SUMMARYRecent development in the multiscale method based on the bridging scale concept is presented with an emphasis on complex material systems. The bridging scale method (BSM) was originally proposed by Wagner and Liu (J. Comput. Phys. 2003; 190:249-274) as an effective way of treating the interface in coupled atomistic/continuum simulation. Since its publication, the BSM has become a very useful paradigm that has been applied to solve a host of problems in mechanical sciences of complex material systems. In this paper, we present a review on the recent developments. We first describe the application of BSM to the coupled atomistic/continuum simulation of dynamic fracture. The important extensions within the framework of space-time method and multiscale non-equilibrium molecular dynamics are then presented. We then focus on the multiresolution continuum theory that inherits the BSM concept in the analysis of heterogeneous material structures. Recent work of incorporating statistical factors into this model based on the concurrent nested homogenization of randomness of the material structures is highlighted. Finally, we present the use of the bridging scale concept in resolving the electron-mechanical coupling mechanism. The robustness of the BSM is demonstrated through many benchmark problems and application examples.

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...

SUMMARYA multiscale method that couples the space‐time Finite Element Method with molecular dynamics is developed for the simulation of dynamic fracture problems. A coarse scale description based on finite element discretization is established in the entire domain. This representation overlaps with a detailed atomistic description employed in the region immediately surrounding the crack tip with the goal to capture the initiation and propagation of the fracture. On the basis of the crack evolution informed by the atomistic simulation, an enrichment function is introduced to represent the fracture path. The space‐time framework further enables flexible choice of time steps in different regions of interest. Coupling between the fine and coarse scale simulation is achieved with the introduction of a projection operator and bridging scale treatment. The main feature of the work is that the evolution of the crack is adaptively tracked and the enrichment in the coarse scale simulation evolves along with it. As a result, we have a moving atomistic region in the concurrent simulation scheme. The robustness of the method is illustrated through examples involving crack propagation in hexagonal lattices with different orientations and loading conditions. Copyright © 2013 John Wiley & Sons, Ltd.