The immersed boundary method for advection-electrodiffusion

University of Michigan

PETSc 20 workshop

Pilhwa Lee

Matus, et al. 2006

2

Free boundary problems in physiology

Embryonic cardiac myocytes contraction vs. microenvironment

Engler, et al. 2008

Pulmonary arterial network, blood flow profiling

Vanderpool, et al. 2011

Pannabecker and Dantzler, 2007

PETSc 20 workshop

Renal inner medulla, solute concentration

3

The immersed boundary method

Fluid-structure interactionCharlie Peskin and David McQeen

http://www.math.nyu.edu/faculty/peskin/myo3D/config12_animation.html

PETSc 20 workshop

Boyce GriffithAdaptive mesh refinement

3D distributed memory parallel computing

4

The immersed boundary method with advection-electrodiffusion

Fluid-solute-structure interactionElectrodiffusion

Osmotic effects

Electroneutrality - space charge layer

PETSc 20 workshop

hypertonic

Membrane swellingCa++,impermeable

Cl-,semi-permeable

Electrical charge density

5

Computational issues

Immensely stiff in hyperbolic systemThin membrane, local mesh refinementSemi-implicit time stepping

0 1 2 3 4−10

0

10

20

30

40

50

60

µm

ψC

a / K BT

PETSc 20 workshopLee, Griffith, Peskin, J. Comp. Phys. 2010

i(x, t) =

Z

⌦L

(x�X(s, t))Ai(s, t)ds

Numerical architecture

PETSc-CUDA

SAMRAI - multilevel adaptive mesh refinement

Hypre - algebraic multigrid

6PETSc 20 workshop

The Stokes equations with permeable membrane

Fast adaptive composite (FAC) method: preconditioner

one layer of ghost cells, bottom solver (PFMG)

Krylov subspace GMRES: main solver

Cell-centered approximate projection method

7

⇢u� un

�t+Dhp

n� 12 = µLh

u+ un

2+ SnF

n+1mf + fnb

un+1 = P u

PETSc 20 workshop

pn+1/2 = pn�1/2 + (⇢

�tL�1 � µ

2I)Dh · u

⇣(Xn+1 �Xn

�t� Un+1 +Un

2) = (Fn+1

mf ·Nn)Nn

Advection-electrodiffusion with immersed boundary

Fast adaptive composite (FAC) method, preconditionertwo layers of ghost cellsbottom solver (PFMG), first order upwindGMRES, main solver

8PETSc 20 workshop

cn+1i � cni

�t+Dh · Jn+1

i = 0

Lni c

n+1i = cniLni = L(un,'n, n

i )

Jn+1i = �Di(Dhc

n+1i ) + [

Di

KBT(�qziDh'

n �Dh ni ) + un]cn+1

i

�Lh'n = (

X

i

qzni cni + ⇢b)/✏

9PETSc 20 workshop

Advection-electrodiffusion with immersed boundary

10

Voltage-sensitive calcium ion channels

PETSc 20 workshop

80 100 120 140 160−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

transmembrnae voltage (mV)

i Ca (A

/m2 )

Goldman, Hodgkin, and KatzIB method

Rasmusson, et al. 2004

11

Voltage-sensitive calcium ion channels

PETSc 20 workshop

i(K

BT)

12

Concentration dependent contraction

PETSc 20 workshop

Fiber / membrane configuration Electrical charge density

[Ca++]

membrane is mostly impermeable to Ca++

[Cl-]

membrane is freely permeable to Cl-

13PETSc 20 workshop

An immersed boundary method for two-phase fluids and gels

@�

@t+r · (�vp) = 0

du

dt=

@u

@t+ vp ·ru = vp

r · {(1� �)vf + �vp} = 0 F p · T = ⌅TS⇤(vf � vp) · T

F p ·N = ⌅NS⇤(vf � vp) ·N

F f + F p = � �E

�X

⇢f@vf

@t+ �(vf � vp) = �(1� �)rp+ ⌘fr · (rvf +rvT

f ) + SF f

⇢p@vp

@t+ �(vp � vf) = ��rp� �0r� +⌘pr · (rvp +rvT

p ) + µr · (ru+ruT) + SF p.

14PETSc 20 workshop

An immersed boundary method for two-phase fluids and gels

1.5 2 2.5 3 3.5 4 4.51.5

2

2.5

3

3.5

x (mm)

y (m

m)

1.5 2 2.5 3 3.5 4 4.51.5

2

2.5

3

3.5

x (mm)

y (m

m)

(a) (b)

t (s)0 0.1 0.2 0.3 0.4

posi

tion

(µm

)

30.5

30.55

30.6

30.65

30.7

30.75Non-NewtonianNewtonian

Lee and Wolgemuth, submitted

15

3D solute concentration of inner medulla in renal peristaltic contraction

PETSc 20 workshop

Schmidt-Nielsen, et al. 2011

Multi-scale, large-scale computational framework

Arteries

Arterioles 1

Arterioles 2

Domain decomposition

Krylov subspace iteration PETSc-CUDAParallelized cellular transport

Plug cell models CellML

Vessel-Vessel coupling

Cell physiology

Cell-Cell interaction

16PETSc 20 workshop

17

3D solute concentration of inner medulla in renal peristaltic contraction

PETSc 20 workshop

!Luminal Interstitial

Epithelial

Hypothesis1. Rectified epithelial transport with electrolytes in interstitial matrix2. Rectified epithelial transport with viscoelasticity in interstitial matrix

18

3D peristaltic contraction

PETSc 20 workshop

FP =X

i

@(F iT⌧i)

@si

F iT = Ksi(|

@Xp

@si� 1), ⌧i =

@Xp/@si|@Xp/@si|

Ks1 = K0s1 sin(!t), Ks2 = K0

s2

⇣IuI = �rpI +

Z

⌦pL

�(x�Xp(s))Fpds

r · uI = 0

19PETSc 20 workshop

@cIi@t

+r · JIi =

Z

⌦L

�(x�X(s)�R

BE(s, ✓))p

EI,i(s)(c

Ii � cEi )dsd✓

J Ii = �DI

ircIi

� DIi

KBT(

Z

⌦L

r RBE(x�X(s))A(s)dsd✓)cIi

+uIcIi

JLi = cLi uL

@q

@t

+@

@x

(q

2

A

) +A

⇢

@pL

@x

= �2⇡⌫r

�

q

A

pL = pI(X+RBE) + pE

@cLi@t

+@JL

i

@s= �pEL,i(c

Li � cEi )

@cEi@t

= �pIE,i(cEi � cIi(X+RB

E))� pLE,i(cEi � cLi )

⇣A@RA

E

@t= ⇧ART

X

i

(cLi � cEi )

⇣B@RB

E

@t= ⇧BRT

X

i

(cEi � cIi(X+RBE))

3D solute concentration

Luminal Interstitial

Epithelial

20PETSc 20 workshop

Future work

Advection-electrodiffusion

3D IBAMR

Coupling with vasculature/tubular network

Two-phase fluids and gels

Acknowledgement

Courant Institute, New York University

Charlie Peskin, David McQueen

Marsha Berger, Olof Widlund

University of North Carolina

Boyce Griffith

Mount Sinai School of Medicine

Eric Sobie

University of Michigan

Daniel Beard, Brian Carlson

PETSc team, Barry Smith, Satish Balay

Hypre team, Robert Falgout

SAMRAI team

21

University of Pennsylvania

Dennis Discher

University of Arizona

Charles Wolgemuth

Medical College of Wisconsin

Allen Cowley

University of Wisconsin, Madison

Naomi Chesler

University College of London/ University of Auckland

Nicolas Smith, Taha Sochi

North Carolina State University

Mette Olufsen