A gradient progressive fusion unstructured isogeometric representation and analysis method

Abstract

This application discloses a gradient-progressive fusion method for unstructured isogeometric representation and analysis, relating to the fields of computational mechanics and engineering numerical simulation. The method includes: first, acquiring the geometric information of the computational domain of a T-spline surface and constructing an isogeometric analysis graph neural network model; then, selecting collocation points within the corresponding parameter domain and assembling the interpolation parameter matrix and partial derivative parameter matrix; next, employing a gradient-progressive fusion model training strategy, training the model by determining a model loss value based on the parameter matrix, which is a weighted sum of gradient loss terms, control equation residual loss terms, and boundary condition loss terms, transforming the partial differential equation solution into a neural network parameter optimization problem, while simultaneously fusing multi-level gradient information to improve training stability and computational accuracy; using the trained model to predict the physical parameters of the control vertices, and obtaining the distribution of physical parameters within the computational domain through spline interpolation. This application achieves fast and accurate solutions to partial differential equations in complex geometric domains.

Description

Technical Field

[0001] This application relates to the fields of computational mechanics and engineering numerical simulation technology, and in particular to a gradient progressive fusion method for unstructured geometric characterization and analysis. Background Technology

[0002] Isogeometric analysis is a numerical computation method widely used in engineering physics and computational mechanics in recent years. It has been extensively applied in the analysis and design of complex curved surface structures such as turbine blades, vehicle body structures, and ship hulls. The core difference between isogeometric analysis and traditional finite element analysis lies in the use of high-order continuous spline basis functions to replace the shape functions of traditional finite element analysis in numerical computation. This allows for the accurate description of complex geometric boundaries and enables the integrated application of computer-aided design and analysis.

[0003] Geometric analysis methods are mainly divided into two categories: the Galerkin method and the collocation method. The Galerkin method solves the weak form of the differential equation by integrating elements and assembling a global stiffness matrix, exhibiting good convergence. However, it currently lacks efficient numerical integration methods for unstructured spline functions such as T-splines, PHT splines, and hierarchical B-splines. When facing complex geometric boundary problems, it can only solve based on tensor product spline forms, leading to low computational efficiency. The collocation method, on the other hand, applies the strong form of the differential equation to several collocation points to construct a discrete linear system for direct solution, resulting in higher computational efficiency. However, when the computational domain is complex, it is prone to ill-conditioned matrix problems, leading to insufficient solution stability or even failure to achieve convergence.

[0004] Since the introduction of Physics-Informed Neural Networks (PINN) in 2019, solving differential equations using neural networks driven by physics knowledge has become a research hotspot in computational mechanics. The academic community has begun exploring combining the nonlinear fitting capabilities of neural networks with isogeometric solution frameworks. For example, existing techniques combine machine learning methods with isogeometric computational frameworks based on non-uniform rational B-splines (NURBS). These schemes use feedforward neural networks to predict physical parameters from the parameter coordinates of control grid vertices and obtain the global physical parameter distribution of the computational domain through basis function interpolation. This can be used for the rapid solution of boundary value problems of partial differential equations in engineering. However, such schemes are only suitable for NURBS splines with tensor product structures. For unstructured spline forms such as T-splines and hierarchical B-splines, which have local refinement advantages and are highly adaptable to complex geometries such as turbine blades and ship structures, there is currently a lack of effective analytical paradigms combining machine learning and isogeometric analysis. This makes it impossible to simultaneously consider the computational efficiency, stability, and solution accuracy for solving complex geometric domains. Summary of the Invention

[0005] The purpose of this application is to provide a gradient-incremental fusion method for unstructured geometric representation and analysis, which can achieve fast and accurate solutions to partial differential equations in complex geometric domains.

[0006] To achieve the above objectives, this application provides the following solution: A gradient-progressive fusion method for unstructured geometric representation and analysis includes the following steps: Obtain the geometric information of the target computational domain described by the graph data structure, and construct an isogeometric analysis graph neural network model; the target computational domain is a T-spline surface computational domain.

[0007] Several collocation points are selected within the parameter domain corresponding to the computational domain of the T-spline surface, and the interpolation parameter matrix and partial derivative parameter matrix are assembled based on geometric information.

[0008] A gradient progressive fusion model training strategy is adopted to train the isogeometric analysis graph neural network model to obtain a trained isogeometric analysis graph neural network model. The gradient progressive fusion model training strategy includes: determining the model loss value based on the interpolation parameter matrix and the partial derivative parameter matrix, and updating the parameters of the isogeometric analysis graph neural network model based on the model loss value. The model loss value is determined by a weighted sum of the gradient loss term, the control equation residual loss term, and the boundary condition loss term.

[0009] By using a trained isogeometric analysis graph neural network model, the physical parameters of the control points on the T-spline surface are determined based on the geometric information and boundary condition parameters of the computational domain of the T-spline surface.

[0010] Based on the physical parameters of the control points on the T-spline surface, the distribution of physical parameters within the computational domain of the T-spline surface is determined by spline interpolation.

[0011] Optionally, the geometric information of the target computational domain described by the graph data structure is obtained, and an isogeometric analysis graph neural network model is constructed, specifically including the following steps: Obtain the spatial coordinates of the control vertices, local node vectors, and neighbor information of the computational domain of the T-spline surface as geometric information.

[0012] Graph data is constructed based on geometric information; where nodes in the graph data correspond to control vertices of the T-spline surface, and edges in the graph data correspond to the connection relationships between each control vertex in the parameter domain.

[0013] An isogeometric analysis graph neural network model is constructed based on graph data. The node input features of the isogeometric analysis graph neural network model include the index of the control vertex, spatial coordinates, and local node vectors in the parameter direction. The node output features include physical parameter vectors composed of physical parameters on each control vertex and low-order spatial partial derivative vectors composed of the first-order spatial partial derivatives of the physical parameters on each control vertex.

[0014] Optionally, several collocation points are selected within the parameter domain corresponding to the computational domain of the T-spline surface, and the interpolation parameter matrix and partial derivative parameter matrix are assembled based on geometric information. This specifically includes the following steps: Several collocation points are uniformly sampled within the parameter domain corresponding to the computational domain of the T-spline surface, and the collocation points at the boundary of the parameter domain are densified.

[0015] Based on geometric information, the mixed function values ​​of the basis functions of each control vertex at each collocation point are calculated, and the interpolation parameter matrix is ​​assembled. The interpolation parameter matrix is ​​used to calculate the physical parameters at the collocation points based on the physical parameters at the control vertices.

[0016] Based on geometric information, the partial derivatives of the basis functions of each control vertex at each collocation point are calculated. Combined with the inverse of the Jacobian matrix, the partial derivative parameter matrix of the basis functions of each control vertex at the collocation point with respect to spatial coordinates is assembled. The partial derivative parameter matrix is ​​used to calculate the spatial partial derivatives at the collocation point based on the physical parameters at the control vertex.

[0017] Optionally, the collocation points at the parameter domain boundary are densified, including using a higher collocation point sampling density at the parameter domain boundary than inside the parameter domain.

[0018] Optionally, the model loss value is determined through the following steps: The physical parameter vectors and low-order spatial partial derivative vectors of the physical parameters at each control vertex are predicted using an isogeometric analysis graph neural network model.

[0019] Based on the interpolation parameter matrix and the physical parameter vectors at the control vertices, the predicted physical parameters at the collocation points in the computational domain are determined.

[0020] Based on the interpolation parameter matrix and the vector of low-order spatial partial derivatives of physical parameters at the control vertices, the predicted low-order spatial partial derivatives at the collocation points in the computational domain are determined.

[0021] Based on the partial derivative parameter matrix and the physical parameter vector at the control vertex, the interpolated low-order spatial partial derivatives at the collocation points in the computational domain are determined.

[0022] Based on the partial derivative parameter matrix and the low-order spatial partial derivative vectors of the physical parameters at the control vertices, the predicted high-order spatial partial derivatives at the collocation points in the computational domain are determined.

[0023] Based on the predicted physical parameters, predicted low-order spatial partial derivatives, interpolated low-order spatial partial derivatives, and predicted high-order spatial partial derivatives, the gradient loss term, the governing equation residual loss term, and the boundary condition loss term are calculated respectively.

[0024] Optionally, the gradient loss term is the deviation between the predicted low-order spatial partial derivatives and the interpolated low-order spatial partial derivatives based on the collocation points in the computational domain.

[0025] Optionally, the residual loss term of the governing equation is the residual calculated by substituting at least one of the predicted physical parameters, predicted low-order spatial partial derivatives, and predicted high-order spatial partial derivatives on the collocation points of the computational domain into the governing equation of the T-spline surface computational domain.

[0026] Optionally, the boundary condition loss term is the residual calculated by substituting the predicted physical parameters and / or predicted low-order spatial partial derivatives corresponding to the collocation points located on the boundary of the T-spline surface computational domain into the preset boundary conditions.

[0027] Optionally, using a trained isogeometric analysis graph neural network model, the physical parameters of the control points on the T-spline surface are determined based on the geometric information and boundary condition parameters of the computational domain of the T-spline surface. This specifically includes the following steps: Using the geometric information and boundary condition parameters of the T-spline surface computational domain as input, the model is forward-propagated through a trained isogeometric analysis graph neural network model to obtain the output features of each graph node of the isogeometric analysis graph neural network model.

[0028] Based on the mapping relationship between graph nodes and T-spline surface control points, physical parameters on the T-spline control points are extracted from the output features.

[0029] Optionally, the spatial partial derivatives of the physical parameters include the first-order and second-order partial derivatives of the physical parameters with respect to spatial coordinates.

[0030] According to the specific embodiments provided in this application, the following technical effects are disclosed: This application provides a gradient-progressive fusion method for unstructured isogeometric representation and analysis. First, the geometric information of the computational domain of a T-spline surface described using a graph data structure is acquired to construct an isogeometric analysis graph neural network model. The graph neural network represents the unstructured geometric information of the surface, solving the problem that traditional isogeometric analysis methods are difficult to adapt to T-splines and other non-tensor product splines. Then, collocation points are selected in the parameter domain corresponding to the T-spline surface computational domain, and interpolation parameter matrices and partial derivative parameter matrices are assembled, providing a discrete sampling framework and accurate derivative references based on spline functions for subsequent physical quantity calculations. Finally, a gradient-progressive fusion model training strategy is used to train the isogeometric analysis graph neural network model. This strategy determines the model loss value, which is a weighted sum of gradient loss terms, control equation residual loss terms, and boundary condition loss terms, based on the interpolation parameter matrix and partial derivative parameter matrix. The model parameters are then updated accordingly, transforming the strong-form solution of partial differential equations into a neural network parameter optimization training problem. This avoids the difficulty in solving ill-conditioned matrices in linear systems using the collocation method. Furthermore, multi-level gradient information is used to constrain the training process, improving model convergence stability and solution accuracy. Finally, the trained isogeometric analysis graph neural network model predicts the physical parameters at T-spline control points, and spline interpolation is used to obtain the distribution of physical parameters throughout the computational domain, achieving a fast and accurate solution to partial differential equations in complex geometric domains. Attached Figure Description

[0031] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0032] Figure 1 This is a flowchart illustrating an unstructured geometric characterization and analysis method for gradient progressive fusion, provided as an embodiment of this application.

[0033] Figure 2 This is a schematic diagram of a two-dimensional complex geometric domain solved in a gradient progressive fusion unstructured geometric representation and analysis method provided in an embodiment of this application.

[0034] Figure 3 This is a schematic diagram illustrating the selection of collocation points for a two-dimensional complex geometric domain in a gradient progressive fusion unstructured geometric representation and analysis method provided in an embodiment of this application. Detailed Implementation

[0035] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of this application, and not all embodiments. Based on the embodiments of this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0036] To make the above-mentioned objectives, features and advantages of this application more apparent and understandable, the application will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0037] This application provides a gradient progressive fusion method for unstructured geometric characterization and analysis. In one exemplary embodiment, such as... Figure 1 As shown, it includes the following steps: A1. Obtain the geometric information of the target computational domain described using a graph data structure, and construct an isogeometric analysis graph neural network model; the target computational domain is a T-spline surface computational domain. Specifically, in this embodiment, step A1 includes the following steps: A11. Obtain the spatial coordinates of the control vertices, local node vectors, and neighbor information of the computational domain of the T-spline surface as geometric information. The T-spline surface is defined by a T-mesh and can be represented as: .

[0038] in, For surface parameter coordinates, For the surface parameters are Spatial coordinates of a point at time; This indicates the first definition of the surface. i The spatial coordinates of each control vertex The blending function defined at this control vertex can be decomposed into... The form of multiplying the basis values ​​of the third-order B-spline in two directions; and These represent the local node vectors of the vertex in the two parametric directions, defined from the T-mesh using the ray rule of the T-spline; and For defining the local node vector at the vertex and The univariate third-order B-spline basis functions on the surface, where the local nodal vectors are defined as follows: and ,in, Indicates in The first parameter direction i The first local node vector in this node j Each node value Indicates in The first parameter directioni The first local node vector in this node j Each node value.

[0039] A12. Construct graph data based on geometric information; where nodes in the graph data correspond to control vertices of the T-spline surface, and edges in the graph data correspond to the connection relationships between control vertices in the parameter domain.

[0040] Specifically, each control vertex of the T-spline is defined as a graph node, forming a set of graph nodes. If two control vertices are connected in the parameter domain corresponding to the computational domain of the T-spline surface (i.e., one vertex is a neighbor of the other), then a connecting edge is established between the graph nodes representing these two vertices, forming a set of edges. In this way, the geometric information of the T-spline is transformed into graph data composed of nodes and edges.

[0041] The geometric data of the T-spline surface are expressed using the mathematical definition of a graph. The graph data can be expressed as follows: V is the set of control vertices of the T-spline, E represents the connection relationship between the control vertices, and it is assumed that there is a connecting edge relationship between the graph nodes corresponding to two control vertices with a connection relationship; the graph data can be represented by a feature matrix. and symmetric adjacency matrix Represented as: .

[0042] in, n To control the number of vertices, d For node feature dimensions, Indicates the first The first control vertex Each node's characteristics; Represents graph nodes and The corresponding control vertices are connected in the parameter domain, which is reflected in the graph structure as these two nodes. and There are connecting edges between them; Represents a node and The corresponding control vertices have no connection relationship in the parameter domain, which is reflected in the graph structure as these two nodes. and There are no edges connecting them.

[0043] A13. Constructing an isogeometric analysis graph neural network model based on graph data; The node input features of the isogeometric analysis graph neural network model include the index of the control vertex, spatial coordinates, and local node vectors in the parameter direction; The node output features include a physical parameter vector composed of the physical parameters on each control vertex and a low-order spatial partial derivative vector composed of the first-order spatial partial derivatives of the physical parameters on each control vertex.

[0044] In an exemplary embodiment, the sequence number, spatial coordinates, and location of each control vertex are... Local node vectors in two parametric directions serve as node input features, enabling unstructured representation of the geometric information of complex computational domains; node output features are determined by the problem being solved. Figure 2 Taking the solution of the Laplace equation for the steady-state temperature field in the complex planar domain as an example, the temperature at the control vertex is considered. and temperature in spatial direction partial derivatives This serves as the target for node prediction. In this way, an unstructured representation of the geometric information of complex computational domains is achieved.

[0045] A2. Select several collocation points in the parameter domain corresponding to the computational domain of the T-spline surface, and assemble them based on geometric information to obtain the interpolation parameter matrix and the partial derivative parameter matrix.

[0046] The computational domain information in step A2, in this example... Figure 2 The steady-state temperature conduction in the two-dimensional complex geometric domain shown is presented as a case study. Specifically, neglecting the heat source term, the governing equation of this computational domain, the Laplace equation, can be expressed as: .

[0047] in, Representing temperature with respect to spatial coordinates The second partial derivative in the direction. The computational domain is under fixed temperature conditions on the left and right sides, and adiabatic conditions on the upper and lower boundaries; without loss of generality, the temperature T on the left side is taken as fixed. The temperature T on the right side is fixed. For adiabatic conditions, the following equation applies: .

[0048] Among them, This represents the unit normal vector at any point on the boundary. This indicates the temperature at that point relative to spatial coordinates. The first-order partial derivative in the direction.

[0049] Specifically, in this embodiment, step A2 includes the following steps: A21. Uniformly sample and select several collocation points within the parameter domain corresponding to the T-spline surface computation domain, and refine the collocation points at the boundary of the parameter domain.

[0050] Specifically, regarding the selection of collocation points in step A21, such as Figure 3 As shown, uniform sampling is performed within the analyzed T-grid parameter domain, and the boundaries are refined. Figure 3 Part (a) shows the distribution of various collocation points in the T-grid parameter domain. Figure 3 Part (b) illustrates the distribution of various collocation points in the spatial computational domain. Assume a total of [number] points are selected. There are collocation points, which can be divided into different categories based on their location. A fixed temperature boundary point on the left, There are three fixed temperature boundary points on the right side, and the adiabatic conditions on the upper and lower sides are total. Each boundary point and Each internal collocation point. After selecting the collocation points, ... Calculate the normal vector of each of the points on the upper and lower thermal boundary alignments. And stacked to form a length of Unit normal vector of the upper and lower thermal boundary points x Vector composed of coordinates and y Vector composed of coordinates .

[0051] A22. Based on geometric information, calculate the mixed function values ​​of the basis functions of each control vertex at each collocation point, and assemble them to obtain the interpolation parameter matrix. The interpolation parameter matrix is ​​used to calculate the physical parameters at the collocation points based on the physical parameters at the control vertices. The interpolation parameter matrix can be used to calculate the physical parameters at any collocation point through basis function interpolation based on the physical parameters at the control vertices. Each element represents the function value of the mixed basis function of a certain control vertex at a certain collocation point.

[0052] Specifically, for the selected common The temperature values ​​at each of the collocation points are obtained by interpolation of the temperature coefficients at the T-spline control vertices using basis functions, and can be expressed in matrix form as shown in the following equation: .

[0053] in, The vector representing the predicted temperature composition at each collocation point, whose elements... These are all temperature values ​​at each point; A vector composed of temperature coefficients at control points, whose elements The matrix represents the temperature values ​​at each control vertex. The interpolation parameter matrix has the following elements. Indicates the first The basis functions of the control vertices in the th case... Each point The value of the mixed function at that point.

[0054] A23. Based on geometric information, calculate the partial derivatives of the basis functions of each control vertex at each collocation point, and combine this with the inverse of the Jacobian matrix to assemble a partial derivative parameter matrix of the basis functions of each control vertex at each collocation point with respect to spatial coordinates. This partial derivative parameter matrix is ​​used to calculate the spatial partial derivatives at collocation points based on the physical parameters at the control vertices. The spatial partial derivatives of the physical parameters include the first-order and second-order partial derivatives of the physical parameters with respect to spatial coordinates. The partial derivative parameter matrix can be used to calculate the partial derivatives of the physical parameters at any collocation point with respect to spatial directions using the derivatives of the basis functions and the chain rule, based on the physical parameters at the control vertices. The construction of this matrix involves calculating the partial derivatives of the basis functions at each control vertex and performing operations with the inverse of the Jacobian matrix.

[0055] Temperature in spatial direction The partial derivatives can be calculated using the chain rule as follows: .

[0056] in Temperature in the spatial direction The first-order partial derivative, Let represent the partial derivatives of temperature with respect to the two parameters. By reducing the order of the basis functions and finding the derivatives, we can interpolate to obtain the temperature derivatives with respect to the parameters at any location. Combining the chain rule with the basis function differentiation, we can obtain the partial derivatives of temperature with respect to spatial directions, written in matrix form: .

[0057] in The matrix is ​​a Jacobian matrix. The inverse transpose of has Jacobian matrix of T-spline surface By using spatial coordinates The derivative is: .

[0058] in Indicates the first control points and Coordinate values, The partial derivative parameter matrix of a single collocation point ,according to The shape of its matrix is ​​as follows For batch calculation of partial derivatives of N collocation points, the above relationship can be obtained by converting the matrix of each collocation point. Stacking in the new dimension yields the batch form partial derivative parameter matrix. Its shape is Multiplying the control point physical parameter vector by this matrix yields the partial derivatives of the physical parameters at the collocation points with respect to spatial directions, resulting in a matrix of the following shape: The results are as follows: the first column shows the partial derivatives of the physical parameters at each collocation point with respect to the x-direction, and the second column shows the partial derivatives of the physical parameters at each collocation point with respect to the y-direction. The assembly of the collocation point parameter matrix is ​​then performed on the aforementioned interpolation parameter matrix. and partial derivative parameter matrix The calculations are performed. The two parameter matrices mentioned above are entirely determined by the geometric information of the computational domain and are independent of the specific physical parameter values. They can be assembled all at once before model training.

[0059] A3. A gradient progressive fusion model training strategy is adopted to train the isogeometric analysis graph neural network model to obtain a trained isogeometric analysis graph neural network model. The gradient progressive fusion model training strategy includes: determining the model loss value based on the interpolation parameter matrix and the partial derivative parameter matrix, and updating the parameters of the isogeometric analysis graph neural network model based on the model loss value. The model loss value is determined by a weighted sum of the gradient loss term, the control equation residual loss term, and the boundary condition loss term.

[0060] Specifically, in this embodiment, the model loss value is determined through the following steps: B1. Using the isogeometric analysis graph neural network model, the physical parameter vectors and the low-order spatial partial derivative vectors of the physical parameters at each control vertex are predicted.

[0061] As an optional implementation, the isogeometric analysis graph neural network model constructed in step A1 is used to process the graph data and output the predicted value for each graph node (i.e., each control vertex). These predicted values ​​form multiple vectors. For example, for a temperature field problem, the output includes: the temperature coefficient vector at the control vertex, the partial derivative vector of the temperature coefficient with respect to the x-direction, and the partial derivative vector of the temperature coefficient with respect to the y-direction.

[0062] In a specific embodiment, the isogeometric analysis graph neural network model constructed in step A1 is used to predict the output features of each graph node from the input features of the graph nodes, which are the temperature coefficients controlling the vertices. Temperature coefficient with respect to space Coordinate direction partial derivative coefficients Temperature in space Coordinate direction partial derivative coefficients ;Will The coefficients at each control vertex are respectively composed of vectors. , , .

[0063] B2. Based on the interpolation parameter matrix and the physical parameter vectors at the control vertices, determine the predicted physical parameters at the collocation points in the computational domain. Specifically, use the interpolation parameter matrix established in step A2. Multiply each of these vectors by the vector obtained in step B1 to obtain the vector obtained in step A2. The predicted physical parameters at each calculated collocation point, i.e., the predicted temperature vector in the example. .

[0064] B3. Based on the interpolation parameter matrix and the low-order spatial partial derivative vectors of the physical parameters at the control vertices, determine the predicted low-order spatial partial derivatives at the collocation points in the computational domain. Specifically, multiply the interpolation parameter matrix assembled in step A2 with the predicted low-order spatial partial derivative vectors of the physical parameters at the control vertices obtained in step B1 to obtain the predicted low-order spatial partial derivatives at all collocation points, i.e., the predicted temperature. directional partial derivatives Predicting temperature directional partial derivatives .

[0065] B4. Based on the partial derivative parameter matrix and the physical parameter vectors at the control vertices, determine the interpolated low-order spatial partial derivatives at the collocation points in the computational domain. Specifically, use the partial derivative parameter matrix assembled in step A2. Compared with the control vertex physical parameter vector predicted in step B1 Multiplying them together, we get the shape as follows: The partial derivative matrix of temperature with respect to spatial coordinates at N collocation points, calculated by taking the derivatives of the basis functions, is used as the low-order gradient for collocation point prediction. The first column represents the vector of partial derivatives of temperature with respect to the x-coordinate at the N collocation points, calculated by taking the derivatives of the basis functions. The second column shows the vectors of partial derivatives of the temperature at N collocation points with respect to the y-coordinate in space, calculated using basis functions. .

[0066] B5. Based on the partial derivative parameter matrix and the low-order spatial partial derivative vectors of the physical parameters at the control vertices, determine the predicted high-order spatial partial derivatives at the collocation points in the computational domain. Specifically, using the partial derivative parameter matrix established in step A2... The vector obtained in step B1 Multiplying them together, we get the shape as follows: The matrix of second-order partial derivatives of temperature at N collocation points with respect to spatial coordinates is calculated by differentiating the basis functions. The first column of this matrix represents the vector of second-order partial derivatives of temperature at N collocation points with respect to the x-coordinate. The second column represents the vectors of the second-order partial derivatives of the temperature at N collocation points with respect to the spatial x-coordinate and spatial y-coordinate directions. Using the partial derivative parameter matrix established in step A2 The vector obtained in step B1 Multiply the results, and take the second column as the vector of the second-order partial derivatives of the temperature with respect to the spatial y-coordinate at the N collocation points. .

[0067] B6. Based on the predicted physical parameters, predicted low-order spatial partial derivatives, interpolated low-order spatial partial derivatives, and predicted high-order spatial partial derivatives, calculate the gradient loss term, the governing equation residual loss term, and the boundary condition loss term, respectively.

[0068] The gradient loss term is calculated as the deviation (e.g., root mean square error) between the predicted low-order spatial partial derivatives at the collocation points in the computational domain and the interpolated low-order spatial partial derivatives. This deviation is used as the gradient loss term to ensure that the gradient predicted by the neural network is consistent with the gradient obtained from the precise derivative of the spline function, thus achieving the fusion of low-order gradient information. The predicted temperature partial derivative vector is obtained from the interpolation parameter matrix at the collocation points. and the temperature partial derivative vector obtained through the partial derivative parameter matrix The deviation between them is used as the gradient loss of the model. To achieve low-order gradient information fusion: .

[0069] The residual loss term of the governing equations is calculated by substituting at least one of the predicted physical parameters, predicted low-order spatial partial derivatives, and predicted high-order spatial partial derivatives at the collocation points in the computational domain into the governing equations (e.g., the Laplace equation) of the T-spline surface computational domain, and using this residual as the governing equation residual loss term. This term ensures that the model prediction results satisfy physical laws and achieves the fusion of high-order gradient information. The vector obtained in step B5 is substituted into the Laplace equation, and its residual is taken as the model's governing equation loss. To achieve high-order gradient information fusion: .

[0070] The boundary condition loss term is calculated by substituting the predicted physical parameters and / or predicted low-order spatial partial derivatives corresponding to the collocation points located on the boundary of the T-spline surface computational domain into the preset boundary conditions (e.g., fixed temperature boundary, adiabatic boundary), and using this residual as the boundary condition loss term. This term is used to constrain the model to meet specific boundary constraints. The loss term is derived from the predictions in step B2. Predicted temperature vector of each point In, select the corresponding Figure 2 Fixed temperature boundary on the left side of the middle The vector composed of the predicted temperatures of each point. , and corresponding to Figure 2Fixed temperature boundary on the right side of the middle The vector composed of the predicted temperatures of each point. The residuals at the fixed temperatures on both sides of the boundary are calculated as the fixed boundary condition losses. ; and then from Take the corresponding values ​​from the middle. Figure 2 Common thermal boundary of the upper and lower sides The vector composed of the predicted temperature partial derivatives of each collocation point Substituting these values ​​into the adiabatic condition equations, the residuals are taken as the adiabatic boundary condition losses of the model. ; Fixed boundary condition loss With adiabatic boundary condition loss The sum is used as the model boundary condition loss. : .

[0071] in, It is a unit vector whose elements are all 1s. These are the vectors composed of the x-coordinate and y-coordinate of the unit normal vector of the adiabatic boundary, respectively.

[0072] The gradient loss term obtained from the above calculation Residual loss term in the governing equation and boundary condition loss terms We perform a weighted summation to obtain a fused model loss value: .

[0073] Where TotalLoss is the model loss value of the fusion of the isogeometric analysis graph neural network model training, and w1, w2, and w3 are the weights of the control equation loss term, boundary condition loss term, and model gradient loss term, respectively. The specific weight coefficients can be adjusted based on experience or experimentation.

[0074] Finally, with the goal of minimizing the fusion loss, the parameters of each layer of the isogeometric analysis graph neural network model are optimized and updated through the backpropagation algorithm until the model converges, resulting in a trained isogeometric analysis graph neural network model.

[0075] A4. Using a trained isogeometric analysis graph neural network model, the physical parameters of the control points on the T-spline surface are determined based on the geometric information and boundary condition parameters of the computational domain of the T-spline surface. In this embodiment, step A4 includes the following steps: A41. Using the geometric information and boundary condition parameters of the T-spline surface computational domain as input, and performing forward propagation through a trained isogeometric analysis graph neural network model, the output features of each graph node of the isogeometric analysis graph neural network model are obtained.

[0076] In an optional implementation, the geometric information and boundary condition parameters of the T-spline surface computational domain of the problem to be solved are used as input to the trained isogeometric analysis graph neural network model obtained in step A3. The model can obtain the output features of all graph nodes in its output layer through a single forward propagation.

[0077] A42. Based on the mapping relationship between graph nodes and T-spline surface control points, extract the physical parameters on the T-spline control points from the output features. In this step, based on the one-to-one mapping relationship between graph nodes and T-spline control vertices established in step A1, extract the physical parameters on the T-spline control points from the graph node features output by the model.

[0078] A5. Based on the physical parameters of the control points on the T-spline surface, determine the distribution of physical parameters within the computational domain of the T-spline surface through spline interpolation.

[0079] After obtaining the physical parameters at all control points of the T-spline, the same spline basis functions used to construct the T-spline surface are used to interpolate the physical parameters at any location within the computational domain, ultimately yielding a continuous and smooth physical parameter distribution field throughout the entire computational domain. This physical parameter distribution is the final solution result of the isogeometric analysis.

[0080] By implementing steps A1 to A5 above, the gradient progressive fusion unstructured isogeometric representation and analysis method provided in the above embodiments of this application solves the problem that traditional isogeometric analysis methods are difficult to adapt to T-splines and other non-tensor product splines by utilizing graph neural networks to represent the unstructured geometric information of T-spline surfaces. By combining the isogeometric collocation method framework with graph neural network node-level prediction, the strong-form solution of partial differential equations is transformed into an isogeometric graph neural network parameter optimization training problem, avoiding the problem that the ill-conditioned matrix of linear systems is difficult to solve by the collocation method. By adopting a gradient hierarchical fusion model training strategy, gradient calculation is performed simultaneously using basis function differentiation and neural network prediction, and gradient consistency is added as a loss term to the model training, which significantly improves the stability of the model training process and the accuracy of the final calculation results.

[0081] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0082] This document uses specific examples to illustrate the principles and implementation methods of this application. The descriptions of the above embodiments are only for the purpose of helping to understand the methods and core ideas of this application. Furthermore, those skilled in the art will recognize that, based on the ideas of this application, there will be changes in the specific implementation methods and application scope. Therefore, the content of this specification should not be construed as a limitation of this application.

Claims

1. A gradient-progressive fusion method for unstructured isogeometric representation and analysis, characterized in that, include: Obtain geometric information of the target computational domain described by graph data structures, and construct an isogeometric analysis graph neural network model; The target computational domain is the T-spline surface computational domain; Several collocation points are selected within the parameter domain corresponding to the computational domain of the T-spline surface, and the interpolation parameter matrix and the partial derivative parameter matrix are assembled based on the geometric information. The isogeometric analysis graph neural network model is trained using a gradient progressive fusion model training strategy to obtain a trained isogeometric analysis graph neural network model. The gradient progressive fusion model training strategy includes: determining the model loss value based on the interpolation parameter matrix and the partial derivative parameter matrix, and updating the parameters of the isogeometric analysis graph neural network model based on the model loss value. The model loss value is determined by a weighted sum of the gradient loss term, the control equation residual loss term, and the boundary condition loss term. By using a trained isogeometric analysis graph neural network model, the physical parameters of the control points on the T-spline surface are determined based on the geometric information and boundary condition parameters of the computational domain of the T-spline surface. Based on the physical parameters of the control points on the T-spline surface, the distribution of physical parameters within the computational domain of the T-spline surface is determined by spline interpolation.

2. The unstructured geometric representation and analysis method for gradient progressive fusion according to claim 1, characterized in that, Obtain the geometric information of the target computational domain described by a graph data structure, and construct an isogeometric analysis graph neural network model, specifically including: Obtain the spatial coordinates of the control vertices, local node vectors, and neighbor information of the computational domain of the T-spline surface as geometric information; Graph data is constructed based on the geometric information; wherein, the nodes on the graph data correspond to the control vertices of the T-spline surface, and the edges on the graph data correspond to the connection relationships between the control vertices in the parameter domain; The isogeometric analysis graph neural network model is constructed based on the graph data; the node input features of the isogeometric analysis graph neural network model include the index of the control vertex, spatial coordinates, and local node vectors in the parameter direction, and the node output features include physical parameter vectors composed of physical parameters on each control vertex and low-order spatial partial derivative vectors composed of the first-order spatial partial derivatives of the physical parameters on each control vertex.

3. The unstructured geometric representation and analysis method for gradient progressive fusion according to claim 1, characterized in that, Within the parameter domain corresponding to the computational domain of the T-spline surface, several collocation points are selected, and the interpolation parameter matrix and partial derivative parameter matrix are assembled based on the geometric information, specifically including: Within the parameter domain corresponding to the computational domain of the T-spline surface, several collocation points are uniformly sampled and selected, and the collocation points at the boundary of the parameter domain are densified. Based on the geometric information, the mixture function values ​​of the basis functions of each control vertex at each collocation point are calculated, and the interpolation parameter matrix is ​​assembled; the interpolation parameter matrix is ​​used to calculate the physical parameters at the collocation points according to the physical parameters at the control vertices; Based on the geometric information, the partial derivative values ​​of the basis functions of each control vertex at each collocation point are calculated, and combined with the inverse of the Jacobian matrix, the partial derivative parameter matrix of the basis functions of each control vertex at the collocation point with respect to spatial coordinates is assembled; the partial derivative parameter matrix is ​​used to calculate the spatial partial derivatives at the collocation point according to the physical parameters on the control vertex.

4. The gradient progressive fusion unstructured geometric representation and analysis method according to claim 3, characterized in that, The encryption process for collocation points at the parameter domain boundary includes: using a higher collocation point sampling density at the parameter domain boundary than inside the parameter domain.

5. The unstructured geometric representation and analysis method for gradient progressive fusion according to claim 1, characterized in that, The model loss value is determined through the following steps: The physical parameter vectors and low-order spatial partial derivative vectors of the physical parameters at each control vertex are predicted using the aforementioned isogeometric analysis graph neural network model. Based on the interpolation parameter matrix and the physical parameter vector on the control vertex, the predicted physical parameters on the collocation point in the computational domain are determined; Based on the interpolation parameter matrix and the low-order spatial partial derivative vector of the physical parameters at the control vertex, determine the predicted low-order spatial partial derivative at the collocation point in the computational domain; Based on the partial derivative parameter matrix and the physical parameter vector at the control vertex, determine the interpolated low-order spatial partial derivatives at the collocation points in the computational domain; Based on the partial derivative parameter matrix and the low-order spatial partial derivative vector of the physical parameters at the control vertex, determine the predicted high-order spatial partial derivative at the collocation point in the computational domain; Based on the predicted physical parameters, the predicted low-order spatial partial derivatives, the interpolated low-order spatial partial derivatives, and the predicted high-order spatial partial derivatives, the gradient loss term, the governing equation residual loss term, and the boundary condition loss term are calculated respectively.

6. The unstructured geometric representation and analysis method for gradient progressive fusion according to claim 5, characterized in that, The gradient loss term is the deviation between the predicted low-order spatial partial derivative and the interpolated low-order spatial partial derivative based on the collocation points in the computational domain.

7. The unstructured geometric representation and analysis method for gradient progressive fusion according to claim 5, characterized in that, The residual loss term of the governing equation is the residual calculated by substituting at least one of the predicted physical parameters, the predicted low-order spatial partial derivatives, and the predicted high-order spatial partial derivatives at the collocation points in the computational domain into the governing equation of the T-spline surface computational domain.

8. The gradient progressive fusion unstructured geometric characterization and analysis method according to claim 5, characterized in that, The boundary condition loss term is the residual calculated by substituting the predicted physical parameters and / or predicted low-order spatial partial derivatives corresponding to the collocation points located on the boundary of the T-spline surface computational domain into the preset boundary conditions.

9. The unstructured geometric representation and analysis method for gradient progressive fusion according to claim 1, characterized in that, Using a trained isogeometric analysis graph neural network model, the physical parameters of the control points on the T-spline surface are determined based on the geometric information and boundary condition parameters of the computational domain of the T-spline surface. Specifically, these parameters include: The geometric information and boundary condition parameters of the T-spline surface computational domain are used as input, and forward propagation is performed through a trained isogeometric analysis graph neural network model to obtain the output features of each graph node of the isogeometric analysis graph neural network model. Based on the mapping relationship between the graph nodes and the T-spline surface control points, the physical parameters on the T-spline control points are extracted from the output features.

10. The unstructured geometric representation and analysis method for gradient progressive fusion according to claim 1, characterized in that, The spatial partial derivatives of the physical parameters include the first-order and second-order partial derivatives of the physical parameters with respect to spatial coordinates.

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