^{1,a)}, Chia-Chun Fu

^{1}, M. Scott Shell

^{1}and L. Gary Leal

^{1,b)}

### Abstract

In this work, we consider two issues related to the use of Smoothed Dissipative Particle Dynamics (SDPD) as an intermediate mesoscale model in a multiscale scheme for solution of flow problems when there are local parts of a macroscopic domain that require molecular resolution. The first is to demonstrate that SDPD with different levels of resolution can accurately represent the fluid properties from the continuum scale all the way to the molecular scale. Specifically, while the thermodynamic quantities such as temperature, pressure, and average density remain scale-invariant, we demonstrate that the dynamic properties are quantitatively consistent with an all-atom Lennard-Jones reference system when the SDPD resolution approaches the atomistic scale. This supports the idea that SDPD can serve as a natural bridge between molecular and continuum descriptions. In the second part, a simple multiscale methodology is proposed within the SDPD framework that allows several levels of resolution within a single domain. Each particle is characterized by a unique physical length scale called the smoothing length, which is inversely related to the local number density and can change on-the-fly. This multiscale methodology is shown to accurately reproduce fluid properties for the simple problem of steady and transient shear flow.

This work was supported by the University of California Lab Research Program. Computations were performed at the CNSI High Performance Computing facility at UCSB (NSF Grant No. CNS-0960316 and Hewlett-Packard). We wish to thank Anthony Redondo and Alan Graham at LANL and Gaurav Tomar at I.I.Sc. for useful discussions.

I. INTRODUCTION

II. SMOOTHED DISSIPATIVE PARTICLE DYNAMICS

III. THE EFFECTS OF SDPD PARTICLE SIZE ON FLUID BEHAVIOR

A. Target system: Lennard-Jones fluid

B. Equilibrium properties of SDPD fluid

IV. MULTISCALE MODELING WITH SDPD

A. Adaptive smoothing length

B. Coupling multiple resolutions

C. Multiscale simulations: Equilibrium study

D. Multiscale simulations: Shear flow

1. Boundary treatment

2. Start up shear flow

V. SUMMARY AND DISCUSSION

### Key Topics

- Viscosity
- 19.0
- Diffusion
- 15.0
- Hydrodynamics
- 15.0
- Molecular dynamics
- 15.0
- Lagrangian mechanics
- 14.0

## Figures

Schematic of hydrodynamic models for different length- and time-scales.

Schematic of hydrodynamic models for different length- and time-scales.

The cubic spline smoothing kernel and its derivative.

The cubic spline smoothing kernel and its derivative.

(a) Density profile (computed from Eq. (2) ) in equilibrium SDPD simulations with varying resolution. The filled triangles represent the density profile for h = 3 when sampling points are randomly located and are not coincident with the particle centers. The dotted line is for the input density. (b) Illustration of density computation for a point located on (x) and off (x ′) the particle center. The large solid and dotted circles represent the support domain for x and x ′, respectively.

(a) Density profile (computed from Eq. (2) ) in equilibrium SDPD simulations with varying resolution. The filled triangles represent the density profile for h = 3 when sampling points are randomly located and are not coincident with the particle centers. The dotted line is for the input density. (b) Illustration of density computation for a point located on (x) and off (x ′) the particle center. The large solid and dotted circles represent the support domain for x and x ′, respectively.

Velocity probability density function of SDPD particles with varying smoothing length (resolution).

Velocity probability density function of SDPD particles with varying smoothing length (resolution).

Temperature of the SDPD fluid as a function of time for h = 3.

Temperature of the SDPD fluid as a function of time for h = 3.

Mean square displacement of SDPD fluid particles with varying smoothing length or resolution for the SDPD fluid with μ = 2.0 and c s = 5. For h ⩽ 3.6, the error bars (obtained for three separate runs for each h) are shown at time = 1500; for larger h the error bars are too small to show. The outset shows the enlarged version for h = 4.8.

Mean square displacement of SDPD fluid particles with varying smoothing length or resolution for the SDPD fluid with μ = 2.0 and c s = 5. For h ⩽ 3.6, the error bars (obtained for three separate runs for each h) are shown at time = 1500; for larger h the error bars are too small to show. The outset shows the enlarged version for h = 4.8.

Self-diffusion coefficient and Schmidt number (b) of SDPD fluid particles plotted as a function of varying resolution for c s = 5 (circles) and c s = 1 (squares).

Self-diffusion coefficient and Schmidt number (b) of SDPD fluid particles plotted as a function of varying resolution for c s = 5 (circles) and c s = 1 (squares).

Diffusion coefficient of the SDPD fluid (symbols) vs smoothing length at different thermodynamic state points. The dotted lines are from the MD simulations at the same state points.

Diffusion coefficient of the SDPD fluid (symbols) vs smoothing length at different thermodynamic state points. The dotted lines are from the MD simulations at the same state points.

Diffusion coefficient as a function of coarse-graining parameter obtained using SDPD (this work) and standard DPD models (from Ref. 48 ) that represent Lennard-Jones fluid (dashed line) with ρ = 0.8 and T = 1.

Diffusion coefficient as a function of coarse-graining parameter obtained using SDPD (this work) and standard DPD models (from Ref. 48 ) that represent Lennard-Jones fluid (dashed line) with ρ = 0.8 and T = 1.

The radial distribution function of the SDPD fluid for different resolutions.

The radial distribution function of the SDPD fluid for different resolutions.

Adaptive resolution scheme for SDPD fluid. A diffuse interface consisting of coarsening, refining, and overlap subregions separates the two levels of SDPD particle.

Adaptive resolution scheme for SDPD fluid. A diffuse interface consisting of coarsening, refining, and overlap subregions separates the two levels of SDPD particle.

Schematic of the computational set up for multiscale SDPD simulations under equilibrium conditions. The dashed lines represent the diffuse interface region separating two resolutions.

Schematic of the computational set up for multiscale SDPD simulations under equilibrium conditions. The dashed lines represent the diffuse interface region separating two resolutions.

Smoothing length (a) and mass density (b) profiles in the box for multiscale simulations combining two resolutions. The dashed-dotted lines represent the interface regions.

Smoothing length (a) and mass density (b) profiles in the box for multiscale simulations combining two resolutions. The dashed-dotted lines represent the interface regions.

Illustration of the implementation of a no-slip boundary.

Illustration of the implementation of a no-slip boundary.

Schematic of the computational set up for multiscale SDPD simulations of the shear flow between two parallel plates. The dashed lines represent the diffuse interface region separating two resolutions. The wall and fluid particles are represented by filled and open circles, respectively.

Schematic of the computational set up for multiscale SDPD simulations of the shear flow between two parallel plates. The dashed lines represent the diffuse interface region separating two resolutions. The wall and fluid particles are represented by filled and open circles, respectively.

Mass density (a) and smoothing length (b) profiles in the box for multiscale shear flow simulations combining two resolutions. The dashed-dotted lines represent the interface regions.

Mass density (a) and smoothing length (b) profiles in the box for multiscale shear flow simulations combining two resolutions. The dashed-dotted lines represent the interface regions.

Time evolution of the velocity profile for start-up shear flow obtained using multiscale SDPD (circles), single resolution SDPD (squares), and the analytical solution (lines). In nearly all cases, the squares lay almost exactly behind the circles. The lines from top to bottom represent dimensionless time tν/H 2 of 1.0, 0.3125, 0.1875, and 0.125.

Time evolution of the velocity profile for start-up shear flow obtained using multiscale SDPD (circles), single resolution SDPD (squares), and the analytical solution (lines). In nearly all cases, the squares lay almost exactly behind the circles. The lines from top to bottom represent dimensionless time tν/H 2 of 1.0, 0.3125, 0.1875, and 0.125.

## Tables

SDPD simulation parameters described in LJ units.

SDPD simulation parameters described in LJ units.

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