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.

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