An adaptive, formally second order accurate version of the immersed boundary method

Boyce E. Griffith, Richard D. Hornung, David M. McQueen, Charles Peskin

Research output: Contribution to journalArticle

Abstract

Like many problems in biofluid mechanics, cardiac mechanics can be modeled as the dynamic interaction of a viscous incompressible fluid (the blood) and a (visco-)elastic structure (the muscular walls and the valves of the heart). The immersed boundary method is a mathematical formulation and numerical approach to such problems that was originally introduced to study blood flow through heart valves, and extensions of this work have yielded a three-dimensional model of the heart and great vessels. In the present work, we introduce a new adaptive version of the immersed boundary method. This adaptive scheme employs the same hierarchical structured grid approach (but a different numerical scheme) as the two-dimensional adaptive immersed boundary method of Roma et al. [A multilevel self adaptive version of the immersed boundary method, Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University, 1996; An adaptive version of the immersed boundary method, J. Comput. Phys. 153 (2) (1999) 509-534] and is based on a formally second order accurate (i.e., second order accurate for problems with sufficiently smooth solutions) version of the immersed boundary method that we have recently described [B.E. Griffith, C.S. Peskin, On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems, J. Comput. Phys. 208 (1) (2005) 75-105]. Actual second order convergence rates are obtained for both the uniform and adaptive methods by considering the interaction of a viscous incompressible flow and an anisotropic incompressible viscoelastic shell. We also present initial results from the application of this methodology to the three-dimensional simulation of blood flow in the heart and great vessels. The results obtained by the adaptive method show good qualitative agreement with simulation results obtained by earlier non-adaptive versions of the method, but the flow in the vicinity of the model heart valves indicates that the new methodology provides enhanced boundary layer resolution. Differences are also observed in the flow about the mitral valve leaflets.

Original languageEnglish (US)
Pages (from-to)10-49
Number of pages40
JournalJournal of Computational Physics
Volume223
Issue number1
DOIs
StatePublished - Apr 10 2007

Fingerprint

Blood
heart valves
Mechanics
blood flow
vessels
Incompressible flow
methodology
incompressible flow
theses
Boundary layers
incompressible fluids
three dimensional models
blood
boundary layers
Fluids
simulation
interactions
formulations

Keywords

  • Adaptive mesh refinement
  • Blood flow
  • Cardiac mechanics
  • Convergence
  • Fluid-structure interaction
  • Hemodynamics
  • Immersed boundary method
  • Projection method

ASJC Scopus subject areas

  • Computer Science Applications
  • Physics and Astronomy(all)

Cite this

An adaptive, formally second order accurate version of the immersed boundary method. / Griffith, Boyce E.; Hornung, Richard D.; McQueen, David M.; Peskin, Charles.

In: Journal of Computational Physics, Vol. 223, No. 1, 10.04.2007, p. 10-49.

Research output: Contribution to journalArticle

Griffith, Boyce E. ; Hornung, Richard D. ; McQueen, David M. ; Peskin, Charles. / An adaptive, formally second order accurate version of the immersed boundary method. In: Journal of Computational Physics. 2007 ; Vol. 223, No. 1. pp. 10-49.
@article{15295c8bf61e466da620f30c4877f217,
title = "An adaptive, formally second order accurate version of the immersed boundary method",
abstract = "Like many problems in biofluid mechanics, cardiac mechanics can be modeled as the dynamic interaction of a viscous incompressible fluid (the blood) and a (visco-)elastic structure (the muscular walls and the valves of the heart). The immersed boundary method is a mathematical formulation and numerical approach to such problems that was originally introduced to study blood flow through heart valves, and extensions of this work have yielded a three-dimensional model of the heart and great vessels. In the present work, we introduce a new adaptive version of the immersed boundary method. This adaptive scheme employs the same hierarchical structured grid approach (but a different numerical scheme) as the two-dimensional adaptive immersed boundary method of Roma et al. [A multilevel self adaptive version of the immersed boundary method, Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University, 1996; An adaptive version of the immersed boundary method, J. Comput. Phys. 153 (2) (1999) 509-534] and is based on a formally second order accurate (i.e., second order accurate for problems with sufficiently smooth solutions) version of the immersed boundary method that we have recently described [B.E. Griffith, C.S. Peskin, On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems, J. Comput. Phys. 208 (1) (2005) 75-105]. Actual second order convergence rates are obtained for both the uniform and adaptive methods by considering the interaction of a viscous incompressible flow and an anisotropic incompressible viscoelastic shell. We also present initial results from the application of this methodology to the three-dimensional simulation of blood flow in the heart and great vessels. The results obtained by the adaptive method show good qualitative agreement with simulation results obtained by earlier non-adaptive versions of the method, but the flow in the vicinity of the model heart valves indicates that the new methodology provides enhanced boundary layer resolution. Differences are also observed in the flow about the mitral valve leaflets.",
keywords = "Adaptive mesh refinement, Blood flow, Cardiac mechanics, Convergence, Fluid-structure interaction, Hemodynamics, Immersed boundary method, Projection method",
author = "Griffith, {Boyce E.} and Hornung, {Richard D.} and McQueen, {David M.} and Charles Peskin",
year = "2007",
month = "4",
day = "10",
doi = "10.1016/j.jcp.2006.08.019",
language = "English (US)",
volume = "223",
pages = "10--49",
journal = "Journal of Computational Physics",
issn = "0021-9991",
publisher = "Academic Press Inc.",
number = "1",

}

TY - JOUR

T1 - An adaptive, formally second order accurate version of the immersed boundary method

AU - Griffith, Boyce E.

AU - Hornung, Richard D.

AU - McQueen, David M.

AU - Peskin, Charles

PY - 2007/4/10

Y1 - 2007/4/10

N2 - Like many problems in biofluid mechanics, cardiac mechanics can be modeled as the dynamic interaction of a viscous incompressible fluid (the blood) and a (visco-)elastic structure (the muscular walls and the valves of the heart). The immersed boundary method is a mathematical formulation and numerical approach to such problems that was originally introduced to study blood flow through heart valves, and extensions of this work have yielded a three-dimensional model of the heart and great vessels. In the present work, we introduce a new adaptive version of the immersed boundary method. This adaptive scheme employs the same hierarchical structured grid approach (but a different numerical scheme) as the two-dimensional adaptive immersed boundary method of Roma et al. [A multilevel self adaptive version of the immersed boundary method, Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University, 1996; An adaptive version of the immersed boundary method, J. Comput. Phys. 153 (2) (1999) 509-534] and is based on a formally second order accurate (i.e., second order accurate for problems with sufficiently smooth solutions) version of the immersed boundary method that we have recently described [B.E. Griffith, C.S. Peskin, On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems, J. Comput. Phys. 208 (1) (2005) 75-105]. Actual second order convergence rates are obtained for both the uniform and adaptive methods by considering the interaction of a viscous incompressible flow and an anisotropic incompressible viscoelastic shell. We also present initial results from the application of this methodology to the three-dimensional simulation of blood flow in the heart and great vessels. The results obtained by the adaptive method show good qualitative agreement with simulation results obtained by earlier non-adaptive versions of the method, but the flow in the vicinity of the model heart valves indicates that the new methodology provides enhanced boundary layer resolution. Differences are also observed in the flow about the mitral valve leaflets.

AB - Like many problems in biofluid mechanics, cardiac mechanics can be modeled as the dynamic interaction of a viscous incompressible fluid (the blood) and a (visco-)elastic structure (the muscular walls and the valves of the heart). The immersed boundary method is a mathematical formulation and numerical approach to such problems that was originally introduced to study blood flow through heart valves, and extensions of this work have yielded a three-dimensional model of the heart and great vessels. In the present work, we introduce a new adaptive version of the immersed boundary method. This adaptive scheme employs the same hierarchical structured grid approach (but a different numerical scheme) as the two-dimensional adaptive immersed boundary method of Roma et al. [A multilevel self adaptive version of the immersed boundary method, Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University, 1996; An adaptive version of the immersed boundary method, J. Comput. Phys. 153 (2) (1999) 509-534] and is based on a formally second order accurate (i.e., second order accurate for problems with sufficiently smooth solutions) version of the immersed boundary method that we have recently described [B.E. Griffith, C.S. Peskin, On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems, J. Comput. Phys. 208 (1) (2005) 75-105]. Actual second order convergence rates are obtained for both the uniform and adaptive methods by considering the interaction of a viscous incompressible flow and an anisotropic incompressible viscoelastic shell. We also present initial results from the application of this methodology to the three-dimensional simulation of blood flow in the heart and great vessels. The results obtained by the adaptive method show good qualitative agreement with simulation results obtained by earlier non-adaptive versions of the method, but the flow in the vicinity of the model heart valves indicates that the new methodology provides enhanced boundary layer resolution. Differences are also observed in the flow about the mitral valve leaflets.

KW - Adaptive mesh refinement

KW - Blood flow

KW - Cardiac mechanics

KW - Convergence

KW - Fluid-structure interaction

KW - Hemodynamics

KW - Immersed boundary method

KW - Projection method

UR - http://www.scopus.com/inward/record.url?scp=33947217364&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33947217364&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2006.08.019

DO - 10.1016/j.jcp.2006.08.019

M3 - Article

AN - SCOPUS:33947217364

VL - 223

SP - 10

EP - 49

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 1

ER -