Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
AFEM for Geometric Biomembranes and
Fluid-Membrane Interaction
R.H. Nochetto (1,2)
Joint with A. Bonito (3) and M. S. Pauletti (1)
Department of Mathematics (1) and Institute for Physical Science and Technology (2)
University of Maryland
Department of Mathematics, Texas A&M (3)
New Directions in Computational PDEs
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
1 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Outline
1
Models
2
Gradient (Helfrich) Flows
3
Fluid-Membrane Interaction
4
Geometrically Consistent Refinement
5
Conclusions
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
2 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Basic Models
Refs: Helfrich (73), Jenkins (77), Seifert (97)
Assumptions
Membrane:
Ω
Γ
Layer of Incompressible Fluid
Ω in
Bending Rigidity
Non Permeable
g
Ω out
Surrounding Fluid:
Newtonian, Incompressible.
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
ΓD
Warwick 2009
3 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Basic Models
Refs: Helfrich (73), Jenkins (77), Seifert (97)
Assumptions
Membrane:
Ω
Γ
Layer of Incompressible Fluid
Ω in
Bending Rigidity
Non Permeable
g
Ω out
Surrounding Fluid:
ΓD
Newtonian, Incompressible.
Gradient Flow
Decreasing Energy
Fluid-Membrane Interaction
Fluid Equations of Motion
Area and Volume Constraint
Immerse and Exerts Forces
Non Physical Dynamics
Membrane moves with Fluid
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
3 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Basic Models
Refs: Helfrich (73), Jenkins (77), Seifert (97)
Assumptions
Membrane:
Ω
Γ
Layer of Incompressible Fluid
Ω in
Bending Rigidity
Non Permeable
g
Ω out
Surrounding Fluid:
ΓD
Newtonian, Incompressible.
Gradient Flow
Decreasing Energy
Fluid-Membrane Interaction
Fluid Equations of Motion
Area and Volume Constraint
Immerse and Exerts Forces
Non Physical Dynamics
Membrane moves with Fluid
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
3 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Basic Models
Refs: Helfrich (73), Jenkins (77), Seifert (97)
Assumptions
Membrane:
Ω
Γ
Layer of Incompressible Fluid
Ω in
Bending Rigidity
Non Permeable
g
Ω out
Surrounding Fluid:
ΓD
Newtonian, Incompressible.
Gradient Flow
Decreasing Energy
Fluid-Membrane Interaction
Fluid Equations of Motion
Area and Volume Constraint
Immerse and Exerts Forces
Non Physical Dynamics
Membrane moves with Fluid
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
3 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Basic Models
Refs: Helfrich (73), Jenkins (77), Seifert (97)
Assumptions
Membrane:
Ω
Γ
Layer of Incompressible Fluid
Ω in
Bending Rigidity
Non Permeable
g
Ω out
Surrounding Fluid:
ΓD
Newtonian, Incompressible.
Gradient Flow
Decreasing Energy
Fluid-Membrane Interaction
Fluid Equations of Motion
Area and Volume Constraint
Immerse and Exerts Forces
Non Physical Dynamics
Membrane moves with Fluid
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
3 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometric Model vs. Fluid-Membrane Model
play
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
4 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Bending energy (Refs: Kuwert, Schätzle, Simonett, Willmore)
Z
W (Γ) =
h2
Γ
1 3
δW = ∆Γ h + h − 2kh ν
2
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
5 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Bending energy (Refs: Kuwert, Schätzle, Simonett, Willmore)
Z
h2 + λ
Z
Z
Id · ν
1 3
δJ = ∆Γ h + h − 2kh ν + λh + pν
2
J(Γ) =
Γ
1+p
Γ
Γ
Constraints
Augmented Energy J, Lagrange Multipliers λ (area) and p (volume)
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
5 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Bending energy (Refs: Kuwert, Schätzle, Simonett, Willmore)
Z
h2 + λ
Z
Z
Id · ν
1 3
δJ = ∆Γ h + h − 2kh ν + λh + pν
2
J(Γ) =
Γ
1+p
Γ
Γ
Constraints
Augmented Energy J, Lagrange Multipliers λ (area) and p (volume)
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
5 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Gradient Flow for Biomembrane Modeling
Trajectory
GT := {(x, t) : x ∈ Γ(t), t ∈ [0, T ]}
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
6 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Gradient Flow for Biomembrane Modeling
Trajectory
GT := {(x, t) : x ∈ Γ(t), t ∈ [0, T ]}
Weak Helfrich flow
Given Γ0 and T > 0, find rx : GT → Rd+1 , λ : [0, T ] → R and
p : [0, T ] → R such that Γ(0) = Γ0 and for all t ∈ (0, T ]
Z
ẋ · φ = − δJ(Γ(t); φ)
Γ(t)
for all smooth φ : Γ(t) → Rd+1 and
A(Γ(t)) = A(Γ(0)),
R.H. Nochetto
V (Γ(t)) = V (Γ(0)).
AFEM for Fluid-Membrane Interaction
Warwick 2009
6 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Gradient Flow for Biomembrane Modeling
Trajectory
GT := {(x, t) : x ∈ Γ(t), t ∈ [0, T ]}
Weak Helfrich flow
Given Γ0 and T > 0, find rx : GT → Rd+1 , λ : [0, T ] → R and
p : [0, T ] → R such that Γ(0) = Γ0 and for all t ∈ (0, T ]
Z
ẋ · φ = − δW (Γ(t); φ) − λ δA(Γ(t); φ) − p δV (Γ(t); φ)
Γ(t)
for all smooth φ : Γ(t) → Rd+1 and
A(Γ(t)) = A(Γ(0)),
R.H. Nochetto
V (Γ(t)) = V (Γ(0)).
AFEM for Fluid-Membrane Interaction
Warwick 2009
6 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Dumbbell Bar Shaped Family
Ellipsoid 8x1x1
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
7 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Dumbbell Bar Shaped Family - Ellipsoid 8x1x1: Energies
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
8 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Dumbbell Bar Shaped Family - Ellipsoid 8x1x1: Multipliers
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
8 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Mesh Smoothing: Comparison between P 1 and P 2
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
9 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Red Bood Cell Family
Ellipsoid 3x3x1
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
10 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Red Bood Cell Family (Continued)
Ellipsoid 6x6x1
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
11 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Red Bood Cell Family (Continued)
Ellipsoid 5x5x1
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
12 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Shape Derivative of Willmore Energy δW : Vector Form
Refs: Rusu (2005), Dziuk (2008), Barrett-Garcke-Nürnberg (2008)
Shape Calculus: ν 0 (Γ, φ) = −∇Γ φ, h0 (Γ, φ) = −∆Γ φ, ∂ν h = −|∇Γ ν|2
Z
Z
Z
1
2
h3 φ;
φ=φ·ν
δW (Γ, φ) = ∇Γ φ · ∇Γ h − h|∇Γ ν| φ +
2
Γ
Γ
Γ
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
13 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Shape Derivative of Willmore Energy δW : Vector Form
Refs: Rusu (2005), Dziuk (2008), Barrett-Garcke-Nürnberg (2008)
Shape Calculus: ν 0 (Γ, φ) = −∇Γ φ, h0 (Γ, φ) = −∆Γ φ, ∂ν h = −|∇Γ ν|2
Z
Z
Z
1
2
h3 φ;
φ=φ·ν
δW (Γ, φ) = ∇Γ φ · ∇Γ h − h|∇Γ ν| φ +
2
Γ
Γ
Γ
Z
Z
∇ Γ φ · ∇Γ h =
Γ
Z
∇Γ φ : ∇Γ h −
Γ
Z
(∇Γ x + ∇Γ x )∇Γ φ : ∇Γ h −
Γ
Z
h∆Γ ν · φ
Γ
1
− h∆Γ ν · φ = h|∇Γ ν| −
2
Γ
Γ
R.H. Nochetto
Z
T
2
Z
AFEM for Fluid-Membrane Interaction
∇Γ h 2 · φ
Γ
Warwick 2009
13 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Shape Derivative of Willmore Energy δW : Vector Form
Refs: Rusu (2005), Dziuk (2008), Barrett-Garcke-Nürnberg (2008)
Shape Calculus: ν 0 (Γ, φ) = −∇Γ φ, h0 (Γ, φ) = −∆Γ φ, ∂ν h = −|∇Γ ν|2
Z
Z
Z
1
2
h3 φ;
φ=φ·ν
δW (Γ, φ) = ∇Γ φ · ∇Γ h − h|∇Γ ν| φ +
2
Γ
Γ
Γ
Z
Z
∇ Γ φ · ∇Γ h =
Γ
Z
∇Γ φ : ∇Γ h −
Γ
Z
T
(∇Γ x + ∇Γ x )∇Γ φ : ∇Γ h −
Γ
h∆Γ ν · φ
Γ
Z
1
− h∆Γ ν · φ = h|∇Γ ν| −
∇Γ h 2 · φ
2
Γ
Γ
Γ
Z
Z
Z
1
T
dW (Γ; φ) = ∇Γ φ : ∇Γ h− (∇Γ x+∇Γ x )∇Γ φ : ∇Γ h+
∇Γ ·h∇Γ ·φ
2 Γ
Γ
Γ
R.H. Nochetto
Z
Z
2
AFEM for Fluid-Membrane Interaction
Warwick 2009
13 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Discrete Geometric Scheme: Ingredients
Curvature: h = −∆Γ x,
x = identity on Γ
(Dziuk’ 91)
Semi-implicit Time Discretization (tn → tn+1 ): explicit geometry
(Γ = Γn , ∇Γ = ∇Γn , ν = ν n ) (Dziuk’ 91)
Z
Z
n+1
xn+1 = xn + τ n vn+1
h
·ψ =
∇Γn xn+1 : ∇Γn ψ,
Γn
Γn
Z
Z
Z
n+1
n
n+1
⇒
h
·ψ−τ
∇ Γn v
: ∇ Γn ψ =
∇ Γ n xn : ∇ Γ n ψ
Γn
Γn
Γn
Mixed Method: operator splitting
1
2
R
n
Velocity: ΓRn vn+1 · φ = −δJ n+1
R (Γ ; φ)
R
n+1
n
Curvature: Γn h
· ψ − τ Γn ∇Γn vn+1 : ∇Γn ψ = Γn ∇Γn xn : ∇Γn ψ
Space discretization: linears or quadratics
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
14 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Fully Discrete Geometric Scheme
Z
Γnh
Vn+1 · Φ = −dWhn+1 (Γnh ; Φ) − λn+1 dAnh (Γnh ; Φ) − p n+1 dVhn (Γnh ; Φ)
Z
Γnh
Hn+1 · Ψ − τn
Z
Γnh
∇Γnh Vn+1 : ∇Γnh Ψ =
Z
Γnh
∇Γnh xn : ∇Γnh Ψ,
for all Φ, Ψ ∈ Sn+1
h ,
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
15 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Fully Discrete Geometric Scheme
Z
Γnh
Vn+1 · Φ = −dWhn+1 (Γnh ; Φ) − λn+1 dAnh (Γnh ; Φ) − p n+1 dVhn (Γnh ; Φ)
Z
Γnh
Hn+1 · Ψ − τn
Z
Γnh
∇Γnh Vn+1 : ∇Γnh Ψ =
Z
Γnh
∇Γnh xn : ∇Γnh Ψ,
for all Φ, Ψ ∈ Sn+1
h , where
Z
Z
1
n+1 n
n+1
n
n
: ∇ Γh Φ +
dWh (Γh ; Φ) =
∇ Γh H
∇Γnh · Hn+1 ∇Γnh · Φ
n
n
2
Γh
Γh
Z
−
(∇Γnh xn + ∇Γnh xnT )∇Γnh Φ : ∇Γnh Hn+1 ,
Γnh
dAnh (Γnh ; Φ)
R.H. Nochetto
Z
n
H · Φ,
=
Γnh
dVhn (Γnh ; Φ)
AFEM for Fluid-Membrane Interaction
Z
Φ · ν.
=
Γnh
Warwick 2009
15 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Boomerang Shape
Twisted Banana
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
16 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Boomerang Shape: Full Simulation
Full Simulation
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
17 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Boomerang Shape: Fast Time Scale
Fast time scale
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
18 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Boomerang Shape: Slow Time Scale
Slow time scale
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
19 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Boomerang Shape - Energy Graphs: Full
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
20 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Boomerang Shape - Energy Graphs: Fast Time
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
20 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Boomerang Shape - Energy Graphs: Slow Time
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
20 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Non-axisymmetric Torus
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
21 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Coupled Fluid-Membrane Model
Refs: Coutand-Shkoller
(Incompressible) Navier-Stokes Equations
Ω
ρv̇ − ∇ · (−pI + µD(v)) = b
|
{z
}
in Ωt ,
∇·v =0
in Ωt ,
Σ
Γ
Ω in
g
Ω out
ΓD
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
22 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Coupled Fluid-Membrane Model
Refs: Coutand-Shkoller
(Incompressible) Navier-Stokes Equations
Ω
ρv̇ − ∇ · (−pI + µD(v)) = b
|
{z
}
in Ωt ,
∇·v =0
in Ωt ,
[Σ]ν = δJ
on Γt ,
Σ
Γ
Ω in
g
Ω out
ΓD
Membrane Force: Bending
1 3
δJ = ∆Γ h + h − 2kh ν + λh.
2
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
22 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Coupled Fluid-Membrane Model
Refs: Coutand-Shkoller
(Incompressible) Navier-Stokes Equations
Ω
ρv̇ − ∇ · (−pI + µD(v)) = b
|
{z
}
in Ωt ,
∇·v =0
in Ωt ,
[Σ]ν = δJ
on Γt ,
Σ
Γ
Ω in
v=ϑ
on Γt ,
v(·, 0) = v0
in Ω0 .
g
Ω out
ΓD
Membrane Force: Bending
1 3
δJ = ∆Γ h + h − 2kh ν + λh.
2
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
22 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Fluid Biomembrane: 3D Boomerang
3D Banana
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
23 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Fluid Biomembrane: 3D Boomerang Streamlines
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
24 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Fluid Biomembrane: 3D Boomerang Energies
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
25 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Fluid Biomembrane: 3D Boomerang Multiplier
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
25 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Fluid Red Blood Cell: Ellipsoid 5x5x1
Ellipsoid 5x5x1
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
26 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Fluid Red Blood Cell: Ellipsoid 5x5x1 Streamlines
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
27 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometric vs Fluid Red Cell: 5x5x1 Ellipsoid
play
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
28 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometric vs Fluid Red Cell - 5x5x1 Ellipsoid: Energies
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
29 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Fluid Biomembrane: Ellipsoid 4x1x1
Ellipsoid 4x1x1
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
30 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Refinement Consistent Refinement
Problem
Free boundary problem
Γ is major unknown
Increase local resolution
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
31 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Refinement Consistent Refinement
Problem
Free boundary problem
Γ is major unknown
Increase local resolution
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
31 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Refinement Consistent Refinement
Problem
Free boundary problem
Γ is major unknown
Increase local resolution
Question
How to add local resolution with
incomplete geometric information of Γ?
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
31 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Refinement Consistent Refinement
Problem
Free boundary problem
Γ is major unknown
Increase local resolution
Question
How to add local resolution with
incomplete geometric information of Γ?
Answer 1
Linear interpolation
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
31 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Counterxample to Answer 1
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
32 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Counterxample to Answer 1
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
32 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Counterxample to Answer 1
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
32 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Understanding the Counterexample
Perform linear interpolation:
Smooth curve γ and polygonal
approximation Γ
Refine by linear interpolation
Pass a smooth curve γ̃ through
all interpolation points
For 1D curves the FEM theory in flat domain extends:
Z
Z
Z
H · φ = ∂s X · ∂s φ = − ∂s2 x̃ · φ ∀φ ∈ Sh ⇒
Γ
Γ
R.H. Nochetto
H = Ph (∂s2 x̃)
Γ
AFEM for Fluid-Membrane Interaction
Warwick 2009
33 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Understanding the Counterexample
Perform linear interpolation:
Smooth curve γ and polygonal
approximation Γ
Refine by linear interpolation
Pass a smooth curve γ̃ through
all interpolation points
For 1D curves the FEM theory in flat domain extends:
Z
Z
Z
H · φ = ∂s X · ∂s φ = − ∂s2 x̃ · φ ∀φ ∈ Sh ⇒
Γ
Γ
R.H. Nochetto
H = Ph (∂s2 x̃)
Γ
AFEM for Fluid-Membrane Interaction
Warwick 2009
33 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Understanding the Counterexample
Perform linear interpolation:
Smooth curve γ and polygonal
approximation Γ
Refine by linear interpolation
Pass a smooth curve γ̃ through
all interpolation points
For 1D curves the FEM theory in flat domain extends:
Z
Z
Z
H · φ = ∂s X · ∂s φ = − ∂s2 x̃ · φ ∀φ ∈ Sh ⇒
Γ
Γ
R.H. Nochetto
H = Ph (∂s2 x̃)
Γ
AFEM for Fluid-Membrane Interaction
Warwick 2009
33 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Exact Interpolation in 2D
Assume we know γ and interpolate it exactly
γ smooth curve (surface)
Γ polyhedral approximation of γ
Refine locally γ
Refine Γ by bisection
Project new node to γ
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
34 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Exact Interpolation in 2D
Assume we know γ and interpolate it exactly
γ smooth curve (surface)
Γ polyhedral approximation of γ
Refine locally γ
Refine Γ by bisection
Project new node to γ
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
34 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Exact Interpolation in 2D
Assume we know γ and interpolate it exactly
γ smooth curve (surface)
Γ polyhedral approximation of γ
Refine locally γ
Refine Γ by bisection
Project new node to γ
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
34 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Exact Interpolation in 2D
Assume we know γ and interpolate it exactly
γ smooth curve (surface)
Γ polyhedral approximation of γ
Refine locally γ
Refine Γ by bisection
Project new node to γ
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
34 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Exact Interpolation in 2D
Assume we know γ and interpolate it exactly
γ smooth curve (surface)
Γ polyhedral approximation of γ
Refine locally γ
Refine Γ by bisection
Project new node to γ
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
34 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Exact Interpolation in 2D
Assume we know γ and interpolate it exactly
γ smooth curve (surface)
Γ polyhedral approximation of γ
Refine locally γ
Refine Γ by bisection
Project new node to γ
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
34 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Exact Interpolation in 3D
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
35 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometrically Consistent Algorithm
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
36 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometrically Consistent Algorithm
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
36 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometrically Consistent Algorithm
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
36 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometrically Consistent Algorithm
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
36 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometric Consistency
Geometric identity h = −∆γ x
Discrete geometric identity H = −∆Γ X
Assume Γ, X, H approximate γ, x, h
It may be impossible to satisfy the discrete geometric identity,
Geometric inconsistency
Numerical artifacts
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
37 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometric Consistency
Geometric identity h = −∆γ x
Discrete geometric identity H = −∆Γ X
Assume Γ, X, H approximate γ, x, h
It may be impossible to satisfy the discrete geometric identity,
Geometric inconsistency
Numerical artifacts
Geometric Consistency
A finite element triple (Γ, X, H) is GC if
Z
Z
X, H ∈ V :
H · Φ = ∇Γ X : ∇Γ Φ,
Γ
∀Φ ∈ V,
Γ
and it is an approximation of the exact triplet (γ, x, h)
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
37 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometrically Consistency Refinement
Refinement Algorithm
(Γ∗ , X∗ , H∗ ) = Surf Ref(Γ, H, X, M)
1
Γ∗ = Isoparametric Refinement(Γ)
2
H∗ = Interpolation(H)
3
X∗ = Inverse Laplace(H)
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
38 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometrically Consistency Refinement
Refinement Algorithm
(Γ∗ , X∗ , H∗ ) = Surf Ref(Γ, H, X, M)
1
Γ∗ = Isoparametric Refinement(Γ)
2
H∗ = Interpolation(H)
3
X∗ = Inverse Laplace(H)
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
38 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometrically Consistency Refinement
Refinement Algorithm
(Γ∗ , X∗ , H∗ ) = Surf Ref(Γ, H, X, M)
1
Γ∗ = Isoparametric Refinement(Γ)
2
H∗ = Interpolation(H)
3
X∗ = Inverse Laplace(H)
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
38 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometrically Consistency Refinement
Refinement Algorithm
(Γ∗ , X∗ , H∗ ) = Surf Ref(Γ, H, X, M)
1
Γ∗ = Isoparametric Refinement(Γ)
2
H∗ = Interpolation(H)
3
X∗ = Inverse Laplace(H)
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
38 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometrically Consistency Refinement
Refinement Algorithm
(Γ∗ , X∗ , H∗ ) = Surf Ref(Γ, H, X, M)
1
Γ∗ = Isoparametric Refinement(Γ)
2
H∗ = Interpolation(H)
3
X∗ = Inverse Laplace(H)
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
38 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometrically Consistency Refinement
Refinement Algorithm
(Γ∗ , X∗ , H∗ ) = Surf Ref(Γ, H, X, M)
1
Γ∗ = Isoparametric Refinement(Γ)
2
H∗ = Interpolation(H)
3
X∗ = Inverse Laplace(H)
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
38 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometrically Consistency Refinement
Refinement Algorithm
(Γ∗ , X∗ , H∗ ) = Surf Ref(Γ, H, X, M)
1
Γ∗ = Isoparametric Refinement(Γ)
2
H∗ = Interpolation(H)
3
X∗ = Inverse Laplace(H)
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
38 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometrically Consistency Refinement
Refinement Algorithm
(Γ∗ , X∗ , H∗ ) = Surf Ref(Γ, H, X, M)
1
Γ∗ = Isoparametric Refinement(Γ)
2
H∗ = Interpolation(H)
3
X∗ = Inverse Laplace(H)
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
38 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Geometrically Consistency Refinement
Refinement Algorithm
(Γ∗ , X∗ , H∗ ) = Surf Ref(Γ, H, X, M)
1
Γ∗ = Isoparametric Refinement(Γ)
2
H∗ = Interpolation(H)
3
X∗ = Inverse Laplace(H)
Remarks
Procedure independent of polynomial degree and dimension
Refinement can be replaced by coarsening and mesh smoothing
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
38 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Mathematical Statement
In heuristic terms
1. This refinement guarantees that the errors for position and mean
curvature are of the same order as they were before.
2. Unstable numerical differentiation is replaced by stable interpolation
plus inversion of −∆Γ .
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
39 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Mathematical Statement
In heuristic terms
1. This refinement guarantees that the errors for position and mean
curvature are of the same order as they were before.
2. Unstable numerical differentiation is replaced by stable interpolation
plus inversion of −∆Γ .
Theorem (Geometrically Consistent Refinement)
If the triple (Γ, X, H) is GC and E is a Strang-type upper bound for the
error |x − X|H 1 (Γ) , then the following statements are valid
1
kh − H∗ kL2 (Γ) = kh − HkL2 (Γ) ;
2
|x − X∗ |H 1 (Γ) ≤ E;
3
the triple (Γ∗ , X∗ , H∗ ) is GC.
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
39 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Conclusions
Spherical caps: for shapes with distinctive ends, spherical caps seem
to be most effective to reduce the bending energy
Red cells: for disk-like shapes, there is a thickening of the outer edge
and depression in the center. The fluid membrane dynamics is quite
different from the gradient flow.
Kinetic energy: is decays exponentially for gradient flows (with a
nonobvious dependence of the equilibrium shape), but it oscillates for
fluid membranes due to inertia.
Geometric consistency: this is important for refinement, coarsening
and mesh smoothing to avoid numerical artifacts.
Mesh smoothing: control of mesh distortion due large domain
deformations in a Lagrangian approach.
Time-step control: this accounts for geometry and highly varying time
scales.
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
40 / 43
Models
Gradient (Helfrich) Flows
Fluid-Membrane Interaction
Geometrically Consistent Refinement
Conclusions
Large Deformation: Willmore Flow of Helix
Large Simulation
R.H. Nochetto
AFEM for Fluid-Membrane Interaction
Warwick 2009
41 / 43