A third-order compact reconstruction method based on finite volume method on unstructured grids

Abstract

This disclosure discloses a third-order compact reconstruction method based on the finite volume method on an unstructured mesh, comprising: constructing an unstructured mesh; reading the coordinates of the mesh nodes and the connection relationships between mesh elements in the unstructured mesh; calculating the integral average value of the reconstruction basis functions within the mesh elements based on the mesh node coordinates; calculating the average flow field according to the finite volume method; calculating the reconstruction relationship weighting coefficients based on the density distribution in the flow field; constructing a first-order derivative and second-order derivative relationship matrix based on the weighting coefficients, the integral average value of the reconstruction basis functions, the connection relationships between mesh elements, and the average flow field; constructing a system of equations to solve for the second-order derivative terms of the reconstruction polynomial, obtaining the second-order derivative terms of the reconstruction polynomial, substituting the second-order derivative terms of the reconstruction polynomial into the first-order derivative and second-order derivative relationship matrix, obtaining the first-order derivative terms of the reconstruction polynomial, and completing the reconstruction.

Description

Technical Field

[0001] This disclosure belongs to the field of computational fluid dynamics technology, specifically relating to a third-order compact reconstruction method based on the finite volume method on unstructured meshes. Background Technology

[0002] Computational fluid dynamics (CFD) is a method that uses numerical methods on a computer to discretize the governing equations and boundary conditions of fluid dynamics, thereby obtaining approximate numerical solutions to the flow field. Developing high-precision numerical discretization methods plays a crucial role in improving the accuracy of numerical solutions and further expanding the application scope of CFD. Unstructured meshes, due to their ability to handle complex geometries and achieve flexible automatic generation, have made unstructured mesh-based CFD methods a current research hotspot. The finite volume method (FMD) is the most widely used numerical method in CFD due to its advantages in automatically satisfying flow conservation properties and the upwind characteristics of the convection term. However, the finite volume method currently widely used on unstructured meshes only achieves second-order accuracy, which cannot meet the requirements of high-precision, high-resolution scenarios such as aeroacoustics, vortex-dominated flow, direct numerical simulation of turbulence, and large eddy simulation. Developing higher-order numerical methods is an effective way to solve related problems. Constructing a higher-order finite volume method based on unstructured meshes faces the following important challenges:

[0003] Unstructured meshes lack regular topological structures in the three spatial dimensions, making it impossible to construct high-order methods independently in each dimension like structured meshes. For the finite volume method, the core of constructing high-precision numerical methods on unstructured meshes lies in using the integral average of the reconstructed variables on the reconstruction template to construct high-order reconstruction polynomial functions of the reconstructed variables within the mesh cells. A pioneering work in this area is the k-order accurate reconstruction method, which sets the integral average of the high-order reconstruction function within the cell to the average value calculated by the finite volume method, and then uses the least squares method to find the coefficients of the k-order distribution function. Furthermore, some researchers have extended the weighted, essentially oscillatory, reconstruction method on structured meshes to unstructured meshes. However, since each mesh cell can only provide one reconstruction condition for the conservation of the integral average within the mesh cell, and the number of undetermined coefficients of the reconstruction polynomial increases rapidly with increasing dimension and order, both of these reconstruction methods—the k-order accurate reconstruction method and the weighted, essentially oscillatory, reconstruction method—suffer from the problem of excessively large reconstruction templates. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the purpose of this disclosure is to provide a third-order compact reconstruction method based on the finite volume method on unstructured meshes. This method does not require solving implicit algebraic equations during the reconstruction process, resulting in low computational cost and high solution efficiency. Furthermore, it considers boundary conditions during the reconstruction process, achieving consistent third-order reconstruction accuracy between boundary mesh elements and internal mesh elements.

[0005] To achieve the above objectives, this disclosure provides the following technical solutions:

[0006] A third-order compact reconstruction method based on the finite volume method on unstructured meshes includes the following steps:

[0007] S100: Construct an unstructured mesh;

[0008] S200: Read the coordinates of the grid nodes in the unstructured grid and the connection relationships between the grid cells;

[0009] S300: Calculate the integral average of the reconstructed basis functions within the grid cell based on the grid node coordinates;

[0010] S400: Calculate the mean flow field using the finite volume method;

[0011] S500: Calculates the reconstructed relation weighting coefficients based on the density distribution in the flow field;

[0012] S600: Construct the first-order and second-order derivative relationship matrices based on the weighting coefficients, the integral average of the reconstructed basis functions, the connection relationship between grid cells, and the average flow field;

[0013] S700: Construct the system of equations for the second derivative of the reconstructed polynomial, obtain the second derivative of the reconstructed polynomial, substitute the second derivative of the reconstructed polynomial into the matrix relating the first and second derivatives, obtain the first derivative of the reconstructed polynomial, and complete the reconstruction.

[0014] Preferably, in step S300, the integral average of the reconstructed basis functions within the grid cell is calculated using Gaussian integration:

[0015]

[0016] Among them, Ω i Let N be the volume of grid cell i, N be the number of Gaussian integration points, and ω be the value of ω. j Let the Gaussian integral weight be the weight of the j-th Gaussian integral point. Let i be the l-th reconstruction basis function of grid cell i. To reconstruct basis functions The integral average value within grid cell i, cor i j Let be the coordinates of the j-th Gaussian integration point in grid cell i.

[0017] Preferably, in step S500, the reconstructed relationship weight coefficient is calculated using the following formula:

[0018]

[0019] in, Here, N is the weight of the neighboring cells nb of mesh cell i, Nc is the number of cells adjacent to the face of mesh cell i, and ISS is the weight of the neighboring cells nb of mesh cell i. i j Let I be the smoothness of the j-th neighboring cell of grid cell i, and let I be the smoothness of each grid cell. i Calculated using the following formula:

[0020]

[0021] Among them, ISS i The smoothness of mesh cell i The average density within the j-th cell adjacent to grid cell i. denoted as the average density within grid cell i.

[0022] Preferably, in step S600, the relationship matrix between the first and second derivatives is represented as follows:

[0023]

[0024] in,

[0025]

[0026]

[0027]

[0028]

[0029]

[0030]

[0031] in, Let be the weighting coefficient for the reconstruction relationship within adjacent cells j of grid cell i. This represents the average value of the reconstructed variables within element i, calculated using the finite volume method. x is the average value of the reconstructed variables in element j adjacent to element i, calculated using the finite volume method. i，j，c Let y be the x-coordinate of the center of the j-th grid cell adjacent to grid cell i. i，j，c Let x be the ordinate of the center of the j-th grid cell adjacent to grid cell i, where j is 1-Nc, and Nc is the number of grid cells adjacent to grid cell i. i，c Let y be the x-coordinate of the center of grid cell i. i，c The ordinate of the center of grid cell i. Let f be the integral average of the function f within grid cell i.

[0032] remember, For AuxMat i , For AuxVec i .

[0033] Preferably, in step S700, the solution of the system of equations for the second derivative term of the reconstructed polynomial is expressed as follows:

[0034]

[0035] in,

[0036]

[0037]

[0038] in, Let AuxMat be the k-th row vector of the relation matrix AuxMat of the j-th cell adjacent to grid cell i. Let AuxMat be the k-th row vector of the relation matrix AuxMat for grid cell i. The k-th component of the relation matrix AuxVec of the j-th cell adjacent to grid cell i. Let x be the k-th component of the relation matrix AuxVec for grid cell i. i，j，c Let y be the x-coordinate of the center of the j-th cell adjacent to grid cell i. i，j，c x is the ordinate of the center of the j-th cell adjacent to grid cell i. i，c Let y be the x-coordinate of the center of grid cell i. i，c Let j be the ordinate of the center of grid cell i. The value of j ranges from 1 to Nc, and the value of k ranges from 1 to 2.

[0039] This disclosure also proposes a third-order compact reconstruction device based on the finite volume method on unstructured meshes, comprising:

[0040] Unstructured mesh generation module, used to generate unstructured meshes;

[0041] The preprocessing module is used to read the node coordinates and the connection relationships between grid cells in the unstructured mesh;

[0042] The integral calculation module is used to calculate the integral average value of the reconstructed basis function within the grid cell based on the grid node coordinates;

[0043] The finite volume method calculation module is used to calculate the average flow field based on the finite volume method.

[0044] The weighting coefficient calculation module is used to calculate the reconstructed relationship weighting coefficients based on the density distribution in the flow field.

[0045] The reconstruction relation matrix construction module is used to construct the first-order and second-order derivative relation matrices based on the weight coefficients, the integral average of the reconstruction basis functions, the connection relationship between grid cells, and the average flow field obtained by the finite volume method.

[0046] The module for solving the third-order reconstructed polynomial is used to solve the system of equations based on the second-order derivative terms of the constructed reconstructed polynomial, and to obtain the second-order derivative terms of the reconstructed polynomial. It is also used to substitute the second-order derivative terms of the reconstructed polynomial into the matrix relating the first-order derivative and the second-order derivative to obtain the first-order derivative terms of the reconstructed polynomial.

[0047] Preferably, the third-order reconstruction polynomial solving module includes:

[0048] The second derivative term solving submodule is used to solve the algebraic equation system of the second derivative term of the reconstructed polynomial in order to obtain the second derivative term of the reconstructed polynomial.

[0049] The first derivative term solving submodule is used to substitute the second derivative term of the reconstructed polynomial into the relation matrix between the first and second derivative terms to obtain the first derivative term of the reconstructed polynomial.

[0050] Compared with the prior art, the beneficial effects of this disclosure are as follows:

[0051] 1. Overcame the large template problem of high-order accuracy finite volume methods on unstructured meshes;

[0052] 2. It realizes the explicit solution of third-order reconstructed polynomials, which has less computation and higher solution efficiency compared with implicit solution methods;

[0053] 3. Boundary conditions were considered during the reconstruction process, achieving consistent third-order accuracy across the entire field. Attached Figure Description

[0054] Figure 1 This is a flowchart of a method provided in one embodiment of the present disclosure;

[0055] Figure 2 This is a schematic diagram of a two-dimensional unstructured mesh;

[0056] Figure 3 This is a diagram showing the correspondence between the coordinates of Gaussian integration points within the actual element and the coordinates of the reference element.

[0057] Figure 4 The graph shows the numerical accuracy results of the reconstruction method obtained through testing the test function.

[0058] Figure 5 This is the streamline distribution diagram of the cavity-driven flow with Re=400 calculated by this method. Detailed Implementation

[0059] The following will refer to the appendix. Figures 1 to 5Specific embodiments of this disclosure are described in detail. While specific embodiments of this disclosure are shown in the accompanying drawings, it should be understood that this disclosure may be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of this disclosure to those skilled in the art.

[0060] It should be noted that certain terms are used in the specification and claims to refer to specific components. Those skilled in the art will understand that different terms may be used to refer to the same component. This specification and claims do not distinguish components based on differences in terminology, but rather on differences in function. The terms "comprising" or "including" used throughout the specification and claims are open-ended and should be interpreted as "comprising but not limited to." The following descriptions are preferred embodiments for carrying out this disclosure; however, these descriptions are for the purpose of understanding the general principles of the specification and are not intended to limit the scope of this disclosure. The scope of protection of this disclosure is determined by the appended claims.

[0061] To facilitate understanding of the embodiments of this disclosure, further explanations and descriptions will be provided below with reference to the accompanying drawings and specific embodiments. The accompanying drawings do not constitute a limitation on the embodiments of this disclosure.

[0062] In one embodiment, such as Figure 1 As shown, a third-order compact reconstruction method based on the finite volume method on unstructured meshes includes the following steps:

[0063] S100: Construct an unstructured mesh;

[0064] S200: Read the coordinates of the grid nodes in the unstructured grid and the connection relationships between the grid cells;

[0065] S300: Calculate the integral average of the reconstructed basis functions within the grid cell based on the grid node coordinates;

[0066] S400: Calculate the mean flow field using the finite volume method;

[0067] S500: Calculates the reconstructed relation weighting coefficients based on the density distribution in the flow field;

[0068] S600: Construct the first-order and second-order derivative relationship matrices based on the weighting coefficients, the integral average of the reconstructed basis functions, the connection relationship between grid cells, and the average flow field;

[0069] S700: Construct the system of equations for the second derivative of the reconstructed polynomial, obtain the second derivative of the reconstructed polynomial, substitute the second derivative of the reconstructed polynomial into the matrix relating the first and second derivatives, obtain the first derivative of the reconstructed polynomial, and complete the reconstruction.

[0070] The above embodiments constitute the complete technical solution of this disclosure. Compared with the prior art, the technical solution described in the above embodiments has the following advantages: 1. It overcomes the large template problem of high-order accuracy finite volume method on unstructured mesh; 2. It realizes explicit solution of third-order reconstruction polynomial, which has small computational load and high solution efficiency compared with implicit solution method; 3. It considers boundary conditions in the reconstruction process and achieves consistent third-order accuracy across the entire field.

[0071] In another embodiment, in step S100, an unstructured mesh is generated using an unstructured mesh generation procedure, for example... Figure 2 The unstructured mesh shown contains two mesh cells, five mesh edges (one internal mesh edge and four boundary mesh edges), and four mesh nodes.

[0072] In another embodiment, in step S200, the coordinates of the grid nodes of the unstructured mesh are read, such as... Figure 2 In the unstructured mesh shown, the mesh nodes n1-n4 are numbered 1-4, the mesh cells E1 and E2 are numbered 1 and 2 respectively, and the mesh boundaries e1-e5 are numbered 1-5 respectively.

[0073] The coordinates of the grid nodes are:

[0074] n1: (0, 0)

[0075] n2: (1, 0)

[0076] n3: (0, 1)

[0077] n4: (1, 1)

[0078] The connection relationships of the grid cells include:

[0079] Internal grid edges: 2 3 1 2

[0081] Boundary grid edges: 4 3 2 0 3 1 1 0 1 2 1 0 2 4 2 0

[0086] The rules for cell connection relationships are as follows: The first two numbers represent the numbers of the two endpoints of the grid boundary, where the direction from the first point to the second point is the direction of the grid boundary; the last two numbers represent the numbers of the two grid cells connected by the grid boundary formed by the first two grid points, where the normal direction inside the grid boundary points to the grid cell represented by the third number, and the normal direction outside the grid boundary points to the grid cell represented by the fourth number. Note: Cells outside the computational domain are represented by 0.

[0087] In another embodiment, in step S300, according to the two-dimensional Taylor expansion:

[0088]

[0089] Among them, u i (x, y) is the reconstruction polynomial function within grid cell i, (x i，c y i，c ) represents the coordinates of the center of grid cell i. To reconstruct the polynomial u i The derivatives at the center of the grid cell.

[0090] Taking the average value of the integral of the reconstructed polynomial (1) within grid cell i, we get:

[0091]

[0092] Among them, Ω i Let i be the volume of grid cell i. Let be the integral average of the reconstructed polynomial within grid cell i.

[0093] Substituting equation (1) into equation (2), we get:

[0094]

[0095] in, Let f be the integral average of the function f within grid cell i.

[0096] Substitute equation (3) into equation (1) and replace u in equation (1). i (x i，c y i，c The reconstructed polynomial is obtained as follows:

[0097]

[0098] in, The integral average of the reconstructed variables within grid cell i. Let l be the undetermined coefficient of the reconstructed polynomial within grid cell i. Let i be the l-th reconstruction basis function of the reconstruction polynomial within grid cell i. Let be the integral average of the basis functions in grid cell i. Let x be a zero-mean basis function. i，c Let y be the x-coordinate of the center of grid cell i. i，c The ordinate of the center of grid cell i. Let f be the integral average of the function f within grid cell i.

[0099] The specific form of the reconstruction basis function within the grid cell is:

[0100]

[0101] The integral average of the reconstructed basis functions within each grid cell is calculated using Gaussian integrals:

[0102]

[0103] Among them, Ω i Let N be the volume of grid cell i, N be the number of Gaussian integration points, and ω be the value of ω. j Let the Gaussian integral weight be the weight of the j-th Gaussian integral point. Let i be the l-th reconstruction basis function of grid cell i. Its integral average value within grid cell i, cor i j Let be the coordinates of the j-th Gaussian integration point in grid cell i.

[0104] against Figure 3 Given a reference triangle formed by vertices 1′(0,0), 2′(1,0), and 3′(0,1), the mapping relationship between the actual mesh cells and the reference cells is as follows:

[0105]

[0106] Where (x1, y1), (x2, y2), and (x3, y3) are the physical coordinates of the three vertices 1, 2, and 3 of the actual unit, (x′, y′) are the coordinates of the Gaussian integration points a′, b′, c′, and d′ within the reference unit, and (x, y) are the coordinates of the corresponding Gaussian integration points a, b, c, and d within the actual unit.

[0107] The coordinates of the third-order Gaussian integration points within the reference cell, i.e., the integration weights, are shown in Table 1:

[0108] Table 1

[0109]

[0110] In another embodiment, in step S400, calculating the average flow field using the finite volume method is a commonly used method, which will not be described in detail here. The integral average value within the grid cell can be calculated using the finite volume method.

[0111] In another embodiment, in step S500, the reconstructed relation weight coefficient is calculated using the following formula:

[0112]

[0113] in, Let N be the weight of the neighboring cells nb of grid cell i, and Nc be the number of cells adjacent to the face of grid cell i. Let I be the smoothness of the j-th neighboring cell of mesh cell i, and let I be the smoothness of each cell. i The calculation formula is:

[0114]

[0115] Among them, ISS i The smoothness of element i, The integral average of the density within the j-th cell adjacent to cell i. The integral average value of the density within element i.

[0116] In another embodiment, in step S600, the integral average value of the cells adjacent to the i-th face of the mesh cell is conserved, that is:

[0117]

[0118] Where nb is the cell number adjacent to the i-th face of the mesh cell, Ω nb Let nb be the volume of the nb cells adjacent to the i-th face of the mesh cell. This is the integral average of the reconstructed variables within the nb cells adjacent to the i-th face of the mesh cell.

[0119] Based on the integral average of the weighting coefficients and the reconstructed basis functions, equation (10) can be further expressed as:

[0120]

[0121] in, The integral average of the reconstructed variables within grid cell i. The integral average of the reconstructed variables within grid cells nb adjacent to grid cell i. Let (x) be the l-th undetermined coefficient of the reconstructed polynomial within grid cell i. i，c y i，c (x) represents the center coordinates of grid cell i, and (x) represents the center coordinates of grid cell i. nb，c y nb，c ) represents the center coordinates of grid cell nb. Let f be the average integral of function f within grid cell i. Let f be the integral average of the function f within grid cell nb.

[0122] Based on the connection relationships between mesh cells read in step S200, all mesh boundaries are scanned to obtain the system of algebraic equations constructed from the conservation relationship of the integral average value of the reconstructed variables:

[0123]

[0124] in,

[0125]

[0126]

[0127]

[0128]

[0129]

[0130]

[0131] in, Let be the weighting coefficient for the reconstruction relationship within adjacent cells j of grid cell i. This represents the average value of the reconstructed variables within element i, calculated using the finite volume method. x is the average value of the reconstructed variables in element j adjacent to element i, calculated using the finite volume method. i，j，c Let y be the x-coordinate of the center of the j-th grid cell adjacent to grid cell i. i，j，c Let x be the ordinate of the center of the j-th grid cell adjacent to grid cell i, where j is 1-Nc, and Nc is the number of grid cells adjacent to grid cell i. i，c Let y be the x-coordinate of the center of grid cell i. i，c The ordinate of the center of grid cell i. Let f be the integral average of the function f within grid cell i.

[0132] For boundary elements, boundary conditions also need to be considered during the reconstruction process. The general form of boundary conditions is as follows:

[0133]

[0134] Among them, u i，b c1, c2, and c3 are the values ​​of the reconstruction variable u at the boundary point b on the boundary unit i. c1, c2, and c3 are the control parameters of the general boundary conditions, which are determined according to the boundary condition type. For example, if the reconstruction variable u = U is specified on a certain boundary, then c1 = 0, c2 = 1, and c3 = U in the reconstruction process of the unit i adjacent to that boundary.

[0135] Since this disclosure provides the reconstruction relationship through boundary conditions during the reconstruction process, it is not necessary to reduce the order of the reconstruction polynomial within the boundary element. Therefore, it can achieve the same third-order reconstruction accuracy for both the boundary element and the internal element.

[0136] Based on the algebraic equations constructed above, the relationship between the first and second derivatives is expressed as follows:

[0137]

[0138] in,

[0139]

[0140]

[0141]

[0142]

[0143]

[0144]

[0145] in, Let be the weighting coefficient for the reconstruction relationship within adjacent cells j of grid cell i. This represents the average value of the reconstructed variables within element i, calculated using the finite volume method. x is the average value of the reconstructed variables in element j adjacent to element i, calculated using the finite volume method. i，j，c Let y be the x-coordinate of the center of the j-th grid cell adjacent to grid cell i. i，j，c Let x be the ordinate of the center of the j-th grid cell adjacent to grid cell i, where j is 1-Nc, and Nc is the number of grid cells adjacent to grid cell i. i，c Let y be the x-coordinate of the center of grid cell i. i，c The ordinate of the center of grid cell i. Let f be the integral average of the function f within grid cell i.

[0146] remember,

[0147] The relationship between the first and second derivatives can also be expressed as:

[0148]

[0149] For the traditional high-order finite volume method, reconstruction is performed only by utilizing the conservation relation (10) of the integral average value of the reconstruction variables within the reconstruction unit. For example, in this case, the two-dimensional third-order reconstruction polynomial (4) has five undetermined coefficients. The integral mean conservation relation (10) of the reconstruction variable needs to be applied to at least five grid cells. However, for a two-dimensional triangular grid, each grid cell has only three adjacent grid cells. Therefore, information from non-surface adjacent grid cells is needed during the reconstruction process, meaning that the reconstruction process of the traditional high-order finite volume method is not compact. This invention not only provides the integral mean conservation condition of equation (10) within face adjacent cells, but also provides the conservation condition of the first derivative of the reconstruction polynomial, so that the third-order reconstruction can be completed using only the information from face adjacent cells. The specific steps are described in the following embodiments.

[0150] In another embodiment, in step S700, a system of algebraic equations for solving the second derivative terms is first constructed from the conservation relation of the first derivative of the reconstructed polynomial.

[0151] The first derivative of the reconstructed polynomial (4) with respect to x is as follows:

[0152]

[0153] By analogy with the conservation relation of the integral mean of the reconstructed variables (10), we can obtain the conservation relation of the first derivative of the reconstructed polynomial with respect to x:

[0154]

[0155] Right now:

[0156]

[0157] Where nb is the cell number adjacent to grid cell i. Let (x) be the j-th undetermined coefficient of the reconstructed polynomial within element i. i，c y i，c (x) represents the center coordinates of grid cell i, and (x) represents the center coordinates of grid cell i. nb，c y nb，c ) represents the center coordinates of the nb cells adjacent to grid cell i.

[0158] Combining the first-order derivative and second-order derivative relationship matrix (15) obtained in step S600, equation (18) can be transformed into:

[0159]

[0160] in, Reconstruct the j-th row vector of matrix AuxMat for element i. Reconstruct the j-th component of the matrix AuxVec for element i.

[0161] because and Since all are unknown, the system of equations composed of equation (19) often requires solving all elements in the field simultaneously, i.e., implicit solution. This solution method has a large amount of computation and low parallel solution efficiency. To solve this problem, the present invention explicitly implements the reconstruction solution process through the following method.

[0162] The second derivative term in equation (1) can be approximated by Taylor's expansion, for example:

[0163]

[0164] in, Reconstruct the second-order cross derivative of the polynomial at the center of grid cell i. Reconstruct the second-order cross derivative of the polynomial at the center of the adjacent cell nb of grid cell i, (x i，c y i，c (x) represents the center coordinates of grid cell i, and (x) represents the center coordinates of grid cell i. nb，c y nb，c (x) represents the center coordinates of the nb cells adjacent to grid cell i, and (x) represents the center coordinates of the nb cells adjacent to grid cell i. nb，c -x i，c ) and (y nb，c -y i，c ) represents the grid size h.

[0165] Right now:

[0166]

[0167] Similarly, for the other two second derivative terms:

[0168]

[0169] Substituting equations (21) and (22) into equation (1), we get:

[0170]

[0171] In grid cell i (xx) i，c )∝h,(yy i，c Therefore, expanding equation (23) yields:

[0172]

[0173] remember,

[0174]

[0175] For the original polynomial u i The approximation of (x, y) can be expressed as equation (24):

[0176]

[0177] That is, using approximate u i If (x, y) is used, then the third-order reconstruction accuracy is still maintained. Therefore, it can be concluded that making the coefficients of the highest-order terms of the reconstruction polynomials in adjacent grids equal does not affect the reconstruction accuracy.

[0178] Then let the highest-order terms of adjacent reconstructed polynomials, i.e., the second-order derivative terms, be equal, that is,

[0179]

[0180] Therefore, equation (19) can be transformed into:

[0181]

[0182] Similarly, we obtain the conservation relation for the first derivative of the reconstructed polynomial with respect to y:

[0183]

[0184] Scanning all internal mesh edges yields a system of algebraic equations satisfied by the second derivative terms:

[0185]

[0186] in,

[0187]

[0188]

[0189] in, Let AuxMat be the k-th row vector of the relation matrix AuxMat of the j-th cell adjacent to grid cell i. Let AuxMat be the k-th row vector of the relation matrix AuxMat for grid cell i. The k-th component of the relation matrix AuxVec of the j-th cell adjacent to grid cell i. Let x be the k-th component of the relation matrix AuxVec for grid cell i. i，j，c Let y be the x-coordinate of the center of the j-th cell adjacent to grid cell i. i，j，c x is the ordinate of the center of the j-th cell adjacent to grid cell i. i，c Let y be the x-coordinate of the center of grid cell i. i，c Let j be the ordinate of the center of grid cell i. The value of j ranges from 1 to Nc, and the value of k ranges from 1 to 2.

[0190] Solve the above system of algebraic equations to obtain the second derivative terms. Substituting the first and second derivative relation matrix obtained in step S600, the first derivative term of the reconstructed polynomial is obtained. Refactoring complete.

[0191] The flow field calculation results based on the method described in this disclosure are given below.

[0192] Unstructured meshes of 10×10, 14×14, 20×20, 28×28, and 40×40 are generated within the computational domain (0,1)×(0,1), and the meshes are generated using an exponential function f1 = x. 2 e y As test function 1, with complex functions As test function 2, the test function is reconstructed using this reconstruction method. The difference between the reconstructed variable value at the midpoint of the grid edge and the accurate value is taken as the reconstruction error, according to the error definition:

[0193]

[0194] Where Error is the numerical error, h is the grid size, and n is the precision order.

[0195] As can be seen from the above formula, the slope of the logError-logh curve represents the accuracy of the numerical method. Figure 4 The graph shows the numerical accuracy results of the reconstruction method obtained through testing the test function. Figure 4 In the figure, the average slope of the reconstruction performed by the method described in this disclosure is 3.01, indicating that the method described in this disclosure does indeed have third-order accuracy. Compared with the classical second-order center method, Figure 4 In the figure, the average slope of the logError-logh curve reconstructed by the second-order central scheme is 2.00, which means that the second-order central scheme has second-order reconstruction accuracy. Furthermore, for test function 1, to achieve the reconstruction error of the reconstruction method of the present invention on a 10×10 grid, the second-order central scheme requires a grid of 71×71. For test function 2, to achieve the reconstruction error of the reconstruction method of the present invention on a 10×10 grid, the second-order central scheme requires a grid of 26×26. Therefore, the method of the present invention has higher accuracy than the classical second-order central scheme and requires fewer grids to achieve the same residual calculation.

[0196] Numerical calculations of the driven flow in an incompressible square cavity at a Reynolds number (Re) of 400 were performed using this reconstruction method. The computational domain (0, 1) × (0, 1) was divided into an 80 × 80 unstructured mesh. All four edges of the square cavity were non-slip boundaries, with the upper boundary being the driving edge. The velocity distribution in the x-direction was set as: u(x, 1) = 16x 2 (1-x) 2 All other boundary velocities are 0. Figure 5The streamline distribution diagram of the cavity-driven flow at Re=400 was calculated using this method. The diagram shows that the method of this invention can clearly capture the large eddy in the middle of the flow field and the two smaller eddies at the bottom corners. Table 2 shows a comparison between the vortex center coordinates calculated using this method and the results from the literature, as follows:

[0197] Table 2

[0198]

[0199] The calculation results of the spectral high-precision algorithm for this example show that the center of the large eddy is located in the range of (0.565-0.578, 0.616-0.629), the center of the small eddy at the left bottom corner is located in the range of (0.038-0.048, 0.038-0.043), and the center of the small eddy at the right bottom corner is located in the range of (0.897-0.902, 0.103-0.114). The calculation results of this method show that the center of the large eddy is (0.578, 0.617), the center of the small eddy at the right bottom corner is (0.043, 0.040), and the center of the small eddy at the right bottom corner is (0.901, 0.113). In summary, it is evident that this disclosure can effectively identify the morphology and location of eddies in separated flows.

[0200] In summary, through the three examples with different complexities, the accuracy and effectiveness of the third-order compact reconstruction method based on the finite volume method on unstructured meshes disclosed in this paper are verified, demonstrating its feasibility and necessity for widespread application.

[0201] The technical solutions provided by this disclosure have been described in detail above with reference to specific embodiments. However, the descriptions of the embodiments above are only for the purpose of helping to understand the core ideas of this disclosure. For those skilled in the art, there will be changes in specific implementation methods and application scope based on the ideas of this disclosure. Therefore, the content of this specification should not be construed as a limitation of this disclosure.

Claims

1. A third-order compact reconstruction method based on the finite volume method on unstructured meshes, comprising the following steps: S100: Construct an unstructured mesh; S200: Read the coordinates of the grid nodes in the unstructured grid and the connection relationships between the grid cells; S300: Calculate the integral average of the reconstructed basis functions within the grid cell based on the grid node coordinates; S400: Calculate the mean flow field using the finite volume method; S500: Calculates the reconstructed relation weighting coefficients based on the density distribution in the flow field; S600: Construct the first-order and second-order derivative relationship matrices based on the weighting coefficients, the integral average of the reconstructed basis functions, the connection relationship between grid cells, and the average flow field; S700: Construct the second derivative term of the reconstructed polynomial and solve the system of equations to obtain the second derivative term of the reconstructed polynomial. Substitute the second derivative term of the reconstructed polynomial into the first derivative and second derivative relation matrix to obtain the first derivative term of the reconstructed polynomial, thus completing the reconstruction. In step S600, the relationship matrix between the first and second derivatives is represented as follows: ， in, ， ， in, Let be the weighting coefficient for the reconstruction relationship within adjacent cells j of grid cell i. This represents the average value of the reconstructed variables within element i, calculated using the finite volume method. This represents the average value of the reconstructed variables within element j adjacent to element i, calculated using the finite volume method. Let x be the x-coordinate of the center of the j-th grid cell adjacent to grid cell i. Let j be the ordinate of the center of the j-th grid cell adjacent to grid cell i, where j is 1-Nc, and Nc is the number of grid cells adjacent to grid cell i. Let x be the x-coordinate of the center of grid cell i. The ordinate of the center of grid cell i. The integral average of function f within grid cell i; remember, for , for .

2. The method according to claim 1, wherein, In step S300, the integral average of the reconstructed basis functions within the grid cell is calculated using Gaussian integration: ， in, Let N be the volume of grid cell i, and N be the number of Gaussian integration points. Let the Gaussian integral weight be the weight of the j-th Gaussian integral point. Let i be the l-th reconstruction basis function of grid cell i. To reconstruct basis functions The integral average value within grid cell i, Let be the coordinates of the j-th Gaussian integration point in grid cell i.

3. The method according to claim 1, wherein, In step S500, the reconstructed relation weight coefficients are calculated using the following formula: ， in, Let N be the weight of the neighboring cells nb of grid cell i, and Nc be the number of cells adjacent to the face of grid cell i. Let I be the smoothness of the j-th neighboring cell of grid cell i, and let I be the smoothness of each grid cell. i Calculated using the following formula: ， Among them, ISS i The smoothness of mesh cell i The average density within the j-th cell adjacent to grid cell i. denoted as the average density within grid cell i.

4. The method according to claim 1, wherein, In step S700, the solution of the system of equations for the second derivative term of the reconstructed polynomial is expressed as follows: ， in, ， ， in, The relation matrix of the j-th cell adjacent to grid cell i. The k-th row vector, The relation matrix of grid cell i The k-th row vector, The relation matrix of the j-th cell adjacent to grid cell i. The kth component, The relation matrix of grid cell i The kth component, Let x be the x-coordinate of the center of the j-th cell adjacent to grid cell i. The ordinate of the center of the j-th cell adjacent to grid cell i is given. Let x be the x-coordinate of the center of grid cell i. Let j be the ordinate of the center of grid cell i, j ranges from 1 to Nc, and k ranges from 1 to 2.

5. A third-order compact reconstruction device based on the finite volume method on an unstructured mesh, comprising: Unstructured mesh generation module, used to generate unstructured meshes; The preprocessing module is used to read the node coordinates and the connection relationships between grid cells in the unstructured mesh; The integral calculation module is used to calculate the integral average value of the reconstructed basis function within the grid cell based on the grid node coordinates; The finite volume method calculation module is used to calculate the average flow field based on the finite volume method. The weighting coefficient calculation module is used to calculate the reconstructed relationship weighting coefficients based on the density distribution in the flow field. The reconstruction relation matrix construction module is used to construct the first-order and second-order derivative relation matrices based on the weight coefficients, the integral average of the reconstruction basis functions, the connection relationship between grid cells, and the average flow field obtained by the finite volume method. The module for solving the third-order reconstructed polynomial is used to solve the system of equations based on the second derivative term of the constructed reconstructed polynomial, and to obtain the second derivative term of the reconstructed polynomial. It is also used to substitute the second derivative term of the reconstructed polynomial into the matrix relating the first and second derivatives to obtain the first derivative term of the reconstructed polynomial. In the reconstruction relation matrix construction module, the first-order derivative and second-order derivative relation matrix is ​​represented as follows: ， in, ， ， in, Let be the weighting coefficient for the reconstruction relationship within adjacent cells j of grid cell i. This represents the average value of the reconstructed variables within element i, calculated using the finite volume method. This represents the average value of the reconstructed variables within element j adjacent to element i, calculated using the finite volume method. Let x be the x-coordinate of the center of the j-th grid cell adjacent to grid cell i. Let j be the ordinate of the center of the j-th grid cell adjacent to grid cell i, where j is 1-Nc, and Nc is the number of grid cells adjacent to grid cell i. Let x be the x-coordinate of the center of grid cell i. The ordinate of the center of grid cell i. The integral average of function f within grid cell i; remember, for , for .

6. The apparatus according to claim 5, wherein, The third-order reconstructed polynomial solving module includes: The second derivative term solving submodule is used to solve the algebraic equation system of the second derivative term of the reconstructed polynomial in order to obtain the second derivative term of the reconstructed polynomial. The first derivative term solving submodule is used to substitute the second derivative term of the reconstructed polynomial into the relation matrix between the first and second derivative terms to obtain the first derivative term of the reconstructed polynomial.

Images