A method for secondary topology optimization of composite materials based on Gaussian function filtering

Abstract

The present invention discloses a method for secondary topology optimization of composite materials based on Gaussian function filtering. The method comprises: performing a primary topology optimization on an isotropic material using the SIMP method to obtain an optimized unit density; performing fuzzy processing to obtain a fuzzy unit density, performing finite element analysis to obtain the principal stress direction, assigning anisotropic material properties, performing secondary topology optimization using an improved SOMP method, and filtering the unit density sensitivity and fiber angle using a Gaussian function to obtain optimized unit density and fiber angle. The present invention performs fuzzy processing on the results of the primary topology optimization and obtains the principal stress direction, thereby expanding the optimization space and significantly reducing the risk of generating a local optimal solution; in the secondary parallel optimization, a numbering method suitable for anisotropic materials is adopted, and a Gaussian function is used for filtering, thereby reducing the number of grayscale units and improving the continuity of the fiber angle, thereby ultimately improving the mechanical properties and manufacturability of the parallel optimized structure.

Description

Technical Field

[0001] The invention relates to a composite material secondary topology optimization method, in particular to a composite material secondary topology optimization method based on Gaussian function filtering. Background Art

[0002] In recent years, topology optimization has become the most challenging and practical direction in optimization design methods and has received widespread attention from the academic community. However, faced with the industry's increasingly stringent demand for lightweight equipment structures, the use of a single new material or innovative optimization method can no longer fully meet the above needs. Therefore, introducing continuous fiber composite materials with superior specific stiffness and specific strength into topology optimization has become a more effective design method. However, the optimized topology with continuous fibers will also present discontinuous fiber paths. For example, the fiber angles of adjacent units change suddenly, which leads to stress concentration and poor mechanical properties, and also reduces manufacturability. At the same time, in the parallel optimization of topology and fiber angle, the optimization results are significantly dependent on the initial values, so there is also the risk of falling into a local optimal solution. Summary of the Invention

[0003] In order to solve the problems existing in the background technology, the present invention provides a composite material secondary topology optimization method based on Gaussian function filtering. This method can solve the problems of high dependence of the optimized structure on the initial fiber angle and sudden changes in the fiber angles of adjacent units. On the one hand, in the first optimization, the topological structure of the isotropic material is obtained, and then the mean fuzzification is performed as the initial value for the subsequent parallel optimization of the structural topology and fiber angle; and, finite element analysis is performed in the blurred topological structure to calculate the principal stress direction of all units, and the principal stress direction is used as the initial value of the fiber angle for the second optimization; on the other hand, in the parallel optimization framework of the composite material topology optimization of the second optimization, Gaussian function filtering is used to filter the unit density sensitivity and fiber angle to achieve the purpose of reducing gray units and smoothing fiber angles.

[0004] The technical solution adopted in the present invention is:

[0005] The composite material secondary topology optimization method based on Gaussian function filtering of the present invention comprises:

[0006] 1) The solid isotropic material with penalization method SIMP (Solid Isotropic Material with Penalization) is used to perform a first topology optimization on the topology of the isotropic material. The topology of the isotropic material after the first topology optimization is fuzzy processed to obtain the fuzzy cell density value of each grid cell. The topology of the fuzzy isotropic material is subjected to finite element analysis to obtain the principal stress direction of each grid cell as the fiber angle value. The topology of the isotropic material is then assigned orthotropic material properties to obtain the topology of the anisotropic material.

[0007] Orthotropic material properties include four independent elastic constants E1, E2, v 21 and G 12 , respectively represent the elastic modulus in direction 1, the elastic modulus in direction 2, the Poisson's ratio in direction 12, and the shear modulus in direction 12.

[0008] 2) The improved solid orthogonal anisotropic material penalty method SOMP (Solid Orthogonal Material with Penalization) with the addition of Gaussian function filtering method is used to perform a second topology parallel optimization on the topology structure of the anisotropic material. The Gaussian function filtering method is used to filter the cell density sensitivity and fiber angle of each grid cell of the topology structure of the anisotropic material. After the second topology parallel optimization, the topology structure of the anisotropic material after the second topology parallel optimization and the final optimized cell density value and optimized fiber angle value of each grid cell are finally obtained, completing the secondary topology optimization of the composite material.

[0009] In the initial state, the cell density is set to a preset value, generally set to the same value, such as 0.3, 0.4, etc. The cell density is 0-1, "0" means no material, and "1" means material.

[0010] In the step 1), the fuzzy processing is specifically to use mean fuzzy to process the cell density value of each grid cell of the topological structure of the isotropic material after the first topology optimization, as follows:

[0011]

[0012] Where i is the number of cells in the neighborhood of the central grid cell, j is the number of blurs, and x i,j and x i,j-1 are the cell densities of the grid cells before and after the jth blur, N e is the number of blurred grid cells, rmin is the blur radius, and n is the total number of grid cells.

[0013] In the step 1), when the preset fuzzy index s of the mean fuzzy is reached, the mean fuzzy processing is completed to obtain the fuzzy unit density value; the preset fuzzy index s is specifically as follows:

[0014] s>2N·volfrac,

[0015]

[0016]

[0017] Where N is the number of mesh elements when the element density is greater than 0.5; volfrac is the preset volume fraction during the first topology optimization, and x e,j is the cell density value of the e-th grid cell after the j-th fuzzification. When the fuzzy index is reached, the fuzzy process is stopped and the new cell density is used as the input of the parallel optimization.

[0018] A 5x5 convolution kernel is used to apply mean blurring to the density of the first optimized topological structure. This involves sliding the convolution kernel to calculate the arithmetic mean of the discrete cell densities. Mean blurring, also known as normalized blurring, is a low-pass filter that applies arithmetic mean blurring to the image using a sliding convolution kernel. Its primary functions are to remove noise and blur the image.

[0019] In the step 1), the cell density value of each grid cell of the topological structure of the isotropic material after fuzzy processing is the fuzzy cell density value, and the topological structure of the isotropic material after fuzzy processing is subjected to finite element analysis, specifically, each grid cell of the topological structure of the isotropic material after fuzzy processing and its fuzzy cell density value are subjected to finite element analysis to obtain the three components of stress value σ x , σ y , τ xy , according to the preset fiber angle range and the three components of the stress value of each grid unit, the principal stress direction of each grid unit is obtained as the fiber angle value.

[0020] In the step 2), the improved density-based solid orthotropic material penalty method SOMP is specifically to add Gaussian function filtering method, optimization criterion method OC (Optimality Criteria) and moving asymptotes method MMA (Method of Moving Asymptotes) in sequence after the original solid orthotropic material penalty method SOMP.

[0021] In the parallel optimization of the structural topology and fiber angle of the composite material, the design domain is discretized by four-node elements, the design variables are element density and fiber angle, the objective function is to minimize the structural compliance, and the volume fraction is used as the constraint. A Gaussian filter is used to filter the two design variables, element density sensitivity and fiber angle, and the standard deviation of the Gaussian function is adjusted to obtain a composite topology optimized structure with better mechanical properties and improve the continuity of the fiber layout.

[0022] In the second parallel optimization of topology and fiber angle, the design domain is discretized into four-point elements, and the elements, nodes, and degrees of freedom are numbered to facilitate the assembly of the overall stiffness matrix and the calculation of node displacements; the element stiffness matrix is ​​calculated and the overall stiffness matrix is ​​assembled; finite element analysis is performed; the filtering of design variables is completed, including mesh element filtering of element density sensitivity and Gaussian filtering of fiber angle; the design variables are updated to meet the objective function, and it is continuously iterated until convergence to obtain the parallel optimization results of the final composite material topology and fiber angle.

[0023] During the second topology parallel optimization, the fuzzy cell density value of each grid cell of the topological structure of the anisotropic material is optimized by first using the original solid orthotropic material penalty method SOMP for the fuzzy cell density value of each grid cell. Then, the fuzzy cell density sensitivity value of the optimized grid cell is filtered using the Gaussian function filtering method, and the sensitivity of the fuzzy cell density value of the grid cell is filtered using the Gaussian kernel function, as follows:

[0024]

[0025] Where W1() is the first Gaussian kernel function of the sensitivity of the fuzzy cell density value of the grid cell, r is the distance between the center of the grid cell being filtered and the surrounding grid cells; σ n is the standard deviation of the first Gaussian kernel function in the nth iteration; m is the preset moving step size, which is 1.01.

[0026] Then the sensitivity of the fuzzy cell density value of the filtered grid cell is normalized and calculated to obtain the sensitivity of the fuzzy cell density value of the final filtered grid cell, as follows:

[0027]

[0028] Where c is the overall topological flexibility, x' e is the fuzzy cell density value of the e-th grid cell, and are the sensitivities of the fuzzy cell density values ​​of the e-th grid cell before and after filtering, N eis the neighborhood set of the e-th grid cell.

[0029] Gaussian filtering is performed on the cell density sensitivity. This involves performing a convolution calculation on all cell density sensitivities using a Gaussian kernel to correct for the influence of the central cell on surrounding cells. The standard deviation of the Gaussian kernel function determines the kernel weight. When the weight is less than 0.05, the influence of this cell on the central cell is ignored. Therefore, the standard deviation is gradually increased with a given step size in each iteration to gradually reduce the Gaussian kernel size and increase the weight of the central cell, thereby limiting the generation of meaningless grayscale cells.

[0030] After filtering, the fuzzy cell density sensitivity value of the grid cell is optimized using the optimization criterion method OC until the change value of the fuzzy cell density value of the grid cell converges to obtain the final optimized cell density value.

[0031] During the second topology parallel optimization, the fiber angle value of each grid of the topological structure of the anisotropic material is first optimized using the original solid orthotropic material penalty method SOMP, and then the fiber angle value of the optimized grid unit is filtered using the Gaussian function filtering method. The fiber angle value of each grid unit is traversed and convolved using a 3*3 Gaussian kernel function to obtain a second Gaussian kernel function W2 of the fiber angle value of the grid unit, and then iterative processing is performed. In each iteration, the second Gaussian kernel function W2 is used to filter the fiber angle value of each grid unit to update the fiber angle value. The fiber angle value of each grid unit is specifically as follows:

[0032]

[0033] Among them, θ e and θ e are the fiber angle values ​​of the e-th grid unit before and after filtering; W2 is the second Gaussian kernel function of the fiber angle value of the grid unit; θ e-nelx-1 ,θ e-nelx ,θ e-nelx+1 ,θ e-1 ,θ e+1 ,θ e+nelx-1 ,θ e+nelx and θ e+nelx+1 They represent the fiber angle values ​​of the e-nelx-1, e-nelx, e-nelx+1, e-1, e+1, e+nelx-1, e+nelx and e+nelx+1 grid cells respectively, where nelx is the number of cells in the x-axis direction;

[0034] After filtering, the fiber angle values ​​of the grid cells are optimized using the moving average method (MMA) until the change in the fiber angle value of the grid cells converges to obtain the final optimized fiber angle value.

[0035] A 3*3 Gaussian kernel function is used to perform convolution calculations on the fiber angle values ​​of each grid cell. The standard deviation of the kernel function is 1.5. After the weights are calculated, they are normalized and the 9 weights are divided by 0.4787 to obtain the second Gaussian kernel W2 of the fiber angle value as follows:

[0036]

[0037] In the second optimization, the topology and fiber angles were optimized in parallel using cell density and fiber angle as design variables. In the present invention, the fiber angles were Gaussian filtered, i.e., all fiber angles were traversed and convolved using a Gaussian kernel to enhance the influence of the surrounding cells on the central cell. Considering that the purpose of Gaussian filtering is to increase the continuity of the fiber angles, the present invention sets the standard deviation of the Gaussian kernel function to an appropriate constant value, and filters the fiber angles using a Gaussian kernel of the same size and weight in each iteration to achieve the purpose of improving fiber continuity.

[0038] The cell density sensitivity is filtered using Gaussian filtering technology; the fiber angle is filtered using Gaussian function, which is a constraint method in a local sense. By introducing Gaussian kernel function to perform weighted correction on all cell fiber angles within the filter radius, the smoothing of the design variables is completed and the continuity of the fiber layout is guaranteed. The flexibility value of the optimization result and the curvature of the fiber placement path can be compared through a large number of numerical examples to adjust the most reasonable value range of the Gaussian filtering parameters and obtain the optimal parallel optimization result. First, the present invention starts from the topological optimization structure of the isotropic material and performs mean fuzzification on the cell density under this structure. Then, the blurred cell density is used as the initial value and given the properties of the orthotropic material to further perform parallel optimization of the structural topology and fiber angle of the composite material. In the parallel optimization, the fiber angle is filtered using Gaussian function, and the standard deviation of the Gaussian function is adjusted to a preset value to obtain a topological optimization structure of a composite material with better mechanical properties.

[0039] The beneficial effects of the present invention are:

[0040] 1) The present invention starts with the unit density of the isotropic material topology and performs mean fuzzification on this unit density. Once the fuzzy index is reached, it is used as the initial unit density value for the subsequent parallel optimization of the topology and fiber orientation. Finite element analysis is performed on the blurred topology to calculate the principal stress directions of all units, which are used as the initial values ​​of the fiber angles for the second optimization. This secondary optimization solves the initial dependence problem of the fiber angle. A reasonable initial fiber angle also significantly reduces the risk of falling into a local optimal solution. Mean fuzzification also correspondingly expands the design space, solving the problem of local optimality to a certain extent.

[0041] 2) In the parallel optimization of topology and fiber orientation, Gaussian filtering is performed on the sensitivity of unit density. That is, the traversal convolution calculation of all unit density sensitivities is performed through the Gaussian kernel. The influence of the central unit on the surrounding units is continuously corrected during the iterative update, thereby limiting the generation of meaningless grayscale units, greatly reducing the number of grayscale units, and enhancing the manufacturability of the structure.

[0042] 3) In the concurrent optimization of topology and fiber orientation, a Gaussian function was used to filter fiber angles and iteratively update the results. After convergence, the optimal composite topology was obtained. This significantly improved the continuity of fiber angles in the topology, alleviated stress concentration issues, and ensured the feasibility of subsequent printing and manufacturing.

[0043] The present invention performs fuzzy processing on the results of the first optimization and calculates the principal stress direction of each unit based on it, which reduces the risk of generating a local optimal solution while expanding the optimization space; on the other hand, in the parallel optimization of the topological structure and fiber angle, a numbering method suitable for anisotropic materials is adopted, and the unit density sensitivity and fiber angle are filtered respectively by Gaussian functions, which reduces the number of grayscale units, improves the continuity of the fiber angle, and improves the problem of stress concentration, thereby greatly improving the manufacturability of the composite material topological structure. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 is a flow chart of the method of the present invention;

[0045] Figure 2 Schematic diagram of the numbering of discrete elements, element nodes, and degrees of freedom for topology optimization of anisotropic materials, where: Figure 2 (a) is a schematic diagram of the overall node numbering for anisotropic material topology optimization. Figure 2 (b) is a schematic diagram of the node numbering of the anisotropic material topology optimization unit;

[0046] Figure 3 is a two-dimensional Gaussian function graph for fiber angle filtering, where Figure 3 (a) is the two-dimensional Gaussian function graph when σ is 1, Figure 3 (b) is the two-dimensional Gaussian function graph when σ is 2, Figure 3 (c) is the two-dimensional Gaussian function graph when σ is 3, Figure 3 (d) is the two-dimensional Gaussian function graph when σ is 4;

[0047] Figure 4 This is the topological structure diagram after the first optimization result is blurred and with the principal stress directions;

[0048] Figure 5 This is a comparison chart of four optimization results for the cantilever beam example;

[0049] Figure 6 A comparison chart of four optimization results for an MBB (Messerschmitt-Bolkow-Blohm) beam example;

[0050] Figure 7 This is a comparison chart of four optimization results for the L-beam example;

[0051] Figure 8 A local comparison of the two results for the cantilever beam example;

[0052] Figure 9 A local comparison of two results for the MBB beam example;

[0053] Figure 10 A local comparison of the two results for the L-beam example. DETAILED DESCRIPTION

[0054] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0055] The composite material secondary topology optimization method based on Gaussian function filtering of the present invention comprises:

[0056] 1) The solid orthotropic material penalty method SIMP is used to perform a first topology optimization on the topology structure of the isotropic material. The topology structure of the isotropic material after the first topology optimization is fuzzy processed to obtain the fuzzy unit density value of each grid unit. The topology structure of the fuzzy isotropic material is subjected to finite element analysis to obtain the principal stress direction of each grid unit as the fiber angle value. The topology structure of the isotropic material is then assigned orthotropic material properties to obtain the topology structure of the anisotropic material.

[0057] In the initial state, the cell density is set to a preset value, generally set to the same value, such as 0.3, 0.4, etc. The cell density is 0-1, "0" means no material, and "1" means material.

[0058] The solid orthotropic material penalty method SIMP specifically takes the unit density as the design variable and the flexibility minimization as the objective function. Under the given design domain and volume constraints, the design domain discretization, finite element analysis, sensitivity analysis, mesh unit filtering, and optimization criterion OC method are performed in sequence to optimize the design variables. The topological structure is finally obtained through iterative update and convergence.

[0059] The SIMP method is an interpolation model of the variable density method. The material interpolation model of the variable density theory transforms the discrete problem into a continuous optimization problem by introducing intermediate density units. However, intermediate density units cannot exist and cannot be manufactured. Therefore, it is necessary to avoid the generation of intermediate density units as much as possible and reduce the number of intermediate density units. At this time, the SIMP method is needed to penalize the intermediate density values ​​that appear in the design variables.

[0060] Orthotropic material properties include four independent elastic constants E1, E2, v 21 and G 12 , respectively represent the elastic modulus in direction 1, the elastic modulus in direction 2, the Poisson's ratio in direction 12, and the shear modulus in direction 12.

[0061] In step 1), the fuzzy processing is specifically to use mean fuzzy to process the cell density value of each grid cell of the topological structure of the isotropic material after the first topology optimization, as follows:

[0062]

[0063] Where i is the number of cells in the neighborhood of the central grid cell, j is the number of blurs, and x i,j and x i,j-1 are the cell densities of the grid cells before and after the jth blur, N e is the number of blurred grid cells, rmin is the blur radius, and n is the total number of grid cells.

[0064] In step 1), when the preset fuzzy index s of the mean fuzzy is reached, the mean fuzzy processing is completed to obtain the fuzzy unit density value; the preset fuzzy index s is specifically as follows:

[0065] s>2N·volfrac,

[0066]

[0067]

[0068] Where N is the number of mesh elements when the element density is greater than 0.5; volfrac is the preset volume fraction during the first topology optimization, and x e,j is the cell density value of the e-th grid cell after the j-th fuzzification. When the fuzzy index is reached, the fuzzy process is stopped and the new cell density is used as the input of the parallel optimization.

[0069] A 5x5 convolution kernel is used to apply mean blurring to the density of the first optimized topological structure. This involves sliding the convolution kernel to calculate the arithmetic mean of the discrete cell densities. Mean blurring, also known as normalized blurring, is a low-pass filter that applies arithmetic mean blurring to the image using a sliding convolution kernel. Its primary functions are to remove noise and blur the image.

[0070] In step 1), the cell density value of each grid cell of the topological structure of the isotropic material after fuzzy processing is the fuzzy cell density value, and the topological structure of the isotropic material after fuzzy processing is subjected to finite element analysis, specifically, each grid cell of the topological structure of the isotropic material after fuzzy processing and its fuzzy cell density value are subjected to finite element analysis to obtain the three components of stress value σ x , σ y , τ xy , according to the preset fiber angle range and the three components of the stress value of each grid unit, the principal stress direction of each grid unit is obtained as the fiber angle value.

[0071] 2) The improved solid orthotropic material penalty method SOMP with the addition of Gaussian function filtering method is used to perform a second topology parallel optimization on the topology structure of the anisotropic material. The Gaussian function filtering method is used to filter the cell density sensitivity and fiber angle of each grid cell of the topology structure of the anisotropic material. After the second topology parallel optimization, the topology structure of the anisotropic material after the second topology parallel optimization and the final optimized cell density value and optimized fiber angle value of each grid cell are finally obtained, completing the secondary topology optimization of the composite material.

[0072] In step 2), the improved density-based solid orthotropic material penalty method SOMP is specifically to add Gaussian function filtering method, optimization criterion method OC (Optimality Criteria) and moving asymptotic method MMA (Method of Moving Asymptotes) in sequence after the original solid orthotropic material penalty method SOMP.

[0073] In the parallel optimization of the structural topology and fiber angle of the composite material, the design domain is discretized by four-node elements, the design variables are element density and fiber angle, the objective function is to minimize the structural compliance, and the volume fraction is used as the constraint. A Gaussian filter is used to filter the element density sensitivity and fiber angle, and the standard deviation of the Gaussian function is adjusted to obtain a composite topology optimized structure with better mechanical properties and improve the continuity of the fiber layout.

[0074] In the second parallel optimization of topology and fiber angle, the design domain is discretized into four-point elements, and the elements, nodes, and degrees of freedom are numbered to facilitate the assembly of the overall stiffness matrix and the calculation of node displacements; the element stiffness matrix is ​​calculated and the overall stiffness matrix is ​​assembled; finite element analysis is performed; the filtering of design variables is completed, including mesh element filtering of element density sensitivity and Gaussian filtering of fiber angle; the design variables are updated to meet the objective function, and it is continuously iterated until convergence to obtain the parallel optimization results of the final composite material topology and fiber angle.

[0075] During the second topology parallel optimization, the original solid orthotropic material penalty method SOMP is used to optimize the fuzzy cell density value of each grid cell of the topological structure of the anisotropic material. Then, the Gaussian function filtering method is used to filter the fuzzy cell density sensitivity value of the optimized grid cell. The Gaussian kernel function is used to filter the sensitivity of the fuzzy cell density value of the grid cell, as follows:

[0076]

[0077] Where W1() is the first Gaussian kernel function of the sensitivity of the fuzzy cell density sensitivity value of the grid cell, r is the distance between the center of the grid cell being filtered and the surrounding grid cells; σ n is the standard deviation of the first Gaussian kernel function in the nth iteration; m is the preset moving step size, which is 1.01.

[0078] Then the sensitivity of the fuzzy cell density value of the filtered grid cell is normalized and calculated to obtain the sensitivity of the fuzzy cell density value of the final filtered grid cell, as follows:

[0079]

[0080] Where c is the overall topological flexibility, x' e is the fuzzy cell density value of the e-th grid cell, and are the sensitivities of the fuzzy cell density values ​​of the e-th grid cell before and after filtering, N e is the neighborhood set of the e-th grid cell.

[0081] Gaussian filtering is performed on the cell density sensitivity. This involves performing a convolution calculation on all cell density sensitivities using a Gaussian kernel to correct for the influence of the central cell on surrounding cells. The standard deviation of the Gaussian kernel function determines the kernel weight. When the weight is less than 0.05, the influence of this cell on the central cell is ignored. Therefore, the standard deviation is gradually increased with a given step size in each iteration to gradually reduce the Gaussian kernel size and increase the weight of the central cell, thereby limiting the generation of meaningless grayscale cells.

[0082] After filtering, the fuzzy cell density sensitivity value of the grid cell is optimized using the optimization criterion method OC until the change value of the fuzzy cell density value of the grid cell converges to obtain the final optimized cell density value.

[0083] During the second topology parallel optimization, the fiber angle value of each grid of the topological structure of the anisotropic material is first optimized using the original solid orthotropic material penalty method SOMP. Then, the fiber angle value of the optimized grid unit is filtered using the Gaussian function filtering method. The fiber angle value of each grid unit is traversed and convolved using a 3*3 Gaussian kernel function to obtain the second Gaussian kernel function W2 of the fiber angle value of the grid unit. Then, it is iterated. In each iteration, the second Gaussian kernel function W2 is used to filter the fiber angle value of each grid unit to update the fiber angle value. The fiber angle value of each grid unit is as follows:

[0084]

[0085] Among them, θ e and θ e are the fiber angle values ​​of the e-th grid unit before and after filtering; W2 is the second Gaussian kernel function of the fiber angle value of the grid unit; θ e-nelx-1 ,θ e-nelx ,θ e-nelx+1 ,θ e-1 ,θ e+1 ,θ e+nelx-1 ,θ e+nelx and θ e+nelx+1 They represent the fiber angle values ​​of the e-nelx-1, e-nelx, e-nelx+1, e-1, e+1, e+nelx-1, e+nelx and e+nelx+1 grid cells respectively, where nelx is the number of cells in the x-axis direction;

[0086] After filtering, the fiber angle values ​​of the grid cells are optimized using the moving average method (MMA) until the change in the fiber angle value of the grid cells converges to obtain the final optimized fiber angle value.

[0087] A 3*3 Gaussian kernel function is used to perform convolution calculations on the fiber angle values ​​of each grid cell. The standard deviation of the kernel function is 1.5. After the weights are calculated, they are normalized and the 9 weights are divided by 0.4787 to obtain the second Gaussian kernel W2 of the fiber angle value as follows:

[0088]

[0089] In the second optimization, the topology and fiber angles were optimized in parallel using cell density and fiber angle as design variables. In the present invention, the fiber angles were Gaussian filtered, i.e., all fiber angles were traversed and convolved using a Gaussian kernel to enhance the influence of the surrounding cells on the central cell. Considering that the purpose of Gaussian filtering is to increase the continuity of the fiber angles, the present invention sets the standard deviation of the Gaussian kernel function to an appropriate constant value, and filters the fiber angles using a Gaussian kernel of the same size and weight in each iteration to achieve the purpose of improving fiber continuity.

[0090] Gaussian filtering technology is used to filter the unit density sensitivity; Gaussian function is used to filter the fiber angle. This is a local constraint method. By introducing the Gaussian kernel function to perform weighted correction on the fiber angles of all units within the filtering radius, the design variables are smoothed and the continuity of the fiber layout is ensured. A large number of numerical examples can be used to compare the flexibility values ​​of the optimization results and the curvature of the fiber placement path, and the most reasonable value range of the Gaussian filtering parameters can be adjusted to obtain the optimal parallel optimization results.

[0091] like Figure 1 As shown, the method optimization process of the present invention is as follows:

[0092] In a two-dimensional plane, input material properties, define the unit density x and unit fiber angle θ, and use the density-based solid isotropic material method SIMP with penalty effect according to the given design domain, load distribution and volume constraint. Perform finite element analysis according to the equilibrium equation KU=F e ), calculate the sensitivity of cell flexibility to cell density dc / dx e , perform gradient filtering, and finally use the optimization criterion method to update the design variable x e , optimize the topological structure of the isotropic material and clarify the unit density distribution. Perform image processing on the unit density through mean fuzzification and set the fuzzy index of the mean fuzzification. When the index is reached, complete the image fuzzification and output the unit density. Perform finite element analysis on the blurred topological structure to calculate the principal stress directions of all units, and use the principal stress directions as the initial values ​​of the fiber angles for the second optimization. Construct a parallel optimization framework for the structural topology and fiber orientation of orthotropic materials, and use the unit density and principal stress directions that meet the fuzzy index as the initial values ​​for the parallel optimization.

[0093] In the parallel optimization of topology and fiber angle, the design domain is discretized into four-point units, and the units, nodes and degrees of freedom are numbered; the unit stiffness matrix is ​​calculated and the overall stiffness matrix K(x' e ,θ n+1 ) ; perform finite element analysis; calculate cell density sensitivity dc / dx' e and the sensitivity of the unit fiber angle dc / dθ e , complete the filtering of design variables, including grid filtering of unit density and Gaussian filtering of fiber angle; use OC method and MMA method to update the design variable x' respectively e ,θ e In order to satisfy the objective function, the algorithm is iterated continuously until the convergence condition change < 0.01 is reached, and the parallel optimization results of the topological structure and fiber angle of the final composite material are obtained.

[0094] like Figure 2 (a) and Figure 2 As shown in (b), the numbering method of elements, nodes and degrees of freedom after the design domain is discretized into four-point elements is introduced:

[0095] In the overall number, represents the element number, from x(1,1), x(1,2)...x(nely,1)...x(nely,nelx); n represents the node number, from node 1, node 2...node (nely+1)*(nelx+1); n represents the degree of freedom number in the x and y directions of each node, from 2*1-1, 2*1, to 2*(nely+1)*(nelx+1)-1, 2*(nely+1)*(nelx+1).

[0096] The four nodes in the unit are numbered counterclockwise. To avoid complexity, one unit defines two nodes as n1 = (nely + 1) * (nelx - elx) + ely, and n2 = (nely + 1) * (nelx - elx + 1) + ely. Therefore, the coordinates on one unit are defined as: edof = [2*n1-1; 2*n1; 2*n2-1; 2*n2; 2*n2+1; 2*n2+2; 2*n1+1; 2*n1+2]. Figure 3 As shown, it is a two-dimensional Gaussian function image: when σ is 1, 2, 3, and 4 respectively, the curve is as follows Figure 3 (a) Figure 3 (b) Figure 3 (c) and Figure 3After repeated calculations, it was found that when σ is 1.5, the fiber angle filtering effect in the parallel optimization is the best. Then, the weights of the 3*3 Gaussian kernel function are normalized. The 9 weights need to be divided by 0.4787 to obtain the final Gaussian kernel.

[0097] like Figure 4 As shown in the figure, after the first optimization, the unit density in the optimization result is mean-fuzzified. After the fuzzy index is reached, the topological structure expressed by the blurred unit density is subjected to finite element analysis, the principal stress direction of each unit is calculated, and the blurred unit density and principal stress direction are used as the initial values ​​for the second optimization.

[0098] like Figure 5 As shown in the figure, the isotropic material topology structure, the isotropic material topology mean fuzzy structure, the composite material secondary parallel topology optimization structure based on Gaussian filtering (GS-Aniso_TO, Topological Optimization of Anisotropic Materials with Gaussian Filtering), and the composite material topology structure with 0° initial fiber (Aniso_TO, Topological Optimization of Anisotropic Materials) at different volume fractions (vol = 30%, 40%, 50%, 60%, and 70%) of the cantilever beam example are compared. Through analysis of the compliance data, the compliance of the GS-Aniso_TO result is 1% to 2% higher than the compliance c of the Aniso_TO result, and the overall compliance is similar, but the fiber continuity is significantly improved, which also demonstrates the feasibility of parallel optimization using Gaussian function filtering.

[0099] Figure 6 Comparisons were made for an MBB beam example using different volume fractions (vol = 30%, 40%, 50%, 60%, and 70%) using isotropic material topology, a mean fuzzy isotropic material topology, a Gaussian-filtered composite quadratic parallel topology optimization structure (GS-Aniso_TO), and an Aniso_TO composite topology structure with 0° initial fibers. Analysis of the flexibility data revealed that the flexibility of the GS-Aniso_TO result was approximately 10% higher than the flexibility c of the Aniso_TO result, while generally similar. However, the fiber continuity was significantly improved, and the number of grayscale elements was significantly reduced, demonstrating the feasibility of parallel optimization using Gaussian filtering.

[0100] Figure 7Comparisons were made for an L-beam example using different volume fractions (vol = 30%, 40%, 50%, 60%, and 70%) using isotropic material topology, a mean fuzzy isotropic material topology, a Gaussian-filtered composite quadratic parallel topology optimization structure (GS-Aniso_TO), and an Aniso_TO composite topology structure with 0° initial fibers. Analysis of the flexibility data revealed that the flexibility of the GS-Aniso_TO results was 2% to 8% higher than that of the Aniso_TO results, while generally similar. However, the fiber continuity was significantly improved, and the number of grayscale elements was significantly reduced, demonstrating the feasibility of parallel optimization using Gaussian filtering.

[0101] like Figure 8 、 Figure 9 and Figure 10 The figure shows a schematic diagram of the local structure comparison between the GS-Aniso_TO structure and the Aniso_TO structure, which shows the topology optimization results of the composite material based on Gaussian function filtering and the composite material topology optimization results of the 0° initial fiber. It more intuitively demonstrates the continuity of the fiber angle planning of the GS-Aniso_TO structure, which not only avoids the problem of structural stress concentration, but also improves the manufacturability and provides conditions for subsequent 3D printing manufacturing.

Claims

1. A composite material secondary topology optimization method based on Gaussian function filtering, characterized in that: include: 1) The solid orthotropic material penalty method SIMP is used to perform the first topology optimization on the topology structure of the isotropic material. The topology structure of the isotropic material after the first topology optimization is fuzzy processed to obtain the fuzzy cell density value of each grid cell; The topological structure of the isotropic material after fuzzy processing is subjected to finite element analysis to obtain the principal stress direction of each grid unit as the fiber angle value; Then the topological structure of the isotropic material is assigned the properties of the orthotropic material to obtain the topological structure of the anisotropic material; 2) A second topology parallel optimization of the anisotropic material topology structure is performed using the improved solid orthotropic material penalty method SOMP with the addition of a Gaussian function filtering method. The Gaussian function filtering method is used to filter the cell density sensitivity and fiber angle of each grid cell of the anisotropic material topology structure. After the second topology parallel optimization, the topology structure of the anisotropic material and the final optimized cell density value and optimized fiber angle value of each grid cell after the second topology parallel optimization are finally obtained, completing the secondary topology optimization of the composite material. During the second topology parallel optimization, the fuzzy cell density value of each grid cell of the topological structure of the anisotropic material is optimized by first using the original solid orthotropic material penalty method SOMP for the fuzzy cell density value of each grid cell, and then using the Gaussian function filtering method to filter the fuzzy cell density sensitivity value of the optimized grid cell; During the second topology parallel optimization, the fiber angle value of each grid of the topological structure of the anisotropic material is first optimized using the original solid orthotropic material penalty method SOMP, and then the fiber angle value of the optimized grid unit is filtered using the Gaussian function filtering method. The fiber angle value of each grid unit is traversed and convolved using a 3x3 Gaussian kernel function to obtain the second Gaussian kernel function of the fiber angle value of the grid unit, and then iterative processing is performed. In each iteration, the fiber angle value of each grid unit is filtered using the second Gaussian kernel function to update the fiber angle value.

2. The composite material secondary topology optimization method based on Gaussian function filtering according to claim 1, characterized in that: In the step 1), the fuzzy processing is specifically to use mean fuzzy to process the cell density value of each grid cell of the topological structure of the isotropic material after the first topology optimization, as follows: in, is the number of cells in the neighborhood of the central grid cell, is the number of blurs, and are the cell densities of the grid cells before and after the jth blur, is the number of blurred grid cells, is the blur radius, and n is the total number of grid cells.

3. The composite material secondary topology optimization method based on Gaussian function filtering according to claim 1, characterized in that: In the step 1), when the preset fuzzy index s of the mean fuzzy is reached, the mean fuzzy processing is completed to obtain the fuzzy unit density value; the preset fuzzy index s is specifically as follows: Where N is the number of mesh elements when the element density is greater than 0.5; volfrac is the preset volume fraction during the first topology optimization. For the After the second blur The cell density value of each grid cell.

4. The composite material secondary topology optimization method based on Gaussian function filtering according to claim 1, characterized in that: In the step 1), the unit density value of each grid unit of the topological structure of the isotropic material after fuzzy processing is a fuzzy unit density value, and the topological structure of the isotropic material after fuzzy processing is subjected to finite element analysis. Specifically, each grid unit of the topological structure of the isotropic material after fuzzy processing and its fuzzy unit density value are subjected to finite element analysis to obtain three components of stress value, and the principal stress direction of each grid unit is obtained as the fiber angle value according to the preset fiber angle range and the three components of the stress value of each grid unit.

5. The composite material secondary topology optimization method based on Gaussian function filtering according to claim 1, characterized in that: In the step 2), the improved solid orthotropic material penalty method SOMP specifically adds a Gaussian function filtering method before the optimization criterion method OC and the moving asymptotic method MMA in the original solid orthotropic material penalty method SOMP.

6. The composite material secondary topology optimization method based on Gaussian function filtering according to claim 5, characterized in that: The sensitivity of the fuzzy cell density value of the grid cell is filtered using the Gaussian kernel function as follows: in, is the first Gaussian kernel function of the sensitivity of the fuzzy cell density value of the grid cell, is the distance between the center of the grid cell being filtered and the surrounding grid cells; is the standard deviation of the first Gaussian kernel function at the nth iteration; is the preset moving step length; Then the sensitivity of the fuzzy cell density value of the filtered grid cell is normalized and calculated to obtain the sensitivity of the fuzzy cell density value of the final filtered grid cell, as follows: Where c is the overall topological flexibility, is the fuzzy cell density value of the e-th grid cell, and are the sensitivities of the fuzzy cell density value of the e-th grid cell before and after filtering, is the neighborhood set of the e-th grid cell; After filtering, the fuzzy cell density sensitivity value of the grid cell is optimized using the optimization criterion method OC until the change value of the fuzzy cell density value of the grid cell converges to obtain the final optimized cell density value.

7. The composite material secondary topology optimization method based on Gaussian function filtering according to claim 5, characterized in that: The fiber angle values ​​of each grid unit are as follows: in, and are the fiber angle values ​​of the e-th grid cell before and after filtering, respectively; is the second Gaussian kernel function of the fiber angle value of the grid cell; 、 、 、 、 、 、 and They represent the fiber angle values ​​of the e-nelx-1, e-nelx, e-nelx+1, e-1, e+1, e+nelx-1, e+nelx and e+nelx+1 grid cells respectively, where nelx is the number of cells in the x-axis direction; After the fiber angle values ​​of the grid cells are filtered, they are optimized using the moving average method (MMA) until the changes in the fiber angle values ​​of the grid cells converge to obtain the final optimized fiber angle values.

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