fastcomputa+on%of%fully% resolved%neuromechanically% … · 2015. 6. 23. · kinema+c deformaon...

Fast Computa+on of Fully Resolved Neuromechanically

Simulated Locomo+onNamu Patel & Neelesh A. Patankar

Neuromechanically driven locomo+on

Kinema+c deforma+on

wave

Muscle contrac+on

wave

Neural ac+va+on wave

Neuromechanically driven locomo+on

Video from: hEps://www.youtube.com/watch?v=LDrvbr_CbhE&spfreload=10Malcolm MacIver at Northwestern University Melina Hale at

University of Chicago

https://www.youtube.com/watch?v=LDrvbr_CbhE&spfreload=10

Equa+ons for Fluid-‐Structure Interac+on

ρ DuDt(x,t) = −∇p(x,t)+ µ∇2u(x,t)+ f(x,t)

∇⋅u(x,t) = 0

∂X∂t(s,t) = u(x,t)δ (x −X(s,t))dx

Ub

∫

f(x,t) = F(s,t)δ (x −X(s,t))dxUb

∫

momentum

con+nuity

Lagrangian point velocity

force spreading

Immersed Boundary Equa+ons for Fluid-‐Structure Interac+on

Immersed Boundary Equa+ons for Fluid-‐Structure Interac+on


∇⋅u(x,t) = 0

∂X∂t(s,t) = u(x,t)δ (x −X(s,t))dx

Ub

∫


∫

momentum

con+nuity

Lagrangian point velocity

force spreading


∫

F(s,t) = FE (s,t)+FC (s,t)

Forcing Term

FEn = Ks X

n −X0

n⎡⎣ ⎤⎦Spring force! "## $##

+ Kb κ (Xn )−κ pn⎡⎣ ⎤⎦

Beam force! "## $##

Elas+c body


∫


Forcing Term

FEn = Ks X

n −X0


+ Kb κ (Xn )−κ pn⎡⎣ ⎤⎦

Beam force! "## $##

Elas+c body


∫


Forcing Term

S+ff body

FEn = Ks X

n −X0


+ Kb κ (Xn )−κ pn⎡⎣ ⎤⎦

Beam force! "## $##

Elas+c body


∫


Forcing Term

S+ff body

Ks >O(Δs)−1

Kb >O(Δs)−5

FEn = Ks X

n −X0


+ Kb κ (Xn )−κ pn⎡⎣ ⎤⎦

Beam force! "## $##

Elas+c body


∫


Forcing Term

S+ff body

Ks >O(Δs)−1

Kb >O(Δs)−5

κ n ≈κ pn

FEn = Ks X

n −X0


+ Kb κ (Xn )−κ pn⎡⎣ ⎤⎦

Beam force! "## $##

Elas+c body


∫


Forcing Term

S+ff body

Ks >O(Δs)−1

Kb >O(Δs)−5

κ n ≈κ pn

small +me step

Forcing Term

Solid, deforming body

f(x,t) = λ(x,t) Lagrange multiplierUd = Ur + (Wr ×R)+Uk ∈Ωr

A.P.S. Bhalla, B.E. Griffith, N.A. Patankar (2013) PLoS Comp Biol A.P.S. Bhalla, R. Bale, B.E. Griffith, N.A. Patankar (2013) JCP


f(x,t) = FC (s,t)δ (x −X(s,t))dxUb

∫

Forcing Term

Directly impose kinema+c deforma+ons, NOT the deforma+on velocity

Solid, deforming body

f(x,t) = λ(x,t) Lagrange multiplierUd = Ur + (Wr ×R)+Uk ∈Ωr

A.P.S. Bhalla, B.E. Griffith, N.A. Patankar (2013) PLoS Comp Biol A.P.S. Bhalla, R. Bale, B.E. Griffith, N.A. Patankar (2013) JCP


f(x,t) = FC (s,t)δ (x −X(s,t))dxUb

∫

Constraint Algorithm


!

1. Predict the posi+on of the Lagrangian structure

!

!


!


2. Solve the fluid equa+ons without kinema+c constraints

!


!



3. Impose the deforma+on kinema+cs


!



3. Impose the deforma+on kinema+cs

4. Correct the fluid velocity and pressure and posi+on of Lagrangian structure


X pn+1 = Xn +ΔtUd

n

Step 1. Predict the posi+on of the constrained body


Step 2. Solve the fluid equa+ons without kinema+c constraints

ρu pn+1 − un

Δt+ [u ⋅∇u]p

n+1/2⎛

⎝⎜⎞

⎠⎟= −∇pp

n+1/2 + µ∇2u pn+1

∇⋅u pn+1 = 0


U pn+1 = R[X p

n+1]u pn+1

Step 3. Interpolate the fluid velocity using discrete Delta func+ons


U pn+1 = R[X p

n+1]u pn+1

Step 3a. Interpolate the fluid velocity using discrete Delta func+ons

R[X pn+1]u p

n+1 ≈ u pn+1δ (x −X p

n+1)dxUb

∫


MCUrn+1 = ρU p

n+1

Ωr

∑ ΔV

ICn+1Wr

n+1 = ρRn+1 ×U pn+1

Ωr

∑ ΔV

Rn+1 = X pn+1 −X p,com

n+1

Step 3b. Compute the rigid transla+onal and rota+onal veloci+es


Step 3c. Given the prescribed kinema+c shape , calculate the deforma+on shape , and velocity

χ(s,t)UkXk


x

y

X p,comn+1


χ(s,t)UkXk


x

y

X p,comn+1

Urn+1

Wrn+1


χ(s,t)UkXk


x

y

X p,comn+1

Urn+1

Wrn+1

Udn+1 = Ur

n+1 +Wrn+1 ×Rn+1 +Uk

n+1


χ(s,t)UkXk


Step 4a. Compute the constraint force

ΔUC

n+1 = Udn+1 −U p

n+1

FCn+1 = ρ

ΔtΔU

C

n+1


Step 4b. Spread the corrected Lagrangian velocity

ρun+1 − u p

n+1

Δt= −∇(pn+1/2 − pp

n+1/2 )+ S[X pn+1]FC

n+1

∇⋅un+1 = 0


Step 4c. Correct the posi+on of the Lagrangian structure

Xcomn+1 = Xcom

n + Δt2Ur

n +Urn+1( )

Xn+1 = Xcomn+1 +Xk

n+1

Neuromuscular Kinema+cs for Swimming

Specify kinema+c constraintsNeuromechanical model

motor neuron axon

muscle sarcomere Ca2+

ca+on influx ∴ depolariza+on

time

Neural stimulus

External forcing

on

off


muscle sarcomere

time

Free calcium ionsCalcium−bound filamets

Kine+c Equa+ons

d Ca free2+

dt= f (Ca free

2+ ,Cabound2+ , stim)

d Cabound2+

dt= g(Ca free

2+ ,Cabound2+ )

Ca2+free

Ca2+bound

T. McMillen, T. Williams, P. Holmes (2007) PLoS Comp Biol

motor neuron axon


muscle sarcomere

Ca2+free

Ca2+bound

time

Muscle forceMuscle Force

dPdt

= k(Pc − P)

Pc = Pc(l fiber ,vfiber ,Cabound2+ )


motor neuron axon


motor neuron axon

muscle sarcomere

time


dPdt

= k(Pc − P)



ca+on efflux ∴ repolariza+on


muscle sarcomere

time


dPdt

= k(Pc − P)


Neural stimulus

Free calcium ionsCalcium−bound filamets

Muscle force

time

Preferred curvature


motor neuron axon

ca+on efflux ∴ depolariza+on

!

å!PR##

PL#

rR#

rL#M##

M##

Specify kinema+c constraints

κ pref = h(PR ,PL ,EI )

Preferred curvature used to calculate preferred shape

time

Preferred curvature

Neuromechanical model

Reduced order fluid simula+ons for choosing EI

= 100maxtss

∂θ∂s

−κ p2

∂θ∂s 2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

% Deviation from preferred curvature

Bending modulus E [Mba]1 2 3 4 5 6 7 8 9 10

Nor

mal

ized

sw

imm

ing

velo

city

U/U

w

-0.1

0

0.1

0.2

0.3

0.4

% D

evia

tion

from

pre

ferre

d cu

rvat

ure

0

10

20

30

40

50

60

70

80

90

100

! ! "###### $######floppy

swimmer sufficiently

stiff swimmer

3D Neuromechanically Driven Locomo+on

Fully resolved simula+on of neuromechanically driven locomo+on

Muscle Co-‐contrac+ons

Swimming with & without co-‐contrac+ons

no co-‐contrac+ons

co-‐contrac+ons

Swimming with & without co-‐contrac+ons

time0 5 10 15 20 25

norm

aliz

ed s

wim

min

g ve

loci

ties

0

0.1

0.2

0.3

0.4

0.5

0.6U/UwV/UwUc/UwVc/Uw

Turning with and without co-‐contrac+ons

Fish school

Acknowledgements

Wenjun Kou

Amneet P.S. Bhalla

Boyce Griffith

IBAMRhttps://github.com/

IBAMR/IBAMRNUIT’s Quest

https://github.com/IBAMR/IBAMR

fastcomputa+on%of%fully% resolved%neuromechanically% … · 2015. 6. 23. · kinema+c deformaon...

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