High order finite elements applied to the computation of elastic spring back in sheet metal forming

The finite element analysis of sheet metal forming processes is highly developed and frequently applied in engineering practice. However, the subsequent analysis of the elastic spring back still bears unsolved problems. Numerical examples have shown that the computation of elastic spring back based on explicit finite element codes may yield unreliable results. On the other hand, the analysis with implicit codes is more reliable but in terms of computer resources more demanding. Usually, low order shell-elements yielding an algebraic rate of convergence are applied. Yet, it is questionable if the assumptions of the underlying shell-theory are fulfilled in the whole computational domain. To overcome these problems, we present a new approach which allows to compute efficient and reliable approximations of the elastic spring back. It is based on a strictly three-dimensional high order solid finite element formulation for curved thinwalled structures. A hexahedral element is applied, allowing for an anisotropic Ansatz of the displacement field, where the polynomial degree of each separate component can be chosen individually and may also be varied in the three local directions of the element. The model error, inherent in each shell-theory turns into a discretization error, which can be readily controlled. If the finite element mesh is properly designed the p-version yields an exponential rate of convergence. The extension to geometrically non-linear problems is straightforward, since a fully 3D approach is applied. Curved boundaries are taken care of by applying the blending function method. The paper presents the overall approach and some numerical examples.     «

The finite element analysis of sheet metal forming processes is highly developed and frequently applied in engineering practice. However, the subsequent analysis of the elastic spring back still bears unsolved problems. Numerical examples have shown that the computation of elastic spring back based on explicit finite element codes may yield unreliable results. On the other hand, the analysis with implicit codes is more reliable but in terms of computer resources more demanding. Usually, low orde...     »