Analysis of cold rigid-plastic axisymmetric forging problem by radial basis function collocation method

In the present work, an axi-symmetric cold forging problem is analyzed using radial basis function collocation method. The material is assumed to be rigid-plastic strain hardening. At each increment of the punch displacement, the problem is solved using an Eulerian control volume approach. The mixed pressure-velocity formulation is adopted, in which the hydrostatic stress and velocities are approximated by linear combinations of multiquadrics radial basis functions, the coefficients of which are obtained by satisfying the continuity and equilibrium equations at certain points called collocation points. The resulting non-linear equations are solved using a trust region method available in MATLAB, which is based on interior-reflective Newton method. Because of the nature of the equations, hydrostatic stress values contain spurious terms. To eliminate them, boundary conditions on hydrostatic stress are required, which are not known initially. Therefore the problem is solved in two stages. In the first stage, the problem is solved without any boundary condition for the hydrostatic stress and the forging load is computed by dividing the total power by the punch velocity. The hydrostatic stress at the punch-workpiece interface is obtained from the known forging load. In the second stage, the problem is solved again by putting the additional hydrostatic stress boundary conditions. Computational performance of the proposed method is studied by carrying out parametric study.