光滑双向渐进结构优化法拓扑优化连续体结构频率和动刚度
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摘要
针对双向渐进结构优化(bi-directional evolutionary structural optimization, BESO)方法的单元过删除问题,提出了光滑双向渐进结构优化(smooth bi-directional evolutionary structural optimization, SBESO)方法,通过引入权重函数更新单元的质量与刚度矩阵,控制单元删除率以使低效单元逐渐从设计域中删除。以连续体结构固有频率最大化为目标,提出了一种基于SBESO的频率优化方法,对比分析了常函数、线性函数和正弦函数等不同权重函数对连续体结构优化的影响。将等效静载荷(equivalentstatic loads,ESL)方法与SBESO方法相融合,提出了动载荷作用下连续体结构的动刚度优化方法。数值算例表明,SBESO方法通过调节单元删除率和权重函数,控制低效单元在结构设计域中逐渐被删除,有效抑制了单元的过删除问题。采用线性和正弦权重函数,更有利于获得连续体结构的频率最优拓扑解。随单元删除率的减少,动刚度最优拓扑解的结构边界逐渐光滑,而且逼近于同一构形。由此,所提出的SBESO方法完善了BESO方法的优化准则,对于解决连续体结构动力学优化设计问题具有较为重要的理论意义。
The classic bi-directional evolutionary structural optimization(BESO), which is known as "hard kill" in nature, discretely removes the inefficient material in the continuum structure. Using BESO method, structural stiffness and mass matrix of the elements under low sensitivity and inefficiently utilized suddenly reduce. It means that when solving structural topological optimization, highly efficient elements are possibly incorrectly deleted by original BESO. As a result, the smooth bi-directional evolutionary structural optimization(SBESO), as a variant of the BESO procedure, is proposed to overcome this disadvantage. The introduced SBESO was based on the philosophy that if an element was not really necessary for the structure, its contribution to the structural stiffness and mass would gradually diminish until it no longer influenced the structure. This removal process was thus performed smoothly. This procedure was known as ‘‘soft-kill'' in nature; where not all of the elements removed from the structure domain using the original BESO criterion were rejected. The weighted function was introduced to regulate element's mass and stiffness matrix, combined with controlling the element deletion rate to make inefficient elements gradually deleted. This method provided good conditioning for the new system of equations that could be resolved in the next iteration because these elements were important to the structure. In this paper, the proposed SBESO method was applied to resolve the structural dynamic topology optimization including frequency optimization and dynamic stiffness optimization of continuum structure under harmonic force excitation. The frequency optimization model was developed to increase the fundamental natural frequency of continuum structure which was away from the frequency of the external force. The dynamic stiffness model was derived to perform structural optimal design according to the structural strain energy of the continuum structure under dynamic load. Aiming at maximizing the natural frequency of continuum structure, the SBESO procedure was employed to solve it. Subsequently the effect of various weighted function including constant, linear and trigonometric function on topology optimization was compared and analyzed. Performing the structural topology optimization of dynamic stiffness in the time domain was quite expensive due to heavy computational time for function and sensitivity. Thus the equivalent static loads(ESL) method in the time domain was used to overcome these disadvantages. Combining ESL and SBESO, structural topology optimization of dynamic stiffness was resolved at ease for the continuum structure under dynamic load. Numerical results showed that SBESO could inhibit the incorrect element deletion, by regulating the element deletion rate and weighted function. Therefore the inefficient elements would gradually diminish. It was also found that the employed linear and trigonometric functions more contributed to obtaining the optimal solution of frequency for the continuum structure compared with constant function. In addition, decreasing element deletion rate aroused increasingly smooth boundary of optimal configuration in structural topology which converged to identical configuration by SBESO. Consequently the optimization method of SBESO improves the original optimization criterion of BESO, which is significantly theoretical to address the dynamic optimization design of continuum structures.
The classic bi-directional evolutionary structural optimization(BESO), which is known as "hard kill" in nature, discretely removes the inefficient material in the continuum structure. Using BESO method, structural stiffness and mass matrix of the elements under low sensitivity and inefficiently utilized suddenly reduce. It means that when solving structural topological optimization, highly efficient elements are possibly incorrectly deleted by original BESO. As a result, the smooth bi-directional evolutionary structural optimization(SBESO), as a variant of the BESO procedure, is proposed to overcome this disadvantage. The introduced SBESO was based on the philosophy that if an element was not really necessary for the structure, its contribution to the structural stiffness and mass would gradually diminish until it no longer influenced the structure. This removal process was thus performed smoothly. This procedure was known as ‘‘soft-kill'' in nature; where not all of the elements removed from the structure domain using the original BESO criterion were rejected. The weighted function was introduced to regulate element's mass and stiffness matrix, combined with controlling the element deletion rate to make inefficient elements gradually deleted. This method provided good conditioning for the new system of equations that could be resolved in the next iteration because these elements were important to the structure. In this paper, the proposed SBESO method was applied to resolve the structural dynamic topology optimization including frequency optimization and dynamic stiffness optimization of continuum structure under harmonic force excitation. The frequency optimization model was developed to increase the fundamental natural frequency of continuum structure which was away from the frequency of the external force. The dynamic stiffness model was derived to perform structural optimal design according to the structural strain energy of the continuum structure under dynamic load. Aiming at maximizing the natural frequency of continuum structure, the SBESO procedure was employed to solve it. Subsequently the effect of various weighted function including constant, linear and trigonometric function on topology optimization was compared and analyzed. Performing the structural topology optimization of dynamic stiffness in the time domain was quite expensive due to heavy computational time for function and sensitivity. Thus the equivalent static loads(ESL) method in the time domain was used to overcome these disadvantages. Combining ESL and SBESO, structural topology optimization of dynamic stiffness was resolved at ease for the continuum structure under dynamic load. Numerical results showed that SBESO could inhibit the incorrect element deletion, by regulating the element deletion rate and weighted function. Therefore the inefficient elements would gradually diminish. It was also found that the employed linear and trigonometric functions more contributed to obtaining the optimal solution of frequency for the continuum structure compared with constant function. In addition, decreasing element deletion rate aroused increasingly smooth boundary of optimal configuration in structural topology which converged to identical configuration by SBESO. Consequently the optimization method of SBESO improves the original optimization criterion of BESO, which is significantly theoretical to address the dynamic optimization design of continuum structures.
引文
[1]Sigmund O,Maute K.Topology optimization approaches[J].Structural&Multidisciplinary Optimization,2013,48(6):1031-1055.
[2]Deaton J D,Grandhi R V.A survey of structural and multidisciplinary continuum topology optimization:Post2000[J].Structural&Multidisciplinary Optimization,2014,49(1):1-38.
[3]Munk D J,Vio G A,Steven G P.Topology and shape optimization methods using evolutionary algorithms:Areview[J].Structural&Multidisciplinary Optimization,2015,52(3):613-631.
[4]Xie Y M,Steven G P.A simple evolutionary procedure for structural optimization[J].Computers&Structures,1993,49(5):885-896.
[5]Xia L,Xia Q,Huang X D,et al.Bi-directional evolutionary structural optimization on advanced structures and materials:Acomprehensive review[J].Archives of Computational Methods in Engineering,2018,25(11):437-478.
[6]Zhou M,Rozvany G I N.On the validity of ESO type methods in topology optimization[J].Structural&Multidisciplinary Optimization,2001,21(1):80-83.
[7]Huang X,Xie Y M.A further review of ESO type methods for topology optimization[J].Structural&Multidisciplinary Optimization,2010,41(5):671-683.
[8]贺丹,刘书田.渐进结构优化方法失效机理分析与改进策略[J].计算力学学报,2014,31(3):310-314.He Dan,Liu Shutian.Power flow method used to vibration transmission for two-stage vibration isolation system[J].Journal of Mechanical Engineering,2014,31(3):310-314.(in Chinese with English abstract)
[9]匡兵,李应弟,刘夫云,等.基于单元密度进化步长控制的双向渐进结构优化方法[J].计算力学学报,2016,33(2):15-21.Kuang Bing,Li Yingdi,Liu Fuyun,et al.Bidirectional evolutionary structural optimization method based on control for evolutionary step length of element density[J].Chinese Journal of Computational Mechanics,2016,33(2):15-21.(in Chinese with English abstract)
[10]Valerio S A,Helio L S,Luttgardes O N.Comparative analysis of strut and tie models using smooth evolutionary structural optimization[J].Engineering Structures,2013,56(6):1665-1675.
[11]Helio L S,Valerio S A,Luttardes O N.A smooth evolutionary structural optimization procedure applied to plane stress problem[J].Engineering Structures,2014,75(5):248-258.
[12]Simonetti H L,Almeida V S,Neves F.Smoothing evolutionary structural optimization for structures with displacement or natural frequency constraints[J].Engineering Structures,2018,163(2):1-10.
[13]Munk D J,Vio G A,Steven G P.A simple alternative formulation for structural optimisation with dynamic and buckling objectives[J].Structural and Multidisciplinary Optimization,2017,55(3):969-986.
[14]Xu B,Huang X,Xie Y M.Two-scale dynamic optimal design of composite structures in the time domain using equivalent static loads[J].Composite Structures,2016,142(1):335-345.
[15]Venini,Paolo.Dynamic compliance optimization:Time vs frequency domain strategies[J].Computers&Structures,2016,177(9):12-22.
[16]Zhao J,Wang C.Topology optimization for minimizing the maximum dynamic response in the time domain using aggregation functional method[J].Computers&Structures,2017,190(5):41-60.
[17]Park G J,Kang B S.Validation of a structural optimization algorithm transforming dynamic loads into equivalent static loads[J].Theory Application,2003,118(1):191-200.
[18]Stolpe M.On the equivalent static loads approach for dynamic response structural optimization[J].Structural&Multidisciplinary Optimization,2014,50(6):921-926.
[19]Jang H H,Lee H A,Lee J Y,et al.Dynamic response topology optimization in the time domain using equivalent static loads[J].AIAA Journal,2012,50(1):226-234.
[20]Kim E,Kim H,Baek S,et al.Effective structural optimization based on equivalent static loads combined with system reduction method[J].Structural&Multidisciplinary Optimization,2014,50(5):775-786.
[21]Lee J,Cho M.Efficient design optimization strategy for structural dynamic systems using a reduced basis method combined with an equivalent static load[J].Structural and Multidisciplinary Optimization,2018,58(5):1489-1504.
[22]陈涛,陈自凯,段利,等.针对结构动态非线性优化问题的ESLM梯度优化方法[J].机械工程学报,2015,51(8):116-124.Chen Tao,Chen Zikai,Duan Li,et al.Gradient-based equivalent static loads method for structure nonlinear dynamic optimization problem[J].Journal of Mechanical Engineering,2015,51(8):116-124.(in Chinese with English abstract)
[23]陈涛,戴江璐,陈自凯,等.基于梯度的等效静载荷法的汽车正面碰撞关键结构优化设计[J].中国机械工程,2016,27(24):3396-3401.Chen Tao,Dai Jianglu,Chen Zikai,et al.Design optimization of key structures in frontal crash based on ESLMG[J].China Mechanical Engineering,2016,27(24):3396-3401.
[24]高云凯,田林雳.基于等效静态载荷法的车身碰撞拓扑优化[J].同济大学学报:自然科学版,2017,45(3):391-397.Gao Yunkai,Tian Linli.Topology optimization of automotive body crashworthiness design with equivalent static loads method[J].Journal of Tongji University:Natural Science,2017,45(3):391-397.(in Chinese with English abstract)abstract)
[25]Lee H A,Park G J.Nonlinear dynamic response topology optimization using the equivalent static loads method[J].Computer Methods in Applied Mechanics and Engineering,2015,283(10):956-970.
[26]Ahmad Zeshan,Sultan Tipu,Zoppi Matteo,et al.Nonlinear response topology optimization using equivalent static loads-case studies[J].Engineering Optimization,2017,49(2):252-268.nd dynamic stiffness for continuum
[2]Deaton J D,Grandhi R V.A survey of structural and multidisciplinary continuum topology optimization:Post2000[J].Structural&Multidisciplinary Optimization,2014,49(1):1-38.
[3]Munk D J,Vio G A,Steven G P.Topology and shape optimization methods using evolutionary algorithms:Areview[J].Structural&Multidisciplinary Optimization,2015,52(3):613-631.
[4]Xie Y M,Steven G P.A simple evolutionary procedure for structural optimization[J].Computers&Structures,1993,49(5):885-896.
[5]Xia L,Xia Q,Huang X D,et al.Bi-directional evolutionary structural optimization on advanced structures and materials:Acomprehensive review[J].Archives of Computational Methods in Engineering,2018,25(11):437-478.
[6]Zhou M,Rozvany G I N.On the validity of ESO type methods in topology optimization[J].Structural&Multidisciplinary Optimization,2001,21(1):80-83.
[7]Huang X,Xie Y M.A further review of ESO type methods for topology optimization[J].Structural&Multidisciplinary Optimization,2010,41(5):671-683.
[8]贺丹,刘书田.渐进结构优化方法失效机理分析与改进策略[J].计算力学学报,2014,31(3):310-314.He Dan,Liu Shutian.Power flow method used to vibration transmission for two-stage vibration isolation system[J].Journal of Mechanical Engineering,2014,31(3):310-314.(in Chinese with English abstract)
[9]匡兵,李应弟,刘夫云,等.基于单元密度进化步长控制的双向渐进结构优化方法[J].计算力学学报,2016,33(2):15-21.Kuang Bing,Li Yingdi,Liu Fuyun,et al.Bidirectional evolutionary structural optimization method based on control for evolutionary step length of element density[J].Chinese Journal of Computational Mechanics,2016,33(2):15-21.(in Chinese with English abstract)
[10]Valerio S A,Helio L S,Luttgardes O N.Comparative analysis of strut and tie models using smooth evolutionary structural optimization[J].Engineering Structures,2013,56(6):1665-1675.
[11]Helio L S,Valerio S A,Luttardes O N.A smooth evolutionary structural optimization procedure applied to plane stress problem[J].Engineering Structures,2014,75(5):248-258.
[12]Simonetti H L,Almeida V S,Neves F.Smoothing evolutionary structural optimization for structures with displacement or natural frequency constraints[J].Engineering Structures,2018,163(2):1-10.
[13]Munk D J,Vio G A,Steven G P.A simple alternative formulation for structural optimisation with dynamic and buckling objectives[J].Structural and Multidisciplinary Optimization,2017,55(3):969-986.
[14]Xu B,Huang X,Xie Y M.Two-scale dynamic optimal design of composite structures in the time domain using equivalent static loads[J].Composite Structures,2016,142(1):335-345.
[15]Venini,Paolo.Dynamic compliance optimization:Time vs frequency domain strategies[J].Computers&Structures,2016,177(9):12-22.
[16]Zhao J,Wang C.Topology optimization for minimizing the maximum dynamic response in the time domain using aggregation functional method[J].Computers&Structures,2017,190(5):41-60.
[17]Park G J,Kang B S.Validation of a structural optimization algorithm transforming dynamic loads into equivalent static loads[J].Theory Application,2003,118(1):191-200.
[18]Stolpe M.On the equivalent static loads approach for dynamic response structural optimization[J].Structural&Multidisciplinary Optimization,2014,50(6):921-926.
[19]Jang H H,Lee H A,Lee J Y,et al.Dynamic response topology optimization in the time domain using equivalent static loads[J].AIAA Journal,2012,50(1):226-234.
[20]Kim E,Kim H,Baek S,et al.Effective structural optimization based on equivalent static loads combined with system reduction method[J].Structural&Multidisciplinary Optimization,2014,50(5):775-786.
[21]Lee J,Cho M.Efficient design optimization strategy for structural dynamic systems using a reduced basis method combined with an equivalent static load[J].Structural and Multidisciplinary Optimization,2018,58(5):1489-1504.
[22]陈涛,陈自凯,段利,等.针对结构动态非线性优化问题的ESLM梯度优化方法[J].机械工程学报,2015,51(8):116-124.Chen Tao,Chen Zikai,Duan Li,et al.Gradient-based equivalent static loads method for structure nonlinear dynamic optimization problem[J].Journal of Mechanical Engineering,2015,51(8):116-124.(in Chinese with English abstract)
[23]陈涛,戴江璐,陈自凯,等.基于梯度的等效静载荷法的汽车正面碰撞关键结构优化设计[J].中国机械工程,2016,27(24):3396-3401.Chen Tao,Dai Jianglu,Chen Zikai,et al.Design optimization of key structures in frontal crash based on ESLMG[J].China Mechanical Engineering,2016,27(24):3396-3401.
[24]高云凯,田林雳.基于等效静态载荷法的车身碰撞拓扑优化[J].同济大学学报:自然科学版,2017,45(3):391-397.Gao Yunkai,Tian Linli.Topology optimization of automotive body crashworthiness design with equivalent static loads method[J].Journal of Tongji University:Natural Science,2017,45(3):391-397.(in Chinese with English abstract)abstract)
[25]Lee H A,Park G J.Nonlinear dynamic response topology optimization using the equivalent static loads method[J].Computer Methods in Applied Mechanics and Engineering,2015,283(10):956-970.
[26]Ahmad Zeshan,Sultan Tipu,Zoppi Matteo,et al.Nonlinear response topology optimization using equivalent static loads-case studies[J].Engineering Optimization,2017,49(2):252-268.nd dynamic stiffness for continuum
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