# 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:

- 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) (SC-22); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1369324

- Alternate Identifier(s):
- OSTI ID: 1369577; OSTI ID: 1393928

- 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. Thu .
"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 = {Thu Jun 29 00:00:00 EDT 2017},

month = {Thu Jun 29 00:00:00 EDT 2017}

}