Geometric multigrid for an implicit-time immersed boundary method

Guy, Robert D.; Philip, Bobby; Griffith, Boyce E.

doi:10.1007/s10444-014-9380-1

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

Journal Article · Sun Oct 12 00:00:00 EDT 2014 · Advances in Computational Mathematics

DOI:https://doi.org/10.1007/s10444-014-9380-1· OSTI ID:1163152

Guy, Robert D. ^[1]; Philip, Bobby ^[2]; Griffith, Boyce E. ^[3]

Univ. of California, Davis, CA (United States)
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Univ. of North Carolina, Chapel Hill, NC (United States)

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.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)

Sponsoring Organization:: USDOE

Grant/Contract Number:: AC05-00OR22725

OSTI ID:: 1163152

Journal Information:: Advances in Computational Mathematics, Vol. 41, Issue 3; ISSN 1019-7168

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 8 works

Citation information provided by
Web of Science

References (32)

An implicit numerical method for fluid dynamics problems with immersed elastic boundaries Mayo, Anita A.; Peskin, Charles S. https://doi.org/10.1090/conm/141/1212583	other	January 1993
A Multigrid Tutorial, Second Edition Briggs, William L.; Henson, Van Emden; McCormick, Steve F. Other Titles in Applied Mathematics https://doi.org/10.1137/1.9780898719505	book	January 2000
On the convergence of multi-grid methods with transforming smoothers Wittum, Gabriel Numerische Mathematik, Vol. 57, Issue 1 https://doi.org/10.1007/BF01386394	journal	December 1990
An efficient semi-implicit immersed boundary method for the Navier–Stokes equations Hou, Thomas Y.; Shi, Zuoqiang Journal of Computational Physics, Vol. 227, Issue 20 https://doi.org/10.1016/j.jcp.2008.07.005	journal	October 2008
Numerical analysis of blood flow in the heart Peskin, Charles S. Journal of Computational Physics, Vol. 25, Issue 3 https://doi.org/10.1016/0021-9991(77)90100-0	journal	November 1977
Immersed Boundary Method for Variable Viscosity and Variable Density Problems Using Fast Constant-Coefficient Linear Solvers I: Numerical Method and Results Fai, Thomas G.; Griffith, Boyce E.; Mori, Yoichiro SIAM Journal on Scientific Computing, Vol. 35, Issue 5 https://doi.org/10.1137/120903038	journal	January 2013
On the Volume Conservation of the Immersed Boundary Method Griffith, Boyce E. Communications in Computational Physics, Vol. 12, Issue 2 https://doi.org/10.4208/cicp.120111.300911s	journal	August 2012
Implicit second-order immersed boundary methods with boundary mass Mori, Yoichiro; Peskin, Charles S. Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 25-28 https://doi.org/10.1016/j.cma.2007.05.028	journal	April 2008
An adaptive, formally second order accurate version of the immersed boundary method Griffith, Boyce E.; Hornung, Richard D.; McQueen, David M. Journal of Computational Physics, Vol. 223, Issue 1 https://doi.org/10.1016/j.jcp.2006.08.019	journal	April 2007
A Multigrid Method for a Model of the Implicit Immersed Boundary Equations Guy, Robert D.; Philip, Bobby Communications in Computational Physics, Vol. 12, Issue 2 https://doi.org/10.4208/cicp.010211.070711s	journal	August 2012
Block-implicit multigrid solution of Navier-Stokes equations in primitive variables Vanka, S. P. Journal of Computational Physics, Vol. 65, Issue 1 https://doi.org/10.1016/0021-9991(86)90008-2	journal	July 1986
An immersed boundary method for fluid–flexible structure interaction Huang, Wei-Xi; Sung, Hyung Jin Computer Methods in Applied Mechanics and Engineering, Vol. 198, Issue 33-36 https://doi.org/10.1016/j.cma.2009.03.008	journal	July 2009
A fast, robust, and non-stiff Immersed Boundary Method Ceniceros, Hector D.; Fisher, Jordan E. Journal of Computational Physics, Vol. 230, Issue 12 https://doi.org/10.1016/j.jcp.2011.03.037	journal	June 2011
Stability and Instability in the Computation of Flows with Moving Immersed Boundaries: A Comparison of Three Methods Tu, Cheng; Peskin, Charles S. SIAM Journal on Scientific and Statistical Computing, Vol. 13, Issue 6 https://doi.org/10.1137/0913077	journal	November 1992
An implicit immersed boundary method for three-dimensional fluid–membrane interactions Le, D. V.; White, J.; Peraire, J. Journal of Computational Physics, Vol. 228, Issue 22 https://doi.org/10.1016/j.jcp.2009.08.018	journal	December 2009
A comparison of implicit solvers for the immersed boundary equations Newren, Elijah P.; Fogelson, Aaron L.; Guy, Robert D. Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 25-28 https://doi.org/10.1016/j.cma.2007.11.030	journal	April 2008
Analysis of Stiffness in the Immersed Boundary Method and Implications for Time-Stepping Schemes Stockie, John M.; Wetton, Brian R. Journal of Computational Physics, Vol. 154, Issue 1 https://doi.org/10.1006/jcph.1999.6297	journal	September 1999
On the hyper-elastic formulation of the immersed boundary method Boffi, Daniele; Gastaldi, Lucia; Heltai, Luca Computer Methods in Applied Mechanics and Engineering, Vol. 197, Issue 25-28 https://doi.org/10.1016/j.cma.2007.09.015	journal	April 2008
An Adaptive Level Set Approach for Incompressible Two-Phase Flows Sussman, Mark; Almgren, Ann S.; Bell, John B. Journal of Computational Physics, Vol. 148, Issue 1 https://doi.org/10.1006/jcph.1998.6106	journal	January 1999
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems Saad, Youcef; Schultz, Martin H. SIAM Journal on Scientific and Statistical Computing, Vol. 7, Issue 3 https://doi.org/10.1137/0907058	journal	July 1986
Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method Ceniceros, Hector D.; Fisher, Jordan E.; Roma, Alexandre M. Journal of Computational Physics, Vol. 228, Issue 19 https://doi.org/10.1016/j.jcp.2009.05.031	journal	October 2009
Multigrid and Krylov Subspace Methods for the Discrete Stokes Equations Elman, Howard C. International Journal for Numerical Methods in Fluids, Vol. 22, Issue 8 https://doi.org/10.1002/(SICI)1097-0363(19960430)22:8<755::AID-FLD377>3.0.CO;2-1	journal	April 1996
Analysis of a multigrid strokes solver Niestegge, A.; Witsch, K. Applied Mathematics and Computation, Vol. 35, Issue 3 https://doi.org/10.1016/0096-3003(90)90048-8	journal	February 1990
An Evaluation of Parallel Multigrid as a Solver and a Preconditioner for Singularly Perturbed Problems Oosterlee, C. W.; Washio, T. SIAM Journal on Scientific Computing, Vol. 19, Issue 1 https://doi.org/10.1137/s1064827596302825	journal	January 1998
Removing the stiffness of elastic force from the immersed boundary method for the 2D Stokes equations Hou, Thomas Y.; Shi, Zuoqiang Journal of Computational Physics, Vol. 227, Issue 21 https://doi.org/10.1016/j.jcp.2008.03.002	journal	November 2008
Immersed Boundary Methods Mittal, Rajat; Iaccarino, Gianluca Annual Review of Fluid Mechanics, Vol. 37, Issue 1 https://doi.org/10.1146/annurev.fluid.37.061903.175743	journal	January 2005
Multigrid relaxation methods for systems of saddle point type Oosterlee, C. W.; Gaspar, F. J. Applied Numerical Mathematics, Vol. 58, Issue 12 https://doi.org/10.1016/j.apnum.2007.11.014	journal	December 2008
Viscoelastic Immersed Boundary Methods for Zero Reynolds Number Flow Strychalski, Wanda; Guy, Robert D. Communications in Computational Physics, Vol. 12, Issue 2 https://doi.org/10.4208/cicp.050211.090811s	journal	August 2012
An Immersed Interface Method for Incompressible Navier--Stokes Equations Lee, Long; LeVeque, Randall J. SIAM Journal on Scientific Computing, Vol. 25, Issue 3 https://doi.org/10.1137/S1064827502414060	journal	January 2003
Unconditionally stable discretizations of the immersed boundary equations Newren, Elijah P.; Fogelson, Aaron L.; Guy, Robert D. Journal of Computational Physics, Vol. 222, Issue 2 https://doi.org/10.1016/j.jcp.2006.08.004	journal	March 2007
On the order of accuracy of the immersed boundary method: Higher order convergence rates for sufficiently smooth problems Griffith, Boyce E.; Peskin, Charles S. Journal of Computational Physics, Vol. 208, Issue 1 https://doi.org/10.1016/j.jcp.2005.02.011	journal	September 2005
The immersed boundary method Peskin, Charles S. Acta Numerica, Vol. 11 https://doi.org/10.1017/S0962492902000077	journal	January 2002

Similar Records

A multigrid method for a model of the implicit immersed boundary equations

Journal Article · Sun Jan 01 00:00:00 EST 2012 · Communications in Computational Physics · OSTI ID:1163152

Guy, Robert; Philip, Bobby

A projection preconditioner for solving the implicit immersed boundary equations

Journal Article · Sat Nov 01 00:00:00 EDT 2014 · Numerical Mathematics: Theory, Methods and Applications · OSTI ID:1163152

Zhang, Qinghai; Guy, Robert D.; Philip, Bobby

Scalable smoothing strategies for a geometric multigrid method for the immersed boundary equations

Journal Article · Tue Dec 20 00:00:00 EST 2016 · arXiv.org Repository · OSTI ID:1163152

Bhalla, Amneet Pal Singh; Knepley, Matthew G.; Adams, Mark F.; +2 more

Related Subjects

97 MATHEMATICS AND COMPUTING
fluid-structure interaction
immersed boundary method
Krylov methods
multigrid solvers
multigrid preconditioners

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

Citation Formats

References (32)

Similar Records

Related Subjects