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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.

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- 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"

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