Parallel O(N) Stokes’ solver towards scalable Brownian dynamics of hydrodynamically interacting objects in general geometries
Abstract
An efficient parallel Stokes’s solver is developed towards the complete inclusion of hydrodynamic interactions of Brownian particles in any geometry. A Langevin description of the particle dynamics is adopted, where the longrange interactions are included using a Green’s function formalism. We present a scalable parallel computational approach, where the general geometry Stokeslet is calculated following a matrixfree algorithm using the General geometry Ewaldlike method. Our approach employs a highlyefficient iterative finite element Stokes’ solver for the accurate treatment of longrange hydrodynamic interactions within arbitrary confined geometries. A combination of midpoint time integration of the Brownian stochastic differential equation, the parallel Stokes’ solver, and a Chebyshev polynomial approximation for the fluctuationdissipation theorem result in an O(N) parallel algorithm. We also illustrate the new algorithm in the context of the dynamics of confined polymer solutions in equilibrium and nonequilibrium conditions. Our method is extended to treat suspended finite size particles of arbitrary shape in any geometry using an Immersed Boundary approach.
 Authors:
 Argonne National Lab. (ANL), Lemont, IL (United States). Mathematics and Computer Science Division; Univ. of Chicago, IL (United States). Inst. for Molecular Engineering
 Univ. of Chicago, IL (United States). Inst. for Molecular Engineering
 Argonne National Lab. (ANL), Lemont, IL (United States). Mathematics and Computer Science Division
 Argonne National Lab. (ANL), Lemont, IL (United States). Materials Science Division; Northwestern Argonne Inst. for Science and Engineering, Evanston, IL (United States)
 Univ. of Chicago, IL (United States). Inst. for Molecular Engineering; National Univ. of Columbia, Sede Medellin (Columbia). Dept. of Materials
 Univ. of Chicago, IL (United States). Inst. for Molecular Engineering; Argonne National Lab. (ANL), Lemont, IL (United States). Materials Science Division
 Publication Date:
 Research Org.:
 Argonne National Lab. (ANL), Argonne, IL (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC22); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC21)
 OSTI Identifier:
 1369324
 Grant/Contract Number:
 AC0206CH11357
 Resource Type:
 Journal Article: Published Article
 Journal Name:
 Journal of Chemical Physics
 Additional Journal Information:
 Journal Volume: 146; Journal Issue: 24; Journal ID: ISSN 00219606
 Publisher:
 American Institute of Physics (AIP)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 36 MATERIALS SCIENCE; 97 MATHEMATICS AND COMPUTING; Brownian dynamics; parallel Stokes’ solver; polymer solutions
Citation Formats
Zhao, Xujun, Li, Jiyuan, Jiang, Xikai, Karpeev, Dmitry, Heinonen, Olle, Smith, Barry, HernandezOrtiz, Juan P., and de Pablo, Juan J.. Parallel O(N) Stokes’ solver towards scalable Brownian dynamics of hydrodynamically interacting objects in general geometries. United States: N. p., 2017.
Web. doi:10.1063/1.4989545.
Zhao, Xujun, Li, Jiyuan, Jiang, Xikai, Karpeev, Dmitry, Heinonen, Olle, Smith, Barry, HernandezOrtiz, Juan P., & de Pablo, Juan J.. Parallel O(N) Stokes’ solver towards scalable Brownian dynamics of hydrodynamically interacting objects in general geometries. United States. doi:10.1063/1.4989545.
Zhao, Xujun, Li, Jiyuan, Jiang, Xikai, Karpeev, Dmitry, Heinonen, Olle, Smith, Barry, HernandezOrtiz, Juan P., and de Pablo, Juan J.. 2017.
"Parallel O(N) Stokes’ solver towards scalable Brownian dynamics of hydrodynamically interacting objects in general geometries". United States.
doi:10.1063/1.4989545.
@article{osti_1369324,
title = {Parallel O(N) Stokes’ solver towards scalable Brownian dynamics of hydrodynamically interacting objects in general geometries},
author = {Zhao, Xujun and Li, Jiyuan and Jiang, Xikai and Karpeev, Dmitry and Heinonen, Olle and Smith, Barry and HernandezOrtiz, Juan P. and de Pablo, Juan J.},
abstractNote = {An efficient parallel Stokes’s solver is developed towards the complete inclusion of hydrodynamic interactions of Brownian particles in any geometry. A Langevin description of the particle dynamics is adopted, where the longrange interactions are included using a Green’s function formalism. We present a scalable parallel computational approach, where the general geometry Stokeslet is calculated following a matrixfree algorithm using the General geometry Ewaldlike method. Our approach employs a highlyefficient iterative finite element Stokes’ solver for the accurate treatment of longrange hydrodynamic interactions within arbitrary confined geometries. A combination of midpoint time integration of the Brownian stochastic differential equation, the parallel Stokes’ solver, and a Chebyshev polynomial approximation for the fluctuationdissipation theorem result in an O(N) parallel algorithm. We also illustrate the new algorithm in the context of the dynamics of confined polymer solutions in equilibrium and nonequilibrium conditions. Our method is extended to treat suspended finite size particles of arbitrary shape in any geometry using an Immersed Boundary approach.},
doi = {10.1063/1.4989545},
journal = {Journal of Chemical Physics},
number = 24,
volume = 146,
place = {United States},
year = 2017,
month = 6
}

Parallel O(N) Stokes’ solver towards scalable Brownian dynamics of hydrodynamically interacting objects in general geometries
An efficient parallel Stokes’s solver is developed towards the complete inclusion of hydrodynamic interactions of Brownian particles in any geometry. A Langevin description of the particle dynamics is adopted, where the longrange interactions are included using a Green’s function formalism. We present a scalable parallel computational approach, where the general geometry Stokeslet is calculated following a matrixfree algorithm using the General geometry Ewaldlike method. Our approach employs a highlyefficient iterative finite element Stokes’ solver for the accurate treatment of longrange hydrodynamic interactions within arbitrary confined geometries. A combination of midpoint time integration of the Brownian stochastic differential equation, the parallelmore » 
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