Multi-working-condition structure reliability topological optimization method based on PTO double-layer density filtering
By adopting the PTO-based double-layer density filtering method in topology optimization, the sensitivity calculation and optimization process is simplified, and the complexity of sensitivity derivation and poor performance in the existing topology optimization methods are solved, and faster convergence speed and a simpler topological configuration are achieved.
Patent Information
- Application Number
- CN202510026457.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-01-08
- Publication Date
- 2025-04-15
AI Technical Summary
The existing topology optimization methods are complex in the optimization process, resulting in poor performance of the generated topology structure and a large proportion of grayscale units.
The multi-case structure reliability topology optimization method based on PTO is adopted. Through internal and external circulation double-layer density filtration, sensitivity calculation is simplified to generate a smooth topology optimization structure.
The convergence speed is improved, the number of iterations is reduced, the generated topological configuration is simpler and more reasonable, the material utilization rate is higher, and the adaptability is stronger.
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Figure CN119830667A_ABST
Abstract
Description
Technical Field
[0001] The invention relates to the technical field of structural engineering, and in particular to a multi-operating condition structural reliability topology optimization method based on PTO double-layer density filtering. Background Art
[0002] Topological optimization refers to an optimization method that seeks the best material layout of a structure within a given design domain (such as whether there are holes in the structure, the location of the holes, the connection method of the components, etc.) under certain constraints. The difficulty of topological optimization lies in that there are many design forms that can meet certain requirements, but the topological form is generally difficult to quantitatively describe or parameterize. With the development of additive manufacturing technology, topological optimization has also made great achievements in the field of mechanical parts design. By optimizing the design, such as Figure 1 , topological optimization makes these mechanical parts both beautiful and practical. Topological optimization has also received more and more attention in large-scale construction machinery. The more typical topological optimization structure is the truss structure. Compared with ordinary cranes, truss cranes not only have better mechanical properties, but more importantly, they are smaller in size and lighter in weight, thus greatly reducing material costs.
[0003] In order to solve the defect of traditional topology optimization methods that require complex sensitivity calculations, a topology optimization method, proportional topology optimization (PTO), is disclosed in the prior art. The literature source: Biyikli E, To A C. Proportional topology optimization: a new non-sensitivity method for solving stress constrained and minimum compliance problems and its implementation in MATLAB[J]. PloS one, 2015, 10(12): e0145041. The defect of this topology optimization technology is that during the optimization process, the sensitivity derivation process is relatively complicated, and there is a large proportion of grayscale units, which leads to poor performance of the generated topology structure. Summary of the invention
[0004] In order to overcome or alleviate one or more of the above technical problems, the present invention aims to provide a multi-operating condition structural reliability topology optimization method based on double-layer density filtration of PTO.
[0005] The present invention provides the following technical solutions:
[0006] The multi-condition structural reliability topology optimization method based on double-layer density filtering of PTO includes the following steps:
[0007] S1: Set the initial parameters of topology optimization, including material elastic modulus, Poisson's ratio, penalty coefficient, historical equilibrium coefficient, target material TM and remaining material RM;
[0008] S2: Divide the unit, perform initial setting of finite element, and calculate weight coefficient;
[0009] S3: Enter the main loop, perform finite element analysis, and calculate the objective function value;
[0010] S4: Initialize the target material volume to the volume of the remaining material and calculate the unit compliance ratio;
[0011] S5: Entering the inner loop, allocating the remaining material volume to each unit according to the calculated flexibility ratio, and then performing density filtering;
[0012] S6: Update the unit inner loop density variable to determine whether the inner loop convergence condition is met, the inner loop convergence condition is whether the remaining material is less than 0.0001, the remaining material is the remaining unallocated unit total mass, if so, jump out of the inner loop and execute step S7; if not, continue to execute step S5;
[0013] S7: Perform outer loop density filtering on the density variable calculated and assigned by S6;
[0014] S8: Determine whether the outer loop convergence condition is met, the outer loop convergence condition is that the maximum value coefficient of unit density change change<0.01, if it is met, output the optimization result; if not, continue to jump to S3, enter the main loop again, and update the main loop density variable.
[0015] According to some embodiments, the unit flexibility ratio calculation formula in step S4 is:
[0016]
[0017] In the formula, C e represents the objective function value of unit e, i.e., the flexibility value, v e represents the volume of the unit, and then the remaining volume v RM The calculated flexibility is distributed to the corresponding units according to the formula:
[0018]
[0019] The cell density after allocation is x e pro Density filtering is also required to avoid numerical instability such as checkerboard and obtain a clearer and more reasonable optimization structure. The filtering formula is:
[0020]
[0021] Where, d e,x ,d s,x ,d e,y ,d s,y are the horizontal and vertical coordinates of units e and s, d(e,s) is the distance between the two units, H e,s is the weight coefficient, r min The filter radius is set, and the cell density value after allocation is limited to the range [x min ,x max ], the range is [0,1]:
[0022]
[0023] According to some embodiments, step S7 specifically comprises the following steps:
[0024] S71: The density variable after filtering in the inner loop is filtered again in the outer loop, and the improved expression is:
[0025]
[0026] The double-filtered density variables participate in the subsequent linear mixing calculations.
[0027] S72: After the outer loop density is filtered, a new parameter history balance coefficient α needs to be introduced to linearly mix the unit density obtained after the previous iteration optimization with the current unit density:
[0028]
[0029] in, Represents the cell density obtained after the last iterative optimization. is the current cell density, and the value range of α is 0 to 1.
[0030] Compared with the prior art, the present invention has the following beneficial effects:
[0031] The multi-condition structural reliability topology optimization method based on double-layer density filtering of PTO provided by the present invention adopts SIMP method to establish a proportional topology optimization material interpolation model, takes volume fraction minimization as a constraint, considers flexibility minimization, and obtains the design variable unit density x equivalent to the volume fraction. First, the initial parameters of topology optimization are established, the design domain is divided into unit grids, and the corresponding unit stiffness matrix KE, displacement number matrix, boundary conditions are applied, and density filters are set. Secondly, enter the main loop, perform finite element analysis, calculate the flexibility of each unit according to the calculation, and sum up all unit flexibility. Divide the flexibility of each unit by the total flexibility to calculate the flexibility ratio. Finally, the unit density is allocated according to the flexibility ratio. Compared with the topology optimization method based on sensitivity, the code is simple, the complex derivation process of sensitivity is reduced, and a smooth topology optimization structure can be generated at the same time, the material utilization rate is higher, the convergence speed is faster, the number of iterations is reduced, the adaptability is stronger, and a more concise and reasonable topological configuration can be obtained.
[0032] The advantages of the present invention are as follows:
[0033] (1) The convergence speed is faster, the number of iterations is smaller, and a more concise and reasonable topological configuration can be obtained, but it will lose a certain grayscale unit suppression ability, resulting in unclear topological boundaries, but the overall performance is still better than the original algorithm; (2) The proposed multi-operating condition reliability optimization method for continuum structures based on the proportional topology optimization algorithm is more stable and reasonable than the structure obtained by the conventional deterministic topology optimization method, which conforms to engineering practice. In addition, different values of the reliability index β and the selection of the operating condition weight coefficient will have a certain impact on the optimized structure. BRIEF DESCRIPTION OF THE DRAWINGS
[0034] Figure 1 A schematic diagram of topological optimization of mechanical parts provided as the background technology of the present invention.
[0035] Figure 2 A schematic diagram of a PTO process provided by an embodiment of the present invention.
[0036] Figure 3 A diagram of a two-dimensional cantilever beam structure provided in Example 2 of the present invention.
[0037] Figure 4 This is a comparison table of optimization results of the classic PTO algorithm provided in Example 2 of the present invention and the improved algorithm provided in the present invention.
[0038] Figure 5 This is a schematic diagram of the simply supported beam structure provided in Example 3 of the present invention.
[0039] Figure 6a This is a diagram of the optimization results of working condition 1 provided in Example 3 of the present invention.
[0040] Figure 6b This is a diagram of the optimization results of working condition 2 provided in Example 3 of the present invention.
[0041] Figure 6c This is a diagram of the optimization results of working condition 3 provided in Example 3 of the present invention.
[0042] Figure 7a This is an iteration diagram of the working condition and objective function of the simply supported beam structure provided in Example 3 of the present invention.
[0043] Figure 7b This is an iteration diagram of the objective function of the working condition 2 of the simply supported beam structure provided in Example 3 of the present invention.
[0044] Figure 7c This is an iteration diagram of the three objective functions of the working condition of the simply supported beam structure provided in Example 3 of the present invention.
[0045] Figure 8 This is a schematic diagram of a simply supported beam structure under different working condition weight coefficient combinations provided in Example 3 of the present invention.
[0046] Fig. 9 The reliability optimization structure diagram of the cantilever beam under the working condition weight coefficient of (0.5, 0.3, 0.2) provided in Example 3 of the present invention.
[0047] Fig.10 The reliability optimization structure diagram of the cantilever beam under the working condition weight coefficient of (0.2, 0.4, 0.4) provided in Example 3 of the present invention.
[0048] Fig.11 The reliability optimization structure diagram of the cantilever beam under the working condition weight coefficient of (0.4, 0.4, 0.2) provided in Example 3 of the present invention.
[0049] Fig.12 This is a comparison table of various result data under different reliability indicators β provided in Example 3 of the present invention. DETAILED DESCRIPTION
[0050] The present invention is described in detail below in conjunction with the embodiments and drawings, but it should be understood that the embodiments and drawings are only used to exemplify the present invention and do not constitute any limitation on the protection scope of the present invention. All reasonable changes and combinations within the scope of the inventive concept of the present invention fall within the protection scope of the present invention.
[0051] The material interpolation model of proportional topology optimization also adopts the SIMP method, so the mathematical model of proportional topology optimization with volume fraction as constraint and flexibility minimization is:
[0052]
[0053] In the formula, x represents the design variable unit density, U represents the displacement vector, K represents the overall stiffness matrix, and x e represents the cell density, v e represents the unit volume, M represents the total mass, C represents the flexibility, x min Represents the minimum limit for cell density.
[0054] The present invention will be described below in conjunction with the accompanying drawings.
[0055] like Figure 2 The multi-operating-condition structural reliability topology optimization method based on double-layer density filtration of PTO provided by the present invention includes two-layer inner and outer loops, the core of which lies in the flexibility distribution in the inner loop, and the basic steps are:
[0056] S1: Set the initial parameters of topology optimization, such as material elastic modulus, Poisson's ratio, penalty coefficient, historical equilibrium coefficient (α), target material (TM), remaining material (RM), etc.
[0057] S2: Divide the unit, perform initial setting of finite element, and calculate weight coefficient;
[0058] S3: Enter the main loop, perform finite element analysis, and calculate the objective function value;
[0059] S4: Initialize the target material volume to the volume of the remaining material and calculate the unit compliance ratio C_proportion;
[0060] S5: Entering the inner loop, allocating the remaining material volume to each unit according to the calculated flexibility ratio, and then performing density filtering;
[0061] S6: Update the density variable of the inner loop of the unit to determine whether the inner loop convergence condition of the remaining material (the remaining material is the total mass of the remaining unallocated unit) is less than 0.0001. If it is satisfied, jump out of the inner loop and execute step S7; if not, continue to execute step S5;
[0062] S7: updating the outer loop unit density of the density variable x calculated and allocated in S6;
[0063] S8: Determine whether the outer loop convergence condition is met, the outer loop convergence condition is that the maximum value coefficient of unit density change change<0.01, if it is met, output the optimization result; if not, continue to jump to S3, enter the main loop again, and update the main loop density variable.
[0064] Among them, the calculation formula of the unit flexibility ratio in step S4 is:
[0065]
[0066] In the formula, Ce represents the objective function value of unit e, i.e., the flexibility value, v e represents the volume of the unit, and then the remaining volume v RM The calculated flexibility is distributed to the corresponding units according to the formula:
[0067]
[0068] The cell density after allocation is x e pro Density filtering is also required to avoid numerical instability such as checkerboard and obtain a clearer and more reasonable optimization structure. The filtering formula is:
[0069]
[0070] Where, d e,x ,d s,x ,d e,y ,d s,y are the horizontal and vertical coordinates of units e and s, d(e,s) is the distance between the two units, H e,s is the weight coefficient, r min is the filter radius. However, it should be noted that the cell density value after allocation may exceed its boundary value, so it should be limited to a certain range [x min ,x max ], the range is usually [0,1]:
[0071]
[0072] The purpose of step S7 is to update the outer loop density variable, which includes the following steps:
[0073] In order to further improve the initial grid independence of the structure, this embodiment adds an inner and outer loop double-layer density filtering step based on the above steps, that is, the following step S71 is added after step S6:
[0074] S71: The density variable after filtering in the inner loop is filtered again in the outer loop, and the improved expression is:
[0075]
[0076] The double-filtered density variables participate in the subsequent linear mixing calculations.
[0077] S72: After the outer loop density is filtered, a new parameter history balance coefficient α needs to be introduced to linearly mix the unit density obtained after the previous iteration optimization with the current unit density:
[0078]
[0079] in, Represents the cell density obtained after the last iterative optimization. is the current cell density. α ranges from 0 to 1. It is an important and unique parameter in the proportional topology optimization algorithm. When it is set to 0, it means that the final cell density depends entirely on the density variable in the current inner loop and has nothing to do with the previous iteration. On the contrary, when it is 1, it means that the final cell density is only related to the density value in the previous iteration and has nothing to do with the current density value. When it is equal to 0.5, it means that the updated cell density is a mixture of half of the density in the previous iteration and half of the cell density in the current inner loop. It can be seen that its value will have a great impact on the optimization results.
[0080] Example 1
[0081] This embodiment is used to optimize the processing Figure 3 The two-dimensional cantilever beam structure shown has a structural aspect ratio of 3:1, a geometric size of 120mm×40mm, the left boundary is constrained by the fixed end, a vertical downward unit concentrated force is applied to the lower right corner, the structure is discretized into 120×40 units, the Poisson's ratio is 0.3, the penalty coefficient is set to 3, the volume fraction is set to 0.5, and the filter radius is set to 1.5.
[0082] The optimization results of the existing classic PTO optimization method and the improved PTO optimization method provided by the present invention under different historical balance coefficients are calculated, and the two methods are compared. Figure 4 As shown in Table 1.
[0083] First, by observing the topology optimization results under different historical equilibrium coefficients α in Table 1, it is found that both PTO methods can produce clear and reasonable topological structures, and their basic configurations are similar, but the difference is that the structure diagram optimized by the original PTO algorithm has more internal support structures, especially when the value of α is large, a part of small branching structures also appears. This not only greatly reduces the manufacturability of the structure, but also makes the structure unreasonable. Although the topological boundary of the structure optimized by the improved internal and external loop double-layer filtering method is not clear enough compared with the original PTO method, the overall structure is obviously simpler than the structure optimized by the original algorithm.
[0084] Secondly, when the historical balance coefficient α is the same, the structural flexibility values optimized by the improved method are slightly higher than those of the original method, and when α is different, there will be slight differences in the optimized structure, but overall the optimized structure of the improved double-layer density filtering method is more stable.
[0085] Finally, through Figure 4As shown in Table 1, it can be observed that the optimized structures by the original classical PTO method are not as concise and reasonable as the structures optimized by the improved PTO method.
[0086] Example 2
[0087] Beams are widely used in housing construction, engineering lifting, ships and other fields. If the beam is subjected to complex working conditions at the same time, a large-sized strong beam is required to meet the strength requirements. The beam structure is simplified and introduced into the topological optimization design.
[0088] This embodiment is used to optimize the processing Figure 5 The simply supported beam structure shown has a structural dimension of 240mm×40mm. The lower left corner is constrained by the fixed end and the lower right corner is hinged. A unit concentrated force is applied every 1 / 6 of the upper boundary of the model except the left and right endpoints, for a total of 5 loads, corresponding to three working conditions. The corresponding loads for each working condition are F1, F2, and F3 respectively.
[0089] The structure is discretized into 240 × 40 units, the Poisson’s ratio is 0.3, the volume fraction is set to 0.6, the penalty coefficient is set to 3, the filter radius is 1.5, and the historical equilibrium coefficient α is set to 0.4.
[0090] The optimization results under each single working condition are as follows: Figure 6a-6c As shown in the figure, the iteration diagram of the objective function of each working condition is as follows Figures 7a to 7c shown.
[0091] By observation Figure 6a-6c and Figures 7a to 7c It can be seen that the optimization results of the three single working conditions converge quickly, and a clear and reasonable topological structure appears, which is in full compliance with the law that "the support structure moves with the movement of the working condition load".
[0092] And after calculation, it is known that the number of iterations of working condition 1 is 162 times, the optimized initial flexibility value is 292.51N·mm, the final flexibility value is 85.93N·mm, and the maximum stress is 0.66MPa; the number of iterations of working condition 2 is 102 times, the optimized initial flexibility value is 873.45N·mm, the final flexibility value is 242.22N·mm, and the maximum stress is 1.31MPa; the number of iterations of working condition 3 is 51 times, the optimized initial flexibility value is 369.52N·mm, the final flexibility value is 92.25N·mm, and the maximum stress is 1.31MPa. And the maximum stress of the structure under the three working conditions is located at the constraints of the left and right endpoints of the lower boundary and the load action point of the upper boundary, which conforms to the law of force transfer path. Substituting these data into the multi-working condition calculation model, the optimized structure under four groups of different working condition weight coefficient combinations is obtained, such as Figure 8 As shown:
[0093] from Figure 8It can be seen from the figure that the multi-condition optimization structure is basically the superposition of the single-condition optimization structure characteristics, and the larger the weight coefficient of a condition, the more obvious its characteristics. Figure 8 In the four groups of graphs, from (a) to (d), as the weight coefficient of working condition 1 decreases, the support structure in the middle of the optimized structure becomes thinner and thinner, and the support structure on both sides becomes thicker and thicker. The following selects three groups of working condition weight coefficient combinations and gives their reliability optimization structure diagrams under different β values:
[0094] from Figures 9-11 It can be seen that in this example, the reliability analysis has little effect on the topological configuration of the structure. The structures of the four groups of images are very similar, and only the internal support structure has some changes. After the reliability analysis is introduced, the internal support structure gradually shifts to the middle, and the support structure at the bottom of both sides disappears, which corresponds to the weight coefficient, indicating that the structural material moves to the working position with a higher weight coefficient. Obviously, such a distribution is more reasonable and efficient.
[0095] Fig.12 Table 2 summarizes the various results under different reliability indicators β. It can be seen that with the increase of β value, the aspect ratio of the structure gradually increases, and the structure becomes more "flat". In addition, for the optimization results after considering reliability, the volume fractions are reduced by 5%, 10%, and 15%, respectively, the maximum stresses are increased by 5.1%, 10.2%, and 15.31%, respectively, and the objective function value also increases to a certain extent. Therefore, in summary, reliability analysis will reduce the volume fraction of the optimized structure, reduce manufacturing costs, and make the distribution of materials more reasonable.
[0096] The above embodiments are only preferred implementations of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions under the concept of the present invention belong to the protection scope of the present invention. It should be pointed out that for ordinary technicians in this technical field, improvements and modifications without departing from the principle of the present invention should also be regarded as the protection scope of the present invention.
Claims
1. A multi-condition structural reliability topology optimization method based on double-layer density filtration of PTO, characterized by: The following steps are involved: S1: Set the initial parameters of topology optimization, including material elastic modulus, Poisson's ratio, penalty coefficient, historical equilibrium coefficient, target material TM and remaining material RM; S2: Divide the unit, perform initial setting of finite element, and calculate weight coefficient; S3: Enter the main loop, perform finite element analysis, and calculate the objective function value; S4: Initialize the target material volume to the volume of the remaining material and calculate the unit compliance ratio; S5: Entering the inner loop, allocating the remaining material volume to each unit according to the calculated flexibility ratio, and then performing density filtering; S6: Update the unit inner loop density variable to determine whether the inner loop convergence condition is met, the inner loop convergence condition is whether the remaining material is less than 0.0001, the remaining material is the remaining unallocated unit total mass, if so, jump out of the inner loop and execute step S7; if not, continue to execute step S5; S7: Perform outer loop density filtering on the density variable calculated and assigned by S6; S8: Determine whether the outer loop convergence condition is met, the outer loop convergence condition is that the maximum value coefficient of unit density change change<0.01, if it is met, output the optimization result; if not, continue to jump to S3, enter the main loop again, and update the main loop density variable.
2. According to claim 1, the multi-operating condition structural reliability topology optimization method based on double-layer density filtration of PTO is characterized by: The calculation formula for the element flexibility ratio in step S4 is: In the formula, C e represents the objective function value of unit e, i.e., the flexibility value, v e represents the volume of the unit, and then the remaining volume v RM The calculated flexibility is distributed to the corresponding units according to the formula: Unit density after allocation Density filtering is also required to avoid numerical instability such as checkerboard and obtain a clearer and more reasonable optimization structure. The filtering formula is: Where, d e,x ,d s,x ,d e,y ,d s,y are the horizontal and vertical coordinates of units e and s, d(e,s) is the distance between the two units, H e,s is the weight coefficient, r min The filter radius is set, and the cell density value after allocation is limited to the range [x min ,x max ], the range is [0,1]:
3. The multi-operating condition structural reliability topology optimization method based on double-layer density filtration of PTO according to claim 1 is characterized in that: Step S7 specifically The following steps are involved: S71: The density variable after filtering in the inner loop is filtered again in the outer loop, and the improved expression is: The double-filtered density variables participate in the subsequent linear mixing calculations. S72: After the outer loop density is filtered, a new parameter history balance coefficient α needs to be introduced to linearly mix the unit density obtained after the previous iteration optimization with the current unit density: in, Represents the cell density obtained after the last iterative optimization. is the current cell density, and the value range of α is 0 to 1.