Abstract
Despite using very large parallel computers, numerical simulation of some forming processes such as multi-pass rolling, extrusion or wire drawing, need long computation time due to the very large number of time steps required to model the steady regime of the process. The direct calculation of the steady-state, whenever possible, allows reducing by 10–20 the computational effort. However, removing time from the equations introduces another unknown, the steady final shape of the domain. Among possible ways to solve this coupled multi-fields problem, this paper selects a staggered fixed-point algorithm that alternates computation of mechanical fields on a prescribed domain shape with corrections of the domain shape derived from the velocity field and the stationary condition v.n = 0. It focuses on the resolution of the second step in the frame of unstructured 3D meshes, parallel computing with domain partitioning, and complex shapes with strong contact restraints. To insure these constraints a global finite elements formulation is used. The weak formulation based on a Galerkin method of the v.n = 0 equation is found to diverge in severe tests cases. The least squares formulation experiences problems in the presence of contact restraints, upwinding being shown necessary. A new upwind least squares formulation is proposed and evaluated first on analytical solutions. Contact being a key issue in forming processes, and even more with steady formulations, a special emphasis is given to the coupling of contact equations between the two problems of the staggered algorithm, the thermo-mechanical and free surface problems. The new formulation and algorithm is finally applied to two complex actual metal forming problems of rolling. Its accuracy and robustness with respect to the shape initialization of the staggered algorithm is discussed, and its efficiency is compared to non-steady simulations.
Highlights
Steady-state formulations are applied to two simulations of rolling processes
Results for test cases without contact For test cases with only one degrees of freedom (DoF), a simple criterion is used to quantify solutions accuracy (47), while it is more complicated for the test case with edges as the mesh regularization makes nodes slide on the surface
The Streamline Upwind Petrov– Galerkin (SUPG) formulation (28b) is the most accurate because it has the best nodal accuracy on the sheet edges, whereas least squares methods tend to smooth the solution. These nodal boundary conditions are more favorable to Galerkin-like formulations
Summary
Steady-state formulations are applied to two simulations of rolling processes Their results in terms of thermo-mechanical variables and product shape will be compared to nonsteady simulations using Forge® [24] software as well as steady-state simulations using Lam3 [6], a finite element software based on structured meshes, brick elements, streamline integration and using a dedicated pre-processor providing best possible initial shape for the mesh. For these computations, material consistency is supposed to be constant and tools are assumed to be perfectly rigid.
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