An embedded domain isogeometric analysis method and system based on truncated hierarchical B-spline

Abstract

The application belongs to the field of engineering structure simulation analysis, and particularly discloses an embedded domain isogeometric analysis method and system based on truncated hierarchical B-spline, which comprises the following steps: obtaining a hierarchical B-spline background grid through a hierarchical B-spline base function and embedding an STL model; performing two-dimensional division on each grid unit: first, performing division according to the position relationship between the grid unit and the STL model physical domain; second, activating the control points of the subdivision area, and performing division according to the activation state of the control points in the grid unit; determining the base function, integration point and weight corresponding to the grid unit based on the two division dimensions, and then obtaining the unit stiffness matrix; assembling the unit stiffness matrix of the grid unit into a global stiffness matrix, and solving the balance equation to obtain the total displacement field and stress field. The application realizes the collaborative optimization of the embedded domain isogeometric analysis precision and the calculation efficiency, and simultaneously considers the overall analysis efficiency and the high-precision analysis of the key area.

Description

Technical Field

[0001] This invention belongs to the field of engineering structure simulation analysis, and more specifically, relates to a geometric analysis method and system based on truncated hierarchical B-splines and embedded domains. Background Technology

[0002] With the increasing demands for structural performance and complexity in modern engineering, the finite element analysis method has been widely applied in many high-tech engineering fields such as aerospace, equipment manufacturing, and biomedicine, becoming a core supporting means for the performance analysis of engineering structures.

[0003] Traditional finite element analysis (FEM) methods approximate continuous field variables using piecewise interpolation functions, transforming complex continuous problems into a solvable set of discrete linear equations, thereby enabling accurate calculation of structural performance parameters such as stress, strain, and displacement. However, this method faces significant challenges when dealing with complex geometric design domains, particularly complex surfaces and uneven physical field distributions. FEM relies on high-quality mesh discretization; however, mesh generation for complex structures is not only inefficient but also prone to mesh distortion, affecting analysis accuracy. Furthermore, geometric approximation errors exist between the finite element mesh and the geometric model; the discretized mesh cannot completely reproduce the true shape of complex surfaces, resulting in insufficient boundary accuracy in the analysis results, failing to meet the demands of high-precision engineering analysis. Obtaining smooth boundaries often requires complex post-processing or refinement techniques, increasing computational burden and design cycle time.

[0004] In recent years, isogeometric analysis (IGA), as an emerging numerical method, has attracted widespread attention for its direct use of basis functions such as NURBS (non-uniform rational B-splines) or T-splines commonly used in CAD models for geometric description and analysis. IGA achieves a unification of geometric description and numerical analysis without the need for additional mesh discretization. Furthermore, IGA can provide high-order continuous field variable approximation, generating analysis results with high-precision boundaries, greatly simplifying the process from design and analysis to verification, and effectively improving the accuracy and efficiency of analysis of complex structures.

[0005] However, traditional isogeometric methods rely on accurate NURBS geometric modeling. For CAD models with complex geometric features such as multiple holes, bridging or separation regions, it is extremely difficult to directly construct a single or even multi-faceted NURBS model. In addition, the basis function activation of traditional isogeometric analysis methods often adopts a strategy based on subdivision units, which is prone to invalid subdivision problems, increases computational redundancy, and is difficult to accurately match the analysis needs of complex structures, which seriously limits the flexibility and applicability of isogeometric analysis methods. Summary of the Invention

[0006] In view of the above-mentioned defects or improvement needs of the existing technology, the present invention provides an embedded domain isogeometric analysis method and system based on truncated hierarchical B-splines, the purpose of which is to improve the analysis accuracy while ensuring the efficiency of isogeometric analysis.

[0007] To achieve the above objectives, according to a first aspect of the present invention, a geometric analysis method based on truncated hierarchical B-splines and embedded domains is proposed, comprising the following steps: Different levels of mesh are obtained by using hierarchical B-spline basis functions, thus forming a hierarchical B-spline background mesh; the STL model of the structure to be analyzed is embedded into this hierarchical B-spline background mesh; The grid cells in the hierarchical B-spline background grid are divided into the following two dimensions: First partitioning dimension: Based on the positional relationship between each mesh cell and the physical domain of the STL model, the mesh cells are divided into solid cells, clipped cells, and empty cells; The second dimension of division: Based on the physical field gradient and stress distribution of the structure, the subdivision region of the STL model is selected; the control points of the subdivision region are activated, and the grid cells are divided into complete cells, THB cells and simplified cells according to the activation state of the control points in each grid cell. Calculate the element stiffness matrix of each mesh element: directly assign a very small preset value to the element stiffness matrix of all empty elements; in addition, determine the integration points and their weights of the mesh elements according to the category of the mesh elements in the first partitioning dimension, determine the basis functions corresponding to the mesh elements according to the category of the mesh elements in the second partitioning dimension, and then obtain the element stiffness matrix of the mesh element according to the basis functions, integration points and their weights corresponding to the mesh elements. The element stiffness matrices of each mesh element are assembled into the global stiffness matrix; the equilibrium equations are solved based on the global stiffness matrix to obtain the total displacement field and stress field.

[0008] As a further preferred method, the control points of the subdivided regions are activated, and the grid cells are divided into complete cells, THB cells, and simplified cells based on the activation status of the control points in each grid cell; specifically: For the subdivided region's grid cells, start dividing from the finest layer and proceed sequentially up to the 0th layer; then for any given layer... l When dividing the layer mesh: The first subdivided region l Layer control points are activated, and the activation basis functions corresponding to the activated control points are obtained. The mesh elements affected by the support domain of the activated basis functions are defined as the first... l Layer candidate units; Defined as the first l Each grid cell in the candidate cells of the layer: If all control points corresponding to a grid cell have been activated, then that grid cell is the first... lA complete layer of cells; if only some of the control points corresponding to a grid cell are activated, then the grid cell is the first complete layer. l Layer THB unit; Then obtain the first l Layer candidate units in the first layer l The parent cells corresponding to the -1 layer mesh are then obtained in the -1 layer mesh. l All sub-elements corresponding to the layer mesh; select those sub-elements that do not belong to the first layer mesh. l The grid cell of the candidate cell in the layer is defined as the first... l Layered simplified unit; For the remaining non-subdivided region's grid cells, only the grid cells corresponding to the 0th layer grid are retained, and they are defined as 0th layer complete cells.

[0009] As a further preferred option, for the... l A complete unit, and its corresponding basis functions For the first l Layer B-spline basis functions.

[0010] As a further preferred option, for the... l The basis functions of the THB layer unit. for:

[0011] in, For the first l -1 layer truncated hierarchical basis functions, which are the second-scaled basis functions using B-splines. l Obtained by truncating the B-spline basis functions of the -1 layer. For the first l Layer B-spline basis functions; i For the first l Index of layer basis functions, j For the first l Index of the -1 level basis function p Let the degree of the basis function be . The coordinates of the control points in the parameter space; Indicates the first l The set of base function indices retained after truncation of base functions at level -1. Indicates the first l The set of base function indices of the layer For the first l The set of control points within the THB unit of the layer.

[0012] As a further preferred option, for the... l Layer simplification unit, its corresponding basis function for:

[0013] in, For the first l Layer B-spline basis functions, For the first l The refined representation of the j-th B-spline basis function in layer -1 is as follows: l The two-scale refinement coefficients corresponding to the basis function of the i-th B-spline in the layer. For the first l The j-th basis function of layer -1, after refinement, corresponds to the j-th... l The set of base function indices for the layer; For the current number l The set of control points within a simplified layer unit.

[0014] As a further preferred method, based on the positional relationship between each mesh element and the physical domain of the STL model, the mesh elements are divided into solid elements, clipped elements, and empty elements; specifically: Mesh cells located entirely within the physical domain of the STL model are solid cells, mesh cells located entirely outside the physical domain of the STL model are empty cells, and mesh cells located on the boundary of the STL model are clipped cells.

[0015] As a further preferred option, for solid elements, Gaussian integrals are used to determine their integration points and weights; for clipped elements, moment fitting and point elimination algorithms are used to determine their integration points and weights.

[0016] As a further preferred option, for empty elements, their element stiffness matrix... .

[0017] According to a second aspect of the present invention, a geometric analysis system for embedded domains based on truncated hierarchical B-splines is provided, comprising a processor for executing the above-described geometric analysis method for embedded domains based on truncated hierarchical B-splines.

[0018] According to a third aspect of the present invention, a computer-readable storage medium is provided having a computer program stored thereon, which, when executed by a processor, implements the above-described geometric analysis method such as the embedding domain based on truncated hierarchical B-splines.

[0019] In summary, compared with the prior art, the above-described technical solutions conceived by this invention mainly possess the following technical advantages: 1. This invention selects subdivided regions based on the physical field gradient and stress distribution of the structure. It can precisely set subdivided regions in key analysis areas where stress concentration and physical field gradient are large. Then, combined with the activation strategy based on control points, it can maximize the analysis accuracy of the core areas of interest by increasing only a small number of computational degrees of freedom and computational cost. This solves the technical contradiction of limited accuracy improvement but soaring computational cost caused by blind subdivision in traditional methods. It achieves synergistic optimization of geometric analysis accuracy and computational efficiency such as embedded domains, while taking into account the overall analysis efficiency and the high-precision analysis requirements of key areas.

[0020] 2. This invention designs a control point-based activation strategy. In the region requiring further subdivision, the corresponding control points are first activated, and then the corresponding basis functions are activated based on these activated control points. This method fundamentally solves the problem of computational redundancy and low efficiency caused by numerous invalid integration regions in traditional subdivision-based activation strategies, thus shortening the analysis cycle of engineering simulations.

[0021] 3. This invention innovatively utilizes the principle of truncated hierarchical B-splines to effectively handle the connection and transition relationships between B-spline meshes of different levels. It solves problems such as redundant basis functions, excessively large support domains, and unclear interlayer coupling caused by the direct superposition of coarse and fine meshes during multi-layer local refinement. While maintaining local refinement capabilities, it also considers overall approximation accuracy and computational cost control, allowing computational resources to be more concentrated in key regions with stress concentration, complex boundary cutting, and large physical field gradients.

[0022] 4. This invention employs an embedded domain method, embedding the complex geometric structure defined by the STL model into a hierarchical B-spline background domain, thus overcoming the significant difficulties of NURBS modeling in traditional isogeometric analysis. The underlying B-spline basis function analysis maintains high-order geometric continuity throughout, avoiding the CAD-FEA geometric gap and ensuring accuracy from geometric input to result reconstruction. This invention demonstrates higher computational efficiency and lower computational cost, along with strong reliability, when handling embedded domain analysis problems involving complex STL models. Attached Figure Description

[0023] Figure 1 This is a flowchart of the geometric analysis method for embedded domains based on truncated hierarchical B-splines according to an embodiment of the present invention.

[0024] Figure 2 This is a schematic diagram illustrating the construction of the embedded domain geometric model according to an embodiment of the present invention.

[0025] Figure 3 This is a schematic diagram of the design domain, boundary conditions, and loads of the L-shaped beam provided in an embodiment of the present invention.

[0026] Figure 4The diagram shows the control point distribution and subdivision unit diagram obtained in the embodiments of the present invention.

[0027] Figure 5 The displacement distribution cloud map and Feng's method obtained in the embodiments of the present invention. Mises stress distribution cloud map.

[0028] Figure 6 This is a diagram showing the geometric analysis results of the embedding domain based on the truncated hierarchical B-spline in an embodiment of the present invention.

[0029] Figure 7 The image shows the geometric analysis results of a standard embedded domain with a background mesh size of 10×10×5.

[0030] Figure 8 The image shows the geometric analysis results of a standard embedded domain with a background mesh size of 40×40×20. Detailed Implementation

[0031] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0032] This invention provides a geometric analysis method based on truncated hierarchical B-splines for embedded domains, such as... Figure 1 As shown, it includes the following steps: S1. Based on different preset mesh specifications, meshes of different levels (subdivision specifications) are obtained through hierarchical B-spline basis functions to form hierarchical B-spline background meshes.

[0033] Specifically, to construct a hierarchical B-spline background mesh, it is first necessary to establish multiple regular geometric background domains that contain the design space. Each background domain has a different grid size; for example, three grid sizes can be defined, each derived from a B-spline basis function. , , Control, among which For base function index, The degree of the B-spline basis function. Any physical point in the background domain. It can be obtained by interpolation using its control points and B-spline basis functions:

[0034] in, The coordinates in the parameter space, For three-dimensional B-spline basis functions, It is a three-dimensional control vector.

[0035] After establishing multiple meshes with different subdivision specifications, the meshes are superimposed according to the basic principle of hierarchical B-splines to obtain the hierarchical B-spline background mesh.

[0036] S2. Embed the STL model of the structure to be analyzed into the hierarchical B-spline background mesh; and pre-select the subdivision region of the STL model according to the physical field gradient and stress distribution of the structure.

[0037] Specifically, based on the results of the preliminary analysis or engineering experience, areas in the structure that may have stress concentration, abrupt strain, large displacement gradient, or complex boundary cutting can be identified and defined as areas that need to be subdivided locally. Subsequently, the control points of the mesh in the subdivided area are activated to avoid redundant degrees of freedom caused by full-area subdivision.

[0038] By employing the embedded domain method, arbitrarily complex STL models are used as the actual design domain, and the boundaries of the design domain are described in the form of a regular background mesh. For example... Figure 2 As shown, this demonstrates how the background domain and the embedded STL design domain are constructed. This method fundamentally solves the problem of traditional isogeometric analysis methods requiring the parameterization of complex geometries into single or multi-faceted NURBS models, enabling direct analysis within an isogeometric framework.

[0039] S3. Based on the positional relationship between each grid cell in the hierarchical B-spline background grid and the physical domain of the STL model, divide each grid cell into solid cells, clipped cells, and empty cells to achieve the first dimension division of the grid cells. Then, determine the integration points and their weights based on the grid cell categories divided in the first dimension.

[0040] Specifically, units located only within the physical domain of the model are defined as solid units, units located outside the physical domain of the model are defined as empty units, and units whose parts are located within the physical domain and part are located outside the physical domain due to being cut by the contour of the physical domain of the model are defined as clipping units.

[0041] Furthermore, for the solid element, Gaussian integrals are used to determine its integration points and weights: The solid element performs Gauss integration at standard Gauss integration points across the entire element domain, using the standard isogeometric element integration method. Its element stiffness matrix is ​​calculated accordingly. The expression is:

[0042] in The strain-displacement matrix is... For the material constitutive matrix, Let be the physical domain of the background unit. It can be written in the following numerical integral form:

[0043] in, The number of Gaussian integration points. The Gaussian integral weights on the entity unit. Let this be the Jacobian matrix from the parametric domain to the physical domain. The Gaussian integral points of the standard tensor product and their weights can be obtained here by calculating using the Gauss-Legendal integral formula. This will not be elaborated upon here.

[0044] For each clipping unit, the integral points and weights are determined using moment fitting and point elimination algorithms: The area where the cropping unit actually participates in the integration is not the entire background unit. Rather, it is the unit and the physical domain intersection Its element stiffness matrix Written as:

[0045] Integration region of the clipping unit The irregular regions, typically obtained by STL boundary cutting, cannot be directly integrated using the standard tensor product Gaussian integral. Therefore, the moment fitting approach is used to reconstruct the integral formula, i.e., to find a set of dedicated integration points. and integral weight The goal is to make the discrete integral formula reproduce the integral moments over the clipped region as accurately as possible, while ensuring that the volume integral over the irregular clipped region still has sufficiently high accuracy. This yields the stiffness matrix of the clipped element. for:

[0046] For empty elements: Since their corresponding background elements are essentially outside the real physical domain, they should not contribute to the structural stiffness, and their stiffness matrix can be considered zero. However, if the stiffness of such elements is completely zeroed in numerical implementation, the associated degrees of freedom will lack sufficient constraints, potentially leading to a singular or severely ill-conditioned overall stiffness matrix, affecting the stable solution of the linear equations. Therefore, in practical calculations, empty elements are given a very small equivalent stiffness. The influence of such empty elements on the overall mechanical response is usually negligible, while maintaining the algebraic invertibility of the overall stiffness matrix, thus avoiding the occurrence of singular matrix problems during the solution process.

[0047] S4. Using a control point-based activation strategy, each grid cell is divided into complete cells, THB cells, and simplified cells to achieve the second dimension of grid cell partitioning. The corresponding basis function is determined based on the grid cell category of the second dimension partitioning.

[0048] Specifically, the mesh is divided starting from the finest layer and proceeding sequentially to the coarsest layer (i.e., layer 0). During this process, for any given layer... l When dividing the layer mesh: The first subdivided region l Layer control points are activated, and the corresponding activation basis functions are obtained. The elements affected by the support domain of these basis functions are defined as the first-order control points. l Layer candidate units, and then according to the first l (p+1) corresponding to each grid cell in the candidate cells of the layer. 3 The activation status of each control point is used to determine the cell type: If all control points corresponding to a candidate cell are activated, it means that the grid cell has been selected by the first control point. l The activation basis functions of the layer are fully covered, and the corresponding grid cell is the th layer. l Layer complete unit; If not all control points corresponding to a candidate cell are active, the grid cell is defined as the first... l Layer THB unit; For the l Layer candidate units in the first layer l The parent cells corresponding to the -1 layer mesh are then obtained in the -1 layer mesh. l All sub-elements corresponding to the layer mesh; select those sub-elements that do not belong to the first layer mesh. l The grid cell of the candidate cell in the layer is defined as the first... l Simplified unit of layer.

[0049] The mesh is divided from the finest to the coarsest layer using the method described above. For the coarsest layer 0, only complete cells exist. Furthermore, for mesh cells located in non-subdivided regions, only the corresponding mesh cells in the layer 0 are retained and designated as complete cells of layer 0; mesh cells from other layers are not included in the calculation.

[0050] Furthermore, the corresponding basis functions are determined based on the grid cell categories defined by the second dimension: For a complete unit, by the first l The layer basis functions provide complete coverage, and their corresponding basis functions are standard B-spline basis functions:

[0051] in For the standard l layer p B-spline basis functions This is the set of control points within the current complete unit.

[0052] For the THB unit, only a portion of the first... l When a layer control point is activated, the first layer must be considered simultaneously. lLayer basis functions and the first l The contribution of the -1 layer basis functions. For the -1st layer basis function... l The -1 layer basis functions are truncated using the two-scale relation of B-splines to obtain truncated hierarchical basis functions, which are then used to express the -1 layer basis functions. l The contribution of the -1 layer basis functions to the THB unit.

[0053] The two-scale relationship is as follows: the first... l B-spline basis functions of layer -1 It can be expressed as p +2nd l B-spline basis functions of the layer A linear combination of these. Generalizing this to three-dimensional space using the tensor product form, it can be written as:

[0054] in, For the first l The refined representation of the j-th B-spline basis function in layer -1 is as follows: l The two-scale refinement coefficients corresponding to the basis function of the i-th B-spline in the layer. For the first l The j-th basis function of layer -1, after refinement, corresponds to the j-th... l The set of base function indices for the layer.

[0055] Definition of the first l The set of indices for the layer activation fine basis functions is To eliminate the first in the refined area l -1 layer basis functions and activation of the first l Redundant overlap of layer basis functions, for the first layer l The base functions are truncated to level -1, retaining only the set of base function indices. Defined as:

[0056] The first can be obtained l The truncated hierarchical basis functions obtained after truncating the -1 level basis functions for:

[0057] Therefore, the basis functions of the THB unit are derived from the first... l Layer activation basis functions and the first layer mentioned above l The THB basis functions, consisting of -1 truncated hierarchical basis functions, can be written as follows:

[0058] in, For the first l -1 is the p-th truncated hierarchical basis function of the j-th layer, which is the two-scale relation of the B-spline.l Obtained by truncating the B-spline basis functions of the -1 layer. For the first l The i-th activation basis function of the layer; i For the first l Index of layer basis functions, j For the first l Index of the -1 level basis function p Let the degree of the basis function be . The coordinates of the control points in the parameter space; Indicates the first l The set of base function indices retained after truncation of base functions at level -1. Indicates the first l The set of activation basis function indices for the layer; For the current number l The set of control points within the THB unit of the layer.

[0059] For the simplified unit, utilize its relationship with the corresponding first... l The relationship between parent units in layer -1 also adopts a two-scale relationship, and the first layer... l The basis function of the parent unit at level -1 is represented as the current level... l Several standard units within the layer sub-unit region l A linear combination of layer basis functions, thus equivalently expressing the simplified unit basis functions, yields the following basis functions:

[0060] in, For the current number l The set of control points within a simplified layer unit.

[0061] S5. Based on the mesh element categories divided by S3 and S4, calculate the element stiffness matrix of each mesh element, and then obtain the global stiffness matrix.

[0062] As described in S3, an empty element is directly assigned a very small element stiffness matrix. There is no need to calculate the element stiffness matrix again, and the element type is no longer distinguished. Then, the two types of elements in S3, solid elements and trimmed elements, are combined with the three types of elements in S4 to obtain 6 types of mesh elements, namely solid complete elements, solid simplified elements, solid THB elements, trimmed complete elements, trimmed simplified elements, and trimmed THB elements.

[0063] Specifically, the integration points and their weights for the grid cells are determined based on the grid cell type in the first partitioning dimension, the basis functions corresponding to the grid cells are determined based on the grid cell type in the second partitioning dimension, and then the element stiffness matrix of the grid cell is obtained based on the basis functions, integration points and their weights corresponding to the grid cell.

[0064] Then, based on the element index, the element stiffness matrix is ​​assembled into the corresponding position in the global stiffness matrix, ultimately obtaining the global sparse stiffness matrix. It also outputs the index of the degrees of freedom that are effectively used in the calculation.

[0065] S6. Apply boundary conditions and solve the analysis results. After superimposing the boundary integral terms and extracting the effective degrees of freedom from the global stiffness matrix, perform the equilibrium equations. The solution, U The displacement vector corresponding to the effective degrees of freedom. F The load vectors corresponding to the effective degrees of freedom are used to calculate the displacements of the activated control points. The total displacement field of the model and the von Neumann displacement field are then calculated using the basis functions and the control point displacements. Mises stress field; the results can be further packaged into a standard VTK file and visualized in ParaView software.

[0066] The following are specific examples: like Figure 3 The figure shows a calculation example of an L-shaped beam.

[0067] Design domain and parameters: The model background mesh size is L×W×H=15×15×8. The design domain is defined by an externally imported STL model. The top surface is completely fixed, and a uniformly distributed downward load is applied to the right end face. =0.1N.

[0068] Figure 4 The diagram shows the control point distribution map (left) and subdivision unit map (right) obtained after analysis using the geometric analysis method of embedded domain based on truncated hierarchical B-splines of this invention.

[0069] As can be seen, the area for local subdivision is selected at the inner corner of the L-shaped beam where stress concentration occurs. A three-layer B-spline background mesh is defined, from coarse to fine, layers 0-2, with corresponding mesh sizes of 10×10×5, 20×20×10, and 40×40×20. The fine-layer mesh control points at the inner corner of the L-shaped beam are activated; for other locations, only the coarse-layer mesh control points need to be activated, resulting in a control point distribution map. Further subdivision of the L-shaped beam is then drawn based on these control points, resulting in a subdivided element map.

[0070] Figure 5 The displacement distribution cloud map (left) and Feng's figure are obtained after analysis using geometric analysis methods such as embedded domain based on truncated hierarchical B-splines in this embodiment of the invention. Mises stress distribution cloud map (right).

[0071] As can be seen, under the action of uniformly distributed load, the downward displacement generated on the right end face of the L-shaped beam is the largest, and the displacement decreases smoothly from the free end to the fixed end; a large stress concentration is generated at the inner corner of the L-shaped beam, and the stress gradually decreases from this high-stress corner area to the fixed end and the free end.

[0072] Figures 6 to 8 The results of the embedded domain isogeometric analysis based on truncated hierarchical B-splines are compared with those of the standard embedded domain isogeometric analysis with different background mesh counts. Figure 6 The method of this invention: maximum total displacement =5.58mm, maximum von Mises stress =2.60MPa, solution time t=16.61s. The analysis accuracy is close to that of geometric analyses such as fine mesh standard embedded domain, exhibiting high precision. It can accurately capture the stress concentration at the inner corner of the L-shaped beam, while greatly reducing the analysis time and significantly improving computational efficiency.

[0073] Figure 7 Geometric analysis of a standard embedded domain with a background mesh size of 10×10×5: Maximum total displacement =5.29mm, maximum von Mises stress =1.76MPa, solution time t=1.44ss. The analysis time is the shortest, but the analysis results have obvious errors and cannot accurately capture the stress concentration at the inner corner of the L-shaped beam.

[0074] Figure 8 Geometric analysis of a standard embedded domain with a background mesh size of 40×40×20: Maximum total displacement =5.59mm, maximum von Mises stress =2.61MPa, solution time t=73.90ss. The analysis accuracy is high, but the computational cost and time consumption are much higher than the other two analysis methods. Comparison shows that when analyzing complex simulation problems with stress concentration or rapidly changing physical fields, the method of this invention can maximize the computational accuracy of the relevant region with only a slight increase in computational cost, achieving an optimal balance between accuracy and efficiency.

[0075] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A geometric analysis method based on truncated hierarchical B-splines, characterized in that, Includes the following steps: Different levels of mesh are obtained by using hierarchical B-spline basis functions, thus forming a hierarchical B-spline background mesh; the STL model of the structure to be analyzed is embedded into this hierarchical B-spline background mesh; The grid cells in the hierarchical B-spline background grid are divided into the following two dimensions: First partitioning dimension: Based on the positional relationship between each mesh cell and the physical domain of the STL model, the mesh cells are divided into solid cells, clipped cells, and empty cells; The second dimension of division: Based on the physical field gradient and stress distribution of the structure, the subdivision region of the STL model is selected; the control points of the subdivision region are activated, and the grid cells are divided into complete cells, THB cells and simplified cells according to the activation state of the control points in each grid cell. Calculate the element stiffness matrix of each mesh element: directly assign a very small preset value to the element stiffness matrix of all empty elements; in addition, determine the integration points and their weights of the mesh elements according to the category of the mesh elements in the first partitioning dimension, determine the basis functions corresponding to the mesh elements according to the category of the mesh elements in the second partitioning dimension, and then obtain the element stiffness matrix of the mesh element according to the basis functions, integration points and their weights corresponding to the mesh elements. The element stiffness matrix of each mesh element is assembled into the global stiffness matrix; The equilibrium equations are solved based on the global stiffness matrix to obtain the total displacement field and stress field.

2. The geometric analysis method based on truncated hierarchical B-splines and embedded domains as described in claim 1, characterized in that, Activate the control points in the subdivided region, and divide the grid cells into complete cells, THB cells, and simplified cells based on the activation status of the control points in each grid cell; specifically: For the subdivided region's grid cells, start dividing from the finest layer and proceed sequentially up to the 0th layer; then for any given layer... l When dividing the layer mesh: The first subdivided region l Layer control points are activated, and the activation basis functions corresponding to the activated control points are obtained. The mesh elements affected by the support domain of the activated basis functions are defined as the first... l Layer candidate units; Defined as the first l Each grid cell in the candidate cells of the layer: If all control points corresponding to a grid cell have been activated, then that grid cell is the first... l A complete layer of cells; if only some of the control points corresponding to a grid cell are activated, then the grid cell is the first complete layer. l Layer THB unit; Then obtain the first l Layer candidate units in the first layer l The parent cells corresponding to the -1 layer mesh are then obtained in the -1 layer mesh. l All sub-elements corresponding to the layer mesh; select those sub-elements that do not belong to the first layer mesh. l The grid cell of the candidate cell in the layer is defined as the first... l Layered simplified unit; For the remaining non-subdivided region's grid cells, only the grid cells corresponding to the 0th layer grid are retained, and they are defined as 0th layer complete cells.

3. The geometric analysis method based on truncated hierarchical B-splines and embedded domains as described in claim 2, characterized in that, For the first l A complete unit, and its corresponding basis functions For the first l Layer B-spline basis functions.

4. The geometric analysis method based on truncated hierarchical B-splines and embedded domains as described in claim 2, characterized in that, For the first l The basis functions of the THB layer unit. for: in, For the first l -1 layer truncated hierarchical basis functions, which are the second-scaled basis functions using B-splines. l Obtained by truncating the B-spline basis functions of the -1 layer. For the first l Layer B-spline basis functions; i For the first l Index of layer basis functions, j For the first l Index of the -1 level basis function p Let the degree of the basis function be . The coordinates of the control points in the parameter space; Indicates the first l The set of base function indices retained after truncation of base functions at level -1. Indicates the first l The set of base function indices of the layer For the first l The set of control points within the THB unit of the layer.

5. The geometric analysis method based on truncated hierarchical B-splines and embedded domains as described in claim 2, characterized in that, For the first l Layer simplification unit, its corresponding basis function for: in, For the first l Layer B-spline basis functions, For the first l The refined representation of the j-th B-spline basis function in layer -1 is as follows: l The two-scale refinement coefficients corresponding to the basis function of the i-th B-spline in the layer. For the first l The j-th basis function of layer -1, after refinement, corresponds to the j-th... l The set of base function indices for the layer; For the current number l The set of control points within a simplified layer unit.

6. The geometric analysis method based on truncated hierarchical B-splines for embedded domains as described in any one of claims 1-5, characterized in that, Based on the positional relationship between each mesh element and the physical domain of the STL model, the mesh elements are divided into solid elements, clipped elements, and empty elements; specifically: Mesh cells located entirely within the physical domain of the STL model are solid cells, mesh cells located entirely outside the physical domain of the STL model are empty cells, and mesh cells located on the boundary of the STL model are clipped cells.

7. The geometric analysis method based on truncated hierarchical B-splines and embedded domains as described in claim 6, characterized in that, For solid elements, Gaussian integrals are used to determine their integration points and weights; for clipped elements, moment fitting and point elimination algorithms are used to determine their integration points and weights.

8. The geometric analysis method based on truncated hierarchical B-splines and embedded domains as described in claim 6, characterized in that, For an empty element, its element stiffness matrix .

9. A geometric analysis system for embedded domains based on truncated hierarchical B-splines, characterized in that, Includes a processor for performing geometric analysis methods such as embedding domains based on truncated hierarchical B-splines as described in any one of claims 1-8.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the geometric analysis method such as the embedded domain based on truncated hierarchical B-splines as described in any one of claims 1-8.

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