In the recent years, technology in the field of industrial production had to cope with an increasing need for short production runs, a high product diversification and complexity and thus, continuous developments on metal forming have taken place. Along with aspects from manufacturing the numerical simulation of forming processes has received steady progress and current methods perform well in that field of application [5].
As one of the more recent approaches incremental forming processes try to achieve those industrial demands due to their high flexibility. They are specially designed as a low-cost alternative for difficult product geometries, variable low-series productions or prototyping. In the process, the product is formed in multiple sequentially similar steps of a small and simple die. Plastic forming only occurs in a very small zone of the work piece, while wide regions of it undergo only elastic deformation [2]. Bulk forming as well as sheet metal forming are possible processes for application of the method.
The numerical properties of those simulations obstruct an efficient simulation with current methods, though. Beside a high number of unknowns and time steps, large deformations, unsteady boundary conditions and the need for continual mesh adaption are aspects that have to be taken into account. Moreover, nonlinear constitutive laws and the indispensible presence of multibody contact between dies and work pieces complicate the computations and lead to unacceptably high computation times. The main problem encountered is the conditioning of the system matrices upon which iterative solvers struggle with bad convergence. Algebraic multigrid methods have proven to be very efficient on finite element discretizations and to overcome mesh dependency but maintain the good complexity of multigrid methods [1][4]. It has been shown that those methods perform well on simulations of incremental forming processes compared to other iterative solvers and retain a high robustness and efficiency [3].
This paper investigates the performance of an algebraic multigrid (AMG) solver when different strategies of using certain process characteristics of incremental forming are applied. The main idea of those strategies is to exploit the local restriction of the actual forming on the model, which is the most striking difference to other forming processes. The discussion includes the consideration of those process properties within the AMG setup as well as in an feasible schur complement decomposition. Numerical results are given.


References:

[1] Briggs, W.L., Henson, V.E. and McCormick, S.: Multigrid Tutorial, 2nd Edition, SIAM, Philadelphia, 2000
[2] Groche, P., Heislitz, F., Jöckel, M., Jung, S., Rachor, C. and Rathmann, T., Modelling of Incremental Forming Processes, Proceedings of NAFEMS World Congress. Como, 2001
[3] Schmid, F. and Schäfer, M.: Performance of Algebraic Multigrid Methods for Simulation of Incremental Forming Processes, Proceedings of European Multigrid Conference, Delft, 2005.
[4] Stüben, K., Algebraic Multigrid (AMG), An introduction with applications, GMD Report 70, St. Augustin, 1999
[5] Wagoner, R.H. and Chenot, J.L, Metal Forming Analysis, Cambridge University Press, Cambridge, 2001