STRUCTURAL LIGHT DESIGN FOR STEADY HEAT CONDUCTION USING MULTI-MATERIAL
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摘要: 在多相材料的结构拓扑优化问题中，通常给定各相材料体积约束或材料总重量约束作为材料的控制用量.在结构轻量化设计的实际工程背景下，以结构总重量最小化为目标的优化模型具有明确的工程意义.针对含多相材料的稳态传热结构拓扑优化问题，提出了以结构总重量最小化为目标和给定热柔顺度为约束的多工况连续体结构拓扑优化建模方法.遵循独立连续映射建模方式，采用两类独立拓扑变量分别表征单元热传导矩阵和单元重量状态.推导了热柔顺度和总重量对设计变量的敏度，基于一阶和二阶泰勒展开得到各自的近似表达式.通过求解偏微分方程，实现了约束函数一次项过滤，消除了棋盘格现象和网格依赖性问题，并保证了约束方程在过滤后严格成立.建立的近似优化模型具有二次函数形式的目标函数和一次函数形式的约束函数.基于对偶序列二次规划方法对优化模型进行求解直至收敛.通过四个三维结构数值算例分析对比了热柔顺度约束限值、不同材料混合及多工况、多约束条件对优化结果的影响.数值算例结果表明，本文提出的优化方法在基于多相材料的多工况稳态热传导结构轻量化设计中具有可行性和有效性.Abstract: In topology optimization problems of structures containing multiphase materials, it is common practice to set the volume constraint of each constituent phase or total mass of entire constituent phase constraint to control the final material usage. On the practical engineering background for lightweight design, it is of significance that the minimized weight is taken as the objective in optimal model from the engineering point of view. To solve the topology optimization problem of steady heat conductive with the multiple candidate materials, a new modeling method of weight minimization with the given thermal compliance constraint under multiple load cases is proposed. Following the modeling manner of independent continuous mapping method, two sets of independent topological variables are employed to identify elemental thermal conductive matrix and elemental weight, respectively. The sensitivities of thermal compliance and global weight with respect to the design variable are derived, and their approximate expressions are calculated based on the first-order and second-order Taylor expansion. To eliminate checkerboard patterns and mesh-dependence, the first term of the constraint function is filtered as a solution of the partial differential equation, which also ensures the constraint equation is consistent. The approximate optimal model with the objective and constraint in the form of quadratic and linear function is established. The topological optimization model is solved by dual sequential quadratic programming. Various effects such as the constraint value of thermal compliance, the selection of multiple materials, and the multiple constraints in multiple load cases on the optimal result are discussed in four 3D numerical examples. The results demonstrate the feasibility and effectiveness of the proposed optimization approach regarding structural light design using multi-material in steady heat conduction.
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图 4 不同材料组合优化拓扑构型
Figure 4. Optimal topological configuration under different combination of basematerial
表 1 材料属性
Table 1. Properties of material
表 2 拓扑优化结果
Table 2. Results of topology optimization
表 3 拓扑优化结果
Table 3. Results of topology optimization
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[1] Bendsoe MP, Kikuchi N. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 1988, 71(2):197-224 doi: 10.1016/0045-7825(88)90086-2 [2] Eschenauer HA, Olhoff N. Topology optimization of continuum structures:a review. Applied Mechanics Reviews, 2001, 54(4):331-390 doi: 10.1115/1.1388075 [3] Rozvany GIN. A critical review of established methods of structural topology optimization. Structural and Multidisciplinary Optimiza-tion, 2009, 37(3):217-237 doi: 10.1007/s00158-007-0217-0 [4] Sigmund O, Maute K. Topology optimization approaches. Structural and Multidisciplinary Optimization, 2013, 48(6):1031-1055 doi: 10.1007/s00158-013-0978-6 [5] Deaton JD, Grandhi RV. A survey of structural and multidisciplinary continuum topology optimization:post 2000. Structural and Multidisciplinary Optimization, 2014, 49(1):1-38 doi: 10.1007/s00158-013-0956-z [6] Zhang W, Yuan J, Zhang J, et al. A new topology optimization approach based on Moving Morphable Components (MMC) and the ersatz material model. Structural and Multidisciplinary Optimization, 2016, 53:1243-1260 doi: 10.1007/s00158-015-1372-3 [7] Li Q, Steven GP, Querin OM, et al. Shape and topology design for heat conduction by evolutionary structural optimization. International Journal of Heat & Mass Transfer, 1999, 42(17):3361-3371 http://www.academia.edu/9126072/Shape_and_topology_design_for_heat_conduction_by_Evolutionary_Structural_Optimization [8] Li Q, Steven GP, Xie YM, et al. Evolutionary topology optimization for temperature reduction of heat conducting fields. International Journal of Heat & Mass Transfer, 2004, 47(23):5071-5083 https://www.researchgate.net/publication/222571292_Evolutionary_topology_optimization_for_temperature_reduction_of_heat_conducting_fields [9] Hansen A, Bendsoe M, Sigmund O. Topology optimization of heat conduction problems using the finite volume method. Structural and Multidisciplinary Optimization, 2006, 31(4):251-259 doi: 10.1007/s00158-005-0584-3 [10] Bruns TE. Topology optimization of convection-dominated steadystate heat transfer problems. International Journal of Heat and Mass Transfer, 2007, 50(15-16):2859-2873 doi: 10.1016/j.ijheatmasstransfer.2007.01.039 [11] Zhuang CG, Xiong ZH, Ding H. A level set method for topology optimization of heat conduction problem under multiple load cases. Computer Methods in Applied Mechanics and Engineering, 2007, 196(4-6):1074-1084 doi: 10.1016/j.cma.2006.08.005 [12] He D, Liu S. BESO method for topology optimization of structures with high efficiency of heat dissipation. International Journal for Simulation & Multidisciplinary Design Optimization, 2008, 2(1):43-48 https://www.researchgate.net/publication/245491486_BESO_method_for_topology_optimization_of_structures_with_high_efficiency_of_heat_dissipation [13] 乔赫廷, 张永存, 刘书田.散热结构拓扑优化目标函数的讨论.中国机械工程, 2011, 22(9):1112-1122 http://www.cnki.com.cn/Article/CJFDTOTAL-ZGJX201109023.htmQiao Heting, Zhang Yongcun, Liu Shutian. Discussion of objective functions in heat conduction topology optimization. China Mechanical Engineering, 2011, 22(9):1112-1122(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-ZGJX201109023.htm [14] 陈文炯, 刘书田, 张永存.基于拓扑优化的自发热体冷却用植入式导热路径设计方法.力学学报, 2016, 48(2):406-412 http://lxxb.cstam.org.cn/CN/abstract/abstract145641.shtmlChen Wenjiong, Liu Shutian, Zhang Yongcun. Optimization design of conductive pathways for cooling a heat generating body with high conductive inserts. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(2):406-412(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract145641.shtml [15] Gao T, Zhang WH, Zhu JH, et al. Topology optimization of heat conduction problem involving design-dependent heat load effect. Finite Elements in Analysis and Design, 2008, 44(4):805-813 https://www.researchgate.net/publication/222962873_Topology_optimization_of_heat_conduction_problem_involving_design-dependent_heat_load_effect [16] Zhuang CG, Xiong ZH, Ding H. Topology optimization of the transient heat conduction problem on a triangular mesh. Numerical Heat Transfer, Part B, 2013, 64(3):239-262 doi: 10.1080/10407790.2013.791785 [17] Zhuang CG, Xiong ZH. A global heat compliance measure based topology optimization for the transient heat conduction problem. Numerical Heat Transfer, Part B, 2014, 45(5):445-471 https://www.researchgate.net/publication/262985830_A_Global_Heat_Compliance_Measure_Based_Topology_Optimization_for_the_Transient_Heat_Conduction_Problem [18] Zhuang CG, Xiong ZH. Temperature-constrained topology optimization of transient heat conduction problems. Numerical Heat Transfer, Part B, 2015, 68(4):366-385 doi: 10.1080/10407790.2015.1033306 [19] Thomsen J. Topology optimization of structures composed of one or two materials. Journal of Structural Optimization, 1992, 5(1-2):108-115 doi: 10.1007/BF01744703 [20] Sigmund O, Torquato S. Design of materials with extreme thermal expansion using a three-phase topology optimization method. Journal of Mechanics and Physics of Solids, 1997, 45(6):1037-1067 doi: 10.1016/S0022-5096(96)00114-7 [21] Wang MY, Wang X. "Color" level sets:a multi-phase method for structural topology optimization with multiple materials. Computer Methods in Applied Mechanics and Engineering, 2004, 193(6):469-496 https://www.researchgate.net/publication/223614704_Color_level_sets_A_multi-phase_method_for_structural_topology_optimization_with_multiple_materials [22] Zhuang C, Xiong Z, Ding H. Topology optimization of multimaterial for the heat conduction problem based on the level set method. Engineering Optimization, 2010, 42(9):811-831 doi: 10.1080/03052150903443780 [23] Cui M, Chen H, Zhou J. A level-set based multi-material topology optimization method using a reaction diffusion equation. ComputerAided Design, 2016, 73:41-52 https://www.researchgate.net/publication/288919715_A_level-set_based_multi-material_topology_optimization_method_using_a_reaction_diffusion_equation [24] Wang Y, Luo Z, Kang Z, et al. A multi-material level set-based topology and shape optimization method. Computer Methods in Applied Mechanics and Engineering, 2015, 283:1570-1586 doi: 10.1016/j.cma.2014.11.002 [25] Liu P, Luo Y, Kang Z. Multi-material topology optimization considering interface behavior via XFEM and level set method. Computer Methods in Applied Mechanics and Engineering, 2016, 308:113-133 doi: 10.1016/j.cma.2016.05.016 [26] Wang MY, Zhou SW. Synthesis of shape and topology of multimaterial structures with a phase-field method. Journal of ComputerAided Materials Design, 2004, 11(2-3):117-138 doi: 10.1007/s10820-005-3169-y [27] Zhou SW, Wang MY. Multimaterial structural topology optimization with a generalized Cahn-Hilliard model of multiphase transition. Structural and Multidisciplinary Optimization, 2007, 33(2):89-111 [28] Huang X, Xie YM. Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Computational Mechanics, 2009, 43(3):393-401 doi: 10.1007/s00466-008-0312-0 [29] Ghabraie K. An improved soft-kill BESO algorithm for optimal distribution of single or multiple material phases. Structural and Multidisciplinary Optimization, 2015, 52(4):773-790 doi: 10.1007/s00158-015-1268-2 [30] Yin L, Ananthasuresh GK. Topology of compliant mechanisms with multiple materials using a peak function material interpolation scheme. Structural and Multidisciplinary Optimization, 2001, 23(1):49-62 doi: 10.1007/s00158-001-0165-z [31] 高彤, 张卫红, Pierre D.多相材料结构拓扑优化:体积约束还是重量约束?力学学报, 2011, 43(2):296-305 http://lxxb.cstam.org.cn/CN/abstract/abstract142018.shtmlGao Tong, Zhang Weihong, Pierre D. Topology optimization of structures designed with multiphase materials:volume constraint or mass constraint? Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(2):296-305(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract142018.shtml [32] Gao T, Zhang W. A mass constraint formulation for structural topology optimization with multiphase materials. International Journal for Numerical Methods in Engineering, 2011, 88(8):774-796 doi: 10.1002/nme.v88.8 [33] Gao T, Xu P, Zhang W. Topology optimization of thermo-elastic structures with multiple materials under mass constraint. Computers and Structures, 2016, 173:150-160 doi: 10.1016/j.compstruc.2016.06.002 [34] Jia J, Cheng W, Long K, et al. Hierarchical design of structures and multiphase materials cells. Computers and Structures, 2016, 165:136-144 doi: 10.1016/j.compstruc.2015.12.001 [35] Takakoli R, Mohseni SM. Alternating active-phase algorithm for multimaterial topology optimization problems:a 115-line MATLAB implementation. Structural and Multidisciplinary Optimization, 2014, 49(4):621-642 doi: 10.1007/s00158-013-0999-1 [36] Ramani A. A pseudo-sensitivity based discrete-variable approach to structural topology optimization with multiple materials. Structural and Multidisciplinary Optimization, 2010, 41(6):913-934 doi: 10.1007/s00158-009-0455-4 [37] Ramani A. Multi-material topology optimization with strength constraints. Structural and Multidisciplinary Optimization, 2011, 43(5):597-615 doi: 10.1007/s00158-010-0581-z [38] Zuo W, Saitou K. Multi-material topology optimization using ordered SIMP interpolation. Structural and Multidisciplinary Optimization, 2016:1-15 [39] 隋允康.建模变换优化——结构综合方法新进展.大连:大连理工大学出版社, 1996Sui Yunkang. Modelling, Transformation and Optimization——New Developments of Structural Synthesis Method. Dalian:Dalian University of Technology Press, 1996(in Chinese) [40] 隋允康, 宣东海, 尚珍.连续体结构拓扑优化的高精度逼近ICM方法.力学学报, 2011, 43(4):716-724 http://lxxb.cstam.org.cn/CN/abstract/abstract142369.shtmlSui Yunkang, Xuan Donghai, Shang Zhen. ICM method with high accuracy approximation for topology optimization of continuum structures. Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(4):716-724(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract142369.shtml [41] 隋允康, 叶红玲, 彭细荣.应力约束全局化策略下的连续体结构拓扑优化.力学学报, 2006, 38(3):364-370 http://lxxb.cstam.org.cn/CN/abstract/abstract141416.shtmlSui Yunkang, Ye Hongling, Peng Xirong. Topological optimization of continuum structure under the strategy of globalization of stress constraints. Chinese Journal of Theoretical and Applied Mechanics, 2006, 38(3):364-370(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract141416.shtml [42] 隋允康, 叶红玲, 彭细荣等.连续体结构拓扑优化应力约束凝聚化的ICM方法.力学学报, 2007, 39(4):554-563 http://lxxb.cstam.org.cn/CN/abstract/abstract141522.shtmlSui Yunkang, Ye Hongling, Peng Xirong, et al. The ICM method for continuum structural topology optimization with condensation of stress constraints. Chinese Journal of Theoretical and Applied Mechanics, 2007, 39(4):554-563(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract141522.shtml [43] 隋允康, 叶红玲.连续体结构拓扑优化的ICM方法.北京:科学出版社, 2013Sui Yunkang, Ye Honglin. Continuum Topology Optimization Methods ICM. Beijing:Science Press, 2013(in Chinese) [44] 隋允康, 彭细荣.结构拓扑优化ICM方法的改善.力学学报, 2005, 37(2):190-198 http://lxxb.cstam.org.cn/CN/abstract/abstract141361.shtmlSui Yunkang, Peng Xirong. The improvement for the ICM method of structural topology optimization. Acta Mechanica Sinica, 2005, 37(2):190-198(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract141361.shtml [45] Sui Yunkang, Peng Xirong. The ICM method with objective function transformed by variable discrete condition for continuum structure. Chinese Journal of Theoretical and Applied Mechanics, 2006, 22(1):68-75 https://www.researchgate.net/publication/225800906_The_ICM_method_with_objective_function_transformed_by_variable_discrete_condition_for_continuum_structure [46] Lazarov BS, Sigmund O. Filters in topology optimization based on Helmholtz-type differential equations. International Journal for Numerical Methods in Engineering, 2011, 86(6):765-781 doi: 10.1002/nme.v86.6 -
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