基于变密度胞元的热传导结构层级拓扑优化

王钦; 刘利阳; 强博; 魏衍强

doi:10.21656/1000-0887.430383

基于变密度胞元的热传导结构层级拓扑优化

doi: 10.21656/1000-0887.430383

沈阳飞机设计研究所，沈阳 110035

详细信息

通讯作者:
王钦(1985—)，男，高级工程师(通讯作者. E-mail: 2636991085@qq.com)

中图分类号: O414
计量
- 文章访问数: 560
- HTML全文浏览量: 208
- PDF下载量: 63
- 被引次数: 0
出版历程
- 收稿日期: 2022-11-29
- 修回日期: 2023-09-13
- 刊出日期: 2023-09-01

Topology Optimization Design of Heat Convection Problems With Variable-Density Cells

Shenyang Aircraft Design & Research Institue, Shenyang 110035, P.R.China

摘要

摘要: 为获得优异的散热结构设计，发展了一种基于腐蚀-扩散算子的变密度胞元层级结构设计方法. 通过腐蚀-扩散算子得到了一系列拓扑相似但体积分数不同的变密度微结构，计算并拟合得到变密度微结构等效热传导系数曲线. 在此基础上，采用移动渐近线法更新宏观设计变量，将变密度微结构植入相应体积分数的宏观单元中完成装配. 通过数值算例对不同优化方法下温度场的热柔顺度、平均温度、方差等参数进行了比较分析，结果表明，变密度胞元层级结构比传统单尺度胞元结构和周期胞元结构具有更好的散热性能.
- 拓扑优化 /
- 变密度胞元 /
- 腐蚀-扩散算子 /
- 渐近均匀化方法 /
- 热传导问题
Abstract: The method of designing cell structures with variable-density cells based on erode-dilate operators was applied to the optimization of thermal structures. A series of variable-density microstructures with the same topology definition but different volume fractions were obtained with erode-dilate operators. Then, the thermal conductivity of the variable-density cells was extracted and the equivalent thermal conductivity fitting curve was obtained. After this, the method of moving asymptotes was used to update macro design variables, and the variable-density microstructure was implanted into the macroscopic unit of the corresponding volume fraction to complete the assembly. Numerical examples were given to compare and analyze the thermal compliance, the mean temperature, the variance and other parameters of the temperature fields with different optimization methods. The results show that, variable-density cell structures have better heat dissipating performance than traditional single-scale cell structures and periodic cell structures.
- topology optimization /
- variable-density cell /
- erode-dilate operator /
- asymptotic homogenization method /
- heat convection problem

HTML全文

图 1 变密度胞元结构及其几何参数与密度的关系

Figure 1. Illustration of the variable density cellular structure

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图 2 腐蚀-扩散算子

Figure 2. The process of the "erode" and "dilate" operators

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图 3 提出算法的优化流程图

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 3. The optimization flowchart of the proposed method

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图 4 设计域及边界条件

Figure 4. The design domain and boundary conditions

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图 5 基础微结构(策略1)

Figure 5. The elementary microstructure for strategy 1

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图 6 网格细化示意图

Figure 6. Mesh refinement

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图 7 准周期微结构库①

Figure 7. The microstructural library ①

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图 8 基础微结构(策略2)

Figure 8. The elementary microstructure for strategy 2

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图 9 准周期微结构库②

Figure 9. Microstructural library ②

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图 10 等效热传导系数曲线

Figure 10. Fitting curves of D₁₁ and D₂₂ with the variation of ρ_e^ma

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图 11 迭代过程

Figure 11. The iteration process of example 1

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图 12 优化结果

Figure 12. The optimized structures for different methods

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图 13 准周期微结构库③

Figure 13. Microstructural library ③

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图 14 准周期微结构库③的等效热传导系数曲线

Figure 14. Fitting curves of D₁₁ and D₂₂ with the variation of ρ_e^ma in library ③

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图 15 准周期微结构库③的优化结果

Figure 15. The optimized structures for library ③

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图 16 宏观单尺度拓扑优化结果

Figure 16. The topology optimization result of the single macro-scale design

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图 17 精细网格下的温度场优化结果

Figure 17. Temperature fields of the 4 optimization results with the fine mesh

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表 1 不同策略优化结果的结构散热性能

Table 1. Thermal performances of structures optimized by different strategies

	compliance	average temperature	maximum temperature	variance
microstructural library ①	3.541 1E4	3.374 1E2	1.094 8E3	4.751 1E4
microstructural library ②	3.133 7E4	2.977 8E2	1.043 3E3	4.033 0E4
microstructural library ③	3.303 3E4	3.126 1E2	1.001 9E3	4.125 8E4
the single macro-scale design	4.963 6E4	4.757 3E2	1.170 6E3	7.221 5E4

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参考文献(32)

[1]	彭世彬, 郭瑞, 冯上升, 等. 主动冷却点阵夹层防热结构温度响应计算模型[J]. 应用数学和力学, 2022, 43(5): 477-489. doi: 10.21656/1000-0887.420405 PENG Shibin, GUO Rui, FENG Shangsheng, et al. A calculation model for temperature responses of active cooling lattice sandwich structures for thermal protection[J]. Applied Mathematics and Mechanics, 2022, 43(5): 477-489. (in Chinese) doi: 10.21656/1000-0887.420405
[2]	ZHU J H, ZHANG W H, XIA L. Topology optimization in aircraft and aerospace structures design[J]. Archives of Computational Methods in Engineering, 2015, 23: 595-622.
[3]	LIU S, HU R, LI Q, et al. Topology optimization-based lightweight primary mirror design of a large-aperture space telescope[J]. Applied Optics, 2014, 53(35): 8318-8325.
[4]	张东东, 栾福强, 赵礼辉, 等. 基于变体积约束的阻尼材料微结构拓扑优化研究[J]. 应用数学和力学, 2022, 43(6): 648-659. doi: 10.21656/1000-0887.420206 ZHANG Dongdong, LUAN Fuqiang, ZHAO Lihui, et al. Research on topology optimization of damping material microstructures with varied volume constraints[J]. Applied Mathematics and Mechanics, 2022, 43(6): 648-659. (in Chinese) doi: 10.21656/1000-0887.420206
[5]	RODRIGUES H, GUEDES J M, BENDSOE M P. Hierarchical optimization of material and structure[J]. Structural and Multidisciplinary Optimization, 2002, 24(1): 1-10. doi: 10.1007/s00158-002-0209-z
[6]	XIA L, BREITKOPF P. Concurrent topology optimization design of material and structure within FE² nonlinear multiscale analysis framework[J]. Computer Methods in Applied Mechanics and Engineering, 2014, 278: 524-542. doi: 10.1016/j.cma.2014.05.022
[7]	XIA L, BREITKOPF P. Multiscale structural topology optimization with an approximate constitutive model for local material microstructure[J]. Computer Methods in Applied Mechanics and Engineering, 2015, 286: 147-167. doi: 10.1016/j.cma.2014.12.018
[8]	LIU L, YAN J, CHENG G. Optimum structure with homogeneous optimum truss-like material[J]. Computers & Structures, 2008, 86(13/14): 1417-1425.
[9]	YAN J, CHENG G D, LIU L. A uniform optimum material based model for concurrent optimization of thermoelastic structures and materials[J]. International Journal for Simulation and Multidisciplinary Design Optimization, 2008, 2(4): 259-266. doi: 10.1051/ijsmdo/2008035
[10]	LONG K, WANG X, GU X. Concurrent topology optimization for minimization of total mass considering load-carrying capabilities and thermal insulation simultaneously[J]. Acta Mechanica Sinica, 2018, 34(2): 315-326. doi: 10.1007/s10409-017-0708-1
[11]	YAN X, HUANG X, SUN G, et al. Two-scale optimal design of structures with thermal insulation materials[J]. Composite Structures, 2015, 120: 358-365. doi: 10.1016/j.compstruct.2014.10.013
[12]	YAN J, GUO X, CHENG G. Multi-scale concurrent material and structural design under mechanical and thermal loads[J]. Computational Mechanics, 2016, 57(3): 437-446. doi: 10.1007/s00466-015-1255-x
[13]	ZHANG P, TOMAN J, YU Y, et al. Efficient design-optimization of variable-density hexagonal cellular structure by additive manufacturing: theory and validation[J]. Journal of Manufacturing Science & Engineering, 2015, 137(2): 021004-021013.
[14]	WANG B, CHENG G. Design of cellular structures for optimum efficiency of heat dissipation[J]. Structural and Multidisciplinary Optimization, 2005, 30(6): 447-458. doi: 10.1007/s00158-005-0542-0
[15]	CHENG L, LIU J, LIANG X, et al. Coupling lattice structure topology optimization with design-dependent feature evolution for additive manufactured heat conduction design[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 332: 408-439. doi: 10.1016/j.cma.2017.12.024
[16]	CHENG L, LIU J, TO A C. Concurrent lattice infill with feature evolution optimization for additive manufactured heat conduction design[J]. Structural and Multidisciplinary Optimization, 2018, 58(2): 511-535. doi: 10.1007/s00158-018-1905-7
[17]	WU Z, XIA L, WANG S, et al. Topology optimization of hierarchical lattice structures with substructuring[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 345: 602-617. doi: 10.1016/j.cma.2018.11.003
[18]	WANG Y, CHEN F, WANG M Y. Concurrent design with connectable graded microstructures[J]. Computer Methods in Applied Mechanics and Engineering, 2017, 317: 84-101. doi: 10.1016/j.cma.2016.12.007
[19]	WANG Y, ZHANG L, DAYNES S, et al. Design of graded lattice structure with optimized mesostructures for additive manufacturing[J]. Materials & Design, 2018, 142: 114-123.
[20]	ZONG H, LIU H, MA Q, et al. VCUT level set method for topology optimization of functionally graded cellular structures[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 354: 487-505. doi: 10.1016/j.cma.2019.05.029
[21]	ZHANG Y, LI H, XIAO M, et al. Concurrent topology optimization for cellular structures with nonuniform microstructures based on the Kriging metamodel[J]. Structural and Multidisciplinary Optimization, 2019, 59(4): 1273-1299. doi: 10.1007/s00158-018-2130-0
[22]	ZHANG Y, XIAO M, GAO L, et al. Multiscale topology optimization for minimizing frequency responses of cellular composites with connectable graded microstructures[J]. Mechanical Systems and Signal Processing, 2020, 135: 106369. doi: 10.1016/j.ymssp.2019.106369
[23]	SIGMUND O. Tailoring materials with prescribed elastic properties[J]. Mechanics of Materials, 1995, 20(4): 351-368. doi: 10.1016/0167-6636(94)00069-7
[24]	OSANOV M, GUEST J K. Topology optimization for architected materials design[J]. Annual Review of Materials Research, 2016, 46(1): 211-233. doi: 10.1146/annurev-matsci-070115-031826
[25]	CHEN W, LIU S. Topology optimization of microstructures of viscoelastic damping materials for a prescribed shear modulus[J]. Structural and Multidisciplinary Optimization, 2014, 50(2): 287-296. doi: 10.1007/s00158-014-1049-3
[26]	WANG Y, WANG M Y, CHEN F. Structure-material integrated design by level sets[J]. Structural and Multidisciplinary Optimization, 2016, 54(5): 1145-1156. doi: 10.1007/s00158-016-1430-5
[27]	杨航, 马力. 多材料点阵结构的热可编程力学行为[J]. 应用数学和力学, 2022, 43(5): 534-552. doi: 10.21656/1000-0887.430104 YANG Hang, MA Li. Multimaterial lattice structures with thermally programmable mechanical behaviors[J]. Applied Mathematics and Mechanics, 2022, 43(5): 534-552. (in Chinese) doi: 10.21656/1000-0887.430104
[28]	BOURDIN B. Filters in topology optimization[J]. International Journal for Numerical Methods in Engineering, 2001, 50(9): 2143-2158. doi: 10.1002/nme.116
[29]	PAPANICOLAU G, BENSOUSSAN A, LIONS J L. Asymptotic Analysis for Periodic Structures[M]. Elsevier, 1978.
[30]	WANG F, LAZAROV B S, SIGMUND O. On projection methods, convergence and robust formulations in topology optimization[J]. Structural and Multidisciplinary Optimization, 2011, 43(6): 767-784. doi: 10.1007/s00158-010-0602-y
[31]	BENDSØE M P, SIGMUND O. Material interpolation schemes in topology optimization[J]. Archive of Applied Mechanics, 1999, 69(9/10): 635-654.
[32]	SIGMUND O. Morphology-based black and white filters for topology optimization[J]. Structural and Multidisciplinary Optimization, 2007, 33(4/5): 401-424.