A 3D high-order grid generation method based on solving partial differential equations

Abstract

The present invention discloses a three-dimensional high-order grid generation method based on partial differential equation solution. At present, generating high-precision and high-quality high-order grids is still a difficult problem. The present invention first generates a linear coarse grid based on the three-dimensional geometric model of the aircraft wing and body, divides the surface grid, and establishes a mapping relationship between the linear coarse grid boundary and the boundary of the three-dimensional geometric model of the aircraft wing and body; then, the linear coarse grid is upgraded to obtain an upgraded linear grid; secondly, the upgraded surface grid patch is projected onto the corresponding three-dimensional surface of the aircraft wing and body, and on the corresponding parameter domain, with the boundary offset vector as the constraint condition, the partial differential equation that controls the internal grid offset vector field is established and solved to obtain the upgraded surface grid point offset vector; with the upgraded tetrahedral grid boundary offset vector as the constraint condition, the partial differential equation that controls the internal grid point offset vector field is established and solved to offset all grid points, thereby obtaining a high-order grid.

Description

Technical Field

[0001] The present invention relates to a high-precision grid generation process in the pre-processing of numerical simulation field, and in particular to a three-dimensional high-order grid generation method based on solving partial differential equations. Background Art

[0002] Mesh generation is a pre-processing process in numerical simulation technology, including the finite element method, finite volume method and finite difference method. This process divides the continuous geometric area into a finite number of basic geometric shapes, which are also called mesh units, such as triangular units, quadrilateral units, tetrahedral units and hexahedral units. The number and quality of mesh units directly affect the accuracy and efficiency of numerical calculations. In order to meet the needs of high-precision simulation of complex models, large-scale mesh generation is usually adopted in industry. Although this brings huge time and computing resource overhead, there is still a loss of geometric accuracy. A good way to solve this problem is to use high-order numerical methods with high-order meshes as input for numerical simulation.

[0003] Compared to low-order numerical methods, high-order numerical methods can effectively reduce computational errors and improve computational accuracy. While maintaining the same computational accuracy, they also require a lower mesh size, thereby improving computational efficiency. To fully exploit the advantages of high-order numerical methods, the straight-edge meshes used in low-order numerical methods must be replaced with curved-edge meshes, which must approximate the geometry at its boundaries. This type of mesh is called a high-order mesh, and its elements are called high-order elements.

[0004] However, generating high-order meshes is not an easy task. High-order elements need to be represented by quadratic, cubic or even higher-order geometric curves and surfaces, which increases the difficulty of generation. In addition, compared with linear meshes, high-order meshes are more prone to low-quality mesh elements such as distortion and intersecting elements, which affect subsequent numerical analysis. At present, generating high-precision, high-quality high-order meshes is still a difficult problem. Most existing high-order mesh generation methods rely on global iterative solutions, which consume a lot of time and computing resources and are difficult to adapt to the large-scale high-order mesh generation needs in industry. Summary of the Invention

[0005] In order to meet the high-order grid generation requirements of three-dimensional geometric models, the present invention provides a three-dimensional high-order grid generation method based on solving partial differential equations. Under the condition of the same grid accuracy, the number of high-order grids is less than the number of linear grids. High-order grids are generated for three-dimensional geometric models to reduce grid calculation time and the use of computing resources.

[0006] In a first aspect, the present invention provides a three-dimensional high-order grid generation method based on solving partial differential equations, which comprises the following steps:

[0007] Step 1: Based on the target 3D geometric model, a linear coarse grid is generated, and a mapping relationship between the linear coarse grid boundary and the 3D geometric model boundary is established in sequence.

[0008] Step 2: Upgrade the order of each volume mesh element in the linear coarse mesh to obtain an upgraded tetrahedral mesh.

[0009] Step 3: Establish a two-dimensional offset vector field optimization model.

[0010] Step 4: Solve the two-dimensional offset vector field optimization model established in step 3 to obtain each surface S i The offset vector field of the corresponding triangular patch mesh on the parameter domain; the parameter domain and the surface S i There is a mapping relationship, calculate the parameter offset vector of the grid point on the triangular patch mesh on the parameter domain; offset the grid point on the parameter domain according to the parameter offset vector; map the offset parameter coordinates back to the surface S i , calculate the grid point surface S i The new physical coordinates on the triangle patch grid point are used as the offset coordinates of the triangle patch grid point to obtain the physical offset vector of the triangle patch grid point. Traverse all surfaces to obtain the physical offset vectors of all triangle patch grid points.

[0011] Step 5: Establish a three-dimensional offset vector field optimization model:

[0012]

[0013] The constraints of the three-dimensional offset vector field optimization model are:

[0014]

[0015] Where Ω represents the inner region of the raised-order tetrahedral mesh, is the physical offset vector of the grid point p located at the boundary of the raised-order tetrahedral mesh, express The 2-norm of the gradient, and Respectively represent the Dirichlet boundary information and Neumann boundary information in the raised-order tetrahedral mesh boundary, and n represents the Neumann boundary The outward unit normal of the upper grid point p, The physical offset vector representing the mesh point p on the boundary of the raised-order tetrahedral mesh to the boundary of the 3D geometric model, express Partial derivative with respect to the outward unit normal n.

[0016] Step 6: Solve the three-dimensional offset vector field optimization model to obtain the optimal offset vector field of the raised-order tetrahedral mesh; then offset each grid point of the raised-order tetrahedral mesh according to the optimal offset vector field to obtain a high-order mesh.

[0017] Preferably, in step 5, the optimal offset vector field solution of the three-dimensional offset vector field optimization model is equivalent to the solution of the following Laplace boundary value problem:

[0018]

[0019] The Laplace boundary value problem is solved by the boundary element method.

[0020] As a preference, in step 6, any internal point p of the raised tetrahedral mesh i The physical offset vector Solved by the following boundary integral formula:

[0021]

[0022] Where Ω represents the inner region of the raised-order tetrahedral mesh, Represents the boundary of the raised-order tetrahedral mesh. p j represents a field point, which is any grid point on the boundary of the raised-order tetrahedral grid. Represents point p j The differential of the neighborhood, G(p i ,p j ) represents the basic solution of the Laplace equation Δu(p)=0, F(p i ,p j ) represents G(p i ,p j )’s normal partial derivative, in three dimensions, G(p i ,p j ) and F(p i ,p j ) is defined as follows:

[0023]

[0024] Where n represents the field point p j The outward unit normal at the source point p i and field point p j The distance between them.

[0025] As a preference, in step 6, when the Dirichlet boundary condition of each point on the boundary surface element of the raised-order tetrahedral mesh is and Neumann boundary condition q(p j ) When one conditional value is known, the remaining unknown conditional value can be obtained by the following integral equation:

[0026]

[0027] Among them, the source point p i With field point p j are located on the boundary surface elements of the raised-order tetrahedral mesh, and the constant c(p i ) represents the source point p i The smoothness of the neighborhood.

[0028] Preferably, the process of establishing the mapping relationship between the linear coarse mesh boundary and the three-dimensional geometric model boundary in step 1 is as follows:

[0029] (1) The boundary of the linear coarse mesh is divided into several patches, and the boundary of the three-dimensional geometric model is divided into several surfaces. A one-to-one mapping relationship is established between the patches and the surfaces.

[0030] (2) A single face is divided into several face units, and a single surface is divided into several surface blocks. A one-to-one mapping relationship is established between the face units and the surface blocks.

[0031] (3) The boundary of a single face is divided into several line units, and the boundary of a single surface is divided into several curve segments. A one-to-one mapping relationship is established between line units and curve segments.

[0032] Preferably, the specific process of step 2 is: adding new grid points in the line units, surface units and body of each volume grid unit as high-order points, and expressing each surface unit in the form of a high-order geometric surface through the Lagrange interpolation method to obtain an elevated tetrahedral grid.

[0033] Preferably, the two-dimensional offset vector field optimization model in step 3 is as follows:

[0034]

[0035] The constraints of the two-dimensional offset vector field optimization model are:

[0036]

[0037] Where V represents the inner area of the triangular patch mesh in the parameter domain, is the parameter offset vector of the grid point p on the triangular patch mesh, express The 2-norm of the gradient, and Respectively represent the Dirichlet boundary information and Neumann boundary information in the patch boundary, and n represents the Neumann boundary The outward unit normal of the upper grid point p, The grid point p representing the boundary of the triangular patch mesh to the surface S i The offset vector of the boundary, express Partial derivative with respect to the outward unit normal n.

[0038] In a second aspect, the present invention provides a method for simulating and collecting aerodynamic indicators of an aircraft wing and body, comprising the following steps:

[0039] (1) Construct a three-dimensional model of the wing and body of the aircraft under test.

[0040] (2) According to the aforementioned three-dimensional high-order grid generation method, a three-dimensional high-order grid of the aircraft wing body is generated.

[0041] (3) The obtained three-dimensional high-order mesh is imported into industrial simulation software. The industrial simulation software uses the high-precision geometric representation of the three-dimensional high-order mesh to perform numerical simulation and calculate the physical characteristics of the aircraft wing-body structure analysis; the physical characteristics include stiffness and strength.

[0042] In a third aspect, the present invention provides a computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the memory stores the computer program; and the processor executes the aforementioned three-dimensional high-order grid generation method based on solving partial differential equations.

[0043] In a fourth aspect, the present invention provides a readable storage medium storing a computer program; when the computer program is executed by a processor, it is used to implement the aforementioned three-dimensional high-order grid generation method based on solving partial differential equations.

[0044] The beneficial effects of the present invention are:

[0045] The present invention establishes a high-order offset vector field physical model covering the three-dimensional geometric model area of the aircraft wing and body, and uses the high-order boundary element method for numerical solution, effectively improving the accuracy of the offset vector field. Subsequently, the positions of all grid points are updated according to the offset vector field, and a high-order grid is generated through interpolation fitting. This achieves the generation of a high-order tetrahedral grid with a smaller number of grids while maintaining the same grid accuracy in the context of the aircraft wing and body model. This invention addresses the needs of high-precision numerical simulation and can be used to analyze three-dimensional wing and body geometric structures, generate corresponding high-order grids, and is equally applicable to other complex three-dimensional geometric models. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 Schematic diagram of the boundary of the three-dimensional geometric model of the aircraft wing and body in the present invention.

[0047] Figure 2 Schematic diagram of the linear tetrahedron mesh covering the three-dimensional geometric model of the aircraft wing and body in the present invention.

[0048] Figure 3 This is the 50th surface diagram of the three-dimensional geometric model of the aircraft wing and body in the present invention.

[0049] Figure 4 This is a schematic diagram of the 50th triangular patch mesh corresponding to the 50th curved surface of the aircraft wing and body three-dimensional geometric model according to the present invention.

[0050] Figure 5 for Figure 2 Schematic diagram of the raised tetrahedral mesh obtained by raising the linear tetrahedral mesh.

[0051] Figure 6 for Figure 4 Schematic diagram of a parametric triangle mesh of a triangular patch mesh on a corresponding surface.

[0052] Figure 7 for Figure 6 Schematic diagram of the offset vector field of the boundary line elements of the parametric triangular mesh.

[0053] Figure 8 for Figure 6 Schematic diagram of the offset vector field inside the parametric triangular mesh.

[0054] Figure 9 is the offset vector field diagram of the raised-order tetrahedral boundary surface element.

[0055] Figure 10(a) shows Figure 9 A partial enlarged view of part A.

[0056] Figure 10(b) shows Figure 9 A partial enlarged view of part B.

[0057] Figure 11 is the offset vector field diagram inside the raised-order tetrahedron.

[0058] Figure 12(a) shows Figure 11 A partial enlarged view of part A.

[0059] Figure 12(b) shows Figure 11 A partial enlarged view of part B.

[0060] Figure 13 The present invention is aimed at Figure 1 Schematic diagram of high-order mesh generation for the 3D geometric model of the aircraft wing and body.

[0061] Figure 14(a) shows Figure 13 A partial enlarged view of part A.

[0062] Figure 14(b) shows Figure 13 A partial enlarged view of part B.

[0063] Figure 14(c) shows Figure 13 A partial enlarged view of part C.

[0064] Figure 15 The present invention is aimed at Figure 1 Schematic diagram of the internal structure of the high-order mesh generated from the 3D geometric model of the aircraft wing and body.

[0065] Figure 16(a) shows Figure 15 A partial enlarged view of part A.

[0066] Figure 16(b) shows Figure 15 A partial enlarged view of part B. DETAILED DESCRIPTION

[0067] The present invention will be further described below with reference to the accompanying drawings.

[0068] A three-dimensional high-order grid generation method based on solving partial differential equations, the technical solution adopted is: inputting a three-dimensional geometric model of an aircraft wing and body, generating a linear coarse grid based on the three-dimensional geometric model of the aircraft wing and body, and sequentially establishing a mapping relationship between the linear coarse grid boundary and the three-dimensional geometric model boundary of the aircraft wing and body; then, the linear coarse grid is upgraded to obtain a upgraded linear grid; secondly, based on the surface, the upgraded linear surface grid is divided into a number of facets; the upgraded surface grid facets are projected onto the corresponding three-dimensional surface of the aircraft wing and body, and on the corresponding parameter domain, the boundary offset vector is used as the approximate Based on the constraint condition, the partial differential equations controlling the internal mesh offset vector field are established, and the high-order boundary element method is used to solve the equations. According to the offset amount, the parametric mesh is back-projected onto the three-dimensional surface to obtain the boundary offset vectors of the raised-order tetrahedral mesh. With the boundary offset vectors of the raised-order tetrahedral mesh as the constraint condition, the partial differential equations controlling the internal mesh point offset vector field are established, and the high-order boundary element method is used to solve the equations. Finally, all mesh points of the raised-order tetrahedral mesh are offset according to their corresponding offset vectors, and each face element is represented in a high-order geometric form, thereby obtaining a high-order mesh.

[0069] The specific steps of the three-dimensional high-order grid generation method are as follows:

[0070] Step 1: Generate a linear coarse mesh based on the 3D geometric model of the aircraft wing and body. Then, establish a mapping relationship between the boundaries of the linear coarse mesh and the boundaries of the 3D geometric model of the aircraft wing and body. The details are as follows:

[0071] like Figure 1 As shown in , the aircraft wing-body 3D geometric model is input, and the currently mature linear mesh generation method is used to generate a linear tetrahedral mesh covering the aircraft wing-body 3D geometric model, as shown in Figure 2 As shown, the surface of the three-dimensional geometric model is a number of curved surfaces, denoted as S i ,i∈[1,M], M is the number of surfaces; the corresponding surface mesh is called a patch. Since the surfaces are all closed and the boundaries are curves, for each surface S iThe mapping relationship between the linear coarse mesh boundary and the corresponding surface boundary is established:

[0072] like Figure 3 、 4 As shown, record the current surface S i The corresponding line element at the boundary of the surface linear triangle mesh is L j ,j∈[1,N], N is the number of line units; record the current surface S i The boundary is Since the boundary points on the patch are all located on the surface boundary when the mesh is constructed, the grid points on the line unit at the boundary of the surface linear triangle mesh are projected to the surface S closest to the line unit. i On the boundary, the surface S i The boundary is divided into N segments, recorded as geometric boundary segments Create line unit L j To geometry boundary segment Mapping

[0073]

[0074] For each surface S i For a 3D model of an aircraft wing and fuselage, a curve is a boundary and a boundary line element is boundary information. For a 3D model of an aircraft wing and fuselage, a surface is a boundary and a boundary surface element is boundary information. Therefore, it is necessary to establish a mapping relationship between the linear coarse mesh boundary and the boundary surface for the 3D model of the aircraft wing and fuselage:

[0075] The surface unit of the linear triangular mesh is E j , j∈[1,A], A is the number of surface elements; the boundary of the three-dimensional geometric model of the aircraft wing body is Since all surface grid points are located on the curved surface when constructing the mesh, the grid points on the surface linear triangular mesh are projected onto the boundary of the aircraft wing-body 3D geometric model closest to the surface unit, and then the boundary of the aircraft wing-body 3D geometric model is divided into A blocks, which are recorded as geometric boundary segments. Create surface element E j To geometry boundary segment Mapping

[0076]

[0077] Step 2: Raise the order of each volume mesh element in the linear coarse mesh to obtain a raised-order tetrahedral mesh, as follows:

[0078] New grid points are added to the line elements, surface elements and body of each volume mesh unit as high-order points, and each surface unit is represented by a high-order geometric surface through the Lagrange interpolation method to obtain an elevated tetrahedral mesh, such as Figure 5 As shown; the Lagrange interpolation method is as follows:

[0079] In the two-dimensional parameter domain, assuming that each line element has m grid points, it is necessary to use the m-1 function N α (ξ) is used as the interpolation Lagrangian value basis function, so the high-order geometric curve form of the line element is:

[0080]

[0081] Where p α is the two-dimensional parameter coordinate of the αth grid point; p(ξ) is the general expression of any point on the line element, each line element has m grid points, N α (ξ) represents the Lagrange interpolation basis function corresponding to the αth grid point, ξ is the parameter of the interpolation basis function, and the range is [-1,1].

[0082] It is known that each side of each face unit has m grid points, so each face unit has grid points, we need to use m-1 function N α (ξ) is used as the interpolation Lagrangian value basis function, so the high-order geometric surface form of the surface element is:

[0083]

[0084] Where p α is the physical coordinate of the αth grid point; p(ξ) is the general expression of any point on the surface element, and each surface element has grid points, N α (ξ) represents the Lagrange interpolation basis function corresponding to the αth grid point, ξ is the two-dimensional parameter vector of the interpolation basis function, and the range of any one-dimensional parameter is [0,1].

[0085] Step 3: Establish a two-dimensional offset vector field optimization model, as follows:

[0086] Establish a two-dimensional offset vector field optimization model, such as Figure 6 As shown in the figure, taking the first surface S0 and the corresponding triangular patch mesh as an example, a two-dimensional offset vector field optimization model is established with the goal of satisfying the minimum change when the offset vector of the line element on the boundary of the triangular patch mesh on the parameter domain moves toward the interior of the patch:

[0087]

[0088] The constraints are:

[0089]

[0090] Where V represents the inner area of the triangular patch mesh in the parameter domain, is the parameter offset vector of the grid point p on the triangular patch mesh, express The 2-norm of the gradient, and Respectively represent the Dirichlet boundary information and Neumann boundary information in the patch boundary, and n represents the Neumann boundary The outward unit normal of the upper grid point p, The grid point p representing the boundary of the triangular patch mesh to the surface S i The offset vector of the boundary, express Partial derivative with respect to the outward unit normal n.

[0091] The optimal offset vector field solution for the 2D optimization model is equivalent to the solution to the following Laplace boundary value problem:

[0092]

[0093] According to the line unit L j To surface boundary segment The above constraints are calculated using the Lagrange interpolation in step 2 (see formula (3)). The line unit offset vector at the boundary of the triangular patch mesh is and the offset vector normal derivative q(ξ) are expressed in high-order geometric form, as Figure 7 As shown, we get:

[0094]

[0095] p α represents the αth grid point on the line element.

[0096] Step 4: Solve the two-dimensional offset vector field optimization model established in step 3 to obtain each surface S i The offset vector field of the corresponding triangular patch mesh on the parameter domain; the parameter domain and the surface S i There is a mapping relationship, calculate the parameter offset vector of the grid point on the triangular patch mesh on the parameter domain; offset the grid point on the parameter domain according to the parameter offset vector; map the offset parameter coordinates back to the surface S i , calculate the grid point surface S i The new physical coordinates on the triangle patch are used as the offset coordinates of the triangle patch grid points to obtain the physical offset vector of the triangle patch grid points. Traverse all surfaces to obtain the physical offset vectors of all triangle patch grid points. The specific process is as follows:

[0097] According to formula (7), any internal point p of the triangular patch mesh i Parameter offset vector on the parameter domain It can be solved by the following boundary integral formula:

[0098]

[0099] Where V represents the inner area of the triangular patch mesh in the parameter domain, Represents the boundary of the triangular patch mesh on the parameter domain. p i Represents the source point, which is any grid point inside the triangle mesh; p j Represents a field point, which is any grid point on the boundary of a triangular patch mesh. Represents point p j The differential of the neighborhood, G(p i ,p j ) represents the basic solution of the Laplace equation Δu(p)=0, F(p i ,p j ) represents G(p i ,p j )’s normal partial derivative, in two dimensions, G(p i ,p j ) and F(p i ,p j ) is defined as follows:

[0100]

[0101] Where n represents the field point p j The outward unit normal at the source point p i and field point p j The distance between them.

[0102] According to formula (7), we can also know that when the Dirichlet boundary condition of each point on the boundary line unit of the triangular face mesh is and Neumann boundary condition q(p j ) when there is a conditional value known in (for example It can be calculated by formula (8), and the remaining unknown condition value can be obtained by the following integral equation:

[0103]

[0104] Among them, the source point p i With field point p j are all located on the boundary line units of the triangular patch mesh, and the constant c(p i ) represents the source point p i The smoothness of the neighborhood, when the source point p i When the neighborhood is smooth, c(p i)=0.5. For formula (11), a high-order boundary element numerical method can be used to solve it. Combined with formula (8), the following discrete form of the boundary integral equation is obtained:

[0105]

[0106] Among them, L k Indicates the kth line unit, field point Indicates the location of line unit L k The αth grid point on the top, Indicates line unit L k The αth Lagrangian basis function, Indicates the field point Differentiation of the neighborhood. When the source point p i and field points When they coincide, the left and right sides of formula (12) can be The coefficients are combined to form a linear equation system consisting of N*m unknowns and N*m equations. Solving this linear equation system can calculate the unknown boundary conditions on the boundary line elements of the triangular patch mesh. Or q(p). Then, the boundary conditions on all the triangular patch mesh boundary line elements are Substituting q(p) into formula (11), we can obtain the grid point parameter offset vector inside the triangular patch mesh, as Figure 8 shown.

[0107] Because the above operations are all performed on the parameter domain of the triangular patch mesh, it is necessary to map the offset internal grid point parameter coordinates back to the surface to obtain the physical coordinates on the surface. By calculating the difference between the actual physical coordinates of all grid points in the triangular patch mesh and the corresponding physical coordinates mapped back to the surface, the physical offset vectors of all grid points in the triangular patch mesh are obtained.

[0108] Traverse all surfaces and the corresponding triangular patch meshes to obtain the physical offset vectors of the grid points of all triangular patch meshes.

[0109] Step 5: Establish a three-dimensional offset vector field optimization model, as follows:

[0110] A three-dimensional offset vector field optimization model is established with the goal of ensuring that the offset vectors of the surface elements at the model boundary meet the minimum change when moving into the model interior.

[0111]

[0112] The constraints are:

[0113]

[0114] Where Ω represents the inner region of the raised-order tetrahedral mesh, is the physical offset vector of the grid point p located at the boundary of the raised-order tetrahedral mesh, express The 2-norm of the gradient, and Respectively represent the Dirichlet boundary information and Neumann boundary information in the raised-order tetrahedral mesh boundary, and n represents the Neumann boundary The outward unit normal of the upper grid point p, The physical offset vector representing the mesh point p on the boundary of the raised-order tetrahedral mesh to the boundary of the aircraft wing-body three-dimensional geometric model, express Partial derivative with respect to the outward unit normal n.

[0115] The optimal offset vector field solution for the 3D optimization model is equivalent to the solution to the following Laplace boundary value problem:

[0116]

[0117] According to the surface element E j To the boundary segment of the aircraft wing-body 3D geometric model The above constraints are calculated using the Lagrange interpolation in step 2 (see formula (4)); thus, we can obtain: for any point ξ on the surface element, the surface element physical offset vector at the boundary of the aircraft wing-body three-dimensional geometric model is and the physical offset vector normal derivative q(ξ) are expressed in high-order geometric form, as Figure 9 、 10(a) , as shown in 10(b), we get:

[0118]

[0119] p α Represents the αth grid point on the surface element.

[0120] Utilizes the physical offset vectors of all triangular face elements and the normal derivative q(ξ) of the physical offset vector, which can represent the physical offset vector and its partial derivatives

[0121] Step 6. Solve to obtain the offset vector field of the raised-order tetrahedral mesh, and then offset each grid point of the raised-order tetrahedral mesh along its corresponding physical offset vector. After the offset, each face element can be represented in a high-order geometric form, thus obtaining a high-order mesh. The specific process is as follows:

[0122] According to formula (15), any internal point p of the raised-order tetrahedral mesh iThe physical offset vector It can be solved by the following boundary integral formula:

[0123]

[0124] Where Ω represents the inner region of the raised-order tetrahedral mesh, Represents the boundary of the raised-order tetrahedral mesh. p i represents the source point, which is any grid point inside the raised-order tetrahedral grid; p j Represents a field point, which is any grid point on the boundary of the raised-order tetrahedral grid. Represents point p j The differential of the neighborhood, G(p i ,p j ) represents the basic solution of the Laplace equation Δu(p)=0, F(p i ,p j ) represents G(p i ,p j )’s normal partial derivative, in three dimensions, G(p i ,p j ) and F(p i ,p j ) is defined as follows:

[0125]

[0126] Where n represents the field point p j The outward unit normal at the source point p i and field point p j The distance between them.

[0127] According to formula (17), it can also be known that when the Dirichlet boundary condition of each point on the boundary surface element of the raised-order tetrahedral mesh is and Neumann boundary condition q(p j ) when there is a conditional value known in (for example It can be calculated by formula (16), and the remaining unknown condition value can be obtained by the following integral equation:

[0128]

[0129] Among them, the source point p i With field point p j are located on the boundary surface elements of the raised-order tetrahedral mesh, and the constant c(p i ) represents the source point p i The smoothness of the neighborhood, when the source point p i When the neighborhood is smooth, c(p i)=0.5. For formula (19), a high-order boundary element numerical method can be used to solve it. Combined with formula (16), the following discrete form of the boundary integral equation is obtained:

[0130]

[0131] Among them, E k Represents the kth surface element, field point Indicates that the element E is located k The αth grid point on the top, Represents surface element E k The αth Lagrangian basis function, Indicates the field point Differentiation of the neighborhood. When the source point p i and field points When they coincide, the left and right sides of formula (20) can be The coefficients are combined to form a linear system of A*n unknowns and A*n equations. Solving this linear system of equations can calculate the unknown boundary conditions on the boundary surface elements of the raised-order tetrahedral mesh. Then, the boundary conditions on all boundary surface elements of the raised-order tetrahedral mesh are Substituting q(p) into formula (19), we can obtain the physical offset vector of the grid point inside the raised-order tetrahedral grid. Figure 11 、 12(a) , as shown in 12(b).

[0132] Each grid point of the raised tetrahedral mesh is offset along its corresponding physical offset vector. In this way, each face element can be represented by a higher-order geometric form after the offset, thereby obtaining a higher-order mesh. Figure 13 、 14(a) , 14(b), 14(c), 15, 16(a), and 16(b).

[0133] Based on the three-dimensional high-order grid obtained above, numerical simulation can be performed using industrial simulation software to calculate the physical characteristics of the aircraft wing-body structure analysis; the physical characteristics include stiffness and strength.

Claims

1. A three-dimensional high-order grid generation method based on solving partial differential equations, characterized by: The following steps are involved: Step 1: Generate a linear coarse grid based on the target 3D geometric model, and sequentially establish a mapping relationship between the linear coarse grid boundary and the 3D geometric model boundary; Step 2: Raise the order of each volume mesh element in the linear coarse mesh to obtain a raised-order tetrahedral mesh; Step 3: Establish a two-dimensional offset vector field optimization model; Step 4: Solve the two-dimensional offset vector field optimization model established in step 3 to obtain each surface S i The offset vector field of the corresponding triangular patch mesh on the parameter domain; the parameter domain and the surface S i There is a mapping relationship, calculate the parameter offset vector of the grid point on the triangular patch mesh on the parameter domain; offset the grid point on the parameter domain according to the parameter offset vector; map the offset parameter coordinates back to the surface S i , calculate the grid point surface S i The new physical coordinates on the surface are used as the offset coordinates of the triangular patch grid points, thereby obtaining the physical offset vectors of the triangular patch grid points; traversing all surfaces to obtain the physical offset vectors of all triangular patch grid points; Step 5: Establish a three-dimensional offset vector field optimization model: The constraints of the three-dimensional offset vector field optimization model are: Where Ω represents the interior area of the tetrahedral mesh, is the physical offset vector of the grid point p located at the boundary of the tetrahedral mesh, express The 2-norm of the gradient, and Represents the Dirichlet boundary information and Neumann boundary information in the tetrahedral mesh boundary, n represents the Neumann boundary The outward unit normal of the upper grid point p, The physical offset vector representing the mesh point p on the tetrahedral mesh boundary to the boundary of the 3D geometric model, express The partial derivative with respect to the outward unit normal n; Step 6: Solve the three-dimensional offset vector field optimization model to obtain the optimal offset vector field of the raised-order tetrahedral mesh; then offset each grid point of the raised-order tetrahedral mesh according to the optimal offset vector field to obtain a high-order mesh.

2. The method for generating a three-dimensional high-order grid based on solving partial differential equations according to claim 1, characterized in that: In step 5, the optimal offset vector field solution of the three-dimensional offset vector field optimization model is equivalent to the solution of the following Laplace boundary value problem: The Laplace boundary value problem is solved by the boundary element method.

3. The method for generating a three-dimensional high-order grid based on solving partial differential equations according to claim 1, characterized in that: In step 6, any internal point p of the tetrahedral mesh i The physical offset vector Solved by the following boundary integral formula: Where Ω represents the interior area of the tetrahedral mesh, represents the boundary of the tetrahedral mesh; p j Represents a field point, which is any grid point on the boundary of the tetrahedral grid. Represents point p j The differential of the neighborhood, G(p i ,p j ) represents the basic solution of the Laplace equation Δu(p)=0, F(p i ,p j ) represents G(p i ,p j )’s normal partial derivative, in three dimensions, G(p i ,p j ) and F(p i ,p j ) is defined as follows: Where n represents the field point p j The outward unit normal at the source point p i and field point p j The distance between them.

4. The method for generating a three-dimensional high-order grid based on solving partial differential equations according to claim 3, characterized in that: In step 6, when the Dirichlet boundary condition at each point on the tetrahedral mesh boundary surface element and Neumann boundary condition q(p j ) When one conditional value is known, the remaining unknown conditional value is obtained by the following integral equation: Among them, the source point p i With field point p j are located on the tetrahedral mesh boundary surface elements, the constant c(p i ) represents the source point p i The smoothness of the neighborhood.

5. The method for generating a three-dimensional high-order grid based on solving partial differential equations according to claim 1, characterized in that: The process of establishing the mapping relationship between the linear coarse mesh boundary and the 3D geometric model boundary in step 1 is as follows: (1) The boundary of the linear coarse mesh is divided into several patches, and the boundary of the three-dimensional geometric model is divided into several surfaces. A one-to-one mapping relationship is established between the patches and the surfaces; (2) A single face is divided into several face units, and a single surface is divided into several surface blocks. A one-to-one mapping relationship is established between face units and surface blocks. (3) The boundary of a single face is divided into several line units, and the boundary of a single surface is divided into several curve segments. A one-to-one mapping relationship is established between line units and curve segments.

6. The method for generating a three-dimensional high-order grid based on solving partial differential equations according to claim 1, characterized in that: The specific process of step 2 is as follows: new grid points are added to the line elements, surface elements and body of each volume mesh element as high-order points, and each surface element is represented in the form of a high-order geometric surface through the Lagrange interpolation method to obtain an elevated tetrahedral mesh.

7. The method for generating a three-dimensional high-order grid based on solving partial differential equations according to claim 1, characterized in that: The two-dimensional offset vector field optimization model described in step 3 is as follows: The constraints of the two-dimensional offset vector field optimization model are: Where V represents the inner area of the triangular patch mesh in the parameter domain, is the parameter offset vector of the grid point p on the triangular patch mesh, express The 2-norm of the gradient, and Respectively represent the Dirichlet boundary information and Neumann boundary information in the patch boundary, and n represents the Neumann boundary The outward unit normal of the upper grid point p, The grid point p representing the boundary of the triangular patch mesh to the surface S i The offset vector of the boundary, express Partial derivative with respect to the outward unit normal n.

8. A method for simulating and collecting aerodynamic indicators of an aircraft wing and body, comprising the following steps: (1) Construct a three-dimensional model of the wing and body of the aircraft under test; (2) generating a three-dimensional high-order mesh of an aircraft wing body using a three-dimensional high-order mesh generation method based on solving partial differential equations according to any one of claims 1 to 7; (3) The obtained three-dimensional high-order mesh is imported into industrial simulation software. The industrial simulation software uses the high-precision geometric representation of the three-dimensional high-order mesh to perform numerical simulation and calculate the physical characteristics of the aircraft wing-body structure analysis.

9. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the computer program is executed by a processor, it is used to implement the three-dimensional high-order grid generation method based on solving partial differential equations as described in any one of claims 1 to 7.

10. A readable storage medium storing a computer program; characterized in that: When the computer program is executed by a processor, it is used to implement the three-dimensional high-order grid generation method based on solving partial differential equations as described in any one of claims 1 to 7.

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