The finite element method achieves an approximate solution by subdividing the domain of interest into a number of smaller sub-domains, called finite elements, and then approximating the solution by using local, piecewise continuous polynomial functions within each element. The accuracy of the solution may be improved in two ways. The first is the h-version to refine the finite element mesh and the second is the p-version to increase the order of polynomial shape functions for a fixed mesh. Zienkiewicz and Taylor [1] concluded that, in general, p convergence is more rapid per degree of freedom introduced. Central to the hierarchical concept is the ability to enrich the polynomial content of selected elements within the mesh. Polynomial functions are well known to be ill-conditioned, e.g., the computer can hardly find the difference between x 10 and x 11 within 0 Q x Q 1. West et al. [2] showed recently that, by reference to an appropriate family of K-orthogonal polynomials, numerical rounding errors associated with floating point arithmetic prescribe the maximum available degree of polynomial enrichment. The principal source of these errors can be traced to the widely ranged coefficients that define a given K-orthogonal polynomial. They concluded that in h-p applications, one has to restrict the degree of polynomial enrichment to: (1) 24 or less in 1-D applications, (2) 14 or less in 2-D applications, and (3) 8 or less in 3-D applications. This severely limits the use of the p-version of finite elements.
Houmat [3] used a quintic polynomial plus some sine terms, i.e., w(x) = c 1 + c 2 x + c 3 x 2 + c 4 x 3 + c 5 x 4 + c 6 x 5 + a r c r + 6 sin rpx to study the free vibration of rectangular plates. Beslin and Nocolas [4] used the trigonometric sets w r (x) = sin (a r x + b r ) sin (c r x + d r ) as shape functions to study the same problem. Bardell et al. [5] suggested the use of mixing hermite cubics and trigonometric functions to analyse coplanar sandwich panels. All authors found that trigonometric functions are more effective in predicting the medium frequency natural modes than polynomials.
In this paper, the use of products of polynomials and Fourier series instead of polynomials alone in the p-element shape functions is recommended. Due to the fact that Fourier series are well behaved, the limitation of the polynomial functions disappears. When applied to the natural vibration analysis of structures, it is found as a bonus that higher modes converge much faster than when using polynomials alone. The concept is not new. Leung [6] enriched the standard beam finite element by means of eigenfunctions (beam functions) and predicted increased accuracy before the name p-version was used by Babuska et al. [7]. This involved the integration of products of polynomials and beams functions. Although closed form integration formulae can be found, e.g., Leung [8,9], the complexity increased rapidly when the products of say three, beam functions are involved,