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The Level Set Method and its Applications
Hongkai Zhao
Department of Mathematics
University of California, Irvine
Outline
• Introduction to the level set method.
• Survey of some of the research and applications.
• Specific topic: solving PDEs on moving interfaces.
• Open discussions.
2
Moving interface problem
The simplest setting: given the motion law of a moving interface
dΓ(t)
dt= v(x, t) or
dΓ(t)
dt= vn(x, t)n
V(X, Γ)
n
Γ
Question: How to represent and track or capture Γ numericaly?
Moreover, v(x, t) or vn(x, t) may depend on:
• ambient velocity (convection)
• geometry of Γ
• global quantity (which depends on Γ)
3
Other approaches
• Particle/mesh (tracking) method:
Parametric (explicit) representation of Γ using particles/triangular
meshes and track the motion by solving a system of ODEs.
+: explicit representation; good efficiency and accuracy.
-: parametrization in higher dimensions; reparametrization and
reconnection for large deformation and topological changes.
• Volume of fluid method:
Implicit representation of Γ using fraction of volumes and track
the volume fraction using conservation form.
+: good conservation property; easy to handle topological changes
-: restricted to conservative type of equation; reconstruction of
interface and computation of geometrical quantities.
4
The level set method (Osher and Sethian, 88)
Step 1: Embed the interface Γ into a level set function φ(x)(implicit representation):
Γ = {x : φ(x) = 0}.The location and geometric quantities of Γ can be extractedfrom φ easily. For examples,
unit normal n =∇φ
|∇φ|, mean curvature κ = ∇ · ∇φ
|∇φ|.
Step 2: Embed the motion of Γ(t):
φ(Γ(t), t) = 0 ⇔ φt + ∇φ · dΓ
dt= 0.
The evolution PDE for φ(x, t) is:
φt + v · ∇φ = 0 or φt + vn|∇φ| = 0
Note: The level set function φ and the velocity field v or vn canbe defined arbitrarily off the zero level set Γ.
5
Morphological interpretation of the level set method
n(x) = ∇φ(x)|∇φ(x)| and κ(x) = ∇ · ∇φ(x)
|∇φ(x)| is the normal and mean
curvature at x of the level set that passes through x.
Φ=Φ(
x
x )
n(x)κ( x)
For examples:
1. φt + v · ∇φ = 0 means every level set of φ is convected
by the velocity field v.
2. φt + (∇ · ∇φ|∇φ|)|∇φ| = 0 means every level set of φ moves
normal to itself by its mean curvature.
6
An example
φt + |∇φ| = 0, φ(x,0) = φ0(x)
Denote p = ∇φ. The Hamiltonian is H(p, x) = |p|. The charac-teristic equation is⎧⎪⎪⎨
⎪⎪⎩p(t) = −∇xH(p, x) = 0x(t) = ∇pH(p, x) = p
|p|φ(t) = ∇pH(p, x) · p − H(p, x) = 0
which can be solved explicitly⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
∇φ(x(t), t) = p(t) = p(0) = ∇φ0(x(0))
x(t) = x(0) + tp(t)|p(t)| = x(0) + t ∇φ(x(t),t)
|∇φ(x(t),t)|φ(x, t) = φ0
(x − t ∇φ(x,t)
|∇φ(x,t)|)
If φ0(x) = |x| − r0 then
φ(x, t) = φ0
(x − t
x
|x|
)= |x| − (t + r0)
So the zero level set for φ(x, t) = 0 is |x| = r0 + t.
7
Mathematical advantages
• A geometric problem becomes a PDE problem. PDE tools,
such as viscosity solution, can be used.
• Singularities and topological changes in Γ can be handled more
easily in φ space.
X
Y
Φ
Φ=0 X
Y
Φ
Φ=0 Φ=0
Φ=0Φ=0
Φ=0
t
t
t 2
3
1
(a) evolution of a curve (b) topological changes
8
Numerical advantages
• Eulerian formulation gives a simple data structure.
The PDE for the level set function is solved on a fixed grid. No
remeshing and surgery is needed for dynamic deformations or
topological changes.
• The formulation is the same in any number of dimensions.
• Efficient numerical algorithms for PDEs are available and can
handle shocks and entropy conditions properly.
Remark: The extra dimension of computation cost can be re-
duced by restricting the computation in a narrow band around
the zero level set.
9
Numerical schemes
Numerical methods for conservation laws and Hamilton-Jacobi
equations play a crucial role.
φt + F(x, φ,∇φ, . . .) = 0
Spatial discretization on rectangular grids:
For hyperbolic terms, such as v · ∇φ, vn|∇φ|: upwind (W)ENO
schemes (Shu, Osher...), Godnov schemes, ...
For parabolic terms, such as ∇ · ∇φ|∇φ|: central difference scheme.
Time discretization:
TVD or TVB Runge-Kutta method.
Spatial discretization on triangulated mesh: Petrov-Galerkin type
of monotone scheme (Barth & Sethian), discontinuous Galerkin
method, ...
10
Reinitialization and extension
• Reinitialization:The desirable level set function is the signed distance function:
|∇φ| = 1, φ(x ∈ Γ) = 0. (1)
Even if |∇φ0| = 1,
|∇φ| = 1, t > 0 iff ∇vn · ∇φ = 0
In general, reinitialization is needed to enforce (1).
• Extension of velocity:
∇vn · ∇φ = 0, vn(x ∈ Γ) is fixedΦ
x
|Φx|=1
Φ=0
Φ>0
Φ<0
V
11
The effect of curvature
Motion by mean curvature is the gradient flow for decreasing |Γ|,which is a regularization that prevents oscillations along Γ and
enforce the entropy condition when singularity develops.
|∇φ|∇· ∇φ
|∇φ| =
⎧⎪⎪⎨⎪⎪⎩
∆φ (if |∇φ| = 1)
∆φ − ∇φ|∇φ|D
2(φ) ∇φ|∇φ| (diffusion along the interface
If numerical viscosity is present, ∼ hα∆φ, curvature effect is
also present, which may cause the decrease of both |Γ| and the
volume enclosed by Γ.
12
Resolution analysis
κ=
κ=
κ= 1r+δ
1r
1r−δ
If vn = vn(κ), e.g. motion by mean curvature, neighboring levelsets of the zero level set evolves with the same law. We have
1
r + δ>
1
r>
1
r + δ,
1
2(
1
r − δ+
1
r + δ) =
r
r2 − δ2>
1
r
Concavity (the inner level set) wins, which causes the loss of area.
To interpolate 1r accurately, the grid size h has to resolve the
finest feature. The error is O(
hrmin
)αfor a method of order α.
13
Static Hamilton-Jacobi equation
Eikonal equation:
|∇u(x)| = f(x) > 0, u(x ∈ Γ0) = 0
u(x) is the first arrival time at x for the wave front starting at
Γ0 with normal velocity 1f(x), i.e.,
{x : u(x) = T} = Γ(T), wheredΓ
dt=
1
f(x)n, Γ(0) = Γ0
or
φt +1
f(x)|∇φ| = 0, {x : φ(x,0) = 0} = Γ0.
14
Fast sweeping method
After upwind differencing following the causality, we have the
following nonlinear system to solve
max{(Dx−ui,j)+, (Dx
+ui,j)−}2+max{(Dy
−ui,j)+, (Dy
+ui,j)−}2 = h2f2
i,j
or
[(ui,j − uxmin)+]2 + [(ui,j − uymin)
+]2 = f2i,jh
2
i = 1,2, . . . , j = 1,2, . . .
where uxmin = min(ui−1,j, ui+1,j), uymin = min(ui,j−1, ui,j+1)
• Fast marching method: following the characteristics sequen-
tially. (Tsitsiklis, Sethian, Sethian & Vladimirsky).
• Fast sweeping method: an iterative method following the char-
acteristics in parallel. (Boue & Dupuis, Zhao, Tsai, et al)
15
Variational level set formulation (Zhao, et al)
• Express the energy functional in terms of the level set function.
volume enclosed by the surface (φ < 0), V =∫Rn
H(−φ)dx
surface area S =∫Rn
δ(φ)|∇φ|dx
• Derive E-L equation/gradient flow for the level set function.
The gradient flow that minimizes the enclosed volume:
φt + δ(−φ) = 0 ⇒ φt + |∇φ| = 0, i.e. vn = −1
The gradient flow that minimizes the surface area:
φt−δ(φ)∇· ∇φ
|∇φ| = 0 ⇒ φt−|∇φ|∇· ∇φ
|∇φ| = 0, i.e. vn = ∇· ∇φ
|∇φ| = κ
16
A level set formulation for two phase flow
by Sussman, Smereka and Osher. Let φ be the level set function
for the moving interface Γ(t) between the two fluids.
• Distributional Navier-Stokes equation:{ρ(ut + (u · ∇)u) = ρg + ∇ · Λ + σκδ(φ)∇φ∇ · u = 0
where Λ = −pI + µi(∇u + ∇uT), (ρ, µ) =
{(ρ1, µ1), φ < 0(ρ2, µ2), φ > 0
n = ∇φ|∇φ|, κ = ∇ · ∇φ
|∇φ|
• The evolution of the interface:
φt + u · ∇φ = 0
Note: The Delta function is numerically smeared over a few
grids.
17
A sharp interface formulation
applied to the Hele-Shaw flow by Hou, Li, Osher and Zhao.
u = −β∇p, ∇u = f
The Poisson equation for the pressure
∇ · (∇p) = −f
with jump conditions at the interface
[p] = σκ, [βpn] = 0
• Solve the pressure equation using the immersed interface method
and the level set function on rectangular grids (LeVeque & Li).
• Evolve the interface:
φt + u · ∇φ = 0
Note: The jump condition is explicitly enforced.
18
Applications of the level set method
• Multiphase fluids
• Materials
• Image processing
• Computer graphics
• Inverse problem
• Shape optimization
• Whereever there is a moving interface and free boundary in
your problem.
19
Some recent development
• Level set formulation for manifolds with higher co-dimensions
(Cheng et al).
• Level set method + Volume of Fluid Method (Pucket and
Sussman).
• Adaptive level set method (Cristini and Lowengrub).
• Particle level set method (Enwright and Fedkiw).
• Solving PDEs on moving interfaces.
20
Open problems
• Method itself.
More rigorous numerical analysis.
Moving mesh for the level set method.
Coupling of tracking method with the level set method.
• Applications.
. . . . . .
21
Two Books
Level Set Methods and Fast Marching Methods (1996, 1999),
by J. Sethian.
Level Set Method and Dynamic Implicit Surfaces (2003), by S.
Osher and R. Fedkiw.
22
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