Speaker: Julia E. Samson Date: 18-12-2015 Abstract: The immersed boundary method is a numerical approach to solving fluid-structure interactions. By decoupling the mesh for the object or boundary (a beating heart, a pulsing jellyfish, a leaf flapping in the wind…) and the fluid grid, the immersed boundary enables one to model both the effects of the fluid on the boundary and the effects of the boundary on the fluid. After explaining the concepts and mechanisms behind this method, examples of current research projects will be used to illustrate the variety of problems that can be addressed using the immersed boundary method.

1. The Immersed Boundary Method!
simulating fluid-structure interactions,!
from 2D fibers to 3D finite elements !
Julia E. Samson, Nick A. Battista, Laura A. Miller"
University of North Carolina at Chapel Hill"
December 18th, 2015"

2. Overview"
1. The immersed boundary method: when,
who, what, and why?"
2. The immersed boundary method: how?
(2D)"
3. Beyond the basics: 3D, IBAMR, and IBFE"
"
Alex Hoover
Tulane University

3. The IB method: a brief history"
Charles S. Peskin
Courant Institute, NY
Flow patterns around
heart valves: a
digital computer
method for solving
the equations of
motion. PhD thesis,
1972.

4. The IB method: a brief history"
Laura A. Miller
UNC Chapel Hill
Boyce E. Griffith
UNC Chapel Hill
Charles S. Peskin
Courant Institute

5. The IB method: definition"
Viscous
fluid?
Fluid grid
generated from
boundary shape?
IB!!!
J
Not IB
L
Not IB
L

6. The IB method: definition"
"
"
A numerical method that allows us to simulate
boundaries (objects) in viscous flows, and in
which the fluid grid is not fitted to the boundary
shape."
"

7. The IB method: definition"
The fluid is modeled on a fixed Cartesian mesh."
"
"
"
"
"
The boundary is modeled on a curvilinear
Lagrangian mesh that moves freely through the
fixed Cartesian mesh."

8. The IB method: applications"
Alex Hoover
Tulane University
Nick Battista
UNC Chapel Hill
Laura Miller
UNC Chapel Hill

9. IB: the math below the surface"
1) Fluid"
2) Structure/boundary"
3) Interactions"
We need to know:"
- how the fluid moves"
- how the boundary moves"
- how the boundary impacts the fluid"
- how the fluid impacts the boundary"

10. The Navier-Stokes equations"
This is the equation of motion for viscous
fluids."

11. The Navier-Stokes equations"
It basically follows Newton’s Second Law:"
F = m * a"
mass * acceleration
pressure
forces
viscous
forces
other body
forces

12. The Navier-Stokes equations"
Now, we add the equation for
incompressible flow."
mass * acceleration
pressure
forces
viscous
forces
other body
forces
the fluid is incompressible

13. Fluid mesh"
The fluid is represented
by a fixed (Eulerian)
Cartesian grid."
"
At each point, we solve
for the pressure and
velocity of the fluid
using the Navier-
Stokes equations. The
body forces will be
given by the boundary."

14. IB: the math below the surface"
1) Fluid"
2) Structure/boundary"
3) Interactions"
We need to know:"
- how the fluid moves"
- how the boundary moves"
- how the boundary impacts the fluid"
- how the fluid impacts the boundary"

15. IB: the math below the surface"
1) Fluid"
2) Structure/boundary"
3) Interactions"
We need to know:"
- how the fluid moves ✔"
- how the boundary moves"
- how the boundary impacts the fluid"
- how the fluid impacts the boundary"

16. Boundary"
The boundary is represented by a curvilinear Lagrangian
mesh that can move around in the fluid."
"
At each time step, we solve for the position of each
boundary point and for the forces at that point."

17. IB: the math below the surface"
1) Fluid"
2) Structure/boundary"
3) Interactions"
We need to know:"
- how the fluid moves ✔"
- how the boundary moves"
- how the boundary impacts the fluid"
- how the fluid impacts the boundary"

18. IB: the math below the surface"
1) Fluid"
2) Structure/boundary"
3) Interactions"
We need to know:"
- how the fluid moves ✔"
- how the boundary moves ✔"
- how the boundary impacts the fluid"
- how the fluid impacts the boundary"

19. Combining fluid and structure"
+ =
+ interactions!"

20. Combining fluid and structure"
Fluid
(fixed Cartesian mesh)
Structure
(moving curvilinear mesh)
moves at local
fluid velocity
exerts
forces on

21. exerts
forces on
Combining fluid and structure"
Fluid
(fixed Cartesian mesh)
Structure
(moving curvilinear mesh)
Spread the elastic force density from curvilinear
mesh onto Cartesian grid.

22. Combining fluid and structure"
Delta function weights are used to determine how much force is applied
from the elastic boundary to nearby fluid grid cells."

23. IB: the math below the surface"
1) Fluid"
2) Structure/boundary"
3) Interactions"
We need to know:"
- how the fluid moves ✔"
- how the boundary moves ✔"
- how the boundary impacts the fluid"
- how the fluid impacts the boundary"

24. IB: the math below the surface"
1) Fluid"
2) Structure/boundary"
3) Interactions"
We need to know:"
- how the fluid moves ✔"
- how the boundary moves ✔"
- how the boundary impacts the fluid ✔"
- how the fluid impacts the boundary"

25. Combining fluid and structure"
Fluid
(fixed Cartesian mesh)
Structure
(moving curvilinear mesh)
moves at local
fluid velocity
Interpolate the velocity field from the
Cartesian grid onto the curvilinear mesh.

26. Combining fluid and structure"
Delta function is used again to determine the velocity at the boundary
point q from fluid velocities near that point."

27. IB: the math below the surface"
1) Fluid"
2) Structure/boundary"
3) Interactions"
We need to know:"
- how the fluid moves ✔"
- how the boundary moves ✔"
- how the boundary impacts the fluid ✔"
- how the fluid impacts the boundary"

28. IB: the math below the surface"
1) Fluid"
2) Structure/boundary"
3) Interactions"
We need to know:"
- how the fluid moves ✔"
- how the boundary moves ✔"
- how the boundary impacts the fluid ✔"
- how the fluid impacts the boundary ✔"

29. IB: the math below the surface"
1) Fluid"
2) Structure/boundary"
3) Interactions"
We need to know:"
- how the fluid moves ✔"
- how the boundary moves ✔"
- how the boundary impacts the fluid ✔"
- how the fluid impacts the boundary ✔"

30. IB: the math below the surface"
We now have a complete formulation for the
immersed boundary method."
" mass * acceleration
pressure
forces
viscous
forces
other body
forces
the fluid is incompressible
Spread the elastic force density from
curvilinear mesh onto Cartesian grid.
Interpolate the velocity
field from the Cartesian
grid onto the
curvilinear mesh.

31. IB: the math below the surface"
We now have a complete formulation for the
immersed boundary method."
"

32. IB time stepping"
At each time step:"
1) Compute the elastic force
density F on the boundary
mesh."
2) Spread the elastic force from the
deformed boundary to the
underlying fluid (this is f)."
3) Solve the equations of fluid
motion defined on the fluid grid
using the elastic body force
density f(x,t) and update the
velocity field."
4) Move the boundary at the local
fluid velocity. Determine the
velocity at each Lagrangian
point through interpolation."

33. Making boundaries flexible (or not)"
There are a lot of fiber models to control
boundary characteristics like elasticity,
stretchiness, porosity, mass…"
"
3 examples in 2D:"
- Springs"
- Torsional springs"
- Target points"
Nick Battista
UNC Chapel Hill

34. github.com/nickabattista/IB2d
Nick Battista
UNC Chapel Hill

35. Springs"
Springs allow longitudinal motion between
two coupled Lagrangian nodes."
ad
RL
RL+d
elastic potential energy
force from
deformation

36. Springs: the rubber band example"
All Lagrangian
points are
connected by
springs with
resting length 0."
"
Colormap shows
vorticity."
ad

37. Torsional springs"
Torsional springs allow transversal motion
between three coupled Lagrangian nodes."
ad
θ If θdesired = 180
and C = 0

38. Torsional springs"
Torsional springs allow transversal motion
between three coupled Lagrangian nodes."
ad
θ If θdesired = 180
and C = 0
elastic potential energy
curvature

39. Torsional springs"
Torsional springs allow transversal motion
between three coupled Lagrangian nodes."
ad
θ If θdesired = 180
and C = 0
deformation forces

40. Torsional springs: the wobbly beam
example"
All Lagrangian
points are
connected by
beams with
curvature 0."
"
Colormap shows
magnitude of
velocity."
ad

41. Target points"
Target points are used to prescribe motion of
Lagrangian points or make boundary rigid."
ad

42. Target points: the pulsing heart
example"
Target point
positions are
updated by
interpolating
between two
positions."
"
Only target points,
no beams or
springs."
"
Colormap shows
pressure."
ad

43. Pushing the boundary…"
2D IB is where it all started, but newer (and
more complex) methods are available:"
- 3D IB"
- IBAMR (IB with Adaptive Mesh
Refinement)"
- IBFE (IB with Finite Elements)"

44. 3D immersed boundary"
Basically the same as 2D but adding a third
dimension."
"
Greatly increases computational cost but
this might be offset by the generation of
more realistic models. "

45. Collective pulsing in xeniid corals"
Xeniid corals are soft corals that form
pulsing colonies. The pulsing increases local
flow and thus mass transfer."

46. Collective pulsing in xeniid corals"
"
"
This pulsing behavior
seems to be coordinated
and we want to know how
local flow and pulsing
behavior are connected."
Collective
pulsing behavior
Water flow

47. Collective pulsing in xeniid corals"

48. IB with Adaptive Mesh Refinement"
Boyce E. Griffith
UNC Chapel Hill
Simulating the blood-
muscle-valve mechanics of
the heart by an adaptive
and parallel version of
the immersed boundary
method. PhD thesis, 2005.

49. Heart valves and blood flow"
Generate 3D simulations
of the interactions
between blood flow and
heart valves to better
understand heart
physiology and to assess
the functioning of
prosthetic valves."
from http://anatomyandphysiologyi.com/heart-anatomy-
chambers-vessels-valves/

50. IB with Adaptive Mesh Refinement"
A more refined grid will give a better
resolution to the simulation. But it also
greatly increases the computational cost…"
ad
25 x 25 50 x 50 100 x 100 200 x 200

51. IB with Adaptive Mesh Refinement"
"
So how to have your cake and eat it too???"
ad
25 x 25 50 x 50 100 x 100 200 x 200

52. IB with Adaptive Mesh Refinement"
Only refine the fluid grid where needed:
close to the boundary and in regions of high
vorticity è Adaptive Mesh Refinement"
ad
25 x 25 50 x 50 100 x 100 200 x 200

53. ad
Heart development in zebrafish"
4 days post
fertilization"
"
Blood cells and
endocardium are
colored"
"
Two chambers: one
atrium and one
ventricle"
Courtesy of Leigh Ann Samsa and Dr. Jiandong Liu
School of Medicine, UNC Chapel Hill

54. Heart development in zebrafish"
Ventricle
Atrium
75 um
Courtesy of Leigh Ann Samsa and Dr. Jiandong Liu
School of Medicine, UNC Chapel Hill

55. ad
Heart development in zebrafish"
Ventricle
Atrium
AV
Canal
Ventricle
IBAMR model

56. ad
Heart development in zebrafish"
Trabeculae appear to shield the endocardium
from higher shearing forces
velocity field + vorticity map streamlines (after atrium finishes
contraction)

57. IB with Finite Elements"
A completely different beast…"
ad
Un
Un-1
Un-2
Un+1
Un+2
Un+3
Un-2
Un-1
Un
Un+1
Un+2
Un+3
Un+4
Un+5 Un+6
A collection of
single nodal points
(= fiber)
A collection of
polygonal pieces
(= elements)

58. IB with Finite Elements"
Generating finite element meshes is hard
(although there are software packages
available)."
"
But the benefits are enormous:"
- Simulations run way faster"
- The FE mesh allows for a more accurate
structure geometry"
- Material properties are captured way better"
- Boundaries are less leaky"
- The models are more stable"

59. Jellyfish locomotion"
Alexander Hoover
Tulane University
From pacemaker to
vortex ring: modeling
jellyfish propulsion
and turning. PhD
thesis, 2015

60. Jellyfish locomotion"

61. Jellyfish locomotion"

62. Jellyfish locomotion"

63. Jellyfish locomotion"

64. Resources"
Code"
2D code examples in MatLab (Nick Battista): github.com/nickabattista/IB2d"
IBAMR code: https://github.com/ibamr/ibamr"
"
Papers"
Griffith, B. E., 2005. Simulating the blood-muscle-valve mechanics of the heart by an adaptive
and parallel version of the immersed boundary method. Ph.D. thesis, New York University."
Mittal, R., Iaccarino, G., 2005. Immersed boundary methods, Annual Review of Fluid Mechanics,
37, 239-261"
Peskin, C. S., McQueen, D. M., 1996. Fluid dynamics of the heart and its valves, In Case Studies
in Mathematical Modeling: Ecology, Physiology, and Cell Biology, Pearson, 313-342"
Peskin, C. S., 2002. The immersed boundary method, Acta Numerica, 11, 1-39"
"
Webpages"
Boyce Griffith: http://griffith.web.unc.edu/ and http://cims.nyu.edu/~griffith/"
Laura Miller: http://miller.web.unc.edu/"
Nick Battista: http://battista.web.unc.edu/"
Alex Hoover: http://hooverap.web.unc.edu/ or email ahoover2@tulane.edu"
"

65. Acknowledgements"
At UNC"
Laura Miller"
Nick Battista"
Shannon Jones"
Boyce Griffith"
"
"
"
Elsewhere"
Alex Hoover"
Shilpa Khatri"
Uri Shavit"
Roi Holzman"
Funding"
The Company of Biologists"
NSF"

66. Questions?!
julia@unc.edu"
@juliaesamson"