1. Interaction of Immersed Boundaries in Complex Fluids
Hector D. Ceniceros
Department of Mathematics, USCB
2011 AMS Spring Western Sectional Meeting, Las Vegas

2. Collaborators Jordan E. Fisher (Ph.D. 2011) Courant
Alexandre M. Roma (USP, Brazil)

3. Outline Introduction: elastic structures in fluids.
The Immersed Boundary (IB) Method.
Recent progress on IB Method (2D and 3D).
Complex Fluids.

4. Applied Math at UCSB
Computational Science and Engineering Graduate Emphasis
Interdisciplinary Complex Fluids/Soft Materials Group
Copolymers
Liquid Crystalline Polymers
Bio-Polymers
Polymeric Solar Cells

5. Flow-Structure Interaction
Structures can be simple or complex
Flexible rod
Heart Model
Jellyfish
Flow-structure interaction: structures react to flow and
flow is affected by structure forces.

6. Flow-Structure Interaction Relevant to Reproduction
Free swimmers
(e.g. spermatozoa)
Ciliary motion
(e.g. in airways, oviduct)
Channel with elastic
contracting walls (Peristalsis)
There is a very nice survey article on the fluid mechanics of reproduction by Fauci and Dillon.
In these examples the fluid is viscoelastic (non-Newtonian)
Fauci and Dilon, Ann. Rev. Fluid Mech. 38: , 2006

7. Immersed Boundary Setting
C. Peskin (70’s)
The crux of the IB method and much of its versatility is the seamless connection of the Lagrangian representation of the immersed structure with the Eulerian representation of the flow.
Eulerian-Lagrangian representation

8. Immersed Boundary Method
C. Peskin, 70’s
Spreading
Interpolation

9. Versatility of the IB Method
Vast structure-building capability:
from a simple link to complex fiber architecture
Easy implementation
readily available flow solvers and simple tracking
Has been used in many applications:
Cardiac fluid dynamics, swimming, insect flight, locomotion of cilia and flagella, peristalsis, particulate flows, bio-films, complex fluids, etc.

10. An Old Problem: Stiffness
Immersed structures can be very stiff and induce severe time-step restrictions for explicit methods (Peskin 77, Stockie and Wetton 95, 99).
Fully implicit discretizations seem too expensive for any practical use (Tu and Peskin 92, Mayo and Peskin 93).
Recent progress with semi-implicit method (Hou and Shi 2008) but limited to periodic interfaces.
Despite its success the IB method has suffered from a notorious problem. In many applications immersed structures can be very stiff and induce severe time step restrictions for explicit methods. This was recognized by Peskin, since the inception of the method and a later study by Stockie and Wetton showed that it was strong tangential forces what gives rise to the numerical stiffness.

11. Peskin’s lagged operators discretization, 1977
with mesh size
Cartesian grids
Peskin’s lagged operators discretization, 1977

12. Stability and Robustness
Neglect advection,
Linear and self-adjoint negative def
Numerical experiment suggest the robustness of this discretization extends to the inertial case with nonlinear interfacial force and immersed boundary structures.
Unconditionally stable
Newren, Fogelson, Guy, Kirby
JCP, 222, 2007

13. Stiffness Problem Solved?
Stiffness can be removed with suitable implicit discretization
e.g. Peskin’s lagged operators discretization
Caveat: solving the implicit discretization even in the linear case is
too costly, impractical
This has been known to the community for almost 40 years
The problem has received renewed attention recently
(Peskin & Mori, Newren et al, Hou & Shi, Griffith, Layton &Beale)

14. Recasting the Equations
Fluid solve

15. Eliminating un+1 In fact unless
One can easily show that Mn positive semi-def.
Positive if there are not too many Lagrangian points per Eulerian cell (injectivity of Sn lost) or SnF not a gradient field. Due to spurious currents Mn is positive def.
In fact
unless
is in the kernel of the projection, i.e. a gradient field
Due to the spreading, the IB method fails to yield discrete gradient

16. Forward Euler/Backward Euler (FE/BE)
Efficiency
How to solve
economically to produce non-stiff integration of IB Method
for a wide range of practical immersed structure situations?
Forward Euler/Backward Euler (FE/BE)
Main cost is fluid solve. Any method requires bn which involves a fluid solve, appropriate to measure the cost relative to one FE/BE step
There are really two interrelated problems:
1. Efficient computation of the flow-structure interaction Mn f
2. Efficient iterative solution methods for Xn+1
Removes implementation dependency to measure performance

17. Costs (2D)
In the design of efficient iterative methods it is crucial to streamline the computation of quantities of the form
i.e. Flow-Structure Interaction Operation
Spreading +fluid solve + interpolation
Matrix-vector multiplication
Caveat: We need a matrix representation of Mn
which is too costly to obtain directly

18. Matrix Approximation correspond to velocity that is obtained
The entries
by interpolating the values produced at Xi by spread unit forces
located at Xj
At the continuum level G(Xi -Xj)
Not true at discrete level due to spreading and interpolation
Tremendous savings if we assume

19. Shifting to the Origin Cost
Fix the point where force is applied and evaluate effect on each Eulerian
grid point. These Eulerian values can be precomputed (2 fluid solves!)
For a each given Xj-Xi the corresponding velocity is obtained from
interpolation of the Eulerian values

20. Accuracy of Matrix Approximation
The estimate follows from estimates on Gh
and an identity for the discrete delta.
Idea of Proof
This error for the proposed, approximated matrix
is asymptotically smaller than the $O(h)$ error of the IB Method for points within a distance $O(h)$ of the immersed boundary~\cite{Mori2008}. Thus, the approximated matrix can be use without any deterioration of the overall accuracy of the IB Method.

21. Initially elliptical interface
The Prototypical Test
Initially elliptical interface
Relaxes to a circle

22. Solving the Linear System
Since the matrix is available it is easy to construct a wide class
of iterative methods (e.g. weighted Jacobi, Gauss-Seidel, etc)
Standard algebraic multigrid works the best in this case
Example: Stokes flow
MG is about 4 FE/BE per time step. At the modest resolution of N=256 we already achieve more than two orders of magnitude in speed-up

23. 2D Model of a Heart Valve
The immersed structure consists of The valve is indirectly restricted in motion by the two hinges but is allowed to rotate. The wall and the hinges are modeled as tethers. This is an application of the IB method where there are rigid structures, tethered points and cross links.
1. Valve. 2. Cushions 3. Hinges 4. Artery wall
There are rigid structures, tethered points and crossed links

24. The Nonlinear System Challenges
Removing the stiffness for this more prototypal IB method application is considerably more challenging than in the simple elliptical interface case.
A_{hB} is nonlinear due to the nonzero resting lengths.
Because of the lack of positive definiteness CG does not converge
and BiCG takes over 100 iterations.
Difficult to find effective preconditioners

25. Solving the Nonlinear System
Fixed point iteration
Fails miserably!
The eigenvalues of J are huge. One could consider the reversed iteration
Additional iterations exacerbate the instability.
Can be done very efficiently as Mn is pos def we could use linear MG
However for 2 there is not a unique solution in general. We can select the correct solution by taking into account translation and rotation invariance with a modified Newton’s method. 1. 2.

26. Numerical Results Imposed horizontal flow Re=50 Time step for FE/BE
We use an imposed flow to move the valve back and forth. Re=50.
CPU time FE/BE: *
CPU time Semi-implicit 1907
Four orders of magnitude faster!

27. Thus our matrix approach is impractical in 2D!
Recall there are two main difficulties:
1. The heavy cost of computing the flow-structure interaction Mnf
2. Solving the nonlinear system
Thus our matrix approach is impractical in 2D!
Our solution:
Adapt a Fast MultiPole Method (treecode) approach to the IB method

28. Two main ideas
The idea is to use far field (multi-pole) expansions of Gh to
compress the effect of clusters of fiber points

29. Treecode Approach

30. How to Select Win and Wout?
Solution: binary partitioning, quadtree (2D), octree (3D)
To evaluate
loop over each panel P
and calculate the far field expansion of all poles in P

31. Expansions

32. Results
Flow past an immersed plate. Each fiber point X is tethered to a
corresponding point T:
Flow is induced with a time periodic forcing term. Re=10
We solve implicit system via CG
CPU time in hours
For implicit the cpu time does not vary much with \sigma in contrast to the explicit one. Computation that would take over a month can be now done in minutes!!

33. Depiction of the flow using streamlines
Flow past a plate
Geometry of the immersed structure is trivial and the deformations are negligible. Of course, the true power of the IB method lies in its seamless handling of both rigid and dynamic, flexible structures. The following example, while simple, tests the method on a dynamic, flexible interface.
Depiction of the flow using streamlines

34. Oscillating Immersed Spheroid
Velocity magnitude
Six order of magnitude faster

35. Complex (Non-Newtonian) Fluids
Polymeric liquids
Gels, sols, emulsions
Foams
Liquid crystalline materials
Granular materials
Swimmers in vicoelastic fluid
We’re particularly interested in investigating the dynamics of immersed flexible structures interacting with these viscoelastic flows.
There is a microstructure (e.g. long molecules) whose interaction
with a flow leads to many phenomena not observed in Newtonian fluids

36. Generic Framework Low Re, Stokes approximation Polymeric stress Q
e.g. molecules modeled as dumbbells
we can get an evolution eq for If
We eliminate configuration space. This is called Oldroyd B model

37. F.E.N.E. Model
Hooke’s law implies dumbbells could be extended without limit!!!!!
Finitely Extensible Nonlinear Elastic
No longer possible to eliminate configuration space to compute stress
Pre-averaging (FENE-P) and closure approximations are sometimes used
to reduce the computational complexity
We are working on multiscale approaches to effectively compute
these type of flows in the presence of immersed flexible boundaries

38. Peristaltic Pumping in Viscoelastic Fluid
Peristalsis: fluid transport that occurs when waves of contraction
are passed along a fluid bearing tube
Stokes-Oldroyd B system
(Teran, Fauci, Shelley 2008)
Flux very different from Newtonian
We’re investigating for larger We=tp/tf
and longer times both in 2D and 3D

39. Conclusions There are two main difficulties:
computing the flow-structure interaction (Mn F)
solving the implicit system.
It is possible to expedite the computation of Mn F and to arrive a efficient solutions for the implicit system to produce enormous savings in 2D & 3D.
Current work is the application of these techniques to investigate the motion of immersed flexible boundaries in complex fluids.

40. Acknowledgements Partial support by National Science Foundation:
DMS and DMS
Special thanks to the American Mathematical Society
for sponsoring this talk