Solution of the equations of rigid–plastic FE analysis by shifted incomplete Cholesky factorisation and the conjugate gradient method in metal forming processes

In general, some iterative methods, such as the standard Newton–Raphson method, the modified Newton–Raphson method and so on, have been employed to solve the system of simultaneous equations obtained in rigid–plastic FE analysis. If the stiffness matrix is symmetric and positive definite, it can be solved by the conjugate-gradient method. In this paper, shifted incomplete Cholesky decomposition of the stiffness matrix is combined with the conjugate-gradient method, designated the Shifted ICCG method, to solve the equations arising in the rigid–plastic FEM. The performance of the algorithm in terms of shift parameter ψ, and CPU time under a range of friction conditions has been assessed. The use of higher order time integration is also investigated. The CPU times required for calculation employing the diagonal matrix method, the shifted ICCG method (ψ=0.0) and the Newton–Raphson method are compared, it being found that the stability is not always good for the diagonal matrix method, and the CPU time required for calculation is very large. Numerical tests show that the shifted ICCG method is stable, and can be used successfully in the application of the rigid–plastic finite element method to the solution of metal forming problems.