arXiv:0803.3719v1 [cond-mat.soft] 26 Mar 2008
Remarks on the accuracy of algorithms for motion by mean curvature
in bounded domains
S.J. Cox??, G. Mishuris??
? Institute of Mathematics and Physics, Aberystwyth University, UK
? Wales Institute of Mathematical and Computational Sciences
Abstract Simulations of motion by mean curvature in bounded domains, with applications to bubble motion
and grain growth, rely upon boundary conditions that are only approximately compatible with the equation of
motion. Three closed form solutions for the problem exist, governing translation, rotation and expansion of a
single interface [1], providing the only benchmarks for algorithm verification. We derive new identities for the
translation solution. Then we estimate the accuracy of a straightforward algorithm to recover the analytical
solution for different values of the velocity V given along the boundary. As expected, for large V the error can
reach unacceptable levels especially near the boundary. We discuss factors influencing the accuracy and propose
a simple modification of the algorithm which improves the computational accuracy.
1 Introduction
Foams are ubiquitous in everyday life [2]. They are used daily in the home, in both food and cleaning products.
Moreover, their industrial uses are varied and contribute significantly to the UK economy. For example, foams
are used to separate ores (e.g. zinc, lead) from the rock in which they are found, and to push oil out of porous
rock. They are also used in the decontamination cleaning of vessels, and in firefighting. In all these applications,
it is the flow of foam that is driving the process. Therefore, if it were possible to predict how a foam would
behave in this range of scenarios, when subjected to a complex collection of deformations, each process could
be made more efficient, either in terms of yield/output or cost-effectiveness.
Aqueous foams and concentrated emulsions, which are very similar to foams, have peculiar and remarkable
properties: they are complex fluids whose properties lie between the familiar extremes of liquid and solid. For
small strains, a foam behaves as an elastic solid while at large strains (or strain-rate) a foam moves like a liquid.
They therefore generate a rich range of behaviours, but with a well-defined local structure that obeys Plateau?s
geometric laws and the Laplace-Young law that balances interface curvatures with bubble pressure differences.
Our idea is to use the precise, known, structure of a foam to predict its rheological response.
One of the leading tools for the analysis of foam structure is the Surface Evolver developed by Ken Brakke
[3]. This consists of software expressly designed for the modelling of soap bubbles, foams, and other liquid
surfaces shaped by minimizing energy (such as surface tension and gravity), and subject to various constraints
(such as bubble volumes). Originally designed to model grain growth in metals, it is easily extended to the case
of foams, and has been used in a number of engineering disciplines to look at, for example, capillary surfaces,
solder joints and fluid behaviour in microgravity. A surface is represented as a collection of triangles, so that
the complicated topologies found in foams are routinely handled. In particular, the Evolver can deal with the
topological changes encountered during foam flow.
Here, we concentrate on a two-dimensional model of a foam, for ease of analysis. We use the following
?viscous froth model? (VFM) [4] to examine the evolution of foam films under shear:
pi ?pj =? ?ij +?vij (1)
where ? is the surface tension in the films (assumed constant), ?ij is the curvature of the film separating
bubble i from bubble j and vij its normal velocity. Each bubble has a well-defined pressure pi. In essence, the
model augments the (equilibrium) Laplace-Young law with a term proportional to the velocity; the constant
of proportionality ? is a drag coefficient, representing the external dissipation due to friction with the walls of
1
the container, allowing the investigation of strain-rate effects. If the external dissipation is negligible, then we
recover the Laplace-Young law for a foam in equilibrium and a model for quasi-static evolution.
A Surface Evolver simulation with the VFM proceeds in the following way. Each film is discretised into short
straight segments and curvatures ?ij are calculated pointwise at the intersections of these segments. Bubble
pressures are obtained by deriving a matrix equation based upon an integral of (1) around each cell. Then
the endpoints of the segments are moved according to the motion equation (1) using a small time step ?t.
Boundary conditions are applied according to the problem under consideration.
In the ideal model of the evolution of crystalline grains in a polycrystalline metal, known as normal grain
growth, the size of each grain evolves due to the normal motion of each of its boundaries [5]. Each boundary
has a certain ?mobility? ?, and moves in such a way as to reduce the total perimeter of the pattern, as in foams.
However, without area (volume in 3D) constraints, this is motion by mean curvature. As noted above, this is
a well-posed limit of the VFM (1) when bubble pressure differences are negligible, such as in freely translating
films (grain boundaries) and ordered (hexagonal) foams. The grain growth model requires that we have 120?
at vertices, justifying a posteriori that assumption in the VFM.
Here, we ask how such a solution can be commensurate with the boundary of the domain and how well this
solution can be calculated numerically. To the best of our knowledge, answers to this question are not available
in the literature.
2 Curvature-driven motion of a bounded interface
2.1 Problem formulation
In vector form, the motion of an interface in the model of ideal grain growth (1) can be described by
v = ?n, (2)
where n and s are the normal and tangential unit vectors to the interface (see Fig. 1):
s = [n1,n2], n = [?n2,n1]. (3)
If the representation of the interface is taken in the form:
x= x(y,t), y ? [y(t),y(t)], (4)
then the vector components n1,n2 are calculated as follows
n1 = dxds = cos? = x
?y
radicalBig
1+(x?y)2
, n2 = dyds = sin? = 1radicalBig
1+(x?y)2
, (5)
where ds =radicalbig(dx)2 +(dy)2 and ? is the tangential angle to the interface (see Fig. 1).
x
y
y
y n s
?
Figure 1: The bounded interface considered here.
Finally, the vector v = [v1,v2] is the instantaneous velocity of the point (x,y) lying on the interface at time t
and ? is the curvature of the interface at that point:
? = d?ds = ?x
??yy
radicalBig
(1+(x?y)2)3
. (6)
2
Equation (2) can be also written in component form:
vn = vn = ?v1n2 +v2n1 =?, (7)
vs = vs =v1n1 +v2n2 = 0. (8)
In this paper we will consider only Mullins? solution [1], also known as the grim reaper (see below), which
is symmetrical with respect to the x-axis. Taking into account the direction of the interface motion, we can
assume in what follows that:
n1 > 0, n2 >0, y?x >0, 0 0, (v2 < 0, v1 >0). (10)
Equation (7) is widely discussed in the literature [1, 6], while the second equation (8) is somehow usually
forgotten in this context. If one is only interested in reconstruction of the interface position at any time step
an approach based only on equation (7) is sufficient. However, if it required to control the position of each
tessellation point along the interface, as in the case of numerical computation, then both equations are equally
important. In particular, equation (8) allows us to find relation between the two unknown components of the
velocity vector v and the normal vector n in the form:
n1 = ?v2v
1
n2. (11)
This allows us to eliminate components of the normal vector n from equation (7) to give:
?
parenleftbigg
v1 + v
22
v1
parenrightbigg
= d?dy. (12)
2.2 Mullins? solution for translation revisited
Let us assume that the interface conserves its shape but moves in the x-direction with a constant speed V. We
consider two points A and C having the same y-coordinate y =y0 at two consecutive time steps t0 and t0 +dt
(see Fig. 2). It is clear that these two points correspond to two different material points. Namely, there exists
a point B on the interface at time t0 which moves according to the curvature law (2) to the point C for an
infinitesimally small time step dt. If the coordinates of the point A are (x0,y0) then the coordinates of B and
C can be written (x0 +dx,y0 +dy) and (x0 +Vdt,y0).
A
B
C
y
x
y0
y + dy0
x0 x + dx0
t + dt0t 0
?
Figure 2: Interface under translational motion at two consecutive instants in time t0 and t0 +dt.
Note that
BC = v(x0 +dx,y0 +dy)dt = v(x0,y0)dt+O(dtds).
On the other hand, |BC| = V sin?dt, tan? = v1/|v2|. As a result one can conclude:
V = 1v
1(y)
parenleftbigv2
1(y)+v
2
2(y)
parenrightbig, y ? [y(t),y(t)]. (13)
3
This relation immediately follows in case of the translation movement of the interface in x-direction from (12).
Now, to reconstruct the solution obtained by Mullins [1] it is sufficient to substitute (2) into (12) to have:
pi/2?Vy = ?. (14)
Here we have taken into account the second of the two symmetry conditions at the point y = 0:
v2(0) = 0, ?(0) =pi/2. (15)
Equation (14) can be written in the form
cotVy = y?x, (16)
which after direct integration leads to Mullins? solution [1]:
x(y) = x(0) +Vt? 1V logcos(Vy), y ? [0,y(t)]. (17)
This solution exists only under the condition y(t) 0. (25)
4
Note that the condition (13) may be not valid at all inside the interval y ? (y,h) as it was for Mullins? solution;
as a result, F(y) is not a constant, in general. Equation (12) can be integrated to give:
?
yintegraldisplay
y
F(?)d? = ?(y)??, (26)
and should be considered together with equation (11) which, in this case, takes the form:
dx
dy = ?
v2
v1 = cot?, (27)
or
x(y) = x?
yintegraldisplay
y
v2(?)
v1(?)d?. (28)
Equations (26) and (27) together indicate that the functions v1 and v2 cannot be chosen arbitrarily to satisfy
the vectorial equation (2) as one might expect. Instead, the following identity has to be satisfied:
arctan v1v
2
=
yintegraldisplay
y
F(?)d? +?, (29)
or writing w =v1/v2, equation (29) becomes:
w?
1+w2 =
(1+w2)v2
w , (30)
which leads to the following identity valid within the entire interval y ? (y,h):
v21(y) = ?v22(y)
?
??
?? 1
2
yintegraltext
y
v2(?)d?+C
+1
?
??
??. (31)
Note that any solution of equations (7) - (8) satisfies this additional relation, which makes sense only under the
additional constraint:
?1 ? 2
yintegraldisplay
y
v2(?)d? +C ? 0. (32)
As 00, one can show that
C = 0, v2(y) ? ?W2y, y ? 0. (36)
Note here that the value W = W(V,h) should be found from the constructed solution and is not an additional
(arbitrary or given) constant.
Finally, both restrictions (32) and (33) should be valid for the symmetrical interface:
hintegraldisplay
0
v2(?)d? ? ?1/2,
hintegraldisplay
0
F(?)d? ?pi/2. (37)
Note that the tangential angle ? for this solution is a monotonically decreasing function in the interval y ? (0,h)
so that
?(y) ? (?min,pi/2), ?min = pi/2?
hintegraldisplay
0
F(?)d?. (38)
It is straightforward to see that Mullins? solution (20) satisfies all these relationships with C = 0, ? = pi/2
and W =V, as expected. In the Appendix, we construct analytical examples of the symmetrical instantaneous
solutions which are different from the Mullins one. Those solutions are not generally speaking steady-state ones.
This means that they can be reached for some time step t, given the interface boundary velocity V(t) and the
position of the ends h(t)), but all these parameters may later change with time. What is extremely interesting
about these solutions is that some of them are well-defined for any velocity V > 0 and arbitrary position of
the boundary y = h. This shows a ?rich behaviour? of possible instantaneous solutions. It is also clear that
there is an infinite number of admissible instantaneous solutions. Some of them can be realised during some
specific non steady-state interface motion. For example, any instantaneous solution obtained during a numerical
computation, for any particular time step, boundary velocity and topology, has to satisfy all the relations (25)
- (35). This will allow us to use the relations as indicators of the accuracy of computations. Moreover, they
could provide a means to improve the accuracy of the algorithms.
Note also that a family of arbitrary (asymmetric) instantaneous solutions constitutes an even larger set
in comparison to the symmetric case. In fact, the family of symmetrical solutions has one degree of freedom
(since the constant C is equal to zero in this case) and correspondingly one less boundary condition (compare
(35) and (15)). Moreover, in the context of further applications of this result to a given algorithm, where the
angle-type boundary condition has to be preserved at the interface intersection point, it is worth mentioning
that the boundary condition (35) is therefore more important for application than the symmetry condition. On
the other hand, symmetrical instantaneous solutions can also be considered a subset of the asymmetric solutions
if one considers the interval (h0,h) instead of (0,h); (0 1.560796, and
lost physical meaning since the interface had a branch lying outside the external boundary y = h. All this
illustrates that the existing algorithm is well organised and works according to expectations but it is naturally
sensitive to the value of the boundary velocity V. Thus it makes sense to ask about algorithm accuracy for a
specific velocity versus space and time steps.
As the exact analytical solution to the Mullins problem is known, we can estimate errors in the computations
for all the physical and geometrical quantities: position of the interface x(y), curvature ?(y) and the velocity
components v1(y),v2(y). Corresponding relative errors for all solution parameters are presented in fig. 4 for
different applied velocities.
As expected, the least accurate solutions are those for the largest velocity V = 1.560796. The error can
be as large as 90% near the boundary (y = h) in the x component of velocity v1 and 60% near the symmetry
axis for the y component. These errors are a consequence of the phenomena discussed in the two observations
above. As V decreases, the accuracy increases for given bounds on the edge lengths L.
At first glance, it would appear that the position of the interface, x(y), should be computed with better
accuracy than all other solution parameters, which are, in fact, results of some derivative procedure. However,
our computations show that this is not in the case and the relative error for x(y) varies from 6% to 28% for the
7
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Relative error in x position
y position
V=0.001
V=0.1
V=0.5
V=1.0
V=1.5
V=1.560796
(a)
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.2 0.4 0.6 0.8 1
Relative error in curvature
y position
V=0.001
V=0.1
V=0.5
V=1.0
V=1.5
V=1.560796
(b)
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 0.2 0.4 0.6 0.8 1
Relative error in x velocity
y position
V=0.001
V=0.1
V=0.5
V=1.0
V=1.5
V=1.560796
(c)
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Relative error in y velocity
y position
V=0.001
V=0.1
V=0.5
V=1.0
V=1.5
V=1.560796
(d)
Figure 4: Relative error in (a) the position of the interface x(y), (b) its curvature ?(y), and the velocity
components (c) v1(y) and (d) v2(y), for different velocities V of the external boundary.
velocity V = 1.560796 while the curvature error is lower. Note also that the error for smaller velocities reaches
a few percent near the boundary or symmetry axis.
The maximal absolute errors for all solution parameters mostly appear near the external boundary y = h(=
1). This highlights that the implementation of the boundary condition in the existing algorithm cannot be
considered as sufficient and should be improved.
Moreover, in the case of Mullins? solution an additional simple local indicator defined by identity (13)
(independent of the integration of the solution variables) could equally be considered. It is clear from the
results presented in fig. 5(a), that the error in this condition is not localized near the ends of the interface, as
one might expect from the above. Moreover, this ?internal error? is present for all values ofV and is comparable
with that near the interface ends.
To investigate accurately this ?internal? error we repeated the computations for a specific velocityV = 1 and
decreased both the time increment, ?t, and the minimum edge length, Lmin (fig. 5(b)). This has improved the
quality of the computations, but there is still an error at some internal points of the interval that is comparable
with the error near the ends. Unfortunately, it considerably increases the computational time by several hours.
The obvious route of decreasing ?t but fixing Lmin, to ameliorate this effect, leads to greater error in the
solution (fig. 5(b)). The accuracy of the solution can be improved internal by making the line segments of equal
length (fig. 5(b)), but this doesn?t prevent an error at the boundary.
Possible sources of the ?internal? error include: a) non-optimal distribution of the tessellation points along
the interface after some time; b) imperfections in the correction procedure (which adds and eliminates points
from the interface at some prescribed time); c) point-to-point error variation related to the fact that the
?diffusion?-type coefficient changes from point to point along the interface (recall that equation (2) is a nonlinear
parabolic equation which is solved by a direct FD scheme with a fixed time step).
In the last computation in figure 5(b), for the line segments of equal length, we have redistributed points
to make the segments equal at every time step. Apart from the fact that the number of tessellation points is
smaller than for the standard algorithm, such a comparison is not absolutely fair as the redistribution in the
8
-0.01
-0.005
0
0.005
0.01
0.015
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Relative error in total velocity
y position
V=0.001
V=0.1
V=0.5
V=1.0
V=1.5
V=1.560796
(a)
-0.004
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
0.005
0.006
0 0.2 0.4 0.6 0.8 1
Relative error in total velocity
y position
Reference case
Decrease timestep by factor 10
Halve timestep and minimum length
Constant length
(b)
Figure 5: Relative error for the characteristic relation (13) for (a) varying velocity V, and (b) varying numerical
parameters (?t,L). The best accuracy is obtained by keeping the line segments of equal length.
standard algorithm was done every 20 time steps. To discover if there is an effect of redistribution frequency
on the accuracy, we have tested these two algorithms under the same strategy by redistributing the points (in
a different way) every time step and every 20th time step. The error in the function F defined in (25), which
is a constant in the case of Mullins solution, are presented in 6(a). It is evident that the standard algorithm is
quite sensitive to the chosen strategy. For a given position of the points on the interface, the relative error may
differ by as much as two orders of magnitude, whereas this is not the case for the equal segment strategy. In
this case, only near the external boundary is there some small fluctuation in the accuracy. Comparing the two
redistribution algorithms for the same frequency of redistribution, the largest error always appears in the case
of the standard algorithm ? by up to two orders of magnitude ? despite the fact that the number of tessellation
points was greater. Moreover, in the standard algorithm this error is irregularly distributed along the interface.
Recall that the number of tessellation points in the standard algorithm changes during the computations from
25 initially to around 150 (for V = 1) in the steady-state regime, while the number of the points in the second
(equal length) algorithm remains constant. Therefore the computational time for the second algorithm was less
by a factor of approximately two. On the other hand, the difference in the computational time between the
different frequencies for redistribution for the equal length algorithm was only a few percent. This indicates
the further possibility to optimise this algorithm by redistributing points every M time steps. It is clear that
M =M(V) and this needs further investigation.
Note that the function F from (25) can be used as an indicator of the accuracy of the computation only
for the Mullins solution. However, there are three universal indicators which can be helpful to estimate the
accuracy for any computations. Namely, the relative errors of the numerical representations of the identities
(26), (28) and (31). The respective results are shown in 6(b)-(d)).
As in the case of the specific indicator discussed above (Fig. 6(a)), all general indicators presented in Fig. 6
(b) ? (d) indicate that the equi-distant distribution of the tessellation points is much better than the standard
algorithm, regardless of the chosen strategy. Moreover, even near the symmetry point, y = 0, where the value of
the indicators all tend to zero and have a large influence on the relative errors, the accuracy of the computations
for the second algorithm is extremely high. This is not the case for the standard algorithm.
In Fig. 7, the relative errors of the solution variables are presented for both algorithms: the standard one
and the equi-distant distribution. The result shown in Fig. 7(a) looks surprisingly at first glance: although the
accuracy of the computations performed with these two algorithms is of the same order and the error related to
the new algorithm is distributed more uniformly, it appears that the accuracy of the standard algorithm is better
than the equal-segment-length algorithm, at least with respect to the accuracy of the position of the interface.
However, this not in the case. In fact, as was shown above, the computational error for the standard algorithm is
redistributed along the interface irregularly whereas that for the equal-segment algorithm is practically uniform.
As a result, the criterion to stop the iteration process to find the steady state solution works differently for the
two algorithms. The prescribed maximal step-to-step incremental growth 10?8 is reached more quickly for the
new algorithm. This is the second reason (together with number of tessellation points) why this algorithm is
faster. If one were to run both algorithms for the same time, or for the same number of iterations, and compare
9
1e-06
1e-05
1e-04
0.001
0.01
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
magnitude of relative error in F
y position
tl1
tl20
V1
V20
(a)
1e-05
1e-04
0.001
0.01
0.1
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Magnitude of relative error in intF
y position
tl1
tl20
V1
V20
(b)
1e-06
1e-05
1e-04
0.001
0.01
0.1
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Magnitude of relative error in intv
y position
tl1
tl20
V1
V20
(c)
1e-04
0.001
0.01
0.1
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Magnitude of relative error in intv2
y position
tl1
tl20
V1
V20
(d)
Figure 6: Relative errors in the computations shown by the integral measures for unit velocity of the external
boundary, V = 1.0, obtained with four different computational strategies: the standard redistribution of the
tessellation points, tl1 and tl20; uniform length segment redistribution, V1 and V20, at every time step and
every 20th step, respectively. (a) the function F defined in (25) (as in Fig. 5); (b,c,d) the three general internal
measures; for the notation used in the graphs see Appendix A.4. All integrals were been computed with the
trapeze rule.
the corresponding results, the discussed paradox should not appear and the new algorithm always provides
better accuracy by as much as two orders of magnitude.
However, for the accuracy of other problem variables, the film curvature, ?, and the interface velocities, v1
and v2, cf. in Fig. 7 (b)-(d), the new algorithm is more accurate, notwithstanding the above argument.
Finally, let us stress again that the proposed three indicators are better measures than a comparison of
numerical steady state solution with the analytical one, since the latter comparison includes an additional error
related to the determination of the steady state regime, while the indicators show us accuracy of the solution
at any time step.
4 Discussion and Conclusions
All these results clearly indicate that the existing algorithm should be used with caution, especially when
investigating foam behaviour near the critical velocity. Moreover, when there are many bubbles in a simulation
(fig. 3(c)), the user is restricted to some critical number of tessellation points M, which gives a limitation on
accuracy even for low velocity. In fact, the foam structure is highly non-uniform, in the sense that the bubbles
may have different sizes. Effectively this means that every interface has its own critical velocity and the bigger
bubbles thus introduce larger errors. This creates the following duality: to accurately describe the process
of bubble motion (for example a neighbour switching event [2]) it is necessary to have the computational
error as small as possible, while the error generated near the critical velocity takes its greatest value. This
is true for boundary or internal bubbles equally. Problems requiring high accuracy of the solution near the
external boundary are related to the investigation of the boundary effects describing the total phenomenological
10
1e-04
0.001
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
magnitude of relative error in x position
y position
tl1
tl20
V1
V20
(a)
1e-07
1e-06
1e-05
1e-04
0.001
0.01
0.1
1
0 0.2 0.4 0.6 0.8 1
magnitude of relative error in curvature
y position
tl1
tl20
V1
V20
(b)
1e-06
1e-05
1e-04
0.001
0.01
0.1
1
0 0.2 0.4 0.6 0.8 1
magnitude of relative error in v1
y position
tl1
tl20
V1
V20
(c)
1e-06
1e-05
1e-04
0.001
0.01
0.1
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
magnitude of relative error in v2
y position
tl1
tl20
V1
V20
(d)
Figure 7: Relative error in (a) the position of the interface x(y), (b) the curvature ?(y), and (c) and (d) the
velocity components v1(y) and v2(y), for velocity of the external boundary V = 1 and different numerical
algorithms.
behaviour of the foam structures.
To revise and improve the existing numerical algorithm, we propose to use another strategy for the redistri-
bution of the tessellation points: an equal-segment-length distribution of the tessellation points is much more
favourable.
Additionally, taking advantage of the auxiliary identities (26), (28) and (31), we may correct the instanta-
neous solution obtained within any algorithm at any or even every time step without time-consuming computa-
tions, as the identities are valid for any instantaneous solution. They also make possible further investigation of
the asymptotic behaviour of the bounded interface solution near the ends. For example, any possible solution
behaves at the symmetry axis according to (36), which allows us to tackle the error in the solution near the
symmetry axis. On the other hand, the results obtained in section 2.3 may allow us to construct and implement
a new numerical procedure/elements tackling the boundary condition in a more accurate way (without losing
any near-boundary points).
Finally, as we have shown, the identities (26), (28) and (31) may be used to probe the accuracy of compu-
tations. These indicators are extremely helpful as they are not based on information about the exact solution
and can therefore illuminate inaccuracy of the numerical solution without any preliminary knowledge about the
exact solution itself.
Summarizing, we have shown that an improvement of the numerical algorithm is highly desirable and
possible. Apart from the fact that some of the improvements have been indicated and proven in this paper,
there is still an open question how to deal with accuracy of the computations near the critical velocities and
near the external boundaries. We have also detected possible directions for future investigation: improved
implementation of the boundary condition, creation of additional near-boundary points. Further, to check new
results related to the numerical algorithm we need a larger set of analytical benchmarks.
11
References
[1] W.W. Mullins, Two-Dimensional Motion of Idealized Grain Boundaries. J. Appl. Phys., 1956, 27:900?904.
[2] D. Weaire and S. Hutzler, The Physics of Foams. Clarendon Press, Oxford, 1999.
[3] K. Brakke, The Surface Evolver. Exp. Math., 1992, 1:141?165.
[4] N. Kern, D. Weaire, A. Martin, S. Hutzler, and S.J. Cox, Two-dimensional viscous froth model for foam
dynamics, Phys. Rev. E, 2004, 70:041411.
[5] D. Weaire and S. McMurry, Some Fundamentals of Grain Growth. Solid State Physics, 1996, 50:1?36.
[6] A. Peleg, B. Meerson, A. Vilenkin Area-preserving dynamics of a long slender finger by curvature:A test
case for globally conserved phase ordering, Phys. Rev. E, 2001, 63:066101.
[7] S.J. Cox, A Viscous Froth Model for Dry Foams in the Surface Evolver, Coll. Surf. A, 2005, 263:81?89.
A Appendix: A family of symmetrical instantaneous solutions
In this section we presentanalytical representationsof some instantaneoussymmetricalsolutions for the interface
satisfying the same boundary (23) and symmetry (15) conditions as Mullins? solution.
A.1 First example
Let we consider the following simple combination of compatible velocities
v2(y) = ?W2y, v1(y) =W
radicalbig
1?W2y2, W = V?1+V2h2, (39)
which satisfy (31) with C = 0 and, as a result, can be used to construct a symmetrical instantaneous solution.
Here W is the same constant as in (36). Natural restrictions (37) for the existence of such solution give the
same estimate:
W < 1h, or V?1+V2h2 < 1h, (40)
which, remarkably, holds true for any values of V and h. The shape of the interface is described by equation
(28)
x= x0(0,t)?
radicalbig
1?W2y2. (41)
The tangential angle ? for this solution is a monotonically decreasing function in the interval y ? (0,h) and
?(y) ? (?min,pi/2), ?min = pi/2?arcsin(Wh)> 0. (42)
A.2 Second example
Let we now consider another specific instantaneous solution assuming that v1 = W < V. Then the second
component of the velocity satisfies the equation
W2
v22(y) = ?
1
2
yintegraltext
0
v2(?)d?
?1. (43)
To find v2(y) it is more convenient to return to the differential equation (30) rather than working with the
nonlinear integral equation (43). After integration it takes the form
?
parenleftBigv2
W
parenrightBig
= ?Wy, (44)
where the odd function ? is defined as
?(?) = 12
parenleftbigg
arctan?+ ?1+?2
parenrightbigg
,
parenleftbigg
??(?) = 1(1+?2)2
parenrightbigg
. (45)
12
Note that ? : R+ ? [0,pi/2) is a monotonic function. Moreover, one can easily obtain the constraintWhpi/2?
pi/4integraldisplay
0
parenleftbigg
1+
parenleftBig
??1(?)
parenrightBig2parenrightbigg
d? = pi/2?
?integraldisplay
0
parenleftbig1+?2parenrightbig??(?)d? = pi/2? ?integraldisplay
0
d?
1+?2 = 0. (50)
It is interesting to note that in the case V << 1 both of the instantaneous solutions constructed above
coincide with Mullins? solution to within an accuracy of O(V2) for any fixed value of h.
A.3 General case
The second example above indicates how to build a wider class of symmetrical instantaneous solutions. Let us
introduce the following set A ? C2([?a,a]), (a > 0) of even functions, ?(?) = ?(??), satisfying the following
four conditions:
?(?) = 12?2 +O(?4), ? ? 0; ?(a) ? 12; ?? > 0;
parenleftBigg
??radicalbig
?(1?2?)
parenrightBigg?
? 0, ? ? (0,a). (51)
Note that a may differ from function to function, but it is necessary that for every function there exists some
a>0 for which all conditions (51) hold. For some functions it may happen that a = ?.
For example, the following three functions belong to the set A:
?1(?) = 12 sin2?, ?2(?) = 12?2, ?3(?) =
?integraldisplay
0
??1(?)d? = 12
parenleftBigg
1? 1
1+bracketleftbig??1(?)bracketrightbig2
parenrightBigg
, (52)
with a1 = pi/2, a2 = 1 and a3 = ? respectively. These three functions have been collected from Mullins?
solution and two previous examples. Thus, the set A is not empty.
Using any function from this set we can construct a symmetrical instantaneous solution with velocity com-
ponents in the form:
v2(y) = ?W??(Wy), v1(y) = W??(Wy)
radicalBigg
1?2?(Wy)
2?(Wy) , (53)
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that identically satisfies equation (31) with C = 0. Then the unknown constant W should be taken to be of the
form W = W?(Vh)/h where W?(b) >0 is a solution of the implicit equation:
??(W?)radicalbig
2?(W?)(1?2?(W?)) =
b
W?. (54)
Because of the last condition in (51), there may exist only one solution of this equation. If, in addition, the
left-hand side of (51) tends to infinity as W? ?a, than the solution always exists and W? 0 as W? ? a, than the solution exist only under additional
condition
Vh0, from (55) was never used. We added it only to have a convex instanta-
neous solution; without it, we can construct non-convex interfaces.
A.4 Integral measures
Here we define explicitly the integral measures of problem quantities, used for assessing the accuracy of the
computations, and their relative errors.
After (26):
intF = ?
yintegraldisplay
0
F(?)d?; relative error = ?
?
?
yintegraldisplay
0
F(?)d?
?
?/
parenleftbigg
?(y)? 12pi
parenrightbigg
?1. (56)
After (28):
intv =
yintegraldisplay
0
v2(?)
v1(?)d?; relative error =
?
?
yintegraldisplay
0
v2(?)
v1(?)d?
?
?/(x(0)?x(y))?1. (57)
After (31):
intv2 =
yintegraldisplay
0
v2(?)d?; relative error =
?
?
yintegraldisplay
0
v2(?)d?
?
?/
parenleftbigg v2
2(y)
2(v22(y)?v21(y))
parenrightbigg
?1. (58)
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