A nodal-integration-based finite element method for solving steady-state nonlinear problems in the loading’s comoving frame

Many thermomechanical processes, such as rolling, turning, grinding, welding or additive manufacturing, involve either a material flowing through a fixed load system or a heat source moving with respect to the material. In many situations, these processes involve a constant speed translational, rotational or helical movement of the loading with respect to the material so that a (quasi-) steady thermo-mechanical state is achieved quickly. Classical Lagrangian steady state finite element simulation of these processes in the material’s frame is a heavy task requiring large meshes refined all along the load path. This article presents a nodal-integration-based finite element method for solving transient and steady-state elastoplastic problems associated with these processes. The simulation is carried out step by step in a frame linked to the loading. As the nodes of the mesh do not represent material points, the computation procedure requires determining the position at the previous time step of the material point associated with each node (anterior point) in order to perform the time-integration of the constitutive equations. The anterior points are located anywhere in the mesh and therefore interpolation techniques are required to get the previous mechanical state there. As all the mechanical variables are calculated at nodes with the method proposed, this approach makes the interpolation more straightforward. Applications to 3D forming and welding are presented to illustrate the efficiency of the proposed method. The results of finite element simulations in the frame tied to the loading are compared to those of Lagrangian calculations simulating the load motion in the material’s frame. The applications demonstrate that the proposed model can significantly accelerate simulations, achieving a maximum acceleration of around 40 in 3D forming and about 4 in welding. These results highlight the substantial efficiency improvements enabled by the proposed method.