University of British Columbia
Vancouver, British Columbia, Canada
The traditional Lagrangian (LF) and Eulerian (EF) formulations in finite element posses some inherent difficulties when used in simulation of nonlinear and large deformation problems such as the case in metal forming processes or general finite strain problems. In LF, these difficulties may include excessive mesh distortion as well as the difficulty of applying contact and varying kinematic and traction boundary conditions. An Eulerian approach, on the other hand, introduces other difficulties like appropriate representation of free boundary and simulation of the material deformation history. A more general ethod of formulation, the Arbitrary Lagrangian-Eulerian (ALE), is developed to over come such difficulties. The key point in differentiating the ALE formulation from Lagrangian or Eulerian type of formulations is that in ALE we introduce a reference computational domain that can move arbitrarily and independently of the material. In this paper an incremental approach is used to derive the ALE finite element formulation including a complete expression of the external virtual work. A detailed discussion on the loading correction analysis is given based the degenerated updated Lagrangian description from ALE, and a modified expression for the surface loading correction contribution to the system stiffness matrix is presented. Discussion of pertinent difference between the presented formulation and similar ones in the literature is provided. Some of the numerical difficulties with the ALE formulation, namely the mesh motion and the stress integration scheme, are discussed and specific applications are given. Sample numerical examples from a developed computer program are presented. More practical example of large deformation and metal forming problems are also discussed and the results are compared with other formulations and experimental solutions in the literature.