A mixed finite element based inverse approach for residual stress reconstruction

In this paper a full-field residual stress identification procedure is proposed and explored from partially measured residual elastic strain or stress within a PDE constrained optimization framework. At first, we have constructed a penalized least-square cost functional with an appropriate regularization term. We have taken data discrepancy term in energy norm while the regularization term is a complementary strain energy functional. We then construct an appropriate Lagrangian to impose the optimization constraint. It turns out that using first order optimality condition leads to standard two field mixed, of Hellinger-Reissner type, variational formulation with residual stress and Lagrangian variable as independent field variables. This allows us to use standard two and three dimensional mixed finite elements for approximation of the stress and Lagrangian variable. Using mixed finite element set-up, we then solved these equations to obtain full-field residual stress distribution. One key issue of the proposed formulation, as in case of any inverse problem, is the choice of penalization parameter. In this context, we have used modified L-curve based approach to choose optimal penalty parameter. Finally, we assess the numerical effectiveness of the proposed procedure against a few problems of engineering interest.