A Spacetime, Balance-Law Formulation of Coupled Atomistic and Continuum Dynamics for Solids

Abstract

Coupled dynamic atomistic and continuum computational methods for solids have received much interest recently, because many problems are not addressed well by either model alone. In most coupled methods more emphasis has been placed on damping spurious reflections than on balancing momentum and energy. I present a new method for concurrent coupling of dynamic atomistic and continuum simulations of solids that enforces these balance laws on the atom/element level while minimizing spurious reflections. The coupled formulation is composed of the continuum spacetime discontinuous Galerkin (SDG) method and the mathematically consistent, time finite element, atomistic discontinuous Galerkin (ADG) method. On the continuum side I develop a two- and three-field SDG formulations for linearized elastodynamics to illuminate the mathematical structure of the original one-field SDG formulation and to assist in making connections to the atomistic formulation. On the atomistic side I examine connections between the SDG and ADG methods, and then extend this to relationships with the Velocity Verlet integrator. The component SDG and ADG methods are coupled using the same Godunov flux solution as is used by the SDG method, to enforce weakly the jump conditions on momentum balance and kinematic compatibility. To obtain compatible fluxes on the atomistic side of the coupling boundary I define a boundary atomistic trace that can be optimized to minimize boundary reflections. The coupled SDG--ADG formulation preserves the characteristic structure of the hyperbolic problem, guarantees element/atom-wise momentum balance to machine precision and yields energy error that is small, dissipative and controllable. The flux-based coupling can also be used with the Velocity Verlet method in place of the ADG, although the SDG--VV method suffers from uncontrolled energy error for long-time simulations due to the mismatch in the mathematical models. I present the formulations in spacetime, with one spatial dimension, employing linear springs and a nonlinear potential, demonstrating their efficacy through numerical examples.