Atomistic-to-continuum coupling via a spacetime discontinuous Galerkin method.

摘要

Accurate simulations across multiple spatial and temporal scales are required for design and analysis purposes in an expanding range of applications. Beyond addressing the high computational cost intrinsic to resolving multiple scales, simulations that span from continuum to atomistic scales must also resolve discrepancies between distinct physical models. This thesis presents a concurrent, atomistic-to-continuum (AtC) coupling method that is based on the spacetime discontinuous Galerkin (SDG) finite element method. The proposed AtC method couples an SDG method for continuum elastodynamics to an atomistic discontinuous Galerkin (ADG) finite element method in time for molecular dynamics. The formulation couples nonoverlapping continuum and atomistic models across a sharp interface in a mathematically consistent fashion by weakly enforcing momentum balance and kinematic compatibility across the interface. Mechanical signals from the atomistic region are fully resolved in the continuum through the use of h-adaptive causal meshing for the SDG solution.;Two- and three-field formulations are presented for the SDG continuum formulation. Both are unconditionally stable and dissipate energy. Convergence studies show the relative performance of each formulation. A special spacetime meshing procedure constructs unstructured, causal grids that support an adaptive patch-by-patch solution process with O (N) computational complexity. An error indicator based on energy dissipation drives a dynamic h-adaptive solution process that controls the total numerical dissipation. A weakened form of the ADG method balances energy to within the accuracy of the numerical integration of the interatomic forces within the atomistic domain. Especially when combined with high-order elements in time, the weakened ADG method can reduce the dissipation within the atomistic part of the model to levels consistent with the machine precision. The h-adaptive solution process in the continuum domain ensures that mechanical signals from the atomistic region are fully resolved in the continuum domain. Further, optimization of an atomic scale configuration parameter virtually eliminates spurious reflections at the coupling interface.;The SDG and ADG methods are both implicit procedures. This is not an issue in the continuum part of the model where the causal solution scheme delivers O (N) scaling. However, due to the non-local interaction between atoms, most molecular dynamics simulations use explicit integrators. An iterative solution scheme developed for the implicit ADG and coupled SDG--ADG models is shown to scale linearly with problem size and to outperform the popular Velocity Verlet integrator for this class of problems.;Atomistic-to-continuum coupling at finite temperature will require a thermomechanical continuum model. The length and time scales under consideration at an AtC coupling interface require a hyperbolic heat conduction model for the continuum. An SDG model for hyperbolic conduction based on the Maxwell-Cattaneo-Vernotte (MCV) model is developed as a first step towards a full thermomechanical model. The MCV model generates a finite signal speed, enabling the use of a causal solution process. Numerical results demonstrate the differences between the MCV and Fourier models of heat conduction.