Three-dimensional mesh quality optimization method based on wolfe criterion modified newton method

Abstract

The present application belongs to the technical field of mesh quality optimization of three-dimensional tetrahedral mesh generation, and relates to a three-dimensional mesh quality optimization method based on Wolfe criterion modified Newton method. The present application firstly performs initial finite element modeling on a target aircraft to obtain mesh information, then marks the current mesh as an initial mesh, determines mesh element quality measurement criteria and an objective function, marks low-quality mesh elements as poor-quality elements, and respectively constructs local optimization domains; the local optimization domains are optimized by using the Wolfe criterion-based modified Newton method to obtain high-quality meshes. The present application uses the reciprocal of the radius ratio and as the objective function to calculate the optimal mesh node positions of the entire local optimization domain, ensuring the calculation convergence and constraint integrity of the entire local optimization domain, reducing the calculation amount while achieving a higher optimization rate under the condition of ensuring robustness, and finally realizing mesh quality optimization with high robustness and high optimization rate.

Description

Technical Field

[0001] This invention belongs to the field of mesh quality optimization technology for three-dimensional tetrahedral Delaunay mesh generation, and relates to a three-dimensional mesh quality optimization method based on Wolfe criterion modified Newton method. Background Technology

[0002] Mesh generation is a crucial branch of numerical computation, especially in computational fluid dynamics (CFD), which demands even higher mesh quality. As the first, most fundamental, and critical step in CFD numerical computation and simulation, its automation level and mesh quality directly impact the total computation time, accuracy, and efficiency of the numerical simulation. In recent years, with the rapid development of high-performance computers and CFD, CFD applications have become increasingly widespread and the problems increasingly complex. Existing mesh generation techniques seem insufficient for high-quality numerical simulations of complex shapes; therefore, high-quality mesh generation techniques remain a research hotspot. In the field of mesh generation, the Delaunay triangulation method has strong and comprehensive theoretical support, and due to the superior quality, efficiency, and overall performance of the meshes generated by this method, it has become one of the commonly used fully automated mesh generation methods.

[0003] Three-dimensional tetrahedral finite element meshing is far more likely to produce poorly shaped elements than two-dimensional triangular meshing because tetrahedral elements exhibit more forms of distortion and deformation than triangular elements. Therefore, three-dimensional tetrahedral finite element mesh optimization is a problem worthy of attention. Existing mesh optimization algorithms can be divided into three types: point deletion and insertion operations, mesh topology transformation, and mesh smoothing. Mesh smoothing can generally be further divided into Laplacian smoothing, intelligent Laplacian smoothing, and optimization-based smoothing.

[0004] Among them, optimization-based smoothing utilizes the theory of "optimization" to determine the optimal position of each grid node by finding the optimal solution to the objective function. Therefore, theoretically, without considering spatiotemporal efficiency, optimization-based smoothing alone can achieve the maximum optimization effect on the initial grid. However, the entire process of grid quality optimization usually combines various algorithms, and optimization-based smoothing is one of the most computationally and memory-intensive steps in this process.

[0005] In mesh optimization, optimization-based smoothing refers to applying optimization algorithms to improve and optimize the topology and geometry of the mesh to obtain a smoother and more natural mesh shape. It involves several core issues, including the selection of cell metric criteria, the establishment of objective functions, and the implementation of optimization algorithms. Among these, the implementation of optimization algorithms is a major challenge, as it largely determines the optimization efficiency of optimization-based smoothing and the optimization rate of poor-quality meshes. Therefore, a good optimization algorithm is crucial.

[0006] An existing optimization algorithm based on smoothing (Dong Liang, Liu Houlin, Dai Cui, et al. Smoothing-based mesh optimization algorithm and its key factors analysis [J]. Journal of Jiangsu University: Natural Science Edition, 2012, 33(5): 533-537.) uses the difference between the number 1 and the radius ratio as the objective function. In each iteration, it first calculates the gradient of the objective function at the current point position, and then multiplies it with multiple matrices constructed using previous and subsequent information to obtain the descent direction, and then calculates the optimal position of the point to achieve the optimization effect. The computational load of this method is relatively large, and each iteration is highly dependent on the result of the previous iteration, resulting in a less than ideal final optimization effect. Therefore, it is necessary to construct an optimization algorithm that can adapt to the characteristics of three-dimensional tetrahedral meshes, reduce the computational load while ensuring robustness, and achieve a higher optimization rate, ultimately realizing mesh quality optimization with high robustness and high optimization rate. Summary of the Invention

[0007] To address the aforementioned problems or shortcomings, and to construct an optimization algorithm that can adapt to the characteristics of three-dimensional tetrahedral meshes, thereby reducing computational load while maintaining robustness and achieving a higher optimization rate, this invention provides a three-dimensional mesh quality optimization method based on the Wolfe criterion modified Newton's method.

[0008] A method for optimizing the quality of 3D meshes based on the Wolfe criterion-modified Newton's method includes the following steps:

[0009] Step 1: Perform initial finite element modeling of the target aircraft, mesh generation, and establish a three-dimensional tetrahedral mesh to obtain mesh information. Mesh information includes the coordinate information of all mesh nodes, the triangular mesh information on the model surface, and the tetrahedral mesh information inside the model. The tetrahedral mesh inside the model will be collectively referred to as mesh element in subsequent steps.

[0010] The triangular mesh consists of 3 different mesh nodes, and the tetrahedral mesh consists of 4 different mesh nodes. Each mesh node has a unique and distinct number.

[0011] Step 2: Mark the three-dimensional tetrahedral mesh obtained in Step 1 as the initial mesh, and classify all mesh nodes according to the triangular mesh information on the model surface. The mesh nodes that participate in the formation of the triangular mesh on the model surface are called boundary mesh nodes, and all the remaining mesh nodes are called internal mesh nodes.

[0012] Step 3: Determine the quality metric for the grid cells and set the objective function A(x).

[0013] The mesh cell quality metric used in this invention is the radius ratio ρ, specifically as follows:

[0014]

[0015] Where V is the volume of a tetrahedral mesh element, S m It is the area of ​​the triangle on the m-th face of the tetrahedron, l j It is the length of the j-th edge of the tetrahedron. Under this mesh element quality metric, the regular tetrahedron achieves a maximum value of 1.

[0016] Before performing mesh quality optimization, a local optimization domain is first constructed for each mesh node to be optimized. The specific definition of the local optimization domain is as follows:

[0017] First, find all mesh cells in the current mesh that have the mesh node to be optimized as their vertex. These mesh cells form a three-dimensional entity similar to a sphere. All the mesh surfaces that make up the surface of this three-dimensional entity are called the "shell" of the mesh node to be optimized. The "shell" and the space inside the "shell" are called the local optimization domain of the mesh node to be optimized.

[0018] Next, we define the mesh quality function:

[0019]

[0020] Where x is the coordinate of the grid node to be optimized, and f i (x) is the mesh quality function of the i-th mesh in the local optimization domain.

[0021] The objective function A(x) is defined as follows:

[0022]

[0023] Where e i (x) is f i The reciprocal of (x) is called the error function, and n is the number of grids in the local optimization domain. Since the optimization theory is to find the minimum value of a function, the existence of the error function ensures that the objective function A(x) defined by it perfectly fits the evaluation logic of the optimization theory, so that the coordinate x when the objective function takes the minimum value is the optimal coordinate position of the grid node to be optimized in the local optimization domain.

[0024] Step 4: Based on the initial mesh in Step 2, calculate the quality of all mesh elements using the mesh element quality metric determined in Step 3.

[0025] Step 5: Set the mesh cell quality optimization threshold to between 0.1 and 0.6. The higher the value, the better the optimization effect but the higher the time and space consumption. Based on the mesh cell quality calculation results in Step 4, mark mesh cells below this optimization threshold as inferior cells and place them into the container Φ for mesh cells to be optimized. M .

[0026] Step 6: Create an empty container and define it as the container for the mesh nodes to be optimized, Φ. V Traverse the container Φ of the mesh cells to be optimized M Place the grid nodes on each grid cell into the container Φ of grid nodes to be optimized. V .

[0027] Step 7: Delete the container Φ of the grid node to be optimized based on the unique identification information of the grid node. V Remove duplicate mesh nodes from the container Φ, and then delete all boundary mesh nodes to ensure that the container Φ is clean. V The internal grid nodes are all distinct and are called the grid nodes to be optimized.

[0028] Step 8: Remove the container Φ of the mesh node to be optimized one by one. V For the grid node to be optimized, a local optimization domain is constructed for the grid node to be optimized, the objective function of the local optimization domain is calculated, and the modified Newton method based on the Wolfe criterion is used to solve for the minimum value of the corresponding objective function of the local optimization domain, thereby optimizing the coordinate position of the grid node to be optimized.

[0029] The specific optimization process is as follows:

[0030] Before the optimal value solution iteration for each grid node to be optimized begins, the iteration number k = 0 is set, and the coordinates of the grid node to be optimized before the start of each iteration are x. k The objective function is A(x) k ).

[0031] After the iteration begins, the gradient of the objective function at the current position of the grid node to be optimized is first calculated. With the initial Hessian matrix Each solution and When, for the first-order differential, the following precision is adopted: O(h) 4 The first derivative central difference formula

[0032]

[0033] Where h is the differential of the independent variable x.

[0034] For second-order differentials, the following precision is adopted: O(h) 2 Central difference formula for the second derivative of )

[0035]

[0036] For the specific differentiation process, both the first and second order differentiations employ numerical differentiation methods with adaptive accuracy to ensure... and The accuracy and robustness of the solution.

[0037] when When the matrix is ​​not positive definite, determine a correction matrix E. k For Hessian matrices Make modifications so that the modified Hessian matrix B k Positive definite, the specific formula is as follows:

[0038]

[0039] For the correction matrix E k The selection is to directly choose E. k =τ k I, where τ k It is the condition number, I is the identity matrix, when τ k When the value is sufficiently large, the modified Hessian matrix B can always be guaranteed. k Positive definite. In this invention, to reduce the computational error caused by modifications to the Hessian matrix, τ is... k The selection of values ​​is strictly regulated, firstly through the Hessian matrix. The minimum eigenvalue is used to set τ k Let the initial value be , and substitute it into formula (6). If the matrix B at this time is... k Positive definite, accept the τ k Values ​​and matrix B k Continue with the subsequent calculations; if matrix B at this point... k If it is not a positive definite matrix, then τ is continuously changed. k Add 1 and assign the result to τ. k until matrix B k Positive definite, accept τ at this time. k Values ​​and matrix B k Continue with the subsequent calculations.

[0040] exist and B k After the calculation is complete, the corrected Newton's equations can be solved to obtain the corrected Newton direction d. kThe modified Newton's equation is defined as follows

[0041]

[0042] Newton direction d k is a relatively optimal descent direction of the objective function A(x k ) at the current position. When the grid nodes to be optimized move along d k direction, the value of the objective function A(x k ) always decreases continuously.

[0043] Therefore, after each iteration, the coordinate positions of the grid nodes to be optimized are updated, as shown in the following formula

[0044] x k+1 = x k + α k d k (8)

[0045] where α k is the step size. Unlike the classical Newton method, where the value of step size α k is always 1, the determination of step size α k in this process is based on the Wolfe criterion, and a line search algorithm satisfying the Wolfe criterion is used to find the step size α for each iteration k . Compared with other line search algorithms, the step size α obtained by this method k is closer to the optimal step size. The formula of the Wolfe criterion is defined as follows

[0046]

[0047]

[0048] The step size α satisfying the above formula k satisfies the Wolfe criterion, where c1, c2∈(0,1) are given constants and c1<c2. After the step size α is found k , the coordinate position x of the grid node to be optimized after this iteration can be determined through formula (8) k+1 .

[0049] The iteration termination condition is relatively simple. When any of the following inequalities is not satisfied, the iteration terminates

[0050]

[0051] where α min is the minimum step size, set to 1×10 -6 ; k max is the maximum number of iterations, set to 7; A minThe minimum value of the objective function difference before and after each iteration is set to 1×10. -4 If any of the above inequalities is not satisfied after each iteration, it means that the current coordinate position of the grid node is already very close to the optimal coordinate position, and the time and space consumption of further iteration is not proportional to its optimization effect on the local optimization domain.

[0052] Step 9: For the overall mesh optimized in Step 8, update the quality of all mesh cells to obtain a high-quality mesh with optimized mesh quality and output it.

[0053] Compared with existing optimization algorithms based on smoothing, the advantages of this invention are:

[0054] The proposed 3D mesh quality optimization method based on the Wolfe criterion-modified Newton's method utilizes the reciprocal sum of the radius ratios as the objective function to calculate the optimal mesh node positions across the entire local optimization domain. This ensures the computational convergence and constraint integrity of the entire local optimization domain. In contrast, existing optimization algorithms based on smoothing use the maximum difference between the number 1 and the radius ratio as the objective function. This method cannot guarantee the validity of the calculation results, and because it only calculates data for a single mesh, it often leads to a decrease in the mesh quality of the entire local optimization domain. The proposed optimization method is adaptable to the characteristics of 3D tetrahedral meshes, and because the Hessian matrix is ​​a 3×3 matrix in 3D coordinates, its solution is relatively simple. Existing optimization algorithms, on the other hand, are highly dependent on the results of the previous iteration, resulting in a large computational load and ultimately unsatisfactory optimization results. Therefore, the proposed optimization method can reduce the computational load while maintaining robustness, achieving a higher optimization rate and ultimately realizing highly robust and efficient mesh quality optimization. Attached Figure Description

[0055] Figure 1 This is a flowchart of the present invention;

[0056] Figure 2 This is a three-dimensional model diagram of an embodiment;

[0057] Figure 3 This is a comparison diagram of the embodiment before mesh quality optimization, after optimization using the existing method, and after optimization using the method of the present invention. Detailed Implementation

[0058] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.

[0059] A method for optimizing the quality of 3D meshes based on the Wolfe criterion-modified Newton's method, referring to... Figure 1 This includes the following steps:

[0060] Step 1: Perform initial finite element modeling of the target aircraft, mesh generation, and establish a three-dimensional tetrahedral mesh to obtain mesh information. Mesh information includes the coordinate information of all mesh nodes, the triangular mesh information on the model surface, and the tetrahedral mesh information inside the model. The tetrahedral mesh inside the model is called a mesh element.

[0061] This embodiment takes a classic manned passenger aircraft as an example to establish... Figure 2 The model structure shown is obtained by using a meshing tool to perform surface triangular meshing on the model. Then, based on the surface triangular mesh, the internal tetrahedral mesh is performed using the well-known Delaunay triangulation theory. The model is further improved by using boundary restoration and other algorithms to ensure the integrity of the model boundary and the topological compatibility between different partitioned regions of the model. Finally, the tetrahedral meshing information of the target aircraft model is obtained, including the coordinate information of all mesh nodes, the surface triangular mesh information, and the internal tetrahedral mesh information.

[0062] Step 2: Mark the three-dimensional tetrahedral mesh obtained in Step 1 as the initial mesh, and classify all mesh nodes according to the triangular mesh information on the model surface. The mesh nodes that participate in the formation of the triangular mesh on the model surface are called boundary mesh nodes, and all the remaining mesh nodes are called internal mesh nodes.

[0063] The initial mesh serves as the target aircraft mesh model before quality optimization. Typically, its relevant mesh quality information is stored before optimization begins, facilitating comparison with the mesh quality information of the subsequently optimized target aircraft model.

[0064] To maintain the integrity of the model boundary, it is necessary to record the mesh nodes on the boundary to avoid modification of the entity in the subsequent mesh optimization process.

[0065] Step 3: Determine the quality metric for the grid cells and set the objective function A(x).

[0066] The quality metric for mesh elements is the radius ratio ρ. Under this metric, the regular tetrahedron achieves a maximum value of 1. Before optimizing the mesh quality, a local optimization domain is first constructed for each mesh node to be optimized. The specific definition of the local optimization domain is as follows:

[0067] First, find all mesh cells in the current mesh that have the mesh node to be optimized as their vertex. These mesh cells form a three-dimensional entity similar to a sphere. All the mesh surfaces that make up the surface of this three-dimensional entity are called the "shell" of the mesh node to be optimized. The "shell" and the space inside the "shell" are called the local optimization domain of the mesh node to be optimized.

[0068] The mesh quality function of the i-th mesh within the local optimization domain is f iLet x be the coordinates of the grid node to be optimized; then the objective function A(x) is defined as follows:

[0069]

[0070] Where e i (x) is f i The reciprocal of (x) is called the error function, and n is the number of grids in the local optimization domain. Since the optimization theory is to find the minimum value of a function, the existence of the error function ensures that the objective function A(x) defined by it perfectly fits the evaluation logic of the optimization theory, so that the coordinate x when the objective function takes the minimum value is the optimal coordinate position of the grid node to be optimized in the local optimization domain.

[0071] Step 4: Based on the initial mesh in Step 2, calculate the quality of all mesh elements using the mesh element quality metric determined in Step 3.

[0072] The calculation of mesh cell quality is relatively simple, and the specific formula has been given in step 3. However, the entire process of mesh quality optimization generally involves a combination of multiple mesh quality optimization methods.

[0073] Step 5: Set the mesh cell quality optimization threshold to 0.5 (the higher the value, the better the optimization effect, but the higher the time and space consumption). Based on the mesh cell quality calculation results in Step 4, mark mesh cells below this optimization threshold as inferior cells and place them in the container Φ for mesh cells to be optimized. M .

[0074] The range of the mesh cell quality optimization threshold is set relatively wide here in order to find the optimal position of more mesh nodes and achieve better optimization results. In this embodiment, it is set to 0.5.

[0075] Step 6: Create an empty container and define it as the container for the mesh nodes to be optimized, Φ. V Traverse the container Φ of the mesh cells to be optimized M Place the grid nodes on each grid cell into the container Φ of grid nodes to be optimized. V .

[0076] Step 7: Delete the container Φ of the grid node to be optimized based on the unique identification information of the grid node. V Remove duplicate mesh nodes from the container Φ, and then delete all boundary mesh nodes to ensure that the container Φ is clean. V The internal grid nodes are all distinct and are called the grid nodes to be optimized.

[0077] To ensure the integrity of the model boundaries, the coordinate positions of these boundary mesh nodes cannot be modified during mesh quality optimization.

[0078] Step 8: Remove the container Φ of the mesh node to be optimized one by one. V For the grid node to be optimized, a local optimization domain is constructed for the grid node to be optimized, the objective function of the local optimization domain is calculated, and the modified Newton method based on the Wolfe criterion is used to solve for the minimum value of the corresponding objective function of each local optimization domain, thereby optimizing the coordinate position of the grid node to be optimized.

[0079] Step 9: For the overall mesh optimized in Step 8, update the quality of all mesh cells to obtain a high-quality mesh with optimized mesh quality and output it.

[0080] Figure 3 This is based on a classic manned passenger aircraft as an example (such as...) Figure 2 As shown in the figure, this is a comparison of data before mesh quality optimization, after optimization using existing methods, and after optimization using the method of this invention. Analysis of the data before and after mesh quality optimization of the target aircraft model based on the optimization method of this invention shows a significant improvement in overall mesh quality, with an overall mesh optimization rate of 13%. The lowest mesh quality increased from 0.00419 to 0.01795. Furthermore, comparison with existing optimization algorithms for the same mesh shows that the proposed method achieves better mesh quality optimization and a higher overall mesh optimization rate, approximately 5% higher. The optimized target aircraft mesh can then be used for further modal simulation analysis and design of the aircraft structure.

[0081] As can be seen from the experimental data above, this invention first performs initial finite element modeling of the target aircraft to obtain mesh information. Then, the current mesh is marked as the initial mesh, and all nodes, edges, and faces on the boundaries are marked based on boundary information. Next, the mesh element quality metric and objective function used throughout the process are determined, and mesh elements with quality below a threshold are marked as inferior elements. Local optimization domains are constructed for each domain. The objective function of these local optimization domains is optimized using a modified Newton method based on the Wolfe criterion to obtain the optimal coordinate positions of the involved nodes. Finally, a high-quality mesh after optimization is obtained. This invention utilizes the reciprocal sum of the radius ratios as the objective function to calculate the optimal mesh node positions for the entire local optimization domain, ensuring the computational convergence and constraint integrity of the entire local optimization domain, and the solution is relatively simple. This invention reduces computational load while maintaining robustness, achieving a higher optimization rate, ultimately realizing highly robust and high-optimization-rate mesh quality optimization.

Claims

1. A three-dimensional mesh quality optimization method based on Wolfe's criterion modified Newton's method, characterized in that, Includes the following steps: Step 1: Perform initial finite element modeling of the target aircraft, mesh generation, and establish a three-dimensional tetrahedral mesh to obtain mesh information; mesh information includes the coordinate information of all mesh nodes, the triangular mesh information of the model surface, and the tetrahedral mesh information inside the model, where the tetrahedral mesh inside the model is called a mesh element; The triangular mesh consists of 3 different mesh nodes, and the tetrahedral mesh consists of 4 different mesh nodes. Each mesh node has a unique and distinct number. Step 2: Mark the three-dimensional tetrahedral mesh obtained in Step 1 as the initial mesh, and classify all mesh nodes according to the triangular mesh information of the model surface. Among them, the mesh nodes that participate in the formation of the triangular mesh of the model surface are called boundary mesh nodes, and all the remaining mesh nodes are called internal mesh nodes. Step 3: Determine the quality metric for the grid cells and set the objective function A(x); The quality metric for mesh cells is the radius ratio ρ, specifically as follows: Where V is the volume of a tetrahedral mesh element, S m It is the area of ​​the triangle on the m-th face of the tetrahedron, l j It is the length of the j-th edge of the tetrahedron. Under this mesh element quality metric, the regular tetrahedron achieves a maximum value of 1. For each grid node to be optimized, a local optimization domain is constructed. The specific definition of the local optimization domain is as follows: First, find all the mesh cells in the current mesh that have the mesh node to be optimized as the vertex. These mesh cells form a three-dimensional solid similar to a sphere. All the mesh surfaces that make up the surface of this three-dimensional solid are called the shell of the mesh node to be optimized. The shell and the space inside the shell are called the local optimization domain of the mesh node to be optimized. Next, we define the mesh quality function: Where x is the coordinate of the grid node to be optimized, and f i (x) is the mesh quality function of the i-th mesh in the local optimization domain; The objective function A(x) is defined as follows: Where e i (x) is f i The reciprocal of (x) is called the error function, where n is the number of grid cells in the local optimization domain; Step 4: Based on the initial mesh in Step 2, calculate the quality of all mesh elements using the mesh element quality metric determined in Step 3; Step 5: Set the mesh cell quality optimization threshold to between 0.1 and 0.

6. Based on the mesh cell quality calculation results in Step 4, mark mesh cells below this optimization threshold as inferior cells and place them into the container Φ for mesh cells to be optimized. M ; Step 6: Create an empty container and define it as the container for the mesh nodes to be optimized, Φ. V Traverse the container Φ of the mesh cells to be optimized M Place the grid nodes on each grid cell into the container Φ of grid nodes to be optimized. V ; Step 7: Delete the container Φ of the grid node to be optimized based on the unique identification information of the grid node. V Remove duplicate mesh nodes from the container Φ, and then delete all boundary mesh nodes to ensure that the container Φ is clean. V The internal grid nodes are all distinct and are called the grid nodes to be optimized. Step 8: Remove the container Φ of the mesh node to be optimized one by one. V For the grid node to be optimized, a local optimization domain is constructed for the grid node to be optimized, the objective function of the local optimization domain is calculated, and the modified Newton method based on the Wolfe criterion is used to solve for the minimum value of the corresponding objective function of the local optimization domain, thereby optimizing the coordinate position of the grid node to be optimized. The specific optimization process is as follows: Before the optimal value solution iteration for each grid node to be optimized begins, the iteration number k = 0 is set, and the coordinates of the grid node to be optimized before the start of each iteration are x. k The objective function is A(x) k ); After the iteration begins, the gradient of the objective function at the current position of the grid node to be optimized is first calculated. With the initial Hessian matrix Each solution and When, for the first-order differential, the following precision is adopted: O(h) 4 The first derivative central difference formula Where h is the differential of the independent variable x; For second-order differentials, the following precision is adopted: O(h) 2 Central difference formula for the second derivative of ) For the specific differentiation process, both the first and second order differentiations employ numerical differentiation methods with adaptive accuracy to ensure... and Accuracy and robustness of the solution; when When the matrix is ​​not positive definite, determine a correction matrix E. k For Hessian matrices Make modifications so that the modified Hessian matrix B k Positive definite, the specific formula is as follows: For the correction matrix E k The selection is to directly choose E. k =τ k I, where τ k It is the condition number, I is the identity matrix, when τ k When the value is sufficiently large, the modified Hessian matrix B can always be guaranteed. k Positive definite; firstly, through the Hessian matrix... The minimum eigenvalue is used to set τ k Let the initial value be , and substitute it into formula (6). If the matrix B at this time is... k Positive definite, accept the τ k Values ​​and matrix B k Continue with the subsequent calculations; if matrix B at this point... k If it is not a positive definite matrix, then τ is continuously changed. k Add 1 and assign the result to τ. k until matrix B k Positive definite, accept τ at this time. k Values ​​and matrix B k Continue with the subsequent calculations; exist and B k After the calculation is completed, the corrected Newton's equations are solved to obtain the corrected Newton direction d. k The modified Newton's equations are defined as follows: Newton direction d k The objective function is A(x) k A better descent direction at the current position, when the grid node to be optimized moves along d k When moving in direction, the objective function A(x) k The value of ) is always decreasing; After each iteration, the coordinate positions of the mesh nodes to be optimized are updated, as shown in the following formula. x k+1 =x k +a k d k (8) For step size α k The determination is based on the Wolfe criterion, using a line search algorithm that satisfies the Wolfe criterion to find the step size α for each iteration. k The formula for the Wolfe criterion is defined as follows: wherein c₁, c₂ ∈ (0, 1) are given constants, and c₁ < c₂, step size α k after the search is completed, determining the coordinate position x of the grid node to be optimized after the current iteration through formula (8) k+1 ; The iteration terminates when any of the following inequalities are not satisfied. Where α min The minimum step size is set to 1×10. -6 ;k max The maximum number of iterations is set to 7; A min The minimum value of the objective function difference before and after each iteration is set to 1×10. -4 ; Step 9: For the overall mesh optimized in Step 8, update the quality of all mesh cells to obtain a high-quality mesh with optimized mesh quality and output it.

2. The three-dimensional mesh quality optimization method based on the Wolfe criterion modified Newton's method as described in claim 1, characterized in that: The mesh cell quality optimization threshold set in step 5 is 0.5.

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