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SUMMARYA computational framework is presented to evaluate the shape as well as non-shape (parameter) sensitivity of ÿnite thermo-inelastic deformations using the continuum sensitivity method (CSM). Weak sensitivity equations are developed for the large thermo-mechanical deformation of hyperelastic thermoviscoplastic materials that are consistent with the kinematic, constitutive, contact and thermal analyses used in the solution of the direct deformation problem. The sensitivities are deÿned in a rigorous sense and the sensitivity analysis is performed in an inÿnite-dimensional continuum framework. The e ects of perturbation in the preform, die surface, or other process parameters are carefully considered in the CSM development for the computation of the die temperature sensitivity ÿelds. The direct deformation and sensitivity deformation problems are solved using the ÿnite element method. The results of the continuum sensitivity analysis are validated extensively by a comparison with those obtained by ÿnite di erence approximations (i.e. using the solution of a deformation problem with perturbed design variables). The e ectiveness of the method is demonstrated with a number of applications in the design optimization of metal forming processes.

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