An Enhanced Finite Element method for Two Dimensional Linear Viscoelasticity using Complex Fourier Elements

In this paper, the finite element analysis of two-dimensional linear viscoelastic problems is performed using quadrilateral complex Fourier elements and, the results are compared with those obtained by quadrilateral classic Lagrange elements. Complex Fourier shape functions contain a shape parameter which is a constant unknown parameter adopted to enhance approximation’s accuracy. Since the iso-parametric formulation utilized in the finite element code, based on the experience of 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 complex Fourier element. Several numerical examples solved, and the results showed that the finite element solutions using complex Fourier elements have excellent agreement with analytical solutions, even though noticeable fewer elements than classic Lagrange elements are employed. Furthermore, the run-times of the executions of the developed finite element code to obtain accurate results, in the same personal computer, using classic Lagrange and complex Fourier elements compared. Run-times indicate that in the finite element analysis of viscoelastic problems, complex Fourier elements reduce computational cost efficiently in comparison to their classic counterpart.

Keywords:

Viscoelasticity , finite element method , Complex Fourier shape functions , Shape parameters , Complex Fourier Lagrange Elements