INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng (2012) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4323 Enriched space–time finite element method: a new paradigm for multiscaling from elastodynamics to molecular dynamics Ye Yang 1 , Shardool Chirputkar 1 , David N. Alpert 1 , Thomas Eason 2 , Stephen Spottswood 2 and Dong Qian 1, * ,† 1 Mechanical Engineering Program, School of Dynamic Systems, University of Cincinnati, Cincinnati, OH 45221, USA 2 Air Force Research Laboratory, Air Vehicles Directorate, Wright-Patterson Air Force Base, Fairborn, OH 45433, USA SUMMARY The main objective of this paper is to present an enriched version of the space–time FEM method to incor- porate multiple temporal scale features with a focus on dynamics problems. The method is established by integrating the basic framework of the space–time discontinuous Galerkin method with the extended finite element method. Two versions of the method have been developed: one at the continuum scale (elastodynam- ics) and the other focuses on the dynamics at the molecular level. After an initial outline of the formulation, we explore the incorporation of different types of enrichments based on the length scale of interest. The effects of both continuous and discontinuous enrichments are demonstrated through numerical examples involving wave propagations and dynamic fracture in harmonic lattice. The robustness of the method is eval- uated in terms of convergence and the ability to capture the fine scale features. It is shown that the enriched space–time FEM leads to an improvement in the convergence properties over the traditional space–time FEM for problems with multiple temporal features. It is also highly effective in integrating atomistic with continuum representations with a coupled framework. Copyright © 2012 John Wiley & Sons, Ltd. Received 12 August 2011; Revised 20 November 2011; Accepted 9 February 2012 KEY WORDS: space–time FEM; extended FEM; enrichment; Galerkin least-square stabilization; bridging- scale method; lattice fracture 1. INTRODUCTION Many engineering problems are characterized by multiple temporal scale features that arise from the combination of the loading conditions, material structures, and responses. Developing numeri- cal methods for these classes of problems has been a constant interest in the field of computational mechanics. As a single scale method, traditional FEM based on the semi-discrete schemes, is not well suited because it generally solves the time-dependent problems by discretizing the spatial domain using finite elements while the responses in time are traced using the finite difference method. As such, it lacks the flexibility in establishing multiscale approximations in the tempo- ral domain. In addition, use of a specific semi-discrete scheme may further bring up the issues of stability and convergence with the choice of parameters. The immense difficulty associated with the semi-discrete schemes has led to another class of the finite element method that also discretizes the temporal domain with mesh. This method- ology was first proposed in [1–3]. Further developments along the path of variation principles (Galerkin method) have resulted in two classes of methods. In the first class, no discontinuity in time is allowed in the approximation and the method is referred to as the time continuous space–time Galerkin method (TCG). Examples of such can be seen in [4] and [5]. This method can be regarded *Correspondence to: Dong Qian, School of Dynamic Systems, University of Cincinnati, Cincinnati, OH 45221, USA. † E-mail: Dong.Qian@uc.edu Copyright © 2012 John Wiley & Sons, Ltd.