微分求积升阶谱有限元方法及其在薄板自由振动中的应用

A differential quadrature hierarchical finite element method and its application to thin plate free vibration

提出了一种基于Hermite插值的微分求积升阶谱有限元方法。单元在几何映射上采用了混合函数方法,而在形函数的构造上,单元边界上采用非均匀节点的Hermite插值基函数,单元内部升阶谱形函数的构造则采用雅克比正交多项式的张量积形式。将单元形函数与高斯-洛巴托积分法结合起来离散薄板的势能泛函从而得到相应的单元矩阵。提出的薄板单元在单元边界以及单元内部的节点配置完全自由,因而可以用于不同阶次的单元的连接。通过在薄板自由振动中的应用计算以及与精确解的对比,结果表明:提出的微分求积升阶谱有限元方法不仅计算精度高,而且收敛速度快,同时在阶次较高时仍然具有良好的数值稳定性。

A Hermite differential quadrature hierarchical finite element method is proposed and elaborated in this paper.The geometric mapping is based on blending function interpolation,while the shape functions at the element edges are derived from Hermite interpolation bases with non-uniform distributed nodes and the internal hierarchical face functions are obtained through tensor product of one dimensional Jacobi orthogonal polynomials.The shape functions in conjunction with Gauss-Lobatto interpolation are employed in discretizing the potential functional of thin plates to obtain the element matrices.In the present elements,the DOFs(Degree of Freedoms)collocations are free at each side and inside of the element,therefore it is allowed to use elements with different order in assembly.The present elements are used in free vibration analysis of thin plate,and the results are compared with the exact solutions and the numerical results by other methods,which show the high accuracy as well as fast convergence of present elements.Moreover,the stability problems are not suffered even when the element order is very high.