Simple grid discrete electrostatic field simulation method for complex CAD (Computer Aided Design) model

Abstract

The invention provides a simple grid discrete electrostatic field simulation method for a complex CAD model, and the method comprises the steps: carrying out the expansion of electrostatic potential in a physical region omega of the complex CAD model in a simple region sigma; expanding the electrostatic potential model in the physical area omega of the complex CAD model on the simple area sigma; solving on the simple region sigma by adopting a finite element method; and post-processing: limiting the electrostatic potential phi obtained by solving on the simple region sigma to the physical region omega of the complex CAD model, thereby obtaining the electrostatic potential phi on the physical region omega of the complex CAD model, namely obtaining the electrostatic potential distribution on the physical region omega of the complex CAD model. Complex grid generation involved in traditional electromagnetic simulation calculation based on a finite element method and the like is avoided.

Description

Technical Field

[0001] The invention belongs to the technical field of electrostatic field simulation, and in particular relates to an electrostatic field simulation method of a simple grid discretization of a complex CAD model. Background Art

[0002] As a key technology in the field of engineering and scientific research, electrostatic field simulation technology has shown unparalleled advantages in dealing with complex geometric (other geometric shapes except rectangles and cubes) structural design. It enables designers to accurately simulate the distribution of electric fields and their impact on material and structural performance in a virtual environment, greatly promoting the optimization and innovation of a variety of technical products from high-voltage equipment to microelectronic devices. Through high-precision simulation, engineers can predict and solve potential problems such as insulation failure and electromagnetic interference before manufacturing, significantly reducing the need for physical prototyping, accelerating product development cycles, and ensuring the performance and safety of terminal products. In addition, this technology also helps material selection and process optimization, provides a scientific basis for lightning protection, electromagnetic compatibility design, etc., and further expands its application in environmental protection and public safety. In summary, electrostatic field simulation technology has become an indispensable tool for promoting the advancement of modern engineering technology.

[0003] Electrostatic field simulation technology uses numerical methods to solve the electrostatic field. Commonly used numerical methods include the traditional finite difference method (FDM), finite element method (FEM), boundary element method (BEM), moment method (MOM), etc. Among them, the finite difference method and finite element method are common methods in mainstream electromagnetic simulation software. The well-known electromagnetic simulation software HFSS (High Frequency Structure Simulatior) mainly uses the finite element method for electromagnetic simulation, while the electromagnetic simulation software CST integrates a broadband spectrum simulation solver based on the finite difference time domain method (FDTD).

[0004] When these numerical methods are used to solve simulations, the simulated CAD models need to be preprocessed for mesh generation. Due to the "dirty geometry" structure of the CAD models involved in industrial practice in the geometric modeling and CAD model design process, it is difficult to fully automate the mesh generation of complex CAD models. At the same time, the quality of mesh generation has a key impact on the precision and accuracy of the simulation. Summary of the invention

[0005] The object of the present invention is to provide an electrostatic field simulation method for performing simple mesh generation pre-processing on a complex CAD model, thereby avoiding the complex mesh generation involved in traditional electromagnetic simulation calculations based on finite element methods.

[0006] The technical solution to achieve the purpose of the present invention is:

[0007] A simple grid discretized electrostatic field simulation method for a complex CAD model comprises the following steps:

[0008] Step 1: Expand the electrostatic potential in the physical region Ω of the complex CAD model on the simple region Σ;

[0009] Step 2: Expand the electrostatic potential model in the physical region Ω of the complex CAD model on the simple region Σ;

[0010] Step 3, using the finite element method to solve the simple area Σ, including meshing component units, defining boundary conditions of the electrostatic potential model on the expanded simple area Σ, discretizing the electrostatic potential model expanded to the simple area Σ, and solving the electrostatic potential model on the simple area using the finite element method;

[0011] Step 4, post-processing: The electrostatic potential Φ solved on the simple area Σ is restricted to the physical area Ω of the complex CAD model, thereby obtaining the electrostatic potential φ on the physical area Ω of the complex CAD model, that is, obtaining the electrostatic potential distribution on the physical area Ω of the complex CAD model.

[0012] Compared with the prior art, the present invention has the following significant advantages:

[0013] (1) Reduce the complexity of mesh generation: Reduce the complexity of mesh generation by embedding complex CAD into simple geometric shapes.

[0014] (2) Avoid the influence of the "dirty geometry" structure of the CAD model on the numerical solution: Due to the separation of model design and simulation, design engineers did not consider the needs of model simulation during the CAD model design stage. Complex CAD models have "dirty geometry" structures such as gaps and small redundant structures. In order to ensure the accuracy of the simulation, these "dirty geometry" structures need to be repaired during the mesh generation process; this patented method uses model expansion to robustly handle "dirty geometry" structures, thereby avoiding the influence of the "dirty geometry" structure of the CAD model on the numerical solution.

[0015] (3) Improved automation and efficiency: The increased level of automation reduces the need for manual intervention and re-meshing, especially during the design iteration process, and can quickly adapt to changes in geometry. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 The figure is a flow chart of the method of the present invention.

[0017] Figure 2 Graphs for physical region Ω and simple region Σ.

[0018] Figure 3 Distribution diagram of the numerical solution of electrostatic potential of a complex CAD model.

[0019] Figure 4 This is the error diagram of Example 1.

[0020] Figure 5 This is the graph of the variation of the infinite norm of the error with the subdivision accuracy in Example 1.

[0021] Figure 6 This is the error diagram of Example 2.

[0022] Figure 7 This is the graph of the variation of the infinite norm of the error with the subdivision accuracy in Example 2. DETAILED DESCRIPTION

[0023] The present invention is further described below in conjunction with the accompanying drawings and specific embodiments.

[0024] Combination Figure 1 This embodiment provides a simple grid discretized electrostatic field simulation method for a complex CAD model. The electrostatic field model to be solved is:

[0025]

[0026] is the Laplace operator, which is a differential operator used to calculate the second-order derivatives of scalar fields;

[0027] φ is the electrostatic potential in the physical region Ω of the complex CAD model, which is an unknown function;

[0028] It represents the rate of change of electric potential in the physical region Ω of the complex CAD model, i.e., the second-order derivative of the electrostatic potential;

[0029] f = ρε is a known function, ρ is the charge density in the physical region Ω of the complex CAD model, and ε is the dielectric constant of the medium;

[0030] g is the distribution of electrostatic potential on the boundary of the physical region Ω of the complex CAD model;

[0031] inΩ means within the region Ω, It is represented on the boundary of region Ω;

[0032] The solution method includes the following steps:

[0033] Step 1: Expand the electrostatic potential in the physical region Ω of the complex CAD model on the simple region Σ

[0034] Within the simple region Σ, an electrostatic potential Φ can be constructed to ensure that it satisfies the following conditions at the boundary of the physical region Ω of the complex CAD model:

[0035]

[0036] Φ is the electrostatic potential expanded from the physical area Ω of the complex CAD model to the simple area Σ;

[0037] Σ is the simple region after the physical region Ω of the complex CAD model is expanded;

[0038] φ is the electrostatic potential in the physical region Ω of the complex CAD model;

[0039] (x, y) are the coordinates of the physical region Ω of the complex CAD model;

[0040] k represents the order of partial derivatives, l represents the order of the highest-order derivative that needs to be matched, and the value of l determines the continuity of the electrostatic potential and its partial derivatives at the boundary of the physical region Ω of the complex CAD model, thereby ensuring the smoothness of the expanded electrostatic potential Φ at the boundary of the complex CAD model region. Generally, l ≥ 1

[0041] In the actual working environment, the derivative value is directly measured by installing special measuring equipment.

[0042] Step 2: Expand the electrostatic potential model in the physical region Ω of the complex CAD model on the simple region Σ

[0043] The Laplace operator is calculated for the electrostatic potential Φ obtained by expansion in step 1 to obtain the f value outside the complex CAD model physical region Ω and within the simple region Σ, which is recorded as f1. The value of the electrostatic potential Φ on the boundary of the simple region Σ is recorded as G.

[0044] Then the electrostatic potential model expanded to the simple region Σ is

[0045]

[0046] It represents the rate of change of the electrostatic potential Φ in the simple region Σ, that is, the second-order derivative of the electrostatic potential Φ; in represents within the region, that is, inΩ represents within the region Ω, and inΣ-Ω represents the part within the region Σ and outside the region Ω; Represented on the boundary of region Σ.

[0047] Step 3: Use the finite element method to solve the problem in the simple region Σ

[0048] A. Grid Division

[0049] Complex CAD models usually include 2D complex CAD models and 3D complex CAD models:

[0050] A.1 2D complex CAD model

[0051] Assuming that the simple area Σ is a rectangle, the grid can be divided as follows.

[0052] (1) Determine the length L of the rectangular area x and width L y .

[0053] (2) Divide n evenly along the length x direction x lines, evenly divided along the width y direction into n y Line.

[0054] (3) Calculate node position

[0055]

[0056] Among them, x i is the i-th node in the x direction; y j is the jth node in the y direction. min and min Represents the minimum value of the coordinates of the grid points defined in two-dimensional space, that is, x min is the minimum coordinate value along the x-axis, y min is the minimum coordinate value along the y-axis.

[0057] (4) Building units: Each unit is composed of four nodes, forming a rectangle or square. For example, a unit can be composed of nodes (x i ,y j ), (x i+1 ,y j ), (x i ,y j+1 ) and (x i+1 ,y j+1 )composition.

[0058] A.2 3D complex CAD model

[0059] Let's assume that the simple region Σ is a cuboid. Then the grid can be divided as follows.

[0060] (1) Determine the length L of the rectangular area x 、Width L y and height L z .

[0061] (2) Divide n evenly along the length x direction x lines, evenly divided along the width y direction into n y The line is evenly divided along the width z direction into n z Line.

[0062] (3) Calculate node position

[0063]

[0064] Among them, x iis the i-th node in the x direction; y j is the jth node in the y direction; z k is the kth node in the z direction. min ,y min and z min Represents the minimum value of the coordinates of the grid points defined in three-dimensional space, that is, x min is the minimum coordinate value along the x-axis, y min is the minimum coordinate value along the y-axis, z min is the minimum coordinate value along the z-axis.

[0065] (4) Unit construction: Each hexahedral unit is defined by 8 nodes and can be constructed by connecting adjacent nodes. For example, a hexahedral unit can be constructed by connecting nodes (x i ,y j ,z k )、(x i+1 ,y j ,z k )、(x i ,y j+1 ,z k ),

[0066] (x i ,y j ,z k+1 )、(x i+1 ,y j+1 ,z k )、(x i+1 ,y j ,z k+1 )、(x i ,y j+1 ,z k+1 ) and (x i+1 ,y j+1 ,z k+1 )composition.

[0067] B. Boundary Condition Processing

[0068] Define the boundary conditions of the electrostatic potential model on the expanded simple region Σ, including Dirichlet and

[0069] Neumann boundary condition.

[0070] C. Discretization

[0071] The electrostatic potential model expanded to a simple region Σ is discretized using B-splines.

[0072] D. Finite element solution

[0073] Call the solver and use the finite element method to solve the electrostatic potential model problem on a simple area, namely the Poisson equation. You can get the local approximate potential function on each divided grid unit. Then use numerical methods such as LU decomposition or Gauss-Seidel iteration method or conjugate gradient method to solve the global equations and get the potential value at the node.

[0074] Step 4: Post-processing

[0075] The electrostatic potential Φ solved on the simple area Σ is restricted to the physical area Ω of the complex CAD model, thereby obtaining the electrostatic potential φ on the physical area Ω of the complex CAD model, that is, the electrostatic potential distribution on the physical area Ω of the complex CAD model.

[0076] Example 1

[0077] Original Model:

[0078]

[0079] f, g are known functions, φ is the function to be determined, and Ω is the physical area of ​​the complex CAD model.

[0080] like Figure 2 , as shown, in this example:

[0081] Ω:

[0082] Σ: [0,1]×[0,1]

[0083] g = 3sin(x 2 -1)sin(y 2 -1)

[0084] f = sin(x 2 -1)sin(y 2 -1)(2x 3 y(2x-1) 2 +6xy((x-1 / 2) 2 +(y-1 / 2) 2 -1 / 8) 2

[0085] +4x 3 y((x-1 / 2) 2 +(y-1 / 2) 2 -1 / 8)+12x 2 y(2x-1)((x-1 / 2) 2 +(y-1 / 2) 2 -1 / 8))

[0086] +sin(x 2 -1)sin(y 2-1)(4x 3 (2y-1)((x-1 / 2) 2 +(y-1 / 2) 2 -1 / 8)

[0087] +2x 3 and(2y-1) 2 +4x 3 y((x-1 / 2) 2 +(y-1 / 2) 2 -1 / 8))

[0088] +2cos(x 2 -1)without(and) 2 -1)(x 3 y((x-1 / 2) 2 +(y-1 / 2) 2 -1 / 8) 2 +3)

[0089] +2cos(and 2 -1)sin(x 2 -1)(x 3 y((x-1 / 2) 2 +(y-1 / 2) 2 -1 / 8) 2 +3)

[0090] +4xcos(x 2 -1)without(and) 2 -1)(3x 2 y((x-1 / 2) 2 +(y-1 / 2) 2 -1 / 8) 2

[0091] +2x 3 y(2x-1)((x-1 / 2) 2 +(y-1 / 2) 2 -1 / 8))+4ycos(y 2 -1)sin(x 2 -1)(x 3 ((x-1 / 2) 2

[0092] +(y-1 / 2) 2 -1 / 8) 2 +2x 3 y(2y-1)((x-1 / 2) 2 +(y-1 / 2) 2 -1 / 8))

[0093] -4x 2sin(x 2 -1)sin(y 2 -1)(x 3 y((x-1 / 2) 2 +(y-1 / 2) 2 -1 / 8) 2 +3)

[0094] -4y 2 sin(x 2 -1)sin(y 2 -1)(x 3 y((x-1 / 2) 2 +(y-1 / 2) 2 -1 / 8) 2 +3)

[0095] Solution steps:

[0096] Step 1: Electrostatic potential expansion

[0097] On the physical area boundary of the complex CAD model, Φ = g, The electrostatic potential Φ=3sin(x 2 -1)sin(y 2 -1).

[0098] Step 2: Electrostatic potential model expansion

[0099] Furthermore, the second-order derivative of the electrostatic potential Φ is calculated, and it is found that f1=6sin(x 2 +y 2 -2)-12(x 2 +y 2 )sin(x 2 -1)sin(y 2 -1), and substitute the coordinates on the boundary of the simple region Σ into the electrostatic potential Φ to obtain the electrostatic potential on the boundary of the simple region Σ, denoted as G. The expanded model is:

[0100]

[0101] Step 3: Finite element solution

[0102] The model obtained in step 2 is solved by the finite element method to obtain the electrostatic potential value Φ on the simple region Σ.

[0103] Step 4: Post-processing

[0104] The electrostatic potential Φ of the simple region Σ is restricted to the physical region Ω of the complex CAD model, that is, the coordinates in the complex CAD model are substituted into the electrostatic potential function Φ of the simple region Σ, thereby obtaining the electrostatic potential distribution of the complex CAD model.

[0105] Solve the electrostatic potential distribution of complex CAD models Figure 3 ;

[0106] Convergence test of the algorithm:

[0107] To ensure the accuracy and reliability of the simulation results, it is very important to verify the convergence of the electrostatic field simulation technology algorithm. The following is a convergence test of the algorithm on a complex CAD model in a two-dimensional plane and a three-dimensional space.

[0108] Example 1: 2D complex CAD model

[0109]

[0110] M(x,y)=x 3 y

[0111] H(x,y)=-e x +e y

[0112] F(x,y), M(x,y), and H(x,y) are test functions constructed to test the convergence of the algorithm.

[0113] The true solution on the physical domain Ω of the complex CAD model is: φ(x,y)=(F 2 (x,y)M(x,y)+3)H(x,y)

[0114] Expanding the electrostatic potential to a simple region Σ: [-1, 1] × [-1, 1], we have

[0115]

[0116] Call the solver and use the finite element method to solve the following equation to get the numerical solution:

[0117]

[0118] Figure 4 This is the error diagram of the complex CAD model in two-dimensional space when the subdivision accuracy is 15. From this error diagram, we find that the magnitude of the error is 10 -6 , which means that the numerical solution of the algorithm is sufficiently close to the true solution.

[0119] Figure 5 is the infinite norm of the error (||error||∞ ) changes with the increase of subdivision accuracy. From this figure, we can find that as h decreases (that is, the subdivision accuracy increases), the infinite norm of the error (||error|| ∞ ) decreases. This indicates that the numerical solution is converging to the exact solution.

[0120] Where h = 1 / (1 + subdivision precision), which is the distance between two consecutive nodes

[0121] Example 2: 3D complex CAD model

[0122]

[0123] M(x,y,z)=x 3 y

[0124] H(x,y,z)=sin(x)+sin(y)+sin(z)

[0125] F(x,y,z),M(x,y,z),·····Hx(,yz,) are test functions constructed for the convergence test of the algorithm;

[0126] The true solution on the physical region Ω of the complex CAD model is:

[0127] φ(x,y,z)=(F 2 (x,y,z)M(x,y,z)+1)H(x,y,z)

[0128] Expanding the electrostatic potential to a simple region Σ: [0,1]×[0,1]×[0,1], we have

[0129]

[0130] Call the solver and use the finite element method to solve the following equation to get the numerical solution:

[0131]

[0132] Figure 6 This is the error graph of the complex CAD model in 3D space when the subdivision accuracy is 3. From this error graph, we find that the magnitude of the error is 10 -5 , which means that the numerical solution of the algorithm is sufficiently close to the true solution.

[0133] Figure 7 is the infinite norm of the error (||error|| ∞ ) changes with the increase of subdivision accuracy. From this figure, we can find that as h decreases (that is, the subdivision accuracy increases), the infinite norm of the error (||error||∞ ) decreases. This indicates that the numerical solution is converging to the exact solution.

[0134] Wherein, h=1 / (1+subdivision precision) is the distance between two consecutive nodes.

[0135] By analyzing the error infinite norm (||error|| ∞ ) versus subdivision accuracy, we confirm that the algorithm is convergent and the results obtained by this method are effective, reliable, and highly accurate.

Claims

1. A simple grid discretization electrostatic field simulation method for a complex CAD model, characterized in that: The following steps are involved: Step 1: Expand the electrostatic potential in the physical region Ω of the complex CAD model on the simple region Σ; Step 2: Expand the electrostatic potential model in the physical region Ω of the complex CAD model on the simple region Σ; Step 3, using the finite element method to solve the simple region Σ, including meshing and constructing units, defining boundary conditions of the electrostatic potential model on the expanded simple region Σ, discretizing the electrostatic potential model expanded to the simple region Σ, and solving the electrostatic potential model on the simple region using the finite element method; Step 4, post-processing: The electrostatic potential Φ solved on the simple area Σ is restricted to the physical area Ω of the complex CAD model, thereby obtaining the electrostatic potential φ on the physical area Ω of the complex CAD model, that is, obtaining the electrostatic potential distribution on the physical area Ω of the complex CAD model.

2. The electrostatic field simulation method of simple grid discretization of complex CAD models according to claim 1, characterized in that: Expand the electrostatic potential in the physical region Ω of complex CAD models, including: Within the simple region Σ, an electrostatic potential Φ can be constructed to ensure that it satisfies the following conditions at the boundary of the physical region Ω of the complex CAD model: Φ is the electrostatic potential expanded from the physical region Ω of the complex CAD model to the simple region Σ; Σ is the simple region after the expansion of the physical region Ω of the complex CAD model; φ is the electrostatic potential in the physical region Ω of the complex CAD model; (x, y) are the coordinates of the physical region Ω of the complex CAD model; k represents the order of the partial derivative, and l represents the order of the highest-order derivative that needs to be matched.

3. The electrostatic field simulation method of simple grid discretization of complex CAD models according to claim 1, characterized in that: The electrostatic potential model expanded to a simple region Σ is: ▽ 2 Φ represents the rate of change of the electrostatic potential Φ in the simple region Σ; inΩ represents the area within the region Ω, and in Σ-Ω represents the part within the region Σ and outside the region Ω; Represents on the boundary of region Σ; G is the value of the electrostatic potential Φ on the boundary of the simple region Σ; f = ρε is a known function, ρ is the charge density in the physical region Ω of the complex CAD model, and ε is the dielectric constant of the medium.

4. The electrostatic field simulation method of simple grid discretization of complex CAD models according to claim 1, characterized in that: For a two-dimensional complex CAD model, meshing is performed in the following way: (1) Determine the length L of the rectangular area x and width L y ; (2) Divide n evenly along the length x direction x lines, evenly divided along the width y direction into n y Lines; (3) Calculate node position Among them, x i is the i-th node in the x direction; y j is the jth node in the y direction; x min and min Represents the minimum value of the coordinates of the grid points defined in two-dimensional space; (4) Building units: Each unit consists of four nodes, forming a rectangle or square.

5. The electrostatic field simulation method of simple grid discretization of complex CAD models according to claim 1, characterized in that: For complex 3D CAD models, meshing is performed in the following way: (1) Determine the length L of the rectangular area x 、Width L y and height L z ; (2) Divide n evenly along the length x direction x lines, evenly divided along the width y direction into n y The line is evenly divided along the width z direction into n z Lines; (3) Calculate node position Among them, x i is the i-th node in the x direction; y j is the jth node in the y direction; z k is the kth node in the z direction; x min ,y min and z min Represents the minimum value of the coordinates of the grid points defined in three-dimensional space; (4) Construction unit: Each hexahedral unit is defined by 8 nodes and can be constructed by connecting adjacent nodes.

6. The electrostatic field simulation method of simple grid discretization of complex CAD models according to claim 1, characterized in that: Boundary conditions, including Dirichlet and Neumann boundary conditions.

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