Design method of three-order precision nonlinear interpolation smooth factor based on five-point template

Abstract

The invention discloses a three-order precision nonlinear interpolation smooth factor design method based on a five-point template, and belongs to the field of computational fluid mechanics, a high-precision format realizes unification of high precision, no oscillation and multi-scale adaptability in complex flow simulation through nonlinear weighting, multi-order combination and smooth factor design, and the method has the advantages of high precision and high precision. The device is an indispensable numerical tool for aerospace. Wherein the smooth factor is a core mechanism for realizing shock wave capture and high resolution in a high-precision format, and weight distribution is dynamically adjusted through smoothness of a quantitative solution, so that balance among shock wave capture, high precision and low dissipation is achieved. Therefore, the invention provides the Z-type third-order precision nonlinear interpolation smooth factor which is based on the five-point template and meets the condition that the optimal precision of the nonlinear weighting framework of the multi-order candidate template is sufficient. A new consistent high-order smooth factor meeting the optimal interpolation precision is designed by using a consistent high-order smooth factor method.

Description

Technical Field

[0001] This invention relates to the field of computational fluid dynamics, and in particular to a design method for a third-order precision nonlinear interpolation smoothing factor based on a five-point template. Background Technology

[0002] In the field of computational fluid dynamics, high-order precision numerical simulation methods have become indispensable tools for studying complex flow phenomena. They are particularly favored in engineering applications such as aerospace, where extremely high accuracy in numerical simulations is required. These methods are characterized by low dissipation, low dispersion, and high resolution. The core of high-order precision schemes lies in reconstructing flow field physical quantities through specific interpolation methods. Among these, nonlinear weighted schemes such as WENO / TENO / WCNS are widely used because they can effectively capture discontinuous structures such as shock waves. Their basic idea is to divide a high-order precision overall template into several lower-order precision candidate sub-templates and dynamically assign weights based on the smoothness of each sub-template (quantized by a smoothness factor). In smooth regions, the weights approach ideal values, and the combined result approximates a high-order linear scheme. In discontinuous regions, sub-templates containing discontinuities are assigned minimal weights, thereby avoiding cross-discontinuity interpolation and achieving oscillatory shock wave capture. Within this framework, the smoothness factor, as a key parameter for measuring the smoothness of the sub-template, is designed so that its rationality directly determines whether the scheme can achieve an ideal balance between shock wave capture capability, numerical accuracy, and numerical dissipation. It is one of the core mechanisms for the success of multi-scale flow simulation.

[0003] Despite the significant success of nonlinear weighted schemes, the design and application of smoothing factors in existing technologies still suffer from several inherent drawbacks. First, when using non-uniformly wide candidate template sets to pursue greater flexibility or multi-resolution characteristics, the smoothing factors of each sub-template may have different dimensions. This causes the calculated nonlinear weights to depend on the dimensions of the governing equations, potentially leading to discrepancies between the final interpolation results and even incorrect solutions when simulating multi-scale problems. Second, even with a smooth flow field, the inherent differences in smoothing factors between different sub-templates can cause deviations between the assigned dynamic weights and the ideal weights. This deviation introduces unnecessary and excessive numerical dissipation in smooth regions, reducing the scheme's ability to resolve multi-scale structures such as turbulence and hindering long-term, detailed simulations.

[0004] Furthermore, some improved methods, such as the multi-order sub-template weighting method, while focusing on resolution near discontinuities, may not guarantee the achievement of the theoretically designed high-order accuracy across the entire smooth region, affecting the overall accuracy of the simulation. Meanwhile, in the smooth region, the deviation of the nonlinear weights from the ideal weights in the classical weighting method can lead to increased truncation errors. Near discontinuities, excessive compression of the weights of sub-templates containing discontinuities in order to completely suppress non-physical oscillations introduces excessive numerical dissipation, reducing the resolution of shock waves and contact discontinuities.

[0005] Therefore, there is an urgent need for a new smoothing factor design method that can effectively solve the dimensional consistency problem in non-equidistant template combinations, ensure that the nonlinear weights and the final interpolation results are dimensionless, and simultaneously enable the nonlinear weights to fully approximate the ideal weights in smooth regions to reduce dissipation and maintain high accuracy. Furthermore, it can rationally allocate weights in discontinuous regions to maintain sharp shock wave capture capability. The Z-type third-order precision nonlinear interpolation smoothing factor based on a five-point template proposed in this invention is designed precisely to address these challenges. Summary of the Invention

[0006] The purpose of this invention is to provide a design method for a third-order precision nonlinear interpolation smoothing factor based on a five-point template, addressing the above-mentioned shortcomings and solving the problem that high-precision formats in the prior art cannot effectively capture smooth regions and strong discontinuities.

[0007] This invention is achieved through the following scheme: The design method for a third-order precision nonlinear interpolation smoothing factor based on a five-point template includes the following steps: Step S1: Extract the higher-order smoothness information of the sub-template by expanding the five-point template to ensure that the smooth region meets the accuracy conditions of the nonlinear weighted framework of the multi-order candidate template. Step S2: Through adaptive order adjuster and normalization processing, the scaling effect is eliminated to ensure that the order of nonlinear weighted error is not lower than that of the background format. In step S3, when a discontinuity is detected, the smoothing factor automatically degenerates into its conventional form to suppress oscillations and maintain third-order accuracy.

[0008] In step S1, specifically, high-order smoothness information is extracted from the three sub-templates of the extended five-point template information. Defined as:

[0009] in The velocity is indicated by the subscript k = -1, 0, 1, and the subscript j represents the coordinates of any grid point.

[0010] In step S2, the scale regulator Defined as:

[0011] Scale regulator ,and A smoother distribution of variables on the extended template indicates a smoother distribution. The time indicates that the extended template is more likely to contain breaks.

[0012] In step S3, the smoothing factor and :

[0013] Where the subscript k = 0, 1.

[0014] Specifically, in the smooth region, for the smoothness factor and Taylor expansion yields:

[0015] in O The symbol represents the precision, h represents the grid spacing, j represents any grid coordinate point, and the superscript 1 represents the first derivative.

[0016] The Taylor expansion yields a sufficient condition for the smoothing factor to satisfy the requirement that the accuracy of the smooth region in the nonlinear weighted framework of multi-order candidate templates does not decrease: .

[0017] The interpolation accuracy test function was selected for verification. The selected test function is: The interpolation test interval is , It is a smooth function, and in This is a first-order extreme point.

[0018] When k=-1: small amount .

[0019] When k=0: small amount .

[0020] When k=1: small amount .

[0021] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are: 1. This paper proposes a Z-type third-order precision nonlinear weighted interpolation method based on a five-point template, which satisfies the optimal precision condition of a nonlinear weighted framework with multiple order candidate templates, for numerical simulation of complex flows with shock waves and multi-scale structures. This method utilizes an extended template approach and a uniform higher-order smoothing factor method to design a novel uniform higher-order smoothing factor that satisfies the optimal interpolation precision, enabling the high-precision scheme to effectively capture smooth regions and strong discontinuities. Attached Figure Description

[0022] Figure 1 This is a schematic diagram of the template corresponding to the higher-order smoothness factor; Figure 2 This refers to the interpolation accuracy of the smoothing factor in this invention; Figure 3 This invention is used to predict the density distribution of a one-dimensional shock tube. Detailed Implementation

[0023] All features disclosed in this specification, or all steps in all disclosed methods or processes, may be combined in any way, except for mutually exclusive features and / or steps.

[0024] Any feature disclosed in this specification (including any appended claims and abstract) may be replaced by other equivalent or similar features, unless specifically stated otherwise. That is, unless specifically stated otherwise, each feature is merely one example of a series of equivalent or similar features.

[0025] In the description of this invention, it should be understood that the terms "upper," "lower," "left," "right," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a predetermined orientation, or be constructed and operated in a predetermined orientation. Therefore, they should not be construed as limitations on this invention.

[0026] Furthermore, the terms "first," "second," etc., are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Therefore, a feature defined with "first," "second," etc., may explicitly or implicitly include one or more of that feature.

[0027] Example 1 This invention provides a technical solution: like Figure 1 As shown, a method for designing a third-order precision nonlinear interpolation smoothing factor based on a five-point template includes the following steps: Step S1: Extract the higher-order smoothness information of the sub-template by expanding the five-point template to ensure that the smooth region meets the accuracy conditions of the nonlinear weighted framework of the multi-order candidate template. Step S2: Through adaptive order adjuster and normalization processing, the scaling effect is eliminated to ensure that the order of nonlinear weighted error is not lower than that of the background format. In step S3, when a discontinuity is detected, the smoothing factor automatically degenerates into its conventional form to suppress oscillations and maintain third-order accuracy.

[0028] In step S1, specifically, high-order smoothness information is extracted from the three sub-templates of the extended five-point template information. Defined as: (1) in The velocity is indicated by the subscript k = -1, 0, 1, and the subscript j represents the coordinates of any grid point.

[0029] When k=-1:

[0030] When k=0:

[0031] When k=1:

[0032] Small quantity To avoid the denominator of the scale regulator being 0.

[0033] In step S2, the scale regulator Defined as: (2) Scale regulator ,and A smoother distribution of variables on the extended template indicates a smoother distribution. The time indicates that the extended template is more likely to contain breaks.

[0034] In step S3, the smoothing factor and : (3) Where the subscript k = 0, 1; Specifically, in the smooth region, the Taylor expansion of equation (3) yields... (4) In the above formula O The precision symbol is indicated by h, the grid spacing is indicated by h, the coordinates of any grid point are indicated by j, and the superscript 1 indicates the first derivative. From equation (4), we can obtain: (5) Equation (5) shows that the smoothing factor of this invention satisfies the sufficient condition that the accuracy of the smooth region of the nonlinear weighted framework of multi-order candidate templates does not decrease; To verify the effectiveness of the above method, we selected an interpolation accuracy test function for verification. The selected test function is: The interpolation test interval is , It is a smooth function, and in This is a first-order extreme point.

[0035] Figure 2The figure shows the interpolation accuracy of the smoothing factor designed in this invention. It can be seen from the figure that the nonlinear interpolation method of this smoothing factor is effective on test functions containing extreme points. It maintained third-order accuracy in the interpolation accuracy test.

[0036] Figure 2 In this context, N represents the number of grid points, L1, L2, and L3. The statistical norm is used to measure interpolation accuracy and error, where L1_error represents the L1 error, L1_order represents the L1 convergence order, L2_error represents the L2 error, L2_order represents the L2 convergence order, and L_infty_error represents... Error, represented by L_infty_order Convergence order.

[0037] Figure 3 This figure shows the one-dimensional shock tube density distribution calculated by this method. The horizontal axis x represents the one-dimensional coordinate position, and the vertical axis represents... Density. In the figure, Exact represents the exact solution, HWCNS-Z3 represents the third-order accurate Z-weighted result, and the implementation details are in the literature [Borges R, Carmona M, Costa B, et al. An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws[J]. JOURNAL OF COMPUTATIONAL PHYSICS, 2008, 227(6): 3191-3211]. HWCNS-AIA represents the nonlinear weighted result in the literature, and the implementation details are in the literature [Cheng M, Tang L, Liu Y, et al. An improved third-order HWCNS for compressible flow simulation on curvilinear grids[J]. Advances in Aerodynamics, 2021, 3(1): 32.]. Present represents the present method. As can be seen from the figure, the present method can accurately capture the one-dimensional shock tube density distribution.

[0038] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for designing a third-order precision nonlinear interpolation smoothing factor based on a five-point template, characterized in that: The steps include the following: Step S1: Extract the higher-order smoothness information of the sub-template by expanding the five-point template to ensure that the smooth region meets the accuracy conditions of the nonlinear weighted framework of the multi-order candidate template. Step S2: Through adaptive order adjuster and normalization processing, the scaling effect is eliminated to ensure that the order of nonlinear weighted error is not lower than that of the background format. In step S3, when a discontinuity is detected, the smoothing factor automatically degenerates into its conventional form to suppress oscillations and maintain third-order accuracy.

2. The design method for the third-order precision nonlinear interpolation smoothing factor based on a five-point template as described in claim 1, characterized in that: In step S1, specifically, high-order smoothness information is extracted from the three sub-templates of the extended five-point template information. Defined as: in The velocity is indicated by the subscript k = -1, 0, 1, and the subscript j represents the coordinates of any grid point.

3. The design method for the third-order precision nonlinear interpolation smoothing factor based on a five-point template as described in claim 1, characterized in that: In step S2, the scale regulator Defined as: Scale regulator ,and A smoother distribution of variables on the extended template indicates a smoother distribution. The time indicates that the extended template is more likely to contain breaks.

4. The design method for the third-order precision nonlinear interpolation smoothing factor based on a five-point template as described in claim 1, characterized in that: In step S3, the smoothing factor and : Where the subscript k = 0, 1.

5. The design method for a third-order precision nonlinear interpolation smoothing factor based on a five-point template as described in claim 4, characterized in that: Specifically, in the smooth region, for the smoothness factor and Taylor expansion yields: in O The symbol represents the precision, h represents the grid spacing, j represents any grid coordinate point, and the superscript 1 represents the first derivative.

6. The design method for the third-order precision nonlinear interpolation smoothing factor based on a five-point template as described in claim 5, characterized in that: The Taylor expansion yields a sufficient condition for the smoothing factor to satisfy the requirement that the accuracy of the smooth region in the nonlinear weighted framework of multi-order candidate templates does not decrease: 。 7. The design method for a third-order precision nonlinear interpolation smoothing factor based on a five-point template as described in any one of claims 1 to 6, characterized in that: The interpolation accuracy test function was selected for verification. The selected test function is: The interpolation test interval is , It is a smooth function, and in This is a first-order extreme point.

8. The design method for the third-order precision nonlinear interpolation smoothing factor based on a five-point template as described in claim 2, characterized in that: When k=-1: small amount .

9. The design method for a third-order precision nonlinear interpolation smoothing factor based on a five-point template as described in claim 2, characterized in that: When k=0: small amount .

10. The design method for a third-order precision nonlinear interpolation smoothing factor based on a five-point template as described in claim 2, characterized in that: When k=1: small amount .

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