Efficient mesh deformation method based on global-local space nested radial basis functions

Abstract

The application provides a high-efficiency mesh deformation method based on global-local space nested radial basis function, which comprises the following steps: determining a surface deformation control point and a space to-be-deformed mesh node; establishing a global low-precision RBF model; establishing a global-local space nested RBF model; realizing global-local space nested RBF model interpolation; performing RBF sub-model boundary weighted correction; performing far-field fairing; and outputting a deformed mesh. Compared with the traditional radial basis function mesh deformation, the modeling calculation amount of the application is reduced to the original times, and the interpolation calculation amount is reduced to the original n times, wherein n is the RBF sub-model quantity. While ensuring the mesh deformation precision, the mesh deformation efficiency is greatly improved, and the application is a high-efficiency and robust mesh deformation method.

Description

Technical Field

[0001] This invention belongs to the field of computational fluid dynamics technology, specifically relating to an efficient mesh deformation method based on global-local spatial nested radial basis functions. Background Technology

[0002] In aerodynamic optimization design and fluid-structure interaction numerical simulation, the computational mesh needs to be updated in real time according to changes in the aircraft's geometry. However, regenerating a complex mesh is a time-consuming and tedious task. Therefore, mesh deformation methods that do not change the number of mesh points or the topological relationships have become the optimal choice for updating the computational mesh. The quality of the mesh deformation method determines the quality of the updated mesh and the time required for mesh deformation, further affecting the computational accuracy and efficiency of optimization design and fluid-structure interaction numerical simulation. When solving large deformation problems of aircraft, the mesh deformation method can even become a limiting factor for the reliability of flow field numerical simulation. Therefore, developing efficient and robust mesh deformation methods is of great significance.

[0003] The basic idea of ​​mesh deformation technology is to move or rotate all the meshes in space based on the geometric changes of the object's boundary, while ensuring that the number of meshes and the topological relationship remain unchanged, to generate a new computational mesh. Currently, there are various mesh deformation methods. Based on different computational models, mesh deformation methods can be divided into three categories: 1) Physical model methods: spring method, spring-body method, temperature-body method, etc.; 2) Mathematical interpolation methods: transboundary interpolation (TFI), radial basis function interpolation (RBF), inverse distance interpolation (IDW), etc.; 3) Hybrid methods: RBF-DGM, RBF-TFI, etc. Among them, the RBF mesh deformation method has the following advantages: relatively simple deformation principle and application; robustness to complex shape deformation problems and high quality of the deformed mesh; no requirements on mesh topological relationships, applicable to structured meshes, unstructured meshes, and hybrid meshes, and has received widespread attention and research in recent years. However, the disadvantages of the RBF mesh deformation method are also quite obvious: the time for constructing the RBF interpolation model and using this model to realize mesh deformation are respectively... and n s ×n v Proportional (n) s To model the number of control points, n v (This represents the number of spatial grids to be deformed, i.e., the number of interpolations). As the grid size increases, the computational cost of the RBF method grows non-linearly, resulting in extremely low efficiency.

[0004] To improve the efficiency of RBF mesh deformation methods, much current research focuses on two aspects: firstly, employing optimization strategies, such as greedy algorithms, to reduce the number of modeling control points n. s On the other hand, based on the greedy algorithm, the spatial interpolation region is restricted, thereby reducing the number of spatial interpolation control points n. vHowever, methods that improve mesh deformation efficiency by sacrificing the number of modeling points or interpolation points often fall short when higher modeling accuracy is required, resulting in a trade-off between computational accuracy and efficiency. Therefore, there is an urgent need to develop a more efficient and robust RBF mesh deformation method to address this problem. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides an efficient mesh deformation method based on global-local spatial nested radial basis functions, which can effectively solve the above problems.

[0006] The technical solution adopted in this invention is as follows:

[0007] This invention provides an efficient mesh deformation method based on global-local spatial nested radial basis functions, comprising the following steps:

[0008] Step 1: Determine the set of deformation control points S in three-dimensional space. all ={s1,s2,…,s N} and the set of nodes to be deformed, V = {v1, v2, ..., v M}; where s1, s2, ..., s N Let v1, v2, ..., v be the N deformation control points. M , where M are the mesh nodes to be deformed;

[0009] Step 2, given the set of deformation control points S all ={s1,s2,…,s N The deformation amounts at each deformation control point in the equation are: ΔT1, ΔT2, ..., ΔT N ;

[0010] Step 3, determine the modeling method and interpolation algorithm for the RBF model, including:

[0011] Modeling method of RBF model: The set of deformation control points S all ={s1,s2,…,s N A subset S of deformation control points in any combination form within} D ={s D1 ,s D2 ,…,s D d} and the corresponding deformation ΔT D1 ,ΔT D2 ,...,ΔT Dd As input data, these are fed into the RBF initial model, and the model parameters ω={ω D1 ,ω D2 ,…,ω Dd}, thereby establishing the RBF model;

[0012] Interpolation calculation algorithm: Based on the model parameters ω={ω D1 ,ω D2 ,…,ω Dd Using interpolation formulas, the set of nodes in the deformable mesh, V = {v1, v2, ..., v}, is calculated. M The deformation amount of any deformable mesh node in} represents the deformation control point subset S. D ={s D1 ,s D2 ,…,s Dd The amount of deformation caused by the deformation of each deformation control point in the deformation control point;

[0013] Step 4: Establish a global low-precision RBF model:

[0014] A greedy algorithm is used to select from the set of deformation control points S all 10-20 deformation control points are randomly selected from the input data. These selected control points and their corresponding deformation amounts are used as input data. Following the RBF modeling method in step 3, an RBF model is established. The fitting error of the obtained RBF model is then judged to meet the accuracy requirements. If not, deformation control points are continuously selected, increasing the number of deformation control points and their corresponding deformation amounts in the input data. This process is repeated until the accuracy requirements are met. The resulting RBF model is called the global low-precision RBF model. The deformation control points in the input data at this point form the set of deformation control points required for modeling the global low-precision RBF model, denoted as S. Greddy ={s G1 ,s G2 ,…,s Gr};

[0015] Step 5: Establish a global-local space nested RBF model; the global-local space nested RBF model includes n global-local space nested RBF sub-models, which are referred to as RBF sub-models for short.

[0016] Step 5.1: Determine the partitioning direction l and the number of partitions n;

[0017] Step 5.2, set the deformation control points S all ={s1,s2,…,s N Deformation control points s1, s2, ..., s in} N Project onto the partition direction l, and continuously group the deformation control points along the partition direction l according to the projection coordinates of the deformation control points in ascending order of equal interval length or equal number of points, to obtain n sets of deformation control point subsets of different partition intervals; each set of deformation control point subsets has a corresponding partition interval.

[0018] Step 5.3: Combine the set of deformation control points S required for modeling each subset of deformation control points and the global low-precision RBF model. Greddy Merge and remove duplicate deformation control points to obtain a global-local combined deformation control point set; therefore, a total of n sets of global-local combined deformation control points are obtained, denoted as...

[0019] Step 5.4: Combine the global and local deformation control point sets for each group. Using the corresponding deformation as input data, and following the RBF model modeling method in step 3, an RBF model is established, called a global-local spatial nested RBF sub-model, or simply an RBF sub-model; therefore, a total of n RBF sub-models are established; each RBF sub-model has an influence area determined by partitioning intervals;

[0020] Step 6, for the set of nodes of the mesh to be deformed, V = {v1, v2, ..., v...} M Any deformable mesh node v in} k For k = 1, 2, ..., M, the corresponding RBF sub-model is determined based on its spatial location, and is represented as: RBF sub-model model k That is: the mesh node v to be deformed k In the RBF sub-model k Within the affected area;

[0021] Step 7: Using the interpolation calculation algorithm determined in Step 3, the RBF sub-model is used. k The model parameters are used to obtain the deformable mesh node v. k The deformation amount is used to determine the deformation of the mesh node v. k Mesh deformation.

[0022] Preferably, in step 3, the modeling method of the RBF model is as follows:

[0023] Based on the deformation control point subset S D ={s D1 ,s D2 ,…,s Dd The pre-deformation positions of each deformation control point and the corresponding deformation amount ΔT in the diagram. D1 ,ΔT D2 ,...,ΔT Dd Solve the following system of symmetric linear equations to obtain the model parameters ω={ω D1 ,ω D2 ,…,ω Dd}, ω D1 ,ω D2 ,…,ω Dd Also known as the weighting coefficient, it is expressed as: ω Duu = 1, ... d;

[0024]

[0025] in:

[0026] The matrix form of the linear equation system is:

[0027]

[0028] The basis functions generated for the u-th deformation control point and the j-th deformation control point.

[0029] Preferably, the basis function expression is:

[0030]

[0031] Where η is an intermediate variable; r0 is the radius of action of the basis function. This represents the Euclidean distance between the u-th deformation control point and the j-th deformation control point;

[0032] The interpolation calculation algorithm is as follows:

[0033] The set of nodes of the deformable mesh is V = {v1, v2, ..., v} M Any deformable mesh node v in} k k = 1, 2, ..., M, its deformation ΔT k We obtain it from the following formula:

[0034]

[0035] in: Represents the mesh node v to be deformed k With S D ={s D1 ,s D2 ,…,s D The Euclidean distance between the u-th deformation control points in d}.

[0036] Preferably, step 6 specifically includes:

[0037] For the deformable mesh node v k Its projection position in the partition direction l is l k Deformation control point set S all The projection range in the partition direction l is [l min ,l max Deformation control point set S allAfter being divided into n groups, along the partition direction l, they are sequentially named: the 1st deformation control point subset, the 2nd deformation control point subset, ..., the nth deformation control point subset. The RBF submodels corresponding to each deformation control point subset are sequentially named: the 1st RBF submodel, the 2nd RBF submodel, ..., the nth RBF submodel. The distinguishing position between the 1st deformation control point subset and the (i+1)th deformation control point subset is represented by l. i,i+1 ;

[0038] (a) If l k ≤l 1,1+1 Determine v k Under the influence region of the first RBF sub-model;

[0039] (b) If l i-1,i <l k ≤l i,i+1 Determine v k Under the influence region of the i-th RBF sub-model;

[0040] (c) If l k ≥l n-1,n Determine v k Under the influence region of the nth RBF sub-model.

[0041] Preferably, it also includes step 8, the RBF sub-model boundary weighting correction algorithm:

[0042] Step 8.1, along the partition direction l, set the RBF sub-model model. k The front-end adjacent RBF sub-model is model k-1 The adjacent RBF sub-model at the back end is model k+1 ;

[0043] Step 8.2, determine the RBF sub-model. k The weighted adjustment range includes: the beginning weighted adjustment range and the end weighted adjustment range;

[0044] RBF sub-model k The minimum and maximum values ​​of the projection of the subset of deformation control points used in the modeling onto the partition direction l are l. k,min ,l k,max Therefore, its modeling range is l k,max -l k,min Let the dimensionless weighted interval coefficients be p, 0 ≤ p ≤ 1; then the RBF submodel... k The weighted correction range length is p(l) k,max -l k,min );

[0045] Therefore, the two boundary points of the first-end weighted correction range are: l k,minand l k,p =l k,min +p(l k,max -l k,min The weighted correction range for the first end is: [l k,min ,l k,p ];

[0046] The two boundary points of the end-weighted correction range are: l k,b =l k,max -p(l k,max -l k,min ) and l k,max The end-weighted correction range is: [l k,b, l k,max ];

[0047] Step 8.3, if the mesh node v to be deformed k At the projection position l in the partition direction l k ∈[l k,min ,l k,p If the result is negative, then the head-end weighted correction algorithm is adopted, including:

[0048] Through the RBF sub-model k The model parameters are used to obtain the deformable mesh node v. k Deformation amount ΔT k ; through RBF sub-model k-1 The model parameters are used to obtain the deformable mesh node v. k Deformation amount ΔT k-1 ;

[0049] Construct the weighting formula; use trigonometric functions to construct the weighting coefficients to obtain the RBF sub-model. k-1 For the RBF sub-model k Weighted correction of deformable mesh node v k The amount of deformation is The formula is as follows:

[0050]

[0051] in:

[0052] Represents the RBF sub-model. k-1 For the RBF sub-model k The deformed interpolation result obtained after weighted correction, l k,min ,l k,max These are the RBF sub-models. k The minimum and maximum values ​​of the projection in the partition direction l, l k For RBF sub-modelk The RBF sub-model k-1 The projection positions of the points participating in the weighted correction range on the partition direction l, l k,p The boundary values ​​of the correction range are determined for the weighted interval coefficients p, and α and β are the values ​​of the RBF sub-model model. k-1 and RBF sub-model k The weighting coefficients are set, and α+β=1 is guaranteed; ε1 is an intermediate parameter, representing the mesh node v to be deformed. k In the relative position of the weighted adjustment range;

[0053] Step 8.4, if the mesh node v to be deformed k At the projection position l in the partition direction l k ∈[l k,b ,l k,max If the result is negative, then the end-weighted correction algorithm is used, as shown in the following formula:

[0054]

[0055] l k,b =l k,max -p(l k,max -l k,min )

[0056] in:

[0057] Represents the RBF sub-model. k+1 For the RBF sub-model k The deformed interpolation result obtained after weighted correction; ΔT k+1 To use the RBF sub-model k+1 The model parameters are used to obtain the deformable mesh node v. k The amount of deformation; ΔT k To use the RBF sub-model k The model parameters are used to obtain the deformable mesh node v. k The deformation amount; α and γ are the deformation amounts of the RBF submodel. k and RBF sub-model k+1 The weighting coefficients are set, and α+γ=1 is guaranteed; ε2 is an intermediate parameter, representing the value of the mesh node v to be deformed. k The relative position within the weighted adjustment range.

[0058] Preferably, it also includes step 9, far-field smoothing, and the process for determining the full-field interpolation method under far-field smoothing is as follows:

[0059] The calculated value of the deformable mesh node v is obtained. k Distance corresponding RBF sub-modelk The minimum distance between all deformation control points in the deformation control point subset used in the modeling Let the dimensionless smooth interval coefficients of the model be q, 0≤q≤1, and r0 be the radius of action of the basis functions. Then the following criteria are applied:

[0060] (a) If Then the deformable mesh node v k The deformation is determined by the RBF submodel including boundary weighting corrections. k Obtained through interpolation calculation;

[0061] (b) If Then the deformable mesh node v k The deformation is determined by a global low-precision RBF model and an RBF sub-model that includes boundary weighting corrections. k The result was obtained by interpolation after further weighting.

[0062] (c) If Then the deformable mesh node v k The deformation is calculated by interpolation using a global low-precision RBF model, thus enabling far-field smoothing through the global low-precision RBF model.

[0063] Preferred, in At that time, the mesh node v to be deformed k The deformation is obtained by weighting using trigonometric functions, specifically through the following formula:

[0064] ΔT weighted =α w ΔT high +β w ΔT low

[0065]

[0066] in:

[0067] ΔT weighted Represents the mesh node v to be deformed k The corrected deformation amount;

[0068] ΔT high The model represents the RBF submodel with boundary weighting corrections. k The interpolation calculation yields the deformable mesh node v k The amount of deformation; ΔT low The deformable mesh node v is calculated by interpolation from a global low-precision RBF model. k Deformation amount; α w and β w These are the weighting coefficients, and α is guaranteed to be... w +β w=1; ε w The intermediate parameter represents the mesh node v to be deformed. k The relative position within the weighted adjustment range.

[0069] The efficient mesh deformation method based on global-local spatial nested radial basis functions provided by this invention has the following advantages:

[0070] Compared to traditional radial basis function mesh deformation, this invention reduces the computational cost of modeling to only a fraction of the original amount. The computational cost of interpolation is reduced to one-fifth of the original amount. n represents the number of RBF sub-models; therefore, while ensuring mesh deformation accuracy, the efficiency of mesh deformation is greatly improved, making it an efficient and robust mesh deformation method. Attached Figure Description

[0071] Figure 1 A flowchart illustrating the efficient mesh deformation method based on global-local spatial nested radial basis functions provided by this invention;

[0072] Figure 2 This is a schematic diagram of the modeling points of the globally low-precision-locally high-precision spatially nested RBF sub-model in step 5 of the present invention;

[0073] Figure 3 A comparison diagram of the extrapolation rules of the model under the following conditions: local high-precision RBF model, global low-precision + local high-precision RBF model, and global low-precision + local high-precision RBF model + boundary weighting correction provided by the present invention.

[0074] Figure 4 A comparison of the compatibility of the two models provided by this invention—the local high-precision RBF model with weighted correction and the global low-precision RBF model with weighted correction—with the far-field smoothing model;

[0075] Figure 5 A schematic diagram of the interpolation interval of the nested RBF model combining far-field smoothing provided by the present invention;

[0076] Figure 6 This is a schematic diagram of the M6 ​​wing structure grid provided by the present invention;

[0077] Figure 7 A schematic diagram of the wing surface deformation control points provided by the present invention;

[0078] Figure 8 The above is a comparison diagram of the front and rear meshes of the wing before and after deformation, provided by the present invention.

[0079] Figure 9 A comparison diagram of the spatial cross-sectional mesh before and after wing deformation provided for this invention;

[0080] Figure 10 The modeling and interpolation time varies with the number of RBF sub-models as provided in this invention;

[0081] Figure 11 The curves showing the variation of the maximum and average interior angles of the deformed mesh with the number of RBF sub-models provided by this invention. Detailed Implementation

[0082] To make the technical problems solved, the technical solutions, and the beneficial effects 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 are not intended to limit the invention.

[0083] To more clearly illustrate the objectives, technical methods, and advantages of this invention, specific embodiments are described below in detail with reference to specific examples and accompanying drawings. Furthermore, the described embodiments are merely a subset of examples of this invention. Based on the embodiments of this invention, all other embodiments obtained by those skilled in the art without inventive effort are within the scope of protection of this application.

[0084] This invention addresses the problem of the inability to simultaneously achieve high modeling accuracy and efficient modeling and interpolation in traditional radial basis function (RBF) mesh deformation methods. It proposes a highly efficient mesh deformation method based on nested global-local radial basis functions. In traditional RBF mesh deformation methods, establishing a high-precision global RBF model requires solving a massive number of n... s ×n s dimension (n) s The present invention replaces the global high-precision RBF model with a globally low-precision-locally high-precision spatially nested RBF model. The spatial mesh deformation is calculated by interpolation of the corresponding RBF sub-model. On the one hand, it effectively reduces the dimension of the linear equation system required for RBF modeling, significantly improving modeling and interpolation efficiency while ensuring high modeling accuracy. On the other hand, by coupling the sample points of the globally low-precision RBF modeling with the sample points of the locally high-precision RBF modeling to establish RBF sub-models, the differences between each RBF sub-model can be effectively reduced, thereby weakening the boundary discontinuities when extrapolating the RBF sub-models. Combined with weighted correction and far-field smoothing, the problem of the continuous increase of intermediate discontinuities in the model propagation to distant locations can be effectively addressed, achieving smooth and robust mesh deformation.

[0085] like Figure 1 As shown, this invention provides a mesh deformation method based on global-local spatial nested radial basis functions, comprising the following steps:

[0086] Step 1: Determine the set of deformation control points S in three-dimensional space. all ={s1,s2,…,sN} and the set of nodes to be deformed, V = {v1, v2, ..., v M}; where s1, s2, ..., s N Let v1, v2, ..., v be the N deformation control points. M , where M are the mesh nodes to be deformed;

[0087] The specific set of deformation control points S all ={s1,s2,…,s N} and the set of nodes to be deformed, V = {v1, v2, ..., v M A baseline grid is formed, which is determined based on the deformation information of the object surface and the spatial grid data.

[0088] Step 2, given the set of deformation control points S all ={s1,s2,…,s N The deformation amounts at each deformation control point in the equation are: ΔT1, ΔT2, ..., ΔT N ;

[0089] Step 3, determine the modeling method and interpolation algorithm for the RBF model, including:

[0090] Modeling method of RBF model: The set of deformation control points S all ={s1,s2,…,s N A subset S of deformation control points in any combination form within} D ={s D1 ,s D2 ,…,s D d} and the corresponding deformation ΔT D1 ,ΔT D2 ,...,ΔT Dd As input data, these are fed into the RBF initial model, and the model parameters ω={ω D1 ,ω D2 ,…,ω Dd}, thereby establishing the RBF model;

[0091] Interpolation calculation algorithm: Based on the model parameters ω={ω D1 ,ω D2 ,…,ω Dd Using interpolation formulas, the set of nodes in the deformable mesh, V = {v1, v2, ..., v}, is calculated. M The deformation amount of any deformable mesh node in} represents the deformation control point subset S. D ={s D1 ,s D2 ,…,s Dd The amount of deformation caused by the deformation of each deformation control point in the deformation control point;

[0092] In this step, the specific modeling method for the RBF model is as follows:

[0093] Based on the deformation control point subset S D ={s D1 ,s D2 ,…,s Dd The pre-deformation positions of each deformation control point and the corresponding deformation amount ΔT in the diagram. D1 ,ΔT D2 ,…,ΔT Dd Solve the following system of symmetric linear equations to obtain the model parameters ω={ω D1 ,ω D2 ,…,ω Dd}, ω D1 ,ω D2 ,…,ω Dd Also known as the weighting coefficient, it is expressed as: ω Du u = 1, ..., d; the deformation ΔT can be expressed by the deformation components ΔT in the three directions X, Y, and Z. x ΔT y and ΔT z express;

[0094]

[0095] in:

[0096] The matrix form of the linear equation system is:

[0097]

[0098] The basis functions generated for the u-th deformation control point and the j-th deformation control point.

[0099] The basis function used in this mesh deformation method can be Wendland's C2 function, a type of compact function, with the following expression:

[0100]

[0101] Where η is an intermediate variable; r0 is the radius of action of the basis function. This represents the Euclidean distance between the u-th deformation control point and the j-th deformation control point;

[0102] The fitting error D of the RBF model error It can be:

[0103]

[0104] Where ΔT x ΔT y and ΔTz The displacement ΔT is obtained by interpolation calculation of the RBF model. x0 ΔT y0 and ΔT z0 This represents the actual displacement in the X, Y, and Z directions after mesh deformation.

[0105] The interpolation calculation algorithm is as follows:

[0106] The set of nodes of the deformable mesh is V = {v1, v2, ..., v} M Any deformable mesh node v in} k k = 1, 2, ..., M, its deformation ΔT k We obtain it from the following formula:

[0107]

[0108] in: Represents the mesh node v to be deformed k With S D ={s D1 ,s D2 ,…,s D The Euclidean distance between the u-th deformation control points in d}.

[0109] Step 4: Establish a global low-precision RBF model:

[0110] A greedy algorithm is used to select from the set of deformation control points S all Randomly select 10-20 deformation control points from the input data. Use these selected deformation control points and their corresponding deformation amounts as input data. Following the RBF modeling method in step 3, establish the RBF model. Determine if the fitting error of the obtained RBF model meets the accuracy requirements. If not, continue to select deformation control points, increasing the number of deformation control points and their corresponding deformation amounts in the input data. Repeat this process until the accuracy requirements are met. The resulting RBF model is called the global low-precision RBF model. The deformation control points in the input data at this point form the set of deformation control points required for modeling the global low-precision RBF model, denoted as: S. Greddy ={s G1 ,s G2 ,…,s Gr};

[0111] In this step, it is determined whether the fitting error of the obtained RBF model meets the accuracy requirements. Specifically, the maximum fitting error of the RBF model is less than 1% of the maximum deformation.

[0112] Step 5: Establish a global-local space nested RBF model; the global-local space nested RBF model includes n global-local space nested RBF sub-models, which are referred to as RBF sub-models for short.

[0113] Step 5.1: Determine the partitioning direction l and the number of partitions n;

[0114] Step 5.2, set the deformation control points S all ={s1,s2,…,s N Deformation control points s1, s2, ..., s in} N Project onto the partition direction l, and continuously group the deformation control points along the partition direction l according to the projection coordinates of the deformation control points in ascending order of equal interval length or equal number of points, to obtain n sets of deformation control point subsets of different partition intervals; each set of deformation control point subsets has a corresponding partition interval.

[0115] Step 5.3: Combine the set of deformation control points S required for modeling each subset of deformation control points and the global low-precision RBF model. Greddy Merge and remove duplicate deformation control points to obtain a global-local combined deformation control point set; therefore, a total of n sets of global-local combined deformation control points are obtained, denoted as...

[0116] Step 5.4: Combine the global and local deformation control point sets for each group. Using the corresponding deformation as input data, and following the RBF model modeling method in step 3, an RBF model is established, called a global-local spatial nested RBF sub-model, or simply an RBF sub-model; therefore, a total of n RBF sub-models are established; each RBF sub-model has an influence area determined by partitioning intervals;

[0117] Step 6, for the set of nodes of the mesh to be deformed, V = {v1, v2, ..., v...} M Any deformable mesh node v in} k For k = 1, 2, ..., M, the corresponding RBF sub-model is determined based on its spatial location, and is represented as: RBF sub-model model k That is: the mesh node v to be deformed k In the RBF sub-model k Within the affected area;

[0118] Specifically, each RBF sub-model is only suitable for interpolation calculations within its corresponding interval. Therefore, before performing interpolation calculations, the spatial deformable mesh node v should be determined. k It is located within the influence region of the corresponding RBF sub-model.

[0119] Step 6 specifically involves:

[0120] For the deformable mesh node v k Its projection position in the partition direction l is l k Deformation control point set S all The projection range in the partition direction l is [l min ,l max Deformation control point set S all After being divided into n groups, along the partition direction l, they are sequentially named: the 1st deformation control point subset, the 2nd deformation control point subset, ..., the nth deformation control point subset. The RBF submodels corresponding to each deformation control point subset are sequentially named: the 1st RBF submodel, the 2nd RBF submodel, ..., the nth RBF submodel. The distinguishing position between the 1st deformation control point subset and the (i+1)th deformation control point subset is represented by l. i,i+1 ;

[0121] (a) If l k ≤l 1,1+1 Determine v k Under the influence region of the first RBF sub-model;

[0122] (b) If l i-1,i <l k ≤l i,i+1 Determine v k Under the influence region of the i-th RBF sub-model;

[0123] (c) If l k ≥l n-1,n Determine v k Under the influence region of the nth RBF sub-model.

[0124] Step 7: Using the interpolation calculation algorithm determined in Step 3, the RBF sub-model is used. k The model parameters are used to obtain the deformable mesh node v. k The deformation amount is used to determine the deformation of the mesh node v. k Mesh deformation.

[0125] Step 8, RBF sub-model boundary weighting correction algorithm:

[0126] Because the global-local nested RBF model uses all deformation control points for modeling, its interpolation accuracy is quite high. However, during extrapolation (propagation to distant locations), discontinuities appear at the boundaries of the RBF sub-models due to differences between them. To eliminate these boundary discontinuities, the weighted values ​​of two adjacent RBF sub-models are used to correct the interpolation results. The weighted correction method is as follows:

[0127] Step 8.1, along the partition direction l, set the RBF sub-model model. kThe front-end adjacent RBF sub-model is model k-1 The adjacent RBF sub-model at the back end is model k+1 ;

[0128] Step 8.2, determine the RBF sub-model. k The weighted adjustment range includes: the beginning weighted adjustment range and the end weighted adjustment range;

[0129] RBF sub-model k The minimum and maximum values ​​of the projection of the subset of deformation control points used in the modeling onto the partition direction l are l. k,min ,l k,max Therefore, its modeling range is l k,max -l k,min Let the dimensionless weighted interval coefficients be p, 0 ≤ p ≤ 1, for example, p = 0.5; then the RBF submodel... k The weighted correction range length is p(l) k,max -l k,min When p = 0.5, it represents the weighted correction range length of the RBF submodel. k 0.5 times the modeling range;

[0130] Therefore, the two boundary points of the first-end weighted correction range are: l k,min and l k,p =l k,min +p(l k,max -l k,min The weighted correction range for the first end is: [l k,min ,l k,p ];

[0131] The two boundary points of the end-weighted correction range are: l k,b =l k,max -p(l k,max -l k,min ) and l k,max The end-weighted correction range is: [l k,b, l k,max ];

[0132] Step 8.3, if the mesh node v to be deformed k At the projection position l in the partition direction l k ∈[l k,min ,l k,p If the result is negative, then the head-end weighted correction algorithm is adopted, including:

[0133] Through the RBF sub-model k The model parameters are used to obtain the deformable mesh node v. k Deformation amount ΔT k; through RBF sub-model k-1 The model parameters are used to obtain the deformable mesh node v. k Deformation amount ΔT k-1 ;

[0134] Construct the weighting formula; use trigonometric functions to construct the weighting coefficients to obtain the RBF sub-model. k-1 For the RBF sub-model k Weighted correction of deformable mesh node v k The amount of deformation is The formula is as follows:

[0135]

[0136] in:

[0137] Represents the RBF sub-model. k-1 For the RBF sub-model k The deformed interpolation result obtained after weighted correction, l k,min ,l k,max These are the RBF sub-models. k The minimum and maximum values ​​of the projection in the partition direction l, l k For RBF sub-model k The RBF sub-model k-1 The projection positions of the points participating in the weighted correction range on the partition direction l, l k,p The boundary values ​​of the correction range are determined for the weighted interval coefficients p, and α and β are the values ​​of the RBF sub-model model. k-1 and RBF sub-model k The weighting coefficients are set, and α+β=1 is guaranteed; ε1 is an intermediate parameter, representing the mesh node v to be deformed. k In the relative position of the weighted adjustment range;

[0138] Step 8.4, if the mesh node v to be deformed k At the projection position l in the partition direction l k ∈[l k,b ,l k,max If the result is negative, then the end-weighted correction algorithm is used, as shown in the following formula:

[0139]

[0140] l k,b =l k,max -p(l k,max -l k,min )

[0141] in:

[0142] Represents the RBF sub-model. k+1 For the RBF sub-model k The deformed interpolation result obtained after weighted correction; ΔT k+1 To use the RBF sub-model k+1 The model parameters are used to obtain the deformable mesh node v. k The amount of deformation; ΔT k To use the RBF sub-model k The model parameters are used to obtain the deformable mesh node v. k The deformation amount; α and γ are the deformation amounts of the RBF submodel. k and RBF sub-model k+1 The weighting coefficients are set, and α+γ=1 is guaranteed; ε2 is an intermediate parameter, representing the value of the mesh node v to be deformed. k The relative position within the weighted adjustment range.

[0143] Therefore, a weighted calculation formula is used to construct a weighted correction relationship between any two spatially adjacent RBF sub-models in the n RBF sub-models, thereby realizing the boundary discontinuity correction of all RBF sub-models in the global scope.

[0144] Figure 3 The cosine function was modeled using four RBF sub-models, and the extrapolation results were extracted and analyzed at five stations in the Y direction: 0.00, 0.25, 0.50, 1.00, and 1.50. (a) The extrapolation pattern of the local high-precision RBF model is shown. High-precision interpolation can be achieved at the object surface. However, as the model propagates at a distance, obvious discontinuities appear at the boundary and gradually increase. (b) The extrapolation pattern of the global low-precision + local high-precision RBF model is shown. After combining the global low-precision RBF modeling points, the differences between the RBF sub-models are reduced, thus effectively weakening the boundary discontinuities. (c) The extrapolation pattern of the global low-precision + local high-precision RBF model + boundary weighted correction is shown. It can be seen that the discontinuities at the boundaries of each RBF sub-model are completely eliminated, and the model decays smoothly as it moves away from the object.

[0145] Step 9, far-field smoothing:

[0146] As the global-local spatially nested RBF model propagates to the far field, the discontinuities at the boundaries of its RBF sub-models continuously increase, potentially leading to the failure of weighted corrections in handling large deformation problems. Far-field smoothing is performed using the global low-precision RBF model obtained in step 4. Figure 4The compatibility of two RBF models with the far-field smoothing model is demonstrated. Both RBF models divide the cosine function into four RBF sub-models for modeling. It can be seen that the model with global low precision + local high precision RBF model + boundary correction has good compatibility with the far-field smoothing model during far-field propagation.

[0147] The process for determining the full-field interpolation method under far-field smoothing is as follows:

[0148] The calculated value of the deformable mesh node v is obtained. k Distance corresponding RBF sub-model k The minimum distance between all deformation control points in the deformation control point subset used in the modeling Let the dimensionless smooth interval coefficients of the model be q, 0≤q≤1, for example, q=0.5, and r0 be the radius of action of the basis functions, representing the minimum distance between the far-field deformable mesh node and the modeling sample point. When the radius of action of the RBF basis function is greater than 0.5 times that of r0, far-field smoothing is performed.

[0149] The following determinations will be made:

[0150] (a) If Then the deformable mesh node v k The deformation is determined by the RBF submodel including boundary weighting corrections. k Obtained through interpolation calculation;

[0151] (b) If Then the deformable mesh node v k The deformation is determined by a global low-precision RBF model and an RBF sub-model that includes boundary weighting corrections. k The result was obtained by interpolation after further weighting.

[0152] Specifically, in At that time, the mesh node v to be deformed k The deformation is obtained by weighting using trigonometric functions, specifically through the following formula:

[0153] ΔT weighted =α w ΔT high +β w ΔT low

[0154]

[0155] in:

[0156] ΔT weighted Represents the mesh node v to be deformed k The corrected deformation amount;

[0157] ΔThigh The model represents the RBF submodel with boundary weighting corrections. k The interpolation calculation yields the deformable mesh node v k The amount of deformation; ΔT low The deformable mesh node v is calculated by interpolation from a global low-precision RBF model. k Deformation amount; α w and β w These are the weighting coefficients, and α is guaranteed to be... w +β w =1; ε w The intermediate parameter represents the mesh node v to be deformed. k The relative position within the weighted adjustment range.

[0158] (c) If Then the deformable mesh node v k The deformation is calculated by interpolation using a global low-precision RBF model, thus enabling far-field smoothing through the global low-precision RBF model.

[0159] The efficient mesh deformation method based on global-local spatial nested radial basis functions provided by this invention has the following advantages:

[0160] The improved RBF mesh deformation method proposed in this invention significantly reduces modeling and interpolation calculation time while maintaining high modeling accuracy. Specific advantages include:

[0161] 1) The computational cost of global-local spatial nested RBF modeling is reduced to that of traditional RBF modeling. The interpolation computation is reduced to one-fifth of its original value. This significantly reduces mesh deformation time.

[0162] 2) By using group modeling, all deformation control points are involved in the modeling of the global-local spatial nested RBF model. Its modeling accuracy is comparable to that of the traditional RBF model, and the model accurately passes through all deformation control points.

[0163] 3) By adopting a global low-precision + local high-precision RBF model and combining boundary weighting correction and far-field smoothing processing, the boundary discontinuities that occur during the extrapolation of the RBF sub-model can be completely eliminated, achieving smooth and robust full-field mesh deformation.

[0164] Here is an example:

[0165] The embodiments of the present invention employ a global-local spatial nested radial basis function mesh deformation method to deform the mesh of the M6 ​​wing structure.

[0166] The M6 ​​wing structure mesh used in this embodiment contains 21 mesh blocks, totaling 2.1 million mesh nodes, such as... Figure 6 As shown, the spatial mesh nodes around the wing surface are the mesh nodes to be deformed. 5000 deformation control points are selected from the wing surface as follows... Figure 7 As shown, it is assumed that the mesh deformation at the deformation control point is (0,0,0.1sin(πz)). The RBF function control radius is 4m, and the mesh is partitioned using an equal-length method.

[0167] The mesh deformation method based on global-local spatial nested radial basis function includes the following steps:

[0168] Determine the control points for surface deformation and the nodes of the spatial mesh to be deformed; establish a global low-precision RBF model; establish a global-local spatial nested RBF model; implement global-local spatial nested RBF model interpolation; perform weighted correction of RBF sub-model boundaries; and perform far-field smoothing.

[0169] Figure 7 Show a comparison image of the mesh before and after deformation. Figure 8 The comparison images of the spatial cross-section mesh before and after deformation are shown. It can be seen that the mesh deformation method based on global-local spatial nested radial basis function can effectively complete the mesh deformation.

[0170] Table 1 shows the statistics of mesh deformation time. The modeling time and deformation interpolation time vary with the number of RBF sub-models as shown in the figure. Figure 10 As shown, modeling using a single model (traditional RBF function mesh deformation method) takes 128.81 seconds, with interpolation and mesh deformation taking 51.21 seconds. However, modeling using nested RBF functions with eight RBF sub-models takes only 0.30 seconds, with interpolation and mesh deformation taking 8.25 seconds. The mesh deformation efficiency is 21.05 times that of the original method. As the number of RBF sub-models increases, the computational load of mesh deformation is significantly reduced, primarily by shortening the time required for modeling and interpolation during the mesh deformation process. Figure 11 The results show that the maximum and average interior angles of the deformed mesh vary with the number of RBF sub-models. It can be seen that within a certain range, increasing the number of RBF sub-models has a relatively small impact on the quality of the deformed mesh. The examples further demonstrate that the global-local spatial nested radial basis function mesh deformation method balances modeling accuracy and modeling and interpolation efficiency, making it an efficient and robust mesh deformation method.

[0171] Table 1. Statistics of Mesh Deformation Time

[0172]

[0173] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. An efficient mesh deformation method based on global-local space nested radial basis functions, characterized in that, Comprising the following steps: Step 1, determining a set of deformation control points in three-dimensional space and a set of to-be-deformed mesh nodes ; wherein, is a deformation control point; is a to-be-deformed mesh node; and is a to-be-deformed mesh node; and the to-be-deformed mesh node is a spatial mesh node on a wing surface Step 2, given a set of deformation control points The deformation amount of each deformation control point in the middle is respectively: ; Step 3, determining the modeling method of the RBF model and the interpolation calculation algorithm, comprising: The modeling method of the RBF model: a set of deformation control points Any combination of the deformation control point subsets And the corresponding deformation amount As input data, input to the RBF initial model, solve the model parameters , so as to establish the RBF model; An interpolation calculation algorithm is used to calculate the deformation of any deformed mesh node in the set of deformed mesh nodes according to the model parameters An interpolation formula is used to calculate the deformation of any deformed mesh node in the set of deformed mesh nodes The deformation of the deformed mesh node represents the deformation caused by the deformation of the deformed control point subset The deformation of the deformed mesh node represents the deformation caused by the deformation of the deformed control point subset Step 4, establishing a global low-precision RBF model: extract 10-20 deformation control points from the deformation control point set using a greedy algorithm The RBF model is established according to the modeling method of the RBF model in step 3, using the extracted deformation control points and the corresponding deformation as input data; it is determined whether the fitting error of the obtained RBF model meets the accuracy requirement; if not, the deformation control points are continuously extracted, and the number of deformation control points and the corresponding deformation in the input data is increased, and the cycle is continuously repeated until the accuracy requirement is met; at this time, the obtained RBF model is called a global low-precision RBF model; at this time, the deformation control points in the input data form a deformation control point set required for modeling the global low-precision RBF model, and is expressed as: ​ Step 5, establishing a global-local space nested RBF model; the global-local space nested RBF model comprises n global-local space nested RBF submodels, which are referred to as RBF submodels in brief; Step 5.1, determining partition direction and the number of partitions ; Step 5.2, grouping the deformation control points in the set of deformation control points according to the projection direction to the projection direction to the projection direction to the projection direction to the projection direction Step 5.

3. Modeling the deformation control point subset of each group and the global low-accuracy RBF model required deformation control point set Merging and removing duplicate deformation control points, a global-local combined deformation control point set is obtained; therefore, a total of group global-local combined deformation control point set is represented as ; Step 5.4, combine each set of global local deformation control points set and the corresponding deformation amount as input data, according to the modeling method of RBF model in step 3, the RBF model is established, called global-local space nested RBF sub model, simply referred to as RBF sub model; therefore, a total of RBF sub model is established; each RBF sub model has an influence area determined by the partition interval; Step 6, for any deformed mesh node in the set of deformed mesh nodes Step 6, for any deformed mesh node in the set of deformed mesh nodes , Step 6, for any deformed mesh node in the set of deformed mesh nodes Step 6, for any deformed mesh node in the set of deformed mesh nodes Step 6, for any deformed mesh node in the set of deformed mesh nodes Step 6, for any deformed mesh node in the set of deformed mesh nodes Step 7, using the interpolation calculation algorithm determined in step 3, the deformation amount of the to-be-deformed mesh node is obtained through the model parameters of the RBF sub-model, and mesh deformation of the to-be-deformed mesh node is realized. ​​​ Step 8, RBF submodel boundary weighting correction algorithm: Step 8.1, along the partition direction , let RBF sub-model of the front-end adjacent RBF sub-model be , and the back-end adjacent RBF sub-model be ; Step 8.2, determining the weighting correction range of the RBF sub-model comprising a head weighting correction range and a tail weighting correction range; RBF sub-model The deformation control point subset used in the modeling is in the partition direction The minimum and maximum values ​​of the projection are Therefore, its modeling scope is Let dimensionless weighted interval coefficients be used. , ; then the RBF sub-model The weighted correction range length is ; Therefore, the two boundary points of the head-end weighting correction range are respectively: and , and the head-end weighting correction range is: ; The two boundary points of the end weighting correction range are respectively: and The end weighting correction range is: ; Step 8.3, if the grid node to be deformed In the direction of the partition of the projection position The first end weighted correction algorithm is adopted, including: Through the RBF sub-model The model parameters are used to obtain the mesh nodes to be deformed. Deformation amount ; through RBF sub-model The model parameters are used to obtain the mesh nodes to be deformed. Deformation amount ; Constructing weighting formula; constructing weighting coefficient by using trigonometric function to obtain RBF sub-model Deforming the RBF sub-model The deformed grid node of the weighted corrected grid node The deformation amount of the deformed grid node of the weighted corrected grid node The formula is as follows: ； Wherein: representing the RBF sub-model representing the RBF sub-model the deformed interpolation result after the weighted correction, representing the RBF sub-model in the partition direction the projection minimum and maximum values, representing the RBF sub-model the projection position of the point participating in the weighted correction range of the RBF sub-model in the partition direction , representing the weighted interval coefficient the correction range boundary value determined, and representing the weight coefficient of the RBF sub-model and the RBF sub-model , and ensuring ; representing the intermediate parameter, representing the relative position of the deformed grid node in the weighted correction range; Step 8.4, if the grid node to be deformed In the direction of the partition of the projection position The end-weighted correction algorithm is adopted, and the formula is as follows: ； Wherein: representing the RBF sub-model representing the RBF sub-model the deformed interpolation result after the weighted correction; representing the deformed amount of the to-be-deformed mesh node by the model parameters of the RBF sub-model representing the deformed amount of the to-be-deformed mesh node by the model parameters of the RBF sub-model representing the deformed amount of the to-be-deformed mesh node by the model parameters of the RBF sub-model and are weight coefficients of the RBF sub-model and the RBF sub-model respectively, and satisfy ; is an intermediate parameter, representing the relative position of the to-be-deformed mesh node in the weighted correction range; Step 9, far field smoothing, and the full-field interpolation mode determination process under far field smoothing is: Calculate the nodes of the mesh to be deformed Distance corresponding RBF sub-model The minimum distance between all deformation control points in the deformation control point subset used in the modeling Assume the model has dimensionless smooth interval coefficients. , , If the radius of action of the basis function is given, then the following determination is made: (a) if , the deformation of the mesh node to be deformed is calculated by the RBF sub-model containing the boundary weight correction. (b) if then the deformation of the mesh node is calculated by the global low-accuracy RBF model and the RBF sub-model with boundary weight correction again after weighting and interpolation; (c) if then the deformation amount of the mesh node to be deformed is calculated by global low-precision RBF model interpolation, and global low-precision RBF model is used for far-field fairing; In the deformation amount of the to-be-deformed mesh node is weighted by a trigonometric function, and is obtained by the following formula: ； Wherein: representative of the nodes of the mesh to be deformed the modified deformation amount of the representative node Represents the RBF submodel with boundary weighting correction. The nodes of the deformable mesh obtained by interpolation calculation The amount of deformation; The nodes of the deformable mesh are represented by those obtained by interpolation from a global low-precision RBF model. The amount of deformation; and These are weighting coefficients, and it is guaranteed that... ; These are intermediate parameters, representing the nodes of the mesh to be deformed. The relative position within the weighted adjustment range.

2. The efficient mesh deformation method based on global-local space nested radial basis functions of claim 1, wherein, In step 3, the modeling method of the RBF model is specifically: According to the subset of deformation control points The pre-deformation positions of the deformation control points in each of the intermediate models and the corresponding deformation amounts , solve the following symmetric linear equation group to obtain the model parameters , Also known as weight coefficients, denoted as: , ; ； Wherein: The matrix form of the linear equation group is: ； a basis function generated for the first deformation control point and the second deformation control point.

3. The efficient mesh deformation method based on global-local space nested radial basis functions of claim 2, wherein, The base function expression is: ； ； in, As an intermediate variable; The radius of action of the basis functions. Representing the The deformation control point and the first Euclidean distance between deformation control points; The interpolation calculation algorithm is: set of any of the nodes of the mesh to be deformed any of the nodes of the mesh to be deformed , the amount of deformation thereof is obtained by the following equation: ； wherein: represent a node of the mesh to be deformed and the Euclidean distance between the i-th control point and the i-th deformed control point.

4. The efficient mesh deformation method based on global-local space nested radial basis functions of claim 1, wherein, Step 6 is specifically: For the nodes of the mesh to be deformed In the partition direction The projection position is Deformation control point set In partition direction The projection range is Deformation control point set Divided into After grouping, along the partition direction They are referred to in sequence as: the first deformation control point subset, the second deformation control point subset, ..., the ... There are several subsets of deformation control points, and the corresponding RBF sub-models for each subset are as follows: the first RBF sub-model, the second RBF sub-model, ..., the third RBF sub-model. The RBF sub-model; the The subset of deformation control points and the first The distinguishing position of the +1 deformation control point subset is represented as follows: ; (a) if , it is determined under the influence of the first RBF sub-model region; (b) if , it is determined under the first RBF sub-model influence region; (c) if , it is determined under the first RBF sub-model influence region.

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