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JOURNAL OF COMPUTATIONAL PHYSICS 127, 179195 (1996)ARTICLE NO. 0167

A Variational Level Set Approach to Multiphase Motion*

HONG-KAI ZHAO, T. CHAN, B. MERRIMAN, AND S. OSHER

Mathematics Department, UCLA, Los Angeles, California 90095-1555

Received July 31, 1995; revised February 22, 1996

The point at which they meet (the triple junctionprescribed angles which can be shown [12] to be definA coupled level set method for the motion of multiple junctions

(of, e.g., solid, liquid, and grain boundaries), which follows the

gradient flow for an energy functional consisting of surface tension

(proportional to length) and bulk energies (proportional to area), is sin 1f23

sin 2

f31

sin 3f12

, developed. Theapproach combines the level set method of S. Osher

and J. A. Sethian with a theoretical variational formulation of the

motion by F. Reitich and H. M. Soner. The resulting method uses

as many level set functions as there are regions and the energy where i is the angle between the two curves ij anfunctional is evaluated entirely in terms of level set functions. The j j; see Fig. 1.gradient projection method leads to a coupled system of perturbed This problem was defined and analyzed clearly in a(by curvature terms) HamiltonJacobi equations. The coupling is

by Reitich and Soner [12], and we base our approaenforced using a single Lagrange multiplier associated with a con-part on their theoretical framework. Their methodstraint which essentially prevents (a) regions from overlapping and

(b) the development of a vacuum. The numerical implementation not lend itself to a direct numerical treatment.is relatively simple and the results agree with (and go beyond) Our objective here is to develop and implement nuthe theory as given in [12]. Other applications of this metho- cal algorithms which capture rather than trackdology, including the decomposition of a domain into subregions

interfaces, based on the level set method of Oshewith minimal interface length, are discussed. Finally, some newSethian [9]. The usual advantages of the level set mtechniques and results in level set methodology are pre-

sented. 1996 Academic Press, Inc. hold (see, e.g., [2, 8, 9, 16]). In the case of a single inteseparating two phases the central idea is to folloevolution of a function , whose zero-level set corres

1. INTRODUCTION to the position of the moving interface. The methomits cusps, corners, and topological changes.

In this article we shall develop an algorithm for the Since its inception, the method has been used to commotion of multiple junctions which is associated with an

and analyze an array of mathematical and physicaenergy functional involving the length of each interface

nomena. See, e.g., [8] and the references theorem.and the area of each subregion. (Three-dimensional ana-

In earlier work [6], Merriman, Bence, and Osherlogues are also easy to implementthe word area re-

extended the level set method to compute the motplaces length and volume replaces area in the

multiple junctions. Also in that paper, and in [5, 7], a sabove). Examples of such motion include solid, liquid,

method based on the diffusion of characteristic fungrain, or multiphase boundaries. These internal interfaces

of each set i , followed by a certain reassignment

are generally out of equilibrium and the resulting motion was shown to be appropriate for the motion of muis driven by decreasing energy. junctions in which the bulk energies are zero (and, hThe simplest model involves three curves meeting at a

the constants ei 0, i 1, ..., n) and the fij are all eqpoint as shown in Fig. 1. Each interface ijseparates regions the same positive constant, i.e., pure mean curvaturei and j and the normal velocity is a positive multiple of More general motion involving somewhat arbitrarythe curvature of the interface plus the difference of the

tions of curvature, perhaps different for each interfacbulk energies.

proposed in [6] as well. This was implemented basby decoupling the motions and then using a reassign

Normal velocity ofij vij fijij (ei ej). (1.1) step. Again each region has its own private level settion. This function moves each level set with a novelocity depending on the proximity to the nearest * Research supported by ARPA/ONR-N00014-92-J-1890, NSF

DMS94-04942, and ARO DAAH04-95-1-0155. face, thus vacuum and overlapping regions general

179


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180 ZHAO ET AL.

The format of our paper is as follows. In the next sewe give the level set formulation for the dynamics motion and discuss other applications such as the opdecomposition of domains. In Section 3 we give the dof the numerical implementation. In Section 4 we prthe numerical results. Finally, in the Appendix we dsome useful new results and techniques concerninlevel set methodology.

FIG. 1. The interfaces ij with normal velocity vij fijij ei ejand angle i . 2. THE LEVEL SET FORMULATION

For a given open region with smooth boundavelop. Then a simple reassignment step is used, removing assume the existence of a level set function (x, y), all the vacuum and overlap. For details see [6]. In that is Lipschitz continuous, satisfyingpaper there was no restriction to gradient flows. However,the general method in [6] lacks (so far) a clean theoretical (x,y) 0 for (x,y)basis to guide the design of numerical algorithms. We rec-

(x,y) 0 for (x,y) tify this with the present method.Here we follow Reitich and Soners variational formula- (x,y) 0 for (x,y)c.

tion in [12]. Given a disjoint family i of regions in R2

with the common boundary between i and j denoted Then we have the simple facts,by ij, we associate to this geometry an energy functionof the form

length () ((x,y)) (x,y) dx dyE E1 E2

area () H((x,y)) dx dy, E1

1ijn

fij length (ij) (1.3)curvature of any level set ofat a point (x,y)

E2 1in

ei area (i) ( / ),

where H(x) is the Heaviside function

where E1 is the energy of the interface (surface tension),E2 is bulk energy, and n is the number of phases. Thegradient flow induces motion such that the normal velocity

H(x) 1, x 0

0, x 0of each interface is defined in (1.1). At triple points (whichcan be seen geometrically by the triangle inequality to bethe only stable junctions if all the fij 0), the angles are and (x) is the Dirac delta functiondetermined by (1.2) throughout the motion. (It is interest-ing that our numerical method, defined in the following

(x) d

dxH(x) (in the sense of distributions)sections, does this automatically. The speed of propagation

of the angle into the equilibrium of this configuration isinfinite.) Reitich and Soner make both these state-

Of course, our numerical simulations involve slmentswe provide the details for (1.1) in the next section,

regularized versions of (x) and H(x)see (3.4) beand the derivation of (1.2) is rather straightforward and Here, and throughout this paper, we definewill not be given here.

To summarize, there are two main points to this paper: 2x 2y.

(1) We develop an efficient and versatile computa-tional algorithm for the theoretical variational problem The statement in (2.2b) is obvious, while that in posed by Reitich and Soner in [12]. was proven in [2], for example.

Given n different regions (phases), which are mov(2) We provide a theoretical basis, as a descent solutiontime, we associate to each regions i(x, y, t) a levto a variational problem, for the time split level set methodfunction i(x, y, t) and defineproposed by Merriman, Bence, and Osher in [6], which in

turn allows us to develop a superior version of that originalad hoc algorithm. E E1 E2


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MULTIPHASE MOTION

i.e., one which is unsuitable for use with Lagrange muers because the gradient of the constraint function vaidentically on the constraint manifold. See the discuafter Remark 2.6. Instead we shall enforce

( H(i(x,y, t)) 1)22

dx dy

for 0, as small as we can manage numericallyfind that corresponds to the area of one grid cell foFIG. 2. Regions of vacuum or overlap.computations. Throughout this paper, 0 will dvarious small positive numbers).

In the first version of this paper we proposed a reaE E1 E2 (2.3b) ment step, similar to that in [6], in which at the e

each calculation we remove the very small vacuumE1 n

i1

i (i(xi ,y, t)) i(x,y, t) dx dy (2.3c) (perhaps) overlap region. We have since found thais unnecessary numerically for situations in which ai 0. If some of the i 0, we set them to the E2 n

i1

ei H(i(x,y, t)) dx dy, Cx, for C an O(1) constant and x grid sizenumerical evidence is that this regularization with ishing viscosity removes the need for reassignmenwheretypically have a one point vacuum region, which lea O(x)2 in (2.6). We justify the need for this exfij i j, 1 i j n. (2.3d)use of vanishing viscosity in the Appendix. We remarthat this is quite unlike the single level set case develoIn the (most interesting) case when n 3 we can solve[9] and discussed in many succeeding papers, wheruniquely for the i . For n 3, (2.3d) restricts our class ofinviscid limit comes automatically through the finite dadmissible surface energiesfij. We shall discuss this relationence approximation of the convection term. See thfurther below in (2.7) and (2.8) and in the remarks whichpendix for a further discussion of this.follow those equations. (We note here that, by allowing

the i to depend on all of the j, we can handle the cases Remark 2.1. The formulations (2.3) and (1.3) cn 4. This will be discussed in future work.) It is clear extended to the case where the ei , i , and fij are funfrom (2.2) that (1.3) and (2.3) are equivalent. Now our of the space variables.problem becomes:

Using the angle relation (1.2) at a triple point, wMinimize E subject to the constraint thatset (normalizing sin 1/f23 1):

ni1

H(i(x,y)) 1 0. (2.4)2 3 sin 1

3 1 sin 2This infinite set of constraints prevents the development

1 2 sin 3 , of overlapping regions and/or vacuum. It requires that thelevel curves (x, y) i(x, y, t) 0 match perfectly. See

leading us toFig. 2.The implementation of (2.3) with the infinite set of con-straints (2.4) is computationally demanding. Instead we try

1 sin 3 sin 2 sin 1

2

sin 3(1 cos 2)

2to replace the constraint (2.4) by a single constraint

sin 2(1 cos 3)

2 ( H(i(x,y, t)) 1)2

2dx dy 0. (2.5)

2 sin 1 sin 3 sin 2

2

sin 1(1 cos 3)

2We shall show below (and it is intuitively clear) that thisis not legitimateone cannot replace an arbitrary numberof constraints by the single constraint obtained by summing

sin 3(1 cos 1)

2their squares. Doing that results in a degenerate constraint,


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182 ZHAO ET AL.

3 sin 2 sin 1 sin 3

2

sin 2(1 cos 1)

2

(2.8c)

ei(i) nj1

H(j) 1 (i) d

sin 1(1 cos 1)

2.

D

(i)

i

i

nds

Thus thei are all positive iff all the angles i are between

D

(i)

i

i

i

ei

0 and 180. The importance of this is seen in the evolutionequation (2.12) we shall derive below.

We now state a first-order necessary condition for our ni1

H(i) 1 dx dyminimization problem.

THEOREM 2.1. The solutions to the minimization

D

(i)

i

i

ndx.

problem:

M: minimize E1 E2 (of2.3) in a fixed domain D subject This expression must vanish for all (x, y). Thto the integral constraint (2.6), obtain (2.9).

We wish to solve this constrained optimization prosatisfy, for i 1, ..., n

by using the gradient projection method of Rosen

where we parametrize the descent direction by timrescale, replacing the common factor (i) by i(i) i i i ei

n

i1

H(i) 1 0 (2.9a) time rescaling does not affect the steady state solutioit does remove stiffness near the zero level sets ofithe speed of descent, not its direction, is affected. Wwith boundary conditionsthe following system of nonlinear evolution equatiothe minimization:

(i)

i

i

n 0 on D, (2.9b)

i

t i i

i

i ei

(where is a Lagrange multiplier.

Proof. Using the Lagrange multiplier, the solution

n

j1 H(j)

1

in D for i

1, 2 ..., nminimizes the functional

with the boundary conditions

f(1 , ..., n) D

ni1

i(i(x,y, t)) i(x,y, t) in

0 on D. (2

ni1

eiH(i(x,y, t)) (2.10) The Lagrange multiplier is updated using Rosenwhich essentially requires that the is, determin(2.12), satisfy the constaint (2.5):

2ni1

H(i(x,y, t)) 12 dx dy.

1

2

d

dt D

n

i1 H(i(x,y, t)) 12

dx dy 0The Frechet derivative of f with respect to i in the (x,y) direction is (2.11)

ni1

D

nj1

H(j) 1 (i)i

tdx dy

fi

, Di (i) i (i)

i

i n

i1

D

nj1

H(j) 1 (i) i

ei(i) nj1

H(j) 1 (i)dx dy i i i ei n

j1

H(j) 1 dx d

D

i

(i) i

(i)

i

i

So we get


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MULTIPHASE MOTION

this acts like surface tension or tangential diffusionthe constant terms try to move the curve normal touniformly. The multiple junction interaction comesthe terms which include the Lagrange multiplier.

Remark 2.3. This motion was analyzed (using a

ni1

D

(i) i i i

i ei

nj1

H(j) 1 dx dy

ni1

D

(i) i nj1

H(j) 12

dx dy

.

(2.14)

stract variational different formulation in terms of cand areas, not level sets, and only for the n 3 cas

Reitich and Soner [12]. If each of the i 0, therunique viscosity solution. Obviously, negative i wiWe need to show that the rescaling works in the follow-disastrous instabilities. If i 0 they demonstrateing sense.uniqueness of this problem, very much in the spinonuniqueness of solutions to scalar HamiltonJLEMMA 2.1. E/t 0, given (2.12) and (2.14).equations without the viscosity solution regulariz

Proof. We use the identity Also in that spirit, they prove uniqueness of the in(i 0) case by letting i be the coefficients with We shall demonstrate numerically that our metho

E

t

Dni1

i (i)t(i) i (i) picks out this unique limit solution, unlike what waposed in [17]. Again this is in the spirit of Sethians en

condition [14] as formulated in [9] for the motioni i

( i)t ei(i)t(i) single front. However, the numerical implementatdifferent from the single front case. Reitich and Soanalysis does not easily lend itself to a numerical imple

ni1

(i)t (i) ei i i

i(2.15) tation. See the Appendix for a further discussion o

i 0 case.

Remark 2.4. Note that, as expected, the equatio ni1

(i) i i i

i ei

2

motion are independent of the choice of level set fun. In particular, if h() is an increasing functionh(0) 0, then the system (2.12), (2.14) is invaria n

j1

H(j) 1 i i

i ei .

h(). This reflects the fact that only leve

matter, not the point values of the representing funWe recognize that every integral above is merely a line

Remark 2.5. If we can set i di at t 0, wherintegral, along the zero level set of the corresponding i ,the signed distance to the boundary (i.e., to the c

of the quantity following i above. The result followspoint on the boundary) then, at least initially, we h

from (2.14) and Schwarz inequalitysee Remark 2.6 be-low for a related, more general argument.

Remark 2.2. The geometric interpretation of the in-i

t i i ei n

i1

H(i) 1duced motion is as follows:

Each level set of each function i moves normal to itselfwith normal velocity:

ni1 D (i)(i i ei)(n

j1 H(j) 1) dx dy

ni1 D (i)(n

j1 H(j) 1)2 dx dy

(2(vi)n i (total curvature of level set) ei

(total amount of overlap among all regions (2.16)In [16] a simple reinitialization procedure was giv

amount of vacuum between all regions). replace each i by di at the beginning of each discretizWe shall use that reinitialization here, describe it next section, and discuss related matters in the AppOf course, we are only interested in the zero level set,This reinitialization corresponds to adding a set ofwhich must coincide with all interfaces ij, j i, if thestraints to the constrained minimization problem in method is to work. The only coupling of this curvaturerem 2.1 of the form:regularized system of HamiltonJacobi equations comes

through the single constraint.The curvature terms tend to straighten out the curves

i 1, i 1, ..., n


6/17

184 ZHAO ET AL.

which is an alternative way of specifying i di . This Minimizeremoves the nonuniqueness of in Remark 2.4 and doesnot change the interface motion.

E ni1

(i(x,y)) i(x,y) (x,y) dx dy, (Remark 2.6. The gradient-projection method minimiz-

ing f((x)) dx, such that g((x)) dx 0, using t Subject totime as the descent variable, leads to

12

n

i1

H(i(x,y)) 1)2 dx dy (2t f g, (2.19a)

H(i(x,y))(x,y) dx dy Ai (i 1, ..., n where (

Using the gradient projection method (after rescleads us to

gf g2

(2.19b)

i

t i i

i

i i n

j1

H(j) 1

with g() 0 at t 0. Thus, Schwarz inequality implies (that f((x, t)) dx is decreasing as long as g 0, since

with n 0, where the Lagrange multipliers , i sa linear system,

d

dtf() f2 ( fg)

2

g2.

(mij)(n)(n)

.

..

n1

b1

.

..

bn , (2

Thus, the constraint term must not be degenerate, i.e.,g 0 if g() 0. This is one of the many equivalentreasons why satisfying the constraint (2.5) is too much to wherehope for; yet (2.6) can be obtained.

m11 n

i1 (i) i (n

j1 H(j) 1)2 dx dy

Remark 2.7. In our calculations we have replaced theboundary condition (2.17b) by nonreflecting boundary mii (i) i1 2 dx dy, i 2conditions approximating (x)2(2/n2) 0 at the bound-

mi1 m1i (i1) i1ary. This minimizes the effects of the boundary.

(n

j1 H(j) 1) dx dy, i 2Remark 2.8. We have begun experimenting with theoriginal descent method, i.e., where (i) is used rather mij 0, othethan i in (2.12a), (2.12b) and in the analogue of (2.14).Surprisingly, preliminary results are excellent. We shall The matrix (mij) is symmetric positive definite bediscuss this in future work. the constraints are independent (a general conseque

the gradient-projection method), andThe framework we have set up here is quite general.

We can easily add more constraints and change the energyb1 n

i1

(i) ifunctional. For example, in a domain decomposition frame-work, we may be given a density function (x, y) 0 forthe density of node points in each subdomain and (x, i i i

n

j1

H(j) 1 dx dy

(y) 0 for the communication cost per unit length of adecomposition cut. The optimal domain decomposition

bi (i1) i1 will have a required number of nodes in each subdomainand will have the minimum communication cost acrossinterfaces. In two dimensions the problem can be formu-

i1

i1

i1

dx dy, i 2, ..., n,

lated as (see [18])


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MULTIPHASE MOTION

where where Mi number of grid points where (i)(m)jk B c x (we have taken the constant c 1 in our ements). IfQ (t)(x)2 then it is stationary and weelse we go back to step 1.i

i

i local curvature of the interface.

This procedure involves a highly nonlinear partial d(2.21d)ential equation restricted to a manifold. There arenontrivial numerical details which we now address.

3. NUMERICAL IMPLEMENTATION FOR

MULTIPHASE MOTION Step 1. We approximate the Heaviside and deltations by C2 (respectively C1) functions as in [11, 16

The numerical implementation of (2.12), (2.13) is simple,but requires much of the modern level set technology. Thealgorithm can be summarized in four steps:

Step 1. Update the Lagrange multiplier by (2.14) H(x) 1, x

0, x

1

21

x

1

sin

x

, x ;

(x) ddx

H(x)

0, x

1

21 cos x , x (m1)

ni1

D

((m)i ) (m)i i

(m)i

(m)i ei

n

j1

H((m)j ) 1 dx dyni1

D

((m)i ) (m)i n

j1

H((m)j ) 12

dx dy

.

(3.1)In our calculations we took x.In (3.1), () ( / ) is the average of

Step 2. Advance i by the evolution PDE (2.12). A curvature at the front. Since the width of the suppsimple time stepping algorithm is our approximate delta function is positive, we wou

some average of curvature of level sets near the frwe were to literally use this formula. We get better r(m1)i

(m)i

t (m)i

ei

(m1)

n

j1H(

(m)j ) 1)

by using

(3.2)

1 .

i (m)i

(m)i

(m)i .

This appears, e.g., in [4]; we discuss it in the AppeWe generally use more accurate and robust RungeKutta- This (nonobvious) formula, when is the distance funtype time discretizations; see, e.g., (3.9). Also, recall that is constant normal to the front and, of course, givein (3.1) and (3.2), if any i 0 we replace it by i xC, correct value at the front.with C O(1). Standard central difference formulae are used for

Step 3. Let d 0i (m1)i ; reinitialize

(m1)i to be the the remaining terms in (3.1).

signed distance function, using several iterations of the Step 2. We view (3.2) as an approximation to a Hfollowing discretized PDE (see [16]): tonJacobi equation with curvature regularization

formd (m1)i d

(m)i

t sign(d (0)i )( d

(m)i 1) 0. (3.3)

t (a(x,y, t))

Step 4. Check whether the solution is stationary,

for 0. High order ENO (essentially nonoscillaapproximations to equations of this type have beetained in [9, 10] and are needed to avoid oscillatioQ

ni1 (i)(m)jk B (i)(m1)jk (i)

(m)jk

n

i1 Mi,

small. We use the following:


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186 ZHAO ET AL.

(i) if ei (n

j1 H(j) 1) 0 then

jk ((max(Dxjk , 0))2 (min(Dxjk , 0))2) (max(Dyjk ,))2 (min(Dyjk , 0))2) (

(ii) ifei (n

j1 H(j) 1) 0 then

jk ((min(Dxjk , 0))2 (max(Dxjk , 0))2) (min(Dyjk , 0))2 (max(Dyjk , 0))2),

where j, k are the index for the x and y coordinates and The curvature term is approximated by using

Dxjk j,k j1,k

x

x

2m

x

x

y

y

so j1,k 2j,k j1,k(x)2

,j,k 2j1,k j2,k

(x)2

jk

x

j1/2,k

x

j1/2,k Dxjk j1,k j,kx

x

2

m

y j,k1/2 y j,k1/ 2 y, j2,k 2j1,k j,k(x)2 , j1,k 2j,k j1,k(x)2 where

x j1/2,k (j1,k j,k)/ x [(j1,k j,k)/ x]2 [(j,k1 j,k1)/ 2 y (j1,k1 j1,k1)/2 y] 2 x j1/2,k (j,k j1,k)/ x [(j

,k j

1,k)/ x]

2

[(j

1,k

1 j

1,k

1)/ 2 y (j

,k

1 j

,k

1)/2 y]

2

y j,k1/ 2 (j,k1 j,k)/ y [(j1,k j1,k)/2 x (j1,k1 j1,k1)/ 2 x] 2 [(j,k1 j,k)/ y]2 y j,k1/ 2 (j,k j,k1)/ y [(j1,k1 j1,k1)/2 x (j1,k j1,k)/ 2 x] 2 [(j,k j,k1)/ y]2 .

To obtain a high order accurate scheme in timDyjk

j,k j,k1

y

y

2m replace the forward Euler-like discretization of (3

using a semi-discrete approximation

j,k1 2j,k j,k1

(y)2 ,

j,k 2j,k1 j,k2

(y)2 tjk L[,j, k]Dyjk

j,k1 j,k

y

y

2m

and certain RungeKutta-type schemes (see ShuOsher [15]). For example, a second-order essentially j,k2 2j,k1 j,k(y)2 , j,k1 2j,k j,k1(y)2 oscillatory RungeKutta algorithm is Heuns metho

m1jk mjk tL[

m,j, k]m[x,y]

x if x y ,x y 0

y if x y ,x y 0.

0 ifx y 0. m1jk

1

2mjk

1

2m1jk

t

2L[m1i ,j, k]


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MULTIPHASE MOTION

which has a slightly reduced CFL (time step) restriction Remark 3.5. A fundamental idea, whose time hasin problems like ours, is to use the level set formufrom the underlying monotone scheme.

At the boundary of our computational domain, we use only near the zero level sets themselves, thus cuttingthe numerical work by an order of magnitude. Thithe nonreflecting boundary condition 2/n2 0.discussed by Adalsteinsson and Sethian in [1], aStep 3. The original level set formulation [9] did notmethod was proposed and successfully implented frequire anything special about the nature of , other thansingle level set case. We have in [18] developed a somthat it be sufficiently smooth. In a number of works [16,simpler method which was used here in this multip

2, 6], it was found to be quite desirable that be constantly case and which was applied to three-dimensional proupdated to be a signed distance, at least near the front. Ain [18]. Typical calculations in the next section invofast reinitialization algorithm was given in [16]; we repeatthree phases, 5 103 iterations and a 100 100 gridit here. In our setting, the singular distributions () and2 h on our SPARC 10 machine. This is a speedupH() are involved in the motion (as was true in [16, 2])least a factor of 5 over the straightforward, global meand this reinitialization step is quite important.More generally, this results in an O(N) speedup foIn (3.3) we approximate the sign function bytime of the method applied to an N N grid.

sign(d 0i ) d (0)i

(d (0)i )2

(x)2. 4. NUMERICAL RESULTS

In all our numerical experiments we use

This equation is of the type (3.6) with

0 anda

(x

,y

,t) sign(d 0i ). Thus the numerical procedure of (3.7), (3.8) D [0, 1] [0, 1], x y 102, t 10is used (with the same boundary conditions, using the ap-propriate a(x, y, t)). In the first experiment, we study the motion of the

The last technical detail involves the constraint near faces under constant velocity caused only by the diffemultiple junctions. Since the numerical Heaviside function of bulk energy (vij ei ej). As is shown by Reitichas the property that H(0) , we require that the con- Soner in [12], there is no unique solution in this casstraint satisfy the VST solution (letting i 0) picks up the u

solution which satisfies their weak angle conditions triple point. We start with a case described in [12]:n

i1

H(i(x)) n

2 0 (3.10)

straight lines meeting at 120 (see the right side of3a,b,c). We set e1 e2 5, e3 1. In our first exper

for i(x) sufficiently small for each i. This is done only displayed in Fig. 3a we use a 100 100 grid and setfor the first few time iterations. Since a triple point is stable, 2 3 0.01. In Fig. 3b we set 1 2 3 we replace n/2 by 3/2 after a few iterations, again, only in and a 150 150 grid. Finally, in Fig. 3c we use 1 regions in which the i(x) are small. This affects only the 3 0.0025 and a 200 200 grid. In all cases the leftvalues of (m1). of these figures agree with the VST solution of [12]

In Fig. 3d we set e1 2.5, e2 1, e3 0.5, 1 Remark 3.1. Using this numerical implementation we3 0.005 on a 200 200 grid. then each interface mfind a very small vacuum region (and almost no overlap)with a different velocity. The 120 relation is maint

and no growth of vacuum or overlap in time.at the triple point which means in this case that wethe VST solution.Remark 3.2. We replaced each i by i , and let

In the second numerical experiment, we start wit0 to see if our numerical scheme picks out the unique VST

4a. In Fig. 4b we set(vanishing surface tension; see [12]) solution computedwith all the i 0. It turns out that it does, but we doneed the numerical lower bound i Cx. 1 2 0, 3 1, e1 e2 e3 0,

Remark 3.3. An implicit in time scheme would be use-which makes 3 180 by the angle relation (2.7) ful to remove the t (max i)(x)

2 parabolic stabilitytriple point. We have zero bulk energy. Thus, we

restriction. However, it should be noted that the nonlinearminimize the length of the boundary of the third do

stiff terms involving H(j) also restrict the time step. Thus At t 0.05, we get Fig. 4(b). Since the starting we have not yet made this method implicit in time.

already has 3 180 (the boundary of the third dois a straight line), we would expect no motion at all Remark 3.4. The angle condition for a triple point,

(1.2), is obtained instantaneously (as predicted), i.e., after triple point. This agrees with our result. This also that our numerical scheme has little artificial dissipaa very small number of iterations.


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188 ZHAO ET AL.

FIG. 3. (a) 1 2 3 0.01; (b) 1 2 3 0.005; (c) 1 2 3 0.0025; (d) 1 2 3 0.005.

In Fig. 4c we set In the most general case, we have both differentenergies and also surface tension between each phaFig. 4d, we use1 ( 3 1)/4, 2 ( 3 1)/4, 3 (3 3)/4,

e1 e2 e3 0.1 ( 3 1)/4, 2 ( 3 1)/4, 3 (3 3

e1 10, e2 5, e3 1.We would expect 1 /2, 2 5/6, 3 2/3 at thetriple point and no convection due to the bulk energy. Thisis shown in Fig. 4c, at t 0.05. So we not only have the same angle condition as in


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MULTIPHASE MOTION

FIG. 4. (a), (b) 1 2 0, 3 1; (c), (d) 1 ( 3 1)/4, 2 ( 3 1)/4, 3 (3 3)/4.

(b), but we also see the convection of the triple point by in this case. Thus, we again recommend using O(x) v

ity in these calculations to be safe.the difference in bulk energy (e1 e2 e3). The area ofthe first domain is shrinking and the area of the second In the next experiment, we see how a multiple jun

evolves into several triple points. We start with Fidomain is growing.In the third experiment, we deal with the case where with all i 1, ei 0. As time goes on, we get Fig. 6

Fig. 6c and Fig. 6d.two interfaces merge and topological change occurs. Westart with two interfaces with sharp corners, Fig. 5a at Figure 7 shows the effect of the curvature term

evolution procedure. We see, for e1 3, e2 5, andt 0. We set e1 1, e2 10, e3 5, and 1 2 3 0. At t 0.02 we get Fig. 5b and at t 0.05 we get Fig. the results with differing s, the same for each inte

Figure 7a shows the result with 1 for a 100 1005(c). We see that this level set approach treats topologicalchanges very easily. The curvature straightens out the initial bump in

yields approximately 120 angles. Figure 7b shows tInFig. 5dwe set1 2 3 0.01. With small viscosity,we get almost the same result as without viscosity at all, sults for 0.1 on the same grid. The bump is less st


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190 ZHAO ET AL.

FIG. 5. (a)(c) 1 2 3 0; (d) 1 2 3 0.01.

and the 120 angle is valid only in a boundary layer near front, i.e., that (x, t) 1. This does not remainin general and is very important, especially when vthe triple point. Figure 7c shows the results for x

0.01. The bump is still visible, and this is clearly regularized involves singularities at the front. Here, we shall giveideas on how to maintain as a distance function. (Imotion by a constant. Finally, Fig. 7d has x/5, with

x 0.005. We are stretching the limits of our slight section the results apply in an arbitrary number l, of dimensions. We simply denote a point in Rl by x).regularization and we see that typical development of a

kink in the inviscid motion case.LEMMA A1. Letvn v be the normalveloc

each level set, and set (x, 0) to be the signed disAPPENDIX: SOME NEW LEVEL SET METHODOLOGY

function. Then remains as a signed distance functvn 0.To make each (x, t) unique and well behaved numeri-

cally, we require that it be the signed distance from the Proof.


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MULTIPHASE MOTION

FIG. 6. i 1; ei 0.

(i) Evolve (x, t), starting from a distance funt

2 2 t 2( vn) (A.1)

old, via

2vn( ). t v 0, (

i.e.,Thus, 1 for later time iff the source term vanishes;i.e., vn 0.

new old vn t O(t)2. (

We show that the previous method of reinitialization(ii) Do the reinitialization on new as usual[16], as described in Section 3, is equivalent to modifying

vn(x, t) off the front. The reinitialization procedurecomes from (m1) (m) sign((0)(1

(m)

) O()2 (


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192 ZHAO ET AL.

FIG. 7. (a) 1 2 3 1; (b) 1 2 3 0.1; (c) 1 2 3 0.01; (d) 1 2 3 0.001.

with on the front.

(0) new. (A.2b) Proof. From (A.2)

We have

LEMMA A2. The reinitialization step (A2) is equivalent (0) new old vn t o((t)2).

to extending vn(x, t) off the front so that

vn 0 We next use Taylor expansion and the fact that


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MULTIPHASE MOTION

old 1

(1) (0) sign((0))(1 (0) ) o(t2)

old vn t sign((0))

(1 ( old vn t) ( old vn t)) o(t)2

old vn t sign((0))

1

old

old vn

oldt O(t)2)

o(t)2

FIG. 8. Corner at the front. old vn sign(

old)old vn

old t O((t)2)

by induction,Thus, if 1 initially, it remains so for a later t

(m) old v(m)n o(t)2,

The PDE (A.3b) is highly nonlinear and difficult towhere lyze. The numerical implementation is difficult to car

in the presence of corners, since the corner can b

preimage of a whole section of a curve under thev(m)n v(m1)n sign(old)

old v

(m1)

n old x x . See Fig. 8.

The formula (A.3a) simplies and becomes useful iwhich means, as m , v(m)n (x, t) goes to the steady-state a function of curvature; see e.g., [4].solution ( independent) of

LEMMA A4. If (x, t) is a signed distancesmooth, thenvn(x, t, )

sign((x, t))

(x, t) vn(x, t, )

(x, t) 0.

(x ) (x)

1 (x) (x). We have not yet tried to implement this procedure.

Another appealing idea involves tracing the velocityback to the frontsee, e.g., [4]. This is the content of Proof. Let x x (x) (x) (see Fig. 9),the following.

LEMMA A3. If (x, 0) is the signed distance, then (x,(x)

1

r(x) (x)

(x)

(x) (x)t) remain a signed distance if we use the following procedure:

Set

r(x) (x) 1

(x),

vn(x, t) vn(x ) (v )(p(x)) (A.3a)

and solve so

t vn(x ) 0, (A.3b)

(x) 1

r(x)

(x)

1 (x) (x).

where the point on the curv

e closest to x is

p(x) x . (A.3c)

Proof.

d

dt 2 2 t

2 vn(x , t)

2 [I ( )] vnFIG. 9. Curvature using distance function.

[2(1

2) (

2)] vn .


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194 ZHAO ET AL.

displayed on the right of that figure. The normal veof the curve 12 is 0 while 23 and 13 have normal vevn , say, equal to one. Thus the Huygens principle appdevised in [17] amounts to moving the boundary normal to itself with unit normal velocity. This meanafter time t the boundary of 3 consists of all pointdistance t from the original boundary, while 12 domove. The result can be achieved by obtaining the vissolution of a single level set function 3 , solving

3 3 0

and the numerical methods of [9, 10] will yield the soin [17]see Fig. (10). Thus even though we compuas the (viscosity) limit as 0 ofFIG. 10. Motion with constant velocity at triple point.

t

,Remark A.1. The resulting evolution equations of

the type

the limit solution is the one in [17], not the regulalimit of Reitich and Soner in [12].

t F

1 (A5)

If we turn to our formulation (2.12) and let all th0, i 1, 2, 3, we arrive at the system

for F(x) 0 are well posed in the sense of viscositysolutions (see, e.g., [3]) if the right-hand side of (A5) is asmooth nondecreasing function of . This is true away

1

t 1 3

j1

H(j) 1 (Afrom 1.

In the special case when F is linear and increasing we 2t

2 3j1

H(j) 1 (Ahave used the following implicit method to integrate (A5)(with approximated by standard central differencing)

3

t 3 1

3

j1

H(j) 1 (A(m1) (m)t

F(m1)

1 (m) (m). (A6)

and in theno overlap, no vacuum case, the system is eqWhen we approach the zeros of the denominator (away lent to (A.8) which again gives the result in [17], nofrom the front) the scheme breaks down. Thus we replace The addition of a x penalty for length reverses ththe denominator by

ACKNOWLEDGMENTS1 min(, 1 ) for small 0. (A7)

The authors thank the referees for some important and constcomments on a first draft and Fernando Reitich for carefully desThis leaves the motion of the front unaffected.

the VST limit of three phase motion to us.Remark A2. The level set method corresponding totwo-phase flow as devised in [9] and in many succeeding REFERENCESpapers treats vanishing surface tension numerically by

1. D. Adalsteinsson and J. A. Sethian, CPAM, U.C. Berkeley computing the viscosity solution of the inviscid problemLBL 34893, PAM-598, 1993 (unpublished).with no explicit numerical viscosity used or needed. In

2. Y.-C. Chang, T.-Y. Hou, B. Merriman, and S. Osher, UCLAthis work, in order to obtain the inviscid limit [12] asReport 94-4, 1994; J. Comput. Phys. 124, 449 (1996).

i 0 we need to explicitly add O(x) times the length3. M. G. Crandall, H. Ishii, and P.-L. Lions, Am. Math. Soc. Bin the variational formulation (2.3b). We now show via a

1 (1992).simple example that this is necessary; otherwise we would

4. L. C. Evans and J. Spruck, Trans. Am. Math. Soc. 330, 321 (compute the inviscid case as in [17].

5. P. Mascarenhas, UCLA CAM Report 92-33, 1992 (unpublis

6. B. Merriman,J. Bence, andS. Osher,J. Comput. Phys. 112, 334Consider the situation in Fig. 10. The initial set up is


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MULTIPHASE MOTION

7. B. Merriman, J. Bence, and S. Osher, Diffusion Generated Motion 12. F. Reitich and H. M. Soner, Proc. Royal Soc. Edinburgh, to by Mean Curvature, inAMS Selected Lectures in Math., The Comput. 13. J. G. Rosen, J. SIAM 9, 514 (1961).Crystal Growers Workshop, edited by J. Taylor (Am. Math. Soc.,

14. J. A. Sethian, Commun. Math. Phys. 101, 487 (1985).Providence, RI, 1993), p. 73.

15. C.-W. Shu and S. Osher, J. Comput. Phys. 83, 32 (1989).8. S. Osher, Proc. Int. Congress Mathematicians, Zurich, 1994, pp. 144916 M. Sussman, P. Smereka, and S. Osher, J. Comput. Phys. 11459.

(1994).9. S. Osher and J. A. Sethian, J. Comput. Phys. 79, 12 (1988).

17. J. E. Taylor, J. Differential Equations, 119, 109 (1995).10. S. Osher and C.-W. Shu, SINUM 28, 907 (1991).

11. C. Peskin, J. Comput. Phys. 25, 220 (1977). 18. H.-K. Zhao,J.-P. Shao, S. Osher, T. Chan, and B. Merriman, p

chan-vese paper

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