An Efficient Analysis Method for Overlapping Finite Element Substructures Based on AI-Fast Prediction of Oversampled Numerical Shape Functions

Abstract

This invention belongs to the field of computational mechanics and structural numerical analysis technology, specifically involving an efficient analysis method for overlapping finite element substructures based on AI-driven rapid prediction of oversampled numerical shape functions. It addresses the following problems in existing technologies: overcoming accuracy loss caused by the linear assumption of boundary displacements while maintaining only a small number of corner degrees of freedom in the substructure; eliminating interface displacement inconsistencies caused by splicing non-overlapping substructures; and achieving rapid construction of high-precision numerical shape functions with low computational cost. This invention aims to provide a novel numerical analysis tool that balances computational accuracy and analytical efficiency for large-scale heterogeneous structure analysis and high-resolution topology optimization.

Description

Technical Field

[0001] This invention belongs to the field of computational mechanics and structural numerical analysis technology, specifically involving an efficient analysis method for overlapping finite element substructures based on AI-based rapid prediction of oversampled numerical shape functions. Background Technology

[0002] In recent years, with the rapid development of machine learning technology, its ability to construct end-to-end mappings in high-dimensional spaces has been widely proven and recognized. In the field of mechanical analysis, researchers have begun to explore and develop machine learning-enhanced computational methods for large-scale structural analysis and topology optimization.

[0003] To improve the efficiency of finite element analysis of large-scale structures, the substructure method is a classic and effective technique. The core idea of ​​the substructure method is to divide the overall structure into multiple subdomains (substructures), independently condense the degrees of freedom of each substructure, and retain only the degrees of freedom at the substructure boundaries for global solution, thereby significantly reducing the size of the linear equation system requiring a solution. However, existing machine learning augmentation analysis techniques based on the substructure method generally suffer from the following three core shortcomings: First, the construction of local numerical shape functions is costly. In traditional methods, the construction of local numerical shape functions relies on complex analytical derivations or tedious numerical iterations, consuming a large amount of computational resources. When dealing with heterogeneous structures containing a large number of heterogeneous microstructures, it is difficult to achieve efficient adaptation and cannot meet the efficiency requirements of large-scale structural analysis.

[0004] Secondly, the traditional linear assumption of boundary displacement is insufficient to accurately characterize the nonlinear displacement of interfaces under strong heterogeneity. The interfaces of strongly heterogeneous structures often exhibit complex nonlinear behaviors such as slip and deformation. Traditional linear assumptions cannot capture these nonlinear characteristics, easily leading to deviations in the characterization of interface displacement and consequently affecting the overall calculation accuracy. In particular, the actual displacements on the boundaries of substructures often exhibit oscillatory distributions, and the linear assumption will produce significant boundary layer errors.

[0005] Third, when only corner degrees of freedom are retained in the substructure, non-overlapping splicing can lead to displacement inconsistencies. Although retaining corner degrees of freedom during substructure order reduction can effectively reduce computational costs, non-overlapping splicing cannot achieve displacement continuity at the interface between adjacent substructures, which can easily cause numerical oscillations or convergence problems, leading to errors in strain energy calculation and thus affecting the accuracy of topology optimization sensitivity.

[0006] In summary, how to overcome the accuracy limitations of the linear boundary displacement assumption while maintaining only a small number of corner degrees of freedom in the substructure, and simultaneously achieve the coordination of displacement fields between substructures, and construct high-precision numerical shape functions with low computational cost, is a key technical problem that urgently needs to be solved in the current intersection of computational mechanics and machine learning. To this end, this invention provides an efficient analysis method for overlapping finite element substructures based on AI-based rapid prediction of oversampled numerical shape functions. Summary of the Invention

[0007] The purpose of this invention is to provide an efficient analysis method for overlapping finite element substructures based on AI-driven rapid prediction of oversampled numerical shape functions, addressing the following problems in existing technologies: overcoming accuracy loss caused by the assumption of linear boundary displacements while maintaining only a small number of corner degrees of freedom in the substructure; eliminating interface displacement inconsistencies caused by splicing non-overlapping substructures; and achieving rapid construction of high-precision numerical shape functions with low computational cost. This invention aims to provide a novel numerical analysis method that balances computational accuracy and analytical efficiency for large-scale heterogeneous structure analysis and high-resolution topology optimization.

[0008] The specific technical solution adopted by this invention is as follows: An efficient analysis method for overlapping finite element substructures based on AI-driven rapid prediction of oversampled numerical shape functions includes the following steps: Step S1: Construction of oversampling numerical shape functions; For a given substructure, solve the local boundary value problem within its extended oversampling region to obtain the mapping relationship from coarse-scale corner point degrees of freedom to fine-scale node degrees of freedom. Construct an oversampling numerical shape function matrix that satisfies the Kronecker-delta property to achieve a reduced-order representation of the substructure that retains only four corner points and a total of eight degrees of freedom without imposing the assumption of substructure boundary displacement. Step S2: Non-uniform substructure mesh construction; Based on the position of the substructure in the overall rectangular arrangement, the internal mesh of the substructure is divided into three categories: internal fine unit, boundary expansion unit, and corner expansion unit. Nine different types of substructures are defined so that adjacent substructures form a regular Cartesian fine mesh after overlapping and splicing. Step S3: Displacement coordination of overlapping substructures; Based on the concept of finite element unit decomposition, a weight function that satisfies the properties of continuity, non-negativity and unit decomposition is constructed in the overlapping area of ​​adjacent substructures. A global coordinated displacement field is constructed by weighted combination of local interpolation of each substructure to solve the interface displacement jump problem in the splicing of traditional non-overlapping substructures. Step S4: Classification and pre-calculation of overlapping fine elements; Based on the number of substructures covering fine elements in the overlapping area, all fine elements are classified into 16 types. The element stiffness matrix under unit density is pre-calculated offline for each type of fine element and stored for reuse. In the online stage, only scalar scaling according to material density is required to assemble the global stiffness matrix. Step S5: U-Net numerical shape function prediction; Construct and train a U-Net convolutional neural network model, taking the material parameter distribution within the substructure oversampling region as input and the corresponding oversampling numerical shape function as output, and establish a nonlinear mapping relationship between input features and numerical shape functions through offline training; Step S6: Online condensation solution; In the online calculation stage, for substructures with uniform material distribution, the pre-calculated uniform density shape function is directly reused. For non-uniform substructures, the U-Net model is called to predict their numerical shape functions in batches. All substructure shape functions are assembled into a block diagonal matrix, the condensation stiffness matrix is ​​constructed and the condensation system is solved to obtain the corner point degrees of freedom displacement. Then, the coarse-scale displacement is restored to the fine-scale nodal degrees of freedom through the predicted shape function. Step S7: Residual Correction; Using the reconstructed fine-scale displacement as the initial solution, the original fine-scale equilibrium equation is corrected by a small number of iterations using the preconditional conjugate gradient method to improve the post-processing accuracy of derivatives such as strain and strain energy.

[0009] Preferably, in step S1, the construction process of the oversampled numerical shape function specifically includes: S101: Select a reasonably large square area Completely cover the square substructure The area occupied, i.e. ,in This is an oversampling region. and All are discretized by uniform square fine-scale finite element elements; S102: As Figure 2 As shown, the oversampling region The four corner points are labeled as follows: By setting the displacement of each corner point in a certain coordinate direction to a unit value, fixing the displacement of the remaining corner points, and ensuring that the displacement on the boundary of the oversampled region is linearly distributed, finite element analysis is performed on the oversampled region under these boundary conditions, resulting in 8 displacement basis vectors. ; S103: Extract the corresponding substructure from each displacement basis vector The elements of the boundary and internal node displacements are used to form 8 columns of displacement vectors, which are then assembled column by column into a preliminary numerical shape function matrix. ,in The number of fine units on one side of the substructure; S104: Through the matrix ,in To restrict operator M from The matrix obtained by extracting the row corresponding to the corner point degrees of freedom makes the final numerical shape function matrix... It satisfies the Kronecker-delta property, ensuring that no strain energy is generated when the substructure undergoes rigid body displacement.

[0010] Preferably, in step S2, the nine types of substructures include: top left substructure, top substructure, top right substructure, left substructure, internal substructure, right substructure, bottom left substructure, bottom substructure, and bottom right substructure. Among them, the top left substructure overlaps with the bottom and right sides; the top substructure overlaps with the bottom, left, and right sides; the top right substructure overlaps with the bottom and left sides; the left substructure overlaps with the top, bottom, and right sides; the internal substructure overlaps with the top, bottom, left, and right sides; the right substructure overlaps with the top, bottom, and left sides; the bottom left substructure overlaps with the top and right sides; the bottom substructure overlaps with the top, left, and right sides; and the bottom right substructure overlaps with the top and left sides.

[0011] Preferably, in step S3, the weighting function The construction of satisfies the following conditions: Condition a: The weight function is continuous and non-negative over the analysis domain; Condition b: Each weight function In fine units The area outside the occupied zone is zero; Condition c: In the global domain The above satisfies the unit decomposition condition ; Specifically, firstly, an intermediate function is defined for each fine unit. ,in For a standard Q4 finite element shape function, if the node is located at... Values ​​are assigned in the overlapping regions with other fine units. Otherwise, assign a value. Then through normalization Obtain the weighting function; global displacement field through The weighted average of the local interpolations is obtained.

[0012] Preferably, in step S4, the 16 types of overlapping fine units are determined according to the number N of substructures covering the fine unit. OV Divided into: N OV =1、N OV =2 and N OV =4; where N OV =1 corresponds to types 1-5 and types 8-11; N OV=2 corresponds to types 6, 7, 12, 13, 14, and 15; and N OV =4 corresponds to type 16; The corresponding element stiffness matrix dimensions are 8×8, 16×16, and 32×32, respectively; due to the uniformity of the Cartesian mesh, all h×h fine elements are in the reference coordinate system. Since the elements are completely isomorphic, the integrand of each type of fine element in the reference coordinate is only a polynomial independent of the density of the artificial material, which can be calculated and stored offline once before the main loop.

[0013] Preferably, in step S5, the U-Net network structure includes an encoder part and a decoder part: the encoder part consists of several levels of convolutional downsampling modules, each level including a convolutional layer, a normalization layer and a GELU activation function, and introduces residual connections; the decoder part gradually recovers the spatial resolution of the feature map through upsampling operations, and at each level, the features of the corresponding encoder layer are passed to the decoder through skip connections. The network output is completed into a complete local numerical shape function matrix through rigid body modal constraints, ensuring that the obtained shape function satisfies the rank invariance through physical hard constraints, and ensuring that the substructure does not generate strain energy when rigid body displacement occurs.

[0014] Furthermore, in step S5, the training employs a composite loss function composed of data error and stiffness fidelity error. L data Using the SmoothL1 loss function, L K The relative error between the reduced stiffness matrix induced by the predicted shape function and the true reduced stiffness matrix is ​​given. The training strategy uses only data terms for the first N epochs for warm-up, and then enables stiffness constraint terms to avoid the stiffness error from causing excessive interference to the optimization direction when the predicted shape function is unstable in the early stage of training.

[0015] The technical effects achieved by this invention are as follows: The efficient analysis method for overlapping finite element substructures based on problem-independent machine learning of the present invention has the following beneficial technical effects: 1. This invention is the first to combine oversampling numerical shape functions with a four-corner point order reduction strategy. While keeping only four corner points (eight degrees of freedom) in each two-dimensional substructure, it breaks through the dependence of traditional substructure methods on the linear assumption of boundary displacement. It can effectively characterize the oscillating boundary response and complex internal deformation modes caused by local material inhomogeneity, and greatly improve the analysis accuracy of strongly heterogeneous structures.

[0016] 2. The unit decomposition-based overlapping coordination mechanism proposed in this invention fundamentally solves the common problem of interface displacement inconsistency in traditional non-overlapping substructure splicing. By designing a specific non-uniform substructure internal mesh, adjacent substructures form a regular Cartesian fine mesh after overlapping, ensuring the continuity of the global displacement field, thereby improving the computational reliability of second-order mechanical response quantities such as strain, strain energy, and topology optimization sensitivity.

[0017] 3. The unified implementation method of the 9 types of substructures and 16 types of overlapping fine elements summarized in this invention fully utilizes the regularity of Cartesian meshes, enabling complex integrals related to weight functions to be pre-calculated offline and reused in the online stage. The global stiffness matrix assembly in the online stage only requires two steps: density scaling and sparse assembly. The complexity is of the same order as the assembly overhead of the classic Q4 finite element method with the same mesh and can be naturally parallelized.

[0018] 4. The problem-independent machine learning prediction model based on U-Net constructed in this invention takes the mapping between local material distribution and local numerical shape functions as the learning objective, rather than the end-to-end relationship from specific structures, loads, or boundary conditions to the global displacement field. Therefore, the trained model can be directly transferred to analysis tasks with different structural sizes, boundary conditions, load forms, and material distributions, exhibiting excellent generalization ability. This model can replace the traditional substructure oversampling shape function solution process with extremely low computational cost, significantly reducing online computation costs.

[0019] 5. Numerical examples show that in high-resolution topology optimization with millions of fine-scale elements, the method of this invention can maintain stable iteration and obtain optimized configurations with rich fine-scale topological features under a small filtering radius, verifying its application potential in large-scale engineering structure analysis and optimization design. Attached Figure Description

[0020] Figure 1 This is the neutron structure of the present invention. and its corresponding oversampling region Schematic diagram; where the red area represents the substructure region. The green area represents the expanded sampling area. ; Figure 2 This is a schematic diagram of the boundary conditions applied in the oversampling region when obtaining the oversampling numerical shape function in this invention; wherein, Figure 2 Display Order Method for setting boundary conditions when the x-direction displacement of the horn point is a unit value; Figure 3 This is an overall flowchart of the PIML-OFEM method in this invention; wherein, Figure 3 It demonstrates the complete process from substructure partitioning, supersampling shape function construction, U-Net prediction, overlap coordination to condensed solution; Figure 4 These are schematic diagrams of nine different types of substructures in this invention; among them, Figure 4 This demonstrates the differences between internal and boundary substructures in non-uniform mesh construction. Figure 5 This is a schematic diagram of a uniform Cartesian grid formed by overlapping substructures in this invention; wherein, Figure 5 This demonstrates the process of obtaining a uniform h×h fine element mesh by overlapping and splicing non-uniform substructure meshes. Figure 6 is a schematic diagram of the classification of 16 types of overlapping fine elements in this invention. It shows the geometric features and node arrangement of each fine element type under three coverage conditions: NOV=1, NOV=2, and NOV=4. Among them, Figure 6(a) shows 16 types of elements with different element stiffness matrices generated according to different overlapping forms; Figure 6(b) shows the position of the 16 types of fine elements in the overall structure, taking the overlapping areas of the four substructures (red, green, orange, and red) as examples. Figure 7 This is a schematic diagram of the U-Net network architecture in this invention; wherein, Figure 7 It demonstrates the three-level architecture of encoder (downsampling), bottleneck layer and decoder (upsampling), as well as the implementation of skip connections; Figure 8 This is a schematic diagram of the overlapping finite element displacement coordination principle in this invention; wherein, Figure 8 Show the overlapping region of two h×2h element units in the x direction, the node assignment method of the weight function, and the node degree of freedom number of the newly added integral element in the overlapping region; Figure 9 is a schematic diagram of the boundary conditions and material distribution of the cantilever beam mechanical sub-example in this invention; wherein, Figure 9(a) shows the boundary conditions of the cantilever beam example. Figure 9(b) shows the material distribution of the complex topology, in which yellow represents holes and black represents solids; Figure 10 These are the mechanical analysis results of complex topological structures under concentrated forces in this invention; among them, (a) deformation diagram and displacement field cloud map obtained by classical FEM analysis; (b) deformation diagram and displacement field cloud map obtained by analysis using the proposed PIML-OFEM method; (c) displacement absolute error cloud map; (d) displacement relative error cloud map; (e) element strain energy cloud map obtained by classical FEM analysis; (f) element strain energy cloud map obtained by PIML-OFEM method; (g) element strain energy relative error cloud map. Figure 11This is a schematic diagram of the Michel beam topology optimization problem in this invention; wherein, (a) the design region and boundary conditions; (b) the topology optimization result obtained by using the Dehomogenization topology optimization method at a resolution of 1600×800; (c) the topology optimization result obtained by directly optimizing the structure by using the proposed PIML-OFEM method instead of the finite element analysis in the classic SIMP algorithm; and (d) the topological details of the structure shown in Figure (c). Detailed Implementation

[0021] To make the objectives and advantages of this invention clearer, the invention will be specifically described below with reference to embodiments. It should be understood that the following text is merely used to describe one or more specific embodiments of the invention and does not strictly limit the scope of protection specifically claimed by the invention.

[0022] like Figures 1-11 As shown, this invention relates to an efficient analysis method for overlapping finite element substructures based on AI-based rapid prediction of oversampled numerical shape functions. This invention belongs to the field of computational mechanics and structural numerical analysis technology, specifically involving a problem-independent machine learning paradigm (a well-trained model can be directly used in mechanical analyses with different boundary conditions, geometries, and load conditions). It is applicable to mechanical analysis of large-scale heterogeneous structures, high-resolution topology optimization, and engineering computation scenarios requiring efficient and high-precision numerical simulation. The specific steps include: Step S1: Construction of oversampling numerical shape functions; like Figure 1-2 As shown, for a given substructure, the local boundary value problem is solved in its extended oversampling region to obtain the mapping relationship from coarse-scale corner point degrees of freedom to fine-scale node degrees of freedom. The oversampling numerical shape function matrix that satisfies the Kronecker-delta property is constructed to realize the reduced-order representation of the substructure that retains only four corner points with a total of eight degrees of freedom without applying the assumption of substructure boundary displacement. Step S2: Non-uniform substructure mesh construction; Based on the position of the substructure within the overall rectangular arrangement, the internal mesh of the substructure is divided into three categories: internal fine elements, boundary expansion elements, and corner expansion elements. Nine different types of substructures are defined, so that adjacent substructures form a regular Cartesian fine mesh after overlapping and splicing, such as... Figure 4-5 As shown; Step S3: Displacement coordination of overlapping substructures; Based on the concept of finite element unit decomposition, a weight function that satisfies the properties of continuity, non-negativity and unit decomposition is constructed in the overlapping area of ​​adjacent substructures. A global coordinated displacement field is constructed by weighted combination of local interpolation of each substructure to solve the interface displacement jump problem in the splicing of traditional non-overlapping substructures. Step S4: Classification and pre-calculation of overlapping fine elements; Based on the number of substructures covering fine elements in the overlapping area, all fine elements are classified into 16 types, as shown in Figure 6. The element stiffness matrix under unit density is pre-calculated offline for each type of fine element and stored for reuse. In the online stage, only scalar scaling according to material density is required to assemble the global stiffness matrix. Step S5: U-Net numerical shape function prediction; Construct and train a U-Net convolutional neural network model, taking the material parameter distribution within the substructure oversampling region as input and the corresponding oversampling numerical shape function as output, and establish a nonlinear mapping relationship between input features and numerical shape functions through offline training; Step S6: Online condensation solution; In the online calculation stage, for substructures with uniform material distribution, the pre-calculated uniform density shape function is directly reused. For non-uniform substructures, the U-Net model is called to predict their numerical shape functions in batches. All substructure shape functions are assembled into a block diagonal matrix, the condensation stiffness matrix is ​​constructed and the condensation system is solved to obtain the corner point degrees of freedom displacement. Then, the coarse-scale displacement is restored to the fine-scale nodal degrees of freedom through the predicted shape function. Step S7: Residual Correction; Using the reconstructed fine-scale displacement as the initial solution, the original fine-scale equilibrium equation is corrected by a small number of iterations using the preconditional conjugate gradient method to improve the post-processing accuracy of derivatives such as strain and strain energy.

[0023] Preferably, in step S1, the construction process of the oversampled numerical shape function specifically includes: S101: As Figure 1 As shown, select a reasonably large square area. Completely cover the square substructure The area occupied, i.e. ,in This is an oversampling region. and All are discretized by uniform square fine-scale finite element elements; S102: As Figure 2 As shown, the oversampling region The four corner points are labeled as follows: By setting the displacement of each corner point in a certain coordinate direction to a unit value, fixing the displacement of the remaining corner points, and ensuring that the displacement on the boundary of the oversampled region is linearly distributed, finite element analysis is performed on the oversampled region under these boundary conditions, resulting in 8 displacement basis vectors. ; S103: Extract the corresponding substructure from each displacement basis vector The elements of the boundary and internal node displacements are used to form 8 columns of displacement vectors, which are then assembled column by column into a preliminary numerical shape function matrix. ,in The number of fine units on one side of the substructure; S104: Through the matrix ,in To restrict operator M from The matrix obtained by extracting the row corresponding to the corner point degrees of freedom makes the final numerical shape function matrix... It satisfies the Kronecker-delta property, ensuring that no strain energy is generated when the substructure undergoes rigid body displacement.

[0024] Preferably, in step S2, the nine types of substructures include: top left substructure, top substructure, top right substructure, left substructure, internal substructure, right substructure, bottom left substructure, bottom substructure, and bottom right substructure. Among them, the top left substructure overlaps with the bottom and right sides; the top substructure overlaps with the bottom, left, and right sides; the top right substructure overlaps with the bottom and left sides; the left substructure overlaps with the top, bottom, and right sides; the internal substructure overlaps with the top, bottom, left, and right sides; the right substructure overlaps with the top, bottom, and left sides; the bottom left substructure overlaps with the top and right sides; the bottom substructure overlaps with the top, left, and right sides; and the bottom right substructure overlaps with the top and left sides. Figure 4 As shown.

[0025] Preferably, in step S3, the weighting function The construction of satisfies the following conditions: Condition a: The weight function is continuous and non-negative over the analysis domain; Condition b: Each weight function In fine units The area outside the occupied zone is zero; Condition c: In the global domain The above satisfies the unit decomposition condition ; Specifically, firstly, an intermediate function is defined for each fine unit. ,in For a standard Q4 finite element shape function, if the node is located at... Values ​​are assigned in the overlapping regions with other fine units. Otherwise, assign a value. Then through normalization Obtain the weighting function; global displacement field through The weighted average of the local interpolations is obtained, such as Figure 8 As shown.

[0026] Preferably, in step S4, the 16 types of overlapping fine units are determined according to the number N of substructures covering the fine unit. OV Divided into: N OV =1、N OV =2 and NOV =4; where N OV =1 corresponds to types 1-5 and types 8-11; N OV =2 corresponds to types 6, 7, 12, 13, 14, and 15; and N OV =4 corresponds to type 16; The corresponding element stiffness matrix dimensions are 8×8, 16×16, and 32×32, respectively; due to the uniformity of the Cartesian mesh, all h×h fine elements are in the reference coordinate system. Since the elements are completely isomorphic, the integrand of each type of fine element in the reference coordinate is only a polynomial independent of the density of the artificial material, which can be calculated and stored offline once before the main loop.

[0027] Preferably, in step S5, the U-Net network structure includes an encoder part and a decoder part: the encoder part consists of several levels of convolutional downsampling modules, each level including a convolutional layer, a normalization layer, and a GELU activation function, and introduces residual connections; the decoder part gradually recovers the spatial resolution of the feature map through upsampling operations, and at each level, the features of the corresponding encoder layer are passed to the decoder through skip connections, such as... Figure 7 As shown, the network output is completed into a complete local numerical shape function matrix after rigid body modal constraints, ensuring that the obtained shape functions satisfy the rank invariance through physical hard constraints, and guaranteeing that the substructure does not generate strain energy when rigid body displacement occurs.

[0028] Furthermore, in step S5, the training employs a composite loss function composed of data error and stiffness fidelity error. L data Using the SmoothL1 loss function, L K The relative error between the reduced stiffness matrix induced by the predicted shape function and the true reduced stiffness matrix is ​​given. The training strategy uses only data terms for the first N epochs for warm-up, and then enables stiffness constraint terms to avoid the stiffness error from causing excessive interference to the optimization direction when the predicted shape function is unstable in the early stage of training.

[0029] In practical implementation, for example: Cantilever Beam Structure Analysis Based on PIML-OFEM This embodiment uses a two-dimensional cantilever beam structure as an example, as shown in Figure 9, to illustrate the complete implementation process of the method of the present invention. The structure has dimensions of 4×1, with the left end completely fixed and a vertical concentrated force applied at the lower right corner. The overall structure is discretized into 3601×901 fine-scale elements.

[0030] Step 1: Construction of oversampling numerical shape functions; For each internal substructure, local finite element analysis is performed within its oversampling region to construct an oversampling numerical shape function matrix that satisfies the Kronecker-delta property. Specifically: Let the fine mesh size parameter m=10 within the substructure, the number of oversampling expansion layers l=6, and the size of the oversampling region be... The four corner points of the oversampling region are labeled as follows: In sequence The displacement of the first corner point in the x-direction is a unit value, and the displacement in the y-direction is zero. The displacements of the other three corner points are all fixed, and the displacements on the boundary of the oversampled region are linearly distributed. Finite element analysis is performed on the oversampled region to obtain displacement basis vectors. Eight similar analyses are performed on the x and y directions of the four corner points to obtain eight displacement basis vectors. After extracting the displacement elements of the corresponding nodes in the substructure, these vectors are assembled into a preliminary numerical shape function matrix Φ. The matrix T = M(Φ) is obtained by extracting the corresponding rows of the corner point degrees of freedom using the constraint operator M. The transformation matrix is ​​then calculated. This yields the oversampled numerical shape function matrix that ultimately satisfies the Kronecker-delta property. .

[0031] Step 2: Construction and overlapping of non-uniform substructure meshes; Based on the position of the substructure in the overall rectangular arrangement (total arrangement is n) slx ×ns ly The type of each substructure is determined. For internal substructures (with neighbors on all four sides), the internal mesh construction is as follows: the central region consists of (m-4)×(m-4) standard fine elements of size h×h; the boundary fine elements on the four sides that overlap with adjacent substructures are expanded from h×h to 2h×h (the expansion direction is perpendicular to the overlap direction); the fine element size of the four corner regions is expanded to 2h×h. Following this construction rule, all substructures are overlapped by a width h and then spliced ​​together to finally obtain (n slx (m-1)+1)×(n sly A regular Cartesian grid consisting of (m-1)+1) uniform h×h fine elements.

[0032] Step 3: Displacement Coordination of Overlapping Substructures In the overlapping region, a weight function is constructed for each fine element. First, for each fine unit Define intermediate functions ,in This is a standard Q4 finite element shape function. If the node is located at... On the boundary of the coverage area of ​​other fine units, assign values Otherwise, assign a value. Then, the weight function is obtained through normalization. The global displacement field is obtained through... The local interpolated displacements of each substructure are obtained by weighted averaging, thereby ensuring the continuity of displacements between substructures.

[0033] Step 4: Classification of Overlapping Fine Elements and Pre-calculation of Stiffness Matrix Based on the number N of substructures covered by each h×h fine unit OV This is classified as one of the 16 types of fine units. (In reference coordinates) Under the condition that the Cartesian mesh is uniform, the Jacobian matrix of all h×h fine elements satisfies J=(h / 2)I, therefore the integrand of each type of fine element is only a polynomial independent of the material density. Before the main loop begins, a 3-point Gaussian quadrature is performed on each of the 16 types of fine elements to calculate the element stiffness matrix under unit density. And it is stored in memory. During the online phase, only scalar scaling is needed for each fine unit. The element stiffness matrix can then be obtained, where Here, p represents the material density of the fine element, and p is the SIMP penalty coefficient. The global stiffness matrix assembly of all fine elements can be completed in one go by grouping by type, scaling by density vectorization, and by predetermined index.

[0034] Step 5: U-Net Numerical Shape Function Prediction Training phase: A dataset containing 20,000 samples was used, with 19,000 for training and 1,000 for validation. Each sample contains the input matrix E_os of the material modulus in the oversampled region and the corresponding numerical shape function output Φ. ML0 The U-Net network employs a three-stage encoder-decoder structure. Each encoder stage contains two convolutional layers (3×3 kernels), a BatchNorm normalization layer, and a GELU activation function, introducing residual connections. The encoder channel counts are 64, 128, and 256 respectively, with the bottleneck layer having 512 channels. Training uses the AdamW optimizer with an initial learning rate of [missing information]. Weight decay The batch size is 64, and a cosine annealing learning rate restart strategy is used. The composite loss function is λ. data =1.0,λ K =0.1, the first 50 epochs only use data items for warm-up. Training is stopped early on the 897th epoch, and the optimal validation MAE is 0.016082.

[0035] Online prediction phase: The oversampled density fields of all substructures are collected and fed into the trained U-Net model as batch inputs. For oversampled regions, the difference between the maximum and minimum density is less than... For approximately uniform substructures, the pre-computed uniform density shape function is directly reused; for non-uniform substructures, U-Net batch prediction is used to obtain the local numerical shape function matrix. The network output is then completed by rigid body modal constraints to obtain the complete shape function matrix Φ. ML .

[0036] Step Six: Online Condensation Solution and Displacement Reconstruction After prediction, all substructure shape functions are assembled into a block diagonal matrix Φ. global Construct the overall stiffness matrix after condensation Solving condensed systems Obtain the displacements of the corner points of the substructure. Using the predicted numerical shape function matrix, matrix multiplication is performed. The coarse-scale corner point displacements are restored to the full range of fine-scale nodal displacements. Taking the cantilever beam with two circular holes as an example: the computational domain is discretized with 1801×901 fine-scale elements. The classical FEM analysis takes 12.39 seconds, and the PIML-OFEM analysis takes 2.64 seconds (of which the solution of the condensed linear equations takes only 1.04 seconds). The average nodal relative displacement error is 0.0140, and the speedup is approximately 4.7 times.

[0037] Step 7: Residual Correction; To further improve local accuracy, the reconstructed fine-scale displacements are used as the initial solution, and the preconditioned conjugate gradient (PCG) method is employed to perform a small number of iterative residual corrections on the original fine-scale equilibrium equations. The diagonal terms of the global stiffness matrix are used as Jacobi preconditioners. For problems involving millions of two-dimensional elements, this correction process typically introduces only about 1-2 seconds of additional computation time, effectively improving the post-processing quality of derivatives such as strain, strain energy, and topology optimization sensitivity.

[0038] Example Effect Verification The overall structure was discretized using 3601×901 fine-scale elements. The classical FEM analysis took 39.85 s; the PIML-OFEM analysis took 12.06 s, with the solution of the condensed linear equations taking 2.60 s. Using the classical FEM as the reference solution, the average nodal relative displacement error of the PIML-OFEM was... At the location of maximum response, the reference solution gives the vertical displacement as follows: The PIML-OFEM results are . Figure 10 The displacement and error contour plots presented in (ad) show that the overall deformation patterns obtained by the two methods are highly consistent; the absolute displacement error is mainly concentrated near the load points and the hole boundaries, while remaining at a low level in most solid areas. The relative displacement error is generally controlled within a small range, indicating that the proposed method can accurately describe the global flexibility and local deformation response in complex porous structures.

[0039]

[0040] Table 1 further compares the computational efficiency and displacement prediction accuracy of different mechanical analysis methods in this complex topology example. The classical FEM, used as the reference solution, has an analysis time of 39.85 s. While the PIML-EMS-LBC method, based on the boundary linearity assumption, reduces the analysis time to 11.14 s, its substructure boundary displacements are still constrained by the linearity assumption, making it difficult to fully characterize nonlinear deformation features in complex hole boundaries and local stress concentration regions, resulting in an average displacement error of 0.0199. In contrast, the proposed PIML-OFEM method reduces the average displacement error to 0.0066 with only a slight increase in analysis time to 12.06 s, demonstrating that the proposed oversampling numerical shape function and overlapping coordination mechanism can significantly improve the displacement recovery accuracy in complex topologies. On the other hand, the OFEM-SUB method (which, compared to PIML-OFEM, uses an online computation method instead of Unet to predict the oversampled numerical shape function) does not employ machine learning acceleration and achieves a lower average displacement error of 0.0046. However, because solving the oversampled numerical shape function requires solving the complete boundary value problem, its analysis time is as high as 146.77 s, significantly higher than other methods. These results demonstrate that PIML-OFEM achieves near-perfect substructure analysis accuracy while significantly reducing online computation costs, thus achieving a better accuracy-efficiency balance in the analysis of complex heterogeneous structures.

[0041] For topology optimization and structural strength assessment, comparing only the displacement field is insufficient. Since strain and strain energy involve the spatial derivatives of the displacement field, they are more sensitive to local errors and interface incompatibilities, and therefore are important indicators for evaluating the accuracy of order reduction methods. Figure 10 (e.g.) The element strain energy distribution and relative error obtained by classical FEM and PIML-OFEM were compared. It can be seen that PIML-OFEM can not only recover the overall strain energy distribution, but also capture better local high-energy regions near the pore boundaries and load transfer paths. The relative error of element strain energy remains at a low level overall. The following shows that the proposed oversampling shape function, overlap coordination mechanism, and small amount of residual correction can jointly improve the post-processing accuracy of displacement derivatives.

[0042] This example further illustrates that the advantages of PIML-OFEM are not limited to displacement response prediction. Because our method maintains the consistency of the global displacement field at substructure joints and does not impose linear assumptions on local boundary displacements, it avoids strain energy deviations caused by interface discontinuities or boundary layer errors, which are present in traditional substructure order reduction methods. This is crucial for subsequent topology optimization, as element strain energy errors directly affect sensitivity calculations and may be amplified into structural configuration errors during iteration.

[0043] After verifying that PIML-OFEM has high accuracy for both displacement and strain energy, it is further embedded into a density-based topology optimization framework. This paper employs the classical SIMP method for compliance minimization, replacing only the finite element analysis module with the proposed PIML-OFEM; optimization variable updates, sensitivity filtering, and volume constraint handling all follow the established SIMP process. Therefore, the results in this section primarily reflect the accuracy, stability, and computational efficiency of the proposed mechanical analysis framework in topology optimization iterations, rather than the impact of introducing new optimization algorithms.

[0044] consider Figure 11 The Michell beam problem shown in (a) has a material volume fraction constraint as follows: The optimization objective is to minimize structural compliance. Figure 5 .6(b) presents the high-resolution topology results obtained from the literature using the homogenization method on a 1600×800 background grid. Previous studies have shown that for this type of single-case problem, a locally rank-2-like microstructure is one of the theoretically reasonable optimal features.

[0045] To verify the applicability of PIML-OFEM in high-resolution topology optimization, this paper discretizes the design domain into 3061×1531 fine-scale elements and uses PIML-OFEM instead of traditional FEM to solve the equilibrium equations at each step. The optimization results are as follows: Figure 11 As shown in (c), a magnified view is available. Figure 11 (d) As can be seen, the final structure exhibits rich fine-scale topological details and displays multi-scale material distribution characteristics consistent with rank-2 microstructures in some regions. This result demonstrates that PIML-OFEM can support topology optimization analysis at resolutions of millions or even higher without significantly sacrificing analytical efficiency.

[0046] It is particularly noteworthy that the filter radius in this example is only about √3 times the fine element size. Many machine learning-enhanced mechanical analysis or low-order substructure methods often require a large filter radius to suppress the accumulation of errors during optimization iterations due to strong high-frequency errors. However, the filter radius directly controls the minimum feature size of the optimized structure. While an excessively large filter radius helps improve numerical stability, it weakens the detail representation capability that high-resolution optimization aims to achieve. PIML-OFEM can maintain stable iterations with a small filter radius, indicating that its displacement field and element strain energy prediction errors do not significantly contaminate the sensitivity update, thus facilitating the release of high-resolution design space.

[0047] In this invention, the oversampled numerical shape function construction method constructs an oversampled numerical shape function matrix that, without imposing assumptions about substructure boundary displacements, solves local boundary value problems within the oversampled region. This matrix relies solely on four corner points (eight degrees of freedom) and exhibits an oscillating displacement distribution on the substructure boundary, while maintaining the consistency of Kronecker-delta properties and rigid body modes. This is a core technical approach that overcomes the limitations of traditional linear assumptions about substructure boundaries.

[0048] In this invention, the non-uniform substructure mesh and overlapping splicing strategy are as follows: based on the positional relationship of the substructure in the overall arrangement, the internal mesh of the substructure is constructed as a non-uniform distribution (three types: internal fine unit, boundary extension unit, and corner extension unit), which is summarized into 9 standard substructure types, so that after the adjacent substructures are overlapped and spliced, an overall uniform Cartesian fine mesh is formed.

[0049] In this invention, an overlapping displacement coordination mechanism based on unit decomposition is constructed: a weight function that satisfies the properties of continuity, non-negativity, and unit decomposition is constructed in the overlapping region of adjacent substructures, and global displacement field coordination is achieved through weighted combination, thereby fundamentally eliminating the interface displacement jump problem in non-overlapping splicing.

[0050] In this invention, a unified framework for 16 types of overlapping fine elements is implemented: based on the number of substructures covered by the fine elements in the overlapping region, all fine elements are categorized into 16 standard types. The integrand function of each type of element is only a polynomial that does not depend on the material density, supporting offline pre-calculation of the unit density stiffness matrix and online scalar scaling reuse.

[0051] In this invention, a problem-independent numerical shape function prediction model based on U-Net is proposed: a U-Net neural network architecture with local material distribution as input and local numerical shape function as output, and a training strategy of composite loss function (data error + stiffness fidelity error) to achieve rapid prediction of shape functions independent of specific structural dimensions, boundary conditions and load forms.

[0052] In this invention, the PIML-OFEM overall analysis framework is a complete and efficient substructure analysis framework that unifies and couples oversampled numerical shape functions, non-uniform substructure overlapping and splicing, unit decomposition displacement coordination, overlapping fine element classification pre-calculation, U-Net shape function prediction, and online condensed solution and residual correction.

[0053] In this invention, alternatives to the U-Net prediction module include: replacing U-Net with ResNet, Transformer, or other convolutional neural network architectures; using graph neural networks to process numerical shape function predictions under unstructured grids; and introducing Physical Information Neural Networks (PINN) to replace the purely data-driven training paradigm.

[0054] In this invention, alternative solutions for substructure overlap coordination include: using the Nitsche method or the Lagrange multiplier method to handle substructure interface displacement constraints, replacing the unit decomposition weighted average method; and in three-dimensional extension, enrichment strategies of the generalized finite element method (GFEM) or the extended finite element method (XFEM) can be used to handle substructure splicing.

[0055] In this invention, alternative solutions to the overall framework include: replacing the SIMP framework with PIML-OFEM embedded in BESO (Bidirectional Progressive Structure Optimization) or the level set method; extending the construction of oversampled shape functions to dynamic and geometric nonlinear problems; and replacing fixed parameter settings with adaptive substructure size and number of oversampled layers.

[0056] The above description is merely a preferred embodiment of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention. Structures, devices, and operating methods not specifically described or explained in this invention are implemented according to conventional methods in the art unless otherwise specified or limited.

Claims

1. An efficient analysis method for overlapping finite element substructures based on AI-based rapid prediction of oversampled numerical shape functions, characterized by: Includes the following steps: Step S1: Construction of oversampling numerical shape functions; For a given substructure, solve the local boundary value problem within its extended oversampling region to obtain the mapping relationship from coarse-scale corner point degrees of freedom to fine-scale node degrees of freedom. Construct an oversampling numerical shape function matrix that satisfies the Kronecker-delta property to achieve a reduced-order representation of the substructure that retains only four corner points and a total of eight degrees of freedom without imposing the assumption of substructure boundary displacement. Step S2: Non-uniform substructure mesh construction; Based on the position of the substructure in the overall rectangular arrangement, the internal mesh of the substructure is divided into three categories: internal fine unit, boundary expansion unit, and corner expansion unit. Nine different types of substructures are defined so that adjacent substructures form a regular Cartesian fine mesh after overlapping and splicing. Step S3: Displacement coordination of overlapping substructures; Based on the concept of finite element unit decomposition, a weight function that satisfies the properties of continuity, non-negativity and unit decomposition is constructed in the overlapping area of ​​adjacent substructures. A global coordinated displacement field is constructed by weighted combination of local interpolation of each substructure to solve the interface displacement jump problem in the splicing of traditional non-overlapping substructures. Step S4: Classification and pre-calculation of overlapping fine elements; Based on the number of substructures covering fine elements in the overlapping area, all fine elements are classified into 16 types. The element stiffness matrix under unit density is pre-calculated offline for each type of fine element and stored for reuse. In the online stage, only scalar scaling according to material density is required to assemble the global stiffness matrix. Step S5: U-Net numerical shape function prediction; Construct and train a U-Net convolutional neural network model, taking the material parameter distribution within the substructure oversampling region as input and the corresponding oversampling numerical shape function as output, and establish a nonlinear mapping relationship between input features and numerical shape functions through offline training; Step S6: Online condensation solution; In the online calculation stage, for substructures with uniform material distribution, the pre-calculated uniform density shape function is directly reused. For non-uniform substructures, the U-Net model is called to predict their numerical shape functions in batches. All substructure shape functions are assembled into a block diagonal matrix, the condensation stiffness matrix is ​​constructed and the condensation system is solved to obtain the corner point degrees of freedom displacement. Then, the coarse-scale displacement is restored to the fine-scale nodal degrees of freedom through the predicted shape function. Step S7: Residual Correction; Using the reconstructed fine-scale displacement as the initial solution, the original fine-scale equilibrium equation is corrected by a small number of iterations using the preconditional conjugate gradient method.

2. The efficient analysis method for overlapping finite element substructures based on AI-based rapid prediction of oversampled numerical shape functions as described in claim 1, characterized in that: In step S1, the construction process of the oversampled numerical shape function specifically includes: S101: Select a reasonably large square area Completely cover the square substructure The area occupied, i.e. ,in This is an oversampling region. and All are discretized by uniform square fine-scale finite element elements; S102: Oversampling region The four corner points are labeled as follows: By setting the displacement of each corner point in a certain coordinate direction to a unit value, fixing the displacement of the remaining corner points, and ensuring that the displacement on the boundary of the oversampled region is linearly distributed, finite element analysis is performed on the oversampled region under these boundary conditions, resulting in 8 displacement basis vectors. ; S103: Extract the corresponding substructure from each displacement basis vector The elements of the boundary and internal node displacements are used to form 8 columns of displacement vectors, which are then assembled column by column into a preliminary numerical shape function matrix. ,in The number of fine units on one side of the substructure; S104: Through the matrix ,in To restrict operator M from The matrix obtained by extracting the row corresponding to the corner point degrees of freedom makes the final numerical shape function matrix... It satisfies the Kronecker-delta property, ensuring that no strain energy is generated when the substructure undergoes rigid body displacement.

3. The efficient analysis method for overlapping finite element substructures based on AI-based rapid prediction of oversampled numerical shape functions as described in claim 2, characterized in that: In step S2, the nine types of substructures include: top left substructure, top substructure, top right substructure, left substructure, internal substructure, right substructure, bottom left substructure, bottom substructure, and bottom right substructure. Among them, the top left substructure overlaps with the bottom and right sides; the top substructure overlaps with the bottom, left, and right sides; the top right substructure overlaps with the bottom and left sides; the left substructure overlaps with the top, bottom, and right sides; the internal substructure overlaps with the top, bottom, left, and right sides; the right substructure overlaps with the top, bottom, and left sides; the bottom left substructure overlaps with the top and right sides; the bottom substructure overlaps with the top, left, and right sides; and the bottom right substructure overlaps with the top and left sides.

4. The efficient analysis method for overlapping finite element substructures based on AI-based rapid prediction of oversampled numerical shape functions as described in claim 3, characterized in that: In step S3, the weight function The construction of satisfies the following conditions: Condition a: The weight function is continuous and non-negative over the analysis domain; Condition b: Each weight function In fine units The area outside the occupied zone is zero; Condition c: In the global domain The above satisfies the unit decomposition condition ; Specifically, firstly, an intermediate function is defined for each fine unit. ,in For a standard Q4 finite element shape function, if the node is located at... Values ​​are assigned in the overlapping regions with other fine units. Otherwise, assign a value. Then through normalization Obtain the weighting function; global displacement field through The weighted average of the local interpolations is obtained.

5. The efficient analysis method for overlapping finite element substructures based on AI-based rapid prediction of oversampled numerical shape functions according to claim 4, characterized in that: In step S4, the 16 types of overlapping fine units are determined according to the number N of substructures covering the fine unit. OV Divided into: N OV =1、N OV =2 and N OV =4; where N OV =1 corresponds to types 1-5 and types 8-11; N OV =2 corresponds to types 6, 7, 12, 13, 14, and 15; and N OV =4 corresponds to type 16; The corresponding element stiffness matrix dimensions are 8×8, 16×16, and 32×32, respectively. All h×h fine elements are in the reference coordinate system. Since the elements are completely isomorphic, the integrand of each type of fine element in the reference coordinate is only a polynomial independent of the density of the artificial material, which can be calculated and stored offline once before the main loop.

6. The efficient analysis method for overlapping finite element substructures based on AI-based rapid prediction of oversampled numerical shape functions as described in claim 5, characterized in that: In step S5, the U-Net network structure includes an encoder part and a decoder part: the encoder part consists of several levels of convolutional downsampling modules, each level including a convolutional layer, a normalization layer and a GELU activation function, and introduces residual connections; The decoder part gradually restores the spatial resolution of the feature map through upsampling operations, and at each level, the features of the corresponding encoder layer are passed to the decoder through skip connections.

7. The efficient analysis method for overlapping finite element substructures based on AI-based fast prediction of oversampled numerical shape functions as described in claim 6, characterized in that: In step S5, the training uses a composite loss function composed of data error and stiffness fidelity error. L data Using the SmoothL1 loss function, L K The relative error between the reduced stiffness matrix induced by the predicted shape function and the true reduced stiffness matrix is ​​given. The training strategy uses only data terms for the first N epochs for warm-up, and then enables stiffness constraint terms.

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