A NONLINEAR VERTEX-BASED MODEL FOR ANIMATION OF
TWO-DIMENSIONAL DRY FOAM
Micky Kelager and Kenny Erleben
eScience Center, Department of Computer Science, University of Copenhagen, Denmark
Keywords:
Dry foam dynamics, Vertex-based model, Nonlinear equations, Nonlinear and nonsmooth systems.
Abstract:
Foam is the natural phenomenon of bubbles that arise due to nucleation of gas in liquids. The current state
of art in Computer Graphics rarely includes foam effects on large scales. In this paper we introduce a vertex-
based, quasi-static equilibrium model from the field of Computational Physics as a new paradigm for foam
effects. Dynamic processes like gas diffusion and bubble collapse are added prior equilibration. Animation-
wise the numerical model is well behaved and stable and can converge even if the foam is locally ill-defined.
A novel contribution is the Ghost-Bubble method that allows foam simulations with free dynamic boundary
conditions. The presented model is interesting and well suited for 2D graphics applications like video games
and procedural or animated textures.
1 FOAM ANIMATION IN
COMPUTER GRAPHICS
Many methods to simulate liquid fluids have been pre-
sented in Computer Graphics (Stam, 1999; Losasso
et al., 2004; Selle et al., 2005; Losasso et al., 2006)
with a common focus on realism. We believe that
the next step in fluids dynamics lies within the sub-
ject of liquid foams, or froths. The scientific work
on foam dynamics in Computer Graphics is sparse.
In our opinion the most promising result to animate
beer froth still has issues with the motion and be-
havior of the foam (Cleary et al., 2007) but the still
frames from the foam simulations seem very convinc-
ing. We argue that the foam animation problems in
Computer Graphics are caused by the fact that foams
are treated as fluids and not as real foams. We be-
lieve that the theoretic foundation of foam dynamics
must be obtained from Computational Physics. In this
paper we revisit a vertex-based foam model from the
field of Computational Physics and derive a mathe-
matical model with a discretization that is suitable for
the purpose of Computer Graphics.
Our focus is directed towards two-dimensional
foams. This dimensional restriction is due to that
physics literature commonly only agrees on the pro-
cesses of two-dimensional foams and how they are
behaving whereas three-dimensional foams are not
yet completely understood. Two-dimensional foams
also exist in the real three-dimensional world, e.g. a
liquid foam constrained between two glass plates or
the single layer of foam resting on a surface. Addi-
tionally, in this paper we only focus on the behav-
ior of dry foams as wet foams are not as accurate
described by current physical models (Weaire et al.,
2003). We handle the dynamics and internal forces
of the dry foam along with external interactive contri-
butions, e.g. topological changes, gas diffusion, and
bubble collapses, but omit external dynamic forces
and collision handling. However, the method is based
on a Lagrangian representation and as such additional
external couplings and interactions can be handled at
vertex level.
This paper introduces the Ghost-Bubble method
which is a novel contribution for the two-dimensional
vertex-based foam model. The Ghost-Bubble method
allows dry foam simulations with free dynamic sur-
faces and finite boundaries. We derive a nonlinear
Newton method for the dry foam model and our dis-
crete model can converge even if the foam is lo-
cally ill-defined. The model has less than 1% to-
tal error even though we use first order finite differ-
ence approximations. The presented physics-based
dry foam model is interesting and well suited for two-
dimensional graphics applications like video games
and procedural or animated textures.
296
Kelager M. and Erleben K. (2010).
A NONLINEAR VERTEX-BASED MODEL FOR ANIMATION OF TWO-DIMENSIONAL DRY FOAM.
In Proceedings of the International Conference on Computer Graphics Theory and Applications, pages 296-303
DOI: 10.5220/0002849002960303
Copyright
c
SciTePress
2 PREVIOUS WORK
Computer Graphics has a greater interest in how bub-
bles and foam simulations are visualized while Com-
putational Physics is more keen on trying to under-
stand, describe, and measure all the different physics
processes within a foam.
Analytic and iterative relaxation-based geometri-
cal methods have been used in Computer Graphics to
render foam and bubbles. For instance in (Icart and
Arqu
`
es, 1999) empirical laws were used to construct
a bubble-bubble interaction model to describe a single
layer of foam bubbles. Constructive solid geometry
(CSG) and analytical solutions were used in (Glass-
ner, 2000a; Glassner, 2000b) to visualize small-sized
bubble clusters of 2-3 bubbles.
Lagrangian elastic models have also been at-
tempted. One such attempt employs an explicit
curvature-driven model where bubble-bubble interac-
tions are modeled using energy potential functions to
penalize unwanted bubble configurations (Durikovic,
2001). In this work gas diffusions and Plateau bor-
ders are ignored and the computational mesh is fixed.
The shown foams consist of no more than 10 bub-
bles. In (Li and Volkov, 2006) a 2D elastic model is
extended with a pressure force model where pressure
variation in the external media is included to model
buoyancy. In (Iwasaki et al., 2004) a mass-spring sys-
tem is employed in a curvature-driven method similar
to (Durikovic, 2001). Results include visualizations
of 1-2 bubbles. The main contribution in this work is
a real-time rendering method that takes light interfer-
ence into account.
Fluid-based methods are another paradigm.
In (Hong and Kim, 2003) the volume-of-fluid method
and the front-tracking method are combined. Inter-
faces are directly adjusted to avoid numerical diffu-
sion and yield mass conservation. This work ad-
dresses bubbles rising in a liquid but does not deal
with foams. A continuous multiphase fluid simulator
is used in (Zheng et al., 2006) where a novel regional
level set method and a semi-implicit surface tension
model are introduced. Bubble thickness is explicit
modeled by a distribution function and the effect of
drainage is included. Gas diffusion is still not mod-
eled nor is it clear how to deal with boundary condi-
tions in the velocity field. Results show 5-10 bubbles.
Lastly bubble-bubble interaction models have also
been addressed. (K
¨
uck et al., 2002) models bubbles
using a mass-less spherical particle system. Springs
are added to model the bubble interactions. Bubble
creation and film rupture are handled by adding and
removing bubbles randomly. Plateau borders are not
modeled but instead special ray-tracing shaders are
used to mimic the real foam structure. Results are
shown with 700-3000 bubbles. A virtual beer was
poured into a glass in (Cleary et al., 2007) using a La-
grangian model of smoothed particle hydrodynamics
(SPH) which makes the interaction between the multi-
phase fluid very convincing. Gas diffusion is handled
and bubble collapsing is dealt with similar to (K
¨
uck
et al., 2002). The foam model omits modeling the
foam films and hence the importance of topological
changes. Rheology and shearing of the bubbles are
handled in a pseudo-physical way by including ex-
plicit cohesive forces between the particles. The mo-
tion of the dry foam on top of the beer is too rigid and
some realism is lost in this account.
In summary the work of foams done in Computer
Graphics are concerned with modeling film thickness
such that the interference patterns can be shown dur-
ing rendering. However, many of the dynamic foam
processes are ignored, such as gas diffusion, foam
rupture, and topological changes. Our method in-
cludes all of these. Lately, the work on bubbles and
foams in Computer Graphics have begun integrating
physically ideas (Cleary et al., 2007; Kim et al., 2007)
which allow foams and bubble clusters to behave
more convincingly. Common for the two-dimensional
foam models in Computational Physics are the re-
quirement of infinite networks which imply periodic
boundary conditions in the numerical simulations.
Distribution functions and statistics are natural inter-
ests within Computational Physics where dynamical
effects come secondary (Weaire and Kermode, 1983;
Kawasaki et al., 1989; Bolton and Weaire, 1992).
Our work introduces the vertex-based foam model
to the field of Computer Graphics. The model in-
cludes effects not previously presented in Computer
Graphics. Further we extend the model to deal with
free moving foam boundaries. We demonstrate exam-
ples of dry foam networks up to 10.000 bubbles. As
far as we know this is an order of magnitude larger
than previous published results on 2D foam simula-
tions in Computer Graphics.
3 A PRIMER ON PHYSICS OF 2D
DRY FOAM
We now briefly review the basics of 2D dry foam. A
complete source of theory can be found in (Weaire
and Hutzler, 1999). A foam is a two-phase system
in which bubbles of gas are enclosed by thin layers
of a liquid substance. Regularly a foam is disordered
which refers to the bubbles within the foam that can
vary greatly in size. Ordered foams usually do not oc-
cur in real life and are therefore a topic of theoretical
A NONLINEAR VERTEX-BASED MODEL FOR ANIMATION OF TWO-DIMENSIONAL DRY FOAM
297
research. Liquid foam has a form that is categorized
as cellular due to that bubbles of gas form cells that
are surrounded by liquid. In a two-dimensional foam
the Plateau borders coincide with the foam films.
Thus, a foam can be structured into three elements;
bubbles/cells, films/Plateau borders, and Plateau junc-
tions. In terms of mesh topology these elements cor-
respond to faces, edges, and vertices, respectively.
This apparently complex two-phase material has a
well-defined local structure which allows progress in
predicting the dynamic properties of the foam. The
different types of foams are generally categorized into
either liquid or solid. A liquid foam is a soft, highly
deformable, substance that range from wet to dry de-
pendent on how much of its volume that contains liq-
uid. A special variant of liquid foam is froth. We con-
centrate on liquid foams with the same characteristics
as soap foam.
A bubble will not shrink and decrease its sur-
face area to zero due to surface tension γ is balanced
against the pressure difference p. This relation is
known as the law of Laplace-Young which is given
by
p = γ
1
R
1
+
1
R
2
=
2γ
r
, (1)
where R
1
and R
2
are the principal radii of curvature
for the surface with p = (p
i
p
j
) as the difference
between two adjacent cells i and j. In two dimensions
1
R
1
+
1
R
2
=
2
r
where r is the local radius of curvature
of a foam film.
Plateau added a set of additional rules to the law of
Laplace-Young. These rules are vital for foam equi-
librium. For a two-dimensional dry foam the law of
Plateau states: The films can intersect only three at a
time and must do so with an angle of φ = 120
.
During equilibration and evolution of a dry foam
the network will undergo different structural changes.
These changes can be categorized into topological
processes entitled T1 and T2. It has been observed
that other topological restructurings in a foam are
combinations of one or more T1 and T2 processes.
In the two-dimensional T1 process the topological
Figure 1: The T1 process on a 2D dry foam. An approach-
ing unstable four-fold vertex will be split into two stable
three-fold vertices.
change consists of exchanging, or flipping, the films
in question as depicted in Figure 1. When two three-
fold vertices are moving towards each other the dis-
tance between the vertices becomes zero and the foam
structure will be in an unstable configuration. If the
distance between the vertices becomes close to zero,
or less than some threshold T
dist
, then a T1 process
takes place,
~
d ·
~
d < T
2
dist
T
dist
> 0, (2)
where
~
d = (~x
i
~x
j
) is the displacement vector be-
tween the two three-fold vertices. Generally, the
length of the film is larger than k
~
d k as the film is
curved due to the law of Laplace-Young. We will ar-
gue that the difference in lengths can be neglected as
T
dist
can be set sufficiently close to zero.
When the T1 condition (2) is satisfied the flipping
strategy we choose is to place the film vertices per-
pendicular to the line segment described by
~
d. If we
assume that the two vertices meet Plateau’s 120
-rule
then due to the geometrical properties of the T1 oper-
ation the vertices will be forced to move apart.
Figure 2: The T2 process for a 2D dry foam. When a three-
sided cell approaches zero area due to gas diffusion it will
perish.
The T2 topological change consists of removing a
three-sided cell, as depicted in Figure 2, and occurs
due to gas diffusion. In a T2 process a foam cell will
collapse into a single vertex. The position of the re-
sulting vertex is set to the average of the three cell
vertices. The T2 process is performed when the cell
area is less or equal to some area threshold T
area
,
n
i
= 3 A
i
T
area
T
area
0, (3)
where n
i
designates the number of sides of the i
th
cell
and A
i
is the current area of the cell. Cells with more
than three sides can also collapse. Here a number of
T1 operations flip the smaller films in the cell until a
three-sided cell is formed.
Gas diffusion is driven by the pressure differences
between adjacent cells across their common bound-
aries. Some cells will grow at the expense of others
but in the end all cells will perish. This evolution of
a foam is called coarsening. By far the most com-
mon applied model for coarsening in an ideal two-
dimensional dry foam is known as von Neumann’s
law,
dA
i
dt
=
2π
3
γκ(n
i
6), (4)
GRAPP 2010 - International Conference on Computer Graphics Theory and Applications
298
where κ is a diffusion constant. This law shows that
cells with more than six sides steadily enlarge while
cells with less than six sides steadily decrease in area.
4 THE VERTEX-BASED DRY
FOAM MODEL
In this section we construct the mathematical equi-
librium model for a two-dimensional dry foam based
on the laws of foam physics. We assume that viscous
forces and inertial effects are negligible. The enclosed
gas is treated as incompressible such that the foam
will minimize the surface energy at all times (Weaire
and Kermode, 1983). For now we assume a foam with
periodic boundaries which implies that each vertex
has three films and three adjacent cells. Let V , E , F
and A be index sets of vertices, edges, faces, and ver-
tex angles, respectively. The dry foam mesh consists
of faces that correspond to bubble cells. The f
th
cell is
labeled with its current area A
f
and pressure p
f
, while
junctions have positions where the v
th
vertex has co-
ordinates ~v
v
=
x
v
y
v
T
. We measure the angle of
intersection between two films, i.e. edges, meeting at
a junction, and denote these vertex angles by φ
a
for
a A. From geometry and the law of Laplace-Young
it follows that the cell areas and angles of film inter-
sections can be written as functions of vertex posi-
tions and cell pressures.
The rule of Plateau results in a constraint for each
vertex angle
2π
3
φ
a
= 0 a A. (5)
The cell areas may be controlled by a similar con-
straint
A
target
f
A
f
= 0 f F . (6)
The target area A
target
f
is coupled with von Neumann’s
law to yield the equilibrium area during coarsening
of the dry foam. Next we agglomerate vertex coordi-
nates and cell pressures into one state vector
~x =
h
~v
T
1
·· · ~v
T
kV k
p
1
·· · p
kF k
i
T
. (7)
The cell area and vertex angle constraints can similar
be agglomerated into a system of nonlinear equations
~
F(~x) =
A
target
1
A
1
(~x)
.
.
.
A
target
kF k
A
kF k
(~x)
2π
3
φ
1
(~x)
.
.
.
2π
3
φ
kAk
(~x)
=
~
0. (8)
A solution ~x
for equilibrium must fulfill
~
F(~x
) = 0.
Our idealized mathematical model may be summa-
rized into,
ideal model
von Neumann’s Law
Topological T1/T2 processes
State changes shear/collapse
~
F(~x) =
~
0
.
(9)
This model consists of a coupling of highly nonlinear
and non-smooth sub processes.
4.1 Discretization of the Mathematical
Model
As the first step we apply a linearization to the whole
model. Essentially, this means that we solve the four
sub processes in an iterative manner until the system
converges. To determine the cell target areas for the
current iteration we apply an Euler discretization of
von Neumann’s Law. Secondly, we apply the topo-
logical processes and state changes as discrete events,
and finally we solve
~
F(~x) =
~
0.
We apply a nonlinear Newton method for
~
F(~x) =
~
0 (Nocedal and Wright, 2006). Given the current it-
erate ~x
k
for some iteration k for which F(~x
k
) 6=
~
0, we
seek to find ~x
k
such that F(~x
k
+ ~x
k
) =
~
0. Taking a
first order Taylor series approximation gives
~
0 = F(~x
k
+ ~x
k
) F(~x
k
) +
dF(~x
k
)
d~x
~x
k
. (10)
Each iteration of the Newton method solves J
~
x
k
=
~
b
where J =
dF(~x
k
)
d~x
and
~
b = F(~x
k
). The Newton up-
date ~x
k+1
= ~x
k
+ ~x
k
is done once a solution for the
step-direction
~
x
k
is found. There are many practi-
cal difficulties in a straightforward implementation.
An analytical solution to the Jacobian J is not feasi-
ble in practice due to the nonlinear implicit coupling
between the state variables. Further, non-singularity
for J is not guaranteed which implies that an inexact
Newton approach must be taken.
To remedy the problems above we apply a blocked
Gauss-Seidel approach to solve for the step-direction
and performing the Newton update. Blocks are cre-
ated from the state variables that are affected by the
vertices of the foam. A block corresponds to the x
and y coordinates of one vertex and the cell pressure
values p
i
, p
j
, and p
k
of the adjacent three cells that
are shared by the vertex. A similar approach is used
in (Kermode and Weaire, 1990). The Newton system
for a single block can now be written as
A NONLINEAR VERTEX-BASED MODEL FOR ANIMATION OF TWO-DIMENSIONAL DRY FOAM
299
A
i
p
i
A
i
p
j
A
i
p
k
A
i
x
A
i
y
A
j
p
i
A
j
p
j
A
j
p
k
A
j
x
A
j
y
A
k
p
i
A
k
p
j
A
k
p
k
A
k
x
A
k
y
∂φ
i
p
i
∂φ
i
p
j
∂φ
i
p
k
∂φ
i
x
∂φ
i
y
∂φ
j
p
i
∂φ
j
p
j
∂φ
j
p
k
∂φ
j
x
∂φ
j
y
p
i
p
j
p
k
x
y
=
A
i
A
j
A
k
∆φ
i
∆φ
j
.
(11)
The right-hand side of (11) consists of the area and
angle constraints
A
f
= A
target
f
A
f
f {i, j,k} F , (12a)
∆φ
a
=
2π
2
φ
a
a {i, j} A. (12b)
The third angle φ
k
is not part of the Jacobian because
it is directly given by
φ
k
= 2π (φ
i
+ φ
j
). (13)
The partial derivatives of the blocked Jacobian ma-
trix are computed numerically by central difference
approximations. To ensure the central difference ap-
proximations are sufficiently accurate we apply a step
size-halving strategy until the absolute differences lie
within a given threshold. To avoid problems with pos-
sible singularities the resulting linear system is solved
using the method of Singular Value Decomposition
(SVD). The topological processes are dealt with di-
rectly in the foam mesh as they correspond to edge
flips and face collapses. Von Neumann’s Law is dis-
cretized as
A
target
f
A
target
f
+
2π
3
γκ(n
f
6), f F . (14)
Since we are dealing with a quasi-static simulation we
have chosen a time step size of unity.
4.2 The Ghost-Bubble Method
A foam with a freely movable surface can not have
periodic boundaries. A new dynamic boundary con-
dition must be applied to the vertex model to allow for
a free foam surface.
We introduce the Ghost-Bubble method which
adds one virtual bubble to the foam network. To make
the ghost bubble move with the foam itself we add it
on the back of the foam mesh. This is illustrated in
Figure 3. We seek to model the ghost bubble as a
natural element of the real world, i.e. a world bubble,
that defines the atmosphere that surrounds the foam.
When gas diffuses through boundary films into the at-
mosphere the change in volume of the world is in-
significant compared to the volume of the atmosphere.
(a) (b)
Figure 3: The ghost bubble. (a) The front of the foam mesh
defines the foam network. (b) The ghost bubble is defined
on the back of the foam mesh where it reuses all existing
boundary vertices of the foam. It acts as a mirrored bubble
that contains all bubbles in the foam network.
From this argument the ghost bubble should retain a
constant area. The same argument holds for a con-
stant ghost bubble pressure. This results in the follow-
ing two rules; the ghost bubble must have constant,
infinite area and constant, unit pressure.
A permutation of indices is performed prior set-
ting up (11) for a boundary block such that the ghost
bubble corresponds to the cell with index k. From our
rules we have A
k
= p
k
= 0 and the changes to the
Newton subsystem become,
A
i
p
i
A
i
p
j
0
A
i
x
A
i
y
A
j
p
i
A
j
p
j
0
A
j
x
A
j
y
0 0 0 0 0
∂φ
i
p
i
∂φ
i
p
j
0
∂φ
i
x
∂φ
i
y
∂φ
j
p
i
∂φ
j
p
j
0
∂φ
j
x
∂φ
j
y
p
i
p
j
0
x
y
=
A
i
A
j
0
∆φ
i
∆φ
j
.
(15)
5 EXPERIMENTS AND RESULTS
We post-process Voronoi diagrams from random gen-
erated point sites to create initial networks for our
foam solver. This is similar to (Weaire and Kermode,
1984) and illustrated in Figure 4.
(a) (b)
Figure 4: In (a) a Voronoi diagram of 500 cells is gener-
ated from random point sites and (b) shows the correspond-
ing equilibrated 2D dry foam network with an area error of
0.001% and an angle error 0.01%.
Our experiments indicate that good threshold val-
GRAPP 2010 - International Conference on Computer Graphics Theory and Applications
300
ues for the topological conditions can be chosen as
0.01 {T
dist
,T
area
} 0.25. (16)
We use the surface tension of γ 0.00025
N
m
. We have
evaluated the presented foam model by running hun-
dreds of simulations with different numbers of cells
and area sizes without any technical instabilities. We
have varied κ from [10
3
..10
3
] and when a high value
of κ is employed the foam is collapsing more rapidly
similar to plain water foam.
We have measured the computation times from
different foam configurations and plotted them in
Figure 5. The plot depicts that the computation
time appears to scale linear in the number of cells.
Figures 4(b), 6, and 7 illustrate the free surface bound-
Figure 5: Time performance plot for a foam network evolv-
ing over time. The initial number of cells is 10k with a
frame time of 39.5s. The last frame contains 11 cells and
took 84ms. The simulation ran on a standard single core
2GHz laptop.
aries that is a result of the Ghost-Bubble method.
Figure 4 establishes that the dry foam model can
equilibrate a newly created foam. Four frames from
a complete foam evolution with initially 10, 000 cells
are shown in Figure 6.
To illustrate the foam bubble collapse mechanism
a random foam mesh of 66 cells have been used. The
dynamic gas diffusion process has been deactivated to
explicitly take control over which bubbles that should
disappear. T2 operations are forced upon the selected
bubbles. The result of a sequence of bubble collapses
is shown in Figure 7 where cells of 4, 5, and 6 edges
have been targeting for deletion.
The foam mesh can be rendered using flat shaded
polygons. Figure 8 illustrates a large example of how
the pressure differences are distributed throughout a
foam. Notice that cells with similar pressures tend to
group together and form walls that almost makes the
image look like a labyrinth.
Figure 6: A foam is evolving with initially 10, 000 cells.
The time step is squared between each shown frame.
A NONLINEAR VERTEX-BASED MODEL FOR ANIMATION OF TWO-DIMENSIONAL DRY FOAM
301
(a) (b)
(c) (d)
(e) (f)
Figure 7: Forced bubble collapse is performed on the dy-
namic foam boundary. Between each sub figure the marked
cell is next to be collapsed.
6 CONCLUSIONS AND FUTURE
WORK
We have introduced a new paradigm for foam simu-
lation from the field of Computational Physics. Fur-
ther, we have presented a novel method to deal with
free moving boundaries in quasi-static simulation of
2D dry foam. Our discrete model has been developed
with focus on stability and robustness that are needed
in the field of Computer Graphics. We believe that the
presented physics-based dry foam model is interest-
ing and well suited for 2D graphics applications like
video games and procedural or animated textures.
For future work we will add dynamic forces to
the foam model. From a performance point of view
we could replace the sequential blocked Gauss-Seidel
variant of the nonlinear Newton solver with a parallel
blocked red-black Gauss-Seidel variant. We may also
exploit that a dual mesh could encode the geometric
invariants of the rules of Plateau. This work may be
extended to 3D where tetrahedra would correspond to
film junctions. However, it is not clear how to deal
with the topological processes in 3D, nor is it clear
how to reconstruct the curved surfaces of the film ge-
ometry in 3D.
Figure 8: The cell pressures are visualized in flat shaded
blue tones. The colors have been exaggerated to make the
difference more visible.
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