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Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects Elijah Newren December 7, 2004 Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects p. 1/16 Outline Example Biological Problems

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SLIDE 1

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects

Elijah Newren

December 7, 2004

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 1/16

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SLIDE 2

Outline

  • Example Biological Problems

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 2/16

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SLIDE 3

Outline

  • Example Biological Problems
  • Equations for Fluid Motion

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 2/16

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SLIDE 4

Outline

  • Example Biological Problems
  • Equations for Fluid Motion
  • Immersed Boundary Method

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 2/16

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SLIDE 5

Outline

  • Example Biological Problems
  • Equations for Fluid Motion
  • Immersed Boundary Method
  • Immersed Interface Method

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 2/16

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SLIDE 6

Outline

  • Example Biological Problems
  • Equations for Fluid Motion
  • Immersed Boundary Method
  • Immersed Interface Method
  • Incoherent Ramblings

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 2/16

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SLIDE 7

Outline

  • Example Biological Problems
  • Equations for Fluid Motion
  • Immersed Boundary Method
  • Immersed Interface Method
  • (Even More) Incoherent Ramblings

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 2/16

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SLIDE 8

Biological Problems

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 3/16

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SLIDE 9

Biological Problems

  • Beating Heart

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 3/16

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SLIDE 10

Biological Problems

  • Beating Heart
  • Platelet Aggregation

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 3/16

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SLIDE 11

Biological Problems

  • Beating Heart
  • Platelet Aggregation
  • Insect Flight

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 3/16

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SLIDE 12

Biological Problems

  • Beating Heart
  • Platelet Aggregation
  • Insect Flight
  • Cochlear Dynamics

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 3/16

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SLIDE 13

Biological Problems

  • Beating Heart
  • Platelet Aggregation
  • Insect Flight
  • Cochlear Dynamics
  • Mechanical Properties of Cells

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 3/16

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SLIDE 14

Biological Problems

  • Beating Heart
  • Platelet Aggregation
  • Insect Flight
  • Cochlear Dynamics
  • Mechanical Properties of Cells
  • Swimming of Organisms

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 3/16

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SLIDE 15

Fluid Motion

ρ(ut + (u · ∇)u) = −∇p + µ∆u + f ∇ · u = 0

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 4/16

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SLIDE 16

Fluid Motion

ρ(ut + (u · ∇)u) = −∇p + µ∆u + f Momentum ∇ · u = 0

  • Change in Momentum (“Mass times acceleration”)

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 4/16

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SLIDE 17

Fluid Motion

ρ(ut + (u · ∇)u) = −∇p + µ∆u + f Pressure Gradient ∇ · u = 0

  • Change in Momentum (“Mass times acceleration”)
  • Pressure Gradient (normal force between volumes)

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 4/16

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SLIDE 18

Fluid Motion

ρ(ut + (u · ∇)u) = −∇p + µ∆u + f Viscosity ∇ · u = 0

  • Change in Momentum (“Mass times acceleration”)
  • Pressure Gradient (normal force between volumes)
  • Viscosity (tangential force between volumes)

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 4/16

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SLIDE 19

Fluid Motion

ρ(ut + (u · ∇)u) = −∇p + µ∆u + f Other Forces ∇ · u = 0

  • Change in Momentum (“Mass times acceleration”)
  • Pressure Gradient (normal force between volumes)
  • Viscosity (tangential force between volumes)
  • Other forces (gravity, psychokinesis, etc.)

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 4/16

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SLIDE 20

Fluid Motion

ρ(ut + (u · ∇)u) = −∇p + µ∆u + f Incompressibility Constraint ∇ · u = 0

  • Change in Momentum (“Mass times acceleration”)
  • Pressure Gradient (normal force between volumes)
  • Viscosity (tangential force between volumes)
  • Other forces (gravity, psychokinesis, etc.)
  • Incompressibility Constraint (Volume doesn’t change)

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 4/16

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SLIDE 21

Navier Stokes Equations

ut + (u · ∇)u = −∇p + ν∆u + f ∇ · u = 0

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 5/16

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SLIDE 22

Navier Stokes Equations

ut + (u · ∇)u = −∇p + ν∆u + f ∇ · u = 0

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 5/16

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SLIDE 23

Navier Stokes Equations

ut + ∇p = (u · ∇)u + ν∆u + f ∇ · u = 0

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 5/16

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SLIDE 24

Navier Stokes Equations

ut + ∇p = (u · ∇)u + ν∆u + f ∇ · u = 0 ⇒ ut = P (−(u · ∇)u + ν∆u + f)

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 5/16

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SLIDE 25

Hodge Decomposition

Given periodic ω, ∃! periodic u & ∇φ such that ω = u + ∇φ ∇ · u = 0

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 6/16

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SLIDE 26

Hodge Decomposition

Given periodic ω, ∃! periodic u & ∇φ such that ω = u + ∇φ ∇ · u = 0 Proof (of existence): Taking the divergence: ∇ · ω = ∇ · u + ∇ · (∇φ) = ∆φ

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 6/16

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SLIDE 27

Hodge Decomposition

Given periodic ω, ∃! periodic u & ∇φ such that ω = u + ∇φ ∇ · u = 0 Proof (of existence): Taking the divergence: ∇ · ω = ∇ · u + ∇ · (∇φ) = ∆φ Thus we merely need to solve ∆φ = ∇ · ω to find φ.

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 6/16

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SLIDE 28

Hodge Decomposition

Given periodic ω, ∃! periodic u & ∇φ such that ω = u + ∇φ ∇ · u = 0 Proof (of existence): Taking the divergence: ∇ · ω = ∇ · u + ∇ · (∇φ) = ∆φ Thus we merely need to solve ∆φ = ∇ · ω to find φ. A solution exists since (using the Fredholm alternative theorem): ∇ · ω, c =

  • Ω

c(∇ · ω) =

  • ∂Ω

c(ω · n) = 0.

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 6/16

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SLIDE 29

Hodge Decomposition

Given periodic ω, ∃! periodic u & ∇φ such that ω = u + ∇φ ∇ · u = 0 Proof (of existence): Taking the divergence: ∇ · ω = ∇ · u + ∇ · (∇φ) = ∆φ Thus we merely need to solve ∆φ = ∇ · ω to find φ. A solution exists since (using the Fredholm alternative theorem): ∇ · ω, c =

  • Ω

c(∇ · ω) =

  • ∂Ω

c(ω · n) = 0. Finally, we just set u = ω − ∇φ.

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 6/16

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SLIDE 30

Navier Stokes Solver

ut + ∇p = −(u · ∇)u + ν∆u + f ∇ · u = 0

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 31

Navier Stokes Solver

ut + ∇p = −(u · ∇)u + ν∆u + f ∇ · u = 0

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 32

Navier Stokes Solver

ut + ∇p = −(u · ∇)u + ν∆u + f ∇ · u = 0 y′ = f(y)

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 33

Navier Stokes Solver

ut + ∇p = −(u · ∇)u + ν∆u + f ∇ · u = 0 y′ = f(y) y2 = y1 + t2

t1

f(y(t))dt

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 34

Navier Stokes Solver

ut + ∇p = −(u · ∇)u + ν∆u + f ∇ · u = 0 y′ = f(y) y2 = y1 + t2

t1

f(y(t))dt y2 = y1 + 1 2∆t(f(y2) + f(y1))

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 35

Navier Stokes Solver

ut + ∇p = −(u · ∇)u + ν∆u + f ∇ · u = 0 y′ = f(y) y2 = y1 + t2

t1

f(y(t))dt y2 = y1 + 1 2∆t(f(y2) + f(y1)) yn+1 − yn ∆t = 1 2(f(yn+1) + f(yn))

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 36

Navier Stokes Solver

un+1 − un ∆t + ∇pn+ 1

2 = −[(u · ∇)u]n+ 1 2 + ν

2∆(un+1 + un) + f n+ 1

2

∇ · un+1 = 0

y′ = f(y) y2 = y1 + t2

t1

f(y(t))dt y2 = y1 + 1 2∆t(f(y2) + f(y1)) yn+1 − yn ∆t = 1 2(f(yn+1) + f(yn))

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 37

Navier Stokes Solver

un+1 − un ∆t + ∇pn+ 1

2 = −[(u · ∇)u]n+ 1 2 + ν

2∆(un+1 + un) + f n+ 1

2

∇ · un+1 = 0

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 38

Navier Stokes Solver

u∗ − un ∆t + 0 = −[(u · ∇)u]n+ 1

2 + ν

2∆(u∗ + un) + f n+ 1

2

∇ · un+1 = 0

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 39

Navier Stokes Solver

u∗ − un ∆t = −[(u · ∇)u]n+ 1

2 + ν

2∆(u∗ + un) + f n+ 1

2

∇ · un+1 = 0

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 40

Navier Stokes Solver

u∗ − un ∆t = −[(u · ∇)u]n+ 1

2 + ν

2∆(u∗ + un) + f n+ 1

2

un+1 − un ∆t + ∇pn+ 1

2 = u∗ − un

∆t ∇ · un+1 = 0

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 41

Navier Stokes Solver

u∗ − un ∆t = −[(u · ∇)u]n+ 1

2 + ν

2∆(u∗ + un) + f n+ 1

2

un+1 − un ∆t +∇pn+ 1

2 = u∗ − un

∆t ∇ · un+1 = 0

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 42

Navier Stokes Solver

u∗ − un ∆t = −[(u · ∇)u]n+ 1

2 + ν

2∆(u∗ + un) + f n+ 1

2

un+1 − un ∆t +∇pn+ 1

2 = u∗ − un

∆t ∇ · un+1 = 0 ∆pn+ 1

2 = 1

∆t∇ · u∗

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 43

Navier Stokes Solver

u∗ − un ∆t = −[(u · ∇)u]n+ 1

2 + ν

2∆(u∗ + un) + f n+ 1

2

un+1 − un ∆t +∇pn+ 1

2 = u∗ − un

∆t ∇ · un+1 = 0 ∆pn+ 1

2 = 1

∆t∇ · u∗

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 44

Navier Stokes Solver

u∗ − un ∆t = −[(u · ∇)u]n+ 1

2 + ν

2∆(u∗ + un) + f n+ 1

2

un+1 − un ∆t +∇pn+ 1

2 = u∗ − un

∆t ∇ · un+1 = 0 ∆pn+ 1

2 = 1

∆t∇ · u∗ un+1 = u∗ − ∆t∇pn+ 1

2

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 45

Navier Stokes Solver

u∗ − un ∆t = −[(u · ∇)u]n+ 1

2 + ν

2∆(u∗ + un) + f n+ 1

2

∆pn+ 1

2 = 1

∆t∇ · u∗ un+1 = u∗ − ∆t∇pn+ 1

2

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 7/16

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SLIDE 46

Immersed <Noun> Methods

Ω

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 8/16

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SLIDE 47

Immersed <Noun> Methods

Ω

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 8/16

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SLIDE 48

Immersed <Noun> Methods

X(s,t) Γ Ω

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 8/16

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SLIDE 49

Immersed <Noun> Formulation

ut + (u · ∇)u = −∇p + ν∆u + f ∇ · u = 0

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 9/16

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SLIDE 50

Immersed <Noun> Formulation

ut + (u · ∇)u = −∇p + ν∆u + f ∇ · u = 0 F(s, t) = Some function of X(s, t) and its derivatives

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 9/16

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SLIDE 51

Immersed <Noun> Formulation

ut + (u · ∇)u = −∇p + ν∆u + f ∇ · u = 0 F(s, t) = Some function of X(s, t) and its derivatives f(x, t) =

  • Γ

F(s, t)δ(x − X(s, t)) ds

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 9/16

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SLIDE 52

Immersed <Noun> Formulation

ut + (u · ∇)u = −∇p + ν∆u + f ∇ · u = 0 F(s, t) = Some function of X(s, t) and its derivatives f(x, t) =

  • Γ

F(s, t)δ(x − X(s, t)) ds dX dt = u(X(s, t), t)

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 9/16

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SLIDE 53

Immersed <Noun> Formulation

ut + (u · ∇)u = −∇p + ν∆u + f ∇ · u = 0 F(s, t) = Some function of X(s, t) and its derivatives f(x, t) =

  • Γ

F(s, t)δ(x − X(s, t)) ds dX dt = u(X(s, t), t) = lim

x→X(s,t) u(x, t)

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 9/16

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SLIDE 54

Immersed <Noun> Formulation

ut + (u · ∇)u = −∇p + ν∆u + f ∇ · u = 0 F(s, t) = Some function of X(s, t) and its derivatives f(x, t) =

  • Γ

F(s, t)δ(x − X(s, t)) ds dX dt = u(X(s, t), t) =

  • Ω

u(x, t)δ(x − X(s, t)) dx

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 9/16

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SLIDE 55

Immersed Boundary Method

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 10/16

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SLIDE 56

Immersed Boundary Method

F(s, t) = Some function of X(s, t) and its derivatives f(x, t) =

  • Γ

F(s, t)δ(x − X(s, t)) ds dX dt = u(X(s, t), t) =

  • Ω

u(x, t)δ(x − X(s, t)) dx

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 11/16

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SLIDE 57

Immersed Boundary Method

Fk = Some function of Xk and its differences f(x, t) =

  • Γ

F(s, t)δ(x − X(s, t)) ds dX dt = u(X(s, t), t) =

  • Ω

u(x, t)δ(x − X(s, t)) dx

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 11/16

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SLIDE 58

Immersed Boundary Method

Fk = Some function of Xk and its differences fij =

  • k

Fkδh(xij − Xk)∆s dX dt = u(X(s, t), t) =

  • Ω

u(x, t)δ(x − X(s, t)) dx

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 11/16

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SLIDE 59

Immersed Boundary Method

Fk = Some function of Xk and its differences fij =

  • k

Fkδh(xij − Xk)∆s dXk dt = Uk =

  • ij

uijδh(xij − Xk)h2

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 11/16

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SLIDE 60

Immersed Boundary Method

Fk = Some function of Xk and its differences fij =

  • k

Fkδh(xij − Xk)∆s dXk dt = Uk =

  • ij

uijδh(xij − Xk)h2

δh(x, y) = δh(x)δh(y) δh(x) =   

1 4h

  • 1 + cos( πx

2h)

  • |x| ≤ 2h

|x| ≥ 2h

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 11/16

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SLIDE 61

Advantages and Shortcomings

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 12/16

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SLIDE 62

Advantages and Shortcomings

Advantages:

  • Applicable to a wide variety of biofluid problems

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 12/16

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SLIDE 63

Advantages and Shortcomings

Advantages:

  • Applicable to a wide variety of biofluid problems
  • Regular Grid

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 12/16

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SLIDE 64

Advantages and Shortcomings

Advantages:

  • Applicable to a wide variety of biofluid problems
  • Regular Grid
  • Simple

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 12/16

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SLIDE 65

Advantages and Shortcomings

Advantages:

  • Applicable to a wide variety of biofluid problems
  • Regular Grid
  • Simple *cough*

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 12/16

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SLIDE 66

Advantages and Shortcomings

Shortcomings:

  • Accuracy

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 12/16

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SLIDE 67

Advantages and Shortcomings

Shortcomings:

  • Accuracy
  • Stability

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 12/16

slide-68
SLIDE 68

Immersed Interface Method

f(x, t) =

  • Γ

F(s, t)δ(x − X(s, t)) ds

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

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SLIDE 69

Immersed Interface Method

f(x, t) =

  • Γ

F(s, t)δ(x − X(s, t)) ds u∗ − un ∆t = −[(u · ∇)u]n+ 1

2 + ν

2∆(u∗ + un) + f n+ 1

2

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

slide-70
SLIDE 70

Immersed Interface Method

f(x, t) =

  • Γ

F(s, t)δ(x − X(s, t)) ds u∗ − un ∆t = −[(u · ∇)u]n+ 1

2 + ν

2∆(u∗ + un) + f n+ 1

2

∆pn+ 1

2 = 1

∆t∇ · u∗

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

slide-71
SLIDE 71

Immersed Interface Method

p (s, t) = fn(s, t) ∂p ∂n

  • (s, t) = ∂

∂sfτ(s, t) u (s, t) = 0 ν ∂u ∂n

  • (s, t) = −fτ(s, t)τ(s, t)

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

slide-72
SLIDE 72

Immersed Interface Method

p (s, t) = fn(s, t) ∂p ∂n

  • (s, t) = ∂

∂sfτ(s, t) u (s, t) = 0 ν ∂u ∂n

  • (s, t) = −fτ(s, t)τ(s, t)

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

slide-73
SLIDE 73

Immersed Interface Method

p (s, t) = fn(s, t) u∗ − un ∆t = −[(u · ∇)u]n+ 1

2 + ν

2∆(u∗ + un) + f n+ 1

2

∆pn+ 1

2 = 1

∆t∇ · u∗ un+1 = u∗ − ∆t∇pn+ 1

2

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

slide-74
SLIDE 74

Immersed Interface Method

p (s, t) = fn(s, t) u∗ − un ∆t = −[(u · ∇)u]n+ 1

2 + ν

2∆(u∗ + un) + f

n+ 1

2

τ

∆pn+ 1

2 = 1

∆t∇ · u∗+∇ · f

n+ 1

2

n

un+1 = u∗ − ∆t∇pn+ 1

2

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

slide-75
SLIDE 75

Immersed Interface Method

u∗ − un ∆t = −[(u · ∇)u]n+ 1

2 + ν

2∆(u∗ + un) + f

n+ 1

2

τ

   ∆pn+ 1

2 =

1 ∆t∇ · u∗

p (s, t) = fn(s, t) un+1 = u∗ − ∆t∇pn+ 1

2

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

slide-76
SLIDE 76

Immersed Interface Method

un+1 = u∗ − ∆t∇pn+ 1

2

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

slide-77
SLIDE 77

Immersed Interface Method

un+1 = u∗ − ∆t∇pn+ 1

2

p = pbelow + pbetween + ∆ypy + O(∆y2)

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

slide-78
SLIDE 78

Immersed Interface Method

un+1 = u∗ − ∆t∇pn+ 1

2

p = pbelow + pbetween + ∆ypy + O(∆y2) B2 = p ∆y

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

slide-79
SLIDE 79

Immersed Interface Method

un+1 = u∗ − ∆t∇pn+ 1

2

p = pbelow + pbetween + ∆ypy + O(∆y2) B2 = p ∆y vn+1 = v∗ − ∆t p − pbelow ∆y − B2

  • Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16
slide-80
SLIDE 80

Immersed Interface Method

un+1 = u∗ − ∆t∇pn+ 1

2

vn+1 = v∗ − ∆t p − pbelow ∆y − B2

  • Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16
slide-81
SLIDE 81

Immersed Interface Method

un+1 = u∗ − ∆t∇pn+ 1

2

vn+1 = v∗ − ∆t p − pbelow ∆y − B2

  • ∇h · un+1 = ∇h · u∗ − ∆t
  • ∆hpn+ 1

2 − ∇h · B

  • Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16
slide-82
SLIDE 82

Immersed Interface Method

un+1 = u∗ − ∆t∇pn+ 1

2

vn+1 = v∗ − ∆t p − pbelow ∆y − B2

  • ∇h · un+1 = ∇h · u∗ − ∆t
  • ∆hpn+ 1

2 − ∇h · B

  • ∆hpn+ 1

2 = 1

∆t∇h · u∗ + ∇h · B

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

slide-83
SLIDE 83

Immersed Interface Method

un+1 = u∗ − ∆t∇pn+ 1

2

vn+1 = v∗ − ∆t p − pbelow ∆y − B2

  • ∇h · un+1 = ∇h · u∗ − ∆t
  • ∆hpn+ 1

2 − ∇h · B

  • ∆hpn+ 1

2 = 1

∆t∇h · u∗ + ∇h · B Cij = (∇h · B)ij

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

slide-84
SLIDE 84

Immersed Interface Method

un+1 = u∗ − ∆t∇pn+ 1

2

vn+1 = v∗ − ∆t p − pbelow ∆y − B2

  • ∇h · un+1 = ∇h · u∗ − ∆t
  • ∆hpn+ 1

2 − ∇h · B

  • ∆hpn+ 1

2 = 1

∆t∇h · u∗ + ∇h · B Cij = (∇h · B)ij ∆hpn+ 1

2 = 1

∆t∇h · u∗ + C

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

slide-85
SLIDE 85

Immersed Interface Method

un+1 = u∗ − ∆t∇pn+ 1

2

vn+1 = v∗ − ∆t p − pbelow ∆y − B2

  • ∇h · un+1 = ∇h · u∗ − ∆t
  • ∆hpn+ 1

2 − ∇h · B

  • ∆hpn+ 1

2 = 1

∆t∇h · u∗ + ∇h · B Cij = (∇h · B)ij ∆hpn+ 1

2 = 1

∆t∇h · u∗ + C    ∆pn+ 1

2 =

1 ∆t∇ · u∗

p (s, t) = fn(s, t)

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 13/16

slide-86
SLIDE 86

My Work

  • Surface Representation – 3D

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 14/16

slide-87
SLIDE 87

My Work

  • Surface Representation – 3D
  • Paralyzing

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 14/16

slide-88
SLIDE 88

My Work

  • Surface Representation – 3D
  • Parallelizing

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 14/16

slide-89
SLIDE 89

My Work

  • Surface Representation – 3D
  • Parallelizing
  • Adaptive Mesh Refinement

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 14/16

slide-90
SLIDE 90

END OF PRESENTATION (You hit down one too many times)

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 15/16

slide-91
SLIDE 91

Delta Functions

δ(x) =    x = 0 ∞ x = 0

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 16/16

slide-92
SLIDE 92

Delta Functions

δ(x) =    x = 0 ∞ x = 0 δ(x) = lim

ℓ→0

1 √ 2πℓe−x2/2ℓ2

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 16/16

slide-93
SLIDE 93

Delta Functions

δ(x) =    x = 0 ∞ x = 0 δ(x) = lim

ℓ→0

1 √ 2πℓe−x2/2ℓ2

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 16/16

slide-94
SLIDE 94

Delta Functions

δ(x) =    x = 0 ∞ x = 0 x2

x1

δ(x) dx =    1 0 ∈ [x1, x2]

  • therwise

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 16/16

slide-95
SLIDE 95

Delta Functions

δ(x) =    x = 0 ∞ x = 0 x2

x1

δ(x) dx =    1 0 ∈ [x1, x2]

  • therwise

x2

x1

f(x)δ(x − a) dx =    f(a) a ∈ [x1, x2]

  • therwise

Numerically Solving the Coupled Motion of Fluid and Contained Elastic Objects – p. 16/16