the immersed boundary method for advection …nicolas smith, taha sochi north carolina state...
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![Page 1: The immersed boundary method for advection …Nicolas Smith, Taha Sochi North Carolina State University Mette Olufsen Title presentation.key Created Date 6/17/2015 5:08:07 PM](https://reader033.vdocuments.us/reader033/viewer/2022060923/60af0ef73cf3d07be8654cb5/html5/thumbnails/1.jpg)
The immersed boundary method for advection-electrodiffusion
University of Michigan
PETSc 20 workshop
Pilhwa Lee
Matus, et al. 2006
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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
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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
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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
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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
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Numerical architecture
PETSc-CUDA
SAMRAI - multilevel adaptive mesh refinement
Hypre - algebraic multigrid
6PETSc 20 workshop
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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
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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)/✏
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9PETSc 20 workshop
Advection-electrodiffusion with immersed boundary
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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
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11
Voltage-sensitive calcium ion channels
PETSc 20 workshop
i(K
BT)
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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-
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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.
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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
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15
3D solute concentration of inner medulla in renal peristaltic contraction
PETSc 20 workshop
Schmidt-Nielsen, et al. 2011
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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
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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
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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
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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
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20PETSc 20 workshop
Future work
Advection-electrodiffusion
3D IBAMR
Coupling with vasculature/tubular network
Two-phase fluids and gels
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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