Analysis of Atomistic/continuum Coupling Using Meshless Methods

In this paper, we compare three interpolation functions in a discretized continuum when used in coupled dynamic atomistic-to-continuum simulations. The focus is on assessing the ability of the discrete continuum model to capture and accurately represent transient effects, namely a travelling longitudinal wave, through both the mixed atomistic-continuum interface and the non-uniform continuum mesh beyond. We specifically examine the differences among BubnovGalerkin, partition of unity, and moving least squares finite element methods in the continuum part of the multiscale model. Our study shows that using partition of unity interpolation functions in the continuum produces superior results compared to the other two approaches. INTRODUCTION It is well known that continuum based techniques such as Lagrangian or Eulerian numerical methods, which use constitutive relations that do not account for the atomistic structure, are invalid beyond the scope of their calibration. In regions containing dislocations, mobile defects, or nonlinear material, these numerical methods have to be modified to capture important phenomena. Molecular dynamics (MD) is an excellent means for predicting interactions on an atomic scale as well as predicting the response when sub-micron scale phenomena occur. However, MD can be computationally expensive beyond small sample sizes and has difficulty implementing boundary conditions applied at a continuum scale. Therefore, to alleviate these problems multiscale methods have been developed in recent years to couple the continuum and atomistic scales together. There has been extensive work on developing novel coupling techniques for linking atomistic and continuum scales. These techniques include the quasicontinuum method [1], bridging domain method [2], bridging scale method [3] and homogenization techniques [4,5], among others. A thorough review of several recent techniques is given in [6]. These techniques have been developed using the finite element method within the continuum scale. Though seemingly well known, to our knowledge, an examination of the level of approximation and choice of interpolation in the continuum region in and around the discrete atomistic domain has not been shown. In this paper we show a comparative study of the quality of interpolation that best suits continuum methods in regions at and near the interface with a molecular dynamics region. We specifically examine interpolation functions prominent in general finite element methods and meshless methods – Bubnov-Galerkin, partition of unity [7], and moving least squares [8] – and assess their ability to capture a travelling wave through a discrete/continuum interface and a non-uniform finite element mesh (increasing element size away from the MD region). Within the interface region, where the continuum and atomistic scales overlap, the displacements on the continuum are dictated by the atomistic results generated from MD. In this study, the forces between the domains are communicated from the atoms to the continuum through ghost nodes. CONTINUUM FORMULATION We begin by reviewing the governing equations on the continuum scale. The conservation of momentum can be defined as: ( ) u f P & & 0 0 0 0 0 0 0 V V V ρ ρ = + ∇ (1) where P is the first Piola-Kirchoff stress tensor, f0 is the body force, ρ0 is the density, V0 is the initial volume and is the acceleration. u& & From classical hyperelastic continuum approximation we can define the first Piola-Kirchoff stress as: F P ∂ ∂ = W V0 1 (2) where W is the potential energy density, and F is the deformation gradient defined as: 1 + ∂ ∂ = ∂ ∂ = X u X x F (3) where X denotes the reference configuration and x denotes the spatial or current configuration. In order to use equation (1) for numerical techniques such as general finite elements we use the principal of virtual work to obtain the variational form: ( ) [ ] 0 0 0 0 0 0 0 0 0 0 = Ω ∂ − + ∇ ∫Ω C V V V C u f P w & & ρ ρ (4) where w is the virtual displacement. In the next two sections we define two different approaches to Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE DEC 2008 2. REPORT TYPE N/A 3. DATES COVERED 4. TITLE AND SUBTITLE Analysis Of Atomistic/Continuum Coupling Using Meshless Methods 5a. CONTRACT NUMBER