Enriched space–time finite element method: a new paradigm for multiscaling from elastodynamics to molecular dynamics

SUMMARY The main objective of this paper is to present an enriched version of the space–time FEM method to incorporate 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 (elastodynamics) 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 evaluated 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.

[1]  Thomas J. R. Hughes,et al.  Space-time finite element methods for second-order hyperbolic equations , 1990 .

[2]  Thomas J. R. Hughes,et al.  A space-time formulation for multiscale phenomena , 1996 .

[3]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[4]  O. C. Zienkiewicz.  A new look at the newmark, houbolt and other time stepping formulas. A weighted residual approach , 1977 .

[5]  P. J. Pahl,et al.  Practical performance of the θ 1 -method and comparison with other dissipative algorithms in structural dynamics , 1988 .

[6]  Alain Combescure,et al.  Automatic energy conserving space–time refinement for linear dynamic structural problems , 2005 .

[7]  T. Hughes,et al.  Space-time finite element methods for elastodynamics: formulations and error estimates , 1988 .

[8]  Chandrakant S. Desai,et al.  Generalized time finite element algorithm for non‐linear dynamic problems , 1984 .

[9]  O. C. Zienkiewicz,et al.  Transient field problems: Two‐dimensional and three‐dimensional analysis by isoparametric finite elements , 1970 .

[10]  Ted Belytschko,et al.  Arbitrary discontinuities in space–time finite elements by level sets and X‐FEM , 2004 .

[11]  Isaac Fried,et al.  Finite-element analysis of time-dependent phenomena. , 1969 .

[12]  Nils-Erik Wiberg,et al.  STRUCTURAL DYNAMIC ANALYSIS BY A TIME‐DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD , 1996 .

[13]  Ted Belytschko,et al.  A finite element method for crack growth without remeshing , 1999 .

[14]  Gregory J. Wagner,et al.  Coupling of atomistic and continuum simulations using a bridging scale decomposition , 2003 .

[15]  J. Tinsley Oden,et al.  A GENERAL THEORY OF FINITE ELEMENTS II. APPLICATIONS , 1969 .

[16]  William W. Hager,et al.  Discontinuous Galerkin methods for ordinary differential equations , 1981 .

[17]  Dong Qian,et al.  A Virtual Atom Cluster Approach to the Mechanics of Nanostructures , 2004 .

[18]  Nils-Erik Wiberg,et al.  Implementation and adaptivity of a space-time finite element method for structural dynamics , 1998 .

[19]  Francesco Costanzo,et al.  Proof of unconditional stability for a single-field discontinuous Galerkin finite element formulation for linear elasto-dynamics , 2005 .

[20]  Gregory J. Wagner,et al.  A multiscale projection method for the analysis of carbon nanotubes , 2004 .

[21]  M. Kawahara,et al.  Periodic Galerkin finite element method of tidal flow , 1978 .

[22]  Ivo Babuška,et al.  Generalized finite element methods for three-dimensional structural mechanics problems , 2000 .

[23]  Donald A. French,et al.  A space-time finite element method for the wave equation* , 1993 .

[24]  John Argyris,et al.  Finite Elements in Time and Space , 1969, The Aeronautical Journal (1968).

[25]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

[26]  T. Hughes,et al.  A new finite element formulation for computational fluid dynamics: VI. Convergence analysis of the generalized SUPG formulation for linear time-dependent multi-dimensional advective-diffusive systems , 1987 .

[27]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[28]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[29]  P. J. Pahl,et al.  Development of an implicit method with numerical dissipation from a generalized ingle-step algorithm for structural dynamics , 1988 .

[30]  Ted Belytschko,et al.  Elastic crack growth in finite elements with minimal remeshing , 1999 .

[31]  W. L. Wood,et al.  A unified set of single step algorithms. Part 1: General formulation and applications , 1984 .

[32]  G. Bazzi,et al.  The ρ‐family of algorithms for time‐step integration with improved numerical dissipation , 1982 .

[33]  G. Hulbert.  Time finite element methods for structural dynamics , 1992 .

[34]  Shardool U. Chirputkar,et al.  Coupled Atomistic/Continuum Simulation based on Extended Space-Time Finite Element Method , 2008 .

[35]  Claes Johnson,et al.  Discontinuous Galerkin finite element methods for second order hyperbolic problems , 1993 .

[36]  Julien Réthoré,et al.  A combined space–time extended finite element method , 2005 .

[37]  Eugenio Oñate,et al.  A general procedure for deriving stabilized space–time finite element methods for advective–diffusive problems , 1999 .