Keywords: shifted ICCG algorithm, Newton-Raphson algorithm, diagonal algorithm, metal forming.

Metal forming is an important for manufacturing, since it can be used to manufacture products for a wide range of industries. The deformation and mechanics of the required products in metal forming are complicated, which requires an accurate mathematical model to control the forming facilities. The continuing industrial needs to improve the quality of products, the level of control in processing and increasing the productivity rate require an efficient numerical modelling scheme [1]. As the size of the problems for practical industrial application increases, it is necessary to use a robust and efficient algorithm [2] in the development of the on-line control model.

In this paper, the authors present the application of an algorithm the shifted ICCG method [4] in the rigid plastic/rigid viscoplastic finite element method analysis of metal forming. A formulation of this method is presented. The performance of this algorithm in terms of CPU time, frictional boundary conditions and memory are discussed. Simulation results indicate that the CPU time is larger than that using the Newton-Raphson algorithm, but there is a significant saving in memory, especially for a large and complicated industrial problem. A comparison of CPU time used in the diagonal algorithm and shifted ICCG algorithm has also been conducted.

The finite element simulation of the slab vertical rolling and horizontal rolling shows that the shifted ICCG method is applicable in the modelling of steel rolling. For slab edging, the rolling force and torque increase with the width reduction and slab thickness. For horizontal rolling, the width reduction in the slab edging affects the rolling force and torque significantly and they increase with the width reduction. The spread of slab increases with the width reduction of slab edging and the slab thickness reduction. The calculated results are in good agreement with the measured values. Numerical simulations were used to demonstrate the efficiency of the shifted ICCG algorithm in the modelling of metal forming.

1

M. Gierzyska-Dolna, P. Lacki, "Some aspects of modelling of metal forming processes", Computers & Structures, 81(8-11), 605-613, 2003. doi:10.1016/S0045-7949(02)00425-X

2

H.R. Le, M.P.F. Sutcliffe, "A robust model for rolling of thin strip and foil", International Journal of Mechanical Sciences, 43(6), 1405-1419, June 2001. doi:10.1016/S0020-7403(00)00092-8

3

J. Xue, "Theorem and Methods of Optimisation", Metallurgical Industry Press, Beijing, China, 1992.

4

T.A. Manteuffel, "Shifted Incomplete Cholesky factorization", in I.S. Duff, G.W. Stewart (eds), Sparse Matrix Proceeding 1978, 41-61, SIAM Publications, Philadelphia, PA, 1978.

5

J.A. Meijerink, H.A. van der Vorst, "An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix", Math. Computation, 32(137), 148-162, 1997. doi:10.2307/2005786

6

T.A. Manteuffel, "An incomplete factorization technique for positive definite linear systems", Math. Computation, 34(150), 473-497, 1980. doi:10.2307/2006097

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