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Title: Geometric multigrid for an implicit-time immersed boundary method

Abstract

The immersed boundary (IB) method is an approach to fluid-structure interaction that uses Lagrangian variables to describe the deformations and resulting forces of the structure and Eulerian variables to describe the motion and forces of the fluid. Explicit time stepping schemes for the IB method require solvers only for Eulerian equations, for which fast Cartesian grid solution methods are available. Such methods are relatively straightforward to develop and are widely used in practice but often require very small time steps to maintain stability. Implicit-time IB methods permit the stable use of large time steps, but efficient implementations of such methods require significantly more complex solvers that effectively treat both Lagrangian and Eulerian variables simultaneously. Moreover, several different approaches to solving the coupled Lagrangian-Eulerian equations have been proposed, but a complete understanding of this problem is still emerging. This paper presents a geometric multigrid method for an implicit-time discretization of the IB equations. This multigrid scheme uses a generalization of box relaxation that is shown to handle problems in which the physical stiffness of the structure is very large. Numerical examples are provided to illustrate the effectiveness and efficiency of the algorithms described herein. Finally, these tests show that using multigridmore » as a preconditioner for a Krylov method yields improvements in both robustness and efficiency as compared to using multigrid as a solver. They also demonstrate that with a time step 100–1000 times larger than that permitted by an explicit IB method, the multigrid-preconditioned implicit IB method is approximately 50–200 times more efficient than the explicit method.« less

Authors:
 [1];  [2];  [3]
  1. Univ. of California, Davis, CA (United States)
  2. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
  3. Univ. of North Carolina, Chapel Hill, NC (United States)
Publication Date:
Research Org.:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1163152
Grant/Contract Number:  
AC05-00OR22725
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Advances in Computational Mathematics
Additional Journal Information:
Journal Volume: 41; Journal Issue: 3; Journal ID: ISSN 1019-7168
Publisher:
Springer
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; fluid-structure interaction; immersed boundary method; Krylov methods; multigrid solvers; multigrid preconditioners

Citation Formats

Guy, Robert D., Philip, Bobby, and Griffith, Boyce E. Geometric multigrid for an implicit-time immersed boundary method. United States: N. p., 2014. Web. doi:10.1007/s10444-014-9380-1.
Guy, Robert D., Philip, Bobby, & Griffith, Boyce E. Geometric multigrid for an implicit-time immersed boundary method. United States. doi:10.1007/s10444-014-9380-1.
Guy, Robert D., Philip, Bobby, and Griffith, Boyce E. Sun . "Geometric multigrid for an implicit-time immersed boundary method". United States. doi:10.1007/s10444-014-9380-1. https://www.osti.gov/servlets/purl/1163152.
@article{osti_1163152,
title = {Geometric multigrid for an implicit-time immersed boundary method},
author = {Guy, Robert D. and Philip, Bobby and Griffith, Boyce E.},
abstractNote = {The immersed boundary (IB) method is an approach to fluid-structure interaction that uses Lagrangian variables to describe the deformations and resulting forces of the structure and Eulerian variables to describe the motion and forces of the fluid. Explicit time stepping schemes for the IB method require solvers only for Eulerian equations, for which fast Cartesian grid solution methods are available. Such methods are relatively straightforward to develop and are widely used in practice but often require very small time steps to maintain stability. Implicit-time IB methods permit the stable use of large time steps, but efficient implementations of such methods require significantly more complex solvers that effectively treat both Lagrangian and Eulerian variables simultaneously. Moreover, several different approaches to solving the coupled Lagrangian-Eulerian equations have been proposed, but a complete understanding of this problem is still emerging. This paper presents a geometric multigrid method for an implicit-time discretization of the IB equations. This multigrid scheme uses a generalization of box relaxation that is shown to handle problems in which the physical stiffness of the structure is very large. Numerical examples are provided to illustrate the effectiveness and efficiency of the algorithms described herein. Finally, these tests show that using multigrid as a preconditioner for a Krylov method yields improvements in both robustness and efficiency as compared to using multigrid as a solver. They also demonstrate that with a time step 100–1000 times larger than that permitted by an explicit IB method, the multigrid-preconditioned implicit IB method is approximately 50–200 times more efficient than the explicit method.},
doi = {10.1007/s10444-014-9380-1},
journal = {Advances in Computational Mathematics},
issn = {1019-7168},
number = 3,
volume = 41,
place = {United States},
year = {2014},
month = {10}
}

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