Three-dimensional rigid–plastic finite element method using diagonal matrix for large-scale simulation of metal-forming processes

A rigid–plastic finite element method using a diagonal matrix is proposed to perform large-scale three-dimensional simulation of metal forming processes in a realistic computing time. Uncoupled equilibrium equations of forces at one nodal point are repeatedly calculated in turn without solving the large global matrix. In the equilibrium equations, three corrective velocities at the nodal point are determined in order to eliminate imbalanced forces for trial velocities. This formulation leads to reduction in the increase rate of the computing time with the number of nodal point. In addition, the program for the developed simulator is much simpler than that of the conventional method and the memory size is very small because the global simultaneous equations are not solved at a time. To examine the convergence of solution in the present method, three-dimensional deformation in upsetting of a rectangular block with flat dies is simulated under non-sliding contact over the die surface. The effects of the initial nodal velocities, the order of nodal points for solving the equilibrium equations and the modification factor for the corrective velocities on the convergence are evaluated.