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

Abstract

The invention discloses a tetrahedral mesh encryption method based on a spatial octree index and error feedback. The method is suitable for the requirement for high-precision local area modeling in the finite element numerical calculation process, and the accuracy and efficiency of simulation calculation can be effectively improved; the method comprises the following steps: acquiring an initial tetrahedral mesh and an error indication returned by a solver; constructing an adjacency relation and a spatial index structure of the tetrahedron units; selecting the unit with the maximum error according to the error value, and inserting a new node into the midpoint of the longest side of the unit; updating a local grid topological structure; and finally, evaluating the grid quality through a volume-to-side ratio index, and carrying out Laplace smoothing operation on units with relatively poor quality to optimize the grid quality. According to the method, numerical solution errors can be dynamically responded, local encryption and structure optimization are cooperatively carried out, the grid quality is remarkably improved, and the method is particularly suitable for finite element self-adaptive modeling tasks in complex geometric scenes.

Description

Technical Field

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

[0002] The finite element method is a numerical simulation technique commonly used in engineering and scientific computing. Its computational accuracy and efficiency depend largely on the quality and distribution density of the mesh. In practical applications, the accuracy requirements often show local variations. Some areas require finer meshes due to drastic changes in physical field gradients, while other areas have lower requirements for mesh accuracy. Therefore, adaptive mesh encryption technology has become a key means to improve computational efficiency and accuracy.

[0003] Traditional mesh adaptation methods can be roughly divided into two categories: one is the encryption method based on geometric features, such as automatic encryption in areas with large boundary curvature; the other is the encryption method based on solution error feedback, that is, after the solver initially solves the problem, the mesh distribution is adjusted according to the error estimate; the latter is considered to be 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 encryption methods based on error feedback still face problems such as low encryption efficiency, significant impact on mesh quality, and inability to guarantee Delaunay properties.

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

[0005] In view of this, an embodiment of the present invention provides a tetrahedral mesh encryption method based on spatial octree indexing and error feedback to solve the problems of low efficiency, poor quality, and difficulty in meeting solver requirements in existing mesh adaptation technologies.

[0006] The technical solutions of the present invention are as follows:

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

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

[0009] S2: establishing an adjacency relationship based on the face-body correspondence in the initial tetrahedral mesh, for identifying cavity areas and updating the topology during the point insertion process;

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

[0011] S4: Selecting a maximum error indicator from the error indicator set, and finding its corresponding tetrahedron using the spatial octree, inserting point p at the midpoint of its longest edge according to the adjacency relationship to obtain an encrypted tetrahedron mesh; if the longest edge is a constrained edge, updating the constrained face and constrained edge;

[0012] S5: S4 is executed repeatedly until the encryption ratio of the number of meshes of the encrypted tetrahedral mesh reaches a specified threshold, thereby obtaining an encrypted tetrahedral mesh;

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

[0014] Optionally, establishing an adjacency relationship based on the face-body correspondence in the initial tetrahedral mesh includes:

[0015] S21: extract all face information of each tetrahedral unit according to the face-body correspondence in the tetrahedral mesh and create a unique identifier for it;

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

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

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

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

[0020] S32: traverse all tetrahedrons in the tetrahedron grid, insert the tetrahedron into the corresponding child node of the spatial octree according to its centroid position, and split it into 8 sub-blocks if the child node element exceeds the threshold;

[0021] S33: After the insertion point, the spatial index structure of the deleted unit and the newly generated unit is updated.

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

[0023] S51: searching for a set of tetrahedral units whose circumscribed circle contains the new point p based on the adjacency relationship and the spatial index to form a cavity area;

[0024] S52: remove all elements in the tetrahedral element set and extract their non-shared triangles as the cavity boundary;

[0025] S53: The new point p and the cavity boundary triangle form a new tetrahedron unit, completing the tetrahedron 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 units with quality lower than a threshold and extract their vertex sets;

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

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

[0032] Through the above steps, the present invention realizes high-quality tetrahedral mesh adaptive encryption based on error feedback, which can effectively focus on high-error areas, improve calculation accuracy, reduce unnecessary mesh redundancy, and improve simulation efficiency;

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

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

[0035] 2. Support efficient search and update: By building adjacency relationships and spatial octrees, the positioning and reconstruction efficiency during the encryption phase is improved.

[0036] 3. Improved quality control mechanism: Quality assessment and smooth optimization mechanisms are introduced after encryption to effectively avoid the generation of thin elements and significantly improve the overall mesh quality.

[0037] 4. Strong versatility: This method does not rely 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, the present invention provides an efficient, stable, and quality-controlled tetrahedral mesh adaptive encryption method, which is particularly suitable for finite element simulation tasks with high requirements on mesh quality and local accuracy, and has important engineering application value and promotion prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in 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 derive other drawings based on these drawings without inventive effort.

[0040] Figure 1 The present invention is a flowchart showing a tetrahedral mesh encryption method based on spatial octree indexing and error feedback according to an exemplary embodiment.

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

[0042] Figure 3 FIG. 4 is a schematic diagram of an initial tetrahedral mesh of a patch antenna according to an exemplary embodiment.

[0043] Figure 4 FIG. 1 is a schematic diagram of a data structure for establishing an adjacency relationship in step S2 according to an exemplary embodiment.

[0044] Figure 5 FIG. 4 is a schematic diagram of a spatial octree spatial index structure in step S3 according to an exemplary embodiment.

[0045] Figure 6 3 is a schematic diagram of Bowyer-Watson interpolation points and cavity reconstruction in step S5 according to an exemplary embodiment.

[0046] Figure 7 3 is a comparison diagram of a patch antenna key area before and after encryption according to an exemplary embodiment.

[0047] Figure 8 FIG. 1 is a comparison diagram of an S-parameter curve of a patch antenna simulation result according to an exemplary embodiment and a result of commercial software HFSS.

[0048] Figure 9 FIG. 4 is a geometric diagram of a horn antenna according to an exemplary embodiment.

[0049] Figure 10 FIG. 4 is an S-parameter curve diagram of a horn antenna simulation result according to an exemplary embodiment. DETAILED DESCRIPTION

[0050] Here, exemplary embodiments will be described in detail, examples of which are shown 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 the present application. Instead, they are only examples of devices and methods consistent with some aspects of the present application as detailed in the appended claims.

[0051] The terms used in this application are for the purpose of describing particular embodiments only and are not intended to be limiting of the present application. The singular forms "a", "an", "said", and "the" used in this application and the appended claims are intended to include plural forms, unless the context clearly indicates otherwise. It should also be understood that the term "and / or" used herein refers to and includes any and all possible combinations of one or more of the associated listed items.

[0052] Figure 1 The overall flow chart of the tetrahedral mesh encryption method based on spatial octree indexing and error feedback proposed by the present invention is shown. The method mainly includes the following steps:

[0053] S1: Get the initial tetrahedral mesh T, constraint face F, constraint edge S, error indicator set {e} returned by the finite element solver, and encryption ratio threshold θ of the model threshold ;

[0054] Specifically, the initial tetrahedral mesh T of the finite element simulation model is first obtained; then, the mesh is initially solved using a finite element solver to obtain an error indicator set {e}, where each element corresponds to a tetrahedral unit, which is used to reflect the size of the local solution error of the unit.

[0055] The finite element simulation model includes but is not limited to electromagnetic simulation, structural mechanics analysis, fluid dynamics simulation, heat conduction analysis, etc. In this embodiment, the microstrip patch antenna is used as the simulation object (refer to Figure 2 ) and horn antenna as simulation objects (reference Figure 9 ), of course not limited to this.

[0056] The error indication set {e} may provide absolute value information of unit errors, or may simply reflect the relative ranking relationship of errors between units, to guide subsequent encryption decisions.

[0057] The error set {e} is the key input of the adaptive encryption method; the error indication can be generated by residual-based, a posteriori estimation-based methods, or directly output by commercial software (such as ANSYS, COMSOL) solvers.

[0058] S2: Based on the face-body correspondence in the initial tetrahedral mesh T, establish an adjacency relationship to identify the cavity area and update the topology during the point insertion process; this step includes the following sub-steps:

[0059] S21: extract all face information of each tetrahedral unit according to the face-body correspondence in the tetrahedral mesh and create a unique identifier for it;

[0060] Specifically, this step aims to make the implicit topological relationship of the mesh explicit, laying the foundation for the efficient construction of the subsequent adjacency relationship. Taking the initial mesh of the microstrip patch antenna as an example, traverse each tetrahedral unit t in the mesh. i Each t i Defined by four vertices {v1, v2, v3, v4}, these form four triangular faces 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, independent of vertex order, the present invention employs a normalization process: the indices of the three vertices that comprise each face are sorted (e.g., in ascending order). For example, a face consisting of vertices with indices {5, 28, 13} is uniquely identified as (5, 13 28).

[0061] This standardized identification ensures that the same shared face is generated regardless of which adjacent tetrahedron is encountered during traversal, even if the local vertex order is different (e.g., {28, 5, 13}). This ensures accurate and unambiguous identification of shared faces in the mesh in subsequent steps, which is the prerequisite and key to establishing accurate adjacency relationships.

[0062] S22: Record the tetrahedral unit to which each face belongs according to 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 the tetrahedral unit to which it belongs; continuing with the patch antenna grid as an example, create a mapping M whose key is the face unique identifier generated in S21 and whose value is the tetrahedral unit index list containing the face.

[0064] Traverse all tetrahedrons t i All sides of J Calculate f j The unique identifier id(f j ). Look up the identifier in the map M:

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

[0066] If id(f j ) already exists, the current tetrahedron t i The index of M[id(f i )] is updated to [ t existing , t i ];

[0067] After the traversal is completed, the mapping M is checked again. If the length of the value list corresponding to a face identifier is 2, the face is an internal shared face.

[0068] All shared and boundary faces can be efficiently identified in a single pass, with a time complexity proportional to the number of mesh faces, making it highly efficient. Furthermore, a clear distinction is made between internal shared faces and model boundary faces, which is crucial for subsequent point insertion algorithms (especially when dealing with constrained edges and faces) and cavity identification, ensuring that the encryption process does not destroy the model's original boundary features.

[0069] S23: Constructing the adjacency graph between tetrahedrons based on shared faces;

[0070] Specifically, based on S22, the final adjacency graph N is constructed. This graph can be directly stored as a neighbor list of each tetrahedral unit; all entries whose values ​​are lists and length is 2 in the map M generated in S22 are traversed again. For each entry, such as M[id(f k )] = [t a , t b ], which shows that the tetrahedron t a and t b By sharing face k and are adjacent to each other. a Add t to the neighbor list b , while at t b Add t to the neighbor list a .

[0071] After the traversal is completed, each tetrahedral cell will have a list of all its direct neighboring cells. This complete set of adjacency data N is the core of local topological operations.

[0072] Precalculating and storing adjacency relationships transforms the time-consuming geometric search problem into an efficient graph traversal problem. In the subsequent Bowyer-Watson insertion step, when searching for the circumscribed sphere containing the newly inserted point (i.e., the cavity), this 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 grid, and significantly enhances the overall efficiency of the encryption algorithm.

[0073] like Figure 4 As shown, all tetrahedral units in the initial tetrahedral mesh T are traversed, their four faces are extracted, and which faces are shared by two units are recorded; the adjacency relationship N between the tetrahedral units is constructed to identify the topological structure 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 Tree based on the vertices V of the initial tetrahedral mesh T oct , used to quickly locate the spatial position of the tetrahedron; this step includes the following sub-steps:

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

[0076] Specifically, before constructing the octree, the spatial range of its root node needs to be determined. This range must be able to completely cover the entire model. Taking the horn antenna model as an example, first traverse all vertices in its initial grid T and find the maximum value of the x, y, and z coordinates of all vertices (x max , y max , z max ) and minimum value (x min , y min , z min ); Thus, a cuboid bounding box that tightly surrounds the model is obtained. In order to simplify the subsequent space division calculation, the cuboid is expanded into a cube; the lengths of the three sides L 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 starting point (x min , y min ,zmin ), with a side length of L max cube.

[0077] The advantage of this design is that using a cube as the root node ensures that each subsequent recursive partitioning generates eight sub-cubes of equal size and regular shape; this greatly simplifies the calculation of the spatial range of sub-nodes and the judgment logic of point or object attribution, making the octree construction and query algorithm more concise and efficient.

[0078] S32: traverse all tetrahedrons in the tetrahedron grid, insert the tetrahedron into the corresponding child node of the spatial octree according to its centroid position, and split it into 8 sub-blocks if the child node element exceeds the threshold;

[0079] Specifically, taking the initial grid of the horn antenna as an example, for each tetrahedral element t i , first calculate the coordinates of its geometric center of mass. Then, starting from the root node of the octree, insert the tetrahedron; the insertion process is a recursive process:

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

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

[0082] 3. Check whether the number of elements in the leaf node exceeds the preset threshold (for example, 10); if so, split the node into 8 child nodes (i.e., 8 child cubes) and move all the original elements in the parent node (including the newly inserted t i ) are redistributed to the corresponding child nodes according to their centroid positions;

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

[0084] This strategy allows the octree to adaptively reflect the spatial density distribution of the grid. In areas with complex model geometry and dense meshes (such as the throat and aperture edge of a horn antenna), the octree hierarchy is deeper and more refined; in areas with sparse meshes, the tree hierarchy is shallower. This makes spatial queries (such as locating cells near specific coordinates) logarithmically efficient (O(logN)), far superior to linear traversal (O(N)), greatly accelerating the process of locating high-error cells.

[0085] S33: After the insertion point, the spatial index structure of the deleted unit and the newly generated unit is updated.

[0086] Specifically, this step ensures the consistency and validity of the spatial index during dynamic mesh updates. After the insertion operation is completed, the local mesh topology changes: some old tetrahedrons (cells forming cavities) are deleted, and a batch of new tetrahedrons are generated. At this time, the octree must be updated synchronously.

[0087] For example, at an insertion point on the edge of a patch antenna, assuming that five old units are deleted and 12 new units are 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, the same insertion logic as S32 is executed: its centroid is calculated, and recursive insertion is performed starting from the root node. If the number of elements in a leaf node exceeds the limit, a split is triggered.

[0090] The benefits of this design are: it ensures that the octree index always accurately reflects the current grid status; this dynamic, real-time update mechanism is the basis for the entire adaptive encryption method to be able to proceed continuously and iteratively; if the index is not updated, subsequent error cell positioning and insertion operations will be based on outdated spatial information, resulting in algorithm errors or even failure; this step ensures the correctness and robustness of the entire encryption process.

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

[0092] S4: Locate high error areas and interpolate points; select the maximum error indicator e from the error indicator set {e} max , and use the spatial octree Tree oct Find its corresponding tetrahedron t i , on its longest side s longest The midpoint of the point p is inserted (based on the Bowyer-Watson interpolation algorithm) to obtain the encrypted tetrahedral mesh T'; if s longest It is a constraint edge, update the constraint surface F and constraint edge S;

[0093] Specifically, select the maximum error value indicator e from the error indicator set {e} max According to e max And use the spatial octree Tree oct Find its corresponding tetrahedron t i, inside the unit, find the longest side s longest , and calculate the midpoint p as the new node to be inserted.

[0094] Insert point p into the current tetrahedral mesh using the Bowyer-Watson interpolation algorithm, which includes:

[0095] 1. Use the spatial index O and adjacency relationship N to find all tetrahedrons whose circumscribed balls contain the point p, forming a cavity S;

[0096] 2. Delete the cells in the cavity and extract their boundary surfaces to form a boundary set B;

[0097] 3. Form a new tetrahedron with point p and each boundary triangle to complete the Delaunay reconstruction;

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

[0099] S5: Loop through S4 until the encrypted tetrahedral mesh's number of meshes reaches the specified threshold θ. threshold (e.g. 30% of the number of tetrahedral elements in the initial mesh) to obtain the encrypted tetrahedral mesh;

[0100] Specifically, calculate the encryption ratio of the current encrypted tetrahedral mesh T'; the encryption ratio calculation formula is θ=(N T’ -N T ) / N T , where N T’ is the current encrypted tetrahedral mesh volume of T', N T is the mesh size of the initial tetrahedral mesh T;

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

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

[0103] To avoid degenerate elements and thin elements introduced by interpolation points, the updated mesh needs to be optimized. The method is as follows:

[0104] 1. Calculate the quality index of all tetrahedral elements , which is defined as:

[0105]

[0106] Where V is the volume of the tetrahedron, l i is the length 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 Laplace smoothing on these vertices, i.e. update them to the geometric centers of their first-order neighbors;

[0109] 4. Repeat the optimization process until the quality of all units is greater than the threshold , or the maximum number of iterations is reached.

[0110] Example 1: Patch Antenna Simulation

[0111] In order to verify the effectiveness of the method described in the present invention, this embodiment uses a microstrip patch antenna as a simulation object; Figure 2 As shown in the figure, it is the geometric model of the patch antenna; Figure 3 Shown is the initial tetrahedral mesh of the model.

[0112] This initial mesh is used as input and adaptively encrypted using steps S1-S6 described in the present invention. During the electromagnetic simulation solution process, errors are usually concentrated near the patch edges, feed points, and radiation boundaries. This method automatically encrypts these key areas based on the error indications returned by the solver, while keeping the mesh in other areas relatively sparse.

[0113] like Figure 7 As shown in the figure, it is a comparison diagram of the grids of the edge area of ​​the patch antenna before and after the adaptive encryption by the method of the present invention; it can be clearly seen from the figure that the encrypted grid ( Figure 7 The density at the edge of the patch is significantly higher than that of the initial mesh ( Figure 7 The left side of the image is shown in Figure 3), while the grid density away from this area does not change much, which intuitively proves the adaptability and accuracy of the method of the present invention.

[0114] like Figure 8 As shown in FIG, the S parameter curve obtained by simulating the mesh encrypted by the method of the present invention is compared with the simulation results of the mainstream commercial software ANSYS HFSS in the industry; it can be seen from the figure that the two curves are highly consistent, which verifies that the method of the present invention can efficiently complete the adaptive mesh encryption while ensuring the calculation accuracy, and its results are accurate and reliable.

[0115] Example 2: Horn Antenna Simulation

[0116] In order to further illustrate the versatility of the present invention, this embodiment uses a horn antenna as a simulation object; Figure 9 Figure 2 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 area with the most drastic field gradient changes is located at the throat and the edge of the aperture.

[0117] The method of the present invention is also used to adaptively encrypt the initial grid of the horn antenna; Figure 10 The figure shows the S-parameter curve obtained by the simulation method of the present invention. The results show that the method can accurately capture the resonant characteristics of the 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 the present application is not limited to the exact structures described above and shown in the drawings, and that various modifications and changes may be made without departing from the scope thereof; the scope of the present 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 faces, constraint edges, error indicator set returned by the finite element solver, and encryption ratio threshold of the model; S2: establishing an adjacency relationship based on the face-body correspondence in the initial tetrahedral mesh, for identifying cavity areas and updating the topology during the point insertion process; S3: Construct a spatial octree based on the vertices of the initial tetrahedron mesh to quickly locate the spatial position of the tetrahedron; S4: Selecting a maximum error indicator from the error indicator set, finding its corresponding tetrahedron using the spatial octree, inserting a point at the midpoint of its longest side according to the adjacency relationship, and obtaining an encrypted tetrahedron mesh; if the longest side is a constrained side, updating the constrained face and constrained side; S5: S4 is executed repeatedly until the encryption ratio of the number of meshes of the encrypted tetrahedral mesh reaches a specified threshold, thereby obtaining an encrypted tetrahedral mesh; S6: Perform quality optimization processing on the encrypted tetrahedral mesh to obtain an adaptively encrypted tetrahedral mesh.

2. The method according to claim 1, characterized in that Establishing an adjacency relationship based on the face-body correspondence in the initial tetrahedral mesh includes: S21: extract all face information of each tetrahedral unit according to the face-body correspondence in the tetrahedral mesh and create a unique identifier for it; S22: Record the tetrahedral unit to which each face belongs according to the unique identifier of each face, and determine whether the face is a shared face; S23: Construct an adjacency graph between tetrahedrons based on shared faces.

3. 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 as the root node of the spatial octree based on the mesh vertices; S32: traverse all tetrahedrons in the tetrahedron grid, insert the tetrahedron into the corresponding child node of the spatial octree according to its centroid position, and split it into 8 sub-blocks if the child node element exceeds the threshold; S33: After the insertion point, the spatial index structure of the deleted unit and the newly generated unit is updated.

4. The method according to claim 1, wherein Inserting point p at the midpoint of its longest side according to the adjacency relationship includes: S41: searching for a set of tetrahedral units whose circumscribed circle contains the new point p based on the adjacency relationship and the spatial index to form a cavity area; S42: remove all elements in the tetrahedral element set and extract their non-shared triangles as the cavity boundary; S43: The new point p and the cavity boundary triangle form a new tetrahedron unit, completing the tetrahedron reconstruction; S44: Update the grid data structure and index structure.

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

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