Fluid-solid coupling method based on compact support type radial basis function

Abstract

The invention discloses a fluid-solid coupling method based on a compact support type radial basis function, and relates to the technical field of numerical simulation, and the method comprises the steps: determining a fluid-solid coupling boundary between a fluid model and a structure model, dividing the fluid-solid coupling boundary into a plurality of sub-domains, and enabling every two adjacent sub-domains to be overlapped with each other; calculating the local coupling matrix of each sub-domain in parallel, and synthesizing the local coupling matrixes into a global coupling matrix; carrying out iterative calculation of load conversion and displacement conversion on the fluid nodes and the structure nodes on the fluid-solid coupling boundary by utilizing the global coupling matrix until a balance state is reached, and obtaining the positions of the structure nodes and the positions of the fluid nodes; the iterative calculation process is repeated at each simulation moment until the maximum simulation moment is reached, and fluid-solid coupling numerical simulation calculation is completed. According to the method, a high-precision and telescopic fluid-solid coupling mapping frame is formed, the method can adapt to anisotropic changes of geometric and physical field variables, and the calculation precision and efficiency of fluid-solid coupling are remarkably improved.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of numerical simulation, in particular to a fluid-structure coupling method based on compactly supported radial basis functions. BACKGROUND

[0002] Fluid-structure interaction is a key problem in engineering field, and the core is to accurately transfer the boundary data of fluid load and structure deformation. High-precision fluid-structure coupling analysis is of great significance for complex three-dimensional engineering problems such as aeroelastic analysis in aerospace field and fluid-structure coupling simulation in ship and ocean engineering field.

[0003] Traditional fluid-structure coupling methods usually use global radial basis functions for interpolation solution, but this method has three significant defects: (1) the influence of global radial basis functions covers all nodes, which will lead to dense coupling matrix, so that the calculation amount increases with the square of the number of nodes, and the calculation efficiency is low; (2) global radial basis functions are prone to smooth local effects, and it is difficult to ensure local precision in complex three-dimensional models; (3) the method of using global radial basis functions for interpolation is limited to two-dimensional space and cannot be directly applied to three-dimensional fluid-structure coupling problems. SUMMARY

[0004] In view of the above problems and technical needs, the application provides a fluid-structure coupling method based on compactly supported radial basis functions, and the technical scheme of the application is as follows: A fluid-structure coupling method based on compactly supported radial basis functions, comprising the following steps: Determine the fluid-structure coupling boundary between the fluid model and the structure model, and divide the fluid-structure coupling boundary into a plurality of subdomains, any two adjacent subdomains overlap each other, and each subdomain contains a plurality of fluid nodes of the fluid model and a plurality of structure nodes of the structure model; For the kth iteration calculation step of the Tth simulation time, the fluid model is solved based on the position of the fluid nodes of the fluid model in the kth iteration calculation step, to obtain the fluid load of all fluid nodes on the fluid-structure coupling boundary in the kth iteration calculation step; Based on the compactly supported radial basis function, the positions of the fluid nodes of the fluid model and the positions of the structure nodes of the structure model in the kth iteration calculation step are calculated in parallel to obtain the local coupling matrix of each subdomain on the fluid-structure coupling boundary in the kth iteration calculation step, and the local coupling matrices of each subdomain are combined to obtain the global coupling matrix; The fluid loads of each fluid node on the fluid-structure interaction boundary are transformed using a global coupling matrix to obtain the structural loads of each structural node on the fluid-structure interaction boundary. The structural loads of each structural node are then substituted into the structural model for solution to obtain the displacements of each structural node on the fluid-structure interaction boundary in the k-th iteration calculation step. The displacements of each structural node on the fluid-structure interaction boundary are then transformed using a global coupling matrix to obtain the displacements of each fluid node on the fluid-structure interaction boundary in the k-th iteration calculation step. When the fluid-structure interaction calculation reaches equilibrium based on the displacements of all structural nodes and fluid nodes on the fluid-structure interaction boundary in each iterative calculation step, the structural node positions of the structural model at the Tth simulation time and the fluid node positions of the fluid model at the Tth simulation time are obtained. Otherwise, the fluid node positions of the fluid model in the (k+1)th iterative calculation step are updated based on the displacements of the fluid nodes on the fluid-structure interaction boundary. Let T = T+1, and proceed to the fluid-structure interaction calculation at the (T+1)th simulation time.

[0005] A further technical solution involves determining the local coupling matrix of any subdomain p on the fluid-structure interaction boundary. include: Based on the positions of each fluid node and each structural node within the subdomain p, a local metric matrix is ​​constructed for each structural node. The local metric matrix characterizes the geometric features and physical field changes at the structural node. Based on the local metric matrix of each structural node, the support radius of the compact support radial basis function in subdomain p is determined according to the positions of each fluid node and each structural node in subdomain p. Based on the support radius of the compactly supported radial basis functions within subdomain p, the local coupling matrix of subdomain p is determined. .

[0006] A further technical solution involves determining the support radius of the compactly supported radial basis function within the subdomain p, including: Based on the local metric matrix of each structural node, the anisotropic distance between each structural node and each fluid node in subdomain p is calculated according to the positions of each fluid node and each structural node in subdomain p. The candidate support radius at each structural node is determined according to the anisotropic distance between each structural node and each fluid node. The anisotropic distance is used to measure the spatial distance in different directions. The candidate structural loads of each structural node in the current iteration calculation step are calculated based on the candidate support radii at each structural node, and the candidate local residuals of each structural node are determined. The candidate local residuals are the absolute values ​​of the difference between the candidate structural loads in the current iteration calculation step and the structural loads in the previous iteration calculation step. Calculate the condition number of the local metric matrix of each structural node, and determine the candidate support radius of the structural node with a condition number less than a predetermined threshold and the smallest candidate local residual as the support radius of the compact support type radial basis function in subdomain p.

[0007] A further technical solution involves constructing arbitrary structural nodes within the subdomain p. Local metric matrix include: Calculate structural nodes The covariance matrix of the position coordinate differences between the covariance matrix and the position coordinate differences of each fluid node in the subdomain p is obtained by performing eigenvalue decomposition on the covariance matrix to obtain the eigenvalues ​​and eigenvectors of the covariance matrix. Structural nodes are constructed based on the eigenvalues ​​and eigenvectors of the covariance matrix. Local metric matrix ,in, R It is an eigenvector matrix and , diagonal matrix and , It is the largest eigenvalue. This is the second largest eigenvalue. It is the smallest eigenvalue.

[0008] A further technical solution involves calculating any structural node within the subdomain p. and arbitrary fluid nodes Anisotropic distance Determine structural nodes Candidate support radius at the location ; in, It is a proportionality coefficient. It is a structural node The local metric matrix, It is a structural node position coordinates and , It is a fluid node position coordinates and , This indicates the matrix transpose.

[0009] A further technical solution involves synthesizing the local coupling matrices of each subdomain into a global coupling matrix. include: The weights of each subdomain are determined based on the anisotropic distance between the structural node with the smallest candidate local residual and the center node of the subdomain, and the support radius of the compact support radial basis function in each subdomain. The global coupling matrix is ​​obtained by weighting and summing the local coupling matrices of each subdomain according to their respective weights. , It is the local coupling matrix of subfield p. It is the weight of subdomain p. This is the total number of subdomains.

[0010] A further technical solution involves determining the weights of any subdomain p. include: Based on the candidate structure node with the smallest local residual within subdomain p With subdomain center node Anisotropic distance Determine the weight of subdomain p. , It is the support radius of the compactly supported radial basis function within the subdomain p. It is the structural node with the minimum candidate local residual within subdomain q. With subdomain center node Anisotropic distance, It is the support radius of the compactly supported radial basis function within the subdomain q. This represents a compact, supported radial basis function. This is the total number of subdomains.

[0011] A further technical solution is that, for any k-th iteration calculation step: Based on the structural nodes on the fluid-structure interaction boundary in the kth iteration calculation step displacement and its corresponding fluid nodes displacement Determine the fluid nodes on the fluid-structure interaction boundary. Interface residuals at the location ; When the interface residuals at all fluid nodes on the fluid-structure interaction boundary do not exceed the iteration error threshold, it is determined that the fluid-structure interaction calculations of the fluid model and the structural model have reached an equilibrium state.

[0012] A further technical solution involves updating any fluid node in the (k+1)th iteration calculation step. The locations include: The IQN multi-secant method is used to generate fluid nodes in the k-th iteration calculation step. Correction amount of interface residual at the location And update the fluid nodes in the (k+1)th iteration calculation step. Location , It is the fluid node in the k-th iteration calculation step. Location, It is a relaxation factor and .

[0013] A further technical solution involves dividing the fluid-structure interaction boundary into multiple subdomains, including: Numerical simulation methods are used to determine the fluid characteristic parameters of the fluid model and the structural characteristic parameters of the structural model, as well as the area of ​​each subdomain and the overlap area of ​​adjacent subdomains. The fluid characteristic parameters of the fluid model and the structural characteristic parameters of the structural model indicate that the more acute the geometry of the fluid model and the more drastic the physical field change at the coupling boundary, the larger the overlap area of ​​adjacent subdomains. The fluid characteristic parameters characterize the geometric features and flow field characteristics of the fluid model, while the structural characteristic parameters characterize the geometric features and structural characteristics of the structural model.

[0014] The beneficial technical effects of this application are: This application discloses a fluid-structure interaction (FSI) method based on compact supported radial basis functions (RBFs). By dividing the FSI boundary into multiple overlapping subdomains, a local coupling matrix is ​​constructed in parallel on each subdomain using compact supported RBFs. Compared to traditional global RBF FSI methods, this application transforms large-scale global solutions into small-region local solutions, with each subdomain computed independently. This effectively avoids the computational inefficiency caused by dense matrices resulting from direct global coupling matrix calculation. Since adjacent subdomains share a region on the FSI boundary, information exchange and adjustments occur within this shared region, resulting in a smoother boundary transition and improved data transfer accuracy at the boundary. Furthermore, a smooth partitioning weight function is used to concatenate the local coupling matrices into a global coupling matrix, further improving the smoothness and accuracy of fluid-structure coupling at subdomain boundaries. This method retains high-order accuracy locally while maintaining global sparsity, thus significantly improving computational efficiency while ensuring FSI computational accuracy.

[0015] By analyzing the geometric features and physical field properties of the fluid and structural models in each subdomain, and adaptively calculating the local metric matrix at each node, combined with local residual constraints, the optimal support radius of the compact support-type radial basis function within the subdomain can be obtained. This method obtains the optimal support radius by fully considering the variation laws of geometric features and physical fields, and specifically designs the support radius for each subdomain, maintaining high accuracy in fluid-structure interaction (FSI) calculations in geometrically complex regions with drastic or sharp physical field changes, such as boundary layers and thin shells. This method can significantly improve the accuracy of FSI calculations for complex structures and effectively solve the local smoothing problem of global radial basis functions, which is particularly significant for large-scale complex three-dimensional FSI simulations in shipbuilding and marine engineering. Attached Figure Description

[0016] Figure 1 This is a flowchart of the fluid-structure interaction method.

[0017] Figure 2 This is a schematic diagram of the flow field mesh.

[0018] Figure 3 This is a schematic diagram of the structural mesh.

[0019] Figure 4 This is a schematic diagram of the coupling boundary of the fluid surface.

[0020] Figure 5 This is a schematic diagram of the coupling boundary of the structural surface.

[0021] Figure 6 This is the pressure contour map of the flow field at time T=0.5.

[0022] Figure 7 This is the velocity contour map of the flow field at time T=0.5.

[0023] Figure 8 This is the displacement contour plot of the structure at time T=0.5. Detailed Implementation

[0024] The specific embodiments of this application will be further described below with reference to the accompanying drawings.

[0025] This application discloses a fluid-structure interaction method based on compact supported radial basis functions. Please refer to [link / reference]. Figure 1 The flowchart shown illustrates the specific steps of this method: Step 1: Determine the fluid-structure interaction boundary between the fluid model and the structural model, and divide the fluid-structure interaction boundary into multiple subdomains. Any two adjacent subdomains overlap with each other. Each subdomain contains several fluid nodes of the fluid model and several structural nodes of the structural model.

[0026] First, based on real-world fluids and structures in practical application scenarios, fluid and structural models are constructed respectively. The fluid model is described using the discrete form of the three-dimensional Euler equations. The continuous conservation equations are transformed into a numerically solvable matrix form using the finite volume method. The core function is to describe the evolution of fluid conservation variables (density, momentum, energy, etc.) in time and space, specifically expressed as follows: (1) Among them, matrix The coefficient matrix is ​​obtained by discretization using the finite volume method, along with the spatial coordinates. Related; , , They are respectively the corresponding x , y , z The flux vector in a direction describes the spatial transport rate of physical quantities, including mass flux, momentum flux, and energy flux. This is the solution vector (conserved variable vector) of the equation, including primitive physical quantities describing the fluid state such as density and pressure (fluid load). For compressible fluids, it is usually defined as... , It is the fluid density. , , It's speed that matters. x , y , z directional components, It represents the total energy per unit mass of the fluid; the subscript t is the partial derivative with respect to time, representing the rate of change of the physical quantity with respect to time. x , y , z Spatial coordinates x , y , z The partial derivative in a direction represents the gradient change of a physical quantity in space.

[0027] The structural model is based on the linear elastic assumption and is described using finite element equilibrium equations, the specific expression of which is: (2) in, It is the stiffness matrix. It is the displacement of structural nodes. It is an external load.

[0028] There is a conversion relationship between the position vector of the fluid node mesh and the displacement vector of the finite element structure node.

[0029] Considering the low computational efficiency and local smoothing issues caused by the dense coupling matrix when using radial basis functions for global interpolation of the fluid-structure interaction (FSI) boundary in traditional methods, this application decomposes the global problem into local problems by dividing the FSI boundary into subdomains. Each subdomain is then computed in parallel, effectively reducing computational efficiency while allowing for individualized processing based on the geometric and physical characteristics of the fluid and structure within each subdomain. Furthermore, overlapping regions are established at the junctions of adjacent subdomains. These overlapping regions provide a shared area between each subdomain and its neighbors, ensuring continuity between them. This design helps address the issue of boundary condition transfer; by exchanging and adjusting information within the shared area, the boundary transition is smoother, avoiding abrupt changes between different subdomains.

[0030] Based on the fluid and structural models, fluid and structural meshes are divided according to the scenario to be simulated. Fluid-structure interaction (FSI) boundaries are selected on both the structural and fluid meshes. Several overlapping subdomains are generated on the fluid-structure FSI boundaries, with a total of Q subdomains. The node set of the FSI boundary is determined in each subdomain. , It includes two types of nodes: a known node set and an evaluation node set. When interpolating to calculate displacement, the known node set consists of structural nodes, and the evaluation node set consists of fluid nodes; when converting loads, the known node set consists of fluid nodes, and the evaluation node set consists of structural nodes.

[0031] Because the flow field characteristics and geometric features differ in different regions within a fluid model, and the structural characteristics and geometric features also vary in different regions within a structural model, it is necessary to consider the characteristics of both the fluid and the structure when dividing the subdomains. This ensures that all subdomains can fully cover the fluid-structure interaction boundary and adapt to complex geometric shapes and changes in the physical field.

[0032] In one embodiment, dividing the fluid-structure interaction boundary into multiple subdomains includes: using numerical simulation to determine the fluid characteristic parameters of the fluid model and the structural characteristic parameters of the structural model, and determining the area of ​​each subdomain and the overlap area of ​​adjacent subdomains based on the fluid characteristic parameters and structural characteristic parameters. The fluid characteristic parameters of the fluid model and the structural characteristic parameters of the structural model indicate that the more acute the geometry of the fluid model and the more drastic the change in the physical field at the coupling boundary, the larger the overlap area of ​​adjacent subdomains.

[0033] When using numerical simulation methods to determine fluid characteristic parameters and structural characteristic parameters, the simplification of fluid-structure interaction can be disregarded, and trial calculations can be performed separately. Fluid characteristic parameters characterize the geometric features and flow field properties of the fluid model, while structural characteristic parameters characterize the geometric features and structural properties of the structural model. Geometric features include mesh scale, geometric curvature, etc., flow field properties include fluid density, velocity, flow field gradient, etc., and structural properties include stiffness, principal strain of the structure, etc.

[0034] Based on the calculated fluid and structural characteristic parameters, regions with complex and irregular geometric shapes, as well as regions with drastic structural deformation or flow field changes, can be identified. This allows for increasing the overlap area in these special regions. The area of ​​each subdomain can be customized based on the fluid and structural characteristic parameters, combined with the actual calculation accuracy requirements. Generally, setting the area of ​​each subdomain to the same size satisfies most needs, but for certain special scenarios, different subdomain sizes can be set.

[0035] Step 2: The fluid-structure interaction process is simulated by partitioning and iterating. For the k-th iteration calculation step at the T-th simulation time, the fluid model is solved based on the fluid node positions of the fluid model in the k-th iteration calculation step to obtain the fluid loads of all fluid nodes on the fluid-structure coupling boundary in the k-th iteration calculation step; T and k are both integer parameters and T≥0, k≥1.

[0036] In this simulation, the positions and fluid characteristic parameters of each fluid node in the fluid model and the positions and structural characteristic parameters of each structural node in the structural model are predetermined at the initial simulation time T=0. The positions of the fluid nodes, fluid characteristic parameters, structural node positions, and structural characteristic parameters in the initial iteration calculation step k=1 of each simulation time are determined based on the calculation results of the previous simulation time.

[0037] In each iterative calculation step, the fluid load of each fluid node can be obtained by solving the fluid model equation of formula (1). The specific solution method can refer to existing technologies, such as using the self-developed structural finite element analysis software SAM.

[0038] Then, the load transformation can be performed using the compact support-type radial basis function fluid-structure interaction data transfer method to obtain the structural loads on each structural node. Since the fluid-structure interaction boundary is usually unstructured (e.g., the fluid mesh is tetrahedral, and the structural mesh is shell elements), the number and distribution of nodes are mismatched. Therefore, the basic idea of ​​fluid-structure interaction data transfer is to use radial basis function interpolation to obtain the data of the corresponding nodes. The data transferred between the fluid and the structure includes loads and displacements. During load transfer, the structural loads of the corresponding structural nodes are interpolated from the fluid loads of the fluid nodes; during displacement transfer, the displacements of the fluid nodes are interpolated from the displacements of the structural nodes. The form of the interpolation function is: (3) in, is the interpolation coefficient of the j-th node, and N is the total number of nodes; It is a radial basis function. independent variable x It is the spatial distance between nodes; It is a polynomial used to eliminate the singularity of interpolation.

[0039] Compactly supported radial basis functions, with their sparse, banded solution matrices, are more advantageous for solving large-scale problems. Therefore, this application adopts compactly supported radial basis functions. Optional compactly supported radial basis functions include Wendland C... 0 Functions and Wendland C 2 Functions, Wendland C 0 The function is: The WendlandC² function is: ,in, This indicates that it is only valid if the value within the parentheses is positive. This is because compact, supported radial basis functions require... The value is greater than zero; therefore, compared to globally supported radial basis functions, compact supported radial basis functions have a supporting radius. When the distance between two nodes is greater than When the function value is zero, only points within a local range affect each other.

[0040] To achieve efficient linear algebraic operations, the rules of the interpolation function are transformed into a matrix form that can be directly called upon in numerical computation. By introducing a global coupling matrix, the computation of continuous interpolation functions is transformed into matrix computation of discrete systems. This global coupling matrix is ​​constructed based on the selected radial basis functions. H This allows the use of the global coupling matrix. H The conversion between displacement and load is realized. The ability to transform continuous interpolation calculations on fluid-structure interaction boundaries into matrix calculations for discrete systems is based on the principle of energy conservation at the interface. The principle of energy conservation states that during the interaction process, the virtual work done by the fluid load (external force) and the structural load (internal force) on the interface displacement is equal, as shown in formula (4): (4) in, It is the virtual work at the fluid-structure interaction boundary. It is the virtual displacement of the structure. It is the virtual displacement of the fluid. It is a structural load. It is a fluid load.

[0041] By introducing a global coupling matrix H The fluid-structure interaction can be approximated as a linear relationship, and this matrix relates the finite element structure and the fluid mesh in... x , y , z If the displacement vector is in the direction, then the virtual displacement relationship between the structure and the fluid mesh can be expressed as: The conversion relationship between fluid load and structural load can be expressed as follows: , It is a global coupling matrix H The transpose of . If there exists a global coupling matrix satisfying the above equation. H It can be used for displacement conversion between structure and fluid, and its transpose can be used for load conversion between fluid and structure.

[0042] Step 3: Based on the compact support type radial basis function, according to the fluid node position of the fluid model and the structural node position of the structural model in the k-th iteration calculation step, calculate the local coupling matrix of each subdomain on the fluid-structure interaction boundary in the k-th iteration calculation step in parallel, and synthesize the local coupling matrices of each subdomain into a global coupling matrix.

[0043] To achieve the transfer of fluid load to structural load, the load conversion formula in step 2 needs to be used. The key lies in determining the global coupling matrix. To address the issue of dense coupling matrices caused by the global computation of traditional global radial basis function methods, this application divides the fluid-structure interaction boundary into multiple subdomains. In each subdomain, a sparse local coupling matrix is ​​constructed using compact, supported local radial basis functions, and these local coupling matrices are then synthesized to obtain the global coupling matrix. By decomposing the global computation into local computations and combining this with parallel acceleration, the computational efficiency of the global coupling matrix is ​​effectively improved.

[0044] The calculation method for the local coupling matrix of each subdomain is similar to that of the conventional global coupling matrix. The main difference lies in the size of the computational domain. Therefore, the calculation process of the conventional global coupling matrix will be used as an example for explanation.

[0045] The derivation of the global coupling matrix is ​​illustrated using the conversion of structural nodal displacements to fluid nodal displacements as an example. It is assumed that there exist [variables / factors] at the fluid-structure interaction boundary. Each structural node and A fluid node, an arbitrary structure node Position coordinates , Arbitrary fluid node Position coordinates , .

[0046] The structural nodal displacements can be expressed as: (5) in, , representing the unit displacement of all degrees of freedom of the structure; P It is a coefficient matrix; It is the interpolation matrix obtained by the radial basis function method, and its elements depend on the selected radial basis functions and structure nodes. The given displacement at that location.

[0047] Fluid nodal displacement It can be obtained through structural node displacement Determined by any linear combination of: (6) in, It is the evaluation matrix of the interpolation function at fluid node j.

[0048] Global Coupling Matrix H Ultimately, it can be expressed as: (7) Among them, the interpolation matrix between structural nodes It can be represented as: (8) Evaluation matrix between fluid and structural nodes It can be represented as: (9) In formula (8) And in formula (9) Both are radial basis functions and , express norm, , express The norm of .

[0049] Based on the above global coupling matrix H As can be seen from the calculation formula, the key to solving the fluid-structure interaction transformation process using the radial basis function method lies in the calculation of the radial basis function. Compared with the traditional global radial basis function, the key influencing factor of the compact support radial basis function is the support radius. Therefore, if you want to calculate the compact support radial basis function at any node, you must first determine the support radius at that node.

[0050] To further improve the computational accuracy of the local coupling matrix, this application analyzes the fluid characteristics of the fluid model and the structural characteristics of the structural model within the subdomain when determining the support radius of the compact support radial basis function, so as to automatically select the optimal support radius, thereby maintaining high resolution in the boundary layer, thin shell or sharp geometry, while controlling the sparsity of the local coupling matrix.

[0051] In one embodiment, the local coupling matrix of any subdomain p on the fluid-structure interaction boundary is determined. include: (1) Based on the positions of each fluid node and each structural node in the subdomain p, construct the local metric matrix of each structural node. The local metric matrix represents the geometric features and physical field changes at the structural node.

[0052] The optimal support radius selection strategy involves identifying the local principal direction at the structural node. The local principal direction typically reflects the changes in geometric features and physical fields within a local region; therefore, determining the local principal direction can help determine the abrupt changes in geometric features and physical fields at the structural node. Considering that the local covariance matrix can reflect changes in the local region, this application constructs a local metric matrix by calculating the local covariance matrix to characterize the local principal direction at the structural node. Specifically, it constructs an arbitrary structural node within the subdomain p. Local metric matrix include: Calculate structural nodes The covariance matrix of the position coordinate differences between the covariance matrix and the position coordinate differences of each fluid node in the subdomain p is obtained, and the eigenvalue decomposition of the covariance matrix is ​​performed to obtain the eigenvalues ​​and eigenvectors of the covariance matrix.

[0053] First, select the node in the current structure within subdomain p. Each fluid node in the neighborhood, and then for any fluid node Calculate structural nodes With fluid nodes The position coordinate difference is a three-dimensional vector. .

[0054] For m neighboring fluid nodes, the covariance matrix C is a 3×3 square matrix, whose elements , , b =1, 2, 3, respectively correspond to x , y , z direction, It is the j-th fluid node With structural nodes exist x The difference in position coordinates of the direction It is the j-th fluid node With structural nodes exist y The difference in position coordinates in the direction.

[0055] Next, eigenvalue decomposition is performed on the covariance matrix C to obtain the eigenvalues. , , and eigenvectors , , : Here, the eigenvalues ​​represent the scale in each direction, the eigenvectors represent the corresponding principal directions, and the principal directions are those with the largest eigenvalue. Related feature vectors , which represents the direction of the greatest change in a local region.

[0056] Then, structural nodes are constructed based on the eigenvalues ​​and eigenvectors of the covariance matrix. Local metric matrix ,in, R It is an eigenvector matrix and , diagonal matrix and , It is the largest eigenvalue. This is the second largest eigenvalue. It is the smallest eigenvalue; It is the main direction, representing the direction of the greatest change in a local area; and Is with Orthogonal secondary principal directions usually represent the secondary directions of deformation or flow.

[0057] (2) Based on the local metric matrix of each structural node, determine the support radius of the compact support radial basis function in subdomain p according to the positions of each fluid node and each structural node in subdomain p.

[0058] In one embodiment, determining the support radius of the compactly supported radial basis function within the subdomain p includes: Based on the local metric matrix of each structural node, the anisotropic distance between each structural node and each fluid node in subdomain p is calculated according to the positions of each fluid node and each structural node in subdomain p. The candidate support radius at each structural node is determined according to the anisotropic distance between each structural node and each fluid node.

[0059] Since the local metric matrix can characterize the geometric features and physical field variations at structural nodes, it is used to define anisotropic distance, which measures spatial distance in different directions. The calculation is performed for any structural node within subdomain p. and arbitrary fluid nodes Anisotropic distance Determine structural nodes Candidate support radius at the location ; in, It is a scaling factor used to control the scaling ratio of the support radius, and can be customized according to the actual application. It is a structural node The local metric matrix, It is a structural node position coordinates and , It is a fluid node position coordinates and , This indicates the matrix transpose.

[0060] Further, the condition number and residual estimates are performed locally on the candidate support radii at each structural node. Based on the candidate support radii at each structural node, the candidate structural loads for each structural node in the current iteration calculation step are calculated, and the candidate local residuals for each structural node are determined. The candidate local residuals are the absolute values ​​of the difference between the candidate structural loads in the current iteration calculation step and the structural loads in the previous iteration calculation step.

[0061] The condition number related to the candidate support radius is the same as the condition number used to calculate the local metric matrix. Therefore, the condition number is calculated for the local metric matrix of each structural node. For a given matrix, the condition number is defined as the ratio of its largest eigenvalue to its smallest eigenvalue; a smaller condition number indicates more stable calculation. Since each structural node within subdomain p corresponds to multiple candidate support radii, and the candidate structural loads predicted using compact support radial basis functions with different support radii are different, to ensure computational stability while maximizing the accuracy of the calculation results, the candidate support radius that meets the condition number requirement and has the smallest candidate local residual is selected as the support radius of subdomain p during the screening process. Finally, the candidate support radius corresponding to the structural node with a condition number less than a predetermined threshold and the smallest candidate local residual is determined as the support radius of the compact support radial basis function within subdomain p. The predetermined threshold for the condition number is customized based on practical application experience.

[0062] (3) Determine the local coupling matrix of subdomain p based on the support radius of the compact support radial basis function within subdomain p. .

[0063] Since the load transfer between fluid and structure and the displacement transfer between structure and fluid are mutual, they can be calculated using the same coupling matrix. Therefore, after determining the support radius, the interpolation matrix, evaluation matrix, and local coupling matrix of subdomain p can be calculated according to formulas (7), (8), and (9) based on the selected compact support radial basis function. .

[0064] Finally, the local coupling matrices of each subdomain are combined to obtain the global coupling matrix of all nodes on the fluid-structure interaction boundary. include: The weights of each subdomain are determined based on the anisotropic distance between the structural node with the smallest candidate local residual and the subdomain center node, and the support radius of the compact support radial basis function in each subdomain.

[0065] To ensure smooth and accurate fluid-structure coupling at the boundaries of adjacent subdomains and avoid abrupt discontinuities during calculation, smoothing weights need to be designed based on the regional characteristics of each subdomain. The weights for any subdomain p are then determined. include: Based on the candidate structure node with the smallest local residual within subdomain p With subdomain center node Anisotropic distance Determine the weight of subdomain p. , It is the support radius of the compactly supported radial basis function within the subdomain p. It is the structural node with the minimum candidate local residual within subdomain q. With subdomain center node Anisotropic distance, It is the support radius of the compactly supported radial basis function within the subdomain q. This represents a compact, supported radial basis function. This is the total number of subdomains; When q=p, = , = The location of the subdomain center node is predetermined when the model is constructed.

[0066] Since the support radius of each subdomain is selected as the support radius corresponding to the structural node with the smallest candidate local residual, this structural node is used as the target point when calculating the weight of the subdomain. The anisotropic distance between the structural node and the central node is calculated to determine the dominant direction and proportion of the local calculation result in the global calculation result. By merging the local coupling matrix in this way, it is possible to ensure that the calculation results in the overlapping areas are continuous and without abrupt changes.

[0067] Finally, the local coupling matrices of each subdomain are weighted and summed to obtain the global coupling matrix. , It is the local coupling matrix of subfield p. It is the weight of subdomain p. This is the total number of subdomains.

[0068] Step 4: Use the global coupling matrix to perform load transformation on the fluid loads of each fluid node on the fluid-structure interaction boundary to obtain the structural loads of each structural node on the fluid-structure interaction boundary. Substitute the structural loads of each structural node into the structural model (Formula 2) for solving to obtain the displacements of each structural node on the fluid-structure interaction boundary in the k-th iteration calculation step. Use the global coupling matrix to perform displacement transformation on the displacements of each structural node on the fluid-structure interaction boundary to obtain the displacements of each fluid node on the fluid-structure interaction boundary in the k-th iteration calculation step.

[0069] When the fluid-structure interaction calculation reaches equilibrium based on the displacements of all structural nodes and fluid nodes on the fluid-structure interaction boundary in each iterative calculation step, the positions of the structural nodes of the structural model at the Tth simulation time and the positions of the fluid nodes of the fluid model at the Tth simulation time are obtained. Otherwise, the positions of the fluid nodes of the fluid model in the (k+1)th iterative calculation step are updated based on the displacements of the fluid nodes on the fluid-structure interaction boundary.

[0070] Equilibrium refers to a state where the forces applied to the surface of a structure are in dynamic equilibrium, and the deformation of the structure does not continue to change or changes very little. The deformation of the structure, in turn, affects the boundary conditions of the fluid, such as altering the fluid load or velocity distribution at the boundary. In equilibrium, the forces exerted by the fluid on the structure and the reaction forces exerted by the structure on the fluid are in balance. In equilibrium, the physical quantities at the interface between the fluid and the structure must match; specifically, the pressure at the fluid boundary must match the displacement at the structural surface; the displacement and deformation of the structure no longer change significantly, and the pressure and velocity of the fluid tend to stabilize.

[0071] For any k-th iteration calculation step: Based on the structural nodes on the fluid-structure interaction boundary in the kth iteration calculation step displacement and its corresponding fluid nodes displacement Determine the fluid nodes on the fluid-structure interaction boundary. Interface residuals at the location In calculating the interface residual, structural nodes are selected based on the preset correspondence between structural nodes and fluid nodes. Corresponding fluid nodes This application selects nodes farthest from the structure according to the nearest neighbor principle. The most recent fluid node as a structural node Corresponding fluid nodes When the interface residuals at all fluid nodes on the fluid-structure interaction boundary do not exceed the iteration error threshold, it is determined that the fluid-structure interaction calculation of the fluid model and the structural model has reached an equilibrium state. The iteration error threshold can be customized according to the calculation accuracy requirements of the actual application.

[0072] If the fluid-structure interaction (FSI) calculation does not reach equilibrium, the fluid node positions in the fluid model need to be updated, and the fluid loads need to be recalculated to proceed to the next iteration of the FSI calculation. To accelerate iteration convergence, the IQN multi-secant method is used to correct the interface residuals. Specifically, the positions of any fluid nodes in the (k+1)th iteration are updated. The locations include: The IQN multi-secant method is used to generate fluid nodes in the k-th iteration calculation step. Correction amount of interface residual at the location And update the fluid nodes in the (k+1)th iteration calculation step. Location , It is the fluid node in the k-th iteration calculation step. Location, It is a relaxation factor and , The specific value can be customized according to the actual application requirements.

[0073] The IQN algorithm works as follows: At the end of the previous iteration, the interface residual is recorded as historical data and stored using a low-rank approximation. In subsequent iterations, the low-rank approximation matrix of the historical data is used. To determine the correction amount for the interface residuals The construction of low-rank matrices is typically based on the relationship between historical interface residuals and corrections. By performing principal component analysis (PCA) or singular value decomposition (SVD) on these data, the most significant influencing factors can be extracted, thus constructing the low-rank matrix. Detailed construction methods can be found in existing techniques. The corrections are used to update the boundary conditions of the fluid and structure, and the new fluid field and structural response are calculated. The equilibrium state is achieved through continuous iteration; as the interface residuals gradually decrease, new interface residuals are used for the next round of low-rank approximation and data updates. Because previous correction information can be utilized in each calculation, the computation no longer starts from scratch, thus accelerating the convergence process. Moreover, since IQN uses low-rank approximation based on local historical data, it ensures that the corrections depend only on local iteration information within the subdomain, avoiding numerical instability caused by global corrections.

[0074] Step 5: Let T = T+1, and enter the fluid-structure interaction calculation at the (T+1)th simulation time. Repeat the above iterative calculation process until the maximum simulation time is reached, and complete the simulation of the fluid-structure interaction calculation.

[0075] In summary, the fluid-structure interaction method proposed in this application is independent of the model topology and can be directly applied to non-planar 3D models, overcoming the limitations of traditional 2D methods. Furthermore, this coupling algorithm is independent of the fluid-structure solver, supporting flexible replacement of fluid and structural models.

[0076] Further applying the fluid-structure interaction method of this application to simulate real-world application scenarios, in a dam-break example, where the water column contacts the flexible baffle, causing the baffle to deform, and the water flows over the baffle, the fluid mesh is generated as follows. Figure 2 As shown, a total of 84,720 grid nodes were created. No-slip boundary conditions were applied to the bottom, left, and right boundaries of the flow field, while a zero-pressure condition was applied to the top boundary. The structured mesh is shown below. Figure 3As shown, a total of 11,070 grid nodes were divided. For the specific grid division method, please refer to the methods in the prior art; the details will not be repeated in this application.

[0077] Based on the fluid and structural models of this dam failure example, the selected fluid-structure interaction boundary on the fluid structure surface is as follows: Figure 4 As shown, the fluid-structure interaction boundary on the surface of the structural model is as follows: Figure 5 As shown. The Wendland C² function is selected as the radial basis function, and the support radius is set to 15% of the baffle height to ensure that the support domain of each structural node contains at least 8 neighboring nodes.

[0078] Initially, the baffle is undeformed and the fluid load is zero. As the water column comes into contact with the baffle, the load and displacement are coupled through a global coupling matrix. H The process is repeated iteratively between the fluid and the structure until the displacement error of the system in adjacent iterations is less than a threshold, reaching an equilibrium state. During the simulation, the velocity and pressure contour plots of the fluid model at time T=0.5 are shown below. Figure 6 , 7 As shown, the displacement contour plot of the structural model at time T=0.5 is as follows. Figure 8 As shown in the figure. Based on the simulation results, the dam-break process can be analyzed, and further guidance can be provided for the optimized design of the baffle.

[0079] The above descriptions are merely preferred embodiments of this application, and this application is not limited to the above embodiments. It is understood that other improvements and variations that can be directly derived or conceived by those skilled in the art without departing from the spirit and concept of this application should be considered to be included within the protection scope of this application.

Claims

1. A fluid-structure interaction method based on compact support-type radial basis functions, characterized in that, The fluid-structure interaction method includes: Determine the fluid-structure interaction boundary between the fluid model and the structural model, and divide the fluid-structure interaction boundary into multiple subdomains. Any two adjacent subdomains overlap with each other. Each subdomain contains several fluid nodes of the fluid model and several structural nodes of the structural model. For the k-th iteration calculation step at the T-th simulation time, the fluid model is solved based on the fluid node positions of the fluid model in the k-th iteration calculation step to obtain the fluid loads of all fluid nodes on the fluid-structure interaction boundary in the k-th iteration calculation step. Based on the compact support type radial basis function, according to the fluid node position of the fluid model and the structural node position of the structural model in the k-th iteration calculation step, the local coupling matrix of each subdomain on the fluid-structure interaction boundary in the k-th iteration calculation step is calculated in parallel, and the local coupling matrices of each subdomain are combined into a global coupling matrix. The fluid loads of each fluid node on the fluid-structure interaction boundary are transformed using the global coupling matrix to obtain the structural loads of each structural node on the fluid-structure interaction boundary. The structural loads of each structural node are then substituted into the structural model for solution to obtain the displacements of each structural node on the fluid-structure interaction boundary in the k-th iteration calculation step. The displacements of each structural node on the fluid-structure interaction boundary are then transformed using the global coupling matrix to obtain the displacements of each fluid node on the fluid-structure interaction boundary in the k-th iteration calculation step. When the fluid-structure interaction calculation reaches equilibrium based on the displacements of all structural nodes and fluid nodes on the fluid-structure interaction boundary in each iterative calculation step, the structural node positions of the structural model at the Tth simulation time and the fluid node positions of the fluid model at the Tth simulation time are obtained. Otherwise, the fluid node positions of the fluid model in the (k+1)th iterative calculation step are updated based on the displacements of the fluid nodes on the fluid-structure interaction boundary. Let T = T+1, and proceed to the fluid-structure interaction calculation at the (T+1)th simulation time.

2. The fluid-structure interaction method according to claim 1, characterized in that, Determine the local coupling matrix of any subdomain p on the fluid-structure interaction boundary. include: Based on the positions of each fluid node and each structural node within the subdomain p, a local metric matrix is ​​constructed for each structural node. The local metric matrix characterizes the geometric features and physical field changes at the structural node. Based on the local metric matrix of each structural node, the support radius of the compact support radial basis function in subdomain p is determined according to the positions of each fluid node and each structural node in subdomain p. Based on the support radius of the compactly supported radial basis functions within subdomain p, the local coupling matrix of subdomain p is determined. .

3. The fluid-structure interaction method according to claim 2, characterized in that, Determining the support radius of the compactly supported radial basis functions within subdomain p includes: Based on the local metric matrix of each structural node, the anisotropic distance between each structural node and each fluid node in subdomain p is calculated according to the positions of each fluid node and each structural node in subdomain p. The candidate support radius at each structural node is determined according to the anisotropic distance between each structural node and each fluid node. The anisotropic distance is used to measure the spatial distance in different directions. The candidate structural loads of each structural node in the current iteration calculation step are calculated based on the candidate support radii at each structural node, and the candidate local residuals of each structural node are determined. The candidate local residuals are the absolute values ​​of the difference between the candidate structural loads in the current iteration calculation step and the structural loads in the previous iteration calculation step. Calculate the condition number of the local metric matrix of each structural node, and determine the candidate support radius of the structural node with a condition number less than a predetermined threshold and the smallest candidate local residual as the support radius of the compact support type radial basis function in subdomain p.

4. The fluid-structure interaction method according to claim 2, characterized in that, Construct arbitrary structural nodes within subdomain p Local metric matrix include: Calculate structural nodes The covariance matrix of the position coordinate differences between the covariance matrix and the position coordinate differences of each fluid node in the subdomain p is obtained by performing eigenvalue decomposition on the covariance matrix to obtain the eigenvalues ​​and eigenvectors of the covariance matrix. Structural nodes are constructed based on the eigenvalues ​​and eigenvectors of the covariance matrix. Local metric matrix ,in, R It is an eigenvector matrix and , diagonal matrix and , It is the largest eigenvalue. It is the second largest eigenvalue. It is the smallest eigenvalue.

5. The fluid-structure interaction method according to claim 3, characterized in that, Calculate any structural node within subdomain p and arbitrary fluid nodes Anisotropic distance Determine structural nodes Candidate support radius at the location ; in, It is a proportionality coefficient. It is a structural node The local metric matrix, It is a structural node position coordinates and , It is a fluid node position coordinates and , This indicates the matrix transpose.

6. The fluid-structure interaction method according to claim 3, characterized in that, Combine the local coupling matrices of each subdomain into a global coupling matrix. include: The weights of each subdomain are determined based on the anisotropic distance between the structural node with the smallest candidate local residual and the center node of the subdomain, and the support radius of the compact support radial basis function in each subdomain. The global coupling matrix is ​​obtained by weighting and summing the local coupling matrices of each subdomain according to their respective weights. , It is the local coupling matrix of subfield p. It is the weight of subdomain p. This is the total number of subdomains.

7. The fluid-structure interaction method according to claim 6, characterized in that, Determine the weight of any subdomain p include: Based on the candidate structure node with the smallest local residual within subdomain p With subdomain center node Anisotropic distance Determine the weight of subdomain p. , It is the support radius of the compactly supported radial basis function within the subdomain p. It is the structural node with the minimum candidate local residual within subdomain q. With subdomain center node Anisotropic distance, It is the support radius of the compactly supported radial basis function within the subdomain q. This represents a compact, supported radial basis function. This is the total number of subdomains.

8. The fluid-structure interaction method according to claim 1, characterized in that, For any k-th iteration calculation step: Based on the structural nodes on the fluid-structure interaction boundary described in the k-th iteration calculation step displacement and its corresponding fluid nodes displacement Determine the fluid nodes on the fluid-structure interaction boundary. Interface residuals at the location ; When the interface residuals at all fluid nodes on the fluid-structure interaction boundary do not exceed the iteration error threshold, it is determined that the fluid-structure interaction calculations of the fluid model and the structural model have reached an equilibrium state.

9. The fluid-structure interaction method according to claim 1, characterized in that, The update yields any fluid node in the (k+1)th iteration calculation step. The locations include: The IQN multi-secant method is used to generate fluid nodes in the k-th iteration calculation step. Correction amount of interface residual at the location And update the fluid nodes in the (k+1)th iteration calculation step. Location , It is the fluid node in the k-th iteration calculation step. Location, It is a relaxation factor and .

10. The fluid-structure interaction method according to claim 1, characterized in that, Dividing the fluid-structure interaction boundary into multiple subdomains includes: Numerical simulation methods are used to determine the fluid characteristic parameters of the fluid model and the structural characteristic parameters of the structural model, as well as the area of ​​each subdomain and the overlap area of ​​adjacent subdomains. The fluid characteristic parameters of the fluid model and the structural characteristic parameters of the structural model indicate that the more acute the geometry of the fluid model and the more drastic the change in the physical field, the larger the overlap area of ​​adjacent subdomains at the coupling boundary. The fluid characteristic parameters characterize the geometric characteristics and flow field properties of the fluid model, and the structural characteristic parameters characterize the geometric characteristics and structural properties of the structural model.

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