Stokesian Dynamics Approach - UPMC · PDF fileNumerical Simulation of Red Blood Cells: a \Stokesian Dynamics" Approach S. Faure, S. Martin, B. Maury & T. Takahashi Groupe de Travail

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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...


Aim of the project





Aim of the project





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)


References





References





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


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





M · dU

dt= Fext , in R3N

Algorithm:


ext

qn+1 = qn + h Un+1






M · dU

dt= Fext , in R3N

Algorithm:


ext

qn+1 = qn + h Un+1




Idea (Maury, 2006): projection onto the set of admissible velocities:

K (qn) ={

V ∈ R3N , Dij(qn) + h Gij(qn) · V ≥ 0, ∀i < j}

Constrained minimization problem: splitting scheme:

Un+1/2 = Un + h M−1 · Fnext

Un+1 = minV∈K(qn)

1

2

∣∣∣V −Un+1/2∣∣∣2



Idea (Maury, 2006): projection onto the set of admissible velocities:

K (qn) ={

V ∈ R3N , Dij(qn) + h Gij(qn) · V ≥ 0, ∀i < j}

Constrained minimization problem: splitting scheme:

Un+1/2 = Un + h M−1 · Fnext

Un+1 = minV∈K(qn)

1

2

∣∣∣V −Un+1/2∣∣∣2



Saddle-point problem: the minimization problem reads asFind (Un+1, µn+1) ∈W such that

L(Un+1, λ) ≤ L(Un+1, µn+1) ≤ L(V, µn+1), ∀(V, λ) ∈W,

with W = R3N × RN(N−1)/2+ and

L(V, λ) = F(V)−∑

1≤i<j≤N

λij (Dij(qn) + h Gij(qn) · V)

where F denotes the unconstrained functional...

Numerical computations by Uzawa algorithm.


How to simulate one single Red Blood Cell ?

Numerical modeling of a Red Blood Cell:

Macro-body made of 11 rigid spheres

Shape of the RBC: two- / three-body interactions between the spheres

“Fake” overlapping of the annular spheres

Film 1 − formation of 1 RBC (with VTK)

Film 2 − formation of 1 RBC (with post-processing: POVRAY)

Film 3 − formation of 30 RBC (with post-processing)
























Formation of 1 RBC (initialization) Formation of 1 RBC (at fixed time)



Formation of 1 RBC (initialization) Formation of 1 RBC (at fixed time)


Fluid / Red Blood Cells interactions (dilute suspensions)

Faxen laws (Faxen, 1922):

4 rigid sphere (radius a, velocity U),4 Stokes flow (viscosity µ, velocity field v∞).

F = −6πµa(

U− v∞|x=0

)+ πµa3 ∇2v∞|x=0

Couette flow (Film 5):

Tank-treading motion (t) and tumbling motion (b)


Fluid / Red Blood Cells interactions (dilute suspensions)

Faxen laws (Faxen, 1922):

4 rigid sphere (radius a, velocity U),4 Stokes flow (viscosity µ, velocity field v∞).

F = −6πµa(

U− v∞|x=0

)+ πµa3 ∇2v∞|x=0

Couette flow (Film 5):

Tank-treading motion (t) and tumbling motion (b)


Fluid / Red Blood Cells interactions (dense suspensions)


M · dU

dt= FH + FP , in R3N

4 FH : hydrodynamic forces4 FP : non hydrodynamic forces

SCoPI algorithm:4 Definition of the so-called “grand resistance” matrix:

FH = −R ·U + F4 Implicit procedure:

(M + h Rn) ·Un+1 = h (Fn + FnP),

qn+1 = qn + h Un+1




M · dU

dt= FH + FP , in R3N

4 FH : hydrodynamic forces4 FP : non hydrodynamic forces

SCoPI algorithm:4 Definition of the so-called “grand resistance” matrix:

FH = −R ·U + F4 Implicit procedure:

(M + h Rn) ·Un+1 = h (Fn + FnP),

qn+1 = qn + h Un+1



Stokesian Dynamics (Brady & Bossis, 1988):

4 suspension of rigid, non-Brownian particles4 bulk macroscopic shear flow4 the Reynolds particle number is small

Relationship between hydrodynamic forces and velocities (Happel &Brenner, 1965 ; Brenner & O’Neill, 1972):

4 resistance formulation: FH = −R · (U− v∞) + Φ : E(v∞)4 mobility formulation: M = R−1

Assumption: pairwise additivity of the interactions

















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



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



Far-field forces (Smoluchowski, 1911 ; Luke, 1989 ; Guazzelli, 2003)

−U2 =1

6πµaF2 +

1

8πµ

(I

r+

r · rr3

)·F1 +

a2

4πµ

(I

3r3− r · r

r5

)·F1 + ...

Lubrication forces (Jeffrey & Onishi, 1984 ; Dance & Maxey, 2003)

Fj→i · ni→j = −3πµa2

2

(Ui −Uj) · ni→j

r+O (ln r)

Intermediate-field forces (Kim & Karrila, 1991)




−U2 =1

6πµaF2 +

1

8πµ

(I

r+

r · rr3

)·F1 +

a2

4πµ

(I

3r3− r · r

r5

)·F1 + ...



2


r+O (ln r)





−U2 =1

6πµaF2 +

1

8πµ

(I

r+

r · rr3

)·F1 +

a2

4πµ

(I

3r3− r · r

r5

)·F1 + ...



2


r+O (ln r)




Film 6 − without lubrication forces Film 7 − with lubrication forces


Conclusion and perspectives

To be improved:4 Far-field forces:

method of reflectionscut-off distance (Satoh, 2001)

4 Lubrication:

method of boxes (complexity in N log N instead of N2)tangential contributionswall-particle lubrication

To be implemented:4 Intermediate-field forces4 Linear systems solvers4 Periodic boundary conditions

To be discussed:4 Lateral migration4 Aggregates (Film 8)





4 Lubrication:








4 Lubrication:






Red blood cell aggregates in a shear flow



Spheres in a Poiseuille flow



Polymer chains in a Poiseuille flow


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Stokesian Dynamics Approach - UPMC · PDF fileNumerical Simulation of Red Blood Cells: a \Stokesian Dynamics" Approach S. Faure, S. Martin, B. Maury & T. Takahashi Groupe de Travail