Tetrahedral mesh refinement method based on spatial octree index and error feedback

Abstract

This invention discloses a tetrahedral mesh refinement method based on spatial octree indexing and error feedback. This method is suitable for high-precision local region modeling in finite element numerical calculations, and can effectively improve the accuracy and efficiency of simulation calculations. The method includes: obtaining the initial tetrahedral mesh and the error indication returned by the solver; constructing the adjacency relationship and spatial index structure of tetrahedral elements; selecting the element with the largest error according to the error value and inserting a new node at the midpoint of its longest side; updating the local mesh topology; and finally evaluating the mesh quality through the volume-to-side-length ratio index and performing Laplace smoothing operation on elements with poor quality to optimize the mesh quality. This method can dynamically respond to numerical solution errors, and significantly improves mesh quality through the synergistic effect of local refinement and structural optimization. It is particularly suitable for adaptive finite element modeling tasks in complex geometric scenes.

Description

Technical Field

[0001] This invention belongs to the field of numerical simulation finite element calculation technology, and in particular relates to a tetrahedral mesh refinement method based on spatial octree indexing and error feedback, which is applicable to various simulation scenarios based on the finite element method, such as structural analysis, electromagnetic simulation, and heat conduction. Background Technology

[0002] The finite element method is a numerical simulation technique commonly used in engineering and scientific computing. Its computational accuracy and efficiency largely depend on the quality and density of the mesh. In practical applications, accuracy requirements often vary locally. Some regions require finer meshes due to drastic changes in the physical field gradient, while other regions have lower requirements for mesh accuracy. Therefore, adaptive mesh refinement techniques have become a key means to improve computational efficiency and accuracy.

[0003] Traditional adaptive mesh methods can be broadly categorized into two types: one is a mesh refinement method based on geometric features, such as automatically refining meshes in regions with high boundary curvature; the other is a mesh refinement method based on solution error feedback, which adjusts the mesh distribution based on error estimation after the solver performs an initial solution. The latter is considered more effective because it can more directly reflect the actual complexity of the physical field solution; most existing commercial software also adopts this approach. However, existing mesh refinement methods based on error feedback still face problems such as low refinement efficiency, significant impact on mesh quality, and inability to guarantee Delaunay properties.

[0004] Therefore, how to efficiently and stably perform tetrahedral mesh adaptive refinement based on the error indication provided by the solver while ensuring mesh quality remains a major challenge in current technology. Summary of the Invention

[0005] In view of this, embodiments of the present invention provide a tetrahedral mesh refinement method based on spatial octree indexing and error feedback, in order to solve the problems of low efficiency, poor quality and difficulty in meeting the requirements of solvers in existing mesh adaptive techniques.

[0006] The technical solution of the present invention is as follows:

[0007] A tetrahedral mesh encryption method based on spatial octree indexing and error feedback includes:

[0008] S1: Obtain the initial tetrahedral mesh, constraint surfaces, constraint edges, error indicator set returned by the finite element solver, and the encryption ratio threshold of the model;

[0009] S2: Based on the face-volume correspondence in the initial tetrahedral mesh, establish adjacency relationships to identify cavity regions and update the topology during the insertion process;

[0010] S3: Construct a spatial octree based on the vertices of the initial tetrahedral mesh for quickly locating the spatial position of the tetrahedron;

[0011] S4: Select the maximum error indicator from the error indicator set, and find its corresponding tetrahedron using the spatial octree. Insert point p at the midpoint of its longest edge according to the adjacency relationship to obtain the encrypted tetrahedral mesh; if the longest edge is a constraint edge, update the constraint surface and constraint edge.

[0012] S5: Repeat S4 until the encryption ratio of the number of meshes in the encrypted tetrahedral mesh reaches the specified threshold, and the encrypted tetrahedral mesh is obtained.

[0013] S6: Perform quality optimization on the encrypted tetrahedral mesh to obtain an adaptively encrypted tetrahedral mesh.

[0014] Optionally, adjacency relationships are established based on the face-volume correspondence in the initial tetrahedral mesh, including:

[0015] S21: Extract all face information of each tetrahedral element based on the face-volume correspondence in the tetrahedral mesh, and create a unique identifier for it;

[0016] S22: Record the tetrahedral element to which each face belongs based on the unique identifier of each face, and determine whether the face is a shared face;

[0017] S23: Construct an adjacency graph between tetrahedra based on shared faces.

[0018] Optionally, a spatial octree is constructed based on the vertices of the initial tetrahedral mesh, including:

[0019] S31: Construct an outer cube based on the grid vertices to serve as the root node of the spatial octree;

[0020] S32: Traverse all tetrahedra in the tetrahedral mesh and insert the tetrahedrons into the corresponding child nodes of the spatial octree according to their centroid positions. If the number of child node elements exceeds the threshold, split it into 8 sub-blocks.

[0021] S33: After insertion, update the spatial index structure for the deleted cell and the newly generated cell.

[0022] Optionally, inserting point p at the midpoint of its longest edge according to the adjacency relationship includes:

[0023] S51: Based on adjacency relations and spatial indexing, find the set of tetrahedral elements whose circumcircle contains the new point p, forming a cavity region;

[0024] S52: Remove all elements from the tetrahedral element set and extract their non-shared triangles as cavity boundaries;

[0025] S53: Combine the new point p with the cavity boundary triangle to form a new tetrahedral unit, thus completing the tetrahedral reconstruction;

[0026] S54: Update the grid data structure and index structure.

[0027] Optionally, the quality optimization process includes:

[0028] S61: Calculate the quality index of all tetrahedral elements;

[0029] S62: Select tetrahedral cells with quality below the threshold and extract their vertex sets;

[0030] S63: Perform a Laplace smoothing operation to update the position of each vertex to the geometric center of its first-order neighbor vertices;

[0031] S64: Repeat the optimization process until the quality of all units is higher than the set threshold or the number of iterations reaches the upper limit.

[0032] Through the above steps, the present invention achieves high-quality tetrahedral mesh adaptive refinement based on error feedback, which can effectively focus on high error regions, improve calculation accuracy while reducing unnecessary mesh redundancy and improving simulation efficiency.

[0033] Compared with the prior art, the present invention has the following beneficial effects:

[0034] 1. Strong adaptability: Based on the error feedback from the solver, it can more accurately locate areas with large errors and avoid unnecessary global encryption.

[0035] 2. Supports efficient search and update: By constructing adjacency relationships and spatial octrees, the efficiency of location and reconstruction during the encryption phase is improved.

[0036] 3. Improved quality control mechanism: After densification, a quality assessment and smoothing optimization mechanism is introduced to effectively avoid the generation of thin elements and significantly improve the overall mesh quality.

[0037] 4. High versatility: This method does not depend on a specific physical field model and is applicable to a variety of finite element simulation scenarios, such as electromagnetic, structural, and thermal field simulations.

[0038] In summary, this invention provides an efficient, stable, and quality-controllable adaptive densification method for tetrahedral meshes, which is particularly suitable for finite element simulation tasks with high requirements for mesh quality and local accuracy, and has significant engineering application value and prospects for promotion. Attached Figure Description

[0039] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the embodiments or the prior art will be described in detail below with reference to the accompanying drawings; obviously, the accompanying drawings are only some embodiments of the present invention, and those skilled in the art can obtain other drawings based on these drawings without creative effort;

[0040] Figure 1 This is a flowchart illustrating a tetrahedral mesh encryption method based on spatial octree indexing and error feedback, according to an exemplary embodiment.

[0041] Figure 2 This is a geometric schematic diagram of a model patch antenna according to an exemplary embodiment.

[0042] Figure 3 This is a schematic diagram of the initial tetrahedral mesh of a patch antenna according to an exemplary embodiment.

[0043] Figure 4 This is a schematic diagram of the data structure for establishing adjacency relationships in step S2, according to an exemplary embodiment.

[0044] Figure 5 This is a schematic diagram of the spatial octree spatial index structure in step S3 according to an exemplary embodiment.

[0045] Figure 6 This is a schematic diagram of the Bowyer-Watson interpolation point and cavity reconstruction in step S5 according to an exemplary embodiment.

[0046] Figure 7 This is a comparison image showing the key areas of a patch antenna before and after encryption, according to an exemplary embodiment.

[0047] Figure 8 This is a comparison of the S-parameter curve of the patch antenna simulation results shown according to an exemplary embodiment and the results of the commercial software HFSS.

[0048] Figure 9 This is a schematic diagram of the antenna geometry of a horn antenna according to an exemplary embodiment.

[0049] Figure 10 This is a graph showing the S-parameter curves of a horn antenna simulation result according to an exemplary embodiment. Detailed Implementation

[0050] Exemplary embodiments will now be described in detail, examples of which are illustrated in the accompanying drawings; when the following description refers to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements; the embodiments described in the following exemplary embodiments do not represent all embodiments consistent with this application; rather, they are merely examples of apparatuses and methods consistent with some aspects of this application as detailed in the appended claims;

[0051] The terminology used in this application is for the purpose of describing particular embodiments only and is not intended to be limiting of the application; the singular forms “a,” “the,” and “the” used in this application and the appended claims are also intended to include the plural forms unless the context clearly indicates otherwise; it should also be understood that the term “and / or” as used herein refers to and includes any or all possible combinations of one or more of the associated listed items;

[0052] Figure 1 This paper presents an overall flowchart of a tetrahedral mesh encryption method based on spatial octree indexing and error feedback proposed in this invention. The method mainly includes the following steps:

[0053] S1: Obtain the initial tetrahedral mesh T, constraint surfaces F, constraint edges S, the error indicator set {e} returned by the finite element solver, and the refinement scale threshold θ of the model. threshold ;

[0054] Specifically, the initial tetrahedral mesh T of the finite element simulation model is first obtained; then, the finite element solver is used to perform an initial solution on the mesh to obtain an error indicator set {e}, where each element corresponds to a tetrahedral element and is used to reflect the magnitude of the local solution error of that element.

[0055] The finite element simulation model includes, but is not limited to, fields such as electromagnetic simulation, structural mechanics analysis, fluid dynamics simulation, and heat conduction analysis. This embodiment uses a microstrip patch antenna as the simulation object (see reference). Figure 2 ) and horn antenna as simulation objects (reference) Figure 9 (Of course, it is not limited to this.)

[0056] The error indication set {e} can provide absolute value information of unit errors, or it can only reflect the relative ordering relationship of errors between units, which can be used to guide subsequent encryption decisions.

[0057] The error set {e} is the key input for the adaptive encryption method; the error indication can be generated by methods based on residuals or posterior estimation, or it can be directly output by commercial software (such as ANSYS, COMSOL) solvers.

[0058] S2: Based on the face-volume correspondence in the initial tetrahedral mesh T, establish adjacency relationships to identify cavity regions and update the topology during the interpolation process; this step includes the following sub-steps:

[0059] S21: Extract all face information of each tetrahedral element based on the face-volume correspondence in the tetrahedral mesh, and create a unique identifier for it;

[0060] Specifically, this step aims to make the implicit topological relationships of the mesh explicit, laying the foundation for the efficient construction of subsequent adjacency relationships. Taking the initial mesh of a microstrip patch antenna as an example, traversing each tetrahedral element t in the mesh... i Each t i Defined by four vertices {v1, v2, v3, v4}, four triangular faces are formed: f1(v1, v2, v3), f2(v1, v2, v4), f3(v1, v3, v4), and f4(v2, v3, v4). To create a unique identifier for each face regardless of the vertex order, this invention employs a normalization method: sorting the indices of the three vertices constituting each face (e.g., ascending order). For example, a face composed of vertices with indices {5, 28, 13} has its unique identifier determined as (5, 13 28).

[0061] This standardized identifier ensures that when encountering the same shared face while traversing any adjacent tetrahedron, even if the order of its local vertices differs (e.g., {28, 5, 13}), the same identifier is generated. This guarantees accurate and unambiguous identification of shared faces in the mesh in subsequent steps, which is a prerequisite and key to constructing precise adjacency relationships.

[0062] S22: Record the tetrahedral element to which each face belongs based on the unique identifier of each face, and determine whether the face is a shared face or a boundary face;

[0063] Specifically, this step uses an efficient data structure (such as a hash table) to establish a mapping relationship between the face identifier and its corresponding tetrahedral element; continuing with the patch antenna mesh as an example, a mapping M is created, whose key is the unique face identifier generated in S21, and whose value is a list of tetrahedral element indices containing that face.

[0064] Traverse all tetrahedrons t i All faces f J Calculate f j The unique identifier id(f j Find the identifier in mapping M:

[0065] If id(f j If ) does not exist in M, then create a new entry in M, M[id(fj )] =[ t j ];

[0066] If id(f j If it already exists, then the current tetrahedron t will be... i The index is appended to its value list, M[id(f i Updated to [t] existing , t i ];

[0067] After the traversal is complete, check the mapping M again. If the list of values ​​corresponding to a face identifier has a length of 2, then the face is an internal shared face.

[0068] All shared and boundary surfaces can be efficiently identified with a single traversal, and the time complexity is proportional to the number of mesh surfaces, making it highly efficient. At the same time, it clearly distinguishes between internal shared surfaces and model boundary surfaces, which is crucial for subsequent interpolation algorithms (especially when dealing with constraint edges and constraint surfaces) and cavity identification, ensuring that the encryption process does not destroy the original boundary features of the model.

[0069] S23: Construct an adjacency graph between tetrahedra based on shared faces;

[0070] Specifically, based on S22, the final adjacency graph N is constructed. This graph can be directly stored as a list of neighbors for each tetrahedral cell; then, all entries in the mapping M generated in S22 whose values ​​are lists of length 2 are traversed again. For each entry, for example, M[id(f k )] = [t a , t b This indicates that tetrahedron t a and t b Through shared surface f k And they are adjacent to each other. Therefore, in t a Add t to the neighbor list b At the same time in t b Add t to the neighbor list a .

[0071] After traversal, each tetrahedral cell will have a list containing all its direct neighbors. This complete set of adjacency relation N data is the core of local topology operations.

[0072] By pre-compiling and storing adjacency relationships, the time-consuming geometric search problem is transformed into an efficient graph traversal problem. In the subsequent Bowyer-Watson insertion step, when it is necessary to find the circumsphere influence region (i.e., cavity) containing the new insertion point, the search can be quickly "flooded" outward by querying the adjacency graph, starting from the initial cell. This greatly improves the speed and robustness of cavity construction, avoids a global search of the entire mesh, and thus significantly improves the overall efficiency of the encryption algorithm.

[0073] like Figure 4 As shown, all tetrahedral elements in the initial tetrahedral mesh T are traversed, their four faces are extracted, and which faces are shared by two elements are recorded; the adjacency relationship N between tetrahedral elements is constructed to identify the topology and determine whether the cavity structure is connected; this adjacency information is used to locate the affected area in the subsequent insertion point and cavity construction process.

[0074] S3: Construct a spatial octree based on the vertices V of the initial tetrahedral mesh T. oct This step is used to quickly locate the spatial position of a tetrahedron; it includes the following sub-steps:

[0075] S31: Construct an outer cube based on the grid vertices to serve as the root node of the spatial octree;

[0076] Specifically, before constructing the octree, the spatial extent of its root node needs to be determined, and this extent must completely encompass the entire model. Taking a horn antenna model as an example, first, traverse all vertices in its initial mesh T and find the maximum value of the x, y, z coordinates of all vertices (x, y, z). max , y max , z max ) and minimum value (x) min , y min , z min This results in a cuboid bounding box that tightly encloses the model. To simplify subsequent spatial partitioning calculations, this cuboid is expanded into a cube; the lengths L of the three sides are calculated. x = x max - x min , L y = y max – y min , L_z = z max - z min Take its maximum value L max = max(L x , L y , L z Finally, the root node of the spatial octree is defined as a node starting at (x...). min , y min ,zmin ), with a side length of L max A cube.

[0077] The advantage of this design is that using a cube as the root node ensures that each subsequent recursive partition generates 8 child cubes of equal size and regular shape. This greatly simplifies the calculation of the spatial range of child nodes and the logic for determining the ownership of points or objects, making the octree construction and query algorithms more concise and efficient.

[0078] S32: Traverse all tetrahedra in the tetrahedral mesh and insert the tetrahedrons into the corresponding child nodes of the spatial octree according to their centroid positions. If the number of child node elements exceeds the threshold, split it into 8 sub-blocks.

[0079] Specifically, taking the initial grid of a horn antenna as an example, for each tetrahedral element t i First, calculate the coordinates of its geometric centroid. Then, starting from the root node of the octree, perform an insertion operation on the tetrahedron; the insertion process is recursive.

[0080] 1. Determine if the current node is a leaf node;

[0081] 2. If the current node is a leaf node, then the tetrahedron t... i Store it in the element list of this node;

[0082] 3. Check if the number of elements in the leaf node exceeds a preset threshold (e.g., 10); if it does, split the node into 8 child nodes (i.e., 8 sub-cubes), and remove all existing elements from the parent node (including the newly inserted t). i Based on its centroid position, it is reassigned to the corresponding child nodes;

[0083] 4. If the current node is not a leaf node (i.e., an internal node), then determine t. i The centroid is located in one of its eight child nodes, and then the insertion operation is recursively performed in that child node.

[0084] Through this strategy, the octree can adaptively reflect the spatial density distribution of the mesh. In regions with complex model geometry and dense mesh (such as the throat and aperture edge of a horn antenna), the octree hierarchy is deeper and the subdivision is finer; while in sparse mesh regions, the tree hierarchy is shallower. This makes the efficiency of spatial queries (such as locating cells near specific coordinates) logarithmic (O(logN)), far superior to linear traversal (O(N)), greatly accelerating the localization process of high-error cells.

[0085] S33: After insertion, update the spatial index structure for the deleted cell and the newly generated cell.

[0086] Specifically, this step ensures the consistency and validity of the spatial index during the dynamic mesh update process; after the interpolation operation is completed, the local mesh topology will change: some old tetrahedrons (the cells that make up the cavity) are deleted, while a batch of new tetrahedrons are generated. At this time, the octree must be updated synchronously.

[0087] Taking an insertion point at the edge of a patch antenna as an example, assuming 5 old elements were deleted and 12 new elements were generated; the update process includes:

[0088] 1. Deletion operation: For each old unit to be deleted, search in the octree according to its previously stored centroid position, locate the leaf node where it is located, and remove it from the element list of that node;

[0089] 2. Insertion operation: For each newly generated cell, perform the same insertion logic as S32: calculate its centroid, recursively insert from the root node, and if this causes the number of elements in a leaf node to exceed the limit, trigger a split.

[0090] The advantages of this design are: it ensures that the octree index always accurately reflects the latest grid state; this dynamic, real-time update mechanism is the basis for the continuous and iterative operation of the entire adaptive encryption method; if the index is not updated, the subsequent error cell location and insertion operations will be based on outdated spatial information, leading to algorithm errors or even failure; this step ensures the correctness and robustness of the entire encryption process.

[0091] like Figure 5 As shown, to accelerate subsequent region search, a spatial octree index is constructed to divide the entire grid space into layers; each tetrahedral cell is inserted into the corresponding node of the spatial octree according to its centroid coordinates to support efficient subsequent positioning.

[0092] S4: Locate the high-error region and insert points; select the maximum error indicator e from the error indicator set {e}. max And utilize the spatial octree oct Find its corresponding tetrahedron t i On its longest side s longest Insert point p at the midpoint of the grid (based on the Bowyer-Watson interpolation algorithm) to obtain the refined tetrahedral mesh T'; if s longest If it is a constraint edge, update the constraint surface F and the constraint edge S;

[0093] Specifically, select the indicator e with the largest error value from the error indicator set {e}. max According to e max And utilize a spatial octree oct Find its corresponding tetrahedron t iWithin this cell, find the longest edge s. longest And calculate its midpoint p, which will be the new node to be inserted.

[0094] The Bowyer-Watson interpolation algorithm is used to insert point p into the current tetrahedral mesh, specifically including:

[0095] 1. Using spatial index O and adjacency relation N, find all tetrahedrons whose circumspheres contain point p, thus forming cavity S;

[0096] 2. Delete the units inside the cavity and extract their boundary surfaces to form boundary set B;

[0097] 3. Combine point p with each boundary triangle to form a new tetrahedron, thus completing the Delaunay reconstruction;

[0098] 4. Update the adjacency structure N and the spatial index structure O to complete the insertion.

[0099] S5: Repeat S4 until the encryption ratio of the encrypted tetrahedral mesh reaches the specified threshold θ. threshold (e.g., 30% of the initial tetrahedral element count) to obtain a refined tetrahedral mesh;

[0100] Specifically, calculate the encryption ratio of the current encrypted tetrahedral mesh T'; the encryption ratio is calculated using the formula θ=(N T’ -N T ) / N T , where N T’ Let T' be the current encrypted tetrahedral mesh size, and N be the size of the mesh. T The mesh size of the initial tetrahedral mesh T;

[0101] If θ≤θ threshold If the encryption process is successful, return to S4; otherwise, terminate the encryption process and proceed to S6.

[0102] S6: Perform quality optimization on the encrypted tetrahedral mesh to obtain an adaptively encrypted tetrahedral mesh;

[0103] To avoid introducing degenerate elements and thin elements through interpolation, quality optimization needs to be performed on the updated mesh; the method is as follows:

[0104] 1. Calculate the quality index of all tetrahedral elements. Its definition is:

[0105]

[0106] Where V is the volume of the tetrahedron, l i This refers to the lengths of the six sides of the unit.

[0107] 2. Sort by quality from worst to best, and select the vertices of the worst units;

[0108] 3. Perform a Laplace smoothing operation on these vertices, which updates them to the geometric centers of their first-order neighbors;

[0109] 4. Repeat this optimization process until the quality of all units is greater than the threshold. Or, it may reach the maximum number of iterations.

[0110] Example 1: Patch Antenna Simulation

[0111] To verify the effectiveness of the method described in this invention, this embodiment uses a microstrip patch antenna as the simulation object; such as Figure 2 The image shows the geometric model of the patch antenna; as shown... Figure 3 The image shows the initial tetrahedral mesh of the model.

[0112] Using this initial mesh as input, adaptive refinement is performed using steps S1-S6 as described in this invention. During the electromagnetic simulation solution process, errors are usually concentrated near patch edges, feed points, and radiation boundaries. This method automatically refines the mesh in these critical areas based on the error indication returned by the solver, while keeping the mesh relatively sparse in other areas.

[0113] like Figure 7 The image shows a comparison of the mesh at the edge of the patch antenna before and after adaptive encryption using the method of this invention; it can be clearly seen from the image that the encrypted mesh ( Figure 7 The density at the patch edge (right side) is significantly higher than the initial mesh ( Figure 7 The grid density changes little in areas far from the left, which intuitively demonstrates the adaptability and accuracy of the method of this invention.

[0114] like Figure 8 As shown in the figure, the S-parameter curves obtained by simulating the mesh encrypted using the method of the present invention are compared with the simulation results of the mainstream commercial software ANSYS HFSS. It can be seen from the figure that the two curves have a high degree of agreement, which verifies that the method of the present invention can efficiently complete adaptive mesh encryption while ensuring calculation accuracy, and the results are accurate and reliable.

[0115] Example 2: Simulation of a Horn Antenna

[0116] To further illustrate the versatility of this invention, this embodiment uses a horn antenna as a simulation object; for example... Figure 9 The figure shows the geometric model of the horn antenna. The electromagnetic field distribution of the horn antenna is different from that of the patch antenna. Its energy is mainly radiated from the horn mouth, and the region with drastic field gradient changes is located in the throat and the edge of the aperture.

[0117] Similarly, using the method described in this invention, the initial grid of the horn antenna is adaptively refined; such as... Figure 10 The figure shows the S-parameter curves obtained by simulation using the method of this invention. The results show that this method can accurately capture the resonance characteristics of a horn antenna, is applicable to different types of electromagnetic simulation problems, and has strong versatility and engineering application value.

[0118] It should be understood that this application is not limited to the precise structure described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope; the scope of this application is limited only by the appended claims.

Claims

1. A tetrahedral mesh encryption method based on spatial octree indexing and error feedback, characterized in that, include: S1: Obtain the initial tetrahedral mesh, constraint surfaces, constraint edges, error indicator set returned by the finite element solver, and the encryption ratio threshold of the model; S2: Based on the face-volume correspondence in the initial tetrahedral mesh, establish adjacency relationships to identify cavity regions and update the topology during the insertion process; S3: Construct a spatial octree based on the vertices of the initial tetrahedral mesh for quickly locating the spatial position of the tetrahedron; S4: Select the maximum error indicator from the error indicator set, find its corresponding tetrahedron using the spatial octree, insert a point at the midpoint of its longest edge according to the adjacency relationship, and obtain the encrypted tetrahedral mesh; if the longest edge is a constraint edge, update the constraint surface and constraint edge. S5: Repeat S4 until the encryption ratio of the number of meshes in the encrypted tetrahedral mesh reaches the specified threshold, and the encrypted tetrahedral mesh is obtained. S6: Perform quality optimization on the encrypted tetrahedral mesh to obtain an adaptively encrypted tetrahedral mesh; The process of establishing adjacency relationships based on the face-volume correspondence in the initial tetrahedral mesh includes: S21: Extract all face information of each tetrahedral element based on the face-volume correspondence in the tetrahedral mesh, and create a unique identifier for it; S22: Record the tetrahedral element to which each face belongs based on the unique identifier of each face, and determine whether the face is a shared face; S23: Construct an adjacency graph between tetrahedra based on shared faces.

2. The method according to claim 1, characterized in that, Construct a spatial octree based on the vertices of the initial tetrahedral mesh, including: S31: Construct an outer cube based on the grid vertices to serve as the root node of the spatial octree; S32: Traverse all tetrahedra in the tetrahedral mesh and insert the tetrahedrons into the corresponding child nodes of the spatial octree according to their centroid positions. If the number of child node elements exceeds the threshold, split it into 8 sub-blocks. S33: After insertion, update the spatial index structure for the deleted cell and the newly generated cell.

3. The method according to claim 1, characterized in that, Inserting point p at the midpoint of its longest edge according to the adjacency relationship includes: S41: Based on adjacency relations and spatial indexing, find the set of tetrahedral elements whose circumcircle contains the new point p, forming the cavity region; S42: Remove all elements from the tetrahedral element set and extract their non-shared triangles as cavity boundaries; S43: Combine the new point p with the cavity boundary triangle to form a new tetrahedral unit, thus completing the tetrahedral reconstruction; S44: Update the grid data structure and index structure.

4. The method according to claim 1, characterized in that, The quality optimization process includes: S61: Calculate the quality index of all tetrahedral elements; S62: Select tetrahedral cells with quality below the threshold and extract their vertex sets; S63: Perform a Laplace smoothing operation to update the position of each vertex to the geometric center of its first-order neighbor vertices; S64: Repeat the optimization process until the quality of all units is higher than the set threshold or the number of iterations reaches the upper limit.

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