116Y. YANG ET AL. as an extension of employing the continuous time function in semi-discrete finite element method, for example, Bazzi and Anderheggen [6], Hoff and Pahl [7,8], and Zienkiewicz and co-workers [9][10][11][12][13]. An extensive review on the subject can be found in [14,15]. The general implementation of TCG involves high computational cost because the entire temporal domain is being discretized.In the second class of the space-time approach, the temporal domains are further divided into 'time slabs' and temporal discontinuities or jumps are allowed between the slabs. Unlike TCG, the time slabs are decoupled from each other with the consideration of the discontinuity conditions. The Galerkin approach is applied in each time slab and the unknowns that are solved in one time slab serve as inputs for the following one. The resulting formulation is called the time-discontinuous Galerkin method (TDG) and is more efficient than TCG for obvious reasons. TDG was originally developed for solving the neutron transport equations by Reed and Hill [16], and Lesaint and Raviart [17]. It has been shown that time-discontinuous Galerkin method leads to solutions to ordinary differential equations that are A-stable and higher-order accurate [17][18][19]. With these salient features, the time-discontinuous Galerkin method has been employed for solving both parabolic and firstorder hyperbolic equations [17,[20][21][22]. Further extension to the field of structural dynamics, for example, elastodynamics, and second-order hyperbolic systems were made by Hughes and colleagues [14,15], Hughes and Stewart [23], and Li and Wiberg [24,25]. Specific implementations can be categorized as either a 'one-field' formulation in which usually the displacement is the only unknown or a 'two-field' formulation in which both the displacement and velocity are considered unknown. To evaluate the convergence properties of the method, error estimates have been developed by French [26], Hughes and Hulbert [27], Hulbert [28], and Johnson [29]. For example, Hughes and Hulbert [27] have augmented their variational formulation with Galerkin least-squares stabilization terms and proved its convergence. Costanzo and Huang [30] expanded on Hulbert and Hughes's [31] formulations to a more general case allowing unstructured finite element grids and provided proofs showing the unconditional stability of the new methods. French [26] included a so-called weighted inner product, which was first used in the space-time methods by Axelsson and Maubach [32]. Johnson provided a priori and a posteriori error estimates for a two-field formulation with linear interpolations [20]. Additional studies have focused on applying adaptive [33], stabilization [34], and multiscale methods [23] to the TDG approach.The idea of capturing multiple temporal responses within the context of TDG has been demonstrated in [27] and [24,25]. Mesh refinement is employed in regions where there is a sharp gradient or discontinuity based on an error estimator (e.g., [35]). Alternatively...
