This article presents a new approach to assess the error in specific quantities of interest in the framework of linear elastodynamics. In particular, a new type of quantities of interest (referred as timeline-dependent quantities) is proposed. These quantities are scalar time-dependent outputs of the transient solution which are better suited to time-dependent problems than the standard scalar ones, frozen in time. The proposed methodology furnishes error estimates for both the standard scalar and the new timeline-dependent quantities of interest. The key ingredient is the modal-based approximation of the associated adjoint problems which allows efficiently computing and storing the adjoint solution.
The approximated adjoint solution is readily post-processed to produce an enhanced solution, requiring only one spatial post-process for each vibration mode and using the time-harmonic hypothesis to recover the time dependence. Thus the proposed goal-oriented error estimate consists in injecting this enhanced adjoint solution into the residual of the direct problem. The resulting estimate is very well suited for transient dynamic simulations because the enhanced adjoint solution is computed before starting the forward time integration of the direct problem. Thus, the cost of the error estimate at each time step is very low.