连续体结构拓扑优化方法介绍 |
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连续体结构拓扑优化方法介绍
材料的有效利用一直是人类追求的目标,也是许多研究领域不变的话题,并伴随着结构优化理论和方法的产生而发展。早期结构优化主要是针对尺寸的优化问题,设计域形状是固定的。后来随着结构优化问题的提出,形状优化方法[1]应运而生。在航空和汽车制造行业,经常应用尺寸和形状优化技术设计结构和零件,形状优化方法也常被用来进行电磁、电化学和声学零件的设计,目前已经有许多成功的算法可以处理这类非线性和有限维的问题[2,3]。 形状优化方法是以边界变分为基础,在结构设计中仍占有重要的位置,设计变量直接控制着结构内外边界的形状。但是其主要的缺点是最终设计结果的拓扑与初始给定的拓扑相同,即便如此,在优化过程中还需要多次重新划分有限元网格,如要在设计过程中改变结构的拓扑,则将使设计问题更加复杂[1]。 为了更有效利用材料,弥补形状优化固定拓扑的局限性,拓扑优化问题被提了出来,对于一个新的设计问题,在优化过程中它要求能够产生新的孔,其拓扑和形状没有先验信息给出,优化算法必须在所谓的参考域内确定材料的最优分布,这种分布能极小化给定的目标函数,并满足强加的约束,因此,拓扑优化问题可以被看作是材料的分配问题,需要确定参考域中每一点材料的特征。由于它的复杂性,拓扑优化是结构优化中更具挑战性的研究课题。 连续体结构拓扑优化的重要发展源于1981年Cheng和Olhoff[4,5]的工作,他们在研究最大刚度变厚板最优设计时,发现最优解中包含许多各种尺寸的加强筋,具有非光滑的特征,这意味着最优设计中必须引入复合材料,拓展最优设计空间。这导致了随后一系列研究进展,包括1984年Lurie等[6]用G-收敛理论解释拓扑优化过程中的非光滑现象;Kohn和Strang等[7]引入松弛概念来处理拓扑优化中的病态变分问题,并论证了这种松弛和均匀化之间的关系;Murat和Tartar[8]引入了特征函数来处理拓扑优化的问题,并指出用均匀化进行松弛处理的必要性;Rozvany等[9]研究了在设计加强筋板中引进松弛的含义。这些研究工作直接导致了1988年Bendsoe和Kikuchi[10]提出了连续体结构拓扑优化的均匀化方法,标志着连续体拓扑优化进入蓬勃发展的阶段。 自从均匀化方法被提出以来,在过去的近三十年中,又有多种连续体结构拓扑优化方法被提出,如“带惩罚指数的固体各向同性微结构模型”(SIMP: Solid Isotropic Microstructures with Penalization)方法[11,12]、结构进化法(ESO)[13,14]、冒泡法(Bubble Method)[15]以及水平集方法(Level Set Method)[16,17]等各种各样的方法。这些方法已经在工程技术领域有着广泛的应用。 1)均匀化方法 均匀化方法的数学理论是20世纪70年代在预测与复合材料等效的均匀化材料的宏观特性时提出来的,其在许多工程领域都有应用,如:多孔介质的流体流动、复合材料中的电磁场等[18]。在均匀化方法中,借助周期微结构的复合材料,将拓扑优化问题转化为复合材料微结构的参数的尺寸设计问题,应用一定的最优化准则或者数学规划法来寻找多孔介质的最优配置。微结构的引入,解决了原来拓扑优化中材料分配中只能在离散集合{0,1}上取值的问题,使其可以在区间[01]上取值,这种设计空间的拓展,保证了拓扑优化最优解的存在。在Bendsøe和Kikuchi[10]1988年首次将均匀化方法成功用于连续体结构的拓扑优化设计中,建立了以结构柔度最小为目标函数,结构体积为约束的连续体结构的拓扑优化设计模型,均匀化方法得到广泛的应用。Suzuki和Kikuchi[19],Guedes和Kikuchi[20],Hassani和Hinton[21],Fernandes等[22]对此方法进行不断完善和发展,并由Diaz和KiKuchi[23]和Ma等[24]进一步推广至特征频率问题的结构拓扑优化,Nishiwaki等[25]将均匀化理论应用于柔性机构的拓扑优化设计中。但这种方法也存在其不足,在优化过程中需要确定微结构和微结构的方向,这样有时显得过于繁琐,同时优化结果中常包含多孔介质材料,也难以制造。同时,因为设计变量多,敏度计算复杂,使得优化求解效率也不高。
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