Numerical Simulation of Red Blood Cells: a “Stokesian Dynamics” Approach S. Faure, S. Martin, B. Maury & T. Takahashi Groupe de Travail “M´ ethodes num´ eriques” Laboratoire Jacques Louis-Lions November 9, 2009 GT M´ ethodes num´ eriques (LJLL) Simulation of Red Blood Cells November 9, 2009 1 / 20
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Numerical Simulation of Red Blood Cells:a “Stokesian Dynamics” Approach
S. Faure, S. Martin, B. Maury & T. Takahashi
Groupe de Travail “Methodes numeriques”Laboratoire Jacques Louis-Lions
November 9, 2009
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 1 / 20
Aim of the project
Aim: Individual and collective behaviour of Red Blood Cells:aggregates, lateral migration...
Numerical tool: C++ program for granular flows (accurate handlingof contacts between rigid spheres)
Modeling: Fluid-RBC interactions, from dilute to densesuspensions...
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 2 / 20
Aim of the project
Aim: Individual and collective behaviour of Red Blood Cells:aggregates, lateral migration...
Numerical tool: C++ program for granular flows (accurate handlingof contacts between rigid spheres)
Modeling: Fluid-RBC interactions, from dilute to densesuspensions...
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 2 / 20
Aim of the project
Aim: Individual and collective behaviour of Red Blood Cells:aggregates, lateral migration...
Numerical tool: C++ program for granular flows (accurate handlingof contacts between rigid spheres)
Modeling: Fluid-RBC interactions, from dilute to densesuspensions...
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 2 / 20
References
Fictitious domain methods:Glowinski et al.: J. Comp. Phys. (2001)Wang, Pan & Glowinski: (2008)
Lattice Boltzmann methods:Ding & Aidun: Proc. A.P.S. (2003)Binder et al.: J. Colloid Interface Sci. (2006)Sun & Munn: Comp. Math. Appl. (2008)Zhang, Johnson & Popel: J. Biomech. (2008)
Other methods :Bagchi, Johnson & Popel: J. Biomech. Eng. (2005)Liu & Liu: J. Comp. Phys. (2006)Secomb, Styp-Rekowska & Pries: Annals Biomed. Eng. (2007)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 3 / 20
References
Fictitious domain methods:Glowinski et al.: J. Comp. Phys. (2001)Wang, Pan & Glowinski: (2008)
Lattice Boltzmann methods:Ding & Aidun: Proc. A.P.S. (2003)Binder et al.: J. Colloid Interface Sci. (2006)Sun & Munn: Comp. Math. Appl. (2008)Zhang, Johnson & Popel: J. Biomech. (2008)
Other methods :Bagchi, Johnson & Popel: J. Biomech. Eng. (2005)Liu & Liu: J. Comp. Phys. (2006)Secomb, Styp-Rekowska & Pries: Annals Biomed. Eng. (2007)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 3 / 20
References
Fictitious domain methods:Glowinski et al.: J. Comp. Phys. (2001)Wang, Pan & Glowinski: (2008)
Lattice Boltzmann methods:Ding & Aidun: Proc. A.P.S. (2003)Binder et al.: J. Colloid Interface Sci. (2006)Sun & Munn: Comp. Math. Appl. (2008)Zhang, Johnson & Popel: J. Biomech. (2008)
Other methods :Bagchi, Johnson & Popel: J. Biomech. Eng. (2005)Liu & Liu: J. Comp. Phys. (2006)Secomb, Styp-Rekowska & Pries: Annals Biomed. Eng. (2007)
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 3 / 20
Outline
1. A brief description of SCoPI
2. How to simulate one single Red Blood Cell ?
3. Fluid / Red Blood Cells interactions in dilute suspensions
4. Fluid / Red Blood Cells interactions in dense suspensions
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 4 / 20
A brief description of SCoPI
SCoPI (simulation of collections of particles in interaction):4 written by Aline Lefebvre4 accurate handling of contacts between rigid spheres
Langevin formulation:
M · dU
dt= Fext , in R3N
Algorithm:
4 Euler schemeM ·Un+1 = M ·Un + h Fn
ext
qn+1 = qn + h Un+1
4 handling of contacts by Uzawa technique
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 5 / 20
A brief description of SCoPI
SCoPI (simulation of collections of particles in interaction):4 written by Aline Lefebvre4 accurate handling of contacts between rigid spheres
Langevin formulation:
M · dU
dt= Fext , in R3N
Algorithm:
4 Euler schemeM ·Un+1 = M ·Un + h Fn
ext
qn+1 = qn + h Un+1
4 handling of contacts by Uzawa technique
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 5 / 20
A brief description of SCoPI
SCoPI (simulation of collections of particles in interaction):4 written by Aline Lefebvre4 accurate handling of contacts between rigid spheres
Langevin formulation:
M · dU
dt= Fext , in R3N
Algorithm:
4 Euler schemeM ·Un+1 = M ·Un + h Fn
ext
qn+1 = qn + h Un+1
4 handling of contacts by Uzawa technique
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 5 / 20
A brief description of SCoPI
Idea (Maury, 2006): projection onto the set of admissible velocities:
Assumption: pairwise additivity of the interactions
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 13 / 20
Fluid / Red Blood Cells interactions (dense suspensions)
For two spheres, all elements of the resistance / mobility matrix for allseparation are known exactly (Jeffrey & Onishi, 1984).
4 At large separations, far-field expressions (method of reflections,...)4 At small separations, analytical results from lubrication theory4 At intermediate separations, tabulated results
Resistance matrix...
4 Resistance / mobility matrix defined for two spheres in interaction.4 The grand resistance matrix is obtained by pairwise additivity...
R = (M∞)−1 +R2B −R∞2B
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 14 / 20
Fluid / Red Blood Cells interactions (dense suspensions)
For two spheres, all elements of the resistance / mobility matrix for allseparation are known exactly (Jeffrey & Onishi, 1984).
4 At large separations, far-field expressions (method of reflections,...)4 At small separations, analytical results from lubrication theory4 At intermediate separations, tabulated results
Resistance matrix...
4 Resistance / mobility matrix defined for two spheres in interaction.4 The grand resistance matrix is obtained by pairwise additivity...
R = (M∞)−1 +R2B −R∞2B
GT Methodes numeriques (LJLL) Simulation of Red Blood Cells November 9, 2009 14 / 20
Fluid / Red Blood Cells interactions (dense suspensions)