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Title: 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 long-range 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 matrix-free algorithm using the General geometry Ewald-like method. Our approach employs a highly-efficient iterative finite element Stokes’ solver for the accurate treatment of long-range hydrodynamic interactions within arbitrary confined geometries. A combination of mid-point time integration of the Brownian stochastic differential equation, the parallel Stokes’ solver, and a Chebyshev polynomial approximation for the fluctuation-dissipation 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 non-equilibrium conditions. Our method is extended to treat suspended finite size particles of arbitrary shape in any geometry using an Immersed Boundary approach.

Authors:
ORCiD logo [1];  [2];  [2];  [3]; ORCiD logo [4];  [3]; ORCiD logo [5];  [6]
  1. Argonne National Lab. (ANL), Lemont, IL (United States). Mathematics and Computer Science Division; Univ. of Chicago, IL (United States). Inst. for Molecular Engineering
  2. Univ. of Chicago, IL (United States). Inst. for Molecular Engineering
  3. Argonne National Lab. (ANL), Lemont, IL (United States). Mathematics and Computer Science Division
  4. Argonne National Lab. (ANL), Lemont, IL (United States). Materials Science Division; Northwestern Argonne Inst. for Science and Engineering, Evanston, IL (United States)
  5. Univ. of Chicago, IL (United States). Inst. for Molecular Engineering; National Univ. of Columbia, Sede Medellin (Columbia). Dept. of Materials
  6. 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) (SC-22); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)
OSTI Identifier:
1369324
Grant/Contract Number:
AC02-06CH11357
Resource Type:
Journal Article: Published Article
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 146; Journal Issue: 24; Journal ID: ISSN 0021-9606
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, Hernandez-Ortiz, 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, Hernandez-Ortiz, 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, Hernandez-Ortiz, 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 Hernandez-Ortiz, 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 long-range 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 matrix-free algorithm using the General geometry Ewald-like method. Our approach employs a highly-efficient iterative finite element Stokes’ solver for the accurate treatment of long-range hydrodynamic interactions within arbitrary confined geometries. A combination of mid-point time integration of the Brownian stochastic differential equation, the parallel Stokes’ solver, and a Chebyshev polynomial approximation for the fluctuation-dissipation 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 non-equilibrium 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
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record at 10.1063/1.4989545

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  • 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 long-range 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 matrix-free algorithm using the General geometry Ewald-like method. Our approach employs a highly-efficient iterative finite element Stokes’ solver for the accurate treatment of long-range hydrodynamic interactions within arbitrary confined geometries. A combination of mid-point time integration of the Brownian stochastic differential equation, the parallelmore » Stokes’ solver, and a Chebyshev polynomial approximation for the fluctuation-dissipation 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 non-equilibrium conditions. Our method is extended to treat suspended finite size particles of arbitrary shape in any geometry using an Immersed Boundary approach.« less
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  • This paper describes a new parallel, scalable and robust finite element based solver for the first-order Stokes momentum balance equations for ice flow. The solver, known as Albany/FELIX, is constructed using the component-based approach to building application codes, in which mature, modular libraries developed as a part of the Trilinos project are combined using abstract interfaces and template-based generic programming, resulting in a final code with access to dozens of algorithmic and advanced analysis capabilities. Following an overview of the relevant partial differential equations and boundary conditions, the numerical methods chosen to discretize the ice flow equations are described, alongmore » with their implementation. The results of several verification studies of the model accuracy are presented using (1) new test cases for simplified two-dimensional (2-D) versions of the governing equations derived using the method of manufactured solutions, and (2) canonical ice sheet modeling benchmarks. Model accuracy and convergence with respect to mesh resolution are then studied on problems involving a realistic Greenland ice sheet geometry discretized using hexahedral and tetrahedral meshes. Also explored as a part of this study is the effect of vertical mesh resolution on the solution accuracy and solver performance. The robustness and scalability of our solver on these problems is demonstrated. Lastly, we show that good scalability can be achieved by preconditioning the iterative linear solver using a new algebraic multilevel preconditioner, constructed based on the idea of semi-coarsening.« less
  • This paper describes a new parallel, scalable and robust finite element based solver for the first-order Stokes momentum balance equations for ice flow. The solver, known as Albany/FELIX, is constructed using the component-based approach to building application codes, in which mature, modular libraries developed as a part of the Trilinos project are combined using abstract interfaces and template-based generic programming, resulting in a final code with access to dozens of algorithmic and advanced analysis capabilities. Following an overview of the relevant partial differential equations and boundary conditions, the numerical methods chosen to discretize the ice flow equations are described, alongmore » with their implementation. The results of several verification studies of the model accuracy are presented using (1) new test cases for simplified two-dimensional (2-D) versions of the governing equations derived using the method of manufactured solutions, and (2) canonical ice sheet modeling benchmarks. Model accuracy and convergence with respect to mesh resolution are then studied on problems involving a realistic Greenland ice sheet geometry discretized using hexahedral and tetrahedral meshes. Also explored as a part of this study is the effect of vertical mesh resolution on the solution accuracy and solver performance. The robustness and scalability of our solver on these problems is demonstrated. Lastly, we show that good scalability can be achieved by preconditioning the iterative linear solver using a new algebraic multilevel preconditioner, constructed based on the idea of semi-coarsening.« less