A boundary consistent discontinuity detection method for globally conservative high-order finite difference schemes

Abstract

The application discloses a boundary consistent discontinuity detection method for a global conservation high-order finite difference format, and comprises the following steps: S1, dividing an interpolation template in a calculation domain; S2, extracting five-point flow field variables, and performing amplitude normalization on the five-point flow field variables; S3, generating a four-dimensional derivative characteristic vector; S4, outputting a discontinuity judgment result; S5, interpolating the discontinuity judgment result; and S6, calculating a flux, and performing numerical simulation according to a semi-discrete control equation until a set calculation termination time or a convergence condition is reached. The application realizes adaptive mixed interpolation on the whole calculation domain including the boundary, retains the shock capturing ability of a nonlinear format, and maintains low dissipation and high efficiency of a linear format in a smooth region.

Description

Technical Field

[0001] This invention relates to the field of computational fluid dynamics simulation technology, specifically to a boundary consistent discontinuity detection method for a globally conserved high-order finite difference scheme. Background Technology

[0002] In the design and performance evaluation of aviation, aerospace, and marine equipment, high-precision computational fluid dynamics simulation technology has become a core tool supporting design decisions, shortening R&D cycles, and reducing testing risks. With the continuous improvement of the performance requirements of aircraft and other equipment, and the increasing complexity of the geometry of the design objects, problems such as shock wave capture in supersonic flow and multi-mesh processing place higher demands on the accuracy, conservation, and efficiency of numerical simulation.

[0003] Higher-order finite difference methods are widely used in engineering fluid simulation due to their advantages of simple mathematical form, high numerical resolution, and ease of engineering implementation. However, in compressible flows containing shock waves and contact discontinuities, linear higher-order schemes can produce non-physical oscillations; and using nonlinear shock wave capture schemes over the entire domain can introduce additional numerical dissipation and increase computational overhead in large smooth regions, making it difficult to simultaneously achieve both accuracy and efficiency.

[0004] Hybrid high-order formats employ partitioned scheduling of different formats using intermittent detectors, enabling nonlinear formats only in discontinuous regions and linear formats in smooth regions. This approach offers an effective solution that balances shock wave capture capability with computational efficiency. The recognition accuracy and global adaptability of the intermittent detectors determine the stability and overall performance of the hybrid format.

[0005] To satisfy discrete conservation constraints, globally conserved high-order finite difference schemes strategically adjust the positions of solution points near the boundary to construct a semi-uniform discrete grid: solution points are uniformly distributed in the internal region, while solution points near the boundary are offset to ensure global conservation. This type of globally conserved scheme has been verified to stably and accurately capture shock waves crossing the boundary.

[0006] Existing discontinuity detection methods are primarily designed for the computational domain. Virtual mesh point techniques are typically used near the boundary to allow the same discontinuity detection algorithm to be used across the entire computational domain. However, for globally conserved high-precision finite difference schemes, the computational domain boundary not only suffers from missing template points, but the solution points at the boundary are also no longer uniformly distributed as within the computational domain. Therefore, interior discontinuity detection methods cannot be directly applied to such boundary regions. In practical applications, the complex shapes of engineering simulations (such as large passenger aircraft, spacecraft, automobiles, etc.) generally rely on multi-mesh techniques, resulting in a very high demand for simulating mesh boundaries. Failure to guarantee the simulation accuracy and reliability at the boundaries will severely impact the credibility of the overall engineering design. Summary of the Invention

[0007] To address the problem that discontinuity detectors are difficult to directly apply to near-boundary non-uniform templates in globally conserved schemes, this invention proposes a boundary-consistent discontinuity detection method for globally conserved high-order finite difference schemes.

[0008] The technical solution of this invention is: a boundary uniform discontinuity detection method for globally conserved high-order finite difference schemes, comprising the following steps:

[0009] S1. Divide the interpolation template within the computational domain;

[0010] S2. Traverse all five-point interpolation templates in the domain according to the index order, extract the five-point flow field variables, and normalize the amplitude of the five-point flow field variables.

[0011] S3. Using the normalized five-point flow field variables, calculate the first to fourth derivatives of the Lagrange interpolation polynomial on each five-point interpolation template, and generate a four-dimensional derivative eigenvector.

[0012] S4. Input the four-dimensional derivative feature vectors of all five-point interpolation templates into the pre-trained classification model and output the discontinuity determination result.

[0013] S5. Interpolate the discontinuity determination results;

[0014] S6. Calculate the flux based on the interpolation results and perform numerical simulation based on the semi-discrete control equations until the set calculation termination time or convergence condition is reached.

[0015] Furthermore, in S1, the interpolation template within the computational domain is divided into an internal uniform template and a near-boundary non-uniform template.

[0016] Furthermore, the near-boundary non-uniform template includes a left-boundary non-uniform template and a right-boundary non-uniform template;

[0017] The non-uniform template on the left boundary includes the first template, the second template, and the third template;

[0018] First Template for ;in, Indicates the left endpoint of the computation domain. This represents the first solution point in the computational domain. This represents the second solution point in the computational domain. This represents the third solution point in the computational domain. This represents the 4th solution point in the computational domain;

[0019] Second template for ;in, This represents the 5th solution point in the computational domain;

[0020] Third Template for ;in, This represents the 6th solution point in the computational domain.

[0021] The right boundary non-uniform template and the left boundary non-uniform template are mirror-symmetric about the center of the computational domain.

[0022] Furthermore, the amplitudes of the five-point flow field variables are normalized as follows:

[0023] ;

[0024] in, This represents the five-point interpolation template. Normalized flow field values ​​at points, This represents the five-point interpolation template. Flow field values ​​at each point This represents the flow field value at the first point of the five-point interpolation template. This represents the flow field value at the second point of the five-point interpolation template. This represents the flow field value at the 3rd point of the five-point interpolation template. This represents the flow field value at the 4th point of the five-point interpolation template. This represents the flow field value at the 5th point of the five-point interpolation template. This indicates taking the maximum value.

[0025] Furthermore, in S3, the first derivative of the Lagrange interpolation polynomial on the internal uniform template... Second derivative Third derivative and fourth derivative They are respectively:

[0026] ;

[0027] ;

[0028] ;

[0029] ;

[0030] in, This represents the normalized flow field value at the first point of the five-point interpolation template. This represents the normalized flow field value at the second point of the five-point interpolation template. This represents the normalized flow field value at the 3rd point of the five-point interpolation template. This represents the normalized flow field value at the 4th point of the five-point interpolation template. This represents the normalized flow field value at the 5th point of the five-point interpolation template.

[0031] Furthermore, in S3, the first... order derivative for:

[0032] ;

[0033] in, Indicates the first Template No. The first derivative corresponds to the second The difference coefficients at each point This represents the five-point interpolation template. Normalized flow field values ​​at points.

[0034] Furthermore, the four-dimensional derivative eigenvector is:

[0035] ;

[0036] in, This represents the first derivative of the Lagrange interpolation polynomial over the five-point interpolation template. This represents the second derivative of the Lagrange interpolation polynomial over the five-point interpolation template. This represents the third derivative of the Lagrange interpolation polynomial over the five-point interpolation template. It represents the fourth derivative of the Lagrange interpolation polynomial over the five-point interpolation template.

[0037] Furthermore, in S5, for templates whose discontinuity determination result is of the discontinuity class, nonlinear weighted interpolation is used to obtain the flow field value at the flux point; for templates whose discontinuity determination result is of the non-discontinuity class, linear Lagrange interpolation is used to obtain the flow field value at the flux point.

[0038] The beneficial effects of this invention are:

[0039] The boundary region derivative feature calculation and interpolation calculation of this invention are both completed entirely based on the real quantities of the physical grid, without the need for virtual point extrapolation; the interpolation polynomial derivative is used as the core detection feature, and its discrimination criterion is derived from the inherent mathematical properties of the flow structure, and the physical semantics of the derivative features inside the computational domain and the boundary template are unified; the classification model trained only on the internal uniform template can be directly transferred to the boundary non-uniform template without the need to supplement boundary samples; the global conservation hybrid scheme constructed based on this detection method realizes adaptive hybrid interpolation in the entire computational domain including the boundary, which not only retains the shock wave capture capability of the nonlinear scheme, but also maintains the low dissipation and high efficiency of the linear scheme in the smooth region. Attached Figure Description

[0040] Figure 1 A flowchart for a boundary uniform discontinuity detection method for globally conserved high-order finite difference schemes;

[0041] Figure 2 A schematic diagram of the interpolation template for a semi-uniform grid and the left boundary of the computational domain;

[0042] Figure 3 This is a schematic diagram of the grid partitioning for the double Mach reflection example;

[0043] Figure 4 This paper compares the discontinuity detection results of the boundary-consistent discontinuity detection method and the comparison method of the present invention in a dual Mach reflection example. Detailed Implementation

[0044] The embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0045] like Figure 1 As shown, this invention provides a boundary uniform discontinuity detection method for globally conserved high-order finite difference schemes, comprising the following steps:

[0046] S1. Divide the interpolation template within the computational domain;

[0047] S2. Traverse all five-point interpolation templates in the domain according to the index order, extract the five-point flow field variables, and normalize the amplitude of the five-point flow field variables.

[0048] S3. Using the normalized five-point flow field variables, calculate the first to fourth derivatives of the Lagrange interpolation polynomial on each five-point interpolation template, and generate a four-dimensional derivative eigenvector.

[0049] S4. Input the four-dimensional derivative feature vectors of all five-point interpolation templates into the pre-trained classification model and output the discontinuity determination result.

[0050] S5. Interpolate the discontinuity determination results;

[0051] S6. Calculate the flux based on the interpolation results and perform numerical simulation based on the semi-discrete control equations until the set calculation termination time or convergence condition is reached.

[0052] In this embodiment of the invention, in S1, the interpolation template within the computational domain is divided into an internal uniform template and a near-boundary non-uniform template.

[0053] To address the positional distribution of grid points in a semi-uniform discrete grid of a globally conserved high-order scheme, all interpolation templates within the computational domain are divided into two categories: internal uniform templates and near-boundary non-uniform templates.

[0054] In this embodiment of the invention, the near-boundary non-uniform template includes a left-boundary non-uniform template and a right-boundary non-uniform template;

[0055] The non-uniform template on the left boundary includes the first template, the second template, and the third template;

[0056] First Template for ;in, Indicates the left endpoint of the computation domain. This represents the first solution point in the computational domain. This represents the second solution point in the computational domain. This represents the third solution point in the computational domain. This represents the 4th solution point in the computational domain;

[0057] Second template for ;in, This represents the 5th solution point in the computational domain;

[0058] Third Template for ;in, This represents the 6th solution point in the computational domain.

[0059] The right boundary non-uniform template and the left boundary non-uniform template are mirror-symmetric about the center of the computational domain.

[0060] In this embodiment of the invention, the flow field variables at each point within the template are divided by the larger of the maximum absolute value of the flow field variables within the template and 1, retaining only the information on the relative changes in the flow field.

[0061] Computational domain Discretized into interleaved flux points and solution points, where the number of flux points is... There are [number] solution points. There are [number] flux points. The flux points are uniformly distributed, and their coordinates are represented as follows: , ,in The solution point lies between two adjacent flux points, with the solution point inside the computational domain located at the midpoint between two adjacent flux points, represented by the coordinates as follows: , The positions of the two solution points near the left and right boundaries shift, no longer located at the midpoint between two adjacent flux points. These points become the conservation solution points, represented by the following coordinates:

[0062] ;

[0063] These flux points and solution points form a semi-uniform grid, and the location of the conservation solution points is derived from global conservation.

[0064] Based on this semi-uniform grid, coexistence exists throughout the entire domain. Each interpolation template consists of five independent five-point interpolation templates, each containing five grid points. Based on the distribution characteristics of the template points, they can be divided into two categories:

[0065] Internal uniform template: Total There are 5 grid points within the template, all of which are uniformly solved internally with consistent spacing.

[0066] Near-boundary non-uniform template: 3 on each of the left and right boundaries, for a total of 6; the template contains offset conservation solution points or boundary flux points, and the points are non-uniformly distributed.

[0067] Taking the left boundary as an example, the three non-uniform templates are as follows:

[0068] First Template : Including the flux point on the left boundary Compared to the first four solution points;

[0069] Second template : Includes the 1st to 5th solution points, of which the first 2 solution points are offset conservation points;

[0070] Third Template : Contains the 2nd to 6th solution points, where the 2nd solution point is the offset conservation point.

[0071] The three non-uniform templates on the right boundary are symmetrically distributed with the left boundary. The semi-uniform mesh and templates near the left and right boundaries are as follows: Figure 2 As shown.

[0072] The amplitude normalization of the five-point flow field variables is as follows:

[0073] ;

[0074] in, This represents the five-point interpolation template. Normalized flow field values ​​at points, This represents the five-point interpolation template. Flow field values ​​at each point This represents the flow field value at the first point of the five-point interpolation template. This represents the flow field value at the second point of the five-point interpolation template. This represents the flow field value at the 3rd point of the five-point interpolation template. This represents the flow field value at the 4th point of the five-point interpolation template. This represents the flow field value at the 5th point of the five-point interpolation template. This indicates taking the maximum value.

[0075] This normalization retains only the relative change information of the flow field, eliminating the influence of the absolute magnitude of physical quantities on the characteristic calculation.

[0076] In this embodiment of the invention, in S3, all computational domains are traversed in index order. Using five interpolation templates and normalized flow field variables, the first to fourth derivative characteristics of the Lagrange interpolation polynomials on each template are calculated. Since the internal template points are uniformly distributed while the boundary template points are non-uniformly distributed, the derivative characteristics are calculated in two parts: internal and boundary.

[0077] For a uniformly distributed five-point template, derive the first to fourth derivatives at the template's center point in a dimensionless unit-spaced index space. Also derive the first derivative of the Lagrange interpolation polynomial over the internal uniform template. Second derivative Third derivative and fourth derivative They are respectively:

[0078] ;

[0079] ;

[0080] ;

[0081] ;

[0082] in, This represents the normalized flow field value at the first point of the five-point interpolation template. This represents the normalized flow field value at the second point of the five-point interpolation template. This represents the normalized flow field value at the 3rd point of the five-point interpolation template. This represents the normalized flow field value at the 4th point of the five-point interpolation template. This represents the normalized flow field value at the 5th point of the five-point interpolation template.

[0083] In this embodiment of the invention, in S3, for each near-boundary non-uniform template, the corresponding first to fourth derivatives are derived by substituting the actual relative coordinates of each point into the Lagrange interpolation formula.

[0084] The first near-boundary non-uniform template order derivative for:

[0085] ;

[0086] in, Indicates the first Template No. The first derivative corresponds to the second The difference coefficients at each point This represents the five-point interpolation template. Normalized flow field values ​​at points.

[0087] In this embodiment of the invention, based on Lagrange interpolation, the first to fourth order derivative values ​​of the internal uniform template and the near-boundary non-uniform template at the midpoint of the template are derived in a dimensionless relative position space. The absolute values ​​of each derivative are taken to form a derivative feature vector with uniform dimension.

[0088] All derivative difference coefficients are derived in dimensionless relative position space. The eigenvalues ​​are not explicitly dependent on the physical grid spacing and can be adapted to different grid spacings.

[0089] The four-dimensional derivative eigenvectors are:

[0090] ;

[0091] in, This represents the first derivative of the Lagrange interpolation polynomial over the five-point interpolation template. This represents the second derivative of the Lagrange interpolation polynomial over the five-point interpolation template. This represents the third derivative of the Lagrange interpolation polynomial over the five-point interpolation template. It represents the fourth derivative of the Lagrange interpolation polynomial over the five-point interpolation template.

[0092] The derivative feature vectors calculated from all templates are input into the same pre-trained classification model, which outputs the discontinuity determination result. In this embodiment, a single-hidden-layer multilayer perceptron is preferred for the classification model. The model is trained only on the internal uniform template samples, and the boundary region is not included in the training. During inference, the near-boundary region is calculated entirely based on the real physical quantities on the physical grid, without any virtual point construction or extrapolation operations. Although the derivative difference coefficients of the internal template and the boundary template are different, both are strictly obtained by Lagrange interpolation and represent the "values ​​of each order of derivative at the center point of the template", with consistent physical semantics. Therefore, the classification model trained based on the derivative features of the internal template can be directly used for the derivative feature classification of the boundary template without retraining.

[0093] In this embodiment of the invention, in S5, for templates whose discontinuity determination result is discontinuity class, nonlinear weighted interpolation (multi-resolution weighted compact nonlinear interpolation is used in this embodiment) is applied to obtain the flow field value at the flux point; for templates whose discontinuity determination result is non-discontinuity class, linear Lagrange interpolation is applied to obtain the flow field value at the flux point.

[0094] Numerical simulations are performed using a time-progression approach based on the semi-discrete control equations until the predetermined computation termination time or convergence condition is reached. Specifically, any suitable time format can be used for time progression, such as the TVD-RK3 format.

[0095] The derivative feature vectors of all templates are input into the same pre-trained classification model, which outputs discontinuity determination results. The classification model only needs to be trained on the internal uniform template samples; no additional training samples are required for the boundary regions. Throughout the entire detection process, the near-boundary region is calculated entirely based on the flow field variables defined on the physical grid, without introducing any virtual point extrapolation operations.

[0096] In this embodiment of the invention, a dual Mach reflection example is used to test the effectiveness of the boundary-consistent discontinuity detection method and the global hybrid high-order conserved finite difference scheme of the present invention. For example... Figure 3 As shown, to verify the effectiveness of discontinuous detection near the boundary, the computational domain along... The region is intentionally divided into two sub-regions. The global hybrid higher-order conserved finite difference scheme of this invention is referred to as TML-WCNS in this example, and it is compared with the MMR-WCNS hybrid finite difference scheme. Figure 4 Shown in Intermittent detection results at different times: MMR-WCNS in smooth regions, near the upper boundary of the computational domain, and at grid block boundaries. The nearby system mistakenly identified smooth regions as discontinuous areas. In contrast, TML-WCNS effectively avoided misidentification of smooth regions and did not generate any false markers at boundaries or sub-region interfaces, verifying its discontinuity detection effect with consistent boundaries.

[0097] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A boundary uniform discontinuity detection method for globally conserved high-order finite difference schemes, characterized in that, Includes the following steps: S1. Divide the interpolation template within the computational domain; S2. Traverse all five-point interpolation templates in the domain according to the index order, extract the five-point flow field variables, and normalize the amplitude of the five-point flow field variables. S3. Using the normalized five-point flow field variables, calculate the first to fourth derivatives of the Lagrange interpolation polynomial on each five-point interpolation template, and generate a four-dimensional derivative eigenvector. S4. Input the four-dimensional derivative feature vectors of all five-point interpolation templates into the pre-trained classification model and output the discontinuity determination result. S5. Interpolate the discontinuity determination results; S6. Calculate the flux based on the interpolation results and perform numerical simulation based on the semi-discrete control equations until the set calculation termination time or convergence condition is reached.

2. The boundary uniform discontinuity detection method for globally conserved high-order finite difference schemes according to claim 1, characterized in that, In step S1, the interpolation template within the computational domain is divided into an internal uniform template and a near-boundary non-uniform template.

3. The boundary uniform discontinuity detection method for globally conserved high-order finite difference schemes according to claim 2, characterized in that, The near-boundary non-uniform template includes a left-boundary non-uniform template and a right-boundary non-uniform template; The non-uniform template at the left boundary includes a first template, a second template, and a third template; First template for ;in, Indicates the left endpoint of the computation domain. This represents the first solution point in the computational domain. This represents the second solution point in the computational domain. This represents the third solution point in the computational domain. This represents the 4th solution point in the computational domain; Second template for ;in, This represents the 5th solution point in the computational domain; The third template for ;in, This represents the 6th solution point in the computational domain; The right boundary non-uniform template and the left boundary non-uniform template are mirror-symmetric about the center of the computational domain.

4. The boundary uniform discontinuity detection method for globally conserved high-order finite difference schemes according to claim 1, characterized in that, The amplitude normalization of the five-point flow field variables is as follows: ； in, This represents the five-point interpolation template. Normalized flow field values ​​at points, This represents the five-point interpolation template. Flow field values ​​at each point This represents the flow field value at the first point of the five-point interpolation template. This represents the flow field value at the second point of the five-point interpolation template. This represents the flow field value at the 3rd point of the five-point interpolation template. This represents the flow field value at the 4th point of the five-point interpolation template. This represents the flow field value at the 5th point of the five-point interpolation template. This indicates taking the maximum value.

5. The boundary uniform discontinuity detection method for globally conserved high-order finite difference schemes according to claim 1, characterized in that, In S3, the first derivative of the Lagrange interpolation polynomial on the internal uniform template Second derivative Third derivative and fourth derivative They are respectively: ； ； ； ； in, This represents the normalized flow field value at the first point of the five-point interpolation template. This represents the normalized flow field value at the second point of the five-point interpolation template. This represents the normalized flow field value at the 3rd point of the five-point interpolation template. This represents the normalized flow field value at the 4th point of the five-point interpolation template. This represents the normalized flow field value at the 5th point of the five-point interpolation template.

6. The boundary uniform discontinuity detection method for globally conserved high-order finite difference schemes according to claim 1, characterized in that, In S3, the first of the near-boundary non-uniform templates order derivative for: ； in, Indicates the first Template No. The first derivative corresponds to the second The difference coefficients at each point This represents the five-point interpolation template. Normalized flow field values ​​at points.

7. The boundary uniform discontinuity detection method for globally conserved high-order finite difference schemes according to claim 1, characterized in that, The four-dimensional derivative eigenvector is: ； in, This represents the first derivative of the Lagrange interpolation polynomial over the five-point interpolation template. This represents the second derivative of the Lagrange interpolation polynomial over the five-point interpolation template. This represents the third derivative of the Lagrange interpolation polynomial over the five-point interpolation template. It represents the fourth derivative of the Lagrange interpolation polynomial over the five-point interpolation template.

8. The boundary uniform discontinuity detection method for globally conserved high-order finite difference schemes according to claim 1, characterized in that, In step S5, for templates whose discontinuity determination result is of the discontinuity class, nonlinear weighted interpolation is used to obtain the flow field value at the flux point; for templates whose discontinuity determination result is of the non-discontinuity class, linear Lagrange interpolation is used to obtain the flow field value at the flux point.

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