Hexahedral Fourier p-elements for Vibration of Prismatic Solids

Fourier p-elements of trapezoidal and cubical hexahedron shapes for the free vibration analysis of 3D elastic solids are presented. Trigonometric functions are used as enriching functions to avoid ill-conditioning problems associated with high order polynomials. The element matrices are analytically integrated in closed form. With the additional Fourier degrees-of-freedom, the accuracy of the computed natural frequencies is greatly improved. As an example, the natural frequencies of a cantilever cube are analyzed by a rectangular hexahedron Fourier p-element, two trapezoidal hexahedron Fourier p-elements and the conventional linear finite elements. The results show that the convergence rate of the present elements is very fast with respect to the number of trigonometric terms. The present elements also produce higher accurate modes than the linear finite elements for the same number of degrees-of-freedom. Furthermore, the first six natural frequencies of a cantilever hexagonal prism and a number of concrete dams with different lengths are given as numerical examples.