基于自然单元法的功能梯度板固有频率优化

Optimum Design of Functionally Graded Plates via Frequency Optimization Using Natural Element Method

对功能梯度板的固有频率优化问题进行研究.从无网格方法自然单元法出发,建立了功能梯度板的一阶剪切变形理论的自由振动分析格式.自然单元法是一种基于自然邻近插值的无网格数值方法.自然邻近插值不涉及矩阵求逆运算,也不需要任何人为的参数.较之于传统复合材料,功能梯度板材料属性沿厚度方向呈梯度连续变化,因此材料加工时产生的残余应力小很多,此外,功能梯度板具有可设计的优点.以第一阶固有频率为优化目标建立优化模型,通过单纯形法搜索较优的功能梯度板的成分分布.最后通过矩形功能梯度板在不同边界条件下的基频优化的实例,验证了整个算法的可行性和有效性.

This paper investigates the natural frequency optimization of the functionally graded （FG） plate. The natural element method （NEM） is employed to conduct the direct free vibration analysis based on the first order shear deformation plate theory. The NEM is a kind of meshless method that is based on the natural neighbour interpolation. The natural neighbour interpolation does not involve the computation of matrix inversion and requires no artificial parameters. Due to the continuous variation of the material property in the FG plate along the thickness, the thermal stress is largely reduced. Furthermore, the FG plate is also advantageous in the optimal tailoring of the volume fraction distribution for a better mechanical performance. In this paper, the fundamental frequency is taken as the optimization objective. A nonlinear optimization problem is formulated, and the optimum solution is obtained by using Nelder-Mead simplex method. Finally, several numerical examples are given to validate the effectiveness and feasibility of the proposed method.