TABLE OF CONTENTS The development of a particular formulation for sheet-forming analysis is presented in this chapter, based on two FEM programs developed at the Ohio State University: SHEET-S and SHEET-3. Examples of simulations carried out with various evolutions of these programs will be presented, but it is useful to first put these programs in the context of other developments in other laboratories.
Sheet-forming FEM codes generally are based on an updated Lagrangian scheme in which nodal displacements, velocities, or accelerations are the primary variables. Programs are often grouped into "implicit" and "explicit" categories, [4] depending on the solution algorithm used at each time step. A more precise terminology makes use of the qualifiers "static" vs. "dynamic" in conjunction with "implicit" and "explicit. " (See Chapter 5 for a more thorough presentation.) Even among these simple divisions, there are many alternate approaches; however, it is useful to focus on the mainstream decisions that are followed by several commercial and research programs. Other classification schemes are by element type and by constitutive equation.
This is the traditional approach to nonlinear, quasi-static problems. Formulation of the equilibrium problem at a given time involves finding a set of nonlinear equations corresponding to each degree of freedom at each node. This set of nonlinear equations is linearized for solution typically by a Newton-Raphson technique, which involves defining derivatives at a current trial state and updating repeatedly until either the force residual or the magnitude of the update variable (or both) falls below...
