【作者】 龚曙光；

【导师】 陈艳萍； 黄云清；

【作者基本信息】 湘潭大学 ， 计算数学， 2005， 博士

【摘要】 结构形状优化就是通过改变区域的几何形状来达到改善结构的受力特性,其中更多的是降低应力集中或改善应力的分布状况,它已越来越受到工程应用的重视。无网格法的最大优势是节点之间摆脱了网格的束缚,将无网格法与形状优化相结合,能够彻底地解决形状优化过程中所出现的网格扭曲或畸变问题。在众多的无网格方法中,无网格Galerkin法是在工程应用中最有发展前景的方法之一。本文首先在移动最小二乘法逼近中,研究了加权函数及其相关参数的选择。利用误差分析讨论了形函数的值与节点影响域大小之间的关系;利用均匀分布和非均匀分布的节点排列方式,探讨了加权函数的分布规律对形函数及其导数的影响;通过建立误差的能量范数,对照分析了权函数曲线的分布规律对逼近结果的精度影响,得到了在移动最小二乘法逼近中选择权函数及其相关参数的建议。利用Lagrange乘子法来施加实质边界条件,通过对Galerkin离散的平衡控制方程直接求导,得到了离散型基于EFG法的灵敏度分析算法,其中为了得到形函数及其导数关于设计变量的偏导数,分别采用了直接微分法和半解析法,并对这两种方法在一致性、计算费用与程序实现等方面进行了讨论,最后的数值实验结果显示上述算法是可行的。无网格Galerkin法的缺点是计算量大、施加边界条件困难。将无网格法与有限元法相耦合应用到形状优化中,则可以发挥两者各自的优势。本文首先在有限元区域和无网格区域之间的等参数四边形界面单元上,利用组配法实现了有限元法与无网格法的耦合,并且界面单元中的插值函数具有线性相容性。然后通过对有限元离散的平衡方程直接求导,得到了离散型的有限元灵敏度分析算法。利用相同的原理实现了界面单元上有限元灵敏度分析和无网格Galerkin法灵敏度分析的耦合,并用实例进行了数值验证。用一种确定的函数来描述边界形状有时会给优化带来一定的难度,虚载荷变量能够克服这个难度。本文在确定节点移动速度域的基础上,建立了一种基于虚载荷变量的无网格Galerkin灵敏度分析算法,提出了基于虚载荷变量的无网格形状优化的流程,该算法的最大优点是它的物理意义很明确,就是现实生产工艺“锻打”的数值模拟。最后利用上述所得到的不同的灵敏度分析方法与优化准则相结合,完成了3个不同类型问题的结构形状优化。更多还原

【Abstract】 Shape optimization has played a very important role in the engineering application. It is a task of structural shape optimization that the performance of structure is improved by changing the geometry shape of structure, in which it is more to minimizing stress concentration of structure and/or to improve the stress-distributing status. The most advantage of meshless method is that it can alleviate the mesh constrained of node, and the distortion of mesh presented in the process of structural shape optimization can be completely avoided by integrating the meshless method into the shape optimization. In many meshless methods, the element-free Galerkin (EFG) method is one of the most promising and well-developed method for engineering application.The selection of the weight function and its parameter are studied firstly in moving least square (MLS) method in this paper. The relation between value of shape function and size of influence domain of a node is discussed by means of error analysis. The influence of weight function on shape function and its derivatives is studied by using uniform and irregular nodal arrangement. The advice on selecting weight function and its parameter in MLS method is presented by analyzing the influence of weight function curve on the precision of computational result on the basis of energy norm proposed. A numerical method for discreteness-based shape design sensitivity analysis (DSA) using EFG method is presented by using direct differential method on the discrete Galerkin control equilibrium equation, and the essential boundary condition is imposed by employing Lagrange-multiplier method. The derivative of shape function with respect to design variable is obtained by utilizing the ideas of direct differential approach and semi-analytical method, and the accuracy and consistency, computational cost and implementations effort of building approaches is discussed. The result obtained in the numerical test show that this approaches can meet demand of engineering application.However the EFG method is marked by some shortcoming, where the primary one is that there is much more computational cost and it is difficult in imposing the essential boundary condition. The advantage in both finite element (FE) method and EFG method can be developed by integrating coupled FE-EFG method into structural shape optimization. A collocation approach is introduced to couple firstly FE and EFG method in isoparametric quadrangle interface element between EF and meshless regions, and the interpolation function for the displacement at the interface element satisfies the linear consistency exactly. By taking更多还原