A TRIANGULAR FOURIER p -ELEMENT FOR THE ANALYSIS OF MEMBRANE VIBRATIONS

A triangular Fourier p -element for the analysis of membrane vibrations is presented. The elementʹs transverse displacement is written in terms of dimensionless area co-ordinates and is described by three linear shape functions plus a variable number of trigonometric shape functions. The three nodal displacements and the amplitudes of the trigonometric functions on the three edges and in the interior of the element, respectively, are used as generalized co-ordinates. Inter-element compatibility is achieved by matching the generalized co-ordinates at the three nodes and the three edges. Results are obtained for a right isosceles triangular and a square membranes and comparisons are made with exact and linear triangular finite element solutions. The results of both membranes confirm that the solutions converge very fast from above to the exact values as the number of trigonometric terms is increased and highly accurate values are obtained with the use of a very few terms. The results also show that the triangular Fourier p -element produces a much higher accuracy than the linear triangular finite element with fewer system degrees of freedom.