CN118153450A - A hybrid mesh quality optimization method based on probability model - Google Patents

Abstract

The present invention belongs to the field of three-dimensional tetrahedral mesh quality optimization, and relates to a hybrid mesh quality optimization method based on a probability model. The present invention first discretizes the target model, generates a three-dimensional tetrahedral mesh, obtains mesh vertex information and corresponding topological information; then determines the quality metric of the mesh unit, establishes an optimization model, sets the error function and the objective function form; then establishes an overall iteration model, sets the maximum overall iteration number; then retrieves low-quality mesh units, and constructs local optimization domains for their vertices; iteratively solves the optimal value of the objective function through a hybrid mesh quality optimization method based on a probability model, and completes mesh quality optimization. The present invention uses a modified gradient descent method to complete a solution, establishes a random direction iterative solution probability model, reduces the amount of calculation while ensuring robustness, and achieves a higher optimization rate, and finally achieves mesh quality optimization with high robustness, high efficiency, and high optimization rate.

Description

A hybrid mesh quality optimization method based on probability model

Technical Field

The invention belongs to the field of three-dimensional tetrahedron mesh quality optimization and relates to a hybrid mesh quality optimization method based on a probability model.

Background technique

Numerical simulation has become a key research method alongside theoretical and experimental methods. This technique is essential for solving partial differential equations (PDEs) and relies on domain discretization through mesh generation. As the cornerstone of numerical computing, mesh generation establishes the initial discretized model required for simulation. Mesh generation is an important branch of numerical computing, especially in the field of computational electromagnetics (CEM), where the requirements for mesh quality are higher. The level of automation and mesh quality are proportional to the duration of the calculation cycle and the accuracy and efficiency of the simulation. However, the quality of the preliminary mesh elements usually produced by automated mesh generators is poor and cannot meet the strict and precise requirements of numerical analysis. This deficiency may lead to decreased simulation stability, amplified numerical errors, reduced efficiency, and unexpected numerical differences. For example, in finite element analysis, too large mesh element angles may aggravate gradient errors, while too small angles may increase the condition number of the stiffness matrix. Therefore, mesh quality optimization is a necessary process after mesh generation.

In the field of mesh generation, mesh vertex smoothing can greatly improve the quality of unstructured meshes, so it is favored by researchers. Mesh vertex smoothing mainly includes optimization-based smoothing methods and heuristic smoothing methods.

Compared with the optimization-based smoothing method, the heuristic smoothing method directly relocates the vertex position through human experience, which is more effective in implementation, but the mesh quality is lower. Among them, the more classic one is Laplacian smoothing. In the implementation process of the optimization-based smoothing method, the knowledge in the field of optimization is generally used to improve the local mesh quality through a series of steps such as establishing an optimization model, setting an objective function, using an optimization algorithm, and solving the optimal vertex position. This type of method has a relatively large improvement in mesh quality, but it also consumes a lot of time. Previous studies have also been pursuing a balance between smoothing efficiency and mesh quality.

The classic gradient descent method iteratively optimizes and solves the vertices to be optimized one by one. Each iteration solves the gradient of the objective function at the current vertex position, then uses the line search method to solve the optimal step size, and finally updates the vertex position to enter the next iteration. The gradient descent method itself has a relatively fast speed and good optimization effect. However, it is not difficult to find that in the complete iteration process of each vertex, only the optimization effect of the first few times is relatively good, and the subsequent iterations are more likely to converge to the local optimal solution, using more time in exchange for the improvement of the upper limit of the overall mesh quality. The balance between smoothing efficiency and mesh quality is not perfect.

Therefore, if the traditional gradient descent method is modified and a three-dimensional tetrahedral mesh optimization algorithm is constructed, the amount of calculation can be reduced while ensuring robustness, while achieving a higher optimization rate, and a better balance between smoothing efficiency and mesh quality can be achieved, ultimately achieving mesh quality optimization with high robustness, high efficiency, and high optimization rate; thereby improving the efficiency and accuracy of electromagnetic simulation, thereby reducing the cost of products in analysis, design, and data verification.

Summary of the invention

In view of the above-mentioned problems or shortcomings, and to solve the problem of poor balance between smoothing efficiency and mesh quality in the traditional gradient descent method, the present invention provides a hybrid mesh quality optimization method based on a probability model. By establishing a probability model, a three-dimensional tetrahedral mesh optimization algorithm is constructed. While ensuring robustness, the amount of calculation is reduced while achieving a higher optimization rate, consuming less time and resources, and achieving a better balance between smoothing efficiency and mesh quality, ultimately achieving mesh quality optimization with high robustness, high efficiency and high optimization rate; thereby reducing the cost of products in analysis, design and data verification.

A hybrid grid quality optimization method based on a probability model comprises the following steps:

Step 1: Model and discretize the target to generate a three-dimensional tetrahedral mesh. Obtain mesh vertex information and corresponding topological information, including boundary vertices, internal vertex information, tetrahedral mesh topological information and adjacent mesh information.

The boundary vertices are mesh vertices that participate in forming the triangular mesh of the model surface, and all the remaining mesh vertices are called internal vertices. The tetrahedral mesh is composed of 4 different mesh vertices, each mesh vertex has a unique and different number, and each tetrahedral mesh also has a unique and different number.

Step 2: Set the grid obtained in step 1 as the initial grid, and use the grid quality criterion ρ to calculate the initial grid quality and store it as subsequent comparison information.

First, a local optimization domain is constructed for each vertex v _free to be optimized. The specific definition of the local optimization domain is as follows:

Search for all mesh cells with the vertex v _free to be optimized as the vertex in the current mesh. These mesh cells constitute a three-dimensional entity similar to a sphere. All mesh faces constituting the surface of the three-dimensional entity are called the shell of the vertex v _free to be optimized. The space inside the shell is called the local optimization domain of the vertex v _free to be optimized, that is, the coordinate constraint space of the vertex v _free to be optimized.

The mesh quality metric used in the present invention is the aspect ratio ρ, which is specifically expressed as follows:

in are the coordinates of the current vertex v _free to be optimized, where x ₁ , x ₂ , and x ₃ represent the positions on the three coordinate axes of the three-dimensional Cartesian coordinate system, respectively (where x ₁ represents the position on the x-axis, x ₂ represents the position on the y-axis, and x ₃ represents the position on the z-axis); _fi (x) is the mesh quality function of the i-th mesh in the local optimization domain of vertex v _free , V is the volume of the corresponding mesh, S _m is the triangular area of the m-th face of the corresponding mesh, and l _h is the length of the h-th edge of the tetrahedron. Under this quality metric, the regular tetrahedron achieves the maximum value of 1.

Step 3: Establish an optimization model and set the error function and objective function.

The optimization algorithm generally seeks the optimal solution for the optimization model, that is, the minimum value of the corresponding objective function. In mesh quality optimization, the objective function is generally established using the mesh quality metric, and the universal optimal mesh type is a regular tetrahedron. However, for the mesh quality metric described in step 2, the maximum value of 1 is obtained under the regular tetrahedron, which does not meet the basic requirements of the optimization model. Therefore, the error function e(x) is established as follows:

At this time, when the corresponding grid is a regular tetrahedron, the error function e(x) takes the minimum value 1, and the optimization model is established.

The objective function A(x) is defined as follows:

Where n is the number of meshes in the local optimization domain of the vertex v _free to be optimized. This objective function form aims to improve the overall mesh quality of the local optimization domain.

Step 4: Establish the overall iterative model, set the maximum overall iteration number as K, and define its current iteration number as k.

The overall iteration model is to optimize the model mesh in multiple rounds. In each round of optimization, the vertices that need to be optimized are first found, and then each vertex is optimized separately. After all vertices are optimized, the current round of optimization is completed and the next round of optimization or exit is entered. The maximum overall iteration number, that is, the number of optimization rounds, is set to K = 3, and the current iteration number is first set to k = 0.

Step 5: For the current model grid, use the grid quality metric criterion ρ described in step 2 to calculate the quality of all grids, set a grid quality optimization threshold τ, and put the grid cells with grid quality lower than the threshold into the grid cell container Φ _G to be optimized.

The mesh quality optimization threshold τ is between 0.1 and 0.6. The larger the value, the more mesh vertices are involved in each round of optimization, the better the optimization effect, but the time and space consumption will also increase.

Step 6: Create a mesh vertex container Φ _V to be optimized, traverse the mesh unit container Φ _G to be optimized, check the four constituent vertices of each mesh unit, and if these vertices are internal vertices described in step 1, add them to the container Φ _V ; if they are boundary vertices, skip them. And remove the duplicate mesh vertices in the container Φ _V to obtain the final mesh vertex container Φ _V to be optimized.

The purpose of skipping boundary vertices is to ensure that the boundary vertices are not moved, thereby maintaining the shape of the model itself and the topological consistency between assemblies.

Step 7: traverse the final mesh vertex container Φ _V to be optimized obtained in step 6, construct a local optimization domain for each vertex v _free to be optimized, establish the objective function A(x), iteratively solve the optimal value of the objective function through the hybrid mesh quality optimization method based on the probability model, and finally complete a complete iteration of mesh quality optimization.

The specific process of the hybrid grid quality optimization method based on the probability model is as follows:

For the optimal value of each vertex v _free to be optimized, first use the modified gradient descent method to complete a solution, so that the vertex is closer to the optimal solution, and then establish a random direction iterative solution probability model to enter the vertex optimal position solution iterative process. Each iteration randomly selects the vertex movement direction according to the probability, and then adjusts the probability of each direction according to the change of the objective function to enter the next iteration. Set the maximum number of iterations of the random direction iterative solution probability model to A = 200, and define its current number of iterations as a.

The gradient descent process before the iterative process of solving the optimal vertex position is as follows:

Calculate the objective function gradient at the initial position of the vertex v _free to be optimized Then the descending direction d is defined as follows:

Then the well-known weak Wolfe-Powell line search criterion is used to find the optimal step size α. The weak Wolfe-Powell line search criterion is defined as follows:

The step length α that satisfies the above formula satisfies the weak Wolfe-Powell criterion, where c ₁ ,c ₂ ∈(0,1) are given constants and c ₁ <c ₂ . After the step length α is found, the position of the vertex v _free to be optimized is updated, and the update formula is as follows:

x _new = x + c ₃ αd (7)

x _new represents the new coordinate position of the vertex v _free to be optimized after an approximate gradient descent is completed. c ₃ is the modified gradient descent weight to prevent the subsequent vertex optimal position solution iteration from falling into the local optimal solution. Among them, c ₃ ∈(0.3,0.8). A larger value can reduce the demand for subsequent iteration steps, but it is also easy to fall into the local optimal solution, resulting in mediocre final optimization effect.

After completing a solution using the modified gradient descent method, a random direction iterative solution probability model is established. Six unit vectors, i.e., six iteration directions ^dj (j=1, 2, 3, 4, 5, 6), are established using the three coordinate axes of the Cartesian coordinate system. They are: ^d1 =(1, 0, 0) ^T , ^d2 =(-1, 0, 0) ^T , ^d3 =(0, 1, 0) ^T , ^d4 =(0, -1, 0) ^T , ^d5 =(0, 0, 1) ^T , and ^d6 =(0, 0, -1) ^T . The selection probability of the six iteration directions is set to ^pj . Initially, the selection probability of each iteration direction is the same, i.e. The vertex density information q is used to set the vertex moving distance L = c ₄ q for each iteration, where c ₄ ∈ (0.05, 0.001). This value defines the moving distance of the vertex for each iteration. A smaller value will achieve better results, but the maximum number of iterations A will also increase. After the ath iteration, the vertex position is:

xa ₊₁ = _xa + _Lda (8)

Where _da is the vertex movement direction in the ath iteration, which is any one of ^dj .

Enter the iterative process of solving the optimal vertex position, and set a = 0. The specific process is as follows:

At the beginning of each iteration, the iteration direction selection probability ^pj is first used to randomly select the moving direction _da of the vertex in this iteration. For this direction _da , the new position xa ₊₁ of the vertex is calculated by equation (8) and accepted. The objective function values A( _xa ) and A( _xa+1 ) before and after the vertex movement are compared. If A( _xa )<A( _xa+1 ), that is, after this iteration, the mesh quality of the local optimization domain is reduced, the selection probability of each iteration direction is updated as follows:

If A(x _a )≥A(x _a+1 ), that is, after this iteration, the mesh quality of the local optimization domain is improved, the selection probability of each iteration direction is updated as follows:

After the selection probability of each iteration direction is updated, the number of iterations a is incremented by 1 and the next iteration is entered. When a == A, the iteration is terminated, the optimization of the current vertex is completed, and the next vertex optimization process is entered until the mesh vertex container Φ _V to be optimized is traversed.

Step 8: The number of iterations k is incremented by 1. If k == K, the overall iterative optimization of the model is terminated and the process goes to step 9. Otherwise, the process goes back to step 5 and the overall iterative optimization of the model is continued.

Step 9: For the entire mesh optimized in step 8, use the mesh quality metric criterion ρ described in step 2 to update the quality of all mesh units, obtain a high-quality mesh after mesh quality optimization and output it, and at the same time output the mesh quality information before and after optimization and the mesh quality optimization time consumption information.

The present invention first uses a modified gradient descent method to complete a solution, so that the vertex is closer to the optimal solution, and then establishes a random direction iterative solution probability model, enters the vertex optimal position solution iterative process, and at the beginning of each iteration, updates the selection probability of each iteration direction of this iteration according to the result of the previous iteration, and randomly selects the vertex movement direction of this iteration according to the selection probability, and then updates the vertex coordinates, calculates the local mesh quality, and enters the next iteration. The classic gradient descent method will solve the gradient of the objective function at the current vertex position at each iteration. The optimization effect of the first few times is relatively good, and the subsequent iterations focus on converging to the local optimal solution, using more time to exchange for the improvement of the upper limit of the overall mesh quality. This idea is not perfect in the balance between smoothing efficiency and mesh quality. The method proposed by the present invention avoids the subsequent solution on the gradient, only uses a modified gradient descent method to optimize the original position, and then continuously determines the optimal solution of the vertex through the probability model iteration, the amount of calculation is greatly reduced, the time resource consumption is less, the efficiency is accelerated, and the mesh quality is also increased, balancing the relationship between smoothing efficiency and mesh quality.

In summary, the present invention reduces the amount of calculation while ensuring robustness, achieves a higher optimization rate, consumes less time resources, achieves a better balance between smoothing efficiency and grid quality, and ultimately achieves grid quality optimization with high robustness, high efficiency, and high optimization rate.

BRIEF DESCRIPTION OF THE DRAWINGS

Fig. 1 is a flow chart of the present invention;

Fig. 2 is a three-dimensional model diagram of an embodiment;

FIG3 is a comparison diagram of average mesh quality before mesh quality optimization, after optimization using the classical gradient descent method, and in the embodiment.

FIG4 is a comparison chart of the time consumed by optimizing using the classical gradient descent method and the embodiment.

Detailed ways

The technical solution of the present invention is described in detail below in conjunction with the accompanying drawings and embodiments.

A three-dimensional mesh quality optimization method based on Wolfe criterion modified Newton method, referring to FIG1, comprises the following steps:

Step 1: Model and discretize the target aircraft, and generate a three-dimensional tetrahedral mesh using the well-known Delaunay triangulation theory to obtain mesh vertex (boundary vertex, internal vertex) information and corresponding topological information (tetrahedral mesh topological information and adjacent mesh information).

This embodiment takes class X-51 as an example, establishes a model structure as shown in Figure 2, uses a mesh generation tool to perform surface triangular mesh generation on the model, and then uses the well-known Delaunay triangulation theory to perform internal tetrahedral mesh generation on the basis of the surface triangular mesh. The model is further improved using algorithms such as boundary recovery to ensure the integrity of the model boundary and the topological compatibility between different divided areas of the model. Finally, the tetrahedral mesh generation information of the class X-51 model is obtained, including the coordinate information of all mesh vertices, the model surface triangular mesh information, and the model internal tetrahedral mesh information.

The boundary vertices are mesh vertices that participate in forming the triangular mesh of the model surface, and all the remaining mesh vertices are called internal vertices. The tetrahedral mesh is composed of 4 different mesh vertices, each mesh vertex has a unique and different number, and each tetrahedral mesh also has a unique and different number.

Step 2: Set the grid obtained in step 1 as the initial grid, and use the grid quality criterion ρ to calculate the initial grid quality and store it as subsequent comparison information.

First, a local optimization domain is constructed for each vertex v _free to be optimized. The specific definition of the local optimization domain is as follows:

Search for all mesh cells with the vertex v _free to be optimized as the vertex in the current mesh. These mesh cells constitute a three-dimensional entity similar to a sphere. All mesh faces constituting the surface of the three-dimensional entity are called the "shell" of the vertex v _free to be optimized. The space inside the "shell" is called the local optimization domain of the vertex v _free to be optimized, that is, the coordinate constraint space of the vertex v _free to be optimized.

The grid quality metric used in the present invention is the aspect ratio ρ, which is expressed as follows:

in are the coordinates of the current vertex v _free to be optimized, where x ₁ , x ₂ , and x ₃ represent the positions on the three coordinate axes of the three-dimensional Cartesian coordinate system, respectively. _fi (x) is the mesh quality function of the i-th mesh in the local optimization domain of vertex v _free . V is the volume of the corresponding mesh. S _m is the triangular area of the m-th face of the corresponding mesh. l _n is the length of the n-th edge of the tetrahedron. Under this quality metric, the regular tetrahedron achieves the maximum value of 1.

Step 3: Establish an optimization model and set the error function and objective function.

The optimization algorithm generally seeks the optimal solution for the optimization model, that is, the minimum value of the corresponding objective function. In mesh quality optimization, the objective function is generally established using the mesh quality metric, and the universal optimal mesh type is a regular tetrahedron. However, for the mesh quality metric described in step 2, the maximum value of 1 is obtained under the regular tetrahedron, which does not meet the basic requirements of the optimization model. Therefore, the error function e(x) is established as follows:

At this time, when the corresponding grid is a regular tetrahedron, the error function e(x) takes the minimum value 1, and the optimization model is established.

The objective function A(x) is defined as follows:

Where n is the number of meshes in the local optimization domain of the vertex v _free to be optimized. This objective function form aims to improve the overall mesh quality of the local optimization domain.

Step 4: Establish the overall iterative model, set the maximum overall iteration number K, and define the current iteration number k.

The overall iteration model is to optimize the model mesh in multiple rounds. In each round of optimization, the vertices that need to be optimized are first found, and then each vertex is optimized separately. After all vertices are optimized, the current round of optimization is completed and the next round of optimization or exit is entered. The maximum overall iteration number, that is, the number of optimization rounds, is set to K = 3, and the current iteration number is first set to k = 0.

Step 5: For the current model grid, the grid quality metric criterion ρ described in step 2 is used to calculate the quality of all grids. In this embodiment, τ=0.25 is set, and grid cells with grid quality lower than the threshold are placed in the grid cell container Φ _G to be optimized.

Step 6: Create a mesh vertex container Φ _V to be optimized, traverse the mesh unit container Φ _G to be optimized, check the four component vertices of each mesh unit, if these vertices are internal vertices described in step 1, add them to the container Φ _V , if they are boundary vertices, skip them. Eliminate duplicate mesh vertices in the container Φ _V to obtain the final mesh vertex container Φ _V to be optimized.

The purpose of skipping boundary vertices is to ensure that the boundary vertices are not moved, thereby maintaining the shape of the model itself and the topological consistency between assemblies.

Step 7, traverse the final mesh vertex container Φ _V to be optimized obtained in step 6, construct a local optimization domain for each vertex v _free to be optimized, establish the objective function A(x), iteratively solve the optimal value of the objective function through the hybrid mesh quality optimization method based on the probability model, and finally complete a complete iteration of mesh quality optimization. In this embodiment, c ₁ =0.001, c ₂ =0.9, c ₃ =0.5, c ₄ =0.002 are set.

Step 8: The number of iterations k is incremented by 1. If k == K, the overall iterative optimization of the model is terminated and the process goes to step 9. Otherwise, the process goes back to step 5 and the overall iterative optimization of the model is continued.

Step 9: For the entire mesh optimized in step 8, use the mesh quality metric criterion ρ described in step 2 to update the quality of all mesh units, obtain a high-quality mesh after mesh quality optimization and output it, and at the same time output the mesh quality information before and after optimization.

FIG3 is a comparison chart of average mesh quality before mesh quality optimization, after optimization using the classical gradient descent method, and after optimization using the method of the present invention, for a model of the X-51 type under different mesh densities (as shown in FIG2 ). The data before and after the optimization of the mesh quality of the X-51 type based on the scheme of the present invention are analyzed, and the overall mesh quality is significantly improved. Under the number of 5 million mesh units, the average mesh quality before optimization is 0.633972, and the average mesh quality after optimization is 0.646157, and the overall mesh optimization rate reaches 1.922%. In addition, compared with the mesh quality optimization effect of the classical gradient descent method on the same mesh, the scheme of the present invention sets the number of iterations for each vertex to 5 using the classical gradient descent method, and the average mesh quality after optimization by this method is 0.646116. The results show that the scheme of the present invention has a better optimization effect on mesh quality and a higher overall mesh optimization rate.

In terms of time, as shown in FIG4, under different grid densities, for the same X-51-like model with 5 million grid units, the solution of the present invention consumes 99.5 seconds, while the classic gradient descent method consumes 140.8 seconds. The results show that the solution of the present invention reduces the time consumption by 41.3 seconds and speeds up the optimization efficiency by 29.33% while the optimization effect is comparable to or even slightly exceeds that of the classic gradient descent method. For the X-51-like grid optimized by the solution of the present invention, the subsequent operation of modal simulation analysis and design of the aircraft structure can be further performed.

It can be seen from the above embodiments that the present invention first discretizes the target model and generates a three-dimensional tetrahedral mesh. Obtain mesh vertex information and corresponding topological information, including boundary vertices, internal vertex information, tetrahedral mesh topological information and adjacent mesh information; determine the quality metric of the mesh unit, establish an optimization model, set the error function and the objective function form; then establish an overall iteration model, set the maximum overall iteration number; retrieve low-quality mesh units, and construct local optimization domains for their vertices respectively; iteratively solve the optimal value of the objective function through a hybrid mesh quality optimization method based on a probability model to complete mesh quality optimization. For the optimal value solution of each vertex to be optimized, the present invention first uses a modified gradient descent method to complete a solution, so that the vertex is closer to the optimal solution, and then establishes a random direction iterative solution probability model, enters the vertex optimal position solution iterative process, consumes less time and resources, and the efficiency is accelerated while the mesh quality is also increased, balancing the relationship between smoothing efficiency and mesh quality, reducing the amount of calculation while ensuring robustness, and achieving a higher optimization rate, and finally achieving high robustness, high efficiency, and high optimization rate mesh quality optimization.

Claims (5)

1. A hybrid grid quality optimization method based on a probability model, characterized by comprising the following steps: Step 1: Model and discretize the target to generate a three-dimensional tetrahedral mesh; obtain mesh vertex information and corresponding topological information, including boundary vertices, internal vertex information, tetrahedral mesh topological information and adjacent mesh information; The boundary vertices are mesh vertices that participate in forming the triangular mesh of the model surface, and all the remaining mesh vertices are called internal vertices; the tetrahedral mesh is composed of 4 different mesh vertices, each mesh vertex has a unique and different number, and each tetrahedral mesh also has a unique and different number; Step 2: The grid obtained in step 1 is set as the initial grid, and the initial grid quality is calculated using the grid quality criterion ρ, which is stored as subsequent comparison information; First, a local optimization domain is constructed for each vertex v _free to be optimized. The specific definition of the local optimization domain is as follows: Find all mesh units with the vertex v _free to be optimized as the vertex in the current mesh. These mesh units form a three-dimensional entity similar to a sphere. All mesh faces constituting the surface of the three-dimensional entity are called the shell of the vertex v _free to be optimized. The space inside the shell is called the local optimization domain of the vertex v _free to be optimized, that is, the coordinate constraint space of the vertex v _free to be optimized. The mesh quality metric is the aspect ratio ρ, which is as follows: in is the coordinate of the current vertex v _free to be optimized, where x ₁ , x ₂ , and x ₃ represent the positions on the three coordinate axes of the three-dimensional Cartesian coordinate system, x ₁ represents the position on the x coordinate axis, x ₂ represents the position on the y coordinate axis, and x ₃ represents the position on the z coordinate axis; _fi (x) is the mesh quality function of the i-th mesh in the local optimization domain of the vertex v _free , V is the volume of the corresponding mesh, S _m is the triangle area of the m-th face of the corresponding mesh, and l _h is the length of the h-th edge of the tetrahedron. Under this quality metric, the regular tetrahedron achieves the maximum value of 1; Step 3: Establish an optimization model and set the error function and objective function form; Establish the error function e(x) as follows: When the corresponding mesh is a regular tetrahedron, the error function e(x) obtains the minimum value 1, and the optimization model is established; The objective function A(x) is defined as follows: Where n is the number of grids in the local optimization domain of the vertex v _free to be optimized; Step 4: Establish an overall iterative model, set the maximum overall iteration number as K, and define its current iteration number as k; The overall iterative model optimizes the model mesh in multiple rounds. In each round of optimization, the vertices that need to be optimized are first found, and then each vertex is optimized separately. After all vertices are optimized, the current round of optimization is completed and the next round of optimization or exit is entered; the maximum overall number of iterations, that is, the number of optimization rounds, is set to K = 3, and the current number of iterations is first set to k = 0; Step 5: For the current model grid, use the grid quality metric criterion ρ described in step 2 to calculate the quality of all grids, set a grid quality optimization threshold τ, and put the grid cells with grid quality lower than the threshold into the grid cell container Φ _G to be optimized, where τ takes a value of 0.1-0.6; Step 6: Create a mesh vertex container Φ _V to be optimized, traverse the mesh unit container Φ _G to be optimized, check the four component vertices of each mesh unit, if these vertices are internal vertices described in step 1, add them to the container Φ _V , if they are boundary vertices, skip them; and remove the repeated mesh vertices in the container Φ _V to obtain the final mesh vertex container Φ _V to be optimized; Step 7, traverse the final mesh vertex container Φ _V to be optimized obtained in step 6, construct a local optimization domain for each vertex v _free to be optimized, establish the objective function A(x), iteratively solve the optimal value of the objective function through the hybrid mesh quality optimization method based on the probability model, and finally complete a complete iteration of mesh quality optimization; The specific process of the hybrid grid quality optimization method based on the probability model is as follows: For the optimal value solution of each vertex v _free to be optimized, first use the modified gradient descent method to complete a solution, so that the vertex is closer to the optimal solution, then establish a random direction iterative solution probability model, and enter the vertex optimal position solution iterative process. Each iteration randomly selects the vertex movement direction according to the probability, and then adjusts the probability of each direction according to the change of the objective function to enter the next iteration; set the maximum number of iterations of the random direction iterative solution probability model to A = 200, and define its current number of iterations as a; The gradient descent process before the iterative process of solving the optimal vertex position is as follows: Calculate the objective function gradient ▽A(x) at the initial position of the vertex v _free to be optimized, and the descent direction d is defined as follows: d＝-▽A(x) (4) The optimal step size α is found using the weak Wolfe-Powell line search criterion, which is defined as follows: f(x+αd)≤f(x)+c ₁ α▽f(x) ^T d (5) ▽f(x+αd) ^T d≥c ₂ ▽f(x) ^T d (6) The step length α that satisfies the above formula satisfies the weak Wolfe-Powell criterion, where c ₁ ,c ₂ ∈(0,1) are given constants and c ₁ ＜c ₂ ; After the step length α is found, the position of the vertex v _free to be optimized is updated, and the update formula is as follows: x _new = x + c ₃ αd (7) x _new represents the new coordinate position of the vertex v _free to be optimized after an approximate gradient descent is completed, c ₃ is the modified gradient descent weight to prevent the subsequent vertex optimal position solution from falling into the local optimal solution, where c ₃ ∈(0.3,0.8); After completing a solution using the modified gradient descent method, a random direction iterative solution probability model is established. Six unit vectors, i.e., six iteration directions ^dj (j=1, 2, 3, 4, 5, 6), are established using the three coordinate axes of the Cartesian coordinate system. They are: ^d1 =(1, 0, 0) ^T , ^d2 =(-1, 0, 0) ^T , ^d3 =(0, 1, 0) ^T , ^d4 =(0, -1, 0) ^T , ^d5 =(0, 0, 1) ^T , and ^d6 =(0, 0, -1) ^T . The selection probability of the six iteration directions is set to ^pj . Initially, the selection probability of each iteration direction is the same, i.e. Using vertex density information q, we set the vertex moving distance L = c ₄ q for each iteration, where c ₄ ∈ (0.05, 0.001). Then, after the ath iteration, the vertex position is: xa ₊₁ = _xa + _Lda (8) Where d _a is the vertex moving direction in the ath iteration, which is any one of d ^j ; Enter the iterative process of solving the optimal vertex position, set a = 0, and the specific process is as follows: At the beginning of each iteration, the iteration direction selection probability ^pj is first used to randomly select the moving direction _da of the vertex in this iteration. For this direction _da , the new position xa ₊₁ of the vertex is calculated by equation (8) and accepted. The objective function values A( _xa ) and A( _xa+1 ) before and after the vertex movement are compared. If A( _xa ) < A( _xa+1 ), after this iteration, the mesh quality of the local optimization domain is reduced, and the selection probability of each iteration direction is updated as follows: If A(x _a )≥A(x _a+1 ), after this iteration, the mesh quality of the local optimization domain is improved, and the selection probability of each iteration direction is updated as follows: After the selection probability of each iteration direction is updated, the number of iterations a is incremented by 1 and the next iteration is entered; and when a == A, the iteration is terminated, the optimization of the current vertex is completed, and the next vertex optimization process is entered until the mesh vertex container Φ _V to be optimized is traversed; Step 8, the number of iterations k is incremented by 1. If k == K, the overall iterative optimization of the model is terminated and the process goes to step 9. Otherwise, the process goes back to step 5 to continue the overall iterative optimization of the model. Step 9: For the entire mesh optimized in step 8, use the mesh quality metric criterion ρ described in step 2 to update the quality of all mesh units, obtain a high-quality mesh after mesh quality optimization and output it, and at the same time output the mesh quality information before and after optimization and the mesh quality optimization time consumption information. 2. The hybrid mesh quality optimization method based on a probability model as claimed in claim 1, characterized in that: in the step 1, Delaunay triangulation theory is used to generate a three-dimensional tetrahedral mesh. 3. The hybrid grid quality optimization method based on probability model according to claim 1, characterized in that: in said step 7, _c1 = 0.001, _c2 = 0.9, _c3 = 0.5, _c4 = 0.002. 4. The hybrid grid quality optimization method based on probability model as claimed in claim 1, characterized in that: τ = 0.25. 5. The hybrid mesh quality optimization method based on probability model as described in claim 1 is characterized in that the high-quality mesh after the mesh quality optimization in step 9 is used for electromagnetic simulation, and then the analysis, design and data verification of the target product are carried out.