A multi-directional equal cross-section feature parallel driving three-dimensional structure topology optimization method

Abstract

The present application belongs to the technical field of structural optimization design, and discloses a three-dimensional structure topology optimization method of multi-directional equal-section characteristic parallel driving, which comprises the following steps: (1) for each equal-section tensile or scanning direction of the structure to be optimized, unidirectional design variables are defined in the corresponding two-dimensional section in a uniform distribution, and the unidirectional design variables are sequentially subjected to density filtering and step projection through a grouping dimension reduction cylindrical filter to obtain unidirectional physical variables; (2) a multi-material interpolation mode is adopted to calculate the synthesized physical variables and synthesized elastic variables of the multi-directional equal-section parallel tensile or scanning; (3) the objective function and constraint function of the topology optimization model of the structure to be optimized and the sensitivity of the relative design variables are calculated, and then a gradient-based optimization algorithm is adopted to solve the optimal topology configuration of the multi-directional equal-section characteristic parallel driving. The present application realizes the collaborative optimization of multi-directional equal-section characteristic parallel tensile or scanning.

Description

Technical Field

[0001] This invention belongs to the technical field of structural optimization design, and more specifically, relates to a three-dimensional structural topology optimization method driven by parallel multi-directional equal cross-sectional features. Background Technology

[0002] As a novel structural optimization design method with high design freedom and independent of human experience, topology optimization can generate novel configurations with optimal material distribution under given working conditions or manufacturing constraints, driven by the performance of multiple physical fields such as mechanics, thermodynamics, acoustics, and optics. It has become one of the most promising advanced structural optimization technologies and has been widely applied in aerospace, marine, transportation, engineering construction, and biomedicine. In recent years, various topology optimization methods have emerged to adapt to different manufacturing or construction process requirements. However, existing topology optimization methods considering manufacturing process constraints mostly focus on emerging additive manufacturing technologies, but their optimization results are often too complex, making it difficult to achieve rapid and accurate geometric reconstruction, which has become a key bottleneck restricting their practical application. Furthermore, research on topology optimization methods for traditional manufacturing processes such as casting, extrusion, rolling, and machining is relatively scarce. Currently, many structures in the manufacturing and construction industries are assembled from profiles, belonging to combinations of multi-directional equal-section features. However, existing extrusion structure topology optimization methods can only achieve equal-section features in a single direction.

[0003] In the field of 3D structural optimization, the mainstream approach for topology optimization design problems with structures featuring uniform cross-sections is currently based on extrusion constraints. However, publicly available solutions are scarce. Early on, scholars like M. Zhou proposed an extrusion structure topology optimization method based on equality constraints and embedded it in the commercial software OptiStruct. However, this method relies on the assumption of a structured mesh and cannot directly adapt to topology optimization scenarios with unstructured meshes. Furthermore, it lacks improvements in density filtering and projection mechanisms, potentially leading to numerical oscillations and blurred boundaries. Another approach, proposed by Hao Li et al., is an extrusion structure topology optimization method based on parametric level sets. This method reduces the dimensionality of 3D optimization to 2D through cross-section projection techniques, improving efficiency and generating smooth geometric boundaries, but it is also limited by mesh type. Bo Wang et al.'s Helmholtz anisotropic filtering method is compatible with structural design problems based on both structured and unstructured meshes, but it cannot reduce the dimensionality of design variables, resulting in high computational costs. It is particularly noteworthy that existing technologies all focus on controlling uniform cross-section features in a single direction and have not yet achieved collaborative optimization through multi-directional parallel stretching or sweeping. However, in engineering practice, there are numerous combinations of multi-directional, uniform cross-section structures, such as building frames, bridge trusses, and furniture frame structures in the construction industry, as well as container housings, aircraft skeletons, and vehicle / ship hulls in the manufacturing industry. Traditional methods can only achieve uniform cross-section characteristics by controlling a single direction step by step. At the intersection of multi-directional materials, problems such as material conflicts and performance coupling failures can easily occur, making it difficult to meet the comprehensive requirements of engineering practice for lightweight structures, efficient manufacturing, and reliable performance. Summary of the Invention

[0004] To address the aforementioned deficiencies or improvement needs of existing technologies, this invention provides a three-dimensional structure topology optimization method driven by parallel multi-directional equal cross-section features. This method aims to solve the problem that existing topology optimization methods cannot achieve collaborative optimization of parallel stretching or sweeping of multi-directional equal cross-section features.

[0005] To achieve the above objectives, according to one aspect of the present invention, a method for topology optimization of three-dimensional structures driven by parallel multi-directional equal cross-sectional features is provided, the method comprising the following steps:

[0006] (1) For each equal cross-section stretching or sweeping direction of the structure to be optimized, a uniformly distributed unidirectional design variable is defined in the corresponding two-dimensional cross-section. The unidirectional design variable is then subjected to density filtering and step projection by a grouped dimension reduction cylindrical filter to obtain the unidirectional physical variable associated with the finite element mesh of the structure to be optimized.

[0007] (2) Based on the obtained unidirectional physical variables, a multi-material interpolation mode is used to calculate the composite physical variables and composite elastic variables of multi-directional equal cross-section parallel stretching or sweeping; wherein, the multi-material interpolation mode regards each directional physical variable as the probability of the corresponding mesh cell being filled with material;

[0008] (3) Based on the synthetic physical variables and synthetic elastic variables, calculate the sensitivity of the objective function and constraint function of the topology optimization model of the structure to be optimized relative to the design variables. Then, use the gradient optimization algorithm to iteratively solve the topology optimization model to obtain the optimal topology configuration driven by the parallel multi-directional equal cross-section features.

[0009] Furthermore, when performing density filtering with the grouped dimension reduction cylindrical filter, the axis of the grouped dimension reduction cylindrical filter is parallel to the stretching direction or the sweeping direction. All design variables on the same stretching path or sweep will always maintain the same value during the optimization process, and will be treated as a group and uniformly represented by a single design variable.

[0010] Furthermore, the formula corresponding to density filtering is:

[0011]

[0012] In the formula, To design a variable array, For density variable arrays, From Transformation from design variable matrix to density variable matrix 3D matrix express elements, For the first The effect of the design variable on the first Weighting coefficients on each density variable They represent the smallest cuboid space enclosing the three-dimensional design domain, respectively. The number of layers to which design variables are evenly distributed. This represents the total number of density variables corresponding to each grid cell. Indicates the first Unit Center coordinates and coordinate, Indicates the first Design variables in the design domain space coordinates and coordinate, For the filter radius, express and The distance between them.

[0013] Furthermore, the formula corresponding to the multi-material interpolation mode is:

[0014]

[0015] In the formula, They represent The first in the stretch structure space The physical variables corresponding to each grid cell are obtained by step projection of the corresponding density variables, and their values ​​tend to be 0 or 1.

[0016] Furthermore, the formula corresponding to the composite physical variable array is:

[0017]

[0018] After further simplification, we get:

[0019]

[0020] In the formula, use They represent The array of physical variables corresponding to the stretched structural mesh elements.

[0021] Furthermore, the formula corresponding to the composite elastic variable matrix is:

[0022]

[0023] In the formula, This represents the material penalty factor, which is an integer not less than 1; here, we take... ; They represent The elastic modulus of the tensile material; This represents the elastic modulus value corresponding to the porous element.

[0024] Furthermore, by increasing the penalty for the air phase material, the formula corresponding to the synthetic elastic variable matrix is:

[0025] .

[0026] Furthermore, the constraint function is a global volume fraction constraint.

[0027] Furthermore, the mathematical expression for the topology optimization model is:

[0028]

[0029] In the formula, They represent Design variable array for constant cross-section stretching, They represent Design variables for tensioning a uniform cross section. They represent the smallest cuboid space enclosing the three-dimensional design domain, respectively. The number of layers to which design variables are evenly distributed. Describe the objective function. Representing the finite element equilibrium equations, This represents the global volume fraction constraint function. The total volume of the design domain. This represents the global volume fraction constraint value for the design domain.

[0030] Furthermore, step (1) includes a step prior to which the low-density cells on the periphery of the initial design domain of the structure to be optimized are removed by conventional topology optimization results.

[0031] In summary, compared with the prior art, the three-dimensional structure topology optimization method driven by multi-directional equal cross-section features provided by the present invention has the following beneficial effects:

[0032] 1. Based on the obtained unidirectional physical variables, a multi-material interpolation model is used to calculate the synthetic physical variables and synthetic elastic variables of multi-directional equal-section parallel stretching or sweeping. Then, based on the synthetic physical variables and synthetic elastic variables, the sensitivity of the objective function and constraint function of the topology optimization model of the structure to be optimized relative to the design variables is calculated. Then, a gradient-type optimization algorithm is used to iteratively solve the topology optimization model to obtain the optimal topology driven by multi-directional equal-section features in parallel. This can realize the structural optimization driven by any or multi-directional equal-section features, including structures obtained by equal-section stretching or equal-section sweeping, which is flexible and diverse in use.

[0033] 2. Unidirectional design variables are sequentially subjected to density filtering and step projection by grouped dimension-reducing cylindrical filters. When performing density filtering by the grouped dimension-reducing cylindrical filters, the axis of the grouped dimension-reducing cylindrical filters is parallel to the stretching or sweeping direction. All design variables on the same stretching path or sweep will always maintain the same value during the optimization process, and are treated as a group and uniformly represented by a single design variable. This transforms the three-dimensional topology optimization problem into a two-dimensional cross-section optimization problem, significantly reducing the number of design variables and making the operation highly efficient.

[0034] 3. The topology optimization method does not require specifying the volume fraction of a single-direction uniform cross-section stretched or swept material. It can obtain optimization results under a global volume fraction constraint, reducing the number of constraint functions and making the method simple and efficient.

[0035] 4. Step (1) includes the step of removing the low-density elements on the periphery of the initial design domain of the structure to be optimized by the traditional topology optimization results. This way, only the necessary part inside is retained as the design domain, avoiding the removal of all low-density elements, so that the space for optimization is not too small, which is convenient for equal section stretching or sweeping, and at the same time makes up for the performance loss caused by the equal section feature.

[0036] 5. Increase the penalty for unused phase materials to ensure the convexity of the optimization problem and reduce the dependence on the initial solution.

[0037] 6. By synthesizing physical variables and elastic variables, problems such as material conflicts and difficulty in coupling properties caused by parallel tensioning or sweeping of isotropic cross-sections can be solved. Attached Figure Description

[0038] Figure 1 This is a flowchart of a three-dimensional structure topology optimization method driven by parallel multi-directional equal cross-section features provided by the present invention;

[0039] Figure 2 This is a schematic diagram of a three-dimensional cylindrical filter provided in an embodiment of the present invention;

[0040] Figure 3 This is a schematic diagram of parallel stretching or sweeping of multi-directional equal cross-section features provided in an embodiment of the present invention;

[0041] Figure 4 This is a schematic diagram of the material intersection under multi-directional equal cross-section parallel tension or sweep provided in the embodiments of the present invention;

[0042] Figure 5 This is a schematic diagram of the three-dimensional cantilever beam design domain model provided in an embodiment of the present invention;

[0043] Figure 6 This is a schematic diagram of the topology optimization results driven by unidirectional equal cross-section features under three-dimensional cylindrical filtering or grouped dimensionality reduction filtering provided in the embodiments of the present invention;

[0044] Figure 7 This is a schematic diagram comparing the operating efficiency of three-dimensional cylindrical filtering and grouped dimensionality reduction filtering. Among them, (a) is a comparison of the time consumption for filter parameter calculation; (b) is a comparison of the time consumption for sensitivity analysis; (c) is a comparison of the time consumption for updating design variables; and (d) is a comparison of the time consumption for topology optimization iteration.

[0045] Figure 8 These are schematic diagrams of topology optimization results driven by unidirectional equal cross-section features; (a) is the traditional topology optimization result; (b) is the topology optimization result driven by equal cross-section features in the z-direction; (c) is the topology optimization result driven by equal cross-section features in the y-direction; and (d) is the topology optimization result driven by equal cross-section features in the x-direction.

[0046] Figure 9 This is a schematic diagram of the topology optimization results driven by parallel cross-sectional features in the z and y directions;

[0047] Figure 10 This is a schematic diagram of the topology optimization results driven by parallel cross-sectional features in the z and x directions;

[0048] Figure 11 This is a schematic diagram of the topology optimization results driven by the parallel cross-sectional features in the y and x directions;

[0049] Figure 12 This is a schematic diagram of the topology optimization results driven by parallel processing of three-dimensional equal cross-section features;

[0050] Figure 13 This is a schematic diagram of the domain reduction result from topology optimization design;

[0051] Figure 14 This is a schematic diagram of the topology optimization results driven by the parallel cross-sectional features in the z and y directions after the design domain is reduced;

[0052] Figure 15 This is a schematic diagram of the topology optimization results driven by the parallel driving of three-dimensional equal cross-section features after the design domain is reduced. Detailed Implementation

[0053] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be 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 illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0054] This invention provides a three-dimensional structure topology optimization method driven by parallel multi-directional isosectional features. For each isosectional stretching or sweeping direction, the method defines uniformly distributed unidirectional design variables within the corresponding two-dimensional cross-section. These variables are then filtered using a grouped dimensionality reduction filter and subjected to step projection to obtain the unidirectional physical variables associated with the finite element mesh. The proposed multi-material interpolation model is used to calculate the composite physical and elastic variables of the multi-directional parallel stretching or sweeping. Furthermore, traditional optimization results are used to remove low-density elements on the periphery of the design domain, compensating for the performance loss caused by the isosectional features. This topology optimization method exhibits good versatility, strong adaptability, high efficiency, and convenient geometric reconstruction.

[0055] This invention pioneers a multi-directional equal-section feature parallel-driven 3D structure topology optimization method, which is equivalent to applying equal-section (extrusion) constraints in multiple directions simultaneously, breaking through the design limitations of the original single-directional equal-section (extrusion) structure topology optimization method. This topology optimization method is applicable to design domains with structured or unstructured mesh partitioning, and can generate configurations with equal-section features without adding special constraint terms to the topology optimization model. A grouped dimensionality reduction cylindrical filter is used to transform 3D structure optimization into 2D section optimization, significantly improving algorithm efficiency and facilitating the application of advanced 2D topology optimization and geometric reconstruction methods. Inspired by discrete independent event probability models, a multi-material interpolation mode effectively solves problems such as material cross-conflict and performance coupling difficulties in multi-directional equal-section parallel stretching or sweeping, and together with the grouped dimensionality reduction filter method, constructs a multi-directional equal-section feature parallel-driven topology optimization method. Furthermore, utilizing the results obtained from traditional topology optimization methods without equal-section feature driving, a design domain reduction strategy that excludes peripheral low-density elements is introduced, further improving the performance of the topology configurations generated by this method. This invention establishes explicit and differentiable expressions for the optimization objective function and constraint function through grouped dimension reduction cylindrical filters, multi-material interpolation modes, and design domain reduction strategies. It also derives the corresponding sensitivity formulas and substitutes them into gradient-based optimization algorithms for solving. This enables the efficient one-time generation of topology optimization results under the parallel driving of multi-directional equal cross-sectional features, which can be achieved under global volume constraints.

[0056] Please see Figure 1 The topology optimization method mainly includes the following steps:

[0057] Step 1: For each uniform cross-section of the structure to be optimized, stretching or sweeping directions, define uniformly distributed unidirectional design variables in the corresponding two-dimensional cross-section. The unidirectional design variables are then subjected to density filtering and step projection through a grouped dimension reduction cylindrical filter to obtain unidirectional physical variables associated with the finite element mesh of the structure to be optimized.

[0058] Specifically, firstly, for each constant cross-section stretching or sweeping direction of the structure to be optimized, uniformly distributed unidirectional design variables are defined within the smallest rectangular space enclosing the three-dimensional design domain of the structure to be optimized, for each constant cross-section stretching or sweeping direction. Taking constant cross-section stretching in the x-direction as an example, a matrix of design variables is defined. , containing elements ,in, They represent The design variable layers for the direction are determined. Then, a finite element mesh is generated for the geometric model of the structure to be optimized, and density variable arrays corresponding to each mesh element are defined. ,Include element .

[0059] Design variable matrix Density filtering is performed to obtain the density variable matrix. The corresponding formula is:

[0060] (1)

[0061] In the formula, To transform from a design variable matrix to a density variable matrix 3D matrix; express Elements in; For the first The effect of the design variable on the first Weighting coefficients on density variables; Indicates the first The center of each grid cell coordinates and coordinate, Indicates the first Design variables in the design domain space coordinates and coordinate, For the filter radius, express and The distance between them.

[0062] Please see Figure 2 The cylindrical filter's axis is parallel to the stretching direction. Through the action of this cylindrical filter, all design variables along the same stretching path will maintain the same value throughout the optimization process, thus producing a uniform cross-sectional feature in that direction in the topology optimization result. Furthermore, the center coordinates of the mesh cells do not need to coincide with the spatial coordinates of the design variables; that is, a one-to-one correspondence between mesh cells and design variables is not required. Therefore, this method is applicable to topology optimization problems with both structured and unstructured meshes.

[0063] Furthermore, due to along For uniform cross-section tension, the design variables along the same tension path satisfy formula (2), and their values ​​are the same. Therefore, they can be grouped together and a single design variable can be used uniformly. express:

[0064] (2)

[0065] At this point, the aforementioned three-dimensional cylindrical filter is converted into a grouped, dimension-reduced cylindrical filter, and the variable matrix is ​​designed. The number of elements in the data is from One reduced to This is equivalent to defining design variables only within the two-dimensional cross-section to be stretched or swept, thus significantly improving the running efficiency of the optimization algorithm. Furthermore, it is also conducive to combining with various advanced two-dimensional topology optimization methods and is easy to perform geometric reconstruction based on the topology optimization results. The specific filtering method is shown in formula (3):

[0066] (3)

[0067] In the formula, To design a variable array, For density variable arrays, From Transformation from design variable matrix to density variable matrix 3D matrix express elements, For the first The effect of the design variable on the first Weighting coefficients on each density variable Indicates the first Unit Center coordinates and coordinate, Indicates the first Design variables in the design domain space coordinates and coordinate, For the filter radius, express and The distance between them.

[0068] Then, based on the Heaviside step function, the density variable array is... Perform a step projection to obtain the physical variable matrix. The corresponding formula is:

[0069] (4)

[0070] In the formula, and These represent the cutoff threshold and steepness of the step projection function, respectively. Through step projection calculations, the physical variable matrix... The elements in the grid gradually approach 0 or 1, thus more realistically reflecting the material filling rate in the grid cells under actual scenarios.

[0071] Step 2: Based on the obtained unidirectional physical variables, a multi-material interpolation mode is used to calculate the composite physical variables and composite elastic variables of multi-directional equal-section parallel stretching or sweeping; wherein, the multi-material interpolation mode regards each directional physical variable as the probability of the corresponding mesh cell being fully filled with material.

[0072] Please see Figure 3 Multi-directional, uniformly stretched or swept structures will intersect in space, with... Taking the parallel tensioning of three-dimensional uniform cross-sections as an example, there will be pairwise intersections or simultaneous intersections of all three directions. Let X, Y, and Z represent the sets of elements included in the simultaneous tensioning or sweeping of three-dimensional uniform cross-sections, and the intersection situations are as follows: Figure 4 As shown.

[0073] If the traditional multi-material interpolation method is used... There is mutual exclusivity among materials in triaxial tension; tension in one direction affects the continuity and symmetry of tension in another direction. In contrast, in multiaxial constant-section tension structures, the materials in each direction are usually the same, and they are stretched independently. Even if they intersect, it is necessary for optimizing structural performance, and mutual exclusivity should be eliminated. Therefore, to make parallel constant-section tension in each direction harmonious, the physical variables in each direction are considered as the probability of the corresponding mesh element being completely filled (the value for complete filling is 1). Thus, formula (5) represents the material filling of a mesh element in space in the stretching or sweeping direction:

[0074] (5)

[0075] In the formula, They represent The first in the stretch structure space The physical variables corresponding to each grid cell.

[0076] use They represent If we extend the physical variable matrix corresponding to the stretched structural mesh element, then during the topology optimization iteration process, the composite physical variable matrix corresponding to the mesh element will be... This can be expressed by formula (6):

[0077] (6)

[0078] Formula (6) can be simplified to obtain formula (7):

[0079] (7)

[0080] Formula (7) perfectly reflects the probability of a cell being filled with material under simultaneous triaxial stretching, which is consistent with the probability model in form. Based on formula (7), a simple multi-material interpolation mode is derived, that is, the composite elastic variable matrix corresponding to the mesh cell is obtained through formula (8). :

[0081] (8)

[0082] In the formula, This represents the material penalty factor, which is an integer not less than 1; here, we take... ; They represent The elastic modulus of a tensile material; The elastic modulus value corresponding to the porous element is much smaller than... The smallest positive number.

[0083] To ensure the convexity of the optimization problem and alleviate the dependence on the initial solution, referring to the form of formula (6), the penalty for the empty phase material is increased, and the improved multi-material interpolation mode is shown in formula (9):

[0084] (9)

[0085] Formula (9) is a general expression. For most scenarios, the materials in triaxial tension or sweep are the same, so the elastic modulus in each direction can be made equal, that is, let Formula (10) can be obtained:

[0086] (10)

[0087] Unlike topology optimization methods driven by unidirectional isotropic cross-section features, topology optimization methods driven by multidirectional isotropic cross-section features rely on synthetic physical variables or synthetic elastic variables to describe the physical performance indices of the elements. Synthetic physical variables primarily reflect the probability of material filling in the mesh elements and are used to calculate the total volume or mass of the solid structure. Synthetic elastic variables, interpolated from the isotropic physical variables, cannot be directly calculated from them. By using synthetic physical variables and synthetic elastic variables, problems such as material conflicts and performance coupling difficulties caused by parallel stretching or sweeping of isotropic cross-sections can be resolved.

[0088] Step 3: Calculate the objective function and constraint function of the topology optimization model of the structure to be optimized, as well as their sensitivity relative to the design variables, based on the synthetic physical variables and synthetic elastic variables. Then, use a gradient-based optimization algorithm to iteratively solve the topology optimization model to obtain the optimal topology configuration driven by the parallel multi-directional equal cross-section features.

[0089] Specifically, for problems such as minimizing the overall structural flexibility (minimizing the overall physical performance by accumulating the physical performance of individual elements), according to the chain rule, the objective function is... Compared to The sensitivity to the design variable can be calculated using formula (11):

[0090] (11)

[0091] objective function relative to The sensitivity to physical variables can be calculated using formula (12):

[0092] (12)

[0093] The sensitivity of the mesh element compliance relative to the interpolated element elastic modulus, obtained by the adjoint method, can be calculated using formula (13):

[0094] (13)

[0095] In the formula, , , They represent the first The elastic modulus, standard stiffness matrix, and nodal displacement array of each mesh element.

[0096] Synthetic elasticity variables relative to The sensitivity to the physical variable is calculated using formula (14):

[0097] (14)

[0098] For the objective function or constraint function using p-norm aggregation, its relative to The sensitivity to the physical variable is calculated using formula (15):

[0099] (15)

[0100] In the formula, It is the physical performance matrix corresponding to the mesh cell, and the objective function or constraint function is relative to the physical performance of that cell. The sensitivity is calculated using formula (16):

[0101] (16)

[0102] Physical properties are typically functions of the synthesized physical variables and synthesized elastic variables of the mesh elements, and are relative to... The sensitivity to the physical variable is calculated using formula (17):

[0103] (17)

[0104] Structural global volume fraction constraints relatively The sensitivity to the physical variable is calculated using formula (18):

[0105] (18)

[0106] In the formula, the global volume fraction is relative to The sensitivity to the physical variable is calculated using formula (19):

[0107] (19)

[0108] In the formula, For the volume array of the grid cells, This represents the total volume of the design domain.

[0109] In a uniform manner, the composite physical variables are relative to The sensitivity to the physical variable is calculated using formula (20):

[0110] (20)

[0111] In a unified manner, relative to physical variables The sensitivity of the filtered variable is calculated using formula (21):

[0112] (twenty one)

[0113] In a unified manner, relative to filter variables The sensitivity to the design variable is calculated using formula (22):

[0114] (twenty two)

[0115] Similarly, following the steps above, it is easy to obtain the relative values ​​of the objective function and the global volume fraction constraint. Towards, Sensitivity of design variables under constant cross-section stretching.

[0116] Finally, with the desired physical performance as the optimization objective and the global volume fraction as the constraint, the following topology optimization model is established, and the mathematical expression of the topology optimization model is shown in formula (23):

[0117] (twenty three)

[0118] In the formula, They represent Design variable array for constant cross-section stretching, They represent Design variables for tensioning a uniform cross section. They represent the smallest cuboid space enclosing the three-dimensional design domain, respectively. The number of layers to which design variables are evenly distributed. Describe the objective function. Representing the finite element equilibrium equations, This represents the global volume fraction constraint function. The total volume of the design domain. This represents the global volume fraction constraint value for the design domain.

[0119] Furthermore, gradient-based algorithms such as the objective function, global volume constraint function, and their sensitivity are substituted into the optimization criteria (OC) and moving asymptote (MMA), combined with finite element analysis, to drive the update of design variables until the convergence condition or the maximum number of iterations is reached. After necessary post-processing, the final topology optimization result driven by the parallel multi-directional equal cross-section features is output.

[0120] For directions where uniform cross-section stretching is not required, simply force the corresponding design variables and sensitivity to 0 during topology optimization. Additionally, it should be noted that although the above implementation uses a uniform cross-section structure stretched along a straight line as an example, this method is fully applicable to uniform cross-section structures swept along a curve; the only difference lies in the minor differences in the grouped filtering calculation.

[0121] As can be seen, this multi-directional, equal-section feature-driven topology optimization method does not require the introduction of additional constraints and only needs a single global volume fraction constraint. Furthermore, the use of a grouped dimensionality reduction filter significantly reduces computational costs. Both the objective function and constraint function are explicit and differentiable, making them easy to solve and integrate into various structural optimization software. Moreover, as a geometry-based feature-driven method, it is applicable to various topology optimization methods such as variable density and level set optimization, as well as a variety of physical field problems.

[0122] Preferably, topology optimization design is performed after removing the low-density cells on the periphery of the initial design domain using traditional topology optimization results, in order to compensate for the performance loss caused by the uniform cross-section feature.

[0123] Specifically, because the constant cross-section stretching or sweeping methods restrict the degrees of freedom in topology optimization, a large number of inefficient elements (elements that contribute little to structural performance) exist in the initial design domain. The constant cross-section stretching or sweeping methods force many inefficient elements to be retained along the stretching path. Therefore, before performing multi-directional parallel constant cross-section stretching topology optimization, the results of traditional topology optimization without constant cross-section features can be used to remove peripheral inefficient elements, retaining only the necessary internal parts as the design domain. This avoids the complete removal of inefficient elements, ensuring the optimizable design space is not too small, and facilitating constant cross-section stretching or sweeping.

[0124] In this regard, taking the cuboid design domain divided by a regular hexahedral structured mesh as an example, we only consider removing such inefficient elements: they are low-density elements, and the perpendicular paths from the mesh element to at least 3 of the 6 faces of the cuboid space are all low-density elements.

[0125] First, under the premise of satisfying the volume constraint, the three-dimensional matrix of the voxels of the traditional topology optimization result is forced to be 0-1.

[0126] Specifically, the following three-dimensional matrix is ​​defined to represent the voxels of the traditional topology optimization results. :

[0127] (twenty four)

[0128] In the formula, Representing a three-dimensional matrix The elements in These represent the design domains of the cuboid. The number of layers in the hexahedral cell mesh in the direction.

[0129] Then, first design the three-dimensional matrix of the initial cuboid design domain. The elements of the matrix are set to all zeros, and then the following filtering algorithm is used to set the elements at the positions of the inefficient elements in the matrix to 1:

[0130] (25)

[0131] Then, during each topology optimization iteration, for directions requiring uniform cross-section stretching or sweeping, the density variables, physical variables, and sensitivity values ​​of the optimization objective and constraints relative to the physical variables at the locations of low-density elements (equivalent to non-design domains) are all set to 0. Taking a constant cross-section tension as an example, the specific calculation formula is as follows:

[0132] (26)

[0133] The present invention will be further described in detail below with reference to specific embodiments.

[0134] Numerical examples were verified using a laptop computer with the following computing environment: a 12th Gen Intel Core i9-12900HX CPU with a clock speed of 2.5GHz, DDR5 4800MHz 32GB × 2 RAM, and Win11 64-bit operating system.

[0135] like Figure 5 As shown, the design domain is a cuboid-shaped cantilever beam with a width of... ,high When testing runtime efficiency, length Lengths of 24cm, 48cm, and 72cm were respectively divided into 24×24×24, 48×24×24, and 72×24×24 hexahedral mesh elements. When testing the multi-directional equal-section feature topology optimization method, the length... The structure is divided into 48×24×24 regular hexahedral element meshes. A resultant force is applied along the lower right edge of the cantilever beam. Distributed load, left end face Three-dimensionally fixed, density filtration radius is Density penalty coefficient The elastic modulus of the material Poisson's ratio With the goal of minimizing structural flexibility, the volume fraction is set to 40% of the total volume of the design domain. The initial values ​​of the design variables for uniaxial, biaxial, and triaxial equal-section tension structures are 1, 1 / 2, and 1 / 3 times the volume fraction constraint values, respectively. The triaxial constant cross-section tensile structure is represented by blue, green, and red, respectively. The color of the intersection between multi-directional materials is a new color created by mixing the colors of the materials in the corresponding directions. The maximum number of iterations is 150, and the cutoff threshold of the step projection function is... Steepness increases with the number of iterations The change is shown in formula (27):

[0136] (27)

[0137] Figure 6 The figures show the results obtained when the cantilever beam lengths are 24cm, 48cm, and 72cm, respectively, using a three-dimensional cylindrical filter and a grouped dimension reduction filter. The topology optimization results driven by the uniform cross-section feature show that the final topology configurations obtained by the two methods are basically the same. Due to the aggregation effect of the grouped dimension reduction cylindrical filter, the compliance value of the configuration obtained by using this filter is lower, demonstrating the superiority of this method. Figure 7 The figure shows the average time consumed for a single filter parameter calculation, sensitivity analysis, design variable update, and topology optimization iteration. It can be seen that, due to the significant reduction in design variables caused by grouped dimensionality reduction filtering, the average time consumed for filter parameter calculation, sensitivity analysis, design variable update, and topology optimization iteration is reduced by 99.98%, 59.88%, 96.57%, and 9.63%, respectively. The program running efficiency is significantly improved, proving the high efficiency of the proposed method.

[0138] Figure 8 The figures show the topology optimization results driven by the uniform cross-section feature in a single direction. The configurations are clear and conform to symmetry. Due to the limitation of a single load condition, the performance of the three topology configurations varies considerably. The flexibility of a structure with a uniform cross-section is the lowest. The flexibility of a structure with a uniform cross-section is second only to that of a structure with a uniform cross-section. The flexibility is greatest for structures with uniform cross-sections.

[0139] Figure 9 , Figure 10 and Figure 11 They are respectively Xianghe Towards, Xianghe Towards, Xianghe The topology optimization results driven by bidirectional equal cross-section features in parallel. Figure 12 The results show the topology optimization results driven by parallel processing of three-dimensional equal cross-section features. It can be seen that this method yields clear and symmetrical usable topological configurations. Under the overall volume constraint, the algorithm automatically allocates the optimal material quantity and distribution for each directional equal cross-section feature structure. Due to the increased design freedom, the materials in different directions achieve a certain degree of complementarity in load-bearing capacity, more effectively covering potential optimal force transmission paths. Therefore, the overall flexibility of the topology optimization results driven by parallel processing of two-dimensional equal cross-section features is generally less than (better than) that driven by unidirectional equal cross-section features, and it is easier to reconfigure, significantly improving manufacturability. Especially... Xianghe The topology optimization results driven by parallel equal cross-sectional features exhibit excellent flexibility and extremely simple and regular structural styles. They are also easy to manufacture using various processes such as 3D printing, multi-axis machining, casting, and sheet metal splicing, demonstrating the superiority of this method.

[0140] For topology optimization driven by parallel triaxial isotropic cross-sectional features, effective complementarity is difficult to achieve due to intensified competition between materials in each direction. During the topology optimization process, the evolution direction of the topology configuration may be influenced by a certain direction (e.g., The material biased towards the (right-hand) direction retained a large number of inefficient elements (the protruding part of the blue cylinder on the right), resulting in a configuration whose performance was actually worse than the topology optimization results driven by the bidirectional uniform cross-section feature parallel process. Therefore, a design domain reduction method that removes peripheral inefficient elements is applied, and the reduced design domain is as follows: Figure 13 As shown, by using this design domain, we obtain Xianghe The topology optimization results driven by parallel processing of axial and triaxial isosectional features are as follows: Figure 14 and Figure 15 As shown, their configuration and flexibility properties are similar to those of traditional topology optimization. With minimal loss of structural performance, the difficulty of geometric reconstruction is greatly reduced (simply draw sketches in each stretching direction, stretch or sweep them into three-dimensional solids, and then remove redundant inefficient elements). This makes them more suitable for construction or manufacturing using traditional equal-material and subtractive processes.

[0141] This invention provides a topology optimization method driven by parallel multi-directional equal cross-sectional features, which is a universal, multifunctional, and easily scalable design method. On the one hand, the proposed grouped dimensionality reduction filtering, material interpolation mode, and design domain reduction strategy are applicable to density-based and boundary evolution-based topology optimization methods, as well as problems in different physical fields such as mechanics and thermodynamics. On the other hand, thanks to the multi-material interpolation mode of this invention, each directional stretching or sweep can be performed independently, improving the flexibility of the method and facilitating the application of many advanced topology optimization and geometric reconstruction methods within each directional two-dimensional cross-section. For example, this method can be combined with topology optimization methods that consider maximum and minimum member dimensions or equal-size constraints, along with certain geometric simplification and reconstruction techniques, to easily obtain multi-directional intersecting variable-thickness and equal-thickness plate splicing structures. This method can be widely applied in multiple fields of manufacturing or construction. Therefore, this method has strong adaptability, is easy to transform and implement, and will spur the emergence of more practical methods, accelerating technological changes in the field of topology optimization.

[0142] The present invention also provides a three-dimensional structure topology optimization system driven by parallel multi-directional equal cross-section features. The system includes a memory and a processor. The memory stores a computer program. When the processor executes the computer program, it performs the three-dimensional structure topology optimization method driven by parallel multi-directional equal cross-section features as described above.

[0143] The present invention also provides a computer-readable storage medium storing machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the three-dimensional structural topology optimization method driven by multi-directional equal cross-sectional features as described above.

[0144] Those skilled in the art will readily understand 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 within the scope of protection of the present invention.

Claims

1. A three-dimensional structure topology optimization method driven by parallel multi-directional equal cross-section features, characterized in that, The method includes the following steps: (1) For each equal cross-section stretching or sweeping direction of the structure to be optimized, a uniformly distributed unidirectional design variable is defined in the corresponding two-dimensional cross-section. The unidirectional design variable is then subjected to density filtering and step projection by a grouped dimension reduction cylindrical filter to obtain the unidirectional physical variable associated with the finite element mesh of the structure to be optimized. (2) Based on the obtained unidirectional physical variables, a multi-material interpolation mode is used to calculate the composite physical variables and composite elastic variables of multi-directional equal cross-section parallel stretching or sweeping; wherein, the multi-material interpolation mode regards each directional physical variable as the probability of the corresponding mesh cell being filled with material; (3) Calculate the objective function and constraint function of the topology optimization model of the structure to be optimized and its sensitivity relative to the design variables based on the synthetic physical variables and synthetic elastic variables respectively. Then, use the gradient optimization algorithm to iteratively solve the topology optimization model to obtain the optimal topology configuration driven by the parallel multi-directional equal cross-section features. When performing density filtering with a grouped dimension reduction cylindrical filter, the axis of the grouped dimension reduction cylindrical filter is parallel to the stretching direction or the sweeping direction. All design variables on the same stretching path or sweeping path will always maintain the same value during the optimization process, and will be treated as a group and represented by a single design variable. The formula corresponding to density filtration is: In the formula, To design a variable array, This refers to the density variable array that corresponds one-to-one with each mesh element after the geometric model of the structure to be optimized is generated by the finite element meshing. From Transformation from design variable matrix to density variable matrix 3D matrix express Element; For the first The effect of the design variable on the first Weighting coefficients on each density variable They represent the smallest cuboid space enclosing the three-dimensional design domain, respectively. The number of layers to which design variables are evenly distributed. This represents the total number of density variables corresponding to each grid cell. Indicates the first The center of each grid cell coordinates and coordinate, Indicates the first Design variables in the design domain space coordinates and coordinate, For the filter radius, express and The distance between them; The formula corresponding to the multi-material interpolation mode is: In the formula, They represent The first in the stretch structure space The physical variables corresponding to each grid cell are obtained by step projection of the corresponding density variables, and their values ​​tend to be 0 or 1.

2. The three-dimensional structure topology optimization method driven by parallel multi-directional equal cross-section features as described in claim 1, characterized in that: The formula corresponding to the composite physical variable array is: After further simplification, we get: In the formula, use They represent The array of physical variables corresponding to the stretched structural mesh elements.

3. The three-dimensional structure topology optimization method driven by parallel multi-directional equal cross-section features as described in claim 2, characterized in that: The formula corresponding to the composite elastic variable matrix is: In the formula, This represents the material penalty factor, which is an integer not less than 1; here, we take... ; They represent The elastic modulus of the tensile material; This represents the elastic modulus value corresponding to the porous element.

4. The three-dimensional structure topology optimization method driven by parallel multi-directional equal cross-section features as described in claim 3, characterized in that: After increasing the penalty for air-phase materials, the formula corresponding to the synthetic elastic variable matrix is: 。 5. The three-dimensional structure topology optimization method driven by parallel multi-directional equal cross-section features as described in any one of claims 1-4, characterized in that: The constraint function is a global volume fraction constraint.

6. The three-dimensional structure topology optimization method driven by parallel multi-directional equal cross-section features as described in claim 2, characterized in that: The mathematical expression for the topology optimization model is: In the formula, They represent Design variable array for constant cross-section stretching, They represent Design variables for tensioning a uniform cross section. They represent the smallest cuboid space enclosing the three-dimensional design domain, respectively. The number of layers to which design variables are evenly distributed. Describe the objective function. Representing the finite element equilibrium equations, This represents the global volume fraction constraint function. The total volume of the design domain. This represents the global volume fraction constraint value for the design domain. This is the volume array of the grid cells.

7. The three-dimensional structure topology optimization method driven by multi-directional equal cross-section features in parallel as described in any one of claims 1-4, characterized in that: Step (1) precedes the step of removing low-density cells from the periphery of the initial design domain of the structure to be optimized, which is a traditional topology optimization result.

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