基于无网格Galerkin法的灵敏度分析与形状优化

结构形状优化已经在工程应用中得到重视,将无网格法与形状优化相结合能够从根本上解决优化过程中出现的有限单元扭曲或畸变问题。为此在无网格Galerkin法的基础上,利用离散导数法,提出一种基于离散型的节点位移灵敏度分析方法,其中采用了拉格朗日乘子法来施加本质边界条件。该方法的最大优势是求解过程与无网格Galerkin法的求解过程相似,容易实现。另外对形状优化的数学模型和节点位移的设计速度域进行了讨论。采用具有解析解的实例,对所提出的灵敏度分析方法进行了验证,所得结果显示两者非常吻合。利用上述所建立的形状优化算法,完成了两个工程实例的形状优化设计。

SHAPE OPTIMIZATION AND SENSITIVITY ANALYSIS BASED ON ELEMENT-FREE GALERKIN METHODS

The shape optimization of structures has played a very important role in the engineering application. The mesh distortion of the finite element in the process of shape optimization can be avoided completely by integrating the meshless method with the shape optimization. A numerical method for discreteness-based design sensitivity analysis is proposed by using the discrete derivatives method in the basis of the element-free Galerkin method, in which the Lagrange multiplier is employed into imposing the essential boundary condition. The main advantage of the approach presented is that the calculating process of sensitivity analysis is similar to the solving of element-free Galerkin method, and the present formulation can be easily implemented. In addition, the mathematics model of shape optimization and design velocity fields are also discussed. An example, which has the exact solution, is presented to testify the sensitivity analysis method derived. The result obtained shows that there exists excellent agreement between the analytical and numerical solution. Finally, by using the presented algorithm of shape optimization based on EFG method, the shape optimization of two engineering examples is achieved.