Modal‐based goal‐oriented error assessment for timeline‐dependent quantities in transient dynamics

Abstract

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 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.\ud 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\ud 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.Peer ReviewedPostprint (published version

Error Assessment in Structural Transient Dynamics

“…However, ifũ d has to be used for a timeline estimates e (t), then, a better option is using modal analysis, see reference [71]. The modal based description ofũ d simplifies the time shift (111) required to assess the error in the timeline quantity and makes the actual computation ofs e (t) more efficient.…”

“…A convenient expression for The aim of reference [71] is assessing the quality of the computed timeline-dependent quantity,s(t) := L O TL (ũ; t), with respect to the exact quantity of interest, s(t) := L O TL (u; t). That is, the goal is to assess the error in the quantity of interest which is now a function of time s e (t) := s(t) −s(t).…”

Goal-oriented space-time adaptivity for transient dynamics using a modal description of the adjoint solution

“…using direct time-integration methods, see reference [2]. An alternative approach proposed in [30] considers modal analysis to compute the adjoint approximation. The modal-based strategy is particularly well suited for some particular quantities of interest and allows effectively computing and storing the adjoint problem.…”

“…The semidiscrete problem is defined using the finite element space V H,p+1 0 (P bg ), that stands for the finite element space associated with the mesh P bg of degree of interpolation p + 1 (a p-refined version of V H 0 (P bg )). Having a p + 1 degree approximation of the adjoint solution,Ũ d , precludes the Galerkin orthogonality effect and the corresponding underestimation of the error, see [30]. Recall that, along the adaptive process, the background mesh is used as the base to build up all the adapted meshes by local refinement.…”

“…Here, the adjoint problem is approximated using modal analysis, as suggested in reference [30], to preclude the costly adjoint approximation and storage. The modal-based adjoint approximation is particularly efficient for some quantities of interest.…”