AN ALTERNATIVE HIERARCHICAL FINITE ELEMENT FORMULATION APPLIED TO PLATE VIBRATIONS

Abstract The plate hierarchical finite element in this paper utilizes trigonometric hierarchical shape functions rather than the more usual forms of orthogonal Legendre polynomials. The new hierarchical finite element is formulated in terms of a fixed number of quintic polynomial shape functions plus a variable number of trigonometric hierarchical shape functions. The polynomial shape functions are used to describe the element's nodal degrees of freedom and the trigonometric hierarchical shape functions are used to give additional freedom to the edges and the interior of the element. The numbers of trigonometric hierarchical terms are allowed to vary in both directions of the element's co-ordinate axes. Results are obtained for a number of plates. The results confirm that the solutions always converge from above as the numbers of hierarchical terms are increased and highly accurate values are obtained with the use of very few hierarchical terms. In comparison with the 36-degree-of-freedom rectangular finite element, the trigonometric hierarchical finite element is found to produce a better accuracy with fewer system degrees of freedom. In comparison with the polynomial hierarchical finite element, the trigonometric hierarchical finite element is found to produce an equivalent accuracy with the same number of system degrees of freedom an fewer numbers of hierarchical terms for a free and a clamped square plate. Furthermore, the trigonometric hierarchical finite element is found to produce a better accuracy with fewer system degrees of freedom and fewer numbers of hierarchical terms for a simply supported square plate and a square plate simply supported on two opposite edges and free on the other two edges.