Adaptive Isogeometric analysis for transient dynamics: Space–time refinement based on hierarchical a-posteriori error estimations

In this paper, based on our proposed IsoGeometric Analysis (IGA)-based adaptivity technique for plate vibration (Yu et al., 2018) the space–time adaptivity strategies used for transient dynamics are established. Specifically, the Geometry Independent Field approximaTion (GIFT) method is applied to discretize the spatial domain, and time discretization is proceeded by Newmark method. In the framework of GIFT/Newmark, three kinds of space–time adaptivity strategies based on hierarchical a-posteriori error estimations are developed successively, that is, Unidirectional Multi-level Space-Time Adaptive GIFT/Newmark (UM-STAGN), Energy-based Space-Time Adaptive GIFT/Newmark (E-STAGN), and Goal-oriented Space-Time Adaptive GIFT/ Newmark (G-STAGN) methods respectively. The main concept of UM-STAGN approach is to get rid of the elements where the error estimators reach the prescribed accuracy at each adaptation step, and then assemble the rest elements as the new subsystems to be solved at the next stage. By reducing the scale of the computational domains gradually, this method achieves efficiency to some degree, though, since it fails to transfer the error information across subsystems, the error of Quantity of Interest (QoI) cannot arrive at an expected precision. For this reason, we introduce E-STAGN methodology, wherein the error indicator of each element is reassessed at every adaptive cycle through solving the whole system again. In this case, the QoI error is able to be convergent to an acceptable accuracy. Nevertheless, as the error is accumulated with time in the time-domain problem, E-STAGN method based on the energy-norm error estimation is unable to expose the source of the error so that it fails to offer an efficient refinement. This is the motivation to establish the G-STAGN technique, where the error estimation drives from our proposed first-order Dual Weight Residual (DWR) method. G-STAGN strategy can detect the origin of QoI error, and hence it leads to a remarkably economical refinement. Numerical examples are carried out in both single-patch and multi-patch structures. It is demonstrated that both UM-STAGN and E-STAGN methods can catch the propagation of stress wave for the primal problem, while G-STAGN technique is capable to track the travel path of dual stress. Therefore, the G-STAGN strategy achieves an optimal convergent rate, compared to that obtained by UM-STAGN, E-STAGN and uniform space–time h-refinement approaches.