template nonlinear weighted interpolation with sixth-order precision adaptive smooth scale separation, characterized in that: The three third-order precision sub-templates K1, K2 and K3 correspond to three template points approximated as and The six template points corresponding to the sixth-order precision sub-template K4 are approximately The calculation method is: K4: Where U is the physical quantity in the flow field, subscripts j+n and jn represent the solution points j+n and jn respectively, U j+n and U j-n Represent the physical quantities at the solution points j+n and jn, n is an integer, U j+1 / 2 Represents the physical quantity at half node j+1 / 2, O(h k ) represents the first small quantity greater than order k, h is the grid scale, and j is the number of the computing node.

3. The method of claim 1 for the multi-order sub-template nonlinear weighted interpolation with sixth-order precision adaptive smooth scale separation, characterized in that: The calculation method of the smoothness metric factors β1, β2, β3 and β4 of the four multi-order sub-templates is: K4:β4=(-U j-2 +4U j-1 -6U j +4U j+1 -U j+2 ) 2 +(You j-2 -5U j-1 +10U j -10U j+1 +5U j+2 -U j+3 ) 2 Where U is the physical quantity in the flow field, subscripts j+n and jn represent the solution points j+n and jn respectively, U j+n and U j-n They represent the physical quantities at the solution points j+n and jn respectively, n is an integer, and j is the number of the calculation node.

4. The method of claim 1 for the multi-order sub-template nonlinear weighted interpolation with sixth-order precision adaptive smooth scale separation, characterized in that: The method of constructing a smooth scale separator based on the smooth metric factors β1, β2, β3 and β4 is: Among them, δ k is the smooth scale separator corresponding to the four multi-order sub-templates, k = 1, 2, 3, 4, h is the grid scale, χ k is the intermediate process variable, C T is the adaptive scale identifier, γ k is the smoothness index, ε is the second smallest quantity, ε=1.0×10 -40 .

5. The method of claim 1 for the multi-order sub-template nonlinear weighted interpolation with sixth-order precision adaptive smooth scale separation, characterized in that: The calculation method for obtaining the first-order derivative, second-order derivative, third-order derivative, and fourth-order derivative of the physical quantity at the half-node based on the discrete calculation of six template points is as follows: Among them, ΔU 1,j+1 / 2 is the first-order derivative of the physical quantity at the half node, ΔU 2,j+1 / 2 is the second-order derivative of the physical quantity at the half node, ΔU 3,j+1 / 2 is the third-order derivative of the physical quantity at the half node, ΔU 4,j+1 / 2 is the fourth-order derivative of the physical quantity at the half node, U is the physical quantity in the flow field, the subscripts j+n and jn represent the solution points j+n and jn respectively, U j+n and U j-n They represent the physical quantities at the solution points j+n and jn respectively, n is an integer, and j is the number of the calculation node.

6. The method of claim 1 for the multi-order sub-template nonlinear weighted interpolation with sixth-order precision adaptive smooth scale separation, characterized in that: Adaptive scale identifier C T The calculation method is: Where h is the grid size, g(k ESW ) is the wave number switching function, k ESW is the effective wave number, Expressed as Gaussian notation, is the magnitude parameter, a1 and a2 are constants, ε1 is the third smallest quantity, ε1=1.0×10 -12 , ΔU 1,i+12 is the first-order derivative of the physical quantity at the half node, ΔU 2,i+1 / 2 is the second-order derivative of the physical quantity at the half node, ΔU 3,i+1 / 2 is the third-order derivative of the physical quantity at the half node, ΔU 4,i+12 is the fourth-order derivative of the physical quantity at the half node, and j is the number of the calculation node.

7. The method of claim 1 for the multi-order sub-template nonlinear weighted interpolation with sixth-order precision adaptive smooth scale separation, characterized in that: The weighting coefficients of the four multi-order sub-templates are calculated as follows: oh k =d k d k / ∑(d k d k ); Among them, ω k are the weighted coefficients of the four multi-order sub-templates, k = 1, 2, 3, 4, d4=100,δ k It is the smooth scale separator corresponding to the four multi-order sub-templates.

8. The method of claim 1 for the multi-order sub-template nonlinear weighted interpolation with sixth-order precision adaptive smooth scale separation, characterized in that: The calculation method of the physical quantity at the half node is: in, is the approximate value of the physical quantity at half node j+1 / 2, k=1,2,3,4,ω k are the weighted coefficients of the four multi-order sub-templates, and There are three third-order precision sub-templates K1, K2 and K3 corresponding to the approximate values ​​of three template points respectively. It is the approximate value of six template points corresponding to the sixth-order precision sub-template K4.

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