带规则几何约束的结构频率拓扑优化方法研究
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摘要
风洞试验是飞机研制过程中的必经环节,因此研制能够准确模拟气动载荷对飞机结构作用机理的风洞模型至关重要。研究表明,只有弹性比例模型才能准确地模拟高速风洞中飞机的振颤特性。弹性比例模型要求在弹性、结构和模态方面与原结构相似,其设计可看成材料与拓扑几何特征并发设计问题,而拓扑优化是解决该问题的有效方法。目前,连续体拓扑优化形成的孔洞多为不规则形状,而飞机机身龙骨结构多为规则栅格状。因此,限制孔洞形状,开展带规则几何约束的连续体频率拓扑优化研究具有重要的工程应用价值。
本文首先对不考虑几何约束的一般连续体拓扑优化进行了研究。结合双向渐进结构法,发展了基于单元特性更改的双向渐进结构法(即双向EPCM);以结构的多阶固有频率为优化目标,建立了优化模型;基于单元敏度分析,以弹性模量和密度很小且相互匹配的空洞单元替代实体单元实现单元的删除;比较了空洞单元的弹性模量与密度的匹配关系对模态的影响;改进了敏度再分配方法,对棋盘格式进行抑制;以壳体平板为优化对象验证了方法的可行性。
其次,在一般连续体拓扑优化的基础上,对孔洞的形状施加几何约束,开展了带规则几何约束的连续体频率拓扑优化研究。利用ANSYS进行建模和模态分析,求出结构的固有频率和振型;给出了规则方孔的几何描述;以结构的多阶固有频率为优化目标,以体积、孔洞形状、最小尺寸和对称性为约束,以孔洞的位置尺寸和单元的存在状态为设计变量,建立了带规则几何约束的连续体结构拓扑优化模型;采用带规则几何约束的双向EPCM方法进行求解;基于目标函数敏度分析进行插孔,并对已插孔的位置尺寸进行步长加速寻优;调整已插孔的位置尺寸,保证单元存在状态为整型变量;利用Matlab编制了通用的优化程序;以双层平板结构为优化对象验证了方法的可行性。
最后,将带规则几何约束的双向EPCM方法应用到圆形机身弹性比例模型的研制中。将机身简化为双层圆柱壳结构,采用局部建模分析再将其扩展成完整结构的专门方案,以圆柱壳一节径下的前三阶频率为优化目标,实现了圆柱壳“变拓扑”的频率优化设计。
Wind tunnel test is an inevitable part in the development process of aircraft, so it's of vital importance to develop models which could simulate the mechanism how aerodynamic loads act on aircraft precisely. Researches indicate that only the elastic scale model could simulate aircraft's vibration in high speed tunnel accurately. Elastic scale model should be similar to original design in both geometry and corresponding modes. It can be seen as a concurrent design problem which concerns material and topology geometric characteristics. And topology optimization is an effective solution. At present, holes generating in topology optimization is mostly irregular geometric. However fuselage keel structure is generally regular grid. Therefore, in order to realize optimization design similar to real aircraft in both structure and modes, restricting hole's shape and making a study on frequency topology optimization of continuum with regular geometrical constraints plays a great value for engineering application.
In this paper, firstly common topology optimization of continuum without geometrical constraints is studied. On the basis of BESO method, Bi-directional Element's Property Changing Method (BEPCM) is developed; optimization model of continuum is constructed with multiple frequencies as objects; based on element sensitivity analysis, element deletion is realized by changing a solid element into a void element with small and mutually matched elastic module and density; how matching relationship of void element's module and density influences modes is given; Sensitivity Redistribution Method is modified to restrain checkerboard; above method is applied to shell plate and verified feasible.
Secondly, based on the common topology optimization of continuum, by restricting hole's shape, frequency topology optimization with regular geometrical constraints is studied. Modeling and modal analyzing is conducted by ANSYS and frequencies and modal shapes of structure are outputted; geometry description of regular square hole is presented; optimization model of continuum is constructed, with multiple frequencies as objects, volume, hole shape, minimum size and symmetry as constraints, position, size of holes and element existing state as design variables; BEPCM with regular geometrical constraints is adopted; holes insertion strategy based on objective analysis is given; accelerative search and adjustment of position and size of inserted holes is adopted; optimization program is compiled using Matlab; above method is applied to double layer flat and proved feasible.
Finally, BEPCM with regular geometrical constraints is applied to topology optimization of circular fuselage elastic scale model. The model is simplified to double cylindrical shell and special strategy that the whole structure could be expanded by local modeling is adopted. With the first three frequencies at the first diameter as objects, frequency topology optimization of cylindrical shell is realized.
本文首先对不考虑几何约束的一般连续体拓扑优化进行了研究。结合双向渐进结构法,发展了基于单元特性更改的双向渐进结构法(即双向EPCM);以结构的多阶固有频率为优化目标,建立了优化模型;基于单元敏度分析,以弹性模量和密度很小且相互匹配的空洞单元替代实体单元实现单元的删除;比较了空洞单元的弹性模量与密度的匹配关系对模态的影响;改进了敏度再分配方法,对棋盘格式进行抑制;以壳体平板为优化对象验证了方法的可行性。
其次,在一般连续体拓扑优化的基础上,对孔洞的形状施加几何约束,开展了带规则几何约束的连续体频率拓扑优化研究。利用ANSYS进行建模和模态分析,求出结构的固有频率和振型;给出了规则方孔的几何描述;以结构的多阶固有频率为优化目标,以体积、孔洞形状、最小尺寸和对称性为约束,以孔洞的位置尺寸和单元的存在状态为设计变量,建立了带规则几何约束的连续体结构拓扑优化模型;采用带规则几何约束的双向EPCM方法进行求解;基于目标函数敏度分析进行插孔,并对已插孔的位置尺寸进行步长加速寻优;调整已插孔的位置尺寸,保证单元存在状态为整型变量;利用Matlab编制了通用的优化程序;以双层平板结构为优化对象验证了方法的可行性。
最后,将带规则几何约束的双向EPCM方法应用到圆形机身弹性比例模型的研制中。将机身简化为双层圆柱壳结构,采用局部建模分析再将其扩展成完整结构的专门方案,以圆柱壳一节径下的前三阶频率为优化目标,实现了圆柱壳“变拓扑”的频率优化设计。
Wind tunnel test is an inevitable part in the development process of aircraft, so it's of vital importance to develop models which could simulate the mechanism how aerodynamic loads act on aircraft precisely. Researches indicate that only the elastic scale model could simulate aircraft's vibration in high speed tunnel accurately. Elastic scale model should be similar to original design in both geometry and corresponding modes. It can be seen as a concurrent design problem which concerns material and topology geometric characteristics. And topology optimization is an effective solution. At present, holes generating in topology optimization is mostly irregular geometric. However fuselage keel structure is generally regular grid. Therefore, in order to realize optimization design similar to real aircraft in both structure and modes, restricting hole's shape and making a study on frequency topology optimization of continuum with regular geometrical constraints plays a great value for engineering application.
In this paper, firstly common topology optimization of continuum without geometrical constraints is studied. On the basis of BESO method, Bi-directional Element's Property Changing Method (BEPCM) is developed; optimization model of continuum is constructed with multiple frequencies as objects; based on element sensitivity analysis, element deletion is realized by changing a solid element into a void element with small and mutually matched elastic module and density; how matching relationship of void element's module and density influences modes is given; Sensitivity Redistribution Method is modified to restrain checkerboard; above method is applied to shell plate and verified feasible.
Secondly, based on the common topology optimization of continuum, by restricting hole's shape, frequency topology optimization with regular geometrical constraints is studied. Modeling and modal analyzing is conducted by ANSYS and frequencies and modal shapes of structure are outputted; geometry description of regular square hole is presented; optimization model of continuum is constructed, with multiple frequencies as objects, volume, hole shape, minimum size and symmetry as constraints, position, size of holes and element existing state as design variables; BEPCM with regular geometrical constraints is adopted; holes insertion strategy based on objective analysis is given; accelerative search and adjustment of position and size of inserted holes is adopted; optimization program is compiled using Matlab; above method is applied to double layer flat and proved feasible.
Finally, BEPCM with regular geometrical constraints is applied to topology optimization of circular fuselage elastic scale model. The model is simplified to double cylindrical shell and special strategy that the whole structure could be expanded by local modeling is adopted. With the first three frequencies at the first diameter as objects, frequency topology optimization of cylindrical shell is realized.
引文
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[3]谢涛,刘静,刘军考.结构拓扑优化综述.机械工程师,2006,8:22-25.
[4]许索强,夏人伟.结构优化方法综述.航空学报,1995,16(4):385-396.
[5]周克民,李俊峰,李霞.结构拓扑优化研究方法综述.力学进展,2005,35(1):69-76.
[6]陈建军,车建文,崔明涛等.结构动力优化设计述评与展望.力学进展,2001,31(2):181-192.
[7]R.Grandhi.Structural Optimization with Frequency Constraints-A Review.AIAA[J],1993,31(12):2296-2306.
[8]Y.M.Xie,G.P.Steven.A simple approach to structural frequency optimization.Computers & Structures,1994,53(6):1487-1492.
[9]Y.M.Xie,G.P.Steven.Evolutionary Structural Optimization for dynamic problems.Computers & Structures,1999,58(6):1067-1073.
[10]C.B.Zhao,G.P.Steven,Y.M.Xie.Simultaneously evolutionary optimization of several natural frequencies of a two dimensional structures.Structural Engineering &Mechanics,1999,7(5):447-456.
[11]向锦武,张呈林,周传荣等.有频率、振型节线位置要求的结构动力学优化设计.计算结构力学及其应用,1995,12(4):401-407.
[12]C.P.Pantelides,S.R.Tzan.Optimal design of dynamically constrained structures.Computers & Structures,1997,62(1):141-150.
[13]M.P.Bendsφe,N.Kikuchi.Generating optimal topologies in structural design using a homogenization method.Computer Methods in Applied Mechanics and Engrgeering,1988,71(2):197-224.
[14]左孔天.连续体结构拓扑优化理论与应用研究:(博士学位论文).武汉:华中科技大学,2004.
[15]O.Sigmund.Materials with prescribed constitutive parameters:an inverse homogenization problems.Int.J.Solid Structures,1994,17(31):2313-2329.
[16]袁振,吴长春.复合材料周期性弹性微结构的拓扑优化.固体力学学报,2003,24(1):40-45.
[17]程耿东.关于桁架结构拓扑优化中的奇异最优解.大连理工大学学报,2000,40(4)379-383.
[18]Y.M.Xie,G.P.Steven.A simple evolutionary procedure for structures using an evolutionary method.Computers & Structures,1993,49(5):885-896.
[19]谢忆民,杨晓英,G.P.Steven等.渐进结构优化法的基本理论及应用.工程力学,1999,16(6):70-81.
[20]X.Y.Yang,Y.M.Xie,G.P.Steven et al.Topology optimization for frequencies using an evolutionary method.Structural Engineering,1999,12:1432-1438.
[21]王伟.机翼结构拓扑优化的一种新渐进结构优化方法.飞机设计,2007,27(4):18-20.
[22]O.M.Querin,V.Young,G.P.Steven et al.Computational efficiency and validation of bi-directional evolutionary structural optimization.Computer Methods in Applied Mechanics and Engrgeering,2000,19(8):559-573.
[23]宿新东.管迪华.利用双方向渐进优化方法对结构固有振型的优化.机械强度,2004,26(5):542-546.
[24]荣见华,唐国金,杨振兴等.一种三维结构拓扑优化设计方法.固体力学学报,2005,26(3):289-296.
[25]庄才敖,荣见华,李东斌.三维结构的频率拓扑优化设计.长沙交通学院学报,2007,23(2):65-68.
[26]杨振兴,荣见华,傅建林.三维结构的频率拓扑优化设计.振动与冲击学报,2006,25(3):44-47.
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