Hybrid model for combined particle and continuum dynamics

Abstract

Differential equations for continuum fields describe many macroscopic phenomena. Hydrodynamics, for example, is described by the Navier-Stokes equations, and their solutions depend on boundary conditions. However, boundary conditions are set by the interactions at the atomistic or molecular scale. We introduce a "hybrid model" that permits a continuum description in one region to be coupled to an atomistic description in another region. The coupling is symmetric in the sense that the fluxes of the conserved quantities are continuous across the particle-field interface. As an example, we couple a Lennard-Jones liquid and the compressible Navier-Stokes equations and show that the hybrid model is consistent with hydrodynamic predictions.

Algorithm Refinement for Stochastic Partial Differential Equations

“…Specifically, the variance in the particle region of the deterministic hybrid is significantly reduced near the interface and decays quickly to zero in the continuum region (see Figs. 5,7,11,and 15). In this section we introduce an approximate model that reproduces this effect.…”

“…Such "hybrid" methods typically couple (at least) two structurally (physically and algorithmically) different computational schemes, which are used in different regions of the problem (e.g., interior and exterior of a shock wave). One class of hybrids involves matching particle methods to continuum partial differential equation (PDE) solvers [1][2][3][4][5][6][7][8][9][10][11][12]; other hybrids combine stochastic systems with deterministic ones, explicit schemes with implicit ones, etc.…”

A constrained particle dynamics for continuum-particle hybrid method in micro- and nano-fluidics

“…At each interval t MD , the nearest integer to n is taken. In order to maintain momentum flux continuity, the following assumptions must be satisfied [9] i…”

“…As a result, the particles are effectively constrained in the MD domain. Equation (9) requires that the average velocities of particles must match the macroscopic velocities in the constrained layer. This requirement has to be extended to the whole overlap region.…”

“…O'Connell and Thompson [7] developed a constrained particle dynamics for stress continuity; Hadjiconstantinou and Patera [8] presented a heterogeneous algorithm for incompressible flows; Flekkoy et al [9] developed a flux-exchange scheme to couple particle dynamics with continuum dynamics. Recently, Nie et al [10] In present paper, we develop a coupling model in terms of a virtual damping force and a virtual added mass force in Sect.…”