Efficient grid deformation method based on global-local space nested radial basis function

Abstract

The invention provides an efficient grid deformation method based on a global-local space nested radial basis function. The method comprises the following steps: determining object plane deformation control points and space to-be-deformed grid nodes; establishing a global low-precision RBF model; establishing a global-local space nested RBF model; global-local space nested RBF (Radial Basis Function) model interpolation is realized; carrying out weighted correction on the boundary of the RBF sub-model; and performing far-field fairing, and outputting a deformed grid. Compared with traditional radial basis function grid deformation, the calculation amount required by modeling is reduced to # imgabs0 # times of the original calculation amount, and the calculation amount required by interpolation is reduced to # imgabs1 # n which is the number of RBF sub-models. The method greatly improves the grid deformation efficiency while guaranteeing the grid deformation precision, and is an efficient and robust grid deformation method.

Description

Technical Field

[0001] The invention belongs to the technical field of computational fluid dynamics, and in particular relates to an efficient grid deformation method based on global-local space nested radial basis functions. Background Art

[0002] In aerodynamic optimization design and fluid-structure interaction numerical simulation, the computational grid needs to be updated in real time according to the changes in the geometric shape of the aircraft. However, regenerating a set of complex grids is a time-consuming and tedious task, so the grid deformation method that does not change the number of grid points and topological relationships has become the best choice for updating the computational grid. The quality of the grid deformation method determines the quality of the updated grid and the time consumption of the grid deformation, which further affects the calculation accuracy and efficiency of the optimization design and fluid-structure interaction numerical simulation. When solving the problem of large deformation of the aircraft, the grid deformation method even becomes a limiting factor for the reliability of the numerical simulation of the flow field. Therefore, it is of great significance to develop efficient and robust grid deformation methods.

[0003] The basic idea of ​​mesh deformation technology is to move or rotate all meshes in space to generate new computational meshes according to the geometric changes of the object surface boundary while ensuring that the number of meshes and topological relationships remain unchanged. There are currently many mesh deformation methods, which can be divided into three categories according to different computational models: 1) Physical model method: spring method, spring body method, temperature body method, etc.; 2) Mathematical interpolation method: transfinite interpolation method TFI, radial basis function interpolation method RBF, inverse distance interpolation method IDW, etc.; 3) Hybrid method: RBF-DGM, RBF-TFI, etc. Among them, the RBF mesh deformation method has the following advantages: the deformation principle and application are relatively simple; the robustness for complex shape deformation problems and the quality of the deformed mesh are high; there are no requirements for the topological relationship of the mesh, and it is suitable for structured meshes, unstructured meshes and hybrid meshes. It has received widespread attention and research in recent years. However, the disadvantages of the RBF mesh deformation method are also very obvious: the time to construct the RBF interpolation model and use the model to achieve mesh deformation are respectively and n s ×n v Proportional (n s is the number of modeling control points, n v is the number of spatial grids to be deformed, i.e. the number of interpolations). As the size of the grid increases, the amount of calculation of the RBF method increases nonlinearly, resulting in extremely low efficiency.

[0004] In order to improve the efficiency of RBF mesh deformation method, a lot of research work is currently done on: on the one hand, using the optimization strategy represented by the greedy algorithm to reduce the number of modeling control points n s On the other hand, the spatial interpolation area is limited based on the greedy algorithm, thereby reducing the number of spatial interpolation control points n vHowever, the method of improving the efficiency of mesh deformation by sacrificing the number of modeling points or interpolation points is often insufficient when higher modeling accuracy is required, and faces the current situation where both computational accuracy and computational efficiency cannot be achieved. Therefore, it is urgent to develop a more efficient and robust RBF mesh deformation method to address this problem. Summary of the invention

[0005] In view of the defects of the prior art, the present invention provides an efficient mesh deformation method based on global-local space nested radial basis functions, which can effectively solve the above problems.

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

[0007] The present invention provides an efficient mesh deformation method based on global-local space nested radial basis function, comprising the following steps:

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

[0009] Step 2: Given a set of deformation control points S all ={s1,s2,…,s N The deformation amounts of each deformation control point in} are: ΔT1, ΔT2, …, ΔT N ;

[0010] Step 3: Determine the modeling method and interpolation calculation algorithm of the RBF model, including:

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

[0012] Interpolation calculation algorithm: According to the model parameter ω={ω D1 ,ω D2 ,…,ω Dd}, and use the interpolation formula to calculate the set of mesh nodes to be deformed V = {v1,v2,…,v M The deformation amount of any mesh node to be deformed in} represents the deformation control point subset S D ={s D1 ,s D2 ,…,s Dd The deformation amount caused by the deformation amount of each deformation control point in};

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

[0014] A greedy algorithm is used to select the deformation control point set S all 10 to 20 deformation control points are randomly selected from the input data, and the selected deformation control points and the corresponding deformation amounts are used as input data. According to the modeling method of the RBF model in step 3, an RBF model is established; it is determined whether the fitting error of the obtained RBF model meets the accuracy requirement. If not, deformation control points are continuously extracted, and the number of deformation control points and the corresponding deformation amounts in the input data is increased. This cycle is repeated until the accuracy requirement is met. The RBF model established at this time is called a global low-precision RBF model; at this time, the deformation control points in the input data form a set of deformation control points required for modeling the global low-precision RBF model, which is expressed as: S Greddy ={s G1 ,s G2 ,…,s Gr};

[0015] Step 5, establishing a global-local spatial nested RBF model; the global-local spatial nested RBF model includes n global-local spatial nested RBF sub-models, and the global-local spatial nested RBF sub-model is referred to as the RBF sub-model;

[0016] Step 5.1, determine the partition direction l and the number of partitions n;

[0017] Step 5.2: Set the deformation control point set S all ={s1,s2,…,s N} in the deformation control points s1,s2,…,s N Projecting to the partition direction l, the projection coordinates of the deformation control points are continuously grouped along the partition direction l in an equal interval length or equal point number manner from small to large, to obtain n groups of deformation control point subsets in different partition intervals; each group of deformation control point subsets has a corresponding partition interval;

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

[0019] Step 5.4: Combine each set of global and local deformation control points and the corresponding deformation as input data, according to the modeling method of the RBF model in step 3, an RBF model is established, which is called the global-local space nested RBF sub-model, referred to as the RBF sub-model; therefore, a total of n RBF sub-models are established; each RBF sub-model has an influence area determined by the partition interval;

[0020] Step 6: For the mesh node set to be deformed V = {v1, v2, ..., v M Any mesh node v to be deformed in k , k = 1, 2, ..., M, according to its spatial position, determine the corresponding RBF sub-model, expressed as: RBF sub-model model k , that is: the mesh node v to be deformed k , in the RBF sub-model model k within the area of ​​influence;

[0021] Step 7: Use the interpolation algorithm determined in step 3 to calculate the value of the RBF sub-model k The model parameters of the mesh node v to be deformed are obtained k The deformation amount of the mesh node v is treated as k The mesh deformation.

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

[0023] According to the deformation control point subset S D ={s D1 ,s D2 ,…,s Dd The pre-deformation position of each deformation control point in} and the corresponding deformation ΔT D1 ,ΔT D2 ,...,ΔT Dd , solve the following symmetric linear equations and obtain the model parameters ω={ω D1 ,ω D2 ,…,ω Dd},ω D1 ,ω D2 ,…,ω Dd Also called the weight coefficient, expressed as: ω Du, u=1,...d;

[0024]

[0025] in:

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

[0027]

[0028] is the basis function generated by the u-th deformation control point and the j-th deformation control point.

[0029] Preferably, the basis function expression is:

[0030]

[0031] Among them, η is the intermediate variable; r0 is the radius of action of the basis function, represents the Euclidean distance between the u-th deformation control point and the j-th deformation control point;

[0032] The interpolation calculation algorithm is:

[0033] The set of mesh nodes to be deformed V = {v1,v2,…,v M Any mesh node v to be deformed in k , k=1,2,...,M, its deformation ΔT k Obtained by the following formula:

[0034]

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

[0036] Preferably, step 6 is specifically:

[0037] For the mesh node v to be deformed 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 successively called: the first deformation control point subset, the second deformation control point subset, ..., the nth deformation control point subset, and the RBF submodels corresponding to each deformation control point subset are successively: the first RBF submodel, the second RBF submodel, ..., the nth RBF submodel; the distinguishing position between the i-th deformation control point subset and the i+1-th deformation control point subset is represented as l i,i+1 ;

[0038] (a) If l k ≤l 1,1+1 , determine v k Under the influence area 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 area of ​​the i-th RBF sub-model;

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

[0041] Preferably, the method further comprises step 8, RBF sub-model boundary weighted correction algorithm:

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

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

[0044] RBF sub-model k The minimum and maximum projection values ​​of the deformation control point subset used in modeling in the partition direction l are l k,min ,l k,max , therefore, its modeling range is l k,max -l k,min ; Assume dimensionless weighted interval coefficient p, 0≤p≤1; then RBF sub-model model k The weighted correction range length is p(l k,max -l k,min );

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

[0046] The two boundary points of the terminal weighted correction range are: 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 The projection position l in the partition direction l k ∈[l k,min ,l k,p ], the head-end weighted correction algorithm is adopted, including:

[0048] Through the RBF sub-model model k The model parameters of the mesh node v to be deformed are obtained k Deformation ΔT k ; Through the RBF sub-model model k-1 The model parameters of the mesh node v to be deformed are obtained k Deformation ΔT k-1 ;

[0049] Construct a weighted formula; use trigonometric functions to construct weighted coefficients to obtain the RBF sub-model model k-1 For RBF sub-model model k Weighted modified mesh node v to be deformed k The deformation amount is The formula is as follows:

[0050]

[0051] in:

[0052] Represents the RBF sub-model model k-1 For RBF sub-model model k The deformation interpolation result obtained after weighted correction, l k,min ,l k,max They are RBF sub-models respectively k The minimum and maximum projection values ​​in the partition direction l, l k is the RBF sub-model modelk In the RBF sub-model model k-1 The projection position of the points participating in the weighted correction range in the partition direction l, l k,p is the boundary value of the correction range determined by the weighted interval coefficient p, α and β are the RBF sub-model model k-1 and RBF sub-model model k The weight coefficient of , and ensure α + β = 1; ε1 is the intermediate parameter, representing the mesh node v to be deformed k relative position within the weighted correction range;

[0053] Step 8.4, if the mesh node v to be deformed k The projection position l in the partition direction l k ∈[l k,b ,l k,max ], the terminal weighted correction algorithm is adopted, and the formula is as follows:

[0054]

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

[0056] in:

[0057] Represents the RBF sub-model model k+1 For RBF sub-model model k Deformation interpolation result after weighted correction; ΔT k+1 To pass the RBF sub-model model k+1 The model parameters of the mesh node v to be deformed are obtained k Deformation; ΔT k To pass the RBF sub-model model k The model parameters of the mesh node v to be deformed are obtained k The deformation of α and γ are the RBF sub-models respectively. k and RBF sub-model model k+1 The weight coefficient of , and ensure α + γ = 1; ε2 is the intermediate parameter, representing the mesh node v to be deformed k Relative position in the weighted correction range.

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

[0059] Calculate the mesh node v to be deformed k RBF sub-model corresponding to the distancek The minimum distance between each deformation control point in the deformation control point subset used for modeling Assume the dimensionless smoothing interval coefficient q of the model, 0≤q≤1, r0 is the radius of action of the basis function, and make the following judgment:

[0060] (a) If Then the mesh node v to be deformed k The deformation is determined by the RBF sub-model model including the boundary weighted correction k The interpolation calculation is obtained;

[0061] (b) If Then the mesh node v to be deformed k The deformation of is composed of the global low-precision RBF model and the RBF sub-model containing boundary weighted correction model k The interpolation calculation is done after weighting again;

[0062] (c) If Then the mesh node v to be deformed k The deformation is calculated by interpolation of the global low-precision RBF model, and far-field smoothing is achieved through the global low-precision RBF model.

[0063] Preferably, When the mesh node v to be deformed k The deformation is weighted by trigonometric function, which is obtained by 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 Represents the RBF sub-model model including boundary weighted correction k The mesh node v to be deformed obtained by interpolation calculation k Deformation; ΔT low Represents the mesh node v to be deformed obtained by interpolation calculation of the global low-precision RBF model k The deformation amount; α w and β w is the weight coefficient, and ensure that α w +β w=1;ε w is an intermediate parameter, representing the mesh node v to be deformed k Relative position in the weighted correction range.

[0069] The present invention provides an efficient mesh deformation method based on global-local space nested radial basis function, which has the following advantages:

[0070] Compared with the traditional radial basis function grid deformation, the amount of calculation required for modeling is reduced to the original times, the amount of calculation required for interpolation is reduced to the original n is the number of RBF sub-models; therefore, while ensuring the accuracy of mesh deformation, the efficiency of mesh deformation is greatly improved, which is an efficient and robust mesh deformation method. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 A schematic flow chart of an efficient mesh deformation method based on global-local space nested radial basis functions provided by the present invention;

[0072] Figure 2 A schematic diagram of modeling points of the global low-precision-local high-precision spatial nested RBF sub-model in step 5 provided by the present invention;

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

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

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

[0076] Figure 6 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 A grid comparison diagram of the front and back surfaces of the wing provided by the present invention;

[0079] Fig. 9 A comparison diagram of the spatial cross-section grids before and after the wing deformation provided by the present invention;

[0080] Fig.10 The curve diagram of the modeling and interpolation time provided by the present invention versus the number of RBF sub-models;

[0081] Fig.11 This is a curve diagram of the maximum inner angle and average inner angle of the deformed grid as a function of the number of RBF sub-models provided by the present invention. DETAILED DESCRIPTION

[0082] In order to make the technical problems, technical solutions and beneficial effects solved by the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0083] In order to more clearly illustrate the purpose, technical methods and advantages of the present invention, the specific implementation of the present invention is described in detail below by combining specific examples and drawings. In addition, the described embodiments are only some examples of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without making creative work belong to the scope of protection of this application.

[0084] In view of the problem that the traditional radial basis function mesh deformation method cannot achieve both modeling accuracy and modeling and interpolation efficiency, this paper proposes an efficient mesh deformation method based on global-local spatial nested radial basis functions. In the traditional radial basis function mesh deformation method, building a global high-precision RBF model requires solving a huge number of n s ×n s Dimension (n s is the number of deformation control points). The present invention replaces the global high-precision RBF model with a global low-precision-local high-precision spatial nested RBF model, and the spatial grid deformation is obtained by interpolation calculation of the corresponding RBF sub-model. On the one hand, it effectively reduces the dimension of the linear equation group that needs to be solved for RBF modeling, and significantly improves the modeling and interpolation efficiency while ensuring high modeling accuracy; on the other hand, by coupling the global low-precision RBF modeling sample points with the local high-precision RBF modeling sample points to establish an RBF sub-model, the difference between each RBF sub-model can be effectively reduced, thereby weakening the boundary discontinuity during the extrapolation of the RBF sub-model. Combined with weighted correction and far-field smoothing processing, it can effectively deal with the problem of increasing discontinuities in the propagation of the model to a distant place, and achieve smooth and robust grid deformation.

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

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

[0087] Specific deformation control point set S all ={s1,s2,…,s N} and the set of mesh nodes to be deformed V = {v1,v2,…,v M}Form a reference grid, which is determined based on the object surface deformation information and spatial grid data.

[0088] Step 2: Given a set of deformation control points S all ={s1,s2,…,s N The deformation amounts of each deformation control point in} are: ΔT1, ΔT2, …, ΔT N ;

[0089] Step 3: Determine the modeling method and interpolation calculation algorithm of the RBF model, including:

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

[0091] Interpolation calculation algorithm: According to the model parameter ω={ω D1 ,ω D2 ,…,ω Dd}, and use the interpolation formula to calculate the set of mesh nodes to be deformed V = {v1,v2,…,v M The deformation amount of any mesh node to be deformed in} represents the deformation control point subset S D ={s D1 ,s D2 ,…,s Dd The deformation amount caused by the deformation amount of each deformation control point in};

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

[0093] According to the deformation control point subset S D ={s D1 ,s D2 ,…,s Dd The pre-deformation position of each deformation control point in} and the corresponding deformation ΔT D1 ,ΔT D2 ,…,ΔT Dd , solve the following symmetric linear equations and obtain the model parameters ω={ω D1 ,ω D2 ,…,ω Dd},ω D1 ,ω D2 ,…,ω Dd Also called the weight coefficient, expressed as: ω Du , u=1,…d; the deformation ΔT can be obtained by the deformation components ΔT in the three directions of XYZ x , ΔT y and ΔT z express;

[0094]

[0095] in:

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

[0097]

[0098] is the basis function generated by 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, which is a kind of compact function, and its expression is:

[0100]

[0101] Among them, η is the intermediate variable; r0 is the radius of action of the basis function, 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 Can be:

[0103]

[0104] Where ΔT x , ΔT y and ΔTz is the displacement calculated by interpolation of the RBF model; ΔT x0 , ΔT y0 and ΔT z0 It is the actual displacement of the mesh in the X, Y, and Z directions after deformation.

[0105] The interpolation calculation algorithm is:

[0106] The set of mesh nodes to be deformed V = {v1,v2,…,v M Any mesh node v to be deformed in k , k=1,2,...,M, its deformation ΔT k Obtained by the following formula:

[0107]

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

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

[0110] A greedy algorithm is used to select the deformation control point set S all A small number of 10 to 20 deformation control points are randomly selected from the data, and the extracted deformation control points and the corresponding deformation amounts are used as input data. According to the modeling method of the RBF model in step 3, an RBF model is established; it is determined whether the fitting error of the obtained RBF model meets the accuracy requirements. If not, deformation control points are continuously extracted, and the number of deformation control points and the corresponding deformation amounts in the input data is increased. This cycle is repeated until the accuracy requirements are met. The RBF model established at this time is called a global low-precision RBF model; at this time, the deformation control points in the input data form a set of deformation control points required for modeling the global low-precision RBF model, which is expressed 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 requirement, which may be specifically: the maximum fitting error of the RBF model is less than 1% of the maximum deformation.

[0112] Step 5, establishing a global-local spatial nested RBF model; the global-local spatial nested RBF model includes n global-local spatial nested RBF sub-models, and the global-local spatial nested RBF sub-model is referred to as the RBF sub-model;

[0113] Step 5.1, determine the partition direction l and the number of partitions n;

[0114] Step 5.2: Set the deformation control point set S all ={s1,s2,…,s N} in the deformation control points s1,s2,…,s N Projecting to the partition direction l, the projection coordinates of the deformation control points are continuously grouped along the partition direction l in an equal interval length or equal point number manner from small to large, to obtain n groups of deformation control point subsets in different partition intervals; each group of deformation control point subsets has a corresponding partition interval;

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

[0116] Step 5.4: Combine each set of global and local deformation control points and the corresponding deformation as input data, according to the modeling method of the RBF model in step 3, an RBF model is established, which is called the global-local space nested RBF sub-model, referred to as the RBF sub-model; therefore, a total of n RBF sub-models are established; each RBF sub-model has an influence area determined by the partition interval;

[0117] Step 6: For the mesh node set to be deformed V = {v1, v2, ..., v M Any mesh node v to be deformed in k , k = 1, 2, ..., M, according to its spatial position, determine the corresponding RBF sub-model, expressed as: RBF sub-model model k , that is: the mesh node v to be deformed k , in the RBF sub-model model k within the area of ​​influence;

[0118] Specifically, each RBF sub-model is only suitable for interpolation calculation in its corresponding interval. Therefore, before interpolation calculation, the spatial mesh node v to be deformed should be determined. k Located under the corresponding RBF sub-model influence area.

[0119] Step 6 is as follows:

[0120] For the mesh node v to be deformed 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 successively called: the first deformation control point subset, the second deformation control point subset, ..., the nth deformation control point subset, and the RBF submodels corresponding to each deformation control point subset are successively: the first RBF submodel, the second RBF submodel, ..., the nth RBF submodel; the distinguishing position between the i-th deformation control point subset and the i+1-th deformation control point subset is represented as l i,i+1 ;

[0121] (a) If l k ≤l 1,1+1 , determine v k Under the influence area 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 area of ​​the i-th RBF sub-model;

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

[0124] Step 7: Use the interpolation algorithm determined in step 3 to calculate the value of the RBF sub-model k The model parameters of the mesh node v to be deformed are obtained k The deformation amount of the mesh node v is treated as k The mesh deformation.

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

[0126] Since the global-local spatial nested RBF model uses all deformation control points for modeling, the interpolation accuracy is quite high. However, when it propagates to a distant place, i.e., when it is extrapolated, due to the differences between the RBF sub-models, discontinuities will appear at the boundaries of the RBF sub-models. In order to eliminate the boundary discontinuities, the weighted values ​​of the two adjacent RBF sub-models are used to correct the interpolation results. The weighted correction method is:

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

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

[0129] RBF sub-model k The minimum and maximum projection values ​​of the deformation control point subset used in modeling in the partition direction l are l k,min ,l k,max , therefore, its modeling range is l k,max -l k,min ; Assume dimensionless weighted interval coefficient p, 0≤p≤1, for example, p=0.5; then RBF submodel model k The weighted correction range length is p(l k,max -l k,min ), when p = 0.5, it means that the weighted correction range length is RBF sub-model model k 0.5 times the modeling range;

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

[0131] The two boundary points of the terminal weighted correction range are: 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 The projection position l in the partition direction l k ∈[l k,min ,l k,p ], the head-end weighted correction algorithm is adopted, including:

[0133] Through the RBF sub-model model k The model parameters of the mesh node v to be deformed are obtained k Deformation ΔT k; Through the RBF sub-model model k-1 The model parameters of the mesh node v to be deformed are obtained k Deformation ΔT k-1 ;

[0134] Construct a weighted formula; use trigonometric functions to construct weighted coefficients to obtain the RBF sub-model model k-1 For RBF sub-model model k Weighted modified mesh node v to be deformed k The deformation amount is The formula is as follows:

[0135]

[0136] in:

[0137] Represents the RBF sub-model model k-1 For RBF sub-model model k The deformation interpolation result obtained after weighted correction, l k,min ,l k,max They are RBF sub-models respectively k The minimum and maximum projection values ​​in the partition direction l, l k is the RBF sub-model model k In the RBF sub-model model k-1 The projection position of the points participating in the weighted correction range in the partition direction l, l k,p is the boundary value of the correction range determined by the weighted interval coefficient p, α and β are the RBF sub-model model k-1 and RBF sub-model model k The weight coefficient of , and ensure α + β = 1; ε1 is the intermediate parameter, representing the mesh node v to be deformed k relative position within the weighted correction range;

[0138] Step 8.4, if the mesh node v to be deformed k The projection position l in the partition direction l k ∈[l k,b ,l k,max ], the terminal weighted correction algorithm is adopted, and the formula is as follows:

[0139]

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

[0141] in:

[0142] Represents the RBF sub-model model k+1 For RBF sub-model model k Deformation interpolation result after weighted correction; ΔT k+1 To pass the RBF sub-model model k+1 The model parameters of the mesh node v to be deformed are obtained k Deformation; ΔT k To pass the RBF sub-model model k The model parameters of the mesh node v to be deformed are obtained k The deformation of α and γ are the RBF sub-models respectively. k and RBF sub-model model k+1 The weight coefficient of , and ensure α + γ = 1; ε2 is the intermediate parameter, representing the mesh node v to be deformed k Relative position in the weighted correction range.

[0143] Therefore, a weighted calculation formula construction method is adopted to establish a weighted correction relationship between any two groups of spatially adjacent RBF sub-models in the n RBF sub-models, so as to realize the discontinuity correction of the boundaries of all RBF sub-models in the global range.

[0144] Figure 3 The cosine function is divided into four RBF sub-models for modeling, and the extrapolation results at five positions in the Y direction, namely 0.00, 0.25, 0.50, 1.00, and 1.50, are extracted for analysis. (a) The extrapolation law of the local high-precision RBF model is shown. High-precision interpolation can be achieved at the object surface. However, in the process of distant propagation, the model has obvious discontinuities at the boundary and gradually increases; (b) The extrapolation law 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, thereby effectively weakening the boundary discontinuities; (c) The extrapolation law after 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 erased, and the model decays smoothly to the distance.

[0145] Step 9, far-field smoothing:

[0146] When the global-local spatial nested RBF model propagates to the far field, the discontinuity at the boundary of its RBF submodel will continue to increase, which may cause the weighted correction to fail when dealing with large deformation problems. The global low-precision RBF model obtained in step 4 is used for far-field smoothing. Figure 4The compatibility of the 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 global low-precision + local high-precision RBF model + boundary correction model is very compatible with the far-field smoothing model during far-field propagation.

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

[0148] Calculate the mesh node v to be deformed k RBF sub-model corresponding to the distance k The minimum distance between each deformation control point in the deformation control point subset used for modeling Assume the dimensionless smoothing interval coefficient q of the model, 0≤q≤1, for example, q=0.5, r0 is the radius of action of the basis function, which represents the minimum value of the distance between the far-field deformed mesh node and the modeling sample point When the radius of action of the RBF basis function is greater than 0.5 times r0, far-field smoothing is performed.

[0149] Make the following determinations:

[0150] (a) If Then the mesh node v to be deformed k The deformation is determined by the RBF sub-model model including the boundary weighted correction k The interpolation calculation is obtained;

[0151] (b) If Then the mesh node v to be deformed k The deformation of is composed of the global low-precision RBF model and the RBF sub-model containing boundary weighted correction model k The interpolation calculation is done after weighting again;

[0152] Specifically, in When the mesh node v to be deformed k The deformation is weighted by trigonometric function, which is obtained by 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 Represents the RBF sub-model model including boundary weighted correction k The mesh node v to be deformed obtained by interpolation calculation k Deformation; ΔT low Represents the mesh node v to be deformed obtained by interpolation calculation of the global low-precision RBF model k The deformation amount; α w and β w is the weight coefficient, and ensure that α w +β w =1;ε w is an intermediate parameter, representing the mesh node v to be deformed k Relative position in the weighted correction range.

[0158] (c) If Then the mesh node v to be deformed k The deformation is calculated by interpolation of the global low-precision RBF model, and far-field smoothing is achieved through the global low-precision RBF model.

[0159] The present invention provides an efficient mesh deformation method based on global-local space nested radial basis function, which has the following advantages:

[0160] The improved RBF mesh deformation method proposed by the present invention greatly reduces the modeling and interpolation calculation time while ensuring high modeling accuracy. The specific advantages include:

[0161] 1) The computational effort required for global-local spatial nested RBF modeling is reduced to that of traditional RBF modeling. times, the amount of interpolation calculation is reduced to the original times, greatly reducing the mesh deformation time.

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

[0163] 3) The global low-precision + local high-precision RBF model is adopted and combined with boundary weighted correction and far-field smoothing processing, which can completely eliminate the boundary discontinuity that occurs when the RBF sub-model is extrapolated and achieve smooth and robust full-field mesh deformation.

[0164] An example is given below:

[0165] The embodiment of the present invention uses a global-local space nested radial basis function grid deformation method to perform grid deformation on the M6 ​​wing structure grid.

[0166] The M6 ​​wing structure grid used in this embodiment includes 21 grid blocks, with a total of 2.1 million grid nodes. Figure 6 As shown in the figure, the spatial grid nodes around the wing surface are the grid nodes to be deformed. 5000 deformation control points are selected from the wing surface as follows Figure 7 As shown in the figure, 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 partition is divided into equal lengths.

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

[0168] Determine the deformation control points of the object surface and the mesh nodes to be deformed in space; 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 on the boundaries of RBF sub-models; and perform far-field smoothing.

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

[0170] The statistics of mesh deformation time are shown in Table 1. The model building time and deformation interpolation time vary with the number of RBF sub-models. Fig.10 As shown in the figure. The modeling time of using one model, namely the traditional RBF function grid deformation method, is 128.81s, and the interpolation and grid deformation time is 51.21s; the modeling time of the nested RBF function of 8 RBF sub-models is only 0.30s, and the interpolation and grid deformation time is 8.25s. The grid deformation efficiency is 21.05 times that of the original method. When the number of RBF sub-models increases, the calculation amount of grid deformation is greatly reduced, which mainly shortens the time required for modeling and interpolation during grid deformation. Fig.11 The maximum inner angle and average inner angle of the deformed mesh change with the number of RBF sub-models. It can be seen that within a certain range, the increase in the number of RBF sub-models has little effect on the quality of the deformed mesh. The embodiment further proves that the mesh deformation method based on the global-local space nested radial basis function takes into account both modeling accuracy and modeling and interpolation efficiency, and is an efficient and robust mesh deformation method.

[0171] Table 1 Mesh deformation time statistics

[0172]

[0173] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be considered as the scope of protection of the present invention.

Claims

1. An efficient mesh deformation method based on global-local spatial nested radial basis functions, characterized in that: The following steps are involved: Step 1: Determine the deformation control point set S in three-dimensional space all ={s1,s2,…,s N } and the set of mesh nodes to be deformed V = {v1,v2,…,v M }; where s1, s2, …, s N , are N deformation control points; v1,v2,…,v M , are M mesh nodes to be deformed; Step 2: Given a set of deformation control points S all ={s1,s2,…,s N The deformation amounts of each deformation control point in} are: ΔT1, ΔT2, …, ΔT N ; Step 3: Determine the modeling method and interpolation calculation algorithm of the RBF model, including: Modeling method of RBF model: The deformation control point set S all ={s1,s2,…,s N } any combination of deformation control point subsets S D ={s D1 ,s D2 ,…,s Dd } and the corresponding deformation ΔT D1 ,ΔT D2 ,...,ΔT Dd As input data, it is input into the RBF initial model and the model parameters ω={ω D1 ,ω D2 ,…,ω Dd }, thus establishing the RBF model; Interpolation calculation algorithm: According to the model parameter ω={ω D1 ,ω D2 ,…,ω Dd }, and use the interpolation formula to calculate the set of mesh nodes to be deformed V = {v1,v2,…,v M The deformation amount of any mesh node to be deformed in} represents the deformation control point subset S D ={s D1 ,s D2 ,…,s Dd The deformation amount caused by the deformation amount of each deformation control point in}; Step 4: Establish a global low-precision RBF model: A greedy algorithm is used to select the deformation control point set S all 10 to 20 deformation control points are randomly selected from the input data, and the selected deformation control points and the corresponding deformation amounts are used as input data. According to the modeling method of the RBF model in step 3, an RBF model is established; it is determined whether the fitting error of the obtained RBF model meets the accuracy requirement. If not, deformation control points are continuously extracted, and the number of deformation control points and the corresponding deformation amounts in the input data is increased. This cycle is repeated until the accuracy requirement is met. The RBF model established at this time is called a global low-precision RBF model; at this time, the deformation control points in the input data form a set of deformation control points required for modeling the global low-precision RBF model, which is expressed as: S Greddy ={s G1 ,s G2 ,…,s Gr }; Step 5, establishing a global-local spatial nested RBF model; the global-local spatial nested RBF model includes n global-local spatial nested RBF sub-models, and the global-local spatial nested RBF sub-model is referred to as the RBF sub-model; Step 5.1, determine the partition direction l and the number of partitions n; Step 5.2: Set the deformation control point set S all ={s1,s2,…,s N } in the deformation control points s1,s2,…,s N Projecting to the partition direction l, the projection coordinates of the deformation control points are continuously grouped along the partition direction l in an equal interval length or equal point number manner from small to large, to obtain n groups of deformation control point subsets in different partition intervals; each group of deformation control point subsets has a corresponding partition interval; Step 5.3: Combine each set of deformation control point subsets with the deformation control point set S required for modeling the global low-precision RBF model. Greddy Merge and remove duplicate deformation control points to obtain a set of global and local combined deformation control points; therefore, a total of n sets of global and local combined deformation control points are obtained, expressed as Step 5.4: Combine each set of global and local deformation control points and the corresponding deformation as input data, according to the modeling method of the RBF model in step 3, an RBF model is established, which is called the global-local space nested RBF sub-model, referred to as the RBF sub-model; therefore, a total of n RBF sub-models are established; each RBF sub-model has an influence area determined by the partition interval; Step 6: For the mesh node set to be deformed V = {v1, v2, ..., v M Any mesh node v to be deformed in k , k = 1, 2, ..., M, according to its spatial position, determine the corresponding RBF sub-model, expressed as: RBF sub-model model k , that is: the mesh node v to be deformed k , in the RBF sub-model model k within the area of ​​influence; Step 7: Use the interpolation algorithm determined in step 3 to calculate the value of the RBF sub-model k The model parameters of the mesh node v to be deformed are obtained k The deformation amount of the mesh node v is treated as k The mesh deformation.

2. The efficient mesh deformation method based on global-local spatial nested radial basis function according to claim 1, characterized in that: In step 3, the modeling method of the RBF model is specifically as follows: According to the deformation control point subset S D ={s D1 ,s D2 ,…,s Dd The pre-deformation position of each deformation control point in} and the corresponding deformation ΔT D1 ,ΔT D2 ,…,ΔT Dd , solve the following symmetric linear equations and obtain the model parameters ω={ω D1 ,ω D2 ,…,ω Dd },ω D1 ,ω D2 ,…,ω Dd Also called the weight coefficient, expressed as: ω Du , u=1,...d; in: The matrix form of the linear equation system is: is the basis function generated by the u-th deformation control point and the j-th deformation control point.

3. The efficient mesh deformation method based on global-local spatial nested radial basis function according to claim 2, characterized in that: The basis function expression is: Among them, η is the intermediate variable; r0 is the radius of action of the basis function, represents the Euclidean distance between the u-th deformation control point and the j-th deformation control point; The interpolation calculation algorithm is: The set of mesh nodes to be deformed V = {v1,v2,…,v M Any mesh node v to be deformed in k , k=1,2,...,M, its deformation ΔT k Obtained by the following formula: in: Represents the mesh node v to be deformed k With S D ={s D1 ,s D2 ,…,s D d} is the Euclidean distance between the u-th deformation control points.

4. The efficient mesh deformation method based on global-local spatial nested radial basis function according to claim 1, characterized in that: Step 6 is as follows: For the mesh node v to be deformed 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 successively called: the first deformation control point subset, the second deformation control point subset, ..., the nth deformation control point subset, and the RBF submodels corresponding to each deformation control point subset are successively: the first RBF submodel, the second RBF submodel, ..., the nth RBF submodel; the distinguishing position between the i-th deformation control point subset and the i+1-th deformation control point subset is represented as l i,i+1 ; (a) If l k ≤l 1,1+1 , determine v k Under the influence area of ​​the first RBF sub-model; (b) If l i-1,i <l k ≤l i,i+1 , determine v k Under the influence area of ​​the i-th RBF sub-model; (c) If l k ≥l n-1,n , determine v k Under the influence area of ​​the nth RBF sub-model.

5. The efficient mesh deformation method based on global-local spatial nested radial basis function according to claim 1, characterized in that: It also includes step 8, RBF sub-model boundary weighted correction algorithm: Step 8.1, along the partition direction l, set the RBF sub-model model k The front adjacent RBF sub-model is model k-1 , the back-end adjacent RBF sub-model is model k+1 ; Step 8.2, determine the RBF sub-model model k The weighted correction range includes: the first end weighted correction range and the end end weighted correction range; RBF sub-model k The minimum and maximum projection values ​​of the deformation control point subset used in modeling in the partition direction l are l k,min ,l k,max , therefore, its modeling range is l k,max -l k,min ; Assume dimensionless weighted interval coefficient p, 0≤p≤1; then RBF sub-model model k The weighted correction range length is p(l k,max -l k,min ); Therefore, the two boundary points of the head-end weighted correction range are: k,min and l k,p = l k,min +p(l k,max -l k,min ), the first end weighted correction range is: [l k,min ,l k,p ]; The two boundary points of the terminal weighted correction range are: 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 ]; Step 8.3, if the mesh node v to be deformed k The projection position l in the partition direction l k ∈[l k,min ,l k,p ], the head-end weighted correction algorithm is adopted, including: Through the RBF sub-model k The model parameters of the mesh node v to be deformed are obtained k Deformation ΔT k ; Through the RBF sub-model model k-1 The model parameters of the mesh node v to be deformed are obtained k Deformation ΔT k-1 ; Construct a weighted formula; use trigonometric functions to construct weighted coefficients to obtain the RBF sub-model model k-1 For RBF sub-model model k Weighted modified mesh node v to be deformed k The deformation amount is The formula is as follows: in: Represents the RBF sub-model model k-1 For RBF sub-model model k The deformation interpolation result obtained after weighted correction, l k,min ,l k,max They are RBF sub-models respectively k The minimum and maximum projection values ​​in the partition direction l, l k is the RBF sub-model model k In the RBF sub-model model k-1 The projection position of the points participating in the weighted correction range in the partition direction l, l k,p is the boundary value of the correction range determined by the weighted interval coefficient p, α and β are the RBF sub-model model k-1 and RBF sub-model model k The weight coefficient of , and ensure α + β = 1; ε1 is the intermediate parameter, representing the mesh node v to be deformed k relative position within the weighted correction range; Step 8.4, if the mesh node v to be deformed k The projection position l in the partition direction l k ∈[l k,b ,l k,max ], the terminal weighted correction algorithm is adopted, and the formula is as follows: in: Represents the RBF sub-model model k+1 For RBF sub-model model k Deformation interpolation result after weighted correction; ΔT k+1 To pass the RBF sub-model model k+1 The model parameters of the mesh node v to be deformed are obtained k Deformation; ΔT k To pass the RBF sub-model model k The model parameters of the mesh node v to be deformed are obtained k The deformation of α and γ are the RBF sub-models respectively. k and RBF sub-model model k+1 The weight coefficient of , and ensure α + γ = 1; ε2 is the intermediate parameter, representing the mesh node v to be deformed k Relative position in the weighted correction range.

6. The efficient mesh deformation method based on global-local spatial nested radial basis function according to claim 5, characterized in that: It also includes step 9, far-field smoothing, and the full-field interpolation method determination process under far-field smoothing is: Calculate the mesh node v to be deformed k RBF sub-model corresponding to the distance k The minimum distance between each deformation control point in the deformation control point subset used for modeling Assume the dimensionless smoothing interval coefficient q of the model, 0≤q≤1, r0 is the radius of action of the basis function, and make the following judgment: (a) If Then the mesh node v to be deformed k The deformation is determined by the RBF sub-model model including the boundary weighted correction k The interpolation calculation is obtained; (b) If Then the mesh node v to be deformed k The deformation of is composed of the global low-precision RBF model and the RBF sub-model containing boundary weighted correction model k The interpolation calculation is done after weighting again; (c) If Then the mesh node v to be deformed k The deformation is calculated by interpolation of the global low-precision RBF model, and far-field smoothing is achieved through the global low-precision RBF model.

7. The efficient mesh deformation method based on global-local spatial nested radial basis function according to claim 6, characterized in that: exist When the mesh node v to be deformed k The deformation is weighted by trigonometric function, which is obtained by the following formula: ΔT weighted =a w ΔT high +b w ΔT low in: ΔT weighted Represents the mesh node v to be deformed k The corrected deformation amount; ΔT high Represents the RBF sub-model model including boundary weighted correction k The mesh node v to be deformed obtained by interpolation calculation k Deformation; ΔT low Represents the mesh node v to be deformed obtained by interpolation calculation of the global low-precision RBF model k The deformation amount; α w and β w is the weight coefficient, and ensure that α w +β w =1;ε w is an intermediate parameter, representing the mesh node v to be deformed k Relative position in the weighted correction range.

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