Simulation of 3D metal-forming using an arbitrary Lagrangian–Eulerian finite element method

The arbitrary Lagrangian–Eulerian (ALE) formulation is an efficient alternative to deal with the finite element mesh distortion that occurs in an updated Lagrangian (UL) formulation when simulating metal-forming processes. Uncoupling mesh movement from material movement, the ALE operator can be split into an UL step followed by an Eulerian step. Deformation due to loading is calculated in the UL step (without convective terms) and the mesh movement (relative to the material) is imposed by nodal relocation techniques in the Eulerian step. In this paper, the authors propose two relocation techniques, similar to an r-refinement, that can move internal and external nodes (including non-planar surface nodes) of eight-node hexahedral elements of structured meshes, while keeping the body volume approximately constant. In this manner, an improved mesh is obtained. Data transfer between the UL mesh and the relocated mesh is performed by using either an expansion of stresses in a Taylor’s series or a new search algorithm that avoids iterative solutions. Two examples, one static and one dynamic, are presented and their results compared with known solutions.