FINITE ELEMENT ANALYSIS OF LINEAR VISCOELASTIC PROBLEMS USING THE PROPOSED GAUSSIAN-FOURIER SHAPE FUNCTIONS

In this paper, an enhanced finite element method is proposed for two-dimensional linear viscoelastic problems using new Gaussian-Fourier elements, and the results are compared with those obtained by classical Lagrange quadrilateral elements. Shape functions in Gaussian-Fourier elements are obtained by the enrichment of the proposed Gaussian-Fourier Radial Basis Functions with polynomial function fields. The idea for Gaussian-Fourier Radial Basis Functions came from the integrating Gaussian Radial Basis Functions with Complex Fourier Radial Basis Functions to be more efficient than each individually. In these proposed shape functions, there is a shape parameter, which is a constant unknown parameter that is selected to increase approximation's accuracy. It is shown that Gaussian-Fourier shape functions over an element satisfy all the requirements necessary for the assurance of convergence to the actual solution as the number of elements is increased, and their size is decreased. The Patch test is performed by utilizing Gaussian-Fourier elements in advance. In this study, based on the experience of the authors, it is proposed that a suitable shape parameter for each problem is adopted based on an acceptable approximation of the problem's geometry by a Gaussian-Fourier element. The Finite element formulation proposed by Zocher for linear viscoelasticity is adopted in this article. In this numerical algorithm, the constitutive equations, expressed in an integral form involving the relaxation moduli, are transformed into an incremental algebraic form prior to the development of the finite element formulation. In order to illustrate the validity and accuracy of the present approach two numerical examples, with available analytical solutions, are examined. The Results showed that Finite element solutions obtained by the proposed approach were in great agreement with analytical solutions even though noticeable fewer elements are required in comparison to the classic Finite Element method; therefore, the computational costs are reduced effectively. This fact may be attributed to the robustness of the proposed shape functions and their efficiency in viscoelasticity.