A Topological Optimization Method Based on Manifold Neural Operators

Abstract

The present invention provides a topology optimization method based on a manifold neural operator, belonging to the technical field of structural design and optimization. The method includes the following steps: Step S1, determine the geometry of the design domain for a given topology optimization task, perform mesh division on it, and solve a set of Laplace operator eigenfunctions defined on the mesh; Step S2, construct a Laplace kernel integral module based on the Laplace operator eigenfunctions, and further construct a manifold neural operator model; Step S3, perform iterative training on the manifold neural operator model according to the optimization objective, optimization constraints, boundary conditions, and material properties of the topology optimization task. By adopting the above method, the present invention realizes continuous field-to-field geometric structure representation by introducing a manifold neural operator, overcomes the problem of poor local continuity of the geometric structure obtained by the traditional discrete point-to-point representation mode based on a fully connected neural network, and realizes a topology optimization design with better manufacturability.

Description

Technical Field

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

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

[0003] Geometric structure representation is the key to performing topology optimization. As an emerging geometric structure representation method, implicit neural representation has attracted wide attention in the field of topology optimization due to its strong representation ability and clear boundary characteristics. However, the current implicit neural representation methods mainly use fully connected neural networks to represent geometric structures by inputting spatial coordinate points and outputting the density value of that point, belonging to a discrete point-to-point representation mode. For example, the patent "Structural Topology Optimization Method Based on Neural Network Adaptive Reparameterization" with the 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 represent geometric structures, thereby reparameterizing design variables as model parameters for topology optimization. The above patent and paper are both based on the discrete point-to-point representation mode of fully connected neural networks, which are 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 a manifold neural operator. By introducing a manifold neural operator, continuous field-to-field geometric structure representation is achieved, overcoming the problem of poor local continuity of the geometric structure 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 purpose, the present invention provides a topology optimization method based on a manifold neural operator, including the following steps:

[0006] Step S1: Determine the geometric shape of the design domain for a given topology optimization task, mesh it, and solve a set of Laplace operator eigenfunctions defined on this mesh.

[0007] Step S2: Construct a Laplace kernel integral module based on the Laplace operator eigenfunctions, and further construct a manifold neural operator model.

[0008] Step S3: Iteratively train the manifold neural operator model according to the optimization objectives, optimization constraints, boundary conditions, and material properties of the topology optimization task. The specific steps are as follows:

[0009] Step S31: Input the field function representing the geometric shape of the design domain into the manifold neural operator model, and output the field function representing the geometric structure.

[0010] Step S32: Conduct a structural response analysis on the output geometric structure and calculate the loss function.

[0011] Step S33: Determine whether the termination condition is satisfied. If it is satisfied, output the geometric structure as the final optimization result and end the loop. If not, execute Step S34.

[0012] Step S34: Update the parameters of the manifold neural operator model according to the loss function and the optimization algorithm, and repeat Steps S31 - S34 until the termination condition is satisfied.

[0013] Preferably, the Laplace operator eigenfunctions are obtained by solving the Laplace operator eigenvalue equation defined on the mesh of the design domain geometric shape. The representation form of the Laplace operator eigenfunctions is the spatial discrete vector of the spatial function, and the vector dimension is the same as the number of nodes in the mesh division.

[0014] Preferably, the manifold neural operator model includes a lifting layer, a projection layer, and multiple Laplace kernel integral modules. Each Laplace kernel integral module contains an encoder, an approximator, and a decoder.

[0015] The encoder performs spectral decomposition on the input function of the Laplace kernel integral module using the Laplace operator eigenfunctions to obtain a weight coefficient vector. The approximator parameterizes the weight coefficient vector using a linear or nonlinear mapping to obtain a parameterized weight coefficient vector. The decoder performs spectral reconstruction on the parameterized weight coefficient vector using the Laplace operator eigenfunctions to obtain the output function of the Laplace kernel integral module.

[0016] Preferably, the optimization objectives include one or more of the optimization objectives of structural load - bearing performance, structural heat transfer performance, structural electromagnetic performance, structural fluid dynamics performance, and structural mass / volume / material usage.

[0017] 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.

[0018] Preferably, the boundary conditions include one or more of mechanical boundary conditions, thermodynamic boundary conditions, fluid dynamics boundary conditions, and electromagnetics boundary conditions.

[0019] Preferably, the field function characterizing the design domain geometry includes a coordinate field function or a Laplace operator eigenfunction; the field function characterizing the geometric structure includes a density field function, a signed distance value function, or a level set function.

[0020] Preferably, the structural response analysis methods include one or more of the finite element method, the finite difference method, solving with commercial simulation software, and solving with software having numerical calculation functions.

[0021] Preferably, the termination conditions include one or more of the number of iterative calculations reaching a preset maximum threshold, the absolute change amount of the structural performance corresponding to the optimization objective being less than the set convergence threshold in consecutive several iterations, and the relative change amount of the structural performance corresponding to the optimization objective being less than the set convergence threshold in consecutive several iterations.

[0022] Preferably, the optimization algorithms include gradient-based optimization algorithms and heuristic optimization algorithms.

[0023] Therefore, the present invention adopts the above-mentioned topology optimization method based on the manifold neural operator, and the beneficial technical effects are as follows: By introducing the manifold neural operator in topology optimization, continuous field-to-geometric structure representation is realized, overcoming the problem of poor local continuity of the traditional representation method based on fully connected neural networks, and improving the manufacturability of the optimized geometric structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Figure 1 is a flowchart of a topology optimization method based on the manifold neural operator of the present invention;

[0025] Figure 2 is the design domain geometry of the topology optimization task selected in the second embodiment of the present invention;

[0026] Figure 3 is the triangular mesh divided in the second embodiment of the present invention;

[0027] Figure 4 is the Laplace eigenfunction (partial) defined on the design domain geometry solved in the second embodiment of the present invention;

[0028] Figure 5 is the finally optimized geometric structure in the second embodiment of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0029] The technical solutions of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0030] Unless otherwise defined, the technical terms or scientific terms used in the present invention shall have the ordinary meanings understood by those of ordinary skill in the field to which the present invention belongs.

[0031] Embodiment 1

[0032] As Figure 1 shown, it is a flowchart of a topology optimization method based on a manifold neural operator of the present invention, including the following steps:

[0033] 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 ;

[0034] Among them, the Laplace operator eigenfunction is a spatial function, represents the independent variable, referring to the spatial position of the design domain geometry; represents the index of the Laplace operator eigenfunction; represents the number of Laplace operator eigenfunctions included.

[0035] The Laplace operator eigenfunction is obtained by solving the Laplace operator eigenvalue equation defined on the mesh of the design domain geometry. The representation form of the Laplace operator eigenfunction is a spatial discrete vector of a spatial function, and the vector dimension is the same as the number of mesh division nodes;

[0036] Among them, the Laplace operator eigenvalue equation is , represents the mathematical symbol of the Laplace operator, represents the th eigenvalue corresponding to the Laplace operator eigenfunction.

[0037] Step S2: Construct a Laplace kernel integral module based on the Laplace operator eigenfunction, and further construct a manifold neural operator model.

[0038] The manifold neural operator model includes a lifting layer, a projection layer, and multiple Laplace kernel integral modules. Each Laplace kernel integral module includes an encoder, an approximator, and a decoder;

[0039] The encoder uses the eigenfunctions of the Laplace operator to perform spectral decomposition on the input function of the Laplace kernel integration module to obtain a weight coefficient vector; the approximator uses a linear or nonlinear mapping to parameterize the weight coefficient vector to obtain a parameterized weight coefficient vector; the decoder uses the eigenfunctions of the Laplace operator to perform spectral reconstruction on the parameterized weight coefficient vector to obtain the output function of the Laplace kernel integration module.

[0040] Step S3: Iteratively train the manifold neural operator model according to the optimization objective, optimization constraints, boundary conditions, and material properties of the topology optimization task. The specific steps are as follows:

[0041] Step S31: Input the field function representing the geometry of the design domain into the manifold neural operator model, and output the field function representing the geometric structure.

[0042] Step S32: Perform a structural response analysis on the output geometric structure and calculate the loss function.

[0043] Step S33: Determine whether the termination condition is satisfied. If it is satisfied, output the geometric structure as the final optimization result and end the loop. If it is not satisfied, execute Step S34.

[0044] Step S34: Update the parameters of the manifold neural operator model according to the loss function and the optimization algorithm, and repeat Steps S31 - S34 until the termination condition is satisfied.

[0045] In Step S3, the optimization objectives include one or more of the optimization objectives of structural load - bearing performance, structural heat - transfer performance, structural electromagnetic performance, structural fluid - dynamics performance, and structural mass / volume / material usage.

[0046] The optimization constraints include one or more of the 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.

[0047] The boundary conditions include one or more of the mechanical boundary conditions, thermodynamic boundary conditions, fluid - dynamics boundary conditions, and electromagnetic boundary conditions.

[0048] The function representing the geometry of the design domain includes a coordinate field function or an eigenfunction of the Laplace operator; the function representing the geometric structure includes a density field function, a signed distance value function, or a level - set function.

[0049] The structural response analysis methods include one or more of the finite - element method, the finite - difference method, solving with commercial simulation software, and solving with software having numerical calculation functions.

[0050] 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 objective is less than the set convergence threshold for several consecutive iterations, and the relative change in the structural performance corresponding to the optimization objective is less than the set convergence threshold for several consecutive iterations;

[0051] The optimization algorithms include gradient-based optimization algorithms and heuristic optimization algorithms.

[0052] Embodiment 2

[0053] Step S1: Determine the geometric shape of the design domain for a given topology optimization task, perform mesh division on it, and solve a set of Laplace operator eigenfunctions defined on this mesh.

[0054] In this embodiment, the selected object is a two-dimensional irregular cantilever beam. The optimization objective is to minimize the structural compliance, that is, to maximize the structural stiffness. The optimization constraint is the topology optimization design task of the volume fraction. The geometric shape of the design domain is as Figure 2 shown. Perform triangular mesh division on the geometric shape of the design domain, which includes 4053 mesh nodes and 7803 mesh elements, as Figure 3 shown. Then, solve the Laplace operator eigenvalue equation by the Galerkin method or the power method to obtain a set of Laplace eigenfunctions defined on the geometric shape of the design domain, as Figure 4 shown. In this embodiment, the first 256 Laplace operator eigenfunctions are selected for subsequent steps.

[0055] Step S2: Construct a Laplace kernel integral module based on the obtained Laplace operator eigenfunctions, and then construct a manifold neural operator model.

[0056] In this implementation, linear mapping layers are used to construct the lifting layer and the projection layer of the manifold neural operator model, and the leaky_relu activation function is used. The manifold neural operator model in this implementation contains 4 identical Laplace kernel integral modules connected in series. The encoder of the kernel integral module uses 256 Laplace operator eigenfunctions to perform spectral decomposition on the input of the kernel integral module to obtain a 256-dimensional weight coefficient vector. The approximator uses a linear transformation, and the decoder uses 256 Laplace operator eigenfunctions to perform spectral reconstruction on the parameterized 256-dimensional weight coefficient vector to obtain the output function.

[0057] Step S3: Iteratively train the manifold neural operator model according to the optimization objective, optimization constraints, boundary conditions, and material properties of the topology optimization task.

[0058] Input the node coordinate field into the manifold neural operator model, and output the node density field (the density value ranges from 0 to 1). Set the left end as a fixed constraint for the geometric structure of the output representation, apply a unit concentrated force load vertically downward to the vertex at the lower right end, with a Poisson's ratio of 0.3 and an elastic modulus of 1. Then perform a static simulation analysis, and calculate the loss function considering the structural stiffness and volume fraction. The formula is as follows:

[0059] ;

[0060] where, represents the loss function, and K represent the displacement matrix and the 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 geometric structure output by the manifold neural operator in the current iteration, represents the artificially set penalty coefficient, with an initial value of 40 in this embodiment, and it increases by 0.06 in each iteration.

[0061] In each iteration training, the automatic differentiation technique is used to calculate the gradient of the loss function with respect to the parameters of the manifold neural operator model, and the model parameters are updated according to the gradient. In this embodiment, the selected termination condition is that the number of iterative calculations reaches a preset maximum threshold, and the threshold is taken as 500. The volume fraction of the final optimization result is 0.404, the compliance is 514.2, and the optimized geometric structure is as shown in Figure 5 shown.

[0062] It should be noted that the topology optimization method based on the manifold neural operator described in the present invention is not limited to the maximum structural stiffness optimization objective and volume fraction optimization constraint in Embodiment 2. Its core lies in the continuous field-to-field geometric structure representation, which can be flexibly extended to various optimization objectives and various optimization constraints.

[0063] It is worth noting that the content not elaborated in detail in the present invention is all prior art and is well known to those skilled in the art.

[0064] Therefore, the present invention adopts the above-mentioned topology optimization method based on the manifold neural operator. By introducing the manifold neural operator, it realizes the continuous field-to-field geometric structure representation, overcomes the problem of poor local continuity of the geometric structure obtained by the traditional discrete point-to-point representation mode based on the fully connected neural network, and realizes a topology optimization design with better manufacturability.

[0065] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions of the present invention or make equivalent replacements, and these modifications or equivalent replacements cannot make the modified technical solutions deviate from the spirit and scope of the technical solutions 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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