In metal forming, an initially simple part—a billet or sheet blank, for example—is plastically deformed between tools (or dies) to obtain the desired final configuration. Thus, a simple part geometry is transformed into a complex one, in a process whereby the tools “store” the desired geometry and impart pressure on the deforming material through the tool-material interface. The physical phenomena constituting a forming operation are difficult to express with quantitative relationships. The metal flow, the friction at the tool-material interface, the heat generation and transfer during plastic flow, and the relationships between microstructure/properties and process conditions are difficult to predict and analyze. Often, in producing discrete parts, several forming operations (preforming) are required to transform the initial “simple” geometry into a “complex” geometry, without causing material failure or degrading material properties. Consequently, the most significant objective of any method of analysis is to assist the forming engineer in the design of forming and/or preforming sequences. For a given operation (preforming or finish-forming), such design essentially consists of (1) establishing the kinematic relationships (shape, velocities, strain-rates, strains) between the deformed and undeformed part, i.e., predicting metal flow; (2) establishing the limits of formability or producibility, i.e., determining whether it is possible to form the part without surface or internal defects; and (3) predicting the forces and stresses necessary to execute the forming operation so that tooling and equipment can be designed or selected. For the understanding and quantitative design and optimization of metal-forming operations it is useful (a) to consider a metal forming process as a system and (b) to classify these processes in a systematic way. A metal-forming system comprises all the input variables relating the billet or blank (geometry and material), the tooling (geometry and material), the conditions at the tool-material interface, the mechanics of plastic deformation, the equipment used, the characteristics of the final product, and finally the plant environment in which the process is being conducted. Such a system is illustrated in Fig. 2.1, using impression die forging as an example.