Topology optimization method based on binary tree search, computer device and storage medium
By optimizing the neighborhood search process using a topology optimization method based on binary tree search, the problem of low computational efficiency of the BESO method in large-scale complex structures is solved, and more efficient topology optimization design is achieved.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- NORTH CHINA ELECTRIC POWER UNIV
- Filing Date
- 2024-12-10
- Publication Date
- 2026-06-23
AI Technical Summary
Existing bidirectional progressive structural optimization (BESO) methods are computationally inefficient when dealing with complex, large-scale structures, making them difficult to meet engineering requirements.
A topology optimization method based on binary tree search is adopted. By establishing a finite element model, the binary tree search algorithm is used to correct and update the sensitivity number of the elements, optimize the neighborhood search process, and reduce redundant calculations.
It improves the computational efficiency of topology optimization and the processing speed of neighborhood search, thereby enhancing the overall optimization efficiency and making it suitable for the optimization design of large-scale complex structures.
Smart Images
Figure CN119761111B_ABST
Abstract
Description
Technical Field
[0001] This application relates to the field of topology optimization technology, specifically to a topology optimization method based on binary tree search, a computer device, and a storage medium. Background Technology
[0002] In recent years, topology optimization techniques, as an important means of structural optimization design, have received widespread attention and in-depth research. Among them, the Bi-directional Evolutionary Structural Optimization (BESO) method has become one of the commonly used algorithms in the field of topology optimization due to its simplicity and efficiency. However, with the increase in the scale and complexity of engineering problems, the computational efficiency and convergence speed of the BESO method have gradually become bottlenecks restricting its application. How to improve the computational efficiency of the BESO method while ensuring optimization accuracy has become a hot research topic.
[0003] Accordingly, a new topology optimization scheme is needed in this field to solve the above problems. Summary of the Invention
[0004] In order to overcome the above-mentioned shortcomings, this application is proposed to solve or at least partially solve the technical problem of low computational efficiency in topology optimization of complex, large-scale structures.
[0005] In a first aspect, a topology optimization method based on binary tree search is provided, the method comprising: S101: establishing a finite element model, defining the design domain of the structure, the physical parameters of the materials, and the boundary conditions of the structure; S102: performing finite element analysis on the finite element model to determine the strain energy parameters of the elements in the finite element model; S103: determining the sensitivity number of each element based on the strain energy parameters and density of the elements; S104: correcting the sensitivity number of the elements based on the binary tree search algorithm and the sensitivity number of each element; S105: updating the design domain of the finite element model based on the corrected sensitivity number of the elements; S106: repeating steps S102 to S105 until the updated finite element model satisfies a preset convergence condition, thereby obtaining the topology optimization result of the finite element model.
[0006] In one technical solution of the aforementioned topology optimization method based on binary tree search, the physical parameters of the material include the elastic modulus and element density; the step of performing finite element analysis on the finite element model to determine the strain energy parameters of the elements in the finite element model includes: determining the element elastic modulus based on the material's elastic modulus, a preset penalty factor, and the element density; defining an objective function based on the principle of static equilibrium and the average compliance characterizing the structural strain energy; and using the volume constraint ratio as a constraint condition, performing finite element analysis on the finite element model based on the element elastic modulus and the objective function to determine the strain energy parameters of the elements.
[0007] In one technical solution of the above-mentioned topology optimization method based on binary tree search, the step of correcting the sensitivity number of a unit based on the binary tree search algorithm and the sensitivity number of each unit includes: using a binary tree search algorithm in a preset storage area based on a preset filtering radius and preset filtering rules to obtain all neighboring units corresponding to the unit; and correcting the sensitivity number of the unit based on the sensitivity numbers of all neighboring units.
[0008] In one technical solution of the above-mentioned topology optimization method based on binary tree search, multiple elements of the finite element model are stored in a preset storage area of a binary tree structure, and each element corresponds to a storage coordinate value. The step of obtaining all neighboring elements of the element in the preset storage area using a binary tree search algorithm based on a preset filtering radius and preset filtering rules includes: using a binary tree search algorithm, based on the storage coordinate value of the element and the preset filtering radius, searching for all neighboring elements within the preset filtering radius range of the element in the preset storage area.
[0009] In one technical solution of the above-mentioned topology optimization method based on binary tree search, the step of correcting the sensitivity number of a unit based on the sensitivity numbers of all neighboring units includes: determining a first sensitivity number of the unit based on the sensitivity number of the neighboring units, the number of neighboring units, the distance between the neighboring units and the unit, and a preset weighting factor; obtaining the sensitivity number of the unit determined in the previous iteration; and correcting the first sensitivity number of the unit based on the sensitivity number of the unit determined in the previous iteration to obtain the corrected sensitivity number of the unit.
[0010] In one technical solution of the above-mentioned topology optimization method based on binary tree search, updating the design domain of the finite element model based on the modified sensitivity number of the element includes: updating the density of the element based on the modified sensitivity number of the element and a preset addition / deletion rule; and adding the element to the design domain of the finite element model in the current iteration based on the density of the element.
[0011] In one technical solution of the aforementioned topology optimization method based on binary tree search, updating the density of the unit based on the corrected sensitivity number of the unit and preset addition / deletion rules includes: determining the target volume of this iteration based on a preset evolution rate parameter; determining a sensitivity number threshold based on the corrected sensitivity number of the unit and the target volume; determining whether the corrected sensitivity number of the unit is greater than the sensitivity number threshold; if yes, updating the density of the unit to a first preset value; if no, updating the density of the unit to a second preset value; wherein the first preset value is greater than the second preset value.
[0012] In one technical solution of the above-mentioned topology optimization method based on binary tree search, repeating steps S102 to S105 until the updated finite element model satisfies the preset convergence condition to obtain the topology optimization result of the finite element model includes: determining whether the target volume of the finite element model in this iteration reaches the preset structural volume; if the target volume in this iteration reaches the preset structural volume, determining whether the preset convergence condition is met based on the change in the objective function and the number of iterations; if so, outputting the finite element model as the topology optimization result; otherwise, repeating steps S102 to S105.
[0013] In a second aspect, a computer device is provided, the computer device including at least one processor; and a memory communicatively connected to the at least one processor; wherein the memory stores a computer program, which, when executed by the at least one processor, implements the method described in any of the above-described technical solutions of the topology optimization method based on binary tree search.
[0014] In a third aspect, a computer-readable storage medium is provided, wherein a plurality of program codes are stored therein, the program codes being adapted to be loaded and run by a processor to perform the method described in any of the above-described technical solutions of the topology optimization method based on binary tree search.
[0015] The above-described technical solutions of this application have at least one or more of the following beneficial effects:
[0016] The topology optimization method based on binary tree search provided in this application includes: S101: establishing a finite element model, defining the design domain of the structure, the physical parameters of the materials, and the boundary conditions of the structure, wherein the finite element model includes multiple elements; S102: performing finite element analysis on the finite element model to determine the strain energy parameters of the elements in the finite element model; S103: determining the sensitivity number of each element based on the strain energy parameters and density of the elements; S104: correcting the sensitivity number of the elements based on the binary tree search algorithm and the sensitivity number of each element; S105: updating the design domain of the finite element model based on the corrected sensitivity number of the elements; S106: repeating steps S102 to S105 until the updated finite element model meets the preset convergence condition, thereby obtaining the topology optimization result of the finite element model. This application optimizes the element selection and sensitivity number update process using a binary tree search algorithm, which can effectively improve the processing speed of the algorithm in neighborhood search, reduce redundant calculations, and improve the overall optimization efficiency. Attached Figure Description
[0017] The disclosure of this application will become more readily understood with reference to the accompanying drawings. It will be readily understood by those skilled in the art that these drawings are for illustrative purposes only and are not intended to limit the scope of protection of this application. Wherein:
[0018] Figure 1 This is a schematic flowchart of the main steps of a topology optimization method based on binary tree search according to an embodiment of this application;
[0019] Figure 2 This is a schematic flowchart illustrating the steps of searching neighborhood units using a binary tree search algorithm according to an embodiment of this application;
[0020] Figure 3 This is a schematic diagram of the co-simulation process of a topology optimization method based on binary tree search according to an embodiment of this application;
[0021] Figure 4 This is a schematic diagram of boundary conditions according to an embodiment of this application;
[0022] Figure 5 is a schematic diagram of topology optimization results according to an embodiment of this application. The regular mesh size of 5(a) is 80*50, the regular mesh size of 5(b) is 160*100, the regular mesh size of 5(c) is 240*150, and the regular mesh size of 5(d) is 320*200.
[0023] Figure 6 This is a schematic diagram of boundary conditions according to another embodiment of this application;
[0024] Figure 7 is a schematic diagram of the material distribution evolution process according to another embodiment of this application, where 7(a) is the result of the 5th iteration, 7(b) is the result of the 10th iteration, 7(c) is the result of the 15th iteration, and 7(d) is the final topology result;
[0025] Figure 8 This is a schematic diagram of boundary conditions according to another embodiment of this application;
[0026] Figure 9 This is a schematic diagram of topology optimization results according to another embodiment of this application;
[0027] Figure 10 This is a schematic diagram illustrating the evolution of the compliance-to-volume ratio according to an embodiment of this application;
[0028] Figure 11 This is a planar symmetrical schematic diagram according to an embodiment of this application;
[0029] Figure 12 This is a rotationally symmetric schematic diagram according to an embodiment of this application;
[0030] Figure 13 This is a schematic diagram of a periodic structure according to an embodiment of this application;
[0031] Figure 14 This is a schematic diagram of boundary conditions according to yet another embodiment of this application;
[0032] Figure 15 is a schematic diagram of topology optimization results according to another embodiment of this application. 15(a) is the optimization result of the general BESO method, 15(b) is the optimization result with single-plane xoz symmetric constraints applied, and 15(c) is the optimization result with biplane xoz and yoz symmetric constraints applied.
[0033] Figure 16 This is a schematic diagram of boundary conditions according to an embodiment of this application;
[0034] Figure 17 is a schematic diagram of the topology optimization results before and after applying planar symmetry constraints according to an embodiment of this application. 17(a) is a material distribution diagram of non-planar symmetry constraints, and 17(b) is a material distribution diagram of planar yoz symmetry constraints.
[0035] Figure 18 This is a schematic diagram of boundary conditions according to another embodiment of this application;
[0036] Figure 19 This is a schematic diagram of the material distribution results according to an embodiment of this application;
[0037] Figure 20 This is a schematic diagram of boundary conditions according to another embodiment of this application;
[0038] Figure 21 is a schematic diagram of the topology optimization results according to another embodiment of the present application. 21(a) is a front view of the topology optimization results of the tripod structure, 21(b) is a top view of the topology optimization results of the tripod structure, and 21(c) is a schematic diagram of the evolution process of the tripod structure's compliance and volume ratio.
[0039] Figure 22 This is a schematic diagram of boundary conditions according to yet another embodiment of this application;
[0040] Figure 23 This is a schematic diagram of the topology optimization results according to yet another embodiment of this application;
[0041] Figure 24 This is a schematic diagram of the main structure of a computer device according to an embodiment of this application.
[0042] Figure label:
[0043] 11: Memory; 12: Processor. Detailed Implementation
[0044] Some embodiments of this application are described below with reference to the accompanying drawings. Those skilled in the art should understand that these embodiments are merely illustrative of the technical principles of this application and are not intended to limit the scope of protection of this application.
[0045] In the description of this application, "module" and "processor" can include hardware, software, or a combination of both. A module can include hardware circuitry, various suitable sensors, communication ports, memory, and may also include software components, such as program code, or a combination of software and hardware. A processor can be a central processing unit, microprocessor, image processor, digital signal processor, or any other suitable processor. The processor has data and / or signal processing capabilities. The processor can be implemented in software, in hardware, or a combination of both. Computer-readable storage media includes any suitable medium capable of storing program code, such as magnetic disks, hard disks, optical disks, flash memory, read-only memory, random access memory, etc. The term "A and / or B" means all possible combinations of A and B, such as only A, only B, or A and B. The terms "at least one A or B" or "at least one of A and B" have a similar meaning to "A and / or B" and can include only A, only B, or A and B. The singular terms "a" or "this" can also include plural forms.
[0046] See appendix Figure 1 , Figure 1 This is a schematic flowchart illustrating the main steps of a topology optimization method based on binary tree search according to an embodiment of this application. Figure 1As shown, the topology optimization method based on binary tree search in this application embodiment mainly includes the following steps S101 to S106.
[0047] S101: Establish a finite element model and define the design domain of the structure, the physical parameters of the materials, and the boundary conditions of the structure.
[0048] S102: Perform finite element analysis on the finite element model to determine the strain energy parameters of the elements in the finite element model.
[0049] S103: Determine the sensitivity number of each unit based on the strain energy parameters and density of the unit.
[0050] S104: Based on the binary tree search algorithm and the sensitivity number of each unit, the sensitivity number of the unit is corrected.
[0051] S105: Update the design domain of the finite element model based on the corrected sensitivity number of the element.
[0052] S106: Repeat steps S102 to S105 until the updated finite element model satisfies the preset convergence condition, and obtain the topology optimization result of the finite element model.
[0053] Based on the methods described in steps S101 to S106 above, this application uses a binary tree search-based filtering method to optimize the process of cell selection and sensitivity number correction, which can effectively improve the processing speed of the algorithm in neighborhood search, reduce redundant calculations, and improve the overall optimization efficiency.
[0054] The following provides further explanation of steps S101 to S106.
[0055] For S101, a finite element model is established, and the design domain of the structure, the physical parameters of the materials, and the boundary conditions of the structure are defined.
[0056] Specifically, a finite element model is established, and the boundary conditions of the structure are defined. The boundary conditions can be constraints or loads. The physical parameters of the material, such as Poisson's ratio, elastic modulus, and density, are also defined. The finite element model contains multiple elements.
[0057] Regarding step S102, in one embodiment, the physical parameters of the material include the elastic modulus and element density; the step of performing finite element analysis on the finite element model to determine the strain energy parameters of the elements in the finite element model includes: determining the element elastic modulus based on the material's elastic modulus, a preset penalty factor, and the element density; defining an objective function based on the principle of static equilibrium and the average compliance characterizing the structural strain energy; and using the volume constraint ratio as a constraint condition, performing finite element analysis on the finite element model based on the element elastic modulus and the objective function to determine the strain energy parameters of the elements.
[0058] Specifically, material interpolation methods include ESO and BESO methods, which typically determine element removal based on the mechanical performance indicators of the structure. Hard element removal methods are also included, which completely remove inefficient elements by setting the relative density to 0. This method does not consider the removed elements during iteration, making its correctness difficult to prove theoretically. Furthermore, there are soft cell removal methods, which replace the density of virtual elements with a very small value, thus solving this problem. Further, there is the BESO method based on a material interpolation model with soft cell removal, introducing a penalty factor p, allowing the removed elements to participate in subsequent sensitivity analysis calculations.
[0059] This embodiment uses the BESO method for soft cell deletion, assuming that the design variables representing the presence or absence of cells are... The formula for determining the elastic modulus of a single element is:
[0060] ;
[0061] in, is the elastic modulus of the material; p is the penalty factor, p≥1; For element i, the element with the highest strain energy density is... For solid elements, the elastic modulus of the solid element is the true elastic modulus of the material; elements with low strain energy density This refers to a virtual element, whose elastic modulus is a very small value, called the virtual Young's modulus, used to avoid singularities in the element stiffness matrix. In this embodiment... .
[0062] The principle of static equilibrium refers to the static equilibrium equations of a continuum structure, expressed as:
[0063] ;
[0064] In the formula, K is the overall stiffness matrix; u is the displacement vector of the node; and f is the load vector of the node.
[0065] The stiffness of a structure is measured by its average compliance, which characterizes its strain energy. The mathematical expression for average compliance is:
[0066] ;
[0067] In the formula, For average softness, Let be the transformation matrix of u.
[0068] Based on the principle of static equilibrium and the average compliance characterizing the strain energy of a structure, it can be determined that the overall stiffness of the structure is inversely proportional to its strain energy. Therefore, with the volume ratio as a constraint and maximizing stiffness as the objective function, the topology optimization formula for the continuum structure is obtained as follows:
[0069] ;
[0070] In the formula, For average softness, Let be the volume of the i-th unit; For the preset structural volume, The ratio of the volume constraint ratio to the total volume V0 is called the volume constraint ratio. The total number of units Let be the displacement vector of element i. for Transform matrix, This is the element stiffness matrix.
[0071] For step S103, the sensitivity number of each unit is determined based on the strain energy parameters and density of the unit.
[0072] The strain energy parameters of the element are That is, the strain energy parameters of the element are determined based on the displacement vector and stiffness matrix.
[0073] The mathematical expression for the sensitivity of the i-th unit with respect to the objective function is: ;
[0074] In the formula, Let be the stiffness matrix of solid element i.
[0075] Using discrete design variables (0 or The sensitivity numbers of physical and virtual units can be expressed as:
[0076] ;
[0077] In the formula, Let be the sensitivity number of unit i. The sensitivity number of the virtual unit is related to the value of the penalty factor p. When the value of p is large, the sensitivity number of the virtual unit tends to 0. At this time, equation (2-6) can be simplified to:
[0078] ;
[0079] Therefore, the sensitivity number of a solid element is equal to the strain energy of the element, while the sensitivity number of a virtual element is zero.
[0080] Regarding step S104, in one embodiment, the step of correcting the sensitivity number of the unit based on the binary tree search algorithm and the sensitivity number of each unit includes: using a binary tree search algorithm in a preset storage area based on a preset filtering radius and preset filtering rules to obtain all neighboring units corresponding to the unit; and correcting the sensitivity number of the unit based on the sensitivity numbers of all neighboring units.
[0081] In one embodiment, multiple elements of the finite element model are stored in a preset storage area of a binary tree structure, with each element corresponding to a storage coordinate value. The step of obtaining all neighboring elements of the element in the preset storage area using a binary tree search algorithm based on a preset filtering radius and preset filtering rules includes: using a binary tree search algorithm to search for all neighboring elements within the preset filtering radius range of the element in the preset storage area based on the storage coordinate value of the element and the preset filtering radius.
[0082] Specifically, a binary tree search algorithm is used to search downwards from the node partitions of each node in a pre-defined storage area of the binary tree structure based on the storage coordinates of the cell, thus determining all neighboring cells. This process includes downward search, leaf node processing, parent node retrieval, and branch pruning steps. A flowchart of the binary tree search algorithm for searching neighboring cells is shown below. Figure 2 As shown.
[0083] First, let L be a list with k empty slots to store the searched neighborhood cells. Construct a kd (k-dimensional tree). If the tree nodes are split according to xr = a, determine whether the x-coordinate of the cell is less than the x-coordinate of the split point. If so, search the left subtree; otherwise, search the right subtree.
[0084] When searching the left subtree, upon reaching a leaf node, the leaf node is marked as visited, and the distance between the coordinates of that leaf node and the coordinates of the cell is calculated. If the distance is greater than or equal to the preset filtering radius... If the distance is less than the preset filtering radius, then there are no cells in the neighborhood of that cell. If so, the cell of the leaf node is determined to be a neighboring cell and stored in the L list.
[0085] Determine if the leaf node is the root node. If it is, complete the search and output the neighboring cells stored in list L. If not, retrieve the parent node of the leaf node and determine if it has been visited. If it has been visited, continue searching upwards to unvisited nodes and determine if the distance between the cell and the unvisited node is less than the preset filtering radius. If so, search the right subtree and determine if the leaf node has been reached.
[0086] If a node has not been visited, determine whether the distance between the cell and the unvisited node is less than the preset filtering radius. If yes, search the right subtree. If no, prune the right subtree of the unvisited node and determine whether the unvisited node is the root node. If yes, output the neighboring cells already stored in list L. If no, continue searching the parent node of the unvisited node.
[0087] In one embodiment, the step of correcting the sensitivity number of a unit based on the sensitivity numbers of all neighboring units includes: determining a first sensitivity number of the unit based on the sensitivity numbers of the neighboring units, the number of neighboring units, the distance between the neighboring units and the unit, and a preset weighting factor; obtaining the sensitivity number of the unit determined in the previous iteration; and correcting the first sensitivity number of the unit based on the sensitivity number of the unit determined in the previous iteration to obtain the corrected sensitivity number of the unit.
[0088] Specifically, taking the storage coordinates of cell i as the center, the sensitivity of all neighboring cells within the preset filtering radius will affect the corrected sensitivity of cell i.
[0089] Based on the sensitivity counts, number of neighboring units of unit i, distance between neighboring units and unit i, and a preset weighting factor, the first sensitivity count of unit i is determined using the following formula:
[0090] ;
[0091] In the formula, It is the number of neighborhood units; This represents the distance from the center of cell j to the center of cell i. For neighborhood units within a preset filtering radius; This is a preset weighting factor. The formula for calculating the preset weighting factor is:
[0092] ;
[0093] The value can be 1 to 3 times the average size of the unit.
[0094] Since the state of the cell is uncertain during each sensitivity calculation, which may cause the objective function to fail to converge, it is necessary to correct the first sensitivity count and obtain the sensitivity count of cell i determined in the previous iteration. Based on the sensitivity number of unit i determined in the previous iteration, the first sensitivity number of unit i is corrected to obtain the corrected sensitivity number of unit i. The calculation formula is as follows: ;
[0095] Regarding step S105, in one embodiment, updating the design domain of the finite element model based on the modified sensitivity number of the element includes: updating the density of the element based on the modified sensitivity number of the element and a preset addition / deletion rule; and adding the element to the design domain of the finite element model in the current iteration based on the density of the element.
[0096] In one embodiment, updating the density of the unit based on the modified sensitivity number of the unit and a preset addition / deletion rule includes: determining the target volume for this iteration based on a preset evolution rate parameter; determining a sensitivity number threshold based on the modified sensitivity number of the unit and the target volume; determining whether the modified sensitivity number of the unit is greater than the sensitivity number threshold; if yes, updating the density of the unit to a first preset value; if no, updating the density of the unit to a second preset value; wherein the first preset value is greater than the second preset value.
[0097] Specifically, before the design domain volume reaches the preset structural volume V*, each iteration will gradually add and delete cells. Before adding or deleting cells, the target volume for this iteration is determined based on a preset evolutionary ratio parameter, ER (evolutionary ratio). The formula for calculating the target volume is: ;
[0098] In the formula, This is the target volume for this iteration. It is the target volume of the previous iteration. It is a preset evolution rate parameter. It represents the number of iterations.
[0099] Based on the corrected sensitivity number of the unit and the target volume, the sensitivity number threshold is determined. Specifically, the sensitivity numbers of all units are sorted, and the initial sensitivity number threshold is defined as the average of the maximum and minimum sensitivity numbers. The sensitivity number threshold is updated using a bisection method based on the current volume ratio and the constraint volume ratio.
[0100] Determine whether the sensitivity number of the corrected cell is greater than the sensitivity number threshold; if so, the cell density is the first preset value, which is 1, i.e., x. i=1; If the sensitivity number of the corrected unit is less than or equal to the sensitivity number threshold, then the unit density is the second preset value, which is... , .
[0101] Then, if the element density is the first preset value, the element is added to the design domain of the finite element model for this iteration; if the element density is the second preset value, the element is deleted from the design domain.
[0102] Regarding step S106, in some embodiments, repeating steps S102 to S105 until the updated finite element model satisfies the preset convergence condition to obtain the topology optimization result of the finite element model includes: determining whether the target volume of the finite element model in this iteration reaches the preset structural volume; if the volume reaches the preset structural volume, determining whether the preset convergence condition is met based on the change in the objective function and the number of iterations; if so, outputting the finite element model as the topology optimization result.
[0103] Specifically, steps S102 to S105 are automatically repeated to perform finite element analysis and add / delete elements on the finite element model in a loop. After the volume of the finite element model in this iteration reaches the preset structural volume, the target volume remains constant, that is:
[0104] ;
[0105] Next, a convergence check is performed to determine whether the change in the objective function is less than the allowable convergence error. The formula for the convergence check is shown below:
[0106] ;
[0107] In the formula, To allow for convergence error, N is an integer.
[0108] In some embodiments, N=5 indicates that at least 10 consecutive iterations are needed to eliminate the error caused by the calculation formula (2-10) for the sensitivity number of the correction unit.
[0109] See appendix Figure 3 , Figure 3 This is a schematic diagram of the co-simulation process of a topology optimization method based on binary tree search according to an embodiment of this application. Figure 3As shown, in some embodiments, the topology optimization method based on binary tree search of this application improves the intuitiveness and operability of the optimization by calling Ansys software for finite element analysis. In each iteration of the optimization process, the finite element analysis results from Ansys are returned to Python for subsequent calculations such as sensitivity number calculation, filtering, sensitivity number correction, and cell addition / removal, generating a new model and importing it into Ansys for the next iteration. The Python program consists of three parts: preprocessing, finite element analysis, and independent mesh filtering and cell addition / removal. Preprocessing includes providing various parameters required for the calculation; finite element analysis, independent mesh filtering, and cell addition / removal are included in the iterative loop and repeated continuously until a given volume is reached and the convergence criterion is met.
[0110] The following section uses specific examples to illustrate the topology optimization method based on binary tree search proposed in this application.
[0111] In one embodiment, the elastic modulus of the material is E = 1.69 × 10⁻⁶. 5 Poisson's ratio ν = 0.3, τ = 0.1%.
[0112] Example 1: This example examines the mesh independence of the program. The simplified cantilever beam structure model is as follows. Figure 4 As shown, the rectangular design domain has dimensions of 80×50, with the left end fixed. A downward vertical load of size 100 is applied at the center of the free end. It is discretized using regular meshes of sizes 32×20, 80×50, 160×100, and 240×150. The parameters of BESO are set as follows: ER=5%, volume ratio V* / V0=0.5. The topology optimization results under different grid densities are shown in Figures 5(a) to 5(d). In Figure 5(a), the regular grid size is 80*50; in Figure 5(b), it is 160*100; in Figure 5(c), it is 240*150; and in Figure 5(d), it is 320*200. As shown in Figures 5(a) to 5(d), the topology optimization results are not affected by the grid size, and the topology optimization results using the binary tree search-based method are grid-independent.
[0113] Example 2: This example is a classic topology optimization problem, with boundary conditions as follows. Figure 6 As shown. A two-dimensional simply supported beam is 120 mm long and 40 mm wide, subjected to a concentrated load of magnitude 100 at the center of its bottom end. It is discretized using a 240 × 80 mesh, and the BESO parameters are set as follows. ER = 5%, V* / V0 = 0.5. The material distribution evolution process of BESO is shown in Figures 7(a) to 7(d), where Figure 7(a) is the result of the 5th iteration, Figure 7(b) is the result of the 10th iteration, Figure 7(c) is the result of the 15th iteration, and Figure 7(d) is the final topology optimization result.
[0114] The following section uses a three-dimensional L-shaped beam as an example to search and filter the neighborhood using both traditional search methods and binary tree search algorithms, and compares the computation time of the two methods. Figure 8 The boundary conditions are for an L-shaped beam. The design domain was discretized using a 1-element mesh, resulting in 40,000 elements and 44,541 nodes. To ensure the fairness and scientific validity of the experimental results, the BESO parameters calculated in both trials were identical. ER=4%, volume ratio V* / V0=0.2. The optimization problem was solved in the same hardware environment, configured with a 12th Gen Intel(R) Core(TM) i9-12900K 3.20 GHz CPU and 64GB of memory to ensure that the comparison of algorithm efficiency was not affected by hardware differences. The maximum number of iterations was set to 80, or terminated early when the objective function converged to below 0.1%. The final topology optimization results and the evolution history of compliance and volume ratio are as follows: Figure 9 , Figure 10 As shown, the time comparison of the two methods is shown in Table 1.
[0115] Table 1. Efficiency Comparison of Two Search Algorithms
[0116] Table 1 shows that the computation time of the binary tree search algorithm is significantly lower than that of traditional methods. In terms of computational complexity, traditional methods, due to searching the neighborhood of each cell individually, experience an exponential increase in time with grid size. In contrast, the binary tree search algorithm, through rapid partitioning and searching, significantly improves the efficiency of neighborhood search, with computation time increasing only slightly with grid size. Comparative experiments using an L-shaped beam example clearly demonstrate the significant advantages of the binary tree search algorithm in topology optimization filtering methods. Compared to traditional search methods, the binary tree search method not only significantly reduces computation time but is also more suitable for optimizing large-scale complex models. Therefore, in practical engineering applications, the binary tree search method provides an efficient and robust solution for solving the neighborhood search problem in topology optimization.
[0117] In the field of topology optimization, introducing engineering constraints can ensure design feasibility. Especially when dealing with complex structures, constraints not only affect the quality of optimization results but also directly relate to the effective utilization of materials and the actual manufacturability of the process. Based on the binary tree search-based topology optimization method of this application and the topology optimization algorithms for plane-symmetric, rotationally symmetric, and periodic structures developed using Python, the practicality and efficiency of this application in implementing engineering constraints are analyzed.
[0118] To satisfy the three engineering constraints mentioned above, the initial design domain is divided into n design sub-regions based on the user-defined constraint pattern and the number of constraint components n, using the Python language. For example... Figure 12 As shown in the rotational symmetry diagram, the annular structure is uniformly divided into 8 design sub-regions, x j,i Let represent the design variable of the i-th element within the j-th design sub-region. To ensure the rotational symmetry of the final design, the design variables of any element within any design sub-region should be strictly the same as those of the corresponding elements in other design sub-regions. The dots in the diagram mark a group of elements with the same position. Therefore, the optimization problem considering engineering constraints can be formulated as follows: ;
[0119] In the formula, V* is the volume of the i-th unit in the j-th design sub-region; V* is the preset structural volume, and the ratio of V* to the total volume V0 is called the volume constraint ratio; n is the total number of design sub-regions; and l is the total number of units in each design sub-region.
[0120] From formulas (2-5), (2-6), (2-7), and (2-8), the compliance sensitivity number of each element considering engineering constraints can be obtained. Let the sensitivity number of each element and its corresponding position be equal to their average sensitivity, then we have...
[0121] ;
[0122] ;
[0123] In the formula, Let i be the sensitivity number of the i-th cell in the j-th design sub-region after filtering. This refers to the element sensitivity number of each element and its corresponding position after considering engineering constraints.
[0124] As shown in equation (3-2), the key to achieving the constraint conditions lies in finding corresponding units with the same position in other design sub-regions for any given unit. Inspired by the binary tree algorithm, if the center coordinates of each unit and its corresponding unit can be aligned or controlled within a certain tolerance through code, the binary tree algorithm can be used to quickly search the neighborhood of unit coordinates, thereby efficiently obtaining the index of each unit and its corresponding unit and storing it as a group of related units, preparing for subsequent sensitivity averaging. The specific implementation method is as follows:
[0125] For planar symmetric constraints, such as Figure 11 The image shows the main view of the 3D design domain. Users can define any one or more planes as symmetric planes. Taking the symmetric plane yoz as an example, it can be seen that the related unit groups have the following characteristics: the y-coordinates and z-coordinates are the same, and the x-coordinates are opposites of each other. By using Python, the x-coordinates of the centers of all units are made to be the absolute values of the original values. At this time, the two design sub-regions constrained by symmetry will coincide in space. Specifically, the center coordinates of each unit and its corresponding unit will coincide. Thus, the binary tree algorithm can be used to extract each related unit group simply and efficiently, achieving sensitivity averaging.
[0126] For rotationally symmetric constraints, such as Figure 12 As shown, considering applicability in three-dimensional cases, the Cartesian coordinates (x, y, z) of the element center are first converted to spherical polar coordinates (r, θ, φ), where the radial distance r is the distance from the coordinate point to the sphere center o, the polar angle θ is the angle between the z-axis and r, and the azimuth angle φ is the angle between the projection line of r on the xoy plane and the positive x-axis. Under this constraint, the associated element group has the following characteristics: r and θ have the same value, and φ increases by an angle kψ in each symmetrical sub-region starting from the first symmetrical sub-region, where k represents the k-th symmetrical sub-region, and ψ represents the angle subtended by each symmetrical sub-region. For any user-defined number of symmetries n, ψ = 2π / n. By traversing the elements whose φ value falls within each sub-region using Python, the angle kψ is reduced accordingly. At this point, all sub-regions are rotated to coincide with the initial sub-region, and each element coincides with its symmetrical element.
[0127] For periodic structure constraints, the number of periodic structures in the x and y directions is m×n, as defined by the user. Figure 13As shown, taking a 3×2 design as an example, the design domain is uniformly divided into 3×2 design sub-regions. Under periodic structure constraints, the associated unit groups have the following characteristics: In the x-direction, the x-coordinate of the unit center increases by a certain value kλ in each design sub-region, starting from the first design sub-region. Here, k represents the k-th design sub-region, and λ represents the length of each design sub-region in the x-direction. For the number of periodic structures m and the total length l of the design domain, λ = l / m. The y-direction follows the same principle as the x-direction, with each sub-region correspondingly increasing by a value kμ. Traversing the units within each sub-region, the center coordinates of each unit are reduced by kλ in the x-direction and by kμ in the y-direction, so that all sub-regions are translated to coincide with the initial sub-region. At this point, each unit coincides with its corresponding periodic unit.
[0128] For the planar symmetric example, Example 1: This example examines the implementation of the program on a planar symmetric plane. For example... Figure 14 The design domain is 120×40×4, fixed at the left end, with a downward distributed load of magnitude 1 applied to the lower right free edge. A regular mesh with element side length 1 is used for discretization, and BESO parameters are set as follows: ER=1%, V* / V0=0.4. The optimization results using the general BESO method are shown in Figure 15(a), the optimization results with single-plane xoz symmetric constraints are shown in Figure 15(b), and the optimization results with biplane xoz and yoz symmetric constraints are shown in Figure 15(c).
[0129] Example 2: The second planar symmetric example considers a double L-shaped beam structure. This example examines the feasibility of symmetry under irregular meshes. The structure is discretized using planar triangular elements with an element size of 2. Its boundary conditions are as follows: Figure 16 As shown, this is similar to joining two L-shaped beams together, with a top constraint and vertically downward loads of magnitude 10 applied to the left and right corners. In addition, a horizontal load of magnitude 3 is applied to the left corner to create the asymmetry of the topology optimization result. An irregular mesh with an element size of 2 is used for discretization, and the BESO parameters are set as follows: ER=2%, V* / V0=0.5. The topology optimization results before and after applying planar symmetry constraints are shown in Figure 17(a) and Figure 17(b). Figure 17(a) is the material distribution diagram of non-planar symmetry constraints, and Figure 17(b) is the material distribution diagram of planar yoz symmetry constraints.
[0130] For the rotationally symmetric examples, Example 1: The first rotationally symmetric example considers a two-dimensional circular torus structure, with boundary conditions as follows: Figure 18As shown: the outer diameter is 50, the inner diameter is 15, the six degrees of freedom of the inner circle are fully constrained, and an external load of magnitude 1 is applied vertically downwards at the tangent of the outer circle profile. The structure is divided into 60 layers of elements radially, each layer consisting of 1440 irregular elements. BESO related parameters are set as follows: ER=4%, V* / V0=0.5. With an arbitrary circumferential symmetry number of n, the BESO material distribution results are as follows: Figure 19 As shown.
[0131] Example 2: The second example, which is rotationally symmetric, considers a tripod support structure for an offshore wind turbine, with boundary conditions as follows: Figure 20 As shown in the figure, the regular triangular prism represents the design domain, with its top surface being an equilateral triangle with a side length of 500 and a height of 300. The base of the tower outside the design domain is connected to the tripod, and a bending moment of magnitude 1×10⁵ in the y-direction acts on the rigid element, connecting it to the top of the tower. An irregular mesh with an element side length of 4.5 is used for discretization, and the relevant BESO parameters are set as follows: ER=5%, V* / V0=0.15. The topology optimization results considering rotational symmetry constraints are shown in Figure 21(a) and Figure 21(b). The evolution history of compliance and volume ratio is shown in Figure 21(c).
[0132] For the periodic structure example, Example 1: This example is a two-dimensional L-shaped beam, with boundary conditions as follows: Figure 22 As shown: The top edge is fixed, and a vertically downward load of magnitude 100 is applied to the top right corner. The period in the x and y directions is defined as m×k, and the design domain is divided into sub-regions of 2×2, 4×4, 6×6, 8×8, 10×10, 12×12, 14×14, and 16×16 respectively. Each sub-region is discretized using a regular 40×40 grid, with a constant filter radius of 4 times the element edge. BESO parameters are set to ER = 5%, V* / V0 = 0.5. The final topology optimization results for a single sub-region and the whole are shown below. Figure 23 As shown.
[0133] It should be noted that although the steps in the above embodiments are described in a specific order, those skilled in the art will understand that in order to achieve the effect of this application, different steps do not necessarily have to be executed in such an order. They can be executed simultaneously (in parallel) or in other orders. These adjusted solutions are equivalent to the technical solutions described in this application and therefore will also fall within the protection scope of this application.
[0134] Those skilled in the art will understand that all or part of the processes in the method of the above-described embodiment can also be implemented by a computer program instructing related hardware. The computer program can be stored in a computer-readable storage medium, and when executed by a processor, it can implement the steps of the various method embodiments described above. The computer program includes computer program code, which can be in the form of source code, object code, executable file, or some intermediate form. The computer-readable storage medium can include any entity or device capable of carrying the computer program code, a medium, a USB flash drive, a portable hard drive, a magnetic disk, an optical disk, a computer memory, a read-only memory, a random access memory, an electrical carrier signal, a telecommunication signal, and a software distribution medium, etc.
[0135] Another aspect of this application provides a computer device.
[0136] In one embodiment of a computer device according to this application, the computer device may include at least one processor; and a memory communicatively connected to the at least one processor; wherein the memory stores a computer program, which, when executed by the at least one processor, implements the method described in any of the above embodiments. See Appendix Figure 24 , Figure 24 The image exemplarily illustrates a communication connection between memory 11 and processor 12 via a bus.
[0137] Another aspect of this application provides a computer-readable storage medium.
[0138] In one embodiment of a computer-readable storage medium according to this application, the computer-readable storage medium may be configured to store a program that performs the topology optimization method based on binary tree search described in the above-described method embodiments. This program may be loaded and run by a processor to implement the aforementioned topology optimization method based on binary tree search. For ease of explanation, only the parts related to the embodiments of this application are shown; for specific technical details not disclosed, please refer to the method section of the embodiments of this application. The computer-readable storage medium may be a storage device comprising various electronic devices. Optionally, in the embodiments of this application, the computer-readable storage medium is a non-transitory computer-readable storage medium.
[0139] The technical solution of this application has been described above with reference to one embodiment shown in the accompanying drawings. However, it will be readily understood by those skilled in the art that the scope of protection of this application is obviously not limited to these specific embodiments. Without departing from the principles of this application, those skilled in the art can make equivalent changes or substitutions to the relevant technical features, and the technical solutions after these changes or substitutions will all fall within the scope of protection of this application.
Claims
1. A topology optimization method based on binary tree search, characterized in that, The method includes: S101: Establish a finite element model and define the design domain of the structure, the physical parameters of the materials, and the boundary conditions of the structure; S102: Perform finite element analysis on the finite element model to determine the strain energy parameters of the elements in the finite element model; S103: Determine the sensitivity number of each unit based on the strain energy parameters and density of the unit; S104: Based on the binary tree search algorithm and the sensitivity number of each unit, the sensitivity number of the unit is corrected; S105: Update the design domain of the finite element model based on the corrected sensitivity number of the element; S106: Repeat steps S102 to S105 until the updated finite element model satisfies the preset convergence condition, and obtain the topology optimization result of the finite element model. The physical parameters of the material include its elastic modulus and element density. The finite element analysis of the finite element model to determine the strain energy parameters of the elements includes: determining the element elastic modulus based on the material's elastic modulus, a preset penalty factor, and element density; defining an objective function based on the principle of static equilibrium and the average compliance characterizing the structural strain energy; and using the volume constraint ratio as a constraint condition, performing finite element analysis on the finite element model based on the element elastic modulus and the objective function to determine the strain energy parameters of the elements. The step of determining the sensitivity number of each unit based on the strain energy parameter and density of the unit includes: determining whether the unit is a solid unit or a virtual unit based on the density of the unit; if the unit is a solid unit, the sensitivity number of the unit is equal to the strain energy parameter of the unit; if the unit is a virtual unit, the sensitivity number of the unit is equal to zero.
2. The topology optimization method based on binary tree search according to claim 1, characterized in that, The correction of the sensitivity number of a unit based on the binary tree search algorithm and the sensitivity number of each unit includes: Based on the preset filtering radius and preset filtering rules, a binary tree search algorithm is used in the preset storage area to obtain all neighboring units corresponding to the unit. The sensitivity number of the unit is corrected based on the sensitivity numbers of all the neighboring units.
3. The topology optimization method based on binary tree search according to claim 2, characterized in that, The finite element model has multiple elements stored in a preset storage area of a binary tree structure, with each element corresponding to a stored coordinate value. Based on a preset filtering radius and preset filtering rules, a binary tree search algorithm is used in the preset storage area to obtain all neighboring elements corresponding to each element, including: Using a binary tree search algorithm, based on the storage coordinates of the cell and the preset filtering radius, all neighboring cells within the preset filtering radius range of the cell are searched in the preset storage area.
4. The topology optimization method based on binary tree search according to claim 3, characterized in that, The step of correcting the sensitivity number of a unit based on the sensitivity numbers of all its neighboring units includes: The first sensitivity number of the unit is determined based on the sensitivity number of the neighboring unit, the number of the neighboring units, the distance between the neighboring units and the unit, and a preset weighting factor. Obtain the sensitivity number of the unit determined in the previous iteration; Based on the sensitivity number of the unit determined in the previous iteration, the first sensitivity number of the unit is corrected to obtain the corrected sensitivity number of the unit.
5. The topology optimization method based on binary tree search according to claim 1, characterized in that, The update of the design domain of the finite element model based on the corrected sensitivity number of the element includes: The density of the units is updated based on the corrected sensitivity number of the units and the preset addition and deletion rules; Based on the density of the element, the element is added to the design domain of the finite element model in this iteration.
6. The topology optimization method based on binary tree search according to claim 5, characterized in that, The step of updating the density of the units based on the corrected sensitivity number of the units and preset addition / deletion rules includes: The target volume for this iteration is determined based on a preset evolution rate parameter; Based on the corrected sensitivity number of the unit and the target volume, a sensitivity number threshold is determined; Determine whether the sensitivity number of the corrected unit is greater than the sensitivity number threshold; If so, update the density of the unit to the first preset value; If not, update the density of the unit to a second preset value; wherein the first preset value is greater than the second preset value.
7. The topology optimization method based on binary tree search according to claim 6, characterized in that, The steps S102 to S105 are repeated until the updated finite element model satisfies the preset convergence condition, and the topology optimization result of the finite element model is obtained, including: Determine whether the target volume of the finite element model in this iteration has reached the preset structural volume; If the target volume of this iteration reaches the preset structural volume, the preset convergence condition is determined based on the change in the objective function and the number of iterations. If so, the finite element model will be output as the topology optimization result; If not, repeat steps S102 to S105.
8. A computer device comprising at least one processor and at least one memory, said memory being adapted to store a plurality of program codes, characterized in that, The program code is adapted to be loaded and run by the processor to perform the topology optimization method based on binary tree search as described in any one of claims 1 to 7.
9. A computer-readable storage medium storing a plurality of program codes, characterized in that, The program code is adapted to be loaded and run by a processor to perform the topology optimization method based on binary tree search as described in any one of claims 1 to 7.