Keywords: algebraic multigrid, incremental forming process, non-linear analysis.

This paper details the application of algebraic multigrid methods in the field of incremental forming processes. This technology is one of the recent approaches to achieve the industrial demands of short production times and highly flexible forming processes. Numerically though, they feature properties which complicate an efficient simulation with current methods [3], the requirement of continual mesh adaption, the indispensible presence of contact and non-linear constitutive laws lead to unacceptably high computation times when classical numerical schemes are applied. The main problem encoutered, is the solution of the ill-conditioned system matrices upon which classical iterative solvers struggle with slow convergence rates. The results presented in the paper show the performance of algebraic multigrid applied those problems. Furthermore improvements are suggested by which the solution method can be enhanced by using certain properties of the underlying process.

The simulation of forming processes feature a large measure of non-linearity mainly caused by material response and the contact between die and workpiece. Discretitzation of the finite-element equations therefore leads to the solution of a set linearized ill-conditioned equation systems due to the use of standard formulations for contact and plasticity. The problem size requires solvers with good complexity, though. Multigrid methods are known to be very robust and efficient for problems arising from finite element discretizations. Especially in connection with large strain plasticity locally refined meshes are widely used and a hierarchical grid structure might be very difficult to achieve. For that reason we concentrate on algebraic multigrid methods. They preserve the main multigrid properties but require only the system matrix and thus, appear to be very attractive to overcome the described problems, see e.g. [2]. Applied upon a test case it is shown, that the presented scheme is performing very well in comparison with classical direct and iterative solvers. It becomes obvious, that even on such ill-conditioned problems, the good complexity of multigrid methods can be achieved.

In contrast to the geometric multigrid, the necessary multilevel components are defined algebraically including the coarse grid operators, as well as the interolation and restriction operators. The method presented focuses on the algebraic coarse grid construction scheme. The underlying principle of strong influence and dependency (see, e.g. [1,4]) offers potential for exploitation. The theory of the coarsening process implies two different strategies. On the one hand, a scalar approach where all degrees of freedom are coarsened in the same loop independently of their physical meaning. On the other hand, the coarsening can be accomplished as unknown-based, i.e. seperately for each physical unknown of the structural problem. The numerical results given in the paper show that in some cases of incremental forming technology this enhances the computations significantly. The basic idea beyond is to exploit the property of a dominating direction within a forming process for the coarsening strategy. Due to the principle of strong dependencies, the representation of the different physical variables on the coarse grids is also different. Large deformations in one spatial direction dominate the others and thus, reveal stronger dependencies among each other. Consequently, those variables are preferred unknowns for the coarse grids. Due to that behaviour, the other physical unkowns tend to be underrepresented and thus, less accounted for the multilevel solution.

Provided that these specific directional properties are given it is shown that the algebraic coarsening process is strongly influenced by these. Thus, the solution method can be improved by taking those process properties into account for the solution of the problem. To show the effect of the suggested enhancements both approaches are investigated on the example of a cupping process. The results reveal thet the overall number of V-cycles per Newton step can be minimized by almost 50 percent by utilizing this property. Also the efficiency in computation time is improved to a substantial amount of up to 40 percent for that problem in an appropriate discretization. It will be part of the future work to exploit similar properties also in other processes of incremental forming, reduce the problem dependency for the enhanced method and regain the black-box character of the algebraic multigrid solver to a higher extent.

1

W.L. Briggs and V.E. Henson and S. McCormick, A Multigrid Tutorial, 2nd Edition, SIAM, Philadelphia, 2000

2

A.J. Cleary and R.D. Falgout and V.E. Henson and J.E. Jones and T.A. Manteuffel and S.F. McCormick and G.N. Miranda and J.W. Ruge, Robustness and scalability of algebraic multigrid, SIAM, J. Sci. Comp., Vol.21, No. 5, pp. 1886-1908, Philadelphia, 2000. doi:10.1137/S1064827598339402

3

P. Groche, F. Heislitz, M. Jöckel, S. Jung, C. Rachor and T. Rathmann, Modelling of Incremental Forming Processes, Proceedings of NAFEMS World Congress. Como, 2001.

4

K. Stüben, Algebraic Multigrid (AMG), An introduction with applications, GMD Report 70, St. Augustin, 1999.

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