The immersed boundary method, from 2D fibres to 3D finite elements

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"

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""

AlexHooverTulaneUniversity

The IB method: a brief history"

CharlesS.PeskinCourantInstitute,NYFlowpatternsaroundheartvalves:adigitalcomputermethodforsolvingtheequationsofmotion.PhDthesis,1972.

The IB method: a brief history"

LauraA.MillerUNCChapelHill

BoyceE.GriffithUNCChapelHill

CharlesS.PeskinCourantInstitute

The IB method: definition"

Viscousfluid?

Fluidgridgeneratedfromboundaryshape?

IB!!!J

NotIBL

NotIBL

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.""

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."

The IB method: applications"AlexHooverTulaneUniversity

NickBattistaUNCChapelHill

LauraMillerUNCChapelHill

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"

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

The Navier-Stokes equations"It basically follows Newton’s Second Law:"F = m * a"

mass*accelerationpressureforces

viscousforces

otherbodyforces

The Navier-Stokes equations"Now, we add the equation for incompressible flow."

mass*acceleration pressureforces

viscousforces

otherbodyforces

thefluidisincompressible

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."

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"

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"

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."

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"

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"

Combining fluid and structure"

+ =

+ interactions!"

Combining fluid and structure"

Fluid(fixedCartesianmesh)

Structure(movingcurvilinearmesh)

movesatlocalfluidvelocity

exertsforceson

exertsforceson

Combining fluid and structure"

Fluid(fixedCartesianmesh)

Structure(movingcurvilinearmesh)

SpreadtheelasticforcedensityfromcurvilinearmeshontoCartesiangrid.

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."

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"

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"

Combining fluid and structure"

Fluid(fixedCartesianmesh)

Structure(movingcurvilinearmesh)

movesatlocalfluidvelocity

InterpolatethevelocityfieldfromtheCartesiangridontothecurvilinearmesh.

Combining fluid and structure"

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

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"

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 ✔"

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 ✔"

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

mass*acceleration pressureforces

viscousforces

otherbodyforces

thefluidisincompressible

SpreadtheelasticforcedensityfromcurvilinearmeshontoCartesiangrid.

InterpolatethevelocityfieldfromtheCartesiangridontothecurvilinearmesh.

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

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."

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"

NickBattistaUNCChapelHill

github.com/nickabattista/IB2d

NickBattistaUNCChapelHill

Springs"Springs allow longitudinal motion between two coupled Lagrangian nodes."

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RLRL+d

elasticpotentialenergy

forcefromdeformation

Springs: the rubber band example"

All Lagrangian points are connected by springs with resting length 0.""Colormap shows vorticity."

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Torsional springs"Torsional springs allow transversal motion between three coupled Lagrangian nodes."

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θ Ifθdesired=180andC=0

Torsional springs"Torsional springs allow transversal motion between three coupled Lagrangian nodes."

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θ Ifθdesired=180andC=0

elasticpotentialenergy

curvature

Torsional springs"Torsional springs allow transversal motion between three coupled Lagrangian nodes."

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θ Ifθdesired=180andC=0

deformationforces

Torsional springs: the wobbly beam example"

All Lagrangian points are connected by beams with curvature 0.""Colormap shows magnitude of velocity."

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

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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."

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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)"

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. "

Collective pulsing in xeniid corals"

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

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."

Collectivepulsingbehavior

Waterflow

Collective pulsing in xeniid corals"

IB with Adaptive Mesh Refinement"

BoyceE.GriffithUNCChapelHillSimulatingtheblood-muscle-valvemechanicsoftheheartbyanadaptiveandparallelversionoftheimmersedboundarymethod.PhDthesis,2005.

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."

fromhttp://anatomyandphysiologyi.com/heart-anatomy-chambers-vessels-valves/

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…"

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25x25 50x50 100x100 200x200

IB with Adaptive Mesh Refinement"

"So how to have your cake and eat it too???"

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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"

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Heart development in zebrafish"4 days post fertilization""Blood cells and endocardium are colored""Two chambers: one atrium and one ventricle"

CourtesyofLeighAnnSamsaandDr.JiandongLiuSchoolofMedicine,UNCChapelHill

Heart development in zebrafish"

Ventricle

Atrium

75um

CourtesyofLeighAnnSamsaandDr.JiandongLiuSchoolofMedicine,UNCChapelHill

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Heart development in zebrafish"

Ventricle

Atrium

AVCanal

Ventricle

IBAMRmodel

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Heart development in zebrafish"

Trabeculaeappeartoshieldtheendocardiumfromhighershearingforces

velocityfield+vorticitymap streamlines(afteratriumfinishescontraction)

IB with Finite Elements"A completely different beast…"

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Un

Un-1

Un-2

Un+1

Un+2

Un+3

Un-2

Un-1Un

Un+1Un+2

Un+3Un+4

Un+5Un+6

Acollectionofsinglenodalpoints(=fiber)

Acollectionofpolygonalpieces(=elements)

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"

Jellyfish locomotion"AlexanderHooverTulaneUniversityFrompacemakertovortexring:modelingjellyfishpropulsionandturning.PhDthesis,2015

Jellyfish locomotion"

Jellyfish locomotion"

Jellyfish locomotion"

Jellyfish locomotion"

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""

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"

Questions?!

julia@unc.edu"@juliaesamson"