Analytic trapezoidal Fourier p-element for vibrating plane problems

Abstract

A trapezoidal Fourier p-element for the in-plane vibration analysis of two-dimensional elastic solids is presented. Trigonometric functions are used as enriching functions instead of polynomials to avoid ill-conditioning problems. The element matrices are analytically integrated in closed form. With the additional Fourier degrees of freedom (d.o.f.s), the accuracy of the computed natural frequencies is greatly increased. One element can predict many modes accurately. Since a triangle can be divided into three trapezoidal elements, the range of application is much wider than the previously derived rectangular Fourier p-element. Numerical examples show that convergence is very fast with respect to the number of trigonometric terms. Comparison of natural modes calculated by the trapezoidal Fourier p-element and the conventional finite elements is carried out. The results show that the trapezoidal Fourier p-element produces much higher accurate modes than the conventional finite elements with the same number of d.o.f.s. For a benchmark problem, the condition number of the mass matrix using Legendre p-element increases rapidly and it becomes non-positive with 22 terms. The condition numbers of the Fourier p-element matrices are consistently much lower than those of the Legendre p-element.