采用改进导重法的拓扑结构灰度单元过滤技术

高翔; 王林军; 杜义贤; 付君健

doi:10.13700/j.bh.1001-5965.2020.0728

采用改进导重法的拓扑结构灰度单元过滤技术

doi: 10.13700/j.bh.1001-5965.2020.0728

高翔^{1, 2},
王林军^{1, 2, ,},
杜义贤^{1, 2},
付君健^{1, 2}

1.
三峡大学水电机械设备设计与维护湖北省重点实验室, 宜昌 443002
2.
三峡大学机械与动力学院, 宜昌 443002

基金项目:

国家自然科学基金 51775308

水电机械设备设计与维护湖北省重点实验室开放基金 2019KJX12

详细信息

通讯作者:
王林军, E-mail: ljwang2006@126.com

中图分类号: TH122
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- 文章访问数: 210
- HTML全文浏览量: 40
- PDF下载量: 16
- 被引次数: 0
出版历程
- 收稿日期: 2021-01-02
- 录用日期: 2021-03-29
- 网络出版日期: 2022-06-20
- 整期出版日期: 2022-06-20

Gray element filtering technology of topology structure based on improved guide-weight method

GAO Xiang^{1, 2},
WANG Linjun^{1, 2
, ,},
DU Yixian^{1, 2},
FU Junjian^{1, 2}

1.
Hubei Key Laboratory of Hydroelectric Machinery Design and Maintenance, China Three Gorges University, Yichang 443002
2.
College of Mechanical and Power, China Three Gorges University, Yichang 443002, China

Funds:

National Natural Science Foundation of China 51775308

Open Foundation of Hubei Key Laboratory of Hydroelectric Machinery Design and Maintenance 2019KJX12

More Information

Corresponding author: WANG Linjun, E-mail: ljwang2006@126.com

摘要

摘要:
针对二分法计算拉格朗日乘子时收敛速度较慢的问题, 提出了拉格朗日乘子计算方法, 应用于优化准则(OC)法和导重(GW)法2种密度更新方法, 并与二分法进行了对比。建立体积约束下柔度最小的拓扑优化模型；通过固体各向同性材料惩罚(SIMP)法或材料属性有理近似(RAMP)法计算单元的弹性模量；通过所提方法计算拉格朗日乘子, 并通过导重法更新单元密度；通过Heaviside投影函数减少灰度单元的数量。计算结果表明：虽然所提方法对有限元分析次数并没有显著改进, 但计算拉格朗日乘子所用CPU时间少于二分法, 且密度更新次数降低至50%以下；在2个数值算例中, 采用SIMP模型时, 导重法所得结构柔度比OC法更小, 能够得到刚度更高的结构。
- 二分法 /
- 优化准则(OC)法 /
- 导重(GW)法 /
- Heaviside投影函数 /
- 拓扑优化
Abstract:
Aiming at the problem of slow convergence speed when calculating Lagrange multiplier by bisection method, the proposed Lagrange multiplier calculation method is applied to the optimal criteria (OC) method and the guide-weight (GW) method to update the density, and the results of the proposed method are compared with the bisection method. The topology optimization model with the smallest compliance under the volume constraint is established. The elastic modulus of element density is calculated by the solid isotropic material with penalization (SIMP) or rational approximation of material properties (RAMP) method. The multiplier is calculated by the proposed method in this paper, and the element density is updated by the GW method. The number of gray element is reduced by the Heaviside projection function. The computational results show that: the proposed method does not significantly reduce the number of finite element analysis, but the CPU time used by the proposed method to calculate the Lagrange multiplier is less than that of the bisection method, and the number of density updates is reduced to less than 50% than before. In addition, in the two numerical examples, when SIMP model is adopted, the structure obtained by the GW method has smaller compliance than the OC method.
- bisection method /
- optimal criteria (OC) method /
- guide-weight (GW) method /
- Heaviside projection function /
- topology optimization

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图 1 SIMP法和RAMP法的弹性模量对比

Figure 1. Elastic module comparison of SIMP and RAMP methods

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图 2 二维简支梁的结构示意图

Figure 2. Schematic structure of two-dimensional simply supported beam

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图 3 OC法求解SIMP模型(简支梁)

Figure 3. OC method for solving SIMP model (simply supported beam)

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图 4 导重法求解SIMP模型(简支梁)

Figure 4. GW method for solving SIMP model (simply supported beam)

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图 5 OC法求解RAMP模型(简支梁)

Figure 5. OC method for solving RAMP model (simply supported beam)

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图 6 导重法求解RAMP模型(简支梁)

Figure 6. GW method for solving RAMP model (simply supported beam)

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图 7 二维悬臂梁的结构示意图

Figure 7. Schematic structure of two-dimensional cantilever beam

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图 8 OC法求解SIMP模型(悬臂梁)

Figure 8. OC method for solving SIMP model (cantilever beam)

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图 9 导重法求解SIMP模型(悬臂梁)

Figure 9. GW method for solving SIMP model (cantilever beam)

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图 10 OC法求解RAMP模型(悬臂梁)

Figure 10. OC method for solving RAMP model (cantilever beam)

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图 11 导重法求解RAMP模型(悬臂梁)

Figure 11. GW method for solving RAMP model (cantilever beam)

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表 1 OC法求解SIMP模型所得结果(简支梁)

Table 1. Results from OC method of solving SIMP model (simply supported beam)

Heaviside 投影函数	OC法(二分法)					OC法(本文方法)
	迭代次数		结构柔度	CPU时间/s	拓扑优化结构	迭代次数		结构柔度	CPU时间/s	拓扑优化结构
	外层	总数	结构柔度	CPU时间/s	拓扑优化结构	外层	总数(改进)	结构柔度	CPU时间/s	拓扑优化结构
式(14)	445	20 761	14.447 6	63.187 0		469	9 392 (54.76%)	14.444 8	62.510 5
式(15)	479	22 332	14.438 5	65.354 4		481	9 813 (56.06%)	14.438 6	64.020 7

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表 2 导重法求解SIMP模型所得结果(简支梁)

Table 2. Results from GW method of solving SIMP model (simply supported beam)

Heaviside 投影函数	导重法(二分法)					导重法(本文方法)
	迭代次数		结构柔度	CPU时间/s	拓扑优化结构	迭代次数		结构柔度	CPU时间/s	拓扑优化结构
	外层	总数	结构柔度	CPU时间/s	拓扑优化结构	外层	总数(改进)	结构柔度	CPU时间/s	拓扑优化结构
式(14)	517	23 821	14.362 4	70.171 0		517	10 470 (56.05%)	14.362 4	68.878 6
式(15)	559	25 780	14.369 8	75.572 8		559	11 389 (55.82%)	14.369 8	74.951 0

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表 3 OC法求解RAMP模型所得结果(简支架)

Table 3. Results from OC method of solving RAMP model (simply supported beam)

Heaviside 投影函数	OC法(二分法)					OC法(本文方法)
	迭代次数		结构柔度	CPU时间/s	拓扑优化结构	迭代次数		结构柔度	CPU时间/s	拓扑优化结构
	外层	总数	结构柔度	CPU时间/s	拓扑优化结构	外层	总数(改进)	结构柔度	CPU时间/s	拓扑优化结构
式(14)	489	23 001	14.463 7	67.421 9		491	10 227 (55.54%)	14.463 5	65.436 4
式(15)	488	22 925	14.618 5	67.756 6		485	9 469 (58.70%)	14.620 0	64.710 9

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表 4 导重法求解RAMP模型所得结果(简支梁)

Table 4. Results from GW method of solving RAMP model (simply supported beam)

Heaviside 投影函数	导重法(二分法)					导重法(本文方法)
	迭代次数		结构柔度	CPU时间/s	拓扑优化结构	迭代次数		结构柔度	CPU时间/s	拓扑优化结构
	外层	总数	结构柔度	CPU时间/s	拓扑优化结构	外层	总数(改进)	结构柔度	CPU时间/s	拓扑优化结构
式(14)	554	25 591	14.552 1	74.083 1		553	11 551 (54.86%)	14.552 1	71.908 0
式(15)	490	22 507	14.509 1	67.468 5		490	10 158 (54.87%)	14.509 1	64.695 8

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表 5 OC法求解SIMP模型所得结果(悬臂梁)

Table 5. Results from OC method of solving SIMP mode (cantilever beam)

Heaviside 投影函数	OC法(二分法)					OC法(本文方法)
	迭代次数		结构柔度	CPU时间/s	拓扑优化结构	迭代次数		结构柔度	CPU时间/s	拓扑优化结构
	外层	总数	结构柔度	CPU时间/s	拓扑优化结构	外层	总数(改进)	结构柔度	CPU时间/s	拓扑优化结构
式(14)	474	20 470	156.289 4	67.346 9		379	7 059 (65.52%)	156.724 5	48.525 3
式(15)	395	17 089	156.632 5	56.269 1		396	7 262 (57.50%)	156.632 5	52.822 6

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表 6 导重法求解SIMP模型所得结果(悬臂梁)

Table 6. Results from GW method of solving SIMP model (cantilever beam)

Heaviside 投影函数	导重法(二分法)					导重法(本文方法)
	迭代次数		结构柔度	CPU时间/s	拓扑优化结构	迭代次数		结构柔度	CPU时间/s	拓扑优化结构
	外层	总数	结构柔度	CPU时间/s	拓扑优化结构	外层	总数(改进)	结构柔度	CPU时间/s	拓扑优化结构
式(14)	519	22 002	156.118 3	70.463 3		519	9 509 (56.78%)	156.117 9	65.420 6
式(15)	528	22 369	156.096 8	68.503 2		528	9 646 (56.88%)	156.096 5	66.566 9

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表 7 OC法求解RAMP模型所得结果(悬臂梁)

Table 7. Results from OC method of solving RAMP model (cantilever beam)

Heaviside 投影函数	OC法(二分法)					OC法(本文方法)
	迭代次数		结构柔度	CPU时间/s	拓扑优化结构	迭代次数		结构柔度	CPU时间/s	拓扑优化结构
	外层	总数	结构柔度	CPU时间/s	拓扑优化结构	外层	总数(改进)	结构柔度	CPU时间/s	拓扑优化结构
式(14)	528	22 913	155.547 8	74.144 5		531	9 680 (57.75%)	155.547 7	67.407 0
式(15)	445	19 299	155.547 6	58.749 6		445	8 447 (56.23%)	155.547 8	56.336 7

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表 8 导重法求解RAMP模型所得结果(悬臂梁)

Table 8. Results from GW method of solving RAMP mode (cantileve beam)

Heaviside 投影函数	导重法(二分法)					导重法(本文方法)
	迭代次数		结构柔度	CPU时间/s	拓扑优化结构	迭代次数		结构柔度	CPU时间/s	拓扑优化结构
	外层	总数	结构柔度	CPU时间/s	拓扑优化结构	外层	总数(改进)	结构柔度	CPU时间/s	拓扑优化结构
式(14)	579	24 683	155.949 9	74.149 1		579	11 262 (54.37%)	155.949 9	71.664 8
式(15)	546	23 200	155.926 1	69.964 2		546	10 469 (54.88%)	155.926 0	67.687 6

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