Fourier continuum finite elements for large deformation problems

Abstract

A procedure to analyze axisymmetric solids undergoing large nonaxisymmetric deformation is presented. The plane of axisymmetry is modeled by isoparametric quadrilateral finite elements with dependence in the circumferential direction determined by a continuous Fourier series. Although the Fourier harmonics are coupled for such analyses, the size of the resulting model can be much smaller than it would be if brick elements were to be used. This model reduction can significantly decrease the amount of computational effort needed for analyzing this class of problems. Different schemes for solving the coupled nonlinear equations are evaluated by analyzing two structures subjected to different loadings and exhibiting various amounts of geometric nonlinearity. The coupled systems of linear simultaneous equations occurring at each Newton-Raphson iteration are solved by both direct and indirect solution schemes. Several types of Gauss-Seidel and conjugate gradient iterations are shown to be quite efficient and robust. The effectiveness of these indirect solution techniques are a result of using the uncoupled stiffness matrices to precondition the coupled systems of linear simultaneous equations. The continuous Fourier series approach also provides a very simple, efficient, and reliable way to conduct adaptive refinement in the circumferential direction.