改进的导重法求解拓扑优化问题及灰度过滤技术
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摘要
为了提高导重法求解拓扑优化问题的计算效果,提出一种改进的导重法,并引入了灰度过滤技术抑制优化过程中灰度单元的产生.首先基于RAMP(the rational approximation of material properties)模型结合导重法求解最小柔度拓扑优化问题的迭代表达式,利用二分法对表达式中的拉格朗日乘子求法进行了改进;为减少优化后结构图像中的灰度单元数量,在迭代表达式中引入灰度过滤函数;最后将上述理论拓展到多工况拓扑优化问题中,采用归一化组合处理方法建立目标函数.对多工况拓扑优化问题的2个典型算例进行计算的结果表明,应用文中理论求解拓扑优化问题具有收敛稳定、求解快速、图像清晰的特点.
An improved guide-weight method is presented to improve the computational efficiency of the guide-weight method on solving topology optimization problems, and the gray-filter method is introduced to restrain the gray-scale element which is generated in the optimization process. Since the optimization model was established based on RAMP(rational approximation of material properties, RAMP) scheme, and the guide-weight method was used as an algorithm combining for proposing the iteration formulas of the least compliance in multi objective topology optimization problems. First, the Lagrange multipliers solution method of the formulas were improved by the bi-sectioning algorithm. Then, with the purpose to suppress the generating of gray-scale element, the gray-scale filtering function was introduced into iterative formulas. Moreover, the proposed theory were extended to multi objective optimization problem, and ill-conditioning were avoided by normalizing single objective. Finally, two examples of multi objective optimizations problems were calculated respectively. The results calculated by examples demonstrate that using the proposed theory above can solve topology optimization problems with steady convergence, fast calculate speed and image sharpening.
An improved guide-weight method is presented to improve the computational efficiency of the guide-weight method on solving topology optimization problems, and the gray-filter method is introduced to restrain the gray-scale element which is generated in the optimization process. Since the optimization model was established based on RAMP(rational approximation of material properties, RAMP) scheme, and the guide-weight method was used as an algorithm combining for proposing the iteration formulas of the least compliance in multi objective topology optimization problems. First, the Lagrange multipliers solution method of the formulas were improved by the bi-sectioning algorithm. Then, with the purpose to suppress the generating of gray-scale element, the gray-scale filtering function was introduced into iterative formulas. Moreover, the proposed theory were extended to multi objective optimization problem, and ill-conditioning were avoided by normalizing single objective. Finally, two examples of multi objective optimizations problems were calculated respectively. The results calculated by examples demonstrate that using the proposed theory above can solve topology optimization problems with steady convergence, fast calculate speed and image sharpening.
引文
[1]Bendsoe M P,Kikuchi N.Generating optimal topologies in structural design using homogenization method[J].Computer Methods Applied Mechanics Engineering,1988,71(2):197-224
[2]Guedes J M,Kikuchi N.Preprocessing and post processing for material based on the homogenization method with adaptive finite element methods[J].Computer Methods in Applied Mechanics and Engineering,1990,83(2):143-198
[3]Zuo Kongtian,Chen Liping,Zhong Yifang,et al.New theory and algorithm research about topology optimization based on artificial material density[J].Chinese Journal of Mechanical Engineering,2004,40(12):31-37(in Chinese)(左孔天,陈立平,钟毅芳,等.基于人工材料密度的新型拓扑优化理论和算法研究[J].机械工程学报,2004,40(12):31-37)
[4]Song Jian,Wen Weidong,Zhang Hongjian.Bi-directional evolutionary structural optimization method based on self-adaption and random sampling sensitivity analysis[J].Journal of Aerospace Power,2013,28(9):2090-2099(in Chinese)(宋健,温卫东,张宏建.基于自适应随机抽样敏度分析的双向渐进结构优化方法[J].航空动力学报,2013,28(9):2090-2099)
[5]van Dijk N P,Maute K,Langelaar M,et al.Level-set methods for structural topology optimization:a review[J].Structural and Multidisciplinary Optimization,2013,48(3):437-472
[6]Allaire G,Dapogny C,Frey P.A mesh evolution algorithm based on the level set method for geometry and topology optimization[J].Structural and Multidisciplinary Optimization,2013,48(4):711-715
[7]Jiao Hongyu,Zhou Qicai,Li Wenjun,et al.Periodic topology optimization using variable density method[J].Journal of Mechanical Engineering,2013,49(13):132-138(in Chinese)(焦洪宇,周奇才,李文军,等.基于变密度法的周期性拓扑优化[J].机械工程学报,2013,49(13):132-138)
[8]Bendsoe M P,Sigmund O.Material interpolation schemes in topology optimization[J].Archive of Applied Mechanics,1999,69(9/10):635-654
[9]Stople M,Svanberg K.An alternative interpolation scheme for minimum compliance topology optimization[J].Structural and Multidisciplinary Optimization,2001,22(2):116-124
[10]Long Kai,Zuo Zhengxing.Independent continuous mapping method for topology optimization of continuum structure based on RAMP interpolation scheme[J].Transactions of Beijing Institute of Technology,2007,27(4):308-311+344(in Chinese)(龙凯,左正兴.基于RAMP模型的ICM连续体拓扑优化方法[J].北京理工大学学报,2007,27(4):308-311+344)
[11]Luo Z,Chen L P,Yang J Z,et al.Compliant mechanism design using multi-objective topology optimization scheme of continuum structures[J].Structural and Multidiscipline Optimization,2005,30(2):142-154
[12]Luo Zhen,Chen Liping,Huang Yuying,et al.Topological optimization using RAMP interpolation scheme[J].Chinese Journal of Computational Mechanics,2005,22(5):585-591(in Chinese)(罗震,陈立平,黄玉盈,等.基于RAMP密度-刚度插值格式的结构拓扑优化[J].计算力学学报,2005,22(5):585-591)
[13]Chen Shuxun,Wei Qifeng,Huang Jincheng.Meaning and rationality of guide-weight criterion for structural optimization[J].Chinese Journal of Solid Mechanics,2013,34(6):628-638(in Chinese)(陈树勋,韦齐峰,黄锦成.结构优化导重准则及其意义与合理性[J].固体力学学报,2013,34(6):628–638)
[14]Chen Shuxun.Analysis,synthesis and optimization of engineering structural systems[M].Beijing:China Science Culture Publishing House,2008(in Chinese)(陈树勋.工程结构系统的分析、综合与优化设计[M].北京:中国科学文化出版社,2008)
[15]Chen Shuxun.Some modern design methods of precise and complex structures[M].Beijing:Beijing University of Aeronautics and Astronautics Press,1992(in Chinese)(陈树勋.精密复杂结构的几种现代设计方法[M].北京:北京航空航天大学出版社,1992)
[16]Liu X J,Li Z D,Chen X.A new solution for topology optimization problems with multiple loads:the guide-weight method[J].Science China Technological Sciences,2011,54(6):1505-1514
[17]Liu X J Li Z D,Wang L P,et al.Solving topology optimization problems by the guide-weight method[J].Frontier Mechanical Engineering,2011,6(1):136-150
[18]Xu Huayang,Guan Liwen,Wang Liping,et al.Topology optimization for the arm of flight simulator under inertial loads[J].Journal of Mechanical Engineering,2014,50(9):14-23(in Chinese)(许华旸,关立文,王立平,等.惯性载荷下飞行模拟器大臂结构的拓扑优化[J].机械工程学报,2014,50(9):14-23)
[19]Li Xiang,Wang Hao,Chen Lifen.Structural frequency response optimization under bandwidth excitation[J].Journal of Vibration and Shock,2012,31(9):113-117(in Chinese)(李翔,王皓,陈力奋.结构频带激励下的整体动响应优化[J].振动与冲击,2012,31(9):113-117)
[20]Zuo Kongtian,Zhao Yudong,Zhong Yifang,et al.Computeraided design of compliant micro-mechanism with multi-objective topology optimization[J].Journal of Computer-Aided Design&Computer Graphics,2006,18(6):854-859(in Chinese)(左孔天,赵雨东,钟毅芳,等.微型柔性机构的多目标计算机辅助拓扑优化设计[J].计算机辅助设计与图形学学报,2006,18(6):854-859)
[21]Min S,Nishiwaki S,Kikuchi N.Unified topology design of static and vibrating structures using multi objective optimization[J].Computers and Structures,2000,75(1):93-116
[2]Guedes J M,Kikuchi N.Preprocessing and post processing for material based on the homogenization method with adaptive finite element methods[J].Computer Methods in Applied Mechanics and Engineering,1990,83(2):143-198
[3]Zuo Kongtian,Chen Liping,Zhong Yifang,et al.New theory and algorithm research about topology optimization based on artificial material density[J].Chinese Journal of Mechanical Engineering,2004,40(12):31-37(in Chinese)(左孔天,陈立平,钟毅芳,等.基于人工材料密度的新型拓扑优化理论和算法研究[J].机械工程学报,2004,40(12):31-37)
[4]Song Jian,Wen Weidong,Zhang Hongjian.Bi-directional evolutionary structural optimization method based on self-adaption and random sampling sensitivity analysis[J].Journal of Aerospace Power,2013,28(9):2090-2099(in Chinese)(宋健,温卫东,张宏建.基于自适应随机抽样敏度分析的双向渐进结构优化方法[J].航空动力学报,2013,28(9):2090-2099)
[5]van Dijk N P,Maute K,Langelaar M,et al.Level-set methods for structural topology optimization:a review[J].Structural and Multidisciplinary Optimization,2013,48(3):437-472
[6]Allaire G,Dapogny C,Frey P.A mesh evolution algorithm based on the level set method for geometry and topology optimization[J].Structural and Multidisciplinary Optimization,2013,48(4):711-715
[7]Jiao Hongyu,Zhou Qicai,Li Wenjun,et al.Periodic topology optimization using variable density method[J].Journal of Mechanical Engineering,2013,49(13):132-138(in Chinese)(焦洪宇,周奇才,李文军,等.基于变密度法的周期性拓扑优化[J].机械工程学报,2013,49(13):132-138)
[8]Bendsoe M P,Sigmund O.Material interpolation schemes in topology optimization[J].Archive of Applied Mechanics,1999,69(9/10):635-654
[9]Stople M,Svanberg K.An alternative interpolation scheme for minimum compliance topology optimization[J].Structural and Multidisciplinary Optimization,2001,22(2):116-124
[10]Long Kai,Zuo Zhengxing.Independent continuous mapping method for topology optimization of continuum structure based on RAMP interpolation scheme[J].Transactions of Beijing Institute of Technology,2007,27(4):308-311+344(in Chinese)(龙凯,左正兴.基于RAMP模型的ICM连续体拓扑优化方法[J].北京理工大学学报,2007,27(4):308-311+344)
[11]Luo Z,Chen L P,Yang J Z,et al.Compliant mechanism design using multi-objective topology optimization scheme of continuum structures[J].Structural and Multidiscipline Optimization,2005,30(2):142-154
[12]Luo Zhen,Chen Liping,Huang Yuying,et al.Topological optimization using RAMP interpolation scheme[J].Chinese Journal of Computational Mechanics,2005,22(5):585-591(in Chinese)(罗震,陈立平,黄玉盈,等.基于RAMP密度-刚度插值格式的结构拓扑优化[J].计算力学学报,2005,22(5):585-591)
[13]Chen Shuxun,Wei Qifeng,Huang Jincheng.Meaning and rationality of guide-weight criterion for structural optimization[J].Chinese Journal of Solid Mechanics,2013,34(6):628-638(in Chinese)(陈树勋,韦齐峰,黄锦成.结构优化导重准则及其意义与合理性[J].固体力学学报,2013,34(6):628–638)
[14]Chen Shuxun.Analysis,synthesis and optimization of engineering structural systems[M].Beijing:China Science Culture Publishing House,2008(in Chinese)(陈树勋.工程结构系统的分析、综合与优化设计[M].北京:中国科学文化出版社,2008)
[15]Chen Shuxun.Some modern design methods of precise and complex structures[M].Beijing:Beijing University of Aeronautics and Astronautics Press,1992(in Chinese)(陈树勋.精密复杂结构的几种现代设计方法[M].北京:北京航空航天大学出版社,1992)
[16]Liu X J,Li Z D,Chen X.A new solution for topology optimization problems with multiple loads:the guide-weight method[J].Science China Technological Sciences,2011,54(6):1505-1514
[17]Liu X J Li Z D,Wang L P,et al.Solving topology optimization problems by the guide-weight method[J].Frontier Mechanical Engineering,2011,6(1):136-150
[18]Xu Huayang,Guan Liwen,Wang Liping,et al.Topology optimization for the arm of flight simulator under inertial loads[J].Journal of Mechanical Engineering,2014,50(9):14-23(in Chinese)(许华旸,关立文,王立平,等.惯性载荷下飞行模拟器大臂结构的拓扑优化[J].机械工程学报,2014,50(9):14-23)
[19]Li Xiang,Wang Hao,Chen Lifen.Structural frequency response optimization under bandwidth excitation[J].Journal of Vibration and Shock,2012,31(9):113-117(in Chinese)(李翔,王皓,陈力奋.结构频带激励下的整体动响应优化[J].振动与冲击,2012,31(9):113-117)
[20]Zuo Kongtian,Zhao Yudong,Zhong Yifang,et al.Computeraided design of compliant micro-mechanism with multi-objective topology optimization[J].Journal of Computer-Aided Design&Computer Graphics,2006,18(6):854-859(in Chinese)(左孔天,赵雨东,钟毅芳,等.微型柔性机构的多目标计算机辅助拓扑优化设计[J].计算机辅助设计与图形学学报,2006,18(6):854-859)
[21]Min S,Nishiwaki S,Kikuchi N.Unified topology design of static and vibrating structures using multi objective optimization[J].Computers and Structures,2000,75(1):93-116