A continuum sensitivity method for finite thermo‐inelastic deformations with applications to the design of hot forming processes

Abstract

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.

Deformation process design for control of microstructure in the presence of dynamic recrystallization and grain growth mechanisms

“…It has been shown that the deformation sensitivity response does not explicitly depend on the history of the temperature sensitivity but is dependent only on the instantaneous temperature sensitivity response h nþ1 (Ganapathysubramanian and Zabaras, 2002). The solution of the sensitivity subproblem is advanced within the incremental solution scheme by integrating the evolution equations for the sensitivity of the plastic and thermal deformation gradients, the evolution equation for the sensitivity of the state variable s and the grain size L. As part of this sub-problem, the linear relationship between T and ½F ; h at time t nþ1 is computed.…”

“…The sensitivity thermal problem and the contact sensitivity problems are dealt the same way as described in Ganapathysubramanian and Zabaras (2002). The next section deals with numerical analysis and involves the comparison of the direct solution with results available experimentally.…”

“…• Sensitivity of the thermal deformation gradient at time t nþ1 (Ganapathysubramanian and Zabaras, 2002):…”

“…Any discontinuity/nondifferentiability of these functions will need to be addressed through regularization. The evolution equation for F h is derived as follows (Ganapathysubramanian and Zabaras, 2002):…”

Inverse geometry heat transfer problem based on a radial basis functions geometry representation

“…Huang et al [4] have proposed to use an IGP to estimate the shape of frost growth on an evaporating tube by using temperature readings. Ganapathysubramanian et al [5] have presented a framework to evaluate the shape sensitivity of finite thermo-inelastic deformations and have applied the method to the design of open-and closed-die forging processes.…”