A multi-level partitioning constraint method and apparatus for general mesh deformation problems
By employing a multi-level partitioning constraint method and apparatus, the problem of high computational cost for radial basis function mesh deformation was solved, achieving efficient computation of large-scale mesh deformation and ensuring mesh quality.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SUN YAT SEN UNIV
- Filing Date
- 2026-01-30
- Publication Date
- 2026-06-02
Smart Images
Figure CN122134971A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of mesh deformation technology in numerical computation, and in particular to a multi-level partitioning constraint method and apparatus for general mesh deformation problems. Background Technology
[0002] In numerical computation fields such as computational fluid dynamics, computational solid mechanics, and multiphysics coupling simulations, the computational mesh is the foundation for numerical solutions, and its quality directly affects the accuracy of the simulation results. In simulations involving geometric changes, such as aeroelasticity of airfoils, fluid-structure interaction, and moving boundary problems, it is often necessary to deform the original computational mesh to adapt to the new boundary shape, while maintaining element quality and mesh consistency as much as possible.
[0003] Radial basis function (RBF) mesh deformation is an important mesh deformation technique. This method constructs a global radial basis function interpolation system to calculate the displacements of other nodes by using the displacement information of moving boundary nodes as known conditions, thus maintaining the quality and consistency of the mesh after deformation. RBF mesh deformation has advantages such as high interpolation accuracy, smooth deformation, and independence from mesh topology, making it widely used in complex geometries and large deformation scenarios.
[0004] However, existing radial basis function mesh deformation methods require solving a system of linear equations with a scale comparable to the number of nodes on the moving boundary surface when solving for interpolation coefficients. The computational cost increases significantly with increasing mesh size. When the number of mesh nodes is large, the matrix solution time increases dramatically, leading to a significant decrease in overall computational efficiency. To address this issue, existing technologies introduce greedy algorithms to reduce the number of nodes involved in coefficient calculation, thereby improving computational efficiency to some extent. However, in large-scale mesh scenarios, the efficiency improvement brought by greedy algorithms remains limited and cannot effectively meet the computational requirements of efficient mesh deformation. Summary of the Invention
[0005] This invention provides a multi-level partitioning constraint method and apparatus for general mesh deformation problems, which solves the technical problem of high computational cost of existing greedy algorithms when calculating radial basis function mesh deformation.
[0006] The first aspect of this invention provides a multi-level partitioning constraint method for a general mesh deformation problem, comprising: Read the mesh data and set the target deformation of the object surface nodes; The object surface is partitioned into multiple levels according to the hierarchy: in the first level, the entire object surface is partitioned as a partitioning object to obtain several coarse partitions; in other levels, each partition of the previous level is further subdivided to obtain the partition array of that level. Based on the common boundary between the sub-partitions obtained by the partition at the next level, the candidate node set of this level is extracted and stored in correspondence with the partitioning result for subsequent construction of the local radial basis function interpolation system. Based on the partitioning results, the following steps are executed sequentially in a loop: Based on the deformation results of the previous level, calculate the target deformation amount of the current level; Construct the local radial basis function interpolation system for each partition based on the partitioning results at the current level: In all levels except the last level, each partition uses a greedy algorithm to select support points within its corresponding candidate node set to construct the radial basis function interpolation system. At this time, the allowable interpolation error is a preset interpolation error to ensure that the candidate node set is accurately deformed. At the final level, each partition selects the support points within its set of nodes using the greedy algorithm to construct the radial basis function interpolation system. At this point, the permissible interpolation error is set to a preset interpolation error to achieve the expected deformation accuracy. Mesh deformation at this level is performed based on the interpolation coefficients of the local radial basis function interpolation system. The radial basis function deformation is applied at the first level; Before mesh deformation, other levels construct the object surface nodes participating in the deformation of this level into a K-dimensional tree for distance search. The volume mesh nodes are pre-screened by comparing the subtree bounding boxes stored in the K-dimensional tree nodes with the distance supported by the double-constrained radial basis function. Two nearest neighbor searches are performed on the pre-screened volume mesh nodes to obtain their distances d(x,Ω1) and d(x,Ω2) to the local deformation Ω1 and the non-deformation region Ω2. Based on this, the double-constrained radial basis function is applied to control the deformation range and avoid mesh intersection. The node positions are updated based on the displacement calculated at this level to obtain the deformed object surface mesh and volume mesh; After all levels have been calculated, the deformed mesh is output.
[0007] Furthermore, the candidate node set is the set of boundary nodes shared by adjacent sub-partitions within a partition at a certain level; at the last level, since there is no next level of subdivision, all nodes within it together constitute the candidate node set.
[0008] Furthermore, the shared boundary node is a node on the common boundary of adjacent sub-partitions at a certain level.
[0009] Furthermore, the dual-restricted radial basis function takes the smaller value of the two restriction functions: Ψ(x) = min(Ψ1(x), Ψ2(x)).
[0010] Further, Ψ1(x) is a function related to the distance d(x, Ω1) from the node coordinate x to the local deformation region Ω1. The definition of Ψ1(x) is as follows: when d(x, Ω1) < D, Ψ1(x) = 1 - d(x, Ω1) / D; in other cases, Ψ1(x) = 0.
[0011] Further, the constraint function Ψ2(x) is a function related to both the distance d(x, Ω1) from the node coordinate x to the local deformation region Ω1 and the distance d(x, Ω2) to the non-deformation region Ω2: Ψ2(x) = min(1, max(0, (α * d(x, Ω2) - d(x, Ω1)) / β)).
[0012] Further, in each level except the first level, the calculation of the distances d(x, Ω1) and d(x, Ω2) from the node coordinate x to the local deformation Ω1 and the non-deformation region Ω2 includes: before the start of each level, the physical surface nodes participating in the deformation of this level are constructed into a single K-dimensional tree. In each node of the K-dimensional tree, the spatial bounding box of the physical surface nodes contained in the corresponding subtree and the maximum support distance of the dual-constrained radial basis function within this subtree are stored. For any volume grid node, starting from the root node, a pre-screening is performed according to the magnitude relationship between the minimum distance from this node to the subtree bounding box and the maximum support distance to prune the subtrees that cannot contribute. For the volume grid nodes passing the pre-screening, a first nearest neighbor search is performed to obtain the distance from this node to the nearest physical surface node and regard its corresponding partition as Ω1, thereby obtaining d(x, Ω1). Then, after excluding the physical surface nodes corresponding to this local deformation region, a second nearest neighbor search is performed to obtain the nearest distance d(x, Ω2) from this node to the remaining physical surface nodes; Further, the update of the node position includes: superimposing the node deformation amount calculated in this level with the node position in the previous level to obtain the updated node position, and using it as the input for constructing the radial basis function interpolation system in the next level; Further, the output of the deformed grid includes: saving the updated node position as an independent data file for visualization and grid quality inspection.
[0013] A multi-level partition constraint device for a general grid deformation problem provided in the second aspect of the present invention includes: A grid reading and deformation amount calculation module, configured to read the grid data, set the target deformation amount of the physical surface nodes, and calculate the target deformation amount of the current level based on the deformation result of the previous level in each level; A multi-level partition module, configured to partition the physical surface by level, extract the set of shared boundary nodes at each level, and store them together with the partition result; A radial basis function interpolation system construction module is used to select candidate nodes according to the method described in the first aspect of the present invention in the first level, intermediate levels and the final level. In the first level, the shared boundary nodes between all coarse partitions are used as candidate nodes. In the intermediate levels, the shared boundary nodes of the sub-partitions of the partitions are used as candidate nodes. In the final level, the object surface nodes inside the partitions are used as candidate nodes. In each level, a greedy algorithm is used to construct a local radial basis function interpolation system in the corresponding candidate node set and solve the interpolation coefficients. The object surface interpolation error threshold of the radial basis function interpolation system is uniformly set to a preset deformation accuracy. The mesh deformation module is used to perform mesh deformation based on the interpolation coefficients of the local radial basis function interpolation system at each level of the calculation. The radial basis function is used at the first level, and the dual-constraint radial basis function is used at other levels to control the deformation range to avoid inter-interval mesh intersections. The distance search module is used to construct the object surface nodes participating in the deformation of the current level into a K-dimensional tree in each level except the first level, and to store the spatial bounding box of the corresponding subtree and the maximum support distance of the double-constrained radial basis function in the nodes of the K-dimensional tree. The subtree is pre-screened by comparing the minimum distance from the volume mesh node to the subtree bounding box with the support distance. The volume mesh nodes that pass the pre-screening are subjected to two nearest neighbor searches to calculate their distances to the moving boundary and the stationary boundary, respectively, so as to provide distance input for the double-constrained radial basis function. The object surface update module is used to update the node position after the calculation is completed at each level, and use the update result as the input for the next level. The mesh output module is used to output the final deformation result after the calculation is completed at all levels, and save it as a separate data file for visualization and mesh quality inspection.
[0014] A computer device provided in a third aspect of the present invention includes a memory and a processor, wherein the memory stores a computer program, and when the program is executed by the processor, it implements the steps of the multi-level partitioning constraint method for the general mesh deformation problem as described in any of the preceding claims.
[0015] The fourth aspect of the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed, implements a multi-level partitioning constraint method for a general mesh deformation problem as described in any of the preceding claims.
[0016] The fifth aspect of the present invention provides a computer program product comprising a computer program / instruction, wherein when the program / instruction is executed by a processor, it implements a multi-level partitioning constraint method for a general mesh deformation problem as described in any of the preceding claims.
[0017] As can be seen from the above technical solutions, the present invention has the following advantages: The above-described solution of the present invention provides a multi-level partitioning constraint method for general mesh deformation problems, comprising: reading mesh data and setting the target deformation amount of the object surface nodes; partitioning the object surface into multiple levels according to the hierarchy, and extracting a candidate node set during the subdivision process for subsequent construction of a local radial basis function interpolation system; constructing a local radial basis function interpolation system in each level partition according to different point selection strategies, wherein, except for the last level, support points are selected within the shared boundary node set with a preset deformation accuracy as the error, and in the last level, support points are selected within the nodes of the partition with a preset deformation accuracy as the error; performing mesh deformation step by step based on the radial basis function interpolation system, and adopting a method in the first level... Radial basis function deformation is performed, with double-constrained radial basis functions applied at other levels to control the deformation range and avoid mesh crossings between partitions. In all levels except the first level, the object surface nodes participating in the deformation at that level are constructed as K-dimensional trees. Nodes are pre-screened by comparing the minimum distance from the volume mesh node to the subtree bounding box with the support distance. Two nearest neighbor searches are performed on the pre-screened volume mesh nodes to obtain their distances to the moving and stationary boundaries, respectively, providing distance inputs for the double-constrained radial basis functions. After the calculation at each level is completed, the node positions are updated and used as inputs for the next level. Finally, the deformed object surface mesh and volume mesh are obtained, and the results are output as independent data files. By employing a multi-level partitioning and hierarchical point selection strategy, this invention effectively reduces the computational complexity of constructing and solving the radial basis function interpolation system, decreasing the global complexity from (ΣN_(k,i))^3 to Σ(N_(k,i) / n_k)^3, where k represents the level, i represents the partition number, and n_k represents the subdivision number of the partition in the next level, thus significantly improving the computational efficiency of large-scale mesh deformation. By using minimal interpolation errors in intermediate levels to ensure accurate deformation of partition boundary nodes, the accumulation of errors across levels is avoided. Pre-set errors are used in the final level to ensure overall accuracy. The dual-constraint radial basis function effectively limits the deformation range and excludes non-deformable regions. Combined with distance calculations based on K-dimensional tree pre-screening and two nearest neighbor searches, the overhead of searching for dynamic and static boundary distances in localized deformation is significantly reduced, ensuring mesh quality and avoiding mesh intersections. Therefore, this invention exhibits excellent performance and reliability in large-scale mesh deformation and other applications requiring large-scale computations using radial basis function interpolation. Attached Figure Description
[0018] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.
[0019] Figure 1A flowchart illustrating the steps of a multi-level partitioning constraint method for a general mesh deformation problem provided in an embodiment of the present invention; Figure 2 This is a schematic diagram of mesh deformation provided in an embodiment of the present invention.
[0020] Figure 3 This is a schematic diagram of the deformed mesh provided in an embodiment of the present invention.
[0021] Figure 4 The diagram above shows the deformation of the object surface mesh at level 1 according to an embodiment of the present invention. The upper part shows the partitioning of the object surface mesh at this level, and the lower part shows the comparison between the object surface mesh after deformation at level 1 and the target deformation result. Figure 5 The diagram above shows the deformation of the object surface mesh at level 2 according to an embodiment of the present invention. The upper part shows the partitioning of the object surface mesh at this level, and the lower part shows the comparison between the object surface mesh after deformation at level 2 and the target deformation result. Figure 6 The diagram above shows the partition deformation area of level 1 provided in the embodiment of the present invention. The diagram below shows the deformation area when the partition of level 1 is deformed. Figure 7 The diagram above shows the deformation of the object surface mesh at level 3, as provided in the embodiment of the present invention. The upper diagram shows the partitioning and deformation of the object surface mesh at this level, and the lower diagram shows the comparison between the object surface mesh after deformation at level 3 and the target deformation result. Figure 8 This is a schematic diagram of the distance search K-dimensional tree plus pre-screening algorithm provided in an embodiment of the present invention; Figure 9 This is a view of the surface of the M6 wing before and after deformation, provided in an embodiment of the present invention. Figure 10 This is a mesh display of the M6 wing after deformation using a multi-level partitioning restriction method, as provided in an embodiment of the present invention. Figure 11 This is a structural block diagram of a multi-level partitioning constraint device for a general mesh deformation problem provided in an embodiment of the present invention. Detailed Implementation
[0022] This invention provides a multi-level partitioning constraint method and apparatus for general mesh deformation problems, which solves the problem of high computational cost of existing greedy algorithms when calculating radial basis function mesh deformation.
[0023] To make the objectives, features, and advantages of this invention more apparent and understandable, the technical solutions of the embodiments of this invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the embodiments described below are only some embodiments of this invention, and not all embodiments. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this invention.
[0024] Terminology Explanation Mesh data: This includes the spatial coordinates of the nodes of the volume mesh and the object surface mesh, as well as their topological connectivity.
[0025] Object surface mesh: The boundary mesh that participates in geometric deformation and provides boundary displacement conditions.
[0026] Volume mesh: refers to the set of mesh elements within the domain enclosed by the surface of an object (surface elements in two dimensions, volume elements in three dimensions).
[0027] Target deformation of surface nodes: The displacement of surface nodes determined by boundary conditions or external inputs, which is used as known input in this method.
[0028] Hierarchy: refers to the hierarchical numbering of the object surface mesh according to the hierarchical partitioning.
[0029] Subpartition: A partition obtained by further subdividing a partition at the next lower level.
[0030] Shared boundary node: A node on the common boundary of adjacent sub-partitions at a certain level.
[0031] Candidate node set: In non-last level, it is the set of shared boundary nodes of adjacent sub-partitions within the partition; in last level, it is the set of all object surface nodes within the partition.
[0032] K-tree (KD-Tree): A tree-like data structure used to spatially organize the object surface nodes participating in the current level of deformation. Each tree node stores the spatial bounding box of the nodes in its subtree and the associated maximum constraint radius, which is used to achieve fast pre-screening and nearest neighbor search.
[0033] Local radial basis function interpolation system: refers to a system of linear equations established within a single partition at a certain level by using a greedy algorithm to select support points from the candidate node set corresponding to that partition, under the condition of satisfying a preset interpolation error threshold; the solution result is the interpolation coefficient of that partition.
[0034] Greedy algorithm: refers to a point selection method that iteratively evaluates the interpolation error on the candidate node set and gradually adds the node with the largest error to expand the support point set until the interpolation error of all candidate nodes does not exceed a set threshold.
[0035] Support point: A set of control nodes selected from the candidate node set by the greedy algorithm for constructing a local radial basis function interpolation system.
[0036] Radial basis function Radial basis function φ(η): A function with the Euclidean distance r = ||x - x_i|| between the node and the support point as the independent variable. In the embodiment of the present invention, the Wendland proposed C2 compactly supported radial basis function is adopted, and its definition is: When η ≤ 1, φ(η) = (1 - η)^4 * (4η + 1); when η > 1, φ(η) = 0; where η = ||x - x_i|| / R, and R is the support radius for controlling the compact support range of the function.
[0037] Interpolation coefficient (denoted as α_i): The weight coefficient obtained by solving the local radial basis function interpolation system, which is used for interpolating and calculating the displacement of the node.
[0038] Radial basis function interpolation formula for node displacement: Δx = Σ_(i = 1)^(N_c) α_i * φ(η_i), where N_c is the total number of support points.
[0039] Doubly constrained radial basis function: A doubly constrained radial basis function that excludes the stationary boundary from the deformation region by introducing a constraint function into the radial basis function interpolation formula for node displacement: Δx = Σ_(i = 1)^(N_c) α_i * Ψ(x) * φ(η_i); where, Ψ(x) = min(Ψ1(x), Ψ2(x)); Ψ1(x) is a constraint function related to the distance d(x, Ω1) from the node coordinate x to the local deformation region Ω1. The definition of Ψ1(x) is: when d(x, Ω1) < D, Ψ1(x) = 1 - d(x, Ω1) / D; in other cases, Ψ1(x) = 0; Ψ2(x) is a constraint function related to both the distance d(x, Ω1) from the node coordinate x to the local deformation region Ω1 and the distance d(x, Ω2) to the non-deformation region Ω2: Ψ2(x) = min(1, max(0, (α * d(x, Ω2) - d(x, Ω1)) / β)).
[0040] Distance function d(x, Ω): The minimum Euclidean distance from point x to set Ω.
[0041] Parameter D: The support distance for controlling the local deformation range in the doubly constrained radial basis function. When the distance from the point to the local deformation region is greater than this parameter, the deformation effect decays to zero.
[0042] Parameters α and β: In the double-constrained radial basis function, these are adjustment parameters used to adjust the constraint function. Their function is to control the position and width of the transition zone between the local deformable domain and the non-deformable domain, so as to ensure that the non-deformable surface is effectively excluded from the deformable region.
[0043] Update node position: After calculating the displacement of the nodes at this level, the displacement of this level is superimposed on the coordinates of the nodes at the previous level to obtain the updated coordinates, and the updated coordinates are used as the input for the next level.
[0044] Please see Figure 1 , Figure 1 A flowchart illustrating the steps of a multi-level partitioning constraint method for a general mesh deformation problem provided in an embodiment of the present invention.
[0045] This embodiment provides a multi-level partitioning constraint method for a general mesh deformation problem, including: Step 101: Read the mesh data and set the target deformation of the object surface nodes.
[0046] Mesh data refers to data including the spatial coordinates and topological connections of nodes in both volumetric and surface meshes. For example... Figure 2 As shown, an example of two-dimensional grid data includes several nodes and line elements formed by connecting the nodes.
[0047] The target deformation of a surface node refers to the surface displacement information determined by boundary conditions or external inputs. For example, in computational fluid dynamics simulations involving wing flutter, the deformation of the wing surface can be obtained through aeroelastic calculations. In mesh deformation problems, the surface deformation is typically used as a known input condition. Figure 2 The diagram illustrates the known target deformation amount and the target deformation result. Figure 3 This illustrates the mesh obtained after deformation of the initial mesh.
[0048] Step 102: Divide the object surface into multi-level partitions according to the hierarchy.
[0049] Multi-level partitioning refers to the process of dividing an object surface step by step according to hierarchy: At the first level, the entire surface is treated as a whole and divided into sections. At other levels, each partition of the previous level is further divided to obtain multiple sub-partitions for that level.
[0050] Figure 4 , Figure 5 and Figure 7 The diagram illustrates the partitioning of the object surface at different levels. Taking the illustrated case as an example, in level 1, the object surface is divided into 2 partitions; in level 2, each partition of the previous level is further divided into two sub-partitions; in level 3, the final level, no further subdivision is performed.
[0051] It should be noted that the partitioning method can be implemented using mature mesh generation tools. In the embodiments of this invention, the mesh partitioning routine METIS_PartMeshDual provided by the open-source software package METIS developed by Karypis et al. is used. This routine is a public calling interface provided by METIS, and its function is to partition based on the dual graph of the mesh. The input includes the number of mesh cells, the number of nodes, and the connection relationship between cells and nodes, and the output is the partition number to which each cell belongs.
[0052] It should be noted that METIS is an existing publicly available technology, and its partitioning algorithm and implementation method have been widely used in the fields of computer science and engineering. This invention uses it only as an optional implementation method to complete the multi-level partitioning process, so as to ensure the rationality of each level of partitioning and computational efficiency.
[0053] Step 103: Extract the candidate node set.
[0054] The candidate node set refers to the set of object surface nodes used to perform greedy point selection at a certain level. In this invention, the candidate node set at the first level is obtained by merging the shared boundary nodes between coarse partitions, the candidate node sets at the intermediate levels are the sets of shared boundary nodes between sub-partitions of each partition, and the candidate node set at the final level is all object surface nodes within the partition.
[0055] The shared boundary node set refers to the set of nodes contained on the common boundary between adjacent sub-partitions in a certain level of partitioning.
[0056] Figure 4 , Figure 5 and Figure 7 This demonstrates the candidate node sets extracted from the object surface at different levels. Taking the illustrated example, Figure 4 The purple dots represent candidate nodes of the first level (composed of boundary nodes shared by the intervals). Figure 5 The green dots represent candidate nodes at level 2 (candidate nodes for each partition are a combination of boundary nodes shared by its subpartitions). Figure 7 The set of candidate nodes at the final level is displayed, with line segments of different colors corresponding to object surface nodes within the corresponding partitions.
[0057] Step 104: Calculate the target deformation amount of the current level based on the deformation results of the previous level.
[0058] The target deformation at the current level refers to: df_k = x_final - x_(k-1) Where x_final represents the node coordinates of the target deformation result; x_(k-1) represents the node coordinates of the previous level deformation result.
[0059] It should be noted that, in this embodiment of the invention, the residual error of the previous level is used as the target deformation amount of the current level, and the final target deformation result is gradually approximated through a layer-by-layer correction method. This method ensures that the deformation error of each level is small compared to the previous level, thereby reducing the complexity of solving the interpolation coefficients and calculating the deformation in a single operation, avoiding the high computational overhead caused by directly solving the interpolation coefficients globally and calculating the overall deformation, and significantly improving the overall computational efficiency while ensuring global accuracy.
[0060] Step 105: Construct a local radial basis function interpolation system.
[0061] Constructing a local radial basis function interpolation system refers to using a greedy algorithm to select support points from the candidate node set corresponding to a certain partition, establishing a set of interpolation equations under the condition of satisfying a preset interpolation error threshold, and obtaining the interpolation coefficients of the partition by solving the set of equations, which are then used to interpolate the displacement of the grid nodes.
[0062] like Figure 4 , Figure 5 As shown, since the greedy algorithm selects support points from the candidate node set under the condition of satisfying the preset interpolation error threshold, all nodes in the candidate node set are adjusted to the target position within the tolerance range. Figure 7 As shown, the candidate nodes of the final stage are the object surface nodes within each partition. Therefore, after the final stage deformation, all object surface nodes are adjusted to the target positions within the tolerance range.
[0063] Greedy algorithms refer to an optimization strategy for traditional radial basis function interpolation methods. They reduce the size of the radial basis function interpolation system by progressively selecting control points, thereby significantly reducing computational overhead while achieving a preset accuracy. The basic process includes: Step 1, Set the interpolation error of the object surface: Set the maximum allowable absolute error between the deformed object surface mesh nodes and the target position based on the radial basis function method; Step 2, Select the initial point set: Specify the initial support point set P0, usually any node can be selected as the starting point; Step 3, Interpolation Calculation and Error Assessment: Calculate the radial basis function based on the current support point set, and calculate the current object surface interpolation error; Step 4, Add support points: Add the node with the largest current error to the support point set; Step 5, terminate iteration: Repeat steps 3 and 4 until the interpolation error of the object surface reaches the value set in step 1.
[0064] Interpolation coefficients refer to a set of weighting coefficients obtained by solving the radial basis function interpolation equations constructed from the support points.
[0065] It should be specifically noted that in the traditional greedy algorithm, in order to evaluate the interpolation error, it is usually necessary to calculate for all the nodes on the physical surface and establish a coefficient matrix of a corresponding scale, resulting in a significant increase in the computational overhead with the increase in the number of nodes. In the embodiments of the present invention, it is only necessary to calculate the interpolation error on the candidate node set of each partition and adjust the support point set accordingly. This improvement not only effectively reduces the scale of the coefficient matrix, but also reduces the computational amount of error evaluation in each iteration, thus significantly improving the overall efficiency of the greedy algorithm in the multi-level partition radial basis function grid deformation.
[0066] Step 106, perform grid deformation based on the local radial basis function interpolation system.
[0067] Grid deformation refers to the process of calculating the displacement of each node using the interpolation coefficients obtained for the partition to which the node belongs in the previous step and updating the node positions accordingly.
[0068] The calculation formula for the node displacement is: Δx=Σ_(i=1)^(N_c) α_i*φ(||(x-x_i) / R||); x refers to the node to be calculated; x_i refers to the support point; α_i refers to the interpolation coefficient obtained through the local radial basis function interpolation system; N_c refers to the total number of support points of the radial basis function interpolation system; φ refers to the radial basis function, with the distance between the node and the support point as the independent variable.
[0069] In non-first-level partitions, the above formula needs to introduce the double-restricted radial basis function Ψ(x): Δx=Σ_(i=1)^(N_c) α_i*Ψ(x)*φ(η_i); Ψ(x) refers to the double-restricted radial basis function, which is used to limit the deformation influence range and exclude the non-deformed area, and is defined as Ψ(x)=min(Ψ1(x), Ψ2(x)).
[0070] Ψ1(x) refers to the restriction function related to the distance d(x,Ω1) from the node coordinate x to the local deformation area Ω1, and is defined as: when d(x,Ω1)<D, Ψ1(x)=1- d(x,Ω1) / D; in other cases, Ψ1(x)=0.
[0071] Ψ2(x) refers to the restriction function related to both the distances d(x,Ω1) and d(x,Ω2) from the coordinate x to the local deformation area Ω1 and the non-deformed area Ω2: Ψ2(x)=min(1,max(0,(α*d(x,Ω2)- d(x,Ω1)) / β)).
[0072] It should be noted that in the first-level partition, the radial basis function interpolation formula is directly used for coarse deformation; while in non-first-level partitions, by introducing double-constrained radial basis functions into the interpolation formula, the local scope of action can be effectively limited, avoiding mutual interference between deformations in different partitions, thereby preventing mesh intersection and maintaining the overall mesh quality.
[0073] The double-restricted radial basis function involves calculating the distance from nodes to local regions. To improve the efficiency of distance search, this invention constructs a global K-dimensional tree for all object surface nodes participating in the current level deformation at the beginning of the level where the double-restricted radial basis function needs to be calculated, and implements the following distance search strategy on this data structure: First, in each node of the K-tree, the axis-aligned bounding box of the object surface nodes contained in its subtree and the maximum support distance of the double-constrained radial basis function within that subtree are stored for subsequent pre-screening. For any volume grid node x, the K-tree is traversed starting from the root node. When the lower bound of the minimum distance from x to the bounding box of a certain subtree is greater than the maximum support distance of that subtree, it can be determined that the deformation contribution of all object surface nodes in that subtree to x is zero, so the subtree is directly pruned, and only the subtrees that may affect x are retained.
[0074] Then, for the pre-screened volume mesh nodes, two nearest neighbor searches are performed: the first searches for the nearest surface node across the entire K-dimensional tree to obtain the nearest distance d(x,Ω1) from the volume mesh node x to the locally deformed region and the corresponding partition number; the second nearest neighbor search is performed again after excluding the surface nodes of that partition to obtain the nearest distance d(x,Ω2) from the volume mesh node x to the non-deformed region. Substituting these two distances into the constraint function Ψ2(x) completes the distance input for the double-constrained radial basis function.
[0075] like Figure 4 As shown, at level 1, the entire surface mesh is partitioned as a whole, and its candidate node set is represented by purple points. A greedy algorithm is used to select points on the candidate node set, with a preset deformation accuracy as the allowable error. The radial basis function interpolation system constructed based on this support point set completes the first coarse deformation of the surface. At this point, all nodes in the candidate node set have reached the target position within the allowable error range, but the deformation results of other nodes still have a large gap compared with the target deformation results.
[0076] like Figure 5 and Figure 7As shown, in levels 2 and 3, the residual deformation error of the previous level is used as the new deformation target. Within each partition, a local radial basis function interpolation system is reconstructed using a set of candidate nodes to gradually correct the deformation of the object surface. The figure shows that the deformation result of level 2 is closer to the target shape than that of level 1, while the result of level 3 has brought the deformation error to within the preset accuracy range.
[0077] Figure 6 The diagram illustrates the deformation region of a certain partition when using double-constrained radial basis functions. It can be seen that the deformation region of this partition does not affect the meshes near other partitions, thus avoiding the problem of mesh intersection between partitions.
[0078] Figure 8 The distance search strategy using a K-dimensional tree with pre-screening is illustrated, showing the processes of K-dimensional tree construction, pre-screening, single-point distance search, and static boundary distance search.
[0079] Step 107: Update node position.
[0080] Updating node positions refers to superimposing the calculated node displacements with the node positions of the previous level after completing the deformation of the current level of the mesh, thereby obtaining the updated node coordinates.
[0081] Step 108: Determine if the current level is the last level.
[0082] Determine if the current level is the last level; if it is the last level, proceed to step 109 to output the result; if it is not the last level, use the node position updated in step 107 as the input for the next level, return to step 104 to continue the calculation of the target deformation of the current level, the construction of the local radial basis function interpolation system and the mesh deformation until the conditions of the last level are met.
[0083] Step 109: Output the results.
[0084] The output mesh refers to the final deformed mesh obtained after updating the node positions at all levels. This mesh includes the final coordinates of the nodes and their topological connectivity.
[0085] It should be noted that the output can include either the intermediate grids at each level for error monitoring and quality verification, or only the final global grid for subsequent numerical simulation calculations and visualization.
[0086] For example, using a 3D M6 wing mesh deformation example, the computational efficiency of the traditional greedy algorithm and the multi-level partitioning constraint algorithm proposed in this invention are compared and verified. This example uses the following surface deformation function: Δy = 0.25 * b * z * [1 - cos(z * π / b)] Where b is the wingspan of the M6 wing. Figure 9 The M6 wing before and after deformation is illustrated. The number of object surface nodes in this example is 47061, and the number of volume mesh nodes is 1638914. Using a traditional greedy algorithm, with an allowable error set to eps=1e-6, the time statistics for each stage of mesh deformation are shown in Table 1.
[0087] Constructing a radial basis function interpolation system / s Calculate deformation amount / s Update node / s Total / s 588.39 8.36 0.01 596.76 Table 1. Time taken at each stage of the traditional greedy algorithm Using the multi-level partitioning constraint method proposed in this invention, the object surface is divided into 30 zones at the first level, and each zone divided at the first level is further subdivided into 8 zones at the second level, for a total of 3 levels. The allowable error for each level is set to eps=1e-6. The time statistics for each level and stage of mesh deformation are shown in Table 2.
[0088] hierarchy Preprocessing / s Constructing a radial basis function interpolation system / s Distance search / s Deformation calculation / s Update node / s Total / s 0 0.00 5.57 0.00 3.20 0.01 1 0.02 2.60 1.72 0.27 0.01 2 0.04 11.65 1.06 0.10 0.01 total 0.06 19.82 2.78 3.57 0.03 26.26 Table 2. Time taken at each level and stage of the multi-level partitioning constraint algorithm. Comparing Tables 1 and 2, in the stage of constructing the radial basis function interpolation system, the traditional greedy algorithm takes approximately 588 seconds, while the multi-level partitioning constraint algorithm of this invention takes approximately 20 seconds for each stage, representing an improvement of approximately 29 times. In the stage of calculating deformation, the traditional greedy algorithm takes approximately 8 seconds, while the multi-level partitioning constraint algorithm of this invention takes approximately 6 seconds for distance search and deformation calculation at each stage, also showing a slight improvement. Figure 10 The deformed M6 wing mesh is shown after calculation using a multi-level partitioning constraint algorithm.
[0089] Please see Figure 11 , Figure 11 A structural block diagram of a multi-level partitioning constraint device for general mesh deformation problems provided in an embodiment of the present invention.
[0090] This embodiment provides a multi-level partitioning constraint device for a general mesh deformation problem, comprising: The mesh reading and deformation calculation module 1101 is used to read mesh data, set the target deformation of the object surface nodes, and calculate the target deformation of the current level based on the deformation result of the previous level at each level. The multi-level partitioning module 1102 is used to partition the object surface according to the level and extract the candidate node set at each level, and store it together with the partitioning result; The radial basis function interpolation system construction module 1103 is used to construct a radial basis function interpolation system based on the candidate node set in each level of calculation. It adopts a greedy algorithm and allows the error to be taken as a preset deformation accuracy. Mesh deformation module 1104 is used to perform mesh deformation based on the interpolation coefficients of the local radial basis function interpolation system at each level of calculation. Radial basis functions are used in the first level, and double-constrained radial basis functions are used in other levels to control the deformation range to avoid inter-interval mesh intersection. The distance search module 1105 is used to construct a global K-dimensional tree for the object surface nodes participating in the deformation of the current level in all levels except the first level, and to calculate the distance from the volume mesh nodes to the moving boundary and the stationary boundary on the K-dimensional tree through pre-screening and two nearest neighbor searches, so as to provide distance input for the dual-restricted radial basis function; The object surface update module 1106 is used to update the node position after each level completes the calculation and use the update result as the input for the next level. The mesh output module 1107 is used to output the final deformation results after all levels have completed the calculations and save them as a separate data file for visualization and mesh quality checking.
[0091] This invention also provides a computer device, including a memory and a processor, wherein the memory stores a computer program; when the computer program is executed by the processor, the processor performs the steps of the multi-level partitioning constraint method for general mesh deformation problems as described in any of the above embodiments.
[0092] This invention also provides a computer-readable storage medium storing a computer program / instructions thereon, which, when executed by a processor, implements the steps of a multi-level partitioning constraint method for a general mesh deformation problem as described in any of the above embodiments.
[0093] This invention also provides a computer program product, including a computer program / instructions, which, when executed by a processor, implement the steps of a multi-level partitioning constraint method for a general mesh deformation problem as described in any of the above embodiments.
[0094] Those skilled in the art will clearly understand that, for the sake of convenience and brevity, the specific working process of the above-described device and module can be referred to the corresponding process in the foregoing method embodiments, and will not be repeated here.
[0095] In the several embodiments provided in this application, it should be understood that the disclosed apparatus and methods can be implemented in other ways. For example, the apparatus embodiments described above are merely illustrative; for instance, the division of units is only a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the coupling or direct coupling or communication connection shown or discussed may be through some interfaces; the indirect coupling or communication connection between apparatuses or units may be electrical, mechanical, or other forms.
[0096] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple network units. Some or all of the units can be selected to achieve the purpose of this embodiment according to actual needs.
[0097] Furthermore, the functional units in the various embodiments of the present invention can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional unit.
[0098] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.
[0099] The above-described embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.
Claims
1. A multi-level partitioning constraint method for general mesh deformation problems, characterized in that, including: Reading grid data and setting the target deformation of the object surface nodes; Using the existing technology to construct multi-level partitions from coarse to fine for the object surface grid. At the first level, the overall object surface is divided into several coarse partitions. At the second level and subsequent levels, each partition of the previous level is further divided into several sub-partitions, and the shared boundary nodes between adjacent sub-partitions are recorded at each level; Performing grid deformation level by level, and calculating the target deformation of the current level according to the difference between the deformation result of the previous level and the target deformation result; Constructing a local radial basis function interpolation system in each partition. Among them, at the first level, the set of shared boundary nodes between all coarse partitions is used as candidate nodes. At the intermediate levels, the shared boundary nodes of the sub-partitions of the partition are used as candidate nodes. At the last level, all object surface nodes within the partition are used as candidate nodes. At each level, the greedy algorithm is used to select support points within the candidate node set of each partition and construct the local radial basis function interpolation system, and the interpolation error threshold is set to the preset interpolation error at each level, and the interpolation coefficients are obtained by solving the radial basis function interpolation system; Performing grid deformation level by level based on the interpolation coefficients. At the first level, the radial basis function interpolation method is used to perform a full-field coarse deformation on the volume grid. At levels other than the first level, a double-limited radial basis function is introduced into the interpolation formula to limit the deformation range and avoid grid crossing, and the distance search between the moving and static boundaries is accelerated by a search strategy based on the K-dimensional tree; Updating the node positions after completing the grid deformation of the partition at the current level and using the updated result as the input for the next level; After the calculations are completed at all levels, the deformed grid is output.
2. The multi-level partitioning constraint method for general mesh deformation problems according to claim 1, characterized in that, The determination method of the set of shared boundary nodes is: at non-final levels, it is extracted from the common boundary of the sub-partitions obtained by subdividing the partitions of this level at the next level.
3. The multi-level partitioning constraint method for general mesh deformation problems according to claim 1, characterized in that, Calculating the target deformation of the current level is to calculate the difference between the deformation result of the previous level and the target deformation result, and its formula is as follows: d_f^k = x_final - x^(k - 1); where, x_final is the node coordinate of the target deformation result; x^(k - 1) is the node coordinate of the deformation result of the previous level.
4. The multi-level partitioning constraint method for general mesh deformation problems according to claim 1, characterized in that, The greedy algorithm includes: Setting the object surface interpolation error threshold; Selecting the initial set of support points; Performing the radial basis function interpolation calculation based on the current set of support points and evaluating the interpolation error; Adding the node with the largest error to the set of support points; Repeating the interpolation calculation and the error evaluation until the interpolation error meets the preset threshold.
5. The multi-level partitioning constraint method for general mesh deformation problems according to claim 1, characterized in that, The double-limited radial basis function Ψ(x) is defined as: Ψ(x)=min(Ψ1(x), Ψ2(x)); where, the definition of Ψ1(x) is: when d(x,Ω1)<D, Ψ1(x)=1 - d(x,Ω1) / D; in other cases, Ψ1(x)=0; Ψ2(x)=min(1,max(0,(α*d(x,Ω2)- d(x,Ω1)) / β)); Wherein, Ψ1(x) is a function related to the distance d(x,Ω1) from the node coordinate x to the local deformation region Ω1, Ψ2(x) is a function related to the distances d(x,Ω1) and d(x,Ω2) from the node coordinate x to both the local deformation region Ω1 and the non-deformation region Ω2, D is a support distance parameter used to control the range of the local deformation, and α and β are adjustment parameters used to adjust the dual-constraint radial basis function.
6. The multi-level partitioning constraint method for general mesh deformation problems according to claim 1, characterized in that, The K-tree-based search strategy includes: at the beginning of each level, constructing all object surface nodes participating in the deformation of that level into a single K-tree; storing the spatial bounding box of the corresponding subtree and the maximum support distance of the double-constrained radial basis function within the subtree in each node of the K-tree; pre-screening any volume mesh node from the root node according to the relationship between the minimum distance from the volume mesh node to the subtree bounding box and the maximum support distance of the subtree to prune subtrees that are unlikely to contribute; performing a first nearest neighbor search on the pre-screened volume mesh nodes to obtain the distance from the node to the nearest object surface node as the moving boundary distance and its corresponding partition for obtaining the corresponding local radial basis function interpolation system; then performing a second nearest neighbor search after excluding the object surface nodes corresponding to the partition to obtain the nearest distance from the node to the stationary boundary, and using the above two distances as the input of the double-constrained radial basis function.
7. A multi-level partitioning constraint device for a general mesh deformation problem, characterized in that, include: The mesh reading and target deformation calculation module is used to read the mesh data, set the target deformation of the object surface nodes, and calculate the target deformation of each level. A multi-level partitioning module is used to partition the object surface according to levels and extract the candidate node set at each level, and store it together with the partitioning result; A radial basis function interpolation system construction module is used to select candidate nodes according to the rules in claim 1 at the first level, intermediate levels and the final level, and to construct a local radial basis function interpolation system using a greedy algorithm and solve for the interpolation coefficients. The mesh deformation module is used to deform the mesh step by step based on the interpolation coefficients. The radial basis function is used in the first level, and the dual-constraint radial basis function is used in other levels to control the deformation range and avoid mesh crossing. The distance search module is used to construct the object surface nodes participating in the deformation of the current level into a K-dimensional tree in all levels except the first level, and calculate the distance from the volume mesh nodes to the moving boundary and the stationary boundary through pre-screening and two nearest neighbor searches; The object surface update module is used to update the node positions after completing the mesh deformation of the current partition, and use the update result as the input for the next level. The mesh output module is used to output the final deformed mesh after calculations have been completed at all levels.
8. A computer device, characterized in that, It includes a memory and a processor, wherein the memory stores a computer program and the processor is configured to execute the computer program to implement a multi-level partitioning constraint method for the general mesh deformation problem as described in any one of claims 1 to 6.
9. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, implements a multi-level partitioning constraint method for the general mesh deformation problem as described in any one of claims 1 to 6.
10. A computer program product comprising a computer program / instructions, characterized in that, When the computer program / instructions are executed by the processor, they implement the steps of the multi-level partitioning constraint method for the general mesh deformation problem as described in any one of claims 1 to 6.