An isogeometric topology optimization method based on online machine learning

Abstract

The present invention belongs to the technical field related to structural optimization design and discloses an isogeometric topology optimization method based on online machine learning. The method comprises the following steps: S1 constructs an optimization model of the structure to be optimized and divides its design domain into a fine geometric grid and a coarse finite element grid; S2 performs regional division, performs regional density mapping based on the density of control points within the sub-regions, and calculates the flexibility of the corresponding sub-regions to obtain the sensitivity of each sub-region; S3 updates the control point and fine element density, extracts sub-region information to form a data set, trains the machine learning model, and adjusts the prediction; S4 determines whether the current number of iterations has reached the total number of iterations. If not, returns to step S2; otherwise, terminates. The present invention solves the problems of low computational efficiency and unsuitability for three-dimensional design in the topology optimization process.

Description

Technical Field

[0001] The present invention belongs to the technical field related to structural optimization design, and more specifically, relates to an isogeometric topology optimization method based on online machine learning. Background Art

[0002] Isogeometric analysis is based on spline geometry, such as non-uniform rational B-splines (NURBS), used to express geometric models. Using its basis functions as shape functions, it leverages its geometric information to directly construct analytical models, achieving a seamless integration of computer-aided design (CAD) and computer-aided engineering (CAE). As a new numerical analysis method, isogeometric analysis offers advantages such as precise geometric discretization, high-order continuity, and integrated CAD and CAE expression, perfectly addressing the shortcomings of finite element analysis (FEA). At the same time, with its high precision and efficiency, it has gradually replaced FEA and has been continuously combined with topology optimization to form a series of isogeometric topology optimization methods, becoming an important new direction in disciplinary research and engineering applications.

[0003] Currently, topology optimization methods are broadly categorized into density description models and boundary description models. The former primarily includes the variable density method (SIMP) and the evolutionary optimization / bidirectional evolutionary optimization method (ESO / BESO), while the latter commonly includes the level set method (Levelset) and the moving deformable component / void method (MMC / MMV). The variable density method is the most widely used due to its intuitive mathematical model, simple implementation, and efficient computation. By combining it with isogeometric analysis, the design variable of the isogeometric variable density method shifts from element density to the density of control points supporting the NURBS curve. The two can construct a mapping relationship through NURBS basis functions, and through continuous optimization, a structure that meets the design requirements is obtained.

[0004] Although the isogeometric variable density method offers more accurate calculation results, as the design scale increases and the two-dimensional problem extends to three-dimensional problems, the iteration time cost becomes increasingly high, ultimately reducing the overall optimization efficiency. Therefore, improving computational efficiency is a key challenge for isogeometric topology optimization in the future.

[0005] To date, machine / deep learning has successfully combined with topology optimization to create numerous methods. While these methods have proven their validity through extensive experimental validation, they have also exposed certain limitations. First, in many methods, the datasets for model training are collected in advance. Topology optimization methods serve merely as a means of acquiring data, leaving them disconnected from machine / deep learning techniques and preventing a close integration between the two. Furthermore, the datasets collected are not solely specific to the current design problem, but rather encompass a vast array of data from a wide range of design problems. The time and expense of collecting data to achieve reasonable predictions is incalculable. Second, some topology optimization methods that incorporate machine / deep learning lack scalability; optimization methods suitable for two-dimensional design problems cannot address three-dimensional ones. Third, mathematical models constructed to accelerate computational speed cannot be adapted to changes in the design problem. Adjusting these "offline" models requires supplementing the currently collected dataset, requiring significant retraining time. Furthermore, as topology optimization is an iterative process, the underlying relationships in the data are constantly evolving, requiring timely adjustments to ensure the effectiveness of the optimized structure. Fourth, most methods remain focused on traditional finite element topology optimization, while derivative methods such as geometric topology optimization remain unresolved.

[0006] Therefore, geometric topology optimization methods such as machine / deep learning that do not require prior data collection and can be adjusted online during iteration still require a lot of research efforts. Summary of the Invention

[0007] In response to the above defects or improvement needs of the existing technology, the present invention provides an isogeometric topology optimization method based on online machine learning to solve the problems of low computational efficiency and unsuitability for three-dimensional design in the topology optimization process.

[0008] To achieve the above object, according to the present invention, a method for isogeometric topology optimization based on online machine learning is provided, which comprises the following steps:

[0009] S1 sets the initial training times r, the machine learning model update frequency f, and the total number of iterations R, constructs an optimization model of the structure to be optimized, and performs isogeometric meshing on the design domain of the model to obtain isogeometric fine elements and control points, and constructs a relationship between the density of isogeometric fine elements and the element stiffness matrix; performs finite element meshing on the design domain to obtain finite element coarse elements, determines the correspondence between coarse elements and fine elements, and constructs a finite element coarse element density relationship;

[0010] S2 iteration number w=1, calculate the flexibility of the coarse unit, divide the fine unit into several regions, each of which is the same size as the coarse unit, obtain the control point density and regional flexibility in each sub-region, and splice the control point density and regional flexibility of any sub-region to form a density matrix P';

[0011] Determine whether the number of iterations satisfies the following conditions: w≤r or w=r+k0f, where k0 is a positive integer greater than 1. If so, the sensitivity g of each sub-region is calculated by solving the equilibrium equation, thereby obtaining the sensitivity of the global isogeometric unit; otherwise, the density matrix ρ* of any sub-region is input into the machine learning prediction model to obtain the sensitivity g corresponding to any sub-region, thereby obtaining the sensitivity of the global isogeometric unit;

[0012] S3 uses the above sensitivity to update the control point and fine unit density, and determines whether the number of iterations meets the following conditions: w = r or w = r + k1f, k1 is a positive integer greater than 0. If w = r is satisfied, the density matrix ρ* obtained for w ≤ r times is used as input, and the sensitivity matrix g is used as output to train the machine learning model, thereby obtaining a machine learning prediction model; otherwise, the density matrix ρ* and sensitivity matrix g corresponding to the current iteration number w = r + k1f are extracted, and the two are respectively input into the current machine learning prediction model as the input and output matrices required for the update, thereby adjusting the machine learning prediction model;

[0013] S4 determines whether the current number of iterations w reaches the total number of iterations R. If not, w=w+1, and returns to step S2; otherwise, the process ends.

[0014] Further preferably, in step S1, the optimization model is performed according to the following relationship:

[0015]

[0016] min:C=F T U

[0017]

[0018] Where D is the area corresponding to the design problem, x is any point in the design domain, and ρ(x) is the density corresponding to the point. The optimization model is aimed at the flexibility minimization problem under a fixed volume constraint, and C is the objective function. In addition, F is the external load vector, K is the global stiffness matrix, U is the overall displacement vector, V=V(ρ) is the sum of the unit volumes in the design domain, V0 is the total volume of the design domain, and f is the volume fraction. The model discretizes the design domain into a total of N units, e is the unit subscript, and ρ e is the cell density corresponding to the subscript.

[0019] Further preferably, in step S1, the iso-geometric fine element stiffness matrix is ​​calculated according to the following relationship:

[0020]

[0021] Among them, ke is the stiffness matrix, B is the displacement-strain matrix of the unit, which is calculated by the NURBS basis function and the coordinates of the equivalent geometric parameters; D is the stress-strain matrix, which is related to the Young's modulus E of the material. e and Poisson's ratio μ; J1 and J2 are the Jacobian transformation matrices from NURBS parameter space to physical space and from integral space to NURBS parameter space, respectively. Ω and They are physical space and integral space respectively.

[0022] Further preferably, in step S1, the density of the coarse unit is calculated according to the following steps:

[0023] S11: dividing the coarse unit into a preset number of units, each unit containing multiple complete fine units;

[0024] S12 determines the number of fine units contained in each coarse unit, thereby establishing a corresponding relationship between the subscripts of the coarse unit and the fine unit;

[0025] S13 constructs a calculation formula for the density of the coarse cells by using the density of the fine cells and the corresponding relationship between the number of coarse cells and fine cells obtained in step S12.

[0026] Further preferably, the density of the coarse unit is carried out according to the following relationship:

[0027]

[0028] Among them, ρ c (e i ) is the i-th coarse unit, and each coarse unit can be divided into Q subsets of equal size. f (e ijl ) is the lth fine unit of the jth subset, ω jl is the density weight of the corresponding fine unit in the subset to which it belongs, i is the number of the coarse unit, j is the number of the subset, and l is the number of the fine unit corresponding to the current subset.

[0029] Further preferably, in step S2, the flexibility of the coarse unit is determined according to the following relationship:

[0030]

[0031] Among them, E e (ρ e ) is the coarse element Young's modulus, ρ e is the coarse element density, u e is the coarse element node displacement vector, k e is the coarse element stiffness matrix, and e is the number of the coarse element.

[0032] Further preferably, in step S2, the unit Young's modulus is calculated according to the following relationship:

[0033]

[0034] Among them, ρ e is the coarse element density, ρ e ∈[0,1], p is the penalty factor of the variable density method (the default value is 3), where the Young's modulus of the solid element is E0=1, and the Young's modulus of the void element is E min =10 -6 , E e (ρ e ) can represent the Young's modulus of any coarse element.

[0035] Further preferably, in step S2, the sensitivity of the fine cells contained in the arbitrary sub-region is calculated according to the following relationship:

[0036]

[0037] Among them, g f (n i ) is the sensitivity of the i-th control point, s i is the unit set affected by the control point, j is any fine unit in the unit set, p is the penalty factor of the variable density method, ρ f (e ij ) is the jth fine unit e corresponding to the i-th control point ij The density of u eij is the displacement vector of the control point in the unit, k eij is the element stiffness matrix of the element, Obtained by NURBS basis functions.

[0038] Further preferably, in step S3, the machine learning model includes an input layer, an output layer and four hidden layers. The neuron feedforward propagation calculation formula is as follows:

[0039] a=σ(z)=σ(∑ω i x i +b)

[0040] Where a is the output of the hidden layer, σ(z) is the calculated linear or nonlinear activation of the previous layer input, ω is the weight matrix of the current layer for the previous layer input, b is the bias, and x is the input.

[0041] Further preferably, in step S3, the control points and fine element density are updated by using sensitivity analysis values ​​and an optimal criterion method.

[0042] In general, the above technical solutions conceived by the present invention have the following beneficial effects compared with the prior art:

[0043] 1. The present invention performs density mapping between fine and coarse elements on a finite element coarse grid. By solving equilibrium equations, information about design variables on the coarse grid is obtained. Fine elements are then optimized on an isogeometric fine grid using a variable density method to obtain information about design variables on the fine grid. The design domain is then segmented into subregions based on the size of the coarse elements. This allows the use of an isogeometric-finite element dual-scale analysis grid to simultaneously extract information from both grids.

[0044] 2. The present invention utilizes an isogeometric-finite element dual-scale analysis grid to extract a large amount of data with global and local information through a small number of iterative steps, thereby obtaining a training set for a machine learning model. The strategy of using iterative historical data to generate a data set online avoids a large amount of time spent on generating an offline data set.

[0045] 3. The present invention uses a dataset to train a DNNs model, which can obtain the mapping relationship between the density of fine-cell control points and the sensitivity analysis value of the fine-cell. This saves the time of solving the equilibrium equation on a fine grid of equal geometry. The predicted cell sensitivity is used to interpolate the control point sensitivity through NURBS basis functions, thereby updating the design variables.

[0046] 4. In the present invention, as iterations proceed, an updated dataset is generated at a certain frequency. Based on the original DNNs model, the weights and biases between neurons are fine-tuned to better capture the numerical relationship between the eigenvalues ​​and target values ​​of the dataset, thereby ensuring the accuracy and convergence of subsequent optimization. In addition, this method is not limited to the dimension of the design problem; two-dimensional design problems can be easily expanded to three-dimensional design problems, significantly shortening the optimization time while ensuring design accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 It is a flow chart of an isogeometric topology optimization method based on online machine learning constructed according to a preferred embodiment of the present invention;

[0048] Figure 2 Schematic diagram of a two-dimensional center-loaded cantilever beam working condition with an analysis grid size of 64×32 in an embodiment of the present invention;

[0049] Figure 3 This is a schematic diagram of density mapping between fine and coarse elements in the isogeometric-finite element dual-scale analysis grid proposed by the present invention;

[0050] Figure 4 This is a schematic diagram of the DNNs neural network and input and output used in the present invention;

[0051] Figure 5 Schematic diagram of the application of the method proposed in the present invention;

[0052] Figure 6 It is a schematic diagram of the two-dimensional topology optimization results of different grids and different boundary conditions implemented in the present invention;

[0053] Figure 7 Schematic diagram of the speedup ratio of the two-dimensional design problem implemented in the present invention, wherein (A) is a schematic diagram of the speedup ratio of the cantilever beam with a middle load at different grid scales, (B) is a schematic diagram of the speedup ratio of the cantilever beam with a bottom load at different grid scales, and (C) is a schematic diagram of the speedup ratio of the MBB beam at different grid scales;

[0054] Figure 8 Schematic diagram of the convergence curve of the optimization of a two-dimensional mid-load cantilever beam with a grid size of 512×256 according to the present invention;

[0055] Figure 9 1. It is a schematic diagram of topology optimization when updating a two-dimensional mid-load cantilever beam with a grid size of 512×256 according to the present invention;

[0056] Figure 10 Schematic diagram comparing the three-dimensional example machine learning implemented by the present invention with the standard method. DETAILED DESCRIPTION

[0057] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely for the purpose of explaining the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0058] See also Figure 1 The isogeometric topology optimization method based on online machine learning provided by the present invention mainly includes the following steps:

[0059] Step 1: Set the framework parameters for online machine learning, namely the iterative historical data window range and model update frequency required for the initial training dataset.

[0060] Specifically, this method differs from other topology optimization methods that combine machine learning and deep learning in that the training dataset does not need to be generated in advance, but is instead generated during the optimization process using historical data. Therefore, it is sufficient to specify a window size for the historical data to ensure that the dataset generated within this range is sufficient to train the initial DNN model. However, the window size for the historical data should not be too large, as this increases iteration time and reduces the speedup ratio.

[0061] In addition to generating datasets online, this method can update the DNN model at a constant frequency. Increasing the update frequency can reduce the number of standard topology optimization calculations and shorten the overall optimization time, but it also reduces topology optimization stability and may cause oscillations that cannot be eliminated. By keeping the update frequency within a certain range, both speedup and stability can be guaranteed.

[0062] In the second step, based on the design problem to be processed (the problem of minimizing the flexibility of the constrained volume), an optimization model of the isogeometric variable density method is constructed. At the same time, the design domain of the problem is divided into isogeometric fine grids using NURBS lines to obtain multiple isogeometric fine units and multiple control points, and the obtained grid is used as the isogeometric background grid.

[0063] In the isogeometric variable density method, the cell density is obtained by accumulating the control point density through interpolation using the NURBS basis function, and the initial control point density is equal to the volume constraint ratio.

[0064] Datasets for machine learning: In this method, online dataset generation relies on a dual-scale isogeometric-finite element analysis grid. The optimization problem is performed only on the fine isogeometric grid, while the coarse finite element grid supplements the local data of the fine grid with global information.

[0065] Specifically, the design variable is the density value ρ(x) at any point in the design domain D. The design goal is to minimize the structural flexibility C to achieve maximum stiffness. The design constraints include the equilibrium equation KU=F, the volume constraint V=V(ρ)≤f·V0, and the design variable in [0,1].

[0066] by Figure 2 For example, this embodiment aims to minimize the structural flexibility of a plane cantilever beam with a point load. In a given design domain D, a displacement constraint is imposed on the left boundary, and a point load F=-1 is imposed on the midpoint area of ​​the right boundary. The above boundary conditions are expanded into a displacement vector U and an external load vector F by applying them to the control points in the design domain, and the cantilever beam is designed to maximize its stiffness. The current grid is subdivided into 64×32 biquad-order isogeometric units and 66×34 control points, which serve as the background grid for isogeometric analysis.

[0067] The mathematical expression of the variable density method structural optimization model is:

[0068]

[0069] min:C=F T U

[0070]

[0071] Where D is the area corresponding to the design problem, x is any point in the design domain, and ρ(x) is the density corresponding to the point. The optimization model is aimed at the flexibility minimization problem under a fixed volume constraint, and C is the objective function. In addition, F is the external load vector, K is the global stiffness matrix, U is the overall displacement vector, V=V(ρ) is the sum of the unit volumes in the design domain, V0 is the total volume of the design domain, and f is the volume fraction. The model discretizes the design domain into a total of N units, e is the unit subscript, and ρ e is the cell density corresponding to the subscript.

[0072] Constructing a relationship:

[0073] (1) The density of the geometrically fine element is obtained by accumulating the density of the control points through interpolation of the NURBS basis function:

[0074]

[0075] Among them, the density of fine element ρ of the i-th f (e i ) is determined by the unit center density ρ(e i ) indicates that the fine unit center density is determined by the preset control point density ρ nij Through the NURBS basis function N ij (e i ) interpolation accumulation; c i is the control point set corresponding to each fine unit, ρ nij is the j-th control point density of the control point set.

[0076] (2) Under this fixed isogeometric analysis grid, the NURBS parameters and control point coordinates corresponding to the unit can be determined, and the stiffness matrix of the isogeometric fine unit can be calculated as follows:

[0077]

[0078] Among them, B is the displacement-strain matrix of the unit, which is calculated by the NURBS basis function and the coordinates of the geometric parameters; D is the stress-strain matrix, which is related to the Young's modulus E of the material. e and Poisson's ratio μ; J1 and J2 are the Jacobian transformation matrices from NURBS parameter space to physical space and from integral space to NURBS parameter space, respectively; Ω and They are physical space and integral space respectively.

[0079] Step 3: Determine the correspondence between the finite element coarse elements and the isogeometric fine elements, and construct a "dimensionality reduction" process for the density mapping relationship. At the same time, meshing is performed using the variable density method and used as the finite element background mesh.

[0080] Specifically, considering that the more geometric fine elements corresponding to the finite coarse element, the more nonlinear the amount of information it contains, the appropriate coarse element size is determined. c , represents the number of fine elements under a coarse element in any direction. Figure 3 , a finite element coarse cell contains 4×4 fine cells of equal geometry. In this case, the density within the coarse cell is not uniform and needs to be homogenized.

[0081] According to the principle of evenly dividing the two-dimensional unit into four sub-areas, the above coarse unit is divided. Figure 3 In each subregion, the fine elements are complete. If a coarse element contains 5×5 fine elements, it is divided into four subregions, each of which contains incomplete elements. According to the degree of bisection, the corresponding weights are assigned: the complete element has a weight of 1, the bisected element has a weight of 0.5, and the quartered element has a weight of 0.25. Thus, the density mapping matrix between the isogeometric fine elements and the finite element coarse elements is established. Therefore, the coarse element density can be calculated as:

[0082]

[0083] Here, since each two-dimensional coarse unit is evenly divided into four sub-regions, Q is 4. c (e i ) is the i-th coarse unit, and each coarse unit can be divided into Q subsets of equal size. f (e ijl ) is the lth fine unit of the jth subset, ω jl is the density weight of the corresponding fine unit in the subset to which it belongs, i is the number of the coarse unit, j is the number of the subset, and l is the number of the fine unit corresponding to the current subset.

[0084] Step 4: According to the density of the finite element coarse element and the element stiffness matrix, the variable density method is followed to couple the global stiffness matrix, and the coarse element node displacement vector is solved to calculate the element flexibility.

[0085] Specifically, in the variable density method, the actual element Young's modulus is related to the element density. The actual element stiffness matrix is ​​equal to the product of the element Young's modulus and the element stiffness matrix (without material properties). In the variable density method, the element Young's modulus is calculated as follows:

[0086]

[0087] Among them, ρ e is the coarse element density, ρ e ∈[0,1], p is the penalty factor of the variable density method (the default value is 3), where the Young's modulus of the solid element is E0=1, and the Young's modulus of the void element is E min =10-6 , so E e (ρ e ) can represent the Young's modulus of any coarse element.

[0088] The finite element stiffness matrix is ​​fixed and is directly given in matrix form in this method. Combined with the above formula, it is coupled into a global stiffness matrix. The equilibrium equation KU=F is solved to obtain the coarse element node displacement vector. The coarse element flexibility is calculated as follows:

[0089]

[0090] Among them, E e (ρ e ) is the coarse element Young's modulus, ρ e is the coarse element density, u e is the coarse element node displacement vector, k e is the coarse element stiffness matrix, and e is the number of the coarse element.

[0091] Step 5: Judge the current iteration step. If it belongs to the initial iteration range or update range, a complete standard calculation needs to be performed under the isogeometric grid. If it does not, a machine learning model can be used for rapid prediction.

[0092] Specifically, isogeometric topology optimization methods based on online machine learning use iterative historical data to generate datasets. A certain number of iterations are required to construct the initial dataset, discarding information from earlier iterations where data features are less pronounced. As iterations progress, to capture current data features and maintain model accuracy, smaller, updated datasets are generated by linking the current mesh information with the previous dataset. The majority of these datasets are derived from the fine isogeometric mesh and must be obtained through standard solvers.

[0093] (1) The formula in step 2 calculates the element density. The Young's modulus of the fine element can be solved according to the formula in step 4. The fine element stiffness matrix is ​​coupled into a global stiffness matrix (including Young's modulus and element stiffness matrix) through the variable density method. The equilibrium equation is solved to obtain the fine element displacement vector, and the displacement vector is used to perform sensitivity analysis. The calculation formula is as follows:

[0094]

[0095] Among them, g f (n i ) is the sensitivity of the i-th control point, s i is the unit set affected by the control point, j is any fine unit in the unit set, p is the penalty factor of the variable density method, ρ f (e ij ) is the jth fine unit e corresponding to the i-th control point ij The density of ueij is the displacement vector of the control point in the unit, k eij is the element stiffness matrix of the element, Obtained by NURBS basis functions.

[0096] If the current iteration step belongs to the initial iteration range or the update range, (1) the fine mesh with equal geometry is divided into several sub-regions with the size of the coarse element as the scale. Figure 4 , (2) The control point density in the sub-region is extracted and combined with the unit flexibility of the coarse unit mapped by the fine unit to form a one-dimensional vector (the size of a sub-region is equal to that of a coarse unit, so the flexibility of the coarse unit is the flexibility of the sub-region), and the one-dimensional vectors of all sub-regions are spliced ​​into a matrix, namely the global density matrix; (3) The unit displacement vector of each isogeometric fine unit can be obtained by solving the equilibrium equation, thereby obtaining the corresponding sensitivity; (4) The input of the data set can be obtained in (1), and the output of the data set can be obtained by extracting the sensitivity of the fine units in the same sub-region and assembling them into a matrix.

[0097] Isogeometric fine grids and finite element coarse grids can divide a large amount of the above types of information, which together constitute the input of the DNNs model.

[0098] The DNNs model used in this method consists of an input layer, an output layer, and four hidden layers. The neuron feedforward propagation calculation formula is as follows:

[0099] a=σ(z)=σ(∑ω i x i +b)

[0100] Where a is the output of the hidden layer, σ(z) is the calculated linear or nonlinear activation of the previous layer input; ω is the weight matrix of the current layer for the previous layer input, b is the bias, and x is the input. The back propagation of the DNNs model is realized through the loss function. This method uses the mean square error (MSE) as the loss function, combined with the Adam optimizer, which is conducive to the convergence of the model parameters. At the same time, the PReLU function is used instead of the ReLU function as the activation function between layers to ensure that the neurons are not deactivated. If the current iteration step does not belong to the initial iteration range or the update range, (1) the geometric fine grid is divided into several sub-regions based on the size of the coarse unit. Reference Figure 4, (2) extract the control point density in the sub-region and the unit flexibility of the coarse unit mapped by the fine unit to form a one-dimensional vector (the size of a sub-region is equal to a coarse unit, so the flexibility of the coarse unit is the flexibility of the sub-region), and splice the one-dimensional vectors of all sub-regions into a matrix, namely the global density matrix; (3) input the above global density matrix into the current machine learning model to directly predict the sensitivity of the fine units in all sub-regions, and then splice to obtain the sensitivity of all fine units that conform to the global numbering order.

[0101] The sensitivity of the control points can be obtained by interpolating and accumulating the sensitivity of the fine elements using the NURBS basis function.

[0102] Step 6: Update the density of control points and iso-geometric cells through sensitivity analysis values ​​and the optimal criterion method, and save historical information for subsequent generation of machine learning training data sets.

[0103] Specifically, the update of design variables is achieved through the optimal criterion method, which is calculated as follows:

[0104]

[0105] Among them, ρ e is the unit density, m (= 0.02) is the forward moving step, η (= 0.5) is the damping coefficient, B e For the optimal case, the calculation is as follows:

[0106]

[0107] Where C is the objective function, V is the volume fraction, and λ is the Lagrangian operator calculated during the optimization process; g has been calculated in step 5 f (n i )get, Calculated using NURBS basis functions.

[0108] The control point density of the unupdated fine element, the element flexibility of the coarse element in step 4, and the sensitivity analysis value of the fine element in step 5 are saved as historical data, and the density of the control points and fine elements is updated.

[0109] Step 7: Determine whether the number of iterations reaches the set initial training number under the current number of iteration steps, and whether the machine learning model needs initial training or updating. If necessary, operate the machine learning model, otherwise skip.

[0110] Specifically, when the machine learning model requires initial training or updating, further judgment is required.

[0111] If a machine learning model does not exist, perform initial training. Based on the initial training range in step 1, extract the data within that range from the historical data and divide it into input and output. To ensure effective training, the input and output are homogenized and shuffled before training the DNN model.

[0112] If a machine learning model exists, the control point density of fine cells and the cell flexibility of coarse cells are extracted from the current isogeometric-finite element dual-scale mesh. This is then divided and expanded into subregions, forming the input and output matrices for the machine learning model update. This small update dataset is also homogenized and shuffled. At this point, the small dataset is not suitable for retraining or resuming the existing DNN model. Instead, the DNN model is updated by adjusting the weights and biases of the neurons between the last hidden layer and the output layer.

[0113] Step 8: Determine whether the current iteration step exceeds the preset maximum number of iteration steps. If it exceeds the upper limit, proceed to step 9; otherwise, jump to step 4.

[0114] Step 9: The optimization process ends and the optimal structure diagram is generated based on the fine element density on the isogeometric fine grid.

[0115] like Figure 5 As shown in the figure, the total number of iterations R is 200, the initial number of iterations r is selected as [10,…,30], the update frequency f is selected as 4, and the initial DNNs model is trained by global and local information on the dual-scale grid from the 10th to the 30th time. The current machine learning model will be updated every 4 times to ensure the prediction accuracy until the end of the optimization;

[0116] See also Figure 6 The images show the results of a two-dimensional example using the method provided by this invention, which optimizes different design problems at different grid sizes. The standard calculation closely matches the optimal structure of machine learning.

[0117] See also Figure 7 , the picture is Figure 6 The standard time for the two-dimensional example is compared with the machine learning time, and the speedup ratio is shown. Except for small-scale problems, the speedup ratio of machine learning is very significant.

[0118] See also Figure 8 and Figure 9 When the grid size is 512×256, the optimization problem of the cantilever beam with central load is used to test the convergence stability of the machine learning method. Figure 8 In the figure, the solid line is the convergence curve of the objective function (structural flexibility), and the hollow dots are the results of the objective function under standard calculation during machine learning updates. Figure 9In the example, the optimized structure calculated by the standard method locally moves from low resolution to high resolution during the machine learning update. This shows that the isogeometric topology optimization method based on online machine learning is applicable to 2D design problems and has stable overall convergence.

[0119] Please refer to Figure 10 To demonstrate the scalability of the method provided by the present invention, the optimization problem is extended from a two-dimensional design problem to a three-dimensional design problem. Figure 10 For three-dimensional design problems with different boundary conditions, the feasibility and scalability of this method are verified through relative error and speedup ratio.

[0120] This method uses the density of any point in the design domain as a discrete design variable and performs optimization analysis using an isogeometric-finite element dual-scale grid. This method allows for the acquisition of large amounts of data online within a certain number of iterations, enabling the training of machine learning DNN models and saving time in solving equilibrium equations for isogeometric topology optimization. Furthermore, as the iterative optimization progresses, an updated dataset can be generated online, enabling fine-tuning of the model and ensuring the certainty and stability of the optimization results.

[0121] It will be easily understood by those skilled in the art that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. An isogeometric topology optimization method based on online machine learning, characterized in that: The method comprises the following steps: S1 sets the initial training times r, the machine learning model update frequency f, and the total number of iterations R, constructs an optimization model of the structure to be optimized, and performs isogeometric meshing on the design domain of the model to obtain isogeometric fine elements and control points, and constructs a relationship between the density of isogeometric fine elements and the element stiffness matrix; performs finite element meshing on the design domain to obtain finite element coarse elements, determines the correspondence between coarse elements and fine elements, and constructs a finite element coarse element density relationship; S2 iteration number w=1, calculate the flexibility of the coarse unit, divide the fine unit into several regions, each of which is the same size as the coarse unit, obtain the control point density and flexibility in each sub-region, and splice the control point density and regional flexibility of any sub-region to form a density matrix ρ*; Determine whether the number of iterations satisfies the following conditions: w≤r or w=r+k0f, where k0 is a positive integer greater than 0. If so, the sensitivity g of each sub-region is calculated by solving the equilibrium equation, thereby obtaining the sensitivity of the global isogeometric unit; otherwise, the density matrix ρ* of any sub-region is input into the machine learning prediction model to obtain the sensitivity g corresponding to any sub-region, thereby obtaining the sensitivity of the global isogeometric unit; S3 uses the sensitivity of the global isogeometric fine cells in step S2 to update the control points and fine cell density, and determines whether the number of iterations satisfies the following conditions: w=r or w=r+k1f, k1 is a positive integer greater than 0. If w=r is satisfied, the density matrix ρ* obtained for w≤r times is used as input, and the sensitivity matrix g is used as output to train the machine learning model, thereby obtaining a machine learning prediction model; otherwise, the density matrix ρ* and sensitivity matrix g corresponding to the current iteration number w=r+k1f are extracted, and the two are respectively input into the current machine learning prediction model as the input and output matrices required for the update, thereby adjusting the current machine learning prediction model; S4 determines whether the current number of iterations w reaches the total number of iterations R. If not, w=w+1 and returns to step S2; otherwise, the process ends. In step S1, the optimization model is performed according to the following relationship: min:C=F T U Where D is the area corresponding to the design problem, x is any point in the design domain, ρ(x) is the density corresponding to the point, the optimization model is for the flexibility minimization problem under a fixed volume constraint, C is the objective function, in addition, F is the external load vector, K is the global stiffness matrix, U is the overall displacement vector, V = V(ρ) is the sum of the unit volumes in the design domain, V0 is the total volume of the design domain, f is the volume fraction, the model discretizes the design domain into a total of N units, e is the unit subscript, ρ e is the cell density corresponding to the subscript.

2. The isogeometric topology optimization method based on online machine learning according to claim 1, characterized in that: In step S1, the iso-geometric fine element stiffness matrix is ​​calculated according to the following relationship: Among them, ke is the stiffness matrix of the isogeometric element, B is the displacement-strain matrix of the element, which is calculated by the NURBS basis function for the coordinates of the isogeometric parameters; D is the stress-strain matrix, which is related to the Young's modulus E of the element. e and the Poisson's ratio μ of the material; J1 and J2 are the Jacobian transformation matrices from NURBS parameter space to physical space and from integral space to NURBS parameter space, respectively. Ω and They are physical space and integral space respectively.

3. The isogeometric topology optimization method based on online machine learning according to claim 1, characterized in that: In step S1, the density of the coarse unit is calculated according to the following steps: S11: dividing the coarse unit into a preset number of units, each unit containing multiple complete fine units; S12 determines the number of fine units contained in each coarse unit, thereby establishing a corresponding relationship between the subscripts of the coarse unit and the fine unit; S13 constructs a calculation formula for the density of the coarse cells by using the density of the fine cells and the corresponding relationship between the number of coarse cells and fine cells obtained in step S12.

4. The isogeometric topology optimization method based on online machine learning according to claim 3, characterized in that: The density of the coarse unit is determined according to the following relationship: Among them, ρ c (e i ) is the i-th coarse unit, each coarse unit can be divided into Q subsets of equal size, ρ f (e ijl ) is the lth fine unit of the jth subset, ω jl is the density weight of the corresponding fine unit in the subset to which it belongs, i is the number of the coarse unit, j is the number of the subset, and l is the number of the fine unit corresponding to the current subset.

5. The isogeometric topology optimization method based on online machine learning according to claim 1 or 2, characterized in that: In step S2, the flexibility of the coarse element is calculated according to the following relationship: Among them, E e (ρ e ) is the coarse element Young's modulus, ρ e is the coarse element density, u e is the coarse element node displacement vector, k e is the coarse element stiffness matrix, and e is the number of the coarse element.

6. The isogeometric topology optimization method based on online machine learning according to claim 5, characterized in that: In step S2, the unit Young's modulus is calculated according to the following relationship: Among them, ρ e is the coarse element density, ρ e ∈[0,1], p is the penalty factor of the variable density method, where the Young's modulus of the solid element is specified as E0=1, and the Young's modulus of the void element is specified as E min =10 -6 , E e (ρ e ) is the Young's modulus of the coarse element.

7. The isogeometric topology optimization method based on online machine learning according to claim 1 or 2, characterized in that: In step S2, the sensitivity of the fine cells contained in the arbitrary sub-region is calculated according to the following relationship: Among them, g f (n i ) is the sensitivity of the i-th control point, s i is the element set affected by the control point, j is any fine element in the element set, p is the penalty factor of the variable density method, ρ f (e ij ) is the jth fine unit e corresponding to the i-th control point ij The density of u eij is the displacement vector of the control point in the unit, k eij is the element stiffness matrix of the element, Obtained by NURBS basis functions.

8. The isogeometric topology optimization method based on online machine learning according to claim 1 or 2, characterized in that: In step S3, the machine learning model includes an input layer, an output layer, and four hidden layers. The neuron feedforward propagation calculation formula is as follows: a=σ(z)=σ(∑ω i x i +b) Where a is the output of the hidden layer, σ(z) is the calculated linear or nonlinear activation of the previous layer input, ω is the weight matrix of the current layer for the previous layer input, b is the bias, and x is the input.

9. The isogeometric topology optimization method based on online machine learning according to claim 1 or 2, characterized in that: In step S3, the control points and fine element density are updated using an optimal criterion method through sensitivity analysis values.

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