A finite element mesh deformation electromagnetic simulation method based on harmonic field gradient lines

Abstract

The application provides a finite element grid deformation electromagnetic simulation method based on harmonic field gradient lines, which comprises the following steps: step 1, taking the grid of an electromagnetic structure before deformation as an initial grid; step 2, defining three types of boundaries according to the deformation of the electromagnetic device geometric boundary and the movement of the grid nodes on the geometric boundary; step 3, establishing a boundary value problem of a harmonic field described by a Laplace equation; step 4, drawing gradient lines passing through each grid node by using the distribution of the obtained scalar field u; step 5, obtaining a new position P2' of a P2 point after deformation according to the property that the relative position of the P2 point is unchanged, performing linear mapping on the gradient lines, and obtaining the gradient lines after deformation with P2' and P3 as end points; step 6, weighting the displacement of the end points of the gradient lines according to the linear mapping relationship and the proportion of the P1 point in dividing the gradient lines, so as to obtain the displacement of the grid node P1; and step 7, moving the grid nodes on the slip boundary Γ d . s ​

Description

TECHNICAL FIELD

[0001] The application belongs to the field of electromagnetic simulation, and particularly relates to a finite element mesh deformation electromagnetic simulation method based on harmonic field gradient lines. BACKGROUND

[0002] With the continuous improvement of computer computing performance, the finite element method is widely used for solving and computing physical problems such as mechanics, electromagnetism, heat transfer and fluid. When the finite element method is used for computation, the solving space needs to be discretized by using a mesh, and the mesh division needs to be adapted to the geometric boundary. In actual problems, the geometric boundary often changes. For geometric optimization problems, the geometric boundary changes with the change of the model geometric parameters. For motion problems, the geometric boundary changes with the change of the position of the object. When the boundary changes, the mesh needs to be adapted to the changed geometric boundary to ensure the normal progress of the computation. The most direct method to change the mesh is to redivide the mesh. However, for geometric optimization problems containing thousands of iterations or motion problems containing a large number of time steps, this method is low in efficiency and time-consuming. In particular, in some applications such as time step discretization and model order reduction, it is required that the mesh topology before and after the geometric deformation remains unchanged to ensure the accuracy of modeling and simulation. Compared with the above method, the mesh deformation algorithm generates the deformed mesh by moving the position of the initial mesh node under the premise of preserving the mesh topology, is high in computational efficiency, and can avoid the numerical error caused by the change of the mesh topology, and is therefore widely used.

[0003] The existing mesh deformation algorithms can be roughly divided into three categories, namely, algebraic interpolation method, physical simulation method and background mesh mapping method.

[0004] The algebraic interpolation method can be further divided into radial basis function, transfinite interpolation, inverse distance interpolation and algebraic damping method. The basic principle is to interpolate the position of the mesh node inside the geometric boundary according to the displacement of the mesh node on the geometric boundary to realize the mesh deformation. The transfinite interpolation can only be used for the deformation of structured mesh, while the radial basis function method, transfinite interpolation method, inverse distance interpolation method and algebraic damping method are suitable for both structured mesh and unstructured mesh.

[0005] The mesh deformation method based on physical simulation associates the mesh deformation with the physical process, and obtains the mesh deformation result by solving the corresponding partial differential equation. The most typical method is the spring simulation method, which equates the edges of the mesh element to springs with certain stiffness, and assumes that the edges of the deformed mesh are in the static equilibrium state of the springs. The deformation of the mesh is obtained by solving the linear equations given by the spring deformation. The early spring simulation method has poor robustness, so improved methods such as tension spring and torsion spring are proposed. In addition, the physical simulation method also includes Laplace equation method, elastic body method and diffusion equation method, which respectively solve different partial differential equations to realize the mesh deformation.

[0006] The mesh deformation method based on background mesh maps the deformation variable to the background mesh, first moves the background mesh, and then obtains the coordinates of the new mesh by interpolation using the relationship between the background mesh and the internal mesh, thereby realizing the mesh deformation. The typical background mesh method is the Delaunay graph mapping method, in addition to the parameterized mesh method.

[0007] The current mesh deformation methods have the following shortcomings which need to be improved. The mesh deformation algorithm based on transfinite interpolation is only applicable to structured meshes. For models with complex geometric structures, unstructured meshes are often used for division, so the application range is limited. The inverse distance interpolation and algebraic damping method determine the influence of the boundary nodes during interpolation according to the distance between the internal mesh nodes and the geometric boundary. The closer the distance, the greater the influence, and the farther the distance, the smaller the influence. Therefore, the mesh deformation quality near the boundary is high, while the mesh deformation quality far from the boundary is poor, and the mesh overlapping problem is prone to occur. The mesh deformation method based on radial basis function determines the coefficients of the radial basis function by solving a linear equation system, and then obtains the deformation result of the mesh node by using the radial basis function. This method has high robustness, and can still obtain high-quality deformed mesh for large deformation problems. However, it has large computational load and is difficult to apply to mesh deformation problems with a large number of nodes.

[0008] In the mesh deformation based on spring simulation, the stiffness coefficient of the spring is the inverse of the mesh edge length, which can avoid the overlapping and interlacing of the edges in one-dimensional case. However, for two-dimensional or three-dimensional cases, when there is torsion or large deformation, it will cause the mesh to overlap. For the elastic body method, it is necessary to solve the equilibrium equation of elastic mechanics. However, it is difficult to determine the appropriate elastic modulus and Poisson's ratio to obtain good mesh deformation results. Using optimization method to find the appropriate elastic modulus distribution consumes a lot of calculation time. The Laplace equation and diffusion equation method move the mesh nodes by solving partial differential equations. However, for large deformation problems, there is a mesh overlapping problem.

[0009] The Delaunay diagram mapping method and parameterized mesh method based on background mesh are relatively complex to implement, and they also suffer from background mesh overlap for large deformations. Summary of the Invention

[0010] This invention proposes a finite element mesh deformation electromagnetic simulation method based on harmonic field gradient lines. This method sets different boundary conditions according to the deformation of the geometric boundaries and defines Neumann boundaries at appropriate boundaries, considering the slippage of mesh nodes on these boundaries, effectively improving the mesh quality near these boundaries under large deformations. Simultaneously, this method introduces the gradient line method, using the length and relative position of the gradient line where the mesh node is located to determine the magnitude and direction of its movement, effectively avoiding mesh overlap under large deformations without changing the mesh topology.

[0011] This invention proposes a finite element mesh deformation electromagnetic simulation method based on harmonic field gradient lines, comprising the following steps:

[0012] Step 1: Use the mesh of the electromagnetic structure before deformation as the initial mesh;

[0013] Step 2: Define three types of boundaries based on the deformation of the geometric boundaries of the electromagnetic device and the movement of the mesh nodes on the geometric boundaries;

[0014] Step 3: Establish the boundary value problem of the harmonic field described by the Laplace equation, and use the finite element method to solve for the value of the scalar field u at each grid node, thereby moving the grid nodes in the grid deformation region Ω.

[0015] Step 4: Using the obtained distribution of the scalar field u, draw the gradient lines passing through each grid node;

[0016] Step 5: Utilize the boundary Γ before and after deformation d By taking advantage of the property that the relative position of point P2 remains unchanged, we obtain the new position P2′ of point P2 after deformation, and perform linear mapping on the gradient line to obtain the deformed gradient line with P2′ and P3 as endpoints.

[0017] Step 6: Based on the linear mapping relationship, the displacements of the two ends of the gradient line are weighted using the proportion of the gradient line segmented by point P1, thereby obtaining the displacement of the grid node P1.

[0018] Step 7, for the slip boundary Γ s Move the grid nodes on it.

[0019] Beneficial effects:

[0020] The application provides a new method for solving the grid overlap problem caused by large geometric deformation in the optimization design of electromagnetic devices. By introducing the gradient line of the harmonic field, the displacement size of the grid node is determined by weighting according to the length of the gradient line where the grid node is located and the corresponding position, which prevents the grid overlap problem caused by directly using the solution of Laplace equation for movement under large deformation. By introducing the Neumann boundary, the sliding of the grid node along the geometric boundary is allowed, which effectively avoids the generation of the grid overlap phenomenon near the boundary. This method only needs to solve the Laplace equation once in the geometric deformation area to realize the movement of the grid node, which has higher calculation efficiency compared with the grid deformation method which needs to solve the Laplace equation in each different dimension direction. BRIEF DESCRIPTION OF DRAWINGS

[0021] Figure 1 Schematic diagram of three boundary conditions on the geometric boundary of an electromagnetic device, (a) schematic diagram of the geometry and grid before deformation, (b) schematic diagram of the geometry and grid after deformation;

[0022] Figure 2 Schematic diagram of gradient line drawing, (a) gradient line before deformation, (b) gradient line after deformation;

[0023] Figure 3 Deformation of a two-dimensional grid using the gradient line method, (a) geometric boundary before deformation, (b) geometric boundary after deformation, (c) grid before deformation, (d) grid after deformation. DETAILED DESCRIPTION

[0024] The technical solutions in the embodiments of the application will be clearly and completely described below with reference to the drawings in the embodiments of the application. Obviously, the described embodiments are only a part of the embodiments of the application, rather than all the embodiments of the application. Based on the embodiments in the application, all other embodiments obtained by those skilled in the art without creative labor fall within the protection scope of the application.

[0025] For the grid overlap problem caused by large deformation in electromagnetic field numerical simulation, the application provides a finite element grid deformation electromagnetic simulation method based on the gradient line of the harmonic field, which comprises the following steps:

[0026] Step 1, taking the grid of the electromagnetic structure before deformation as the initial grid, as shown in Figure 1 (a).

[0027] Step 2, according to the deformation of the geometric boundary of the electromagnetic device and the movement of the grid node on the geometric boundary, three types of boundaries are defined.

[0028] As shown in Figure 1 (a), for the geometric boundary without deformation, the nodes on the boundary do not move, and the boundary is defined as Γf ; for the geometric boundary with deformation, if the nodes on the boundary follow the boundary movement, and the coordinates of the nodes after movement can be determined according to the deformed geometric boundary, the boundary is defined as Γ d ; if the nodes on the boundary slide along the geometric boundary, and the relative position cannot be determined according to the deformed geometric boundary, the boundary is defined as Γ s . Figure 1 (b) is a schematic diagram of the deformed geometry and mesh.

[0029] Step 3, establish the boundary value problem of harmonic field described by Laplace equation, and solve the value of scalar field u at each mesh node by using finite element method, so as to move the mesh nodes in the deformed region Ω. The boundary value problem described by Laplace equation is:

[0030]

[0031] u = 0, on Γ f

[0032] u = 1, on Γ d

[0033]

[0034] In the formula, the boundary Γ f and the boundary Γ d are Dirichlet boundary with value of 0 and 1 respectively, and the boundary Γ s is Neumann boundary, and in and on represent the position of the node.

[0035] Step 4, draw the gradient line passing through each mesh node by using the distribution of the obtained scalar field u. The gradient line passing through any node P1 in the mesh is shown in Figure 2 (a), and the two end points P2 and P3 end in the boundaries Γ d and Γ f respectively.

[0036] Step 5, obtain the new position P2' of P2 point after deformation by using the property that the relative position of P2 point on the boundary Γ d does not change, and perform linear mapping on the gradient line to obtain the deformed gradient line with P2' and P3 (P3 point does not move) as end points, as shown in Figure 2 (b).

[0037] Step 6, according to the linear mapping relationship, the displacement of the two end points of the gradient line is weighted by using the proportion α of P1 point dividing the gradient line, so as to obtain the displacement of the mesh node P1.

[0038] Wherein, the proportion α of P1 point dividing the gradient line is:

[0039]

[0040] In the formula, L′ is the length of curve segment P1P2, and L is the total length of the gradient line.

[0041] The method of weighted shifting of point P1 using the proportion α is as follows:

[0042] u P1 =(1-α)·u P2 +α·u P3 (3)

[0043] In the formula u P1 Let u be the displacement of grid node P1. P2 and u P3 These are the displacements of the two endpoints P2 and P3 of the gradient line as the geometric boundary deforms. These displacements can be obtained by the change in the relative positions of corresponding points on the geometric boundary before and after deformation.

[0044] Step 7, for the slip boundary Γ s Move the grid nodes on it. For example... Figure 2 As shown in (a), the maximum point on the slip boundary is denoted as P. max Draw the gradient line starting from this point and ending at the boundary Γ. d Point P on n Based on the linear transformation relationship of the gradient lines before and after deformation, P can be used. n The displacement is obtained by P max Displacement of a point. Slip boundary Γ s Corresponding to the Neumann boundary condition, it is equivalent to the gradient line, therefore for Γ s For any point P4, its displacement can also be obtained using the gradient line method:

[0045] u P4 =(1-α)·uP max +α·u PE (4)

[0046] In the formula, uP4 is the slip boundary Γ s The displacement of any point P4, where α is the distance between line segment P4 and line segment P. max P E The ratio, u Pmax For P max The displacement of the point, u PE For P E Displacement of a point.

[0047] The following example of two-dimensional mesh deformation in an electromagnetic field numerical simulation demonstrates the mesh deformation effect of this method. This method can be further extended to three-dimensional cases.

[0048] In this invention, scalar values ​​at grid nodes are obtained by solving the Laplace equation. Other partial differential equations, such as the Poisson equation or the diffusion equation, can also be used for solving. The gradient line drawing method in this scheme can draw along the direction perpendicular to the tangent of the equipotential line.

[0049] Depend on Figure 3 As shown in (a)-(b), the inner rectangle undergoes deformation, often corresponding to the core structure in electromagnetic optimization design, while the outer rectangle does not deform, often corresponding to the air domain. First, three different boundary types are determined. The inner rectangle involves translation and stretching transformations; therefore, the boundary is defined as the deformable boundary Γ. d The left and right sides of the outer rectangle did not deform, therefore they are defined as fixed boundaries Γ. f The inner rectangle exhibits significant horizontal deformation. Considering the slippage of mesh nodes along the boundaries of the upper and lower borders of the outer rectangle helps prevent mesh overlap; therefore, the upper and lower boundaries of the outer rectangle are defined as slip boundaries Γ. s The geometry before deformation is meshed to obtain the initial mesh, such as... Figure 3 As shown in (c). The method of this invention is applied to solve the Laplace boundary value problem, and the grid nodes in the field and on the slip boundary are moved using the harmonic field gradient line to obtain... Figure 3 (d) shows the deformed mesh. As can be seen from the figure, despite the large deformation, the deformed mesh does not overlap. This method effectively avoids the mesh overlap problem caused by large deformation by introducing slip boundaries and gradient lines.

[0050] Although the illustrative specific embodiments of the present invention have been described above to enable those skilled in the art to understand the invention, it should be understood that the invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes will be obvious as long as they are within the spirit and scope of the invention as defined and determined by the appended claims, and all inventions utilizing the concept of the present invention are protected.

Claims

1. A finite element mesh deformation electromagnetic simulation method based on harmonic field gradient lines, characterized in that, Includes the following steps: Step 1: Use the mesh of the electromagnetic structure before deformation as the initial mesh; Step 2: Define three types of boundaries based on the deformation of the geometric boundaries of the electromagnetic device and the movement of the mesh nodes on the geometric boundaries; Step 3: Establish the boundary value problem of the harmonic field described by the Laplace equation, and solve the scalar field at each grid node using the finite element method. The value of is used to determine the deformation region of the mesh. Move the grid nodes within; Step 4: Use the obtained scalar field The distribution of the gradient is plotted, and gradient lines are drawn through each grid node. Step 5: Utilize the boundaries before and after deformation superior The property that the relative position of points remains unchanged, after deformation New location of the point And perform a linear mapping on the gradient line to obtain... and The gradient line after deformation at the endpoints; Step 6: Based on the linear mapping relationship, using... The ratio of point-segmented gradient lines is weighted by the displacements of the two endpoints of the gradient line to obtain the mesh nodes. The displacement; Step 7: For the slip boundary Move the grid nodes on the grid; Step 2 includes: For geometric boundaries that do not deform, the nodes on the boundary do not move; this boundary is defined as... For a deformed geometric boundary, if the nodes on the boundary move with the boundary, and the coordinates of the moved nodes can be determined based on the deformed geometric boundary, then this boundary is defined as... If mesh nodes on the boundary slide along the geometric boundary and their relative positions cannot be determined from the deformed geometric boundary, this boundary is defined as... ; in The proportion of point-segmented gradient lines for: In the formula For curve segments Length, This represents the total length of the gradient lines; Utilization ratio right The weighted movement method for points is as follows: In the formula For grid nodes displacement, and These are the two endpoints of the gradient line. and The displacement caused by the deformation of the geometric boundary is obtained by the change in the relative position of corresponding points on the geometric boundary before and after the deformation. Find the maximum point on the slip boundary and denote it as Draw gradient lines starting from this point and ending at the boundary. Points on ; Based on the linear transformation relationship of the gradient lines before and after deformation, using The displacement is obtained Displacement of a point; slip boundary Corresponding to the Neumann boundary condition, it is equivalent to the gradient line, therefore for Take office at one point Similarly, the magnitude of its displacement is obtained using the gradient line method: In the formula u P4 For slip boundary Take office at one point displacement, For line segments Dividing line segments proportion, for The displacement of the point, for Displacement of a point.

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