Integral approximations to classical diffusion and smoothed particle hydrodynamics
Abstract
The contribution of the paper is the approximation of a classical diffusion operator by an integral equation with a volume constraint. A particular focus is on classical diffusion problems associated with Neumann boundary conditions. By exploiting this approximation, we can also approximate other quantities such as the flux out of a domain. Our analysis of the model equation on the continuum level is closely related to the recent work on nonlocal diffusion and peridynamic mechanics. In particular, we elucidate the role of a volumetric constraint as an approximation to a classical Neumann boundary condition in the presence of physical boundary. The volumeconstrained integral equation then provides the basis for accurate and robust discretization methods. As a result, an immediate application is to the understanding and improvement of the Smoothed Particle Hydrodynamics (SPH) method.
 Authors:
 Columbia Univ., New York, NY (United States); Pennsylvania State Univ., University Park, PA (United States)
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Univ. of South Florida, Tampa, FL (United States); Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1235919
 Report Number(s):
 SAND20150849J
Journal ID: ISSN 00457825; PII: S0045782514004988
 Grant/Contract Number:
 AC0494AL85000
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 Computer Methods in Applied Mechanics and Engineering
 Additional Journal Information:
 Journal Volume: 286; Journal Issue: C; Journal ID: ISSN 00457825
 Publisher:
 Elsevier
 Country of Publication:
 United States
 Language:
 English
 Subject:
 42 ENGINEERING; 97 MATHEMATICS AND COMPUTING; smooth particle hydrodynamics; classical diffusion; nonlocal diffusion; nonlocal operator; nonlocal Neumann condition; numerical approximation; integral operators; diffusion; Neumann boundary
Citation Formats
Du, Qiang, Lehoucq, R. B., and Tartakovsky, A. M. Integral approximations to classical diffusion and smoothed particle hydrodynamics. United States: N. p., 2014.
Web. doi:10.1016/j.cma.2014.12.019.
Du, Qiang, Lehoucq, R. B., & Tartakovsky, A. M. Integral approximations to classical diffusion and smoothed particle hydrodynamics. United States. doi:10.1016/j.cma.2014.12.019.
Du, Qiang, Lehoucq, R. B., and Tartakovsky, A. M. 2014.
"Integral approximations to classical diffusion and smoothed particle hydrodynamics". United States.
doi:10.1016/j.cma.2014.12.019. https://www.osti.gov/servlets/purl/1235919.
@article{osti_1235919,
title = {Integral approximations to classical diffusion and smoothed particle hydrodynamics},
author = {Du, Qiang and Lehoucq, R. B. and Tartakovsky, A. M.},
abstractNote = {The contribution of the paper is the approximation of a classical diffusion operator by an integral equation with a volume constraint. A particular focus is on classical diffusion problems associated with Neumann boundary conditions. By exploiting this approximation, we can also approximate other quantities such as the flux out of a domain. Our analysis of the model equation on the continuum level is closely related to the recent work on nonlocal diffusion and peridynamic mechanics. In particular, we elucidate the role of a volumetric constraint as an approximation to a classical Neumann boundary condition in the presence of physical boundary. The volumeconstrained integral equation then provides the basis for accurate and robust discretization methods. As a result, an immediate application is to the understanding and improvement of the Smoothed Particle Hydrodynamics (SPH) method.},
doi = {10.1016/j.cma.2014.12.019},
journal = {Computer Methods in Applied Mechanics and Engineering},
number = C,
volume = 286,
place = {United States},
year = 2014,
month =
}
Web of Science

The contribution of the paper is the approximation of a classical diffusion operator by an integral equation with a volume constraint. A particular focus is on classical diffusion problems associated with Neumann boundary conditions. By exploiting this approximation, we can also approximate other quantities such as the flux out of a domain. Our analysis of the model equation on the continuum level is closely related to the recent work on nonlocal diffusion and peridynamic mechanics. In particular, we elucidate the role of a volumetric constraint as an approximation to a classical Neumann boundary condition in the presence of physical boundary.more »

Integral approximations to classical diffusion and smoothed particle hydrodynamics.
Abstract not provided. 
Smoothed particle hydrodynamics model for LandauLifshitz NavierStokes and advectiondiffusion equations
We propose a novel Smoothed Particle Hydrodynamics (SPH) discretization of the fullycoupled LandauLifshitzNavierStokes (LLNS) and advectiondiffusion equations. The accuracy of the SPH solution of the LLNS equations is demonstrated by comparing the scaling of velocity variance and selfdiffusion coefficient with kinetic temperature and particle mass obtained from the SPH simulations and analytical solutions. The spatial covariance of pressure and velocity fluctuations are found to be in a good agreement with theoretical models. To validate the accuracy of the SPH method for the coupled LLNS and advectiondiffusion equations, we simulate the interface between two miscible fluids. We study the formation ofmore » 
Smoothed particle hydrodynamics model for LandauLifshitzNavierStokes and advectiondiffusion equations
We propose a novel smoothed particle hydrodynamics (SPH) discretization of the fully coupled LandauLifshitzNavierStokes (LLNS) and stochastic advectiondiffusion equations. The accuracy of the SPH solution of the LLNS equations is demonstrated by comparing the scaling of velocity variance and the selfdiffusion coefficient with kinetic temperature and particle mass obtained from the SPH simulations and analytical solutions. The spatial covariance of pressure and velocity fluctuations is found to be in a good agreement with theoretical models. To validate the accuracy of the SPH method for coupled LLNS and advectiondiffusion equations, we simulate the interface between two miscible fluids. We study formationmore » 
METAL DIFFUSION IN SMOOTHED PARTICLE HYDRODYNAMICS SIMULATIONS OF DWARF GALAXIES
We perform a series of smoothed particle hydrodynamics simulations of isolated dwarf galaxies to compare different metal mixing models. In particular, we examine the role of diffusion in the production of enriched outflows and in determining the metallicity distributions of gas and stars. We investigate different diffusion strengths by changing the prefactor of the diffusion coefficient, by varying how the diffusion coefficient is calculated from the local velocity distribution, and by varying whether the speed of sound is included as a velocity term. Stronger diffusion produces a tighter [O/Fe]–[Fe/H] distribution in the gas and cuts off the gas metallicity distributionmore »