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

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

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

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

f(x,t) = F(s,t)δ (x −X(s,t))dxUb

∫

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

f(x,t) = F(s,t)δ (x −X(s,t))dxUb

∫

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

f(x,t) = F(s,t)δ (x −X(s,t))dxUb

∫

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

Forcing  Term

S+ff  body

FEn = Ks X

n −X0

n⎡⎣ ⎤⎦Spring force! "## $##

+ Kb κ (Xn )−κ pn⎡⎣ ⎤⎦

Beam force! "## $##

Elas+c  body

f(x,t) = F(s,t)δ (x −X(s,t))dxUb

∫

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

Forcing  Term

S+ff  body

Ks >O(Δs)−1

Kb >O(Δs)−5

FEn = Ks X

n −X0

n⎡⎣ ⎤⎦Spring force! "## $##

+ Kb κ (Xn )−κ pn⎡⎣ ⎤⎦

Beam force! "## $##

Elas+c  body

f(x,t) = F(s,t)δ (x −X(s,t))dxUb

∫

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

Forcing  Term

S+ff  body

Ks >O(Δs)−1

Kb >O(Δs)−5

κ n ≈κ pn

FEn = Ks X

n −X0

n⎡⎣ ⎤⎦Spring force! "## $##

+ Kb κ (Xn )−κ pn⎡⎣ ⎤⎦

Beam force! "## $##

Elas+c  body

f(x,t) = F(s,t)δ (x −X(s,t))dxUb

∫

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

Forcing  Term

S+ff  body

Ks >O(Δs)−1

Kb >O(Δs)−5

κ n ≈κ pn

small  +me  step  

FEn = Ks X

n −X0

n⎡⎣ ⎤⎦Spring force! "## $##

+ Kb κ (Xn )−κ pn⎡⎣ ⎤⎦

Beam force! "## $##

Elas+c  body

f(x,t) = F(s,t)δ (x −X(s,t))dxUb

∫

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

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  

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

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  

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

f(x,t) = FC (s,t)δ (x −X(s,t))dxUb

∫

Constraint  Algorithm

Constraint  Algorithm

!

1. Predict  the  posi+on  of  the  Lagrangian  structure  

!

!

Constraint  Algorithm

!

1. Predict  the  posi+on  of  the  Lagrangian  structure  

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

!

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  

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  

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

Constraint  Algorithm

X pn+1 = Xn +ΔtUd

n

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

Constraint  Algorithm

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

Constraint  Algorithm

U pn+1 = R[X p

n+1]u pn+1

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

Constraint  Algorithm

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

∫

Constraint  Algorithm

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

Constraint  Algorithm

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

χ(s,t)UkXk

Constraint  Algorithm

x

y

X p,comn+1

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

χ(s,t)UkXk

Constraint  Algorithm

x

y

X p,comn+1

Urn+1

Wrn+1

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

χ(s,t)UkXk

Constraint  Algorithm

x

y

X p,comn+1

Urn+1

Wrn+1

Udn+1 = Ur

n+1 +Wrn+1 ×Rn+1 +Uk

n+1

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

χ(s,t)UkXk

Constraint  Algorithm

Step  4a.  Compute  the  constraint  force  

ΔUC

n+1 = Udn+1 −U p

n+1

FCn+1 = ρ

ΔtΔU

C

n+1

Constraint  Algorithm

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

Constraint  Algorithm

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

Specify  kinema+c  constraintsNeuromechanical  model

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

Specify  kinema+c  constraintsNeuromechanical  model

muscle  sarcomere

Ca2+free

Ca2+bound

time

Muscle forceMuscle  Force

dPdt

= k(Pc − P)

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

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

motor  neuron  axon

Specify  kinema+c  constraintsNeuromechanical  model

motor  neuron  axon

muscle  sarcomere

time

Muscle forceMuscle  Force

dPdt

= k(Pc − P)

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

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

ca+on  efflux  ∴  repolariza+on

Specify  kinema+c  constraintsNeuromechanical  model

muscle  sarcomere

time

Muscle forceMuscle  Force

dPdt

= k(Pc − P)

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

Neural stimulus

Free calcium ionsCalcium−bound filamets

Muscle force

time

Preferred curvature

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

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