Sensitivity of forming processes to shape parameters

Shape sensitivity analysis is presented for metal-forming processes described in terms of the Bow approach [1]. The direct differentiation method is employed to derive the sensitivity expressions. The continuum approach is adopted so that both the equilibrium equations and response functional are differentiated before discretization. The necessary derivatives with respect to shape design variables are calculated making use of the framework already available for isoparametric elements (control volume approach). In finite element implementation a system of equations is obtained which has the same system matrix as the equilibrium problem. The right-hand side is calculated by partially differentiating the internal and external forces with respect to the design parameters. The procedure is illustrated by calculating sensitivities of some independent and dependent variables with respect to the die angle in an extrusion problem and to the roll radius in a plane rolling.