面向应力约束的独立连续映射方法
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摘要
大多数已有的拓扑优化研究为系统刚度最大化设计,尤其以体积比约束下的静态柔顺度最小化问题为典型.从工程角度出发,结构强度设计至关重要.以往的应力研究表明,应力约束拓扑优化存在着奇异性、约束数目庞大、高度非线性特性等诸多数值困难.为了实现应力约束下的拓扑优化设计,采用归一化p范数应力指标以减少单元应力约束数目.遵循独立连续映射建模方式,引入密度变量的倒变量函数作为设计变量.推导了应力约束函数和体积目标函数对设计变量的敏度,并基于一阶和二阶泰勒近似得到各自的显式表达式.通过构造的系列二次规划子问题,原拓扑优化问题采用序列二次规划算法高效求解.二维数值算例考察了结构刚度和强度设计结果的异同,以及不同应力约束上限值对应力约束拓扑优化结果的影响.通过提出方法与传统变密度法结果的比较,说明提出的独立连续映射方法在应力约束下具有可行性和有效性.优化结果也表明了考虑应力约束的连续体拓扑优化具有必要性.
Most existing study on topology optimization have concentrated on maximizing the system stiffness. Especially, the minimization of static compliance subject to the volume fraction is widespread in the formulation. From the engineering point of view, structural strength design is of vital importance. Past study on stress constraint have shown that an amount of numerical difficulties with the stress-constrain topology optimization exist including the so-called singularity, vast of stress constraints, highly nonlinear behavior and so on. To achieve the topological design under stress constraint requirement, the normalized stress measure using p-norm function is adopted for the reduction of stress constraints. Following the modeling manner of independent continuous mapping method, the reciprocal function of relative density is regarded as the design variables. The sensitivities of stress constraint and volume objective with respect to the design variable are derived, and their explicit expressions are formulated based on the first-order and second-order Taylor approximation respectively. By setting up the sub-problem in the form of a quadratic program, the original topology optimization problem is efficiently solved using the sequential quadratic programming approach. The difference between stiffness and strength design, as well as the effect of various upper bounds of stress value on the optimized results for stress constraint are investigated in 2D numerical examples. Through the comparison of the proposed method and traditional variable density method, the feasibility and effectiveness of the proposed optimization approach in stress constrained problems are verified. The results also demonstrate that the consideration of stress constraint in continuum structure is indispensable.
Most existing study on topology optimization have concentrated on maximizing the system stiffness. Especially, the minimization of static compliance subject to the volume fraction is widespread in the formulation. From the engineering point of view, structural strength design is of vital importance. Past study on stress constraint have shown that an amount of numerical difficulties with the stress-constrain topology optimization exist including the so-called singularity, vast of stress constraints, highly nonlinear behavior and so on. To achieve the topological design under stress constraint requirement, the normalized stress measure using p-norm function is adopted for the reduction of stress constraints. Following the modeling manner of independent continuous mapping method, the reciprocal function of relative density is regarded as the design variables. The sensitivities of stress constraint and volume objective with respect to the design variable are derived, and their explicit expressions are formulated based on the first-order and second-order Taylor approximation respectively. By setting up the sub-problem in the form of a quadratic program, the original topology optimization problem is efficiently solved using the sequential quadratic programming approach. The difference between stiffness and strength design, as well as the effect of various upper bounds of stress value on the optimized results for stress constraint are investigated in 2D numerical examples. Through the comparison of the proposed method and traditional variable density method, the feasibility and effectiveness of the proposed optimization approach in stress constrained problems are verified. The results also demonstrate that the consideration of stress constraint in continuum structure is indispensable.
引文
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
2 Eschenauer HA,Olhoff N.Topology optimization of continuum structures:a review.Applied Mechanics Reviews,2001,54(4):331-390
3 Rozvany GIN.A critical review of established methods of structural topology optimization.Structural and Multidisciplinary Optimization,2009,37(3):217-237
4 Sigmund O,Maute K.Topology optimization approaches.Structural and Multidisciplinary Optimization,2013,48(6):1031-1055
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
6 Cheng GD,Guo X.Epsilon-relaxed approach in structural topology optimization.Structural and Multidisciplinary Optimization,1997,10(1):40-45
7 Duysinx P,Bens?e MP.Topology optimization of continuum structures with local stress constraints.International Journal for Numerical Methods in Engineering,1998,43(2):1453-1478
8 Yang RJ,Chen CJ.Stress-based topology optimization.Structural and Multidisciplinary Optimization,1996,12(2):98-105
9 Guo X,Zhang W,Wang Y,et al.Stress-related topology optimization via level set method.Computer Methods in Applied Mechanics and Engineering,2011,200:3439-3452
10 Zhang W,Guo X,Wang Y,et al.Optimal topology design of continuum structures with stress concentration alleviation via level set method.International Journal for Numerical Methods in Engineering,2013,93:942-959
11 Yang D,Liu H,Zhang W,et al.Stress-constrained topology optimization based on maximum stress measures.Computers&Structures,2018,198:23-29
12 Leon DMD,Alexandersen J,Fonseca JSO,et al.Stress-constrained topology optimization for compliant mechanism design.Structural and Multidisciplinary Optimization,2015,52:929-943
13 Cai S,Zhang W,Zhu J,et al.Stress constrained shape and topology optimization with fixed mesh:A B-spline finite cell method combined with level set function.Computer Methods in Applied Mechanics and Engineering,2014,278(7):361-387
14 Cai S,Zhang W.Stress constrained topology optimization with freeform design domains.Computer Methods in Applied Mechanics and Engineering,2015,289:267-290
15 Xia L,Zhang L,Xia Q,et al.Stress-based topology optimization using bi-directional evolutionary structural optimization method.Computer Methods in Applied Mechanics and Engineering,2018,333:356-370
16王选,刘宏亮,龙凯等.基于改进的渐进结构法的应力约束拓扑优化.力学学报,2018,50(2):385-394(Wang Xuan,Liu Hongliang,Long Kai,et al.Stress-constrained topology optimization based on improved bi-directional evolutionary optimization method.Chinese Journal of Theoretical and Applied Mechanics,2018,50(2):385-394(in Chinese))
17 Fan Z,Xia L,Lai W,et al.Evolutionary topology optimization of continuum structures with stress constraints.Structural and Multidisciplinary Optimization,https://doi.org/10.1007/s00158-018-2090-4
18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,1996(Sui Yunkang.Modelling,Transformation and Optimization--New Developments of Structural Synthesis Method.Dalian:Dalian University of Technology Press,1996(in Chinese))
19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,2013(Sui Yunkang,Ye Honglin.Continuum Topology Optimization Methods ICM.Beijing:Science Press,2013(in Chinese))
20 Sui YK,Peng XR.Modeling,Solving and Application for Topology Optimization of Continuum Structures ICM Method Based on Step Function.Cambridge:Elsevier Inc.,2018
21龙凯,王选,韩丹.基于多相材料的稳态热传导轻量化设计.力学学报,2017,49(2):359-366(Long Kai,Wang Xuan,Han Dan.Structural light design for steady heat conduction using multimaterial.Chinese Journal of Theoretical and Applied Mechanics,2017,49(2):359-366(in Chinese))
22 Long K,Wang X,Gu X.Multi-material topology optimization for the transient heat conduction problem using a sequential quadratic programming algorithm.Engineering Optimization,2018,50(2):2091-2107
23 Long K,Wang X,Gu X.Concurrent topology optimization for minimization of total mass considering load-carrying capabilities and thermal insulation simultaneously.Acta Mechanica Sinica,2018,34(2):315-326
24隋允康,杨德庆,王备.多工况应力和位移约束下连续体结构拓扑优化.力学学报,2000,32(2):171-179(Sui Yunkang,Yang Deqing,Wang Bei.Topological optimization of continuum structure with stress and displacement constraints under multiple loading cases.Acta Mechanica Sinica,2000,32(2):171-179(in Chinese))
25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(Sui 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))
26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(Sui 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))
27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(Sui 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))
28宣东海,隋允康,铁军等.结构畸变能处理的应力约束全局化的连续体结构拓扑优化.工程力学,2011,28(10):1-8(Xuan Donghai,Sui Yunkang,Tie Jun,et al.Continuum structural topology optimization with globalized stress constraint treated by structural distotional strain energy density.Engineering Mechanics,2011,28(10):1-8(in Chinese))
29 Fleury C,Braibant V.Structural optimization:a new dual method using mixed variables.International Journal for Numerical Methods in Engineering,1986,23:409-428
30 Svanberg K.The method of moving asymptotes-A new method for structural optimization.International Journal for numerical Methods in Engineering,1987,24:359-373
31 Bruyneel M,Duysinx P,Fleury C.A family of MMA approximations for structural optimization.Structural and Multidisciplinary Optimization,2002,24(4):263-276
32 Sigmund O.Morphology-based black and white filters for topology optimization.Structural and Multidisciplinary Optimization,2007,33:401-424
33 Long K,Wang X,Gu X.Local optimum in multi-material topology optimization and solution by reciprocal variables.Structural and Multidisciplinary Optimization,2018,57(3):1283-1295
2 Eschenauer HA,Olhoff N.Topology optimization of continuum structures:a review.Applied Mechanics Reviews,2001,54(4):331-390
3 Rozvany GIN.A critical review of established methods of structural topology optimization.Structural and Multidisciplinary Optimization,2009,37(3):217-237
4 Sigmund O,Maute K.Topology optimization approaches.Structural and Multidisciplinary Optimization,2013,48(6):1031-1055
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
6 Cheng GD,Guo X.Epsilon-relaxed approach in structural topology optimization.Structural and Multidisciplinary Optimization,1997,10(1):40-45
7 Duysinx P,Bens?e MP.Topology optimization of continuum structures with local stress constraints.International Journal for Numerical Methods in Engineering,1998,43(2):1453-1478
8 Yang RJ,Chen CJ.Stress-based topology optimization.Structural and Multidisciplinary Optimization,1996,12(2):98-105
9 Guo X,Zhang W,Wang Y,et al.Stress-related topology optimization via level set method.Computer Methods in Applied Mechanics and Engineering,2011,200:3439-3452
10 Zhang W,Guo X,Wang Y,et al.Optimal topology design of continuum structures with stress concentration alleviation via level set method.International Journal for Numerical Methods in Engineering,2013,93:942-959
11 Yang D,Liu H,Zhang W,et al.Stress-constrained topology optimization based on maximum stress measures.Computers&Structures,2018,198:23-29
12 Leon DMD,Alexandersen J,Fonseca JSO,et al.Stress-constrained topology optimization for compliant mechanism design.Structural and Multidisciplinary Optimization,2015,52:929-943
13 Cai S,Zhang W,Zhu J,et al.Stress constrained shape and topology optimization with fixed mesh:A B-spline finite cell method combined with level set function.Computer Methods in Applied Mechanics and Engineering,2014,278(7):361-387
14 Cai S,Zhang W.Stress constrained topology optimization with freeform design domains.Computer Methods in Applied Mechanics and Engineering,2015,289:267-290
15 Xia L,Zhang L,Xia Q,et al.Stress-based topology optimization using bi-directional evolutionary structural optimization method.Computer Methods in Applied Mechanics and Engineering,2018,333:356-370
16王选,刘宏亮,龙凯等.基于改进的渐进结构法的应力约束拓扑优化.力学学报,2018,50(2):385-394(Wang Xuan,Liu Hongliang,Long Kai,et al.Stress-constrained topology optimization based on improved bi-directional evolutionary optimization method.Chinese Journal of Theoretical and Applied Mechanics,2018,50(2):385-394(in Chinese))
17 Fan Z,Xia L,Lai W,et al.Evolutionary topology optimization of continuum structures with stress constraints.Structural and Multidisciplinary Optimization,https://doi.org/10.1007/s00158-018-2090-4
18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,1996(Sui Yunkang.Modelling,Transformation and Optimization--New Developments of Structural Synthesis Method.Dalian:Dalian University of Technology Press,1996(in Chinese))
19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,2013(Sui Yunkang,Ye Honglin.Continuum Topology Optimization Methods ICM.Beijing:Science Press,2013(in Chinese))
20 Sui YK,Peng XR.Modeling,Solving and Application for Topology Optimization of Continuum Structures ICM Method Based on Step Function.Cambridge:Elsevier Inc.,2018
21龙凯,王选,韩丹.基于多相材料的稳态热传导轻量化设计.力学学报,2017,49(2):359-366(Long Kai,Wang Xuan,Han Dan.Structural light design for steady heat conduction using multimaterial.Chinese Journal of Theoretical and Applied Mechanics,2017,49(2):359-366(in Chinese))
22 Long K,Wang X,Gu X.Multi-material topology optimization for the transient heat conduction problem using a sequential quadratic programming algorithm.Engineering Optimization,2018,50(2):2091-2107
23 Long K,Wang X,Gu X.Concurrent topology optimization for minimization of total mass considering load-carrying capabilities and thermal insulation simultaneously.Acta Mechanica Sinica,2018,34(2):315-326
24隋允康,杨德庆,王备.多工况应力和位移约束下连续体结构拓扑优化.力学学报,2000,32(2):171-179(Sui Yunkang,Yang Deqing,Wang Bei.Topological optimization of continuum structure with stress and displacement constraints under multiple loading cases.Acta Mechanica Sinica,2000,32(2):171-179(in Chinese))
25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(Sui 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))
26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(Sui 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))
27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(Sui 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))
28宣东海,隋允康,铁军等.结构畸变能处理的应力约束全局化的连续体结构拓扑优化.工程力学,2011,28(10):1-8(Xuan Donghai,Sui Yunkang,Tie Jun,et al.Continuum structural topology optimization with globalized stress constraint treated by structural distotional strain energy density.Engineering Mechanics,2011,28(10):1-8(in Chinese))
29 Fleury C,Braibant V.Structural optimization:a new dual method using mixed variables.International Journal for Numerical Methods in Engineering,1986,23:409-428
30 Svanberg K.The method of moving asymptotes-A new method for structural optimization.International Journal for numerical Methods in Engineering,1987,24:359-373
31 Bruyneel M,Duysinx P,Fleury C.A family of MMA approximations for structural optimization.Structural and Multidisciplinary Optimization,2002,24(4):263-276
32 Sigmund O.Morphology-based black and white filters for topology optimization.Structural and Multidisciplinary Optimization,2007,33:401-424
33 Long K,Wang X,Gu X.Local optimum in multi-material topology optimization and solution by reciprocal variables.Structural and Multidisciplinary Optimization,2018,57(3):1283-1295