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

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.