Topological optimization method based on manifold neural operator

Abstract

The invention provides a topological optimization method based on a manifold neural operator, and belongs to the technical field of structural design and optimization. The method comprises the following steps of S1, determining design domain geometry of a given topological optimization task, performing grid division on the design domain geometry, and solving a group of Laplace operator characteristic functions defined on a grid; s2, constructing a Laplacian kernel integral module based on the Laplacian operator feature function, and further constructing a manifold neural operator model; and S3, carrying out iterative training on the convective neural operator model according to the optimization target, optimization constraint, boundary conditions and material attributes of the topological optimization task. By the adoption of the method, continuous field-to-field geometric structure representation is achieved by introducing the manifold neural operator, the problem that local continuity of a geometric structure obtained through a traditional discrete point-to-point representation mode based on a full-connection neural network is poor is solved, and topological optimization design with better manufacturability is achieved.

Description

Technical Field

[0001] The present invention relates to the technical field of structural design and optimization, and in particular to a topology optimization method based on manifold neural operator. Background Art

[0002] As an important means to achieve lightweight structure and high-performance innovative configuration design, topology optimization is of great significance in modern engineering. Its core is to automatically find the optimal distribution of materials in a given design space through mathematical modeling and algorithm iteration, so as to maximize the structural performance or minimize the amount of material. For example, the weight of an aircraft wing can be reduced by more than 30% through topology optimization while maintaining structural stiffness. At present, topology optimization has been widely used in aerospace, vehicle engineering, architectural design and other fields. Therefore, studying innovative methods of topology optimization has important scientific value and practical significance.

[0003] Geometric structure representation is the key to topology optimization. As an emerging geometric structure representation method, implicit neural representation has attracted widespread attention in the field of topology optimization due to its powerful representation ability and clear boundary characteristics. However, the current implicit neural representation method mainly uses a fully connected neural network to characterize the geometric structure by inputting a spatial coordinate point and outputting the density value of the point, which belongs to a discrete point-to-point representation mode. For example, the patent "Structural Topology Optimization Method Based on Adaptive Reparameterization of Neural Networks" with patent application publication number CN118966031B and the paper "TOuNN: Topology Optimisation using neural networks" both use a fully connected neural network model to map point coordinates (input) to point density (output) to achieve the representation of the geometric structure, thereby reparameterizing the design variables into model parameters for topology optimization. The above patents and papers are both based on a discrete point-to-point representation mode of a fully connected neural network, which makes it difficult to effectively consider the correlation between different regions of the structure, resulting in poor local continuity of the optimized geometric structure. Summary of the invention

[0004] The purpose of the present invention is to provide a topology optimization method based on manifold neural operators. By introducing manifold neural operators, continuous field-to-field geometric structure representation is achieved, overcoming the problem of poor local continuity of geometric structures obtained by the traditional discrete point-to-point representation mode based on fully connected neural networks, and realizing a topology optimization design with better manufacturability.

[0005] To achieve the above object, the present invention provides a topology optimization method based on manifold neural operator, comprising the following steps: Step S1, determining the design domain geometry of a given topology optimization task, meshing it, and solving a set of Laplace operator eigenfunctions defined on the mesh; Step S2, constructing a Laplace kernel integration module based on the Laplace operator characteristic function, and further constructing a manifold neural operator model; Step S3: According to the optimization goal, optimization constraints, boundary conditions, and material properties of the topology optimization task, the manifold neural operator model is iteratively trained. The specific steps are as follows: Step S31, inputting a field function representing the geometry of the design domain into a manifold neural operator model, and outputting a field function representing the geometric structure; Step S32, performing structural response analysis on the output geometric structure and calculating the loss function; Step S33, judging whether the termination condition is met, if so, outputting the geometric structure as the final optimization result and ending the loop, if not, executing step S34; Step S34, update the manifold neural operator model parameters according to the loss function and the optimization algorithm, and repeat steps S31-S34 until the termination condition is met.

[0006] Preferably, the Laplace operator characteristic function is obtained by solving the Laplace operator characteristic equation defined on the design domain geometric grid. The Laplace operator characteristic function is represented as a spatial discrete vector of the spatial function, and the vector dimension is the same as the number of grid division nodes.

[0007] Preferably, the manifold neural operator model includes a lifting layer, a projection layer, and a plurality of Laplace kernel integration modules, each of which includes an encoder, an approximator, and a decoder; The encoder uses the Laplace operator characteristic function to perform spectral decomposition on the input function of the Laplace kernel integral module to obtain a weight coefficient vector; the approximator uses linear or nonlinear mapping to parameterize the weight coefficient vector to obtain a parameterized weight coefficient vector; the decoder uses the Laplace operator characteristic function to perform spectral reconstruction on the parameterized weight coefficient vector to obtain an output function of the Laplace kernel integral module.

[0008] Preferably, the optimization objectives include one or more of a structural load-bearing performance optimization objective, a structural heat transfer performance optimization objective, a structural electromagnetic performance optimization objective, a structural fluid dynamics performance optimization objective, and a structural mass / volume / material usage optimization objective.

[0009] Preferably, the optimization constraints include one or more of structural load-bearing performance constraints, structural heat transfer performance constraints, structural electromagnetic performance constraints, structural fluid dynamics performance constraints, and structural mass / volume / material usage constraints.

[0010] Preferably, the boundary conditions include one or more of mechanical boundary conditions, thermodynamic boundary conditions, fluid dynamic boundary conditions, and electromagnetic boundary conditions.

[0011] Preferably, the field function characterizing the geometry of the design domain includes a coordinate field function or a Laplace operator characteristic function; the field function characterizing the geometric structure includes a density field function, a signed distance value function or a level set function.

[0012] Preferably, the structural response analysis method includes one or more of the finite element method, the finite difference method, commercial simulation software solution, and software solution with numerical calculation function.

[0013] Preferably, the termination conditions include one or more of the following: the number of iterative calculations reaches a preset maximum threshold, the absolute change in the structural performance corresponding to the optimization target in several consecutive iterations is less than a set convergence threshold, and the relative change in the structural performance corresponding to the optimization target in several consecutive iterations is less than a set convergence threshold.

[0014] Preferably, the optimization algorithm includes a gradient optimization algorithm and a heuristic optimization algorithm.

[0015] Therefore, the present invention adopts the above-mentioned topology optimization method based on manifold neural operator, and the beneficial technical effects are as follows: by introducing the manifold neural operator in topology optimization, continuous field-to-field geometric structure characterization is achieved, overcoming the problem of poor local continuity of traditional characterization methods based on fully connected neural networks, and improving the manufacturability of the optimized geometric structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 A flow chart of a topology optimization method based on manifold neural operator of the present invention; Figure 2 Designing domain geometry for the topology optimization task selected in the second embodiment of the present invention; Figure 3 A triangular mesh divided by the second embodiment of the present invention; Figure 4 The Laplace characteristic function (part) defined on the design domain geometry solved by the second embodiment of the present invention; Figure 5 This is the geometric structure after final optimization of the second embodiment of the present invention. DETAILED DESCRIPTION

[0017] The technical solution of the present invention is further described below through the accompanying drawings and embodiments.

[0018] Unless otherwise defined, technical or scientific terms used in the present invention shall have the common meanings understood by one having ordinary skills in the field to which the present invention belongs.

[0019] Embodiment 1 like Figure 1As shown, it is a flow chart of a topology optimization method based on manifold neural operator of the present invention, which includes the following steps: Step S1: Determine the geometry of the design domain for a given topology optimization task, mesh it, and solve a set of Laplace operator eigenfunctions defined on the mesh. ; Among them, the Laplace operator characteristic function is a spatial function, represents the independent variable, which refers to the spatial position of the design domain geometry; represents the index of the characteristic function of the Laplacian operator; Indicates the number of Laplacian operator eigenfunctions included.

[0020] The Laplace operator characteristic function is obtained by solving the Laplace operator characteristic equation defined on the design domain geometric grid. The representation of the Laplace operator characteristic function is a spatial discrete vector of the spatial function, and the vector dimension is the same as the number of grid division nodes; Among them, the characteristic equation of the Laplace operator is , The mathematical symbol for the Laplace operator, Indicates The eigenvalue corresponding to the eigenfunction of the Laplace operator.

[0021] Step S2: construct a Laplace kernel integration module based on the Laplace operator characteristic function, and further construct a manifold neural operator model.

[0022] The manifold neural operator model includes a lifting layer, a projection layer, and multiple Laplace kernel integration modules, each of which includes an encoder, an approximator, and a decoder; The encoder uses the Laplace operator characteristic function to perform spectral decomposition on the input function of the Laplace kernel integral module to obtain a weight coefficient vector; the approximator uses linear or nonlinear mapping to parameterize the weight coefficient vector to obtain a parameterized weight coefficient vector; the decoder uses the Laplace operator characteristic function to perform spectral reconstruction on the parameterized weight coefficient vector to obtain an output function of the Laplace kernel integral module.

[0023] Step S3: According to the optimization goal, optimization constraints, boundary conditions, and material properties of the topology optimization task, the manifold neural operator model is iteratively trained. The specific steps are as follows: Step S31, inputting a field function representing the geometry of the design domain into a manifold neural operator model, and outputting a field function representing the geometric structure; Step S32, performing structural response analysis on the output geometric structure and calculating the loss function; Step S33, judging whether the termination condition is met, if so, outputting the geometric structure as the final optimization result and ending the loop, if not, executing step S34; Step S34, update the manifold neural operator model parameters according to the loss function and the optimization algorithm, and repeat steps S31-S34 until the termination condition is met.

[0024] In step S3, the optimization target includes one or more of a structural load-bearing performance optimization target, a structural heat transfer performance optimization target, a structural electromagnetic performance optimization target, a structural fluid dynamics performance optimization target, and a structural mass / volume / material usage optimization target.

[0025] The optimization constraints include one or more of structural load-bearing performance constraints, structural heat transfer performance constraints, structural electromagnetic performance constraints, structural fluid dynamics performance constraints, and structural mass / volume / material usage constraints.

[0026] The boundary conditions include one or more of mechanical boundary conditions, thermodynamic boundary conditions, fluid dynamic boundary conditions, and electromagnetic boundary conditions.

[0027] Functions that characterize the geometry of the design domain include coordinate field functions or Laplace operator characteristic functions; functions that characterize the geometric structure include density field functions, signed distance value functions or level set functions.

[0028] Structural response analysis methods include one or more of finite element method, finite difference method, commercial simulation software solution, and software solution with numerical calculation function.

[0029] The termination conditions include one or more of the following: the number of iterative calculations reaches a preset maximum threshold, the absolute change of the structural performance corresponding to the optimization target in several consecutive iterations is less than the set convergence threshold, and the relative change of the structural performance corresponding to the optimization target in several consecutive iterations is less than the set convergence threshold; Optimization algorithms include gradient optimization algorithms and heuristic optimization algorithms.

[0030] Embodiment 2 Step S1, determine the design domain geometry of a given topology optimization task, mesh it, and solve a set of Laplace operator eigenfunctions defined on the mesh.

[0031] In this embodiment, the object selected is a two-dimensional irregular cantilever beam, the optimization goal is to minimize the structural flexibility, that is, to maximize the structural stiffness, and the optimization constraint is a topology optimization design task with a volume fraction. The design domain geometry is as follows: Figure 2 As shown in Figure 2, the design domain geometry is triangularly meshed, including 4053 mesh nodes and 7803 mesh elements. Figure 3As shown, the Laplace operator characteristic equation is then solved by the Galerkin method or the power method to obtain a set of Laplace characteristic functions defined on the design domain geometry, such as Figure 4 As shown, this embodiment selects the first 256 Laplace operator characteristic functions for subsequent steps.

[0032] Step S2: construct a Laplace kernel integration module according to the obtained Laplace operator characteristic function, and then construct a manifold neural operator model.

[0033] This implementation uses a linear mapping layer to construct the lifting layer and projection layer of the manifold neural operator model, and uses the leaky_relu activation function; the manifold neural operator model of this implementation includes 4 identical Laplace kernel integration modules connected in series, wherein the encoder of the kernel integration module uses 256 Laplace operator characteristic functions to perform spectral decomposition on the input of the kernel integration module to obtain a 256-dimensional weight coefficient vector, the approximator uses a linear transformation, and the decoder uses 256 Laplace operator characteristic functions to perform spectral reconstruction on the parameterized 256-dimensional weight coefficient vector to obtain the output function.

[0034] Step S3: iteratively train the manifold neural operator model according to the optimization goal, optimization constraints, boundary conditions, and material properties of the topology optimization task.

[0035] Input the node coordinate field to the manifold neural operator model, output the node density field (density value is between 0-1), set the left end of the output representation of the geometric structure as a fixed constraint, apply a vertical unit concentrated force load to the vertex at the lower right end, Poisson's ratio 0.3, elastic modulus 1, and then perform static simulation analysis to calculate the loss function considering the structural stiffness and volume fraction. The formula is as follows: ; in, represents the loss function, and K denote the displacement matrix and stiffness matrix respectively, represents the matrix transpose, represents the predefined volume fraction constraint value, which is set to 0.4 in this embodiment. represents the volume fraction of the output geometry of the manifold neural operator in the current iteration, It represents the artificially set penalty coefficient. In this embodiment, the initial value is 40, and it increases by 0.06 in each iteration.

[0036] In each iteration of training, the automatic differentiation technique is used to calculate the gradient of the loss function to the parameters of the manifold neural operator model, and the model parameters are updated according to the gradient. The termination condition selected in this embodiment is that the number of iterative calculations reaches the preset maximum threshold, which is 500. The volume fraction of the final optimization result is 0.404, the compliance is 514.2, and the optimized geometric structure is as follows Figure 5 shown.

[0037] It should be noted that the topology optimization method based on manifold neural operator described in the present invention is not limited to the maximum structural stiffness optimization target and volume fraction optimization constraint of Example 2. Its core lies in the continuous field-to-field geometric structure characterization, which can be flexibly extended to multiple optimization targets and multiple optimization constraints.

[0038] It is worth noting that the contents not elaborated in detail in the present invention are all prior art and are well known to those skilled in the art.

[0039] Therefore, the present invention adopts the above-mentioned topology optimization method based on manifold neural operator, and realizes continuous field-to-field geometric structure representation by introducing manifold neural operator, thereby overcoming the problem of poor local continuity of geometric structure obtained by the traditional discrete point-to-point representation mode based on fully connected neural network, and realizes topology optimization design with better manufacturability.

[0040] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. A topology optimization method based on manifold neural operator, characterized in that: The following steps are involved: Step S1, determining the design domain geometry of a given topology optimization task, meshing it, and solving a set of Laplace operator eigenfunctions defined on the mesh; Step S2, constructing a Laplace kernel integration module based on the Laplace operator characteristic function, and further constructing a manifold neural operator model; Step S3: According to the optimization goal, optimization constraints, boundary conditions, and material properties of the topology optimization task, the manifold neural operator model is iteratively trained. The specific steps are as follows: Step S31, inputting a field function representing the geometry of the design domain into a manifold neural operator model, and outputting a field function representing the geometric structure; Step S32, performing structural response analysis on the output geometric structure and calculating the loss function; Step S33, judging whether the termination condition is met, if so, the output geometric structure is taken as the final optimization result and the loop ends, if not, executing step S34; Step S34, update the manifold neural operator model parameters according to the loss function and the optimization algorithm, and repeat steps S31-S34 until the termination condition is met.

2. A topology optimization method based on manifold neural operator according to claim 1, characterized in that: The Laplace operator characteristic function is obtained by solving the Laplace operator characteristic equation defined on the design domain geometric grid. The representation of the Laplace operator characteristic function is a spatial discrete vector of the spatial function, and the vector dimension is the same as the number of grid division nodes.

3. The topology optimization method based on manifold neural operator according to claim 1 is characterized in that: The manifold neural operator model includes a lifting layer, a projection layer, and multiple Laplace kernel integration modules, each of which includes an encoder, an approximator, and a decoder; The encoder uses the Laplace operator characteristic function to perform spectral decomposition on the input function of the Laplace kernel integral module to obtain a weight coefficient vector; the approximator uses linear or nonlinear mapping to parameterize the weight coefficient vector to obtain a parameterized weight coefficient vector; The decoder uses the Laplace operator characteristic function to perform spectrum reconstruction on the parameterized weight coefficient vector to obtain the output function of the Laplace kernel integration module.

4. The topology optimization method based on manifold neural operator according to claim 1 is characterized in that: The optimization objectives include one or more of structural load-bearing performance optimization objectives, structural heat transfer performance optimization objectives, structural electromagnetic performance optimization objectives, structural fluid dynamics performance optimization objectives, and structural mass / volume / material usage optimization objectives.

5. The topology optimization method based on manifold neural operator according to claim 1, characterized in that: The optimization constraints include one or more of structural load-bearing performance constraints, structural heat transfer performance constraints, structural electromagnetic performance constraints, structural fluid dynamics performance constraints, and structural mass / volume / material usage constraints.

6. The topology optimization method based on manifold neural operator according to claim 1, characterized in that: The boundary conditions include one or more of mechanical boundary conditions, thermodynamic boundary conditions, fluid dynamic boundary conditions, and electromagnetic boundary conditions.

7. The topology optimization method based on manifold neural operator according to claim 1, characterized in that: The field functions that characterize the geometry of the design domain include coordinate field functions or Laplace operator characteristic functions; the field functions that characterize the geometric structure include density field functions, signed distance value functions or level set functions.

8. The topology optimization method based on manifold neural operator according to claim 1, characterized in that: The structural response analysis method includes one or more of the finite element method, the finite difference method, simulation software solution, and software solution with numerical calculation function.

9. The topology optimization method based on manifold neural operator according to claim 1, characterized in that: The termination conditions include one or more of the following: the number of iterative calculations reaches a preset maximum threshold, the absolute change in the structural performance corresponding to the optimization target in several consecutive iterations is less than the set convergence threshold, and the relative change in the structural performance corresponding to the optimization target in several consecutive iterations is less than the set convergence threshold.

10. The topology optimization method based on manifold neural operator according to claim 1, characterized in that: Optimization algorithms include gradient optimization algorithms and heuristic optimization algorithms.

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