A high-precision interpolation method for flow field data in overlapping grid regions

Abstract

The present invention belongs to the technical field of fluid mechanics, and specifically discloses a high-precision interpolation method for flow field data in overlapping grid areas, comprising the following steps: cutting holes in overlapping grid areas, obtaining interpolation units at the same time, and removing non-nearest interpolation units in the interpolation units to obtain an interpolation interface; searching for contribution units of the interpolation interface to obtain an interpolation template of the interpolation interface; using bilateral interpolation to determine the flux of the interpolation interface according to the flow field variables of each interpolation point in the interpolation template; and using the flux of the interpolation interface as a boundary condition to perform flow field calculations. The present invention can simultaneously utilize the data of two grids, so that the width of the overlapping area of ​​the overlapping grids is reduced, solving the problem of difficulty in assembling overlapping grids when ensuring high interpolation accuracy in the prior art.

Description

Technical Field

[0001] The invention belongs to the technical field of fluid mechanics and relates to an interpolation method, in particular to a high-precision interpolation method for flow field data in an overlapping grid area. Background Art

[0002] It is still challenging to simulate some complex problems using computational fluid dynamics (CFD), such as when considering flows around multiple moving, deforming, and complex geometric objects. If a single mesh is used to discretize the flow region to handle the flow in the complex inner boundary region, mesh motion and mesh re-division will be involved. In recent years, the overlapping mesh method has provided a reliable solution for simulating flows around complex inner boundaries, especially high-speed compressible flows near multiple moving, deforming, and complex-shaped objects. In the calculation process of overlapping meshes, the flow field calculation needs to be exchanged in the overlapping areas of different meshes, which involves the interpolation process of the solution.

[0003] So far, the conventional interpolation strategies for overlapping grid areas almost all use the traditional finite difference method. This traditional interpolation strategy has a big defect, that is, the two overlapping grids must invade each other's area deeply enough to form an overlapping area of ​​considerable width to ensure that the interpolation is carried out normally. In the prior art, if bilinear interpolation is implemented on two layers of interpolation units, the width of the overlapping area between the two grids is very large. If the interpolation accuracy needs to be improved, the overlapping area will be widened several times, causing a sharp increase in the difficulty of grid generation and assembly. On the one hand, the reason is that multiple layers of interpolation objects are required, and on the other hand, a unit or vertex is not allowed to be both an interpolation object and provide interpolation support for other interpolation objects in a certain interpolation template. This defect makes the assembly of overlapping grids still rely more on human experience, which is not as simple as it seems, and often requires repeated modification of the grid to succeed. In addition, the interpolation processing of the discontinuous area of ​​the flow field is more complicated. Although there is no need to consider the discontinuity problem in the case of linear interpolation, when higher precision interpolation is required, simple Lagrangian interpolation cannot be used in the discontinuous area, and it is necessary to introduce WENO interpolation. WENO (Weighted Essentially Non-Oscillatory) interpolation is a numerical method that aims to improve the accuracy and stability of numerical calculations, especially when dealing with simulations with complex flow phenomena. For multi-dimensional problems, the complexity of WENO interpolation is much higher than that of one-dimensional problems, resulting in a significant increase in complexity. Summary of the invention

[0004] In view of the above-mentioned deficiencies in the prior art, the present invention provides a high-precision interpolation method for flow field data in overlapping grid areas, which solves the problem in the prior art that overlapping grids are difficult to assemble while ensuring high interpolation accuracy.

[0005] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is:

[0006] A high-precision interpolation method for flow field data in overlapping grid areas comprises the following steps:

[0007] S1, cutting holes in the overlapping grid area, obtaining interpolation units, and removing non-nearest interpolation units in the interpolation units to obtain an interpolation interface;

[0008] S2, searching for contribution units of the interpolation interface to obtain an interpolation template of the interpolation interface;

[0009] S3, using bilateral interpolation, according to the flow field variables of each interpolation point in the interpolation template, determine the flux of the interpolation interface;

[0010] S4. The flux of the interpolation interface is used as the boundary condition to perform flow field calculation.

[0011] The beneficial effects of the above scheme are:

[0012] (1) The number of interpolation supports in the present invention is greatly increased compared with the traditional strategy. On the one hand, bilateral interpolation can use the data of two grids at the same time, and on the other hand, it can use the grid center and the interface at the same time. Therefore, the number of supports per unit length is significantly increased, which is conducive to improving the interpolation accuracy.

[0013] (2) The bilateral interpolation of the present invention eliminates the ambiguity of interpolation, that is, given a point in the overlapping area, even if it is located in two grids at the same time, there is only one function value, which is conducive to reducing the conservation error.

[0014] (3) The bilateral interpolation of the present invention is combined with flux interpolation, which greatly relaxes the requirement on the width of the overlapping area. As long as the cutting boundaries can touch each other, it is sufficient.

[0015] (4) The flux interpolation of the present invention utilizes both the grid center and the interface numerical flux. Since the interface numerical flux has sufficient dissipation, there is no need to consider discontinuities during interpolation, which greatly simplifies the discontinuity processing in overlapping areas.

[0016] Further, step S2 specifically includes:

[0017] S21, searching for the opposite-side contribution unit of the interpolation interface in the opposite-side grid using ADT recursive search, and calculating the natural coordinates of the interpolation interface in the contribution unit according to Newton iteration method;

[0018] S22, determining the ipsilateral contributing unit of the interpolation interface in the ipsilateral grid;

[0019] S23, determining an interpolation template according to the contralateral contribution unit and the natural coordinates and the ipsilateral contribution unit.

[0020] The beneficial effects of the above further scheme are: ADT recursive search is used to process deeply nested structures, which is very suitable for the search scenario of contribution units, and the convergence rate of Newton iteration method is very fast, which can quickly calculate the natural coordinates of the opposite side.

[0021] Furthermore, step S3 specifically includes:

[0022] S31, calculating the interpolation basis function of each interpolation point in the interpolation template;

[0023] S32, constructing an interpolation matrix according to the interpolation basis function;

[0024] S33, calculating RBF interpolation weights;

[0025] S34. Calculate the flux of the interpolation interface according to the interpolation matrix and the RBF interpolation weights.

[0026] Further, in step S31, the interpolation basis function is a Gaussian basis function;

[0027] Gaussian basis function for:

[0028]

[0029] in, represents a natural constant, represents the standard deviation, express The coordinates of the point.

[0030] Further, in step S32, the interpolation matrix is ​​a symmetric matrix;

[0031] Interpolation Matrix for:

[0032]

[0033] in, represents the interpolation basis function, and , , Respectively Point and The coordinates of the point.

[0034] Furthermore, in step S33, the calculation formula used to calculate the RBF interpolation weight is:

[0035]

[0036] in, express The flux at the point, represents the RBF interpolation weight, represents the interpolation basis function, express The coordinates of the point, express The coordinates of the point.

[0037] The beneficial effects of the above further scheme are: bilateral interpolation is to use the grids of both sides for interpolation. At the same time, the interpolation of the flow field state (conserved variables or original variables) is modified to flux interpolation, so that the flux at the grid center and the flux on the interface can be used as interpolation support at the same time, effectively reducing the interpolation error. In addition, by directly interpolating the flux on the cutting boundary, the number of layers of interpolation objects (ghost units / points) on the boundary can be reduced.

[0038] Further, step S4 specifically includes:

[0039] S41, applying physical boundary conditions;

[0040] S42, interpolating the solution to obtain the flux of the interpolation interface;

[0041] S43, performing inter-process communication of the interpolated solution through MPI to exchange flow field information for different overlapping grid areas;

[0042] S44, using the Runge-Kutta method for time advancement and flow field calculation;

[0043] S45, repeating steps S41 to S44;

[0044] S46, completing the time advancement cycle, storing the flow field data, and completing the flow field calculation.

[0045] The beneficial effect of the above further scheme is that: through the time advancement cycle, the state of the flow field at different time steps can be simulated, so as to more accurately reflect the change process of the flow field. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 A schematic diagram of a flow chart of a high-precision interpolation method for flow field data in overlapping grid areas;

[0047] Figure 2 Schematic diagram of overlapping grid area for traditional interpolation method;

[0048] Figure 3 Schematic diagram of overlapping grid areas for bilateral interpolation;

[0049] Figure 4 Schematic diagram of linear interpolation of grid A and grid B;

[0050] Figure 5 Schematic diagram of bilateral interpolation of grid A and grid B. DETAILED DESCRIPTION

[0051] Now, exemplary embodiments of the present invention will be described in detail with reference to the accompanying drawings. It should be understood that the embodiments shown and described in the accompanying drawings are only exemplary and are intended to explain the principles and spirit of the present invention, rather than to limit the scope of the present invention.

[0052] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0053] like Figure 1 As shown, a high-precision interpolation method for flow field data in overlapping grid areas includes the following steps:

[0054] S1. Cut holes in the overlapping grid area, obtain interpolation units, and remove non-nearest interpolation units in the interpolation units to obtain an interpolation interface.

[0055] For example, after the hole cutting in step S1, several layers of cells or vertices in the overlapping mesh area may be marked as interpolation cells, where the interpolation cells are cells or vertices, depending on the spatial discretization mode, FVM (i.e., finite volume method) or FDM (i.e., finite difference method), etc.

[0056] Figure 2 Figure 1 is a schematic diagram of the overlapping grid area of ​​the traditional interpolation method. For example, the traditional high-order finite difference method uses a large numerical template and requires many interpolation units. For the commonly used second-order FVM method, two layers of interpolation units are generally set (such as Figure 2 The orange and light blue elements shown in the figure are used as valid elements to calculate the boundary conditions.

[0057] Figure 3 Schematic diagram of overlapping grid areas for bilateral interpolation. Since the method of using bilateral interpolation to interpolate boundary fluxes only requires that the cutting boundaries can touch each other, it is necessary to remove redundant interpolation units.

[0058] S2. Searching for contribution units of the interpolation interface to obtain an interpolation template of the interpolation interface.

[0059] In this embodiment, step S2 specifically includes:

[0060] S21, searching for the opposite-side contribution unit of the interpolation interface in the opposite-side grid using ADT recursive search, and calculating the natural coordinates of the interpolation interface in the contribution unit according to Newton iteration method;

[0061] S22, determining the ipsilateral contributing unit of the interpolation interface in the ipsilateral grid;

[0062] S23, determining an interpolation template according to the contralateral contribution unit and the natural coordinates and the ipsilateral contribution unit.

[0063] Exemplarily, the traditional contribution unit search refers to searching for the contribution unit of the interpolation unit in the opposite grid, and selecting the opposite contribution unit and several adjacent interpolation points (grid centers or vertices) as support to construct the interpolation template.

[0064] For example, Figure 2 As shown, Figure 2 B1, B2, B3 and B4 are used as interpolation units O A The contribution unit of O A interpolation template, and A1, A2, A3 and A4 are used as interpolation units O B The contribution unit of O B The interpolation template.

[0065] As for the bilateral interpolation proposed in the present invention, Figure 3 As shown, in step S21, the contribution unit of the interpolation interface in the opposite grid is found by using ADT recursive search. For the interpolation interface O of grid A, A For example, we can find its contribution unit B1 in grid B and calculate O using Newton iteration method. A The natural coordinates in the opposite side contribution unit B1. Then, find the position of the grid center A1 and other contribution interfaces of the contribution unit on the same side as the interpolation interface. Finally, use the interfaces and grid centers found on the same side and the opposite side to construct the interpolation template of the interpolation interface. Figure 3 The interpolation interface O of the grid A is shown A For example, the support units and support interfaces in the overlapping area together constitute a complete interpolation template.

[0066] S3. Using bilateral interpolation, the flux of the interpolation interface is determined according to the flow field variables of each interpolation point in the interpolation template.

[0067] In this embodiment, step S3 specifically includes:

[0068] S31, calculating the interpolation basis function of each interpolation point in the interpolation template;

[0069] S32, constructing an interpolation matrix according to the interpolation basis function;

[0070] S33, calculating RBF interpolation weights;

[0071] S34. Calculate the flux of the interpolation interface according to the interpolation matrix and the RBF interpolation weights.

[0072] In this embodiment, the number of contribution units and specific donor information involved in different interpolation schemes may be different. Figure 4 As shown, Figure 4 This is a schematic diagram of linear interpolation between grid A and grid B. Taking a one-dimensional problem as an example, Figure 4The interpolation process in is a simple linear interpolation, and the interpolation units of grid A and grid B interpolate from each other. The interpolation algorithm used can be a weighted interpolation based on the central variable of the contributing unit:

[0073]

[0074] in, The variable value representing the interpolation region, represents the variable value of the contributing unit, express The interpolation weight of the contributing unit relative to position I, represents the number of contribution units, satisfy .

[0075] exist Figure 4 In the interpolation process shown, both grid A and grid B only use the other grid cells to interpolate, which is called "unilateral interpolation". Grid A and grid B construct their own interpolation functions, which are the blue dotted line and the green dotted line respectively.

[0076] like Figure 5 As shown, Figure 5 is a schematic diagram of bilateral interpolation of grid A and grid B. In step S3, the interpolation method used is bilateral interpolation, and a single interpolation function can be constructed using the grid centers and interfaces in grid A and grid B, that is, Figure 5 For example, in Figure 5 If we want to interpolate the flux value of the interface at j=1 / 2 in grid B, f , the flux values ​​at the interface i=9 / 2, i=7 / 2 and the grid center i=4 in grid A and the flux values ​​at the interface j=3 / 2 and the grid center j=1 in grid B can be used to form an interpolation template. Therefore, the function interpolation based on RBF (radial basis function) can be used to introduce the kernel function to characterize the localized characteristics of the data. The specific method can be: substitute the coordinates of these 5 interpolation nodes and the function values ​​of these 5 interpolation nodes into the calculation of the RBF interpolation weights, and finally use the following calculation formula for calculation:

[0077]

[0078] in, express The flux at the point, represents the RBF interpolation weight, represents the interpolation basis function, express The coordinates of the point, express The coordinates of the point.

[0079] In this embodiment, the interpolation basis function may be a Gaussian basis function;

[0080] Gaussian basis function for:

[0081]

[0082] in, represents a natural constant, represents the standard deviation, express The coordinates of the point.

[0083] S4. The flux of the interpolation interface is used as the boundary condition to perform flow field calculation.

[0084] In this embodiment, step S4 specifically includes:

[0085] S41, applying physical boundary conditions;

[0086] S42, interpolating the solution to obtain the flux of the interpolation interface;

[0087] S43, performing inter-process communication of the interpolated solution through MPI to exchange flow field information for different overlapping grid areas;

[0088] S44, using the Runge-Kutta method for time advancement and flow field calculation;

[0089] S45, repeating steps S41 to S44;

[0090] S46, completing the time advancement cycle, storing the flow field data, and completing the flow field calculation.

[0091] In this embodiment, MPI (Message Passing Interface) is a programming model and communication protocol for writing parallel distributed applications. In the field of high-performance computing, MPI allows program processes on different processors to communicate with each other, exchange data or synchronize their operations. The interpolated solution is passed between processes through MPI as an intermediate result to ensure that all processes can work together to complete the overall computing task. In step S43, in order to ensure the accuracy of flow field calculation, in non-steady-state calculations, flow field information exchange is also performed at each pseudo-time step of each time step according to the time advancement method.

[0092] This embodiment uses the Runge-Kutta method to perform time-marching cycles, which can simulate the state of the flow field at different time steps, thereby more accurately reflecting the change process of the flow field. In addition, real-time data exchange in the overlapping grid area at each time step can ensure the continuity of the solution in the overlapping area.

[0093] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the present invention.

Claims

1. A high-precision interpolation method for flow field data in overlapping grid areas, characterized in that: The method comprises: S1, cutting holes in the overlapping grid area, obtaining interpolation units, and removing non-nearest interpolation units in the interpolation units to obtain an interpolation interface; S2, searching for contribution units of the interpolation interface to obtain an interpolation template of the interpolation interface; S3, using bilateral interpolation to determine the flux of the interpolation interface according to the flow field variables of each interpolation point in the interpolation template; S4, using the flux of the interpolation interface as a boundary condition to perform flow field calculation; The step S2 specifically includes: S21, using ADT recursive search to search for the opposite-side contribution unit of the interpolation interface in the opposite-side grid, and calculating the natural coordinates of the interpolation interface in the contribution unit according to Newton's iteration method; S22, determining the same-side contribution unit of the interpolation interface in the same-side grid; S23, determining the interpolation template according to the contralateral contribution unit and the natural coordinates and the ipsilateral contribution unit; The step S3 specifically includes: S31, calculating the interpolation basis function of each interpolation point in the interpolation template; S32, constructing an interpolation matrix according to the interpolation basis function; S33, calculating RBF interpolation weights; S34, calculating the flux of the interpolation interface according to the interpolation matrix and the RBF interpolation weights; The step S4 specifically includes: S41, applying physical boundary conditions; S42, interpolating the solution to obtain the flux of the interpolation interface; S43, performing inter-process communication of the interpolated solution through MPI to exchange flow field information for different overlapping grid areas; S44, using the Runge-Kutta method for time advancement and flow field calculation; S45, repeating steps S41 to S44; S46, completing the time advancement cycle, storing the flow field data, and completing the flow field calculation.

2. The method according to claim 1, characterized in that In the step S31, the interpolation basis function is a Gaussian basis function; The Gaussian basis function for: in, represents a natural constant, represents the standard deviation, express The coordinates of the point.

3. The method according to claim 1, characterized in that: In the step S32, the interpolation matrix is ​​a symmetric matrix; The interpolation matrix for: in, represents the interpolation basis function, and , , Respectively Point and The coordinates of the point.

4. The method according to claim 1, characterized in that: In step S33, the calculation formula used to calculate the RBF interpolation weights is: in, express The flux at the point, represents the RBF interpolation weight, represents the interpolation basis function, express The coordinates of the point, express The coordinates of the point.

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