ARTICLES

Sci. Rep. Kanazawa Univ.
Vol. 62, pp. 1-22, 2018

Hyperbolic mean curvature flow with an obstacle
Vita K USUMASARI∗
Graduate School of Natural Science and Technology, Kanazawa University
Kanazawa, 920-1192, Japan(Received June 26, 2017 and accepted in revised form February 28, 2018)

Abstract We treat an interface motion with an obstacle according to the hyperbolic mean
curvature flow. In order to realize this motion, we follow the approximation method that is
the so-called Hyperbolic MBO (HMBO) algorithm. In this work, we modify the scheme totreat the obstacle problem. Then, we investigate the behaviour when the interface touches
the obstacle. We consider two cases of the interface motion. In the first case, the interface
stops moving and lies on the obstacle after touching it. For the second case, the interface
reflects after touching the obstacle. We also plot the points when the interface contacts withthe obstacle and call it the free boundary shape.
Keywords.

1

hyperbolic mean curvature flow, obstacle problem.

Introduction

In this paper, we consider an interfacial motion problem called the hyperbolic mean curvature flow[3]. We suppose that the interfaces are given by a parametrized family of curves γ : I ×[0, T ) → R2 ,where I = [a, b] and T > 0. Let us consider the region P ⊂ R2 such that the boundary of P coincides
with the interface. Also, the interface is considered as an oriented curve, such that the region P ison the left side of the curve. Then, the problem of the hyperbolic mean curvature flow is to findγ(s,t) satisfying
 2
∂ γ


(s,t) = −κ(s,t)n(s,t), ∂t 2(1.1)γ(s, 0) = γ0 (s), ∂ γ (s, 0) = v (s)n (s),00∂twith s ∈ I and t ∈ [0, T ). Here, κ(s,t) is the curvature, n(s,t) is the unit outer normal vector to thecurve γ at a point (s,t) and the unit outer normal vector at t = 0 is denoted by n0 (s). This geometricevolution says that the normal acceleration of the interface is proportional to its curvature [4] andgiven bya = −κ.
(1.2)∗ Current address:

Department of Mathematics, Faculty of Mathematics and Science, Universitas Negeri Malang,
Jalan Semarang No. 5 Malang, Jawa Timur, Indonesia. Email: vitakusumasari@gmail.com

2Hyperbolic mean curvature flow with an obstacle

In order to realize the interface motion according to (1.1), we follow the approximation method
introduced by Ginder and Svadlenka [2, 3] that is the so-called Hyperbolic MBO (HMBO) algorithm. In those papers, the authors construct the numerical scheme related to the MBO algorithm.
The MBO algorithm was developed by Merriman, Bence, and Osher [1]. This algorithm applies alevel set approach for realizing the interfacial motion according to the mean curvature flow. Furthermore, [2, 3] develop a modified scheme that can apply to the hyperbolic mean curvature flow
and call it the HMBO algorithm. Also, [3] explains the numerical analysis of this algorithm andpresents the computational results, including multiphase and volume-preserving motions.
In this work, we are interested in the obstacle problem when the interface touches the obstacle.
We use the level set method, so that the interface and the obstacle are expressed as a zero level setof an auxiliary function [4]. The interface model with an obstacle is given in Section 2. The aimsof this paper are to modify the scheme for the obstacle problem using the HMBO algorithm andinvestigate the behaviour of the interface when it touches the obstacle.
We consider two cases of the interface motion based on the choice of the initial curve. In thefirst case, we consider a closed curve as the initial curve and a circle as the obstacle located inside
the curve. The curve stops moving and lies on the obstacle after touching it. In this case, we
compare the result of the HMBO algorithm with the solution of a differential equation describing
circle motion by (1.1) before touching the obstacle. We do this comparison for several mesh sizes
and obtain the convergence order of the HMBO algorithm for the case of a circle evolving by thehyperbolic mean curvature flow.
For the second case, the initial curve is attached to the boundary of the domain and the obstacle
is below the curve. The curve reflects after touching the obstacle. We make a comparison between
the result of the HMBO algorithm for the obstacle problem and the solution of the hyperbolic
obstacle problem based on [7]. At every time we plot the points where the curve contacts the
obstacle, we call it the free boundary shape. Then, we find the slope of the free boundary shapeand present this result for both schemes.
The paper is organized as follows. Section 2 is about the interface model with an obstacle.
The numerical method to simulate this interface motion is given in Section 3. In this section, we
explain the level set method and the HMBO algorithm. Also, we give the scheme to treat the
obstacle problem using the HMBO algorithm. Then, we apply the numerical method and present
the numerical results in Section 4. Finally, the results are summarized in Section 5.

2

Interface model with an obstacle

In this paper, we treat the interface motion with an obstacle. We assume that the obstacle O ⊂ R2
is a convex open set. As we mentioned before, the interface is expressed by a parametrized curveγ(s,t). Here, we suppose that γ is a Lipschitz function and γ ∈ C2 ({γ ∈ Ōc }). According to (1.1),
we formally solve the following obstacle problem

∂ 2γ


= −κn
if γ ∈ Ōc ,


∂t 2




∂γ


if γ ∈ Ō, ∂t · ν∂ O ≤ 0(2.1)
γ ∈ Oc ,γ(s, 0) = γ0 (s),






 ∂ γ (s, 0) = v0 (s)n0 (s),∂tVita K USUMASARI

3

where ν∂ O is the unit outer normal vector to the obstacle. To simulate the interface motion with an
obstacle, we apply the HMBO algorithm using the level set method. We will explain this numericalmethod in the next section.

3

Numerical method

3.1

The level set method

In this paper, we use the level set method to represent the interface motion (1.1). By this method,
we consider the interface described by the zero level set of a function ϕ : R2 × [0, T ) → R. Let theinterface at a time t ∈ [0, T ) be given byΓt = {x ∈ R2 | ϕ(x,t) = 0}.
By the level set method, we can derive the equation for time evolution of Γt as the zero level setof ϕ(x,t). Then, we can express equation (1.2) by the level set representation in terms of ϕ(x,t).
The motion in normal direction by the normal acceleration is
d
(v),
dt
where v is the normal velocity. The level set representation of the normal velocity is
a=

(3.1)

ϕt + v|∇ϕ| = 0.
So, the normal acceleration (3.1) is
d
a=−
dt



ϕt
|∇ϕ|


=−

ϕtt |∇ϕ| − ϕt |∇ϕ|t.
|∇ϕ|2Further, the level set formulation for curvature is∇ϕκ = div.
|∇ϕ|According to equation (1.2), we get
ϕtt |∇ϕ| − ϕt |∇ϕ|t
∇ϕ
= div
.
|∇ϕ|2
|∇ϕ|

(3.2)

In this case, we take the level set function to be the signed distance to the Γt . We define thesigned distance function from a point x to Γt as(infy∈Γt kx − yk
if x ∈ Et ,d(x,t) =(3.3)−infy∈Γt kx − yk otherwise,with Et = {x ∈ R2 |ϕ(x,t) > 0} and Γt = ∂ Et . Since the signed distance function has |∇d| = 1, we
solve (3.2) approximated by [3]ϕtt = ∆ϕ.
T
Therefore, we consider that tk = kτ is k-th time step and τ = M, M is a positive integer. Let
dk (x) denote the signed distance to the interface at time t = tk . According to the HMBO algorithm[2, 3], the implementation of this interfacial motion is to solve the initial problem
2
ϕtt (x,t) = ∆ϕ(x,t) in R × (0, τ),(3.4)ϕ(x, 0) = dk (x)in R2 ,
ϕt (x, 0) = v0 (x)
in R2

4

Hyperbolic mean curvature flow with an obstacle

for k = 0, . . . , M − 1 with M being the number of time steps. But, we solve equation (3.4) only forthe first step, k = 0. For the further steps, k = 1, . . . , M − 1, we modify this initial problem. If wesolve it for all steps, we need to know the velocities along the interface. This can complicate thenumerical solution [3]. Therefore, this difficulty is overcome by using the solution to the following
problemin R2 × (0, τ),ϕtt (x,t) = 2∆ϕ(x,t)ϕ(x, 0) = 2dk (x) − dk−1 (x) in R2 ,ϕt (x, 0) = 0

(3.5)

2

in R .

We explain the intuitive idea of this modification. Let us have a very small time step size andassume that the interface moves smoothly. Therefore, the interface does not change much, so that
κ is approximately constant. Then we can rewrite equation (3.4) depending on time only. Let theinitial value be denoted by ϕ0 and let the initial velocity be v0 . The equation becomes
ϕtt (t) = −κ in (0, τ),(3.6)
ϕ(0) = ϕ0 ,ϕt (0) = v0 .
The solution of equation (3.6) is1
ϕ(t) = ϕ0 + v0t − κt 2 .
2

(3.7)

We denote ϕ(−t) as ϕ−1 and we get1
ϕ−1 = ϕ0 − v0t − κt 2 .
2Then

ϕ0 − ϕ−1 1
− κt.
t
2
Substituting the approximation of v0 above into (3.7) yields
v0 =ϕ(t) = 2ϕ0 − ϕ−1 − κt 2 .

(3.8)

Now, the solution (3.8) does not contain the term v0 . We see that function ϕ(t) in (3.8) is thesolution of the modified equationin (0, τ),ϕtt (t) = −2κ
ϕ(0) = 2ϕ0 − ϕ−1 ,
ϕt (0) = 0.
From this intuitive idea, we deduce the modified equation (3.5) for the further steps with zero
initial velocity by making the following modifications:√
1. Use constant 2 as the wave speed.
2. Combine the initial values from both previous and current time steps, that is 2dk (x) − dk−1 (x).
The detailed explanation of the analysis of this method is given in [3]. Following this idea, we will
describe the HMBO algorithm adopted from [3] in the next section.

5

Vita K USUMASARI

3.2

The HMBO algorithm

3.2.1

The original HMBO algorithm

The HMBO algorithm is a numerical approximation for the interface motion (1.1) [3]. It meansthat we find Γt = {γ(s,t)|s ∈ I} satisfying (1.1). In this case, we consider that the interface willT
evolve up to a time T . We take the time step ∆t = M, where M is a positive integer and 0 < ∆t  1.
Then, the approximation method is as follows:
1. For k = 0, we assume that the initial curve is Γ0 . Then, construct the signed distance function
to Γ0 as defined in (3.3) and denote it by d0 (x). We find u : Ω × (0, ∆t) → R, Ω ⊂ R2 ,satisfyingutt (x,t) = ∆u(x,t) in Ω × (0, ∆t), u(x,t) = d (x)on ∂ Ω × (0, ∆t),
0
(3.9)

u(x,
0)
=
d
(x)
in
Ω,
0




ut (x, 0) = 0
in Ω.
For simplicity, we restrict the initial velocity ut (x, 0) = 0. Then, we define the zero level setof u(x, ∆t) as Γ1 and compute the signed distance function to Γ1 .
2. For k = 1, 2, . . . , M − 1, repeat the following steps(a) Solve the equationutt (x,t) = 2∆u(x,t)
 u(x,t) = d (x)ku(x, 0) = 2dk (x) − dk−1 (x)ut (x, 0) = 0

in Ω × (0, ∆t),on ∂ Ω × (0, ∆t),in Ω,

(3.10)in Ω.

(b) Update the surrounded region and the interface using the zero level set of the solution
to (3.10):
Ek+1 = {x ∈ Ω | u(x, ∆t) > 0},Γk+1 = ∂ Ek+1 .
(c) Compute the signed distance function to Γk+1 .
We apply the finite difference approximation to solve the wave equation in the HMBO algorithm. Hence, utt and ∆u are approximated by the central difference. Here, the domain Ω
is a subset of R2 , Ω = (a, b) × (a, b). Each (a, b) is divided into N equal intervals, that ish = ∆x = ∆y = b−a
N . So, we have xi = a + ih, y j = a + jh for i, j = 0, . . . , N. Also, the timeinterval (0, ∆t) is divided into m equal intervals, that is ∆τ = ∆tm and τl = l∆τ, l = 0, . . . , m. In
this algorithm, we use Dirichlet boundary condition. If the interface touches the boundary of thedomain, then the interface rests at the boundary [6].
3.2.2

The modification of the HMBO algorithm for the obstacle problem

Since the HMBO algorithm requires us to solve the wave equation, we need to modify the wave
equation to treat the obstacle problem. Then, we follow the idea from the hyperbolic obstacle
problem [7].

6Hyperbolic mean curvature flow with an obstacle(1) The hyperbolic obstacle problem
In [7], the author describes the hyperbolic obstacle problem as the string vibration with an
obstacle. In this phenomenon, we can consider two conditions for the string vibration near the
obstacle. First, when the string goes up from the obstacle, the energy conservation law holds.
Second, if the string goes down, the energy is not preserved. Also, we assume that the reflection
constant is zero when the string hits the obstacle. It means that the string stops if it collides
with the obstacle. Here, the shape of the string is described by the graph of a scalar functionv : Ω̂×[0, T ) ≡ Ω̂T → R, where Ω̂ ⊂ Rn and the obstacle is the graph of a fixed function ψ : Ω̂ → R.
In [7], the author considered that ψ is the zero function. Then, we derive the equation when theRenergy conservation law holds. We suppose that the tension energy of the string is Ω̂ |∇v|2 dx and
R 2
the kinetic energy is Ω̂ vt χ{v>ψ} dx.
Therefore, the stationary points of the following action functional describe the motion of thestring:
Z Z
T((vt )2 − |∇v|2 )χ{v>ψ} dxdt,

J(v) =
0Ω̂where χ{v>ψ} is the characteristic function of the set {(x,t) ∈ Ω̂T | v(x,t) > ψ(x)}. We calculatedthe first variation dεJ(v + εφ )|ε=0 = 0, with φ ∈ C0∞ (Ω̂T ∩ {v > ψ}). Then, we get the weakformulation for the wave-type equation
vtt = ∆vin Ω̂T ∩ {v > ψ}.

(3.11)d
From the inner variation dεJ(v ◦ τε−1 )|ε=0 = 0, where τε = Id + εη and η ∈ C0∞ (Ω̂T ; Rn × R), we
obtain the free boundary condition ([7])|∇v|2 − (vt )2 = 0onΩ̂T ∩ ∂ {v > ψ}.

(3.12)

From (3.11) and (3.12), we can derive the equation [5]χ{v>ψ} vtt = ∆v
Hence, we introduce the problem as
χ{v>ψ} vtt = ∆v v(x, 0) = f (x)0vt (x, 0) = g0 (x)v(x,t) = p(x,t)

in Ω̂T .

in Ω̂T ,on Ω̂,(3.13)on Ω̂,on ∂ Ω̂T , with p(x, 0) = f0 (x) on ∂ Ω̂,where the first equation is understood in the sense of distributions. When the string touches the
obstacle, the solution v also satisfies(v ≥ ψ, ∆v ≥ vtt in Ω̂T ,∆v = vtton Ω̂T ∩ {v > ψ}in the sense of distributions.
We solve (3.13) using a finite difference approximation. Consider Ω̂ = (a, b) ⊂ R to be divided
into N equal intervals, so we have h = b−a
N and xi = a + ih, i = 0, . . . , N. For tk = k∆t, k = 0, . . . , M,
we approximate vtt and ∆v by central difference. Also, the characteristic function is defined ([5])
by(
1 if vki−1 > ψi−1 or vki > ψi or vki+1 > ψi+1 ,χ{v>ψ} (xi ,tk ) =0 otherwise,

7

Vita K USUMASARI

where vki = v(xi ,tk ) and ψi = ψ(xi ). Hence, we get the scheme(
2vk+1= 2vki − vk−1
+ ∆th2 (vki+1 + vki−1 − 2vki ) if χ{v>ψ} (xi ,tk ) = 1,
i
i
vk+1
= ψi
i

(3.14)

if χ{v>ψ} (xi ,tk ) = 0

for k = 0, . . . , M − 1 and i = 1, . . . , N − 1.
(2) The HMBO algorithm for the obstacle problem
In the HMBO algorithm, the interface is expressed as the zero level set of a function u : Ω ×(0, ∆t) → R. Similarly, the obstacle is also represented by the zero level set of a fixed function.
We define w : Ω → R such that {x ∈ Ω | w(x) = 0} is the obstacle. Let µ : Ω → R be the signed
distance function to the obstacle. To treat the obstacle problem, we follow the discretization of thehyperbolic obstacle problem given in scheme (3.14) for solving equations (3.9) and (3.10). Letuli, j = u(xi , y j , τl ) and µi, j = µ(xi , y j ). Then, we have the following scheme

l−1
l+1
l


ui, j = 2ui, j − ui, j2+c2 ∆τ
(uli+1, j + uli−1, j + uli, j+1 + uli, j−1 − 4uli, j ) if χ{u>µ} (xi , y j , τl ) = 1, (3.15)h2ul+1 = µif χ(x , y , τ ) = 0,
i, j

{u>µ}

i, j

i

j

l

for l = 0, . . . , m − 1 and i, j = 1, . . . , N − 1. The constant c2 = 1 for equation (3.9) and c2 = 2 forequation (3.10). Here, we define(
1 if uli, j > µi, j ,(3.16)
χ{u>µ} (xi , y j , τl ) =0 otherwise.
By implementing this scheme, we obtainu(x,t) ≥ µ(x) for (x,t) ∈ Ω × (0, T ).
However, scheme (3.15) is still developed. We will investigate the results using this scheme.
Remark. If we follow the characteristic function for the hyperbolic obstacle problem, then thecharacteristic function for the HMBO algorithm should be(
1 if uli±1, j > µi±1, j or uli, j > µi, j or uli, j±1 > µi, j±1 ,χ{u>µ} (xi , y j , τl ) =(3.17)
0 otherwise.
However, using (3.17), the interface motion bounces after touching the obstacle. This behaviour
does not agree with the expected motion. The expected motion based on problem (2.1) is that theinterface can rest on the obstacle after touching it. Hence, to match the expected motion, we use
the characteristic function (3.16). We will present these results in the next section.

4

Numerical results

In this part, we consider two cases of the interface motion. Both cases are based on the choice of
the initial curve. In the first case, we consider a closed curve as the initial curve and the obstacle
is a circle located inside the curve. Applying the HMBO algorithm for the obstacle problem, thecurve stops moving and lies on the obstacle after touching it. For the second case, the initial curveis attached to the boundary of the domain. Also, we position the obstacle so that it is below the
curve. In this case, the curve reflects after touching the obstacle.

8

4.1

Hyperbolic mean curvature flow with an obstacle

First case

For the numerical test, we consider a circle evolving by (1.1) with initial radius r0 and initial
velocity v0 . We give a fixed circle with a smaller radius as an obstacle. The circle will shrink
before touching the obstacle and stop after touching it. Before touching the obstacle, the curve isthe circle with radius r(t) satisfying

1


,
r00 (t) = −


r(t)
(4.1)
r(0) = r0 ,



 0
r (0) = v0 .
Consider a circle with initial radius r0 = 0.35, centered at (x, y) = (0.5, 0.5) on a domain(0, 1) × (0, 1), and zero initial velocity. A circle with radius r = 0.1 at the center of the initialcircle is given as the obstacle. We use mesh size h = N1 wherep π N is grid resolution, time stepte∆t = 29 with te being the extinction time given by te = 0.35 2 [3]. Then, this gives the time
discretization for finite difference approximation to the wave equation ∆τ = 2∆t6 . In this part, we
take the condition that is close to the case of a circle evolving by the hyperbolic mean curvature
flow in [3], so we can compare both results.
The error of the radius of the circle before touching the obstacle is obtained by the comparison
between the result from the HMBO algorithm and the solution of (4.1) using Runge Kutta fourthorder method. The L2 error isvuu 29
e = t∆t ∑ (rrk − rnk )2 ,k=1

where rn is the maximum distance to the center from the HMBO algorithm result and rr is the solution to (4.1). Furthermore, we show the convergence order related to L2 error. The convergenceorder is computed bylog eh2 − log eh1,
log h2 − log h1
where h2 > h1 are two different mesh sizes. The error and the convergence order are shown in thetable below.
Table 1: Error and convergence order using the HMBO algorithmNeconvergence order

16
32
64
128
256
512

0.107927
0.0966846
0.0813308
0.0573378
0.0324356
0.0181595

0.159
0.249
0.504
0.822
0.837

From Table 1, as the mesh size decreases, the error value also decreases. Moreover, the L2 error
and its convergence order agree with the result of the circle case given in [3].
We also tried other conditions when the obstacle position is not at the center of the initial
curve. From the results of this trial, the interfaces shrink and stop moving after touching the
obstacle. Then, the interfaces lie on the obstacle and follow the shape of obstacle. The numericalresults of the interface motion with an obstacle for several times are shown in the figures below.

9Vita K USUMASARI

(a)

(b)

(c)

(d)

Figure 1: The interface motion with an obstacle

4.2

Second caseIn this case, we consider the curve having small displacement, such that the interface motion by
the hyperbolic mean curvature flow coincides with the wave equation. Therefore, we compare theresults of the HMBO algorithm for the obstacle problem and the hyperbolic obstacle problem using
scheme (3.14). For the HMBO algorithm, we consider Ω = (−1, 1) × (0, 2), the mesh size h =2
N , N = 128, 256, 512, 1024, the time step ∆t = 0.001, and time discretization for finite difference
approximation ∆τ = ∆tm , m = 10. Then, for scheme (3.14), we use Ω̂ = (−1, 1), the mesh size2
h = N , N = 128, 256, 512, 1024, and the time step ∆t = 0.001.
In this trial, the initial curves are given by a piece-wise linear function or a quadratic function
with small displacement. Also, we set the position of an obstacle below the curve such that it is
not too close to the curve but the curve motion can touch it. Here, we use more general function
for the obstacle besides the zero function as in [7], namely constant function, linear function, andquadratic function. Then, we will describe the numerical results for each obstacle function.

10

Hyperbolic mean curvature flow with an obstacle

4.2.1

Constant function

In this part, the obstacle is given by a constant function. The initial conditions and the obstacle
for both schemes are described below. Here, u(x, y, 0) and w(x, y) are the initial value and the
obstacle function for the HMBO algorithm, respectively. For scheme (3.14), the initial conditions
are v(x, 0) and vt (x, 0) with the boundary conditions being v(−1,t) ≡ v(−1, 0) and v(1,t) ≡ v(1, 0).
Also, ψ(x) represents the obstacle function. More precisely, we have
• Case 1 (Initial curve is a piecewise linear function)u(x, y, 0) = −0.05|x| − y + 1.05
on Ω,
w(x, y) = 0.975 − yv(x, 0) = −0.05|x| + 1.05, vt (x, 0) = 0
on Ω̂.
ψ(x) = 0.975• Case 2 (Initial curve is a quadratic function)u(x, y, 0) = −0.05x2 − y + 1.05
on Ω,
w(x, y) = 0.975 − yv(x, 0) = −0.05x2 + 1.05, vt (x, 0) = 0
on Ω̂.
ψ(x) = 0.975
The figures below show the curve motions with an obstacle using the HMBO algorithm and

t =0

t = 1.5

t = 1.92

t = 2.88

Figure 2: Curve motion with a constant obstacle for Case 1

11Vita K USUMASARI

scheme (3.14). For instance, we plot each case with the mesh size N = 1024. Since the curve
has small displacement, we use different scales for y−axis and x−axis to make the figure of thecurve motion clearer. In these figures, the curve motion using the HMBO algorithm is indicatedby curve HBMO and curve wave is for the hyperbolic obstacle problem using scheme (3.14).
From Figure 2 and Figure 3, the curve reflects after touching the obstacle and vibrates above
the obstacle. We plot the points when the curve contacts with the obstacle at every time step. To
get the contact points with the obstacle, we find the end points in both sides when the curve isclose enough to the obstacle. We find the left contact point xL as
xL = min{x j | | f (x j ) − g(x j )| < ε}

(4.2)

j∈N

and the right contact point xR as
xR = max{x j | | f (x j ) − g(x j )| < ε},
j∈N

(4.3)

where 0 < ε  1. Then, f (x j ) and g(x j ) are the values of functions describing the curve and the
obstacle, respectively, for every grid point at a certain time.

t =0

t = 1.47

t = 2.04

t = 2.91

Figure 3: Curve motion with a constant obstacle for Case 2

12Hyperbolic mean curvature flow with an obstacle

For the HMBO algorithm, f (x) and g(x) are the functions whose graphs represent the zero levelset of u(x, y) and w(x, y), respectively. Meanwhile, for scheme (3.14), f (x) represents the functionv(x) and g(x) is the obstacle function ψ(x). For the range that determines where the curve touches
the obstacle, that is {x | | f (x) − g(x)| < ε}, we take ε = 0.001 for both schemes. Furthermore,
we plot the contact points, xL and xR , at every time step for each mesh size using both schemes.
These graphs are called the free boundary shape [7] and are shown in the figures below. In thesefigures, the free boundary shape from the curve motion using the HMBO algorithm is formed by
x R HBMO and x L HBMO. Further, x R wave and x L wave denote the free boundary shape forthe hyperbolic obstacle problem.

N = 128

N = 256

N = 512N = 1024

Figure 4: The free boundary shape for Case 1

13Vita K USUMASARI

N = 128

N = 256

N = 512N = 1024

Figure 5: The free boundary shape for Case 2

Moreover, we find the slope of the free boundary shape when the curve is going up for bothsides. This slope represents the free boundary condition (3.12). According to this free boundary
condition, we expect that the slope of the free boundary shape should be ±1. The slopes of thefree boundary shape using both schemes for each mesh size are given in the following tables. By
the curve motion, we consider that t ∗ denotes the time when the curve starts going up.
Table 2: The slope of the free boundary shape for Case 1
Slope
N
128
256
512
1024

HMBO algorithm
right and left sidest∗± 0.76± 0.91± 0.93± 0.99

1.119
1.338
1.425
1.454

scheme (3.14)
right and left sidest∗± 1.01± 1.01±1± 1.01

1.556
1.521
1.518
1.493

14

Hyperbolic mean curvature flow with an obstacle

Table 3: The slope of the free boundary shape for Case 2
Slope
N

HMBO algorithm
right and left sides
t∗

128
256
512
1024

4.2.2

± 0.75± 0.9± 0.93± 0.95scheme (3.14)
right and left sidest∗±1± 0.99± 0.99± 0.99

1.628
1.628
1.61
1.606

1.167
1.392
1.478
1.509

Linear function

We simulate the curve motion with the obstacle given by a linear function using the HMBO algorithm and scheme (3.14). Here, we use the same domain and discretization in space and time as in
the previous part. Also, the initial and boundary conditions are the same as before. We take• Case 3 (Initial curve is a piecewise linear function)u(x, y, 0) = −0.05|x| − y + 1.05
on Ω,
w(x, y) = 0.015x − y + 0.975v(x, 0) = −0.05|x| + 1.05, vt (x, 0) = 0
on Ω̂.
ψ(x, y) = 0.015x + 0.975• Case 4 (Initial curve is a quadratic function)u(x, y, 0) = −0.05x2 − y + 1.05
on Ω,
w(x, y) = 0.015x − y + 0.975v(x, 0) = −0.05x2 + 1.05, vt (x, 0) = 0
on Ω̂.
ψ(x, y) = 0.015x + 0.975
The figures below show the curve motions for each case using both schemes with N = 1024.

t =0

t = 1.53

Figure 6: Curve motion with a linear obstacle for Case 3 (t = 0, 1.53)

15

Vita K USUMASARI

t = 1.89

t = 2.85

Figure 7: Curve motion with a linear obstacle for Case 3 (t = 1.89, 2.85)

t =0

t = 1.41

t = 1.95

t = 2.88

Figure 8: Curve motion with a linear obstacle for Case 4

16Hyperbolic mean curvature flow with an obstacle

Also, we plot the free boundary shape using both schemes for each case. Hence, the freeboundary shapes for each mesh size are shown in the figures below.

N = 128

N = 256

N = 512N = 1024

Figure 9: The free boundary shape for Case 3

N = 128

N = 256

Figure 10: The free boundary shape for Case 4 (N = 128, 256)

17

Vita K USUMASARI

N = 512N = 1024

Figure 11: The free boundary shape for Case 4 (N = 512, 1024)

Moreover, we find the slope of the free boundary shape when the curve is going up. In order toobtain the left contact point, xL , and the right contact point, xR , with the obstacle, we use equations(4.2) and (4.3). The tables below represent the slopes of the free boundary shape for both schemes.
Table 4: The slope of the free boundary shape for Case 3
Slope
N
128
256
512
1024

HMBO algorithm
right side left side
t∗
-0.66
-0.85
-0.92
-0.94

0.8
0.91
0.93
0.94

scheme (3.14)
right side left side

1.187
1.427
1.524
1.557

-1
-1.01
-0.99
-1

1
1
1
1

t∗
1.652
1.65
1.621
1.619

Table 5: The slope of the free boundary shape for Case 4
Slope
N
128
256
512
1024

HMBO algorithm
right side left side
t∗
-0.62
-0.82
-0.91
-0.93

0.66
0.87
0.93
0.94

1.223
1.385
1.473
1.501

scheme (3.14)
right side left side
-0.99
-0.99
-0.99
-0.99

0.99
0.99
0.99
0.99

t∗
1.616
1.604
1.598
1.594

18

Hyperbolic mean curvature flow with an obstacle

4.2.3

Quadratic function

Next, we use a quadratic function as the obstacle that is given below• Case 5 (Initial curve is a piecewise linear function)u(x, y, 0) = −0.05|x| − y + 1.05
on Ω,
w(x, y) = −0.015x2 − y + 0.975v(x, 0) = −0.05|x| + 1.05, vt (x, 0) = 0
on Ω̂.
ψ(x, y) = −0.015x2 + 0.975• Case 6 (Initial curve is a quadratic function)u(x, y, 0) = −0.05x2 − y + 1.05
on Ω,
w(x, y) = −0.015x2 − y + 0.975v(x, 0) = −0.05x2 + 1.05, vt (x, 0) = 0
on Ω̂.
ψ(x, y) = −0.015x2 + 0.975
The figures below show the curve motions for each case using both schemes with N = 1024.

t =0

t = 1.5

t = 1.89

t = 2.88

Figure 12: Curve motion with a quadratic obstacle for Case 519Vita K USUMASARI

t =0

t = 1.44

t = 2.1

t = 2.94

Figure 13: Curve motion with a quadratic obstacle for Case 6

Also, the free boundary shapes for each mesh size are shown in the figures below.

N = 128

N = 256

Figure 14: The free boundary shape for Case 5 (N = 128, 256)

20

Hyperbolic mean curvature flow with an obstacle

N = 512N = 1024

Figure 15: The free boundary shape for Case 5 (N = 512, 1024)

N = 128

N = 256

N = 512N = 1024

Figure 16: The free boundary shape for Case 6

21Vita K USUMASARI

Moreover, we find the slopes of free boundary shape when the curve is going up and they are givenin the tables below.
Table 6: The slope of the free boundary shape for Case 5
Slope
HMBO algorithm
right and left sidest∗N± 0.84± 0.96± 0.98
±1

128
256
512
1024

1.149
1.378
1.475
1.504

scheme (3.14)
right and left sidest∗± 1.04± 1.02± 1.02± 1.03

1.611
1.579
1.579
1.565

Table 7: The slope of the free boundary shape for Case 6
Slope
N
128
256
512
1024

HMBO algorithm
right and left sidest∗± 1.07± 1.22± 1.22± 1.22

1.14
1.341
1.445
1.473

scheme (3.14)
right and left sidest∗± 1.26± 1.27± 1.28± 1.28

1.59
1.578
1.572
1.568

From Case 1 - Case 6, the slopes of free boundary shape obtained by using the HMBO algorithm
and scheme (3.14) coincide as the mesh size becomes smaller. It means that the curve motion
using both schemes gives similar results. Also, the slope of free boundary shape approaches thefree boundary condition.

5

Conclusion

We considered the interface motion with an obstacle according to the hyperbolic mean curvature
flow. In this work, we formally solved the obstacle problem given in equation (2.1). In orderto realize this motion, we used the HMBO algorithm as an approximation method. Then, we
modified the HMBO algorithm to treat the obstacle problem. The HMBO algorithm requires us
to solve the wave equation. Therefore, we modified the wave equation based on the hyperbolic
obstacle problem and we got scheme (3.15).
Moreover, we investigated the behaviour of the interface when it hit the obstacle. We considered two cases of the interface motion based on the choice of the initial curve. In the first case, we
considered that the initial curve was a closed curve and the obstacle was located inside the curve.
In this case, the interface stopped moving and lied on the obstacle after touching it, following theshape of the obstacle. We compared the result from the HMBO algorithm with the solution ofequation (4.1) describing circle motion by (1.1) before touching the obstacle. The L2 error and its
convergence order agreed with the result of the case of a circle evolving by the hyperbolic mean
curvature flow given in [3].

22Hyperbolic mean curvature flow with an obstacle

For the second case, the initial curve was fixed at the boundary of the domain and the obstacle
was below the curve. After touching the obstacle, the interface reflected and vibrated above the
obstacle. At every time we plotted the points where the interface contacted the obstacle, we called
it the free boundary shape. We compared the results of the HMBO algorithm for the obstacle
problem with the solution of the hyperbolic obstacle problem using scheme (3.14). The slope ofthe free boundary shape obtained by using the HMBO algorithm coincided with the one using
scheme (3.14) as the mesh size becomes smaller. This indicated that the curve motion using bothschemes converged to the same result. Also, the slope of the free boundary shape approached thefree boundary condition.

Acknowledgement
The author would like to express her gratitude to Professor Seiro Omata for his support and useful
discussions during preparation of this paper. Also, the author would like to thank the Ministry of
Research, Technology and Higher Education (RISTEKDIKTI) of the Republic of Indonesia forthe support of scholarship.

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[2] E. Ginder, K. Svadlenka. On an algorithm for curvature-dependent interfacial acceleration. Proceedings of Computational Engineering Conference, JSCES, Vol.19 (2014).
[3] E. Ginder, K. Svadlenka. Wave-type threshold dynamics and the hyperbolic mean curvature flow. Japan
Journal of Industrial and Applied Mathematics 33 No.2 (2016), 501-523.
[4] E. Ginder, A. Katayama, K. Svadlenka. On an approximation method for hyperbolic mean curvature
flow, RIMS Kôkyûroku 1995 (2016), 1–8.
[5] F. Makhrus, Y. Akagawa, A. Syahrini. Numerical methods for 1-D hyperbolic-type problems with freeboundary. The Science Reports of Kanazawa University 59 (2015), 27-50.
[6] K. Svadlenka, E. Ginder, S. Omata. A variational method for multiphase volume-preserving interface
motions. Journal of Computational and Applied Mathematics 257 (2014), 157-179.
[7] H. Yoshiuchi. A numerical solution of film vibration with obstacle. Dissertation, Kanazawa University(2007).