【作者】 赵康；

【导师】 郭旭；

【作者基本信息】 大连理工大学 ， 工程力学， 2005， 硕士

【摘要】 连续体结构的拓扑优化是近20年来结构优化领域内的一个热点,被公认为是最具有挑战性的研究课题之一。针对连续体结构的拓扑优化目前采用较多的是均匀化方法和变密度法。本文着重研究基于水平集方法的结构拓扑、形状优化技术。 水平集方法(Level Set Method)是国外学者Sethian和Osher于1988年提出的一种用于追踪运动边界的数值方法,在图像处理、流体力学等方向有着广泛的应用。自从2000年Sethian和Wiegmann把水平集方法引入结构优化领域以来,由于其具有的独特优势,引起了各国学者的高度关注和极大的研究热情。基于Level Set方法的结构优化技术的研究虽然兴起的时间不长,如著名学者Bendson在其2002年的专著中所说的,这一技术还处于初始发展阶段。 本文对基于Level Set方法的结构优化技术进行了研究,总的指导思想是以LevelSet方法为中心,充分挖掘Level Set方法的特性,发挥其独特的优势,以解决传统结构优化方法解决不了或解决得不好的问题。 文中研究了拓扑相关荷载作用下的结构拓扑/形状优化问题,此类问题采用传统的方法处理起来非常困难,我们基于水平集方法提出了解决方案,并得到了不错的结果;此外我们提出了基于拓扑描述函数方法的结构拓扑/形状优化一体化,在拓扑优化中通过引入拓扑导数,大大提高了优化效率;接着我们在水平集框架下,提出了基于理性最优化准则的拓扑优化方法,数值结果表明该方法稳定高效;然后我们成功实现了水平集方法与扩展有限元方法(Extended Finite Element Method)的结合,并将其运用到我们提出的基于欧拉描述的形状优化当中;另外,我们基于拓扑描述函数方法还做了复合材料设计方面的工作。 研究工作表明,水平集方法在结构优化领域的应用,具有很多其他方法所不具备的优势,是一个很有价值的研究方向。更多还原

【Abstract】 Topology optimization of continuum structures has received more and more research attentions in the past twenty years, which is regarded as one of the most challenging research topics. The most popular approaches for topology optimization of continuum structures are homogenization approach and variable density approach. In this paper, the techniques for topology/shape optimization of continuum structures based on Level Set Method are studied.Level Set Method(LSM) was a numerical method for tracing the propagation of dynamic interfaces. Since proposed by Sethian and Osher in 1988, it has had extensive applications in many areas including image processing, fluid mechanics, etc. Sethian and Sethian introduced LSM into the field of structural optimization in 2000. Then it aroused high attention and much interest in researchers around the world for its special advantages. As Bendson said in one of his monographs in 2002, the research work about structural optimization based on LSM was in the initial stage.This thesis focuses on the techniques of structural optimization based on LSM, and by studying and fully utilizing the characteristics and advantages of LSM, we try to solve difficult problems in structural optimization for traditional approaches.In this thesis, topology/shape optimization under topology-dependent loads, which is hard to deal with by traditional approaches, is studied and good results are obtained by LSM. Furthermore simultaneous shape and topology optimization approach based on topology description functions is proposed. Because topological derivative is introduced into the topology optimization, the efficiency of computation is greatly improved. Then the rational evolution method in structural topology optimization is proposed in the frame of LSM, and it’s proved to be stable and efficient by numerical results. Moreover Extended Finite Element Method is successfully implemented and integrated with LSM, and is adopted in our research work about shape optimization based on Euler description. Besides the approach based on topology description function is proposed for the design of the microstructures of composite materials with prescribed properties.As showed in our research work, LSM has unique advantages, which other approaches do not have in structural optimization, and is a valuable research topic.更多还原