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

Abstract

The application discloses a design method of a three-order precision nonlinear interpolation smoothing factor based on a five-point template, belongs to the field of computational fluid dynamics, and realizes the unification of high precision, non-oscillation and multi-scale adaptability in complex flow simulation through nonlinear weighting, multi-order combination and smoothing factor design, and becomes an indispensable numerical tool for aerospace. The smoothing factor is a core mechanism for realizing shock capturing and high resolution of the high-precision format, and the weight distribution is dynamically adjusted through quantitative solution smoothing, so that a balance is achieved between shock capturing, high precision and low dissipation. Therefore, the application proposes a Z-type three-order precision nonlinear interpolation smoothing factor based on a five-point template, which meets the optimal precision sufficient condition of a multi-order candidate template nonlinear weighting framework. The method uses the method of expanding the template, and uses a consistent high-order smoothing factor method to design a new consistent high-order smoothing factor which meets the optimal interpolation precision.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of computational fluid dynamics, and particularly to a design method of third-order accuracy nonlinear interpolation smoothness factor based on five-point stencil. BACKGROUND

[0002] In the field of computational fluid dynamics, high-order numerical simulation methods have become an indispensable tool for studying complex flow phenomena, especially in engineering applications such as aerospace, which require high accuracy of numerical simulation. These methods are favored for their low dissipation, low dispersion, and high resolution. The core of high-order accuracy schemes is to reconstruct the flow field physical quantities through specific interpolation methods. Nonlinear weighted schemes such as WENO / TENO / WCNS are widely used because they can effectively capture discontinuities such as shocks. The basic idea is to divide a high-order overall stencil into several lower-order candidate sub-stencils, and dynamically allocate weights according to the smoothness of each sub-stencil (quantified by the smoothness factor). In smooth regions, the weight is close to the ideal value, and the combined result approximates the high-order linear scheme. In the discontinuity region, the sub-stencil containing the discontinuity is given a small weight, avoiding cross-discontinuity interpolation, and achieving shock capturing without oscillation. In this framework, the smoothness factor, as a key parameter to measure the smoothness of the sub-stencil, directly determines whether the scheme can achieve an ideal balance between shock capturing ability, numerical accuracy, and numerical dissipation. It is one of the core mechanisms for the success of multi-scale flow simulation.

[0003] Although nonlinear weighted schemes have achieved remarkable success, there are still some inherent defects in the design and application of smoothness factors in existing technologies. First, when using non-equal-width candidate stencil sets to pursue higher flexibility or multi-resolution characteristics, the smoothness factors of each sub-stencil may have different dimensions, which leads to the calculated nonlinear weights depending on the dimension of the control equation. This may cause the final interpolation result to deviate from the expected value when simulating multi-scale problems, or even get the wrong solution. Second, even if the flow field is smooth, the inherent differences in smoothness factors between different sub-stencils will cause deviations in the dynamic weights assigned to them from the ideal weights. This deviation will introduce unnecessary high numerical dissipation in the smooth region, reducing the resolution of multi-scale structures such as turbulence, and is not conducive to long-term fine simulation.

[0004] In addition, some improved schemes, such as multi-order sub-stencil weighting methods, may not guarantee high-order accuracy in the entire smooth region, affecting the overall accuracy of the simulation. At the same time, the deviation of the nonlinear weight from the ideal weight in the smooth region may lead to an increase in the truncation error, while in the vicinity of the discontinuity, the weight of the sub-stencil containing the discontinuity is compressed too much to completely suppress non-physical oscillations, introducing excessive numerical dissipation and reducing the resolution of shocks and contact discontinuities.

[0005] Therefore, there is an urgent need for a new smooth factor design method that can effectively solve the dimensional consistency problem in non-equidistant template combination, ensure that the non-linear weight and the final interpolation result are independent of the dimension, while in the smooth area, the non-linear weight can fully approximate the ideal weight to reduce dissipation and maintain high precision characteristics, and in the discontinuous area, the weight can be reasonably distributed to maintain sharp shock capturing ability. The Z-type third-order accuracy nonlinear interpolation smooth factor based on the five-point template proposed in the present application is designed to meet these challenges. SUMMARY

[0006] The present application aims to provide a design method for a third-order accuracy nonlinear interpolation smooth factor based on a five-point template, which solves the problem that the high-precision format in the prior art cannot effectively capture smooth areas and strong discontinuous areas.

[0007] The present application is realized by the following scheme:

[0008] The design method for a third-order accuracy nonlinear interpolation smooth factor based on a five-point template includes the following steps:

[0009] Step S1: Extract high-order smoothness information of sub-templates by expanding the five-point template to ensure that the smooth area meets the multi-order candidate template nonlinear weighting framework accuracy condition;

[0010] Step S2: Eliminate the scale effect by an adaptive order regulator and normalization processing to ensure that the nonlinear weighting error order is not lower than the background format;

[0011] Step S3: When a discontinuity is detected, the smooth factor automatically degenerates into a traditional form to suppress oscillation and maintain third-order accuracy.

[0012] In step S1, the high-order smoothness information of the three sub-templates of the expanded five-point template information is extracted , which is defined as:

[0013]

[0014] wherein represents the velocity, the subscript k=-1, 0, 1, and the subscript j represents the grid point coordinates of any point.

[0015] In step S2, the order regulator is defined as:

[0016]

[0017] The order regulator , and indicates that the smoother the variable distribution on the expanded template is, represents the more likely to contain discontinuities in the extended template.

[0018] In step S3, the smoothing factor and :

[0019]

[0020] wherein subscript k=0, 1.

[0021] Specifically, in the smoothing area, the smoothing factor and is obtained by Taylor expansion:

[0022]

[0023] wherein O represents the precision symbol, h represents the grid spacing, j represents an arbitrary grid coordinate point, and the superscript 1 represents the first derivative.

[0024] The smoothing factor obtained from the Taylor expansion satisfies the sufficient condition of the precision non-decreasing order of the multi-variable order candidate template nonlinear weighted framework smoothing area:

[0025] .

[0026] An interpolation precision test function is selected for verification, the selected test function is , the interpolation test interval is , is a smoothing function, and is a first-order extreme point at .

[0027] When k=-1:

[0028] , the small amount .

[0029] When k=0:

[0030] , the small amount .

[0031] When k=1:

[0032] , the small amount .

[0033] In summary, due to the adoption of the above technical solutions, the beneficial effects of the present application are:

[0034] 1. The method is aimed at numerical simulation of complex flow with shock wave and multi-scale structure, and a Z-type third-order precision nonlinear weighted interpolation method based on a five-point template is proposed, which meets the optimal precision sufficient condition of the multi-order candidate template nonlinear weighted framework. The method uses the method of expanding the template, and uses a consistent high-order smoothing factor method to design a new consistent high-order smoothing factor which meets the optimal interpolation precision, and can realize effective capture of high-precision format to smooth area and strong discontinuity area. BRIEF DESCRIPTION OF DRAWINGS

[0035] Figure 1 A schematic diagram of the high-order smoothing factor corresponding to the template;

[0036] Figure 2 The interpolation precision of the smoothing factor of the present application;

[0037] Figure 3 The predicted one-dimensional shock tube density distribution of the present application. DETAILED DESCRIPTION

[0038] All features disclosed in this specification, and / or all steps of any methods or processes disclosed in this specification, can be combined in any manner, except where features or steps are mutually exclusive.

[0039] Any feature disclosed in this specification, unless stated otherwise, can be replaced by alternative features or equivalents having the same or a similar effect. That is, unless stated otherwise, each feature is one example only of a number of alternative or similar features.

[0040] In the description of the present application, it should be understood that the orientation or positional relationship indicated by the terms "up", "down", "left", "right", etc. is based on the orientation or positional relationship shown in the drawings, and is only for the convenience of describing the present application and simplifying the description, and does not indicate or imply that the device or element referred to must have a predetermined orientation, be constructed and operated in a predetermined orientation, and therefore cannot be understood as a limitation on the present application.

[0041] In addition, the terms "first", "second", etc. are only for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly indicating the number of technical features indicated. Therefore, the features defined with "first", "second", etc. can explicitly or implicitly include one or more of the features.

[0042] Example 1

[0043] The present application provides a technical solution:

[0044] As shown in Figure 1 A design method of a third-order precision nonlinear interpolation smoothing factor based on a five-point template, comprising the following steps:

[0045] Step S1, high-order smoothness information of sub-templates is extracted by extending five-point template to ensure that smooth regions meet the accuracy conditions of multi-order candidate template nonlinear weighted framework;

[0046] Step S2, through adaptive order regulator and normalization processing, scale effect is eliminated to ensure that the non-linear weighted error order is not lower than the background format;

[0047] Step S3, when the discontinuity is detected, the smooth factor is automatically degraded to the traditional form to suppress oscillation and maintain the third-order accuracy.

[0048] In step S1, high-order smoothness information is extracted from three sub-templates of extended five-point template information , which is defined as:

[0049] (1)

[0050] Wherein represents the velocity, subscript k=-1, 0, 1, and subscript j represents the grid point coordinates of any point.

[0051] When k=-1:

[0052]

[0053] When k=0:

[0054]

[0055] When k=1:

[0056]

[0057] Small quantity , in order to avoid the denominator of order regulator being 0.

[0058] In step S2, the order regulator is defined as:

[0059] (2)

[0060] The order regulator , and indicates that the variable distribution on the extended template is smoother, indicates that the extended template is more likely to contain discontinuity.

[0061] In step S3, the smooth factor and :

[0062] (3)

[0063] where subscript k=0,1;

[0064] Specifically, in the smooth region, Taylor expansion of formula (3) obtains

[0065] (4)

[0066] In the above formula O denotes the precision symbol, h denotes the grid spacing, j denotes an arbitrary grid coordinate point, and the superscript 1 denotes the first derivative;

[0067] From formula (4), the following can be obtained:

[0068] (5)

[0069] Formula (5) shows that the smoothing factor of the present application satisfies the sufficient condition of non-decreasing order of accuracy in the smoothing region of the multi-variable order candidate template nonlinear weighted framework;

[0070] In order to verify the effectiveness of the above method, an interpolation accuracy test function is selected for verification, and the test function selected is , the interpolation test interval is , is a smoothing function, and is a first-order extreme point at .

[0071] Figure 2 The interpolation accuracy of the smoothing factor designed in the present application is given in Table 1, and from the figure it can be seen that the nonlinear interpolation method of the smoothing factor maintains third-order accuracy in the interpolation accuracy test of the test function containing the extreme point.

[0072] Figure 2 In Table 1, N represents the number of grid points, L1, L2 and are statistical norms for measuring interpolation accuracy and error, wherein L1_error represents L1 error, L1_order represents L1 convergence order, L2_error represents L2 error, L2_order represents L2 convergence order, L_infty_error represents error, and L_infty_order represents convergence order.

[0073] Figure 3 is the one-dimensional shock tube density distribution calculated by the present method, and in the figure, the horizontal coordinate x represents the one-dimensional coordinate position, and the vertical coordinate represents Density. In the figure, Exact represents the exact solution, HWCNS-Z3 represents the third-order precision Z-type weight result, and the implementation details are described 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 weight result in the literature, and the implementation details are described 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.], and 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.

[0074] The above merely describes preferred embodiments of the present application and is not intended to limit the present application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the protection scope of the present application.

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. 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 index k represents velocity, with subscripts k = -1, 0, and 1, and subscript j representing the coordinates of any grid point. 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. Step S3: When a discontinuity is detected, the smoothing factor automatically degenerates into the conventional form to suppress oscillations and maintain third-order accuracy; In step S3, the smoothing factor and : Where the subscript k = 0, 1; 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.

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 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.

3. 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: 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: 。 4. 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 3, 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.

5. 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: When k=-1: small amount .

6. 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: When k=0: small amount .

7. 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: When k=1: small amount .

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