INTERACTIVE JELLYFISH ANIMATION USING SIMULATION

Dave Rudolf

Department of Computer Science, University of Saskatchewan, Saskatoon, Canada

PhaseSpace Inc., San Leandro, USA

David Mould

Department of Computer Science, University of Saskatchewan, Saskatoon, Canada

School of Computer Science, Carleton University, Ottawa, Canada

Keywords:

Jellyﬁsh, Marine invertebrates, Physics-based animation, Fluid dynamics.

Abstract:

This paper presents an automatic animation system for jellyﬁsh that accounts for interaction between the

organism and its surroundings. We endeavor to model the jellyﬁsh’s morphology, as well as its achieved

thrust. We physically simulate the elastic body of the jellyﬁsh and its surrounding sea water. We use a modiﬁed

immersed boundary method to combine spring-mass systems and a grid-based semi-Lagrangian ﬂuid solver.

The resulting simulations are efﬁcient with an acceptable compromise in physical accuracy. We reduce our

model for axially symmetric species to 2D, and extrapolate the results to 3D. We add detail to the 3D shape

with noise that is inspired by empirical observations of real jellyﬁsh. We also suggest suitable contraction

functions so that our virtual jellyﬁsh propells itself within the water in a manner similar to the real organism.

The resulting system is capable of animating jellyﬁsh in real-time on modest desktop hardware.

1 INTRODUCTION

Seascapes are becoming more common in the com-

puter animation industry following the popularity of

productions such as Pixar’s Finding Nemo (2003). As

well, advances in numerical techniques and hardware

are making physical simulation of virtual environ-

ments more feasible. Simulation has the advantage of

automatically animating interactions between charac-

ters and their environments. This paper proposes an

animation system for jellyﬁsh that uses computational

ﬂuid dynamics so that the virtual organism can affect

its environment, and vice versa.

We are interested in the unique mode of jet-

propelled locomotion exhibited by jellyﬁsh, as illus-

trated in Figure 1. This style of motion is largely not

understood either by computer scientists or by marine

biologists. Beer et al. (1997) suggest that the study of

relatively simple animals such as jellyﬁsh will yield a

better idea of how to animate more complicated de-

formable marine life.

Physiologically, jellyﬁsh are not overly complex:

they are invertebrates with a simple nervous sys-

tem (Arai, 1997). However, their mode of locomotion

is still difﬁcult to model numerically, owing to the

Figure 1: Captured footage of a real jellyﬁsh swimming.

combination of the organism’s elastic body and the

surrounding incompressible ﬂuids (Stockie & Wetton,

1999). We use simulation not for accuracy reasons,

but so that our virtual jellyﬁsh can physically interact

with the water around it. Our simulations use a mass-

spring system (Terzopoulos, Platt, Barr & Fleischer,

1987) to represent the elastic body of the jellyﬁsh, and

a semi-Lagrangian ﬂuid solver (Stam, 1999) for the

surrounding sea water. The two representations are

coupled using the immersed boundary method (Pe-

skin, 2002). This method is effective for our system.

However, a full 3D simulation would still be quite ex-

pensive. Thus, we make reductions to our model.

Many species of jellyﬁsh are axially symmetric,

which we exploit by simulating only a low-resolution

2D slice. By doing so, we are able to animate jelly-

ﬁsh in an appealing, convincing, and efﬁcient manner.

We then extrapolate the 2D results to a higher resolu-

tion 3D model. Since a na

¨

ıve 3D extrapolation would

lack the geometric complexity seen in real jellyﬁsh,

241

Rudolf D. and Mould D.

INTERACTIVE JELLYFISH ANIMATION USING SIMULATION.

DOI: 10.5220/0001792402410248

In Proceedings of the Fourth International Conference on Computer Graphics Theory and Applications (VISIGRAPP 2009), page

ISBN: 978-989-8111-67-8

Copyright

c

2009 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

we also propose means to add variation back into the

3D geometry, based on observations from the biology

community.

Controlling motor functions is a further challenge.

Jellyﬁsh have a simple muscular structure, but use it

in complex ways that are not well studied. We give

one possible approach to motor control for jellyﬁsh,

based on empirical observation. We focus on the res-

onant gait of jellyﬁsh, which is marked by a regular

swimming action at a rate close to a resonant fre-

quency of the organism and its environment (Megill,

2002). We restrict our interest to fully grown adult

jellyﬁsh, as the organism most actively swims in this

phase of its development.

The main contribution of this paper is our auto-

mated system for animating jellyﬁsh, running on con-

ventional desktop hardware at interactive rates, yet

still accounting for physical interactions between the

jellyﬁsh and its surrounding ﬂuid. Our model incor-

porates the biological aspects of the organism that in-

ﬂuence its locomotion. We also show that a simula-

tion model of jellyﬁsh can effectively be reduced to a

coarse 2D model, while providing methods to extrap-

olate back to a detailed and compelling 3D rendering

model. We modify Peskin’s immersed boundary to be

more suitable for coarse models. Lastly, we also pro-

vide a simple procedural mechanism for controlling

the organism’s physiology in order to achieve loco-

motion.

Our work is most interesting to graphics re-

searchers and tool designers. The animations pro-

duced by our model are convincing to a general au-

dience. Although our particular model would not be

accurate enough for experimental biologists, the biol-

ogy community can draw inspiration from our work

in developing their own simulations of jellyﬁsh.

2 PREVIOUS WORK

Physics-based animation techniques have been devel-

oped for a wide variety of characters: snakes, worms,

and caterpillars (Miller, 1988); ﬁsh (Tu & Terzopou-

los, 1994); and, of course, human beings (Hodgins,

Wooten, Brogan & O’Brien, 1995). Jellyﬁsh have

previously been animated through procedural tech-

niques and key-framing (Pixar Animation Studios,

2003). These techniques do not easily model two-

way interactions between the jellyﬁsh and their envi-

ronment.

Particle-based approaches such as mass-spring

systems (Terzopoulos, Platt, Barr & Fleischer, 1987;

Tu & Terzopoulos, 1994) can be used to simulate elas-

tic bodies. Thus, particle-based ﬂuid representations,

such as smoothed particle hydrodynamics (Desbrun &

Gascuel, 1996), might seem tempting. However, SPH

is not as efﬁcient as grid-based methods (Griebel,

Dornseifer & Neunhoeffer, 1998) when simulating

contiguous regions of incompressible ﬂuid, especially

when large forces are acting on the ﬂuid. Stam (1999)

gives an inexpensive grid-based approach that is un-

conditionally stable, though with some numerical

damping.

To couple particle-based solids with grid-based

ﬂuids, the graphics community typically enforces

boundary conditions on the ﬂuid that correspond to

the solid surface’s velocity at each ﬂuid cell. The

most recent work in this vein is that of Robinson-

Mosher et al. (2008), which increases the stability

of simulation by strictly enforcing conservation of

momentum. From the mathematics community, Pe-

skin (2002) uses the immersed boundary method to

couple grid-based ﬂuid representations with particle-

based solids. This method is especially interesting to

us because it has been used to model active elastic

bodies, such as the human heart (McQueen & Peskin,

2000), that exert great force on the ﬂuids around them.

Unlike the work of Robinson-Mosher et al. (2008),

the immersed boundary method does not explicitly

prevent ﬂuid from leaking across thin elastic bound-

aries, and performs poorly when the ﬂuid’s free sur-

face intersects the elastic body. However, our jellyﬁsh

are completely submerged, and leakage is quite mini-

mal (Stockie & Wetton, 1999).

As with all physics-based animation techniques,

we need a way to automatically control the bodies that

are in motion. For some creatures (Wu & Popovi

´

c,

2003; Miller, 1988; Tu & Terzopoulos, 1994), simpli-

ﬁed physical systems are used so that the animation

has some semblance of realism, but the animation can

be controlled with a relatively small number of pa-

rameters. Our system is more complex, in that it has

more parameters. A more adaptive controller, such

as a PID controller (Dean & Wellman, 1991), is also

possible; however, tuning the weights of the control

function is not a trivial task.

From the biology community, Megill (2002)

discusses different gaits of jellyﬁsh. Dabiri and

Gharib (2003) provide empirical data on the large-

scale morphology and kinematics of swimming jel-

lyﬁsh. Gladfelter (1972), and Megill (2002) describe

the elastic properties of the organism, and its distri-

bution of muscle ﬁbres. Gladfelter also gives detailed

data on the deformation of jellyﬁsh as they contract.

The symmetry of jellyﬁsh was previously ex-

ploited by Dabiri and Gharib (2003), though they treat

the organism as perfectly symmetric. To generate a

3D model from a 2D slice, we take inspiration from

GRAPP 2009 - International Conference on Computer Graphics Theory and Applications

242

Rasmussen et al. (2003), who animated 3D explosions

by augmenting the ﬂow of 2D simulations with a 3D

periodic noise ﬁeld. Perlin (Perlin, 2002) gives a sim-

ple means of computing a continuous noise ﬁeld.

3 A NUMERICAL MODEL OF

JELLYFISH

This section describes our numerical model of jelly-

ﬁsh: our mass-spring conﬁguration for of the organ-

ism’s body, and how we incorporate it into a semi-

Lagrangian ﬂuid solver with a modiﬁed immersed

boundary method. This section also discusses the bi-

ological aspects of the organism that are involved in

its locomotion, and lastly how we extrapolate and em-

bellish our 3D model.

3.1 Jellyﬁsh Physiology

Figure 2 highlights the anatomy of the organism that

are of interest to this work. A jellyﬁsh swims by

repeatedly contracting its umbrella, producing thrust

by expelling ﬂuid from the subumbrellar cavity. The

tentacles along the aperture of the umbrella add hy-

drodynamic drag. These tentacles are mostly passive

during normal swimming, but are used when hunt-

ing or moving along the ocean ﬂoor (Megill, 2002).

Many species of jellyﬁsh are approximately axially

symmetric, where the axis of symmetry runs from the

apex of the umbrella through the center of its aperture.

Shih (1977) lists a large number of different species of

jellyﬁsh, whose umbrellas and tentacles vary in size

and physical conﬁguration.

Figure 2 also depicts the different tissues within

the umbrella itself. The circumferential muscle that

lines the subumbrellar wall is chieﬂy involved in lo-

comotion. When this muscle contracts, it pulls the

umbrella inward, creating a ﬂuid jet at the aperture of

the umbrella that propels the organism forward.

Joint Mesoglea Bell Mesoglea

Circumferencial Muscle

Subumbrellar

Cavity

Aperture

Mouth

Apex

Tentacle

Figure 2: Vertical and horizontal cross-sections of a medu-

san umbrella.

Figure 3: The non-linear deformation of the umbrella under

the contraction of the circumferential muscle, shown as a

horizontal cross-section.

Aside from the circumferential muscle, the re-

mainder of the umbrella is categorized into two re-

gions: the bell mesoglea and the joint mesoglea, both

essentially passive. The joint mesoglea’s elastic mod-

ulus is roughly 10% the bell mesoglea’s, so that the

when the umbrella is compressed, the joint mesoglea

deforms into ridges, as illustrated in Figure 3. Be-

cause of the ridges, the elastic properties of the um-

brella are not a linear function of the displacement.

Megill (2002) gives us the shape of the stress-strain

proﬁle.

3.2 2D Simulation Model

We want our simulations to capture three aspects of

the jellyﬁsh’s anatomy: the mesoglea; the circumfer-

ential muscle; and the tentacles. We represent the

subumbrellar surface of the organism with a chain

of Hookean springs, and further enforce the struc-

ture of the umbrella with angular springs. Instead

of angular springs, one might be tempted to enforce

relative orientations with small networks of Hookean

springs, similar to the work of Miller (1988), and Tu

and Terzopoulos (1994). However, we experimentally

found that the large number of short springs (relative

to the resolution of our ﬂuid grid) made the system

less stable. Figure 4 shows our mass-spring conﬁg-

uration. The circumferential muscle is represented

by linear springs going longitudinally across the um-

brella. These springs exert an inward thrust on the

subumbrellar surface, as the circumferential muscle

does in real jellyﬁsh. To mimic a contraction of this

muscle, we reduce the rest lengths of these springs to

exert force on the subumbrellar wall.

Figure 4 also shows tentacles, represented by two

strands of points on either side of the umbrella, con-

nected with both linear and angular springs. The

size and makeup of jellyﬁsh tentacles can vary widely

among different species and we artistically chose the

physical parameters of the tentacles. We give the ten-

tacles the same elastic modulus as the umbrella, but

with 1/100th the cross-sectional area.

We model the ﬂow of the sea water surrounding

the jellyﬁsh using the incompressible Navier-Stokes

INTERACTIVE JELLYFISH ANIMATION USING SIMULATION

243

equations:

∂~u

∂t

= ν∇

2

~u −~u · ∇~u −

∇p

ρ

+

~

F, ∇ ·~u = 0, (1)

where ~u is the ﬂuid’s velocity ﬁeld, p is the pressure

ﬁeld, ρ is the ﬂuid density, ν is the ﬂuid’s kinetic vis-

cosity,

~

F is a ﬁeld of external forces acting on the

ﬂuid, and ∇ and ∇

2

respectively are spatial gradient

and Laplacian operators. Griebel et al. (1998) give a

thorough explanation of Equation 1.

For our 2D simulation, we use the Cartesian for-

mulation of the Navier-Stokes equation. Since we are

modeling axial symmetry, it may be tempting to use

a cylindrical formulation as given by Acheson (Ache-

son, 1990). However, the axis of symmetry can po-

tentially change constantly, and the two halves of the

slice may not mirror each other exactly.

Sea water is essentially incompressible, and a jel-

lyﬁsh can exert large forces on the ﬂuid within its

cavity. Upholding the incompressibility constraint in

Equation 1 is very important to our system. SPH

methods (Desbrun & Gascuel, 1996) approximate in-

compressibility by introducing strong pressure forces

between ﬂuid particles, increasing the numerical stiff-

ness (and thus the computational expense) of the sim-

ulation. Grid-based methods are more appropriate.

We immerse our mass-spring model of the jellyﬁsh in

a square ﬂuid volume that is ten times the jellyﬁsh’s

diameter, and give the ﬂuid cavity free-slip boundary

conditions, as described by Griebel et al. (1998).

We use the immersed boundary method (Peskin,

2002) to couple the particle-based elastic and the grid-

based ﬂuid representations. In Peskin’s method, the

elastic point-masses are advected along the ﬂow ﬁeld

of the ﬂuid grid, and the elastic forces of the Hookean

springs are applied to the ﬂuid grid using the force

term

~

F in Equation 1. Peskin uses a smoothing ker-

nel to distribute the forces to several, possibly dozens,

of grid cells near the point-mass. Doing so increases

the cost of simulation, and effectively puts an up-

per limit on the frequency at which the force proﬁle

can vary over the ﬂuid grid. Since we use relatively

coarse grids (i.e., 50×50 or 100 ×100), the frequency

limitation of Peskin’s smoothing kernel can make the

ﬂuid appear artiﬁcially viscous around the solid point-

masses. We diverge slightly from Peskin’s method,

and instead distribute the elastic force of a point-mass

onto its four closest grid cells with bilinear interpola-

tion to ﬁnd the contributions to each corner.

Peskin’s method is known to be numerically stiff,

and Stockie and Wetton (1999) show that semi-

implicit integration schemes yield roughly an order

of magnitude in efﬁciency. However, we ﬁnd ex-

perimentally that we gain two orders of magnitude

in speed-up over an explicit scheme, simply by us-

ing the semi-Lagrangian method (Stam, 1999). This

method does not gain us larger time-steps, but merely

decreases the cost of each step. Stam’s method is

unconditionally stable when simulating ﬂuids alone,

but large time-steps can still make the mass-spring

network unstable. Similarly, the Courant-Friedrichs-

Lewy stability criterion (Griebel, Dornseifer & Neun-

hoeffer, 1998) does not predict time-steps that keep

the elastic body stable. Lacking a better measure of

stability, we controlled step sizes with the Courant-

Friedrichs-Lewy condition, using a safety factor of

0.001.

3.3 Muscle Activation

Megill (2002) describes several different gaits of the

jellyﬁsh. We aim to animate the resonant gait, which

involves an organism oscillating at or near the reso-

nant frequency of the system. It is the most common

and most studied gait.

To contract the umbrella, we modify the rest

lengths of the subumbrellar springs. Biology liter-

ature provides no evidence that the organism uses

closed-loop controllers such as those described by

Dean (1991) to induce muscle contractions, but rather

suggests a predictable, cyclical pattern (Megill, 2002;

Dabiri & Gharib, 2003). We thus use Hermite splines

to generate each spring’s rest length over time.

When a jellyﬁsh contracts, it propels itself for-

ward. However, when it expands, it also pulls it-

self backward. For a jellyﬁsh to achieve a net posi-

tive movement over a contraction cycle, the organism

must incur less overall drag in the cycle’s expansion

phase than in the contraction phase. We are unaware

of any literature that details how this drag reduction is

achieved. We suspect that jellyﬁsh make themselves

Figure 4: Our mass-spring network for a jellyﬁsh slice, and

the corresponding contraction functions that we use. Lines

represent Hookean springs. Also, angular springs are used

between points on the tentacles, and around the umbrella,

though not across the umbrella. Each spring’s greyscale

level in the left image corresponds to the contraction func-

tion in the right image with the same grey level.

GRAPP 2009 - International Conference on Computer Graphics Theory and Applications

244

more ﬂat to achieve higher drag, and more rounded to

reduce their drag. In Figure 1, the ﬁrst three frames

show the jellyﬁsh expanding, and its umbrella shape

is relatively curved. In the fourth frame, the jellyﬁsh

is contracting, and the umbrella appears less curved

and more conical. In order to achieve this morphol-

ogy, we slow the expansion of subumbrellar springs

close to the aperture. We give the Hermite splines for

the contraction functions different slopes in the ex-

pansion phase. Springs closer to the umbrella’s apex

have steeper slopes than those closer to its aperture.

Figure 4 shows our mass-spring model of the jelly-

ﬁsh, with the corresponding contraction functions for

each spring.

The frequency of the jellyﬁsh’s contractions has

a direct effect on the organism’s achieved thrust, as

well as its morphology. The resonant frequency of

the system depends mostly on the umbrella’s diam-

eter (Megill, 2002). We experimented with different

contraction frequencies to determine which ones were

optimal for our model. Figure 5 shows the transla-

tional motion of a simulated jellyﬁsh with an umbrella

diameter of 40 mm, but with different contraction fre-

quencies. We get a large maxima in total vertical dis-

placement for a frequency of 0.7 Hz, which agrees

with data measured by Dabiri and Gharib (2003).

3.4 3D Rendering Model

The points in our simulation model represent the sub-

umbrellar surface. To account for the thickness of the

umbrella, we generate the exumbrellar side by pro-

jecting each surface point backward along its normal.

Figure 3 gives suitable thickness values. We then re-

sample the coarse set of points to arbitrary resolutions

by interpolating a cubic spline through the original

point-masses and then the newly generated exumbrel-

lar points.

The subumbrellar points may not be symmetric

Distance Travelled for Several Contraction Frequencies

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2018

16

14

12

108

642

0

Time (s)

Distance (m)

f=0.4

f=0.7

f=1.0

f=1.5

f=2.0

Figure 5: Trajectories of several simulated organisms with

different frequencies of contraction.

Figure 6: The results of our 2D simulation; trace particles

show the ﬂow of the ﬂuid around the contracting jellyﬁsh.

within the 2D plane (i.e., the two sides of the um-

brella may not mirror each other). Thus, we cannot

extrapolate the geometry by merely rotating the re-

sults of the slice about the axis of symmetry, as Dibiri

and Gharib (2003) did. Instead, we consider each pair

of points ~x

i

and ~x

j

opposite to each other on the um-

brella. Note that the pairs of points in question re-

sulted from the resampling of the coarse geometry.

We deﬁne an axis of rotation for each pair, which goes

through the center of the line segment between the

points ~x

i

and ~x

j

, and is orthogonal to that line seg-

ment. We use the same disc extrapolation scheme for

generating tentacles along the 3D aperture of the um-

brella. Figure 7 illustrates our process of extrapolat-

ing circular area from pairs of points in our 2D model.

Our disc-based extrapolation generates an artiﬁ-

cially smooth volume. Although a given species may

be roughly symmetric, individual jellyﬁsh are not ex-

actly so, as seen in Figure 8. To add variation to our

rendering model, we perturb each point~x

i, j

on the um-

brella’s surface by some scalar distance c

i, j

in the di-

rection of its surface normal ~n

i, j

, where (i, j) is the

azimuth and elevation indices of the point.

Several factors can cause small-scale asymme-

tries. Periodic features in the underlying umbrella ge-

ometry are particular to the species of jellyﬁsh (Shih,

Figure 7: An example of point pairs along the 2D umbrella

slice that have been extrapolated to 3D discs. For each point

pair, the line between them is bisected orthoganally by the

rotational axis, and a disc results from rotating those points

about that axis.

INTERACTIVE JELLYFISH ANIMATION USING SIMULATION

245

Figure 8: Jellyﬁsh that are not exactly symmetric.

1977), as seen in the right-most two images in Fig-

ure 8. For these structural patterns, we deﬁne func-

tions to generate the desired appearance:

c

str

i, j

= α

str

sin( f σ

j

), or c

str

i, j

= α

str

|sin( f σ

j

)|, (2)

where α

str

is an artistically-chosen scale factor, f is

the frequency of the variation, and σ

j

is the longitu-

dinal angle of points~x

i, j

for all values of i. This angle

can be expressed as:

σ

j

= 2π

j

j

max

. (3)

Another type of variation, seen in Figure 3, is caused

by the nonuniform elastic properties of the jellyﬁsh’s

mesoglea, which produce ripples across the contract-

ing umbrella. This variation is also roughly sinu-

soidal, though its amplitude depends on the amount of

umbrellar contraction. To account for contraction ef-

fects, we use a spring’s uncontracted rest length l

r

and

its current length l

c

to deﬁne a contraction coefﬁcient

that is at 1 when l

c

= l

r

, and 0 when l

c

= (0.46)l

r

.

Then, the overall displacement of the umbrella hull

can be described:

c

cmp

i, j

= α

cmp

l

c

− γl

r

l

r

− γl

r

n

η + κ

sin(σ

j

) + 1

2

o

. (4)

The constants η = (1.34 −1), γ = (1 −0.54), and κ =

(1.51 − 1.34) all come from Figure 3. Note that only

points on the exumbrellar surface are affected in this

manner.

Lastly, asymmetric differences arise between in-

dividuals of the same species. We mimic this varia-

tion artistically using Perlin noise (2002). We control

the frequency of each dimension of the noise func-

tion independently, with the intent of qualitatively ap-

proximating the look of a target jellyﬁsh. Since the

mesoglea is thicker at the top of the umbrella, it is

more resistant to deformation in this region. We at-

tenuate the amplitude of the noise function for points

near the peak of the umbrella. Our expression for

the displacement of the umbrella points due to non-

periodic variation is as follows:

c

noise

i, j

= α

noise

i

i

max

∗ Q

P

2i

i

max

,

8 j

j

max

,

t

100

, (5)

where t is the simulation time, and Q

P

is the Perlin

noise ﬁeld. We apply this displacement to both sub-

umbrellar and exumbrellar points, and also tentacle

points.

Expanded Contracted

3D

Ex-

trap.

Struct.

Compr.

Noise

All

Figure 9: Our 3D extrapolation, and the types of noise that

we apply to the hull of the umbrella. The ﬁrst column of im-

ages shows the jellyﬁsh at its expanded (rest) conﬁguration,

and the bottom row shows the contracted conﬁguration.

4 RESULTS

Figure 10 shows a full cycle of our jellyﬁsh swim-

ming. We render the ﬁnal model with a simple Lam-

bertian surface lighting model. We ﬁnd the motion

of our jellyﬁsh to be quite convincing, comparable

to previous animations of jellyﬁsh such as in Pixar’s

“Finding Nemo” (2003). Our method has the advan-

tage that the resulting animation’s upward movement

is more closely tied to the contraction of the organ-

ism, and the virtual organism interacts with its envi-

ronment directly and automatically. Also, our mor-

phology appears to more closely resemble that of real

GRAPP 2009 - International Conference on Computer Graphics Theory and Applications

246

Figure 10: A sequence of frames from our animation sys-

tem.

jellyﬁsh.

Although accuracy is not the primary concern of

our model, we discuss it brieﬂy here. We found

experimentally that our model has a similar reso-

nant frequency to captured data from Dabiri and

Gharib (2003). However, the motion that our model

generates is noticeably different from empirical data

of jellyﬁsh. Figure 5 shows high-frequency motion

that has a higher amplitude than that shown by Dabiri

and Gharib (2003), meaning that our model’s posi-

tion oscillates up and down more, relative to its over-

all translational motion.

We are able to achieve real-time animations at 30

to 40 frames per second on modest hardware (AMD

Turion 1.6 GHz processor), depending on the reso-

lution of our 2D simulation and of our 3D extrapo-

lation. We experimentally ﬁnd that time-steps of the

order of 0.01 seconds begin to hit the stability lim-

its of the system. Each frame must be broken into

multiple simulation steps, but of course, resampling

and 3D extrapolation only needs to be done once per

frame. Figure 11 shows the parameters of our sim-

ulations, and the values that we used to generate the

animations that are featured in this paper.

Parameter Value

Fluid Grid Height/Width (n) 100

Fluid Viscosity (ν) 1.304 ×10

−3

Fluid Density (ρ) 1

Safety Factor for Integration Step Size 0.001

Jellyﬁsh Contraction Frequency 0.7Hz

Jellyﬁsh Diameter 40mm

Jellyﬁsh Height 28mm

Jellyﬁsh Thickness (ϑ) 4mm

Hookean Elastic Modulus 1186Pa

Angular Elastic Stiffness 1.186 N · m/rad

Tentacle Cross-sectional Area Factor 1/100

Structural Variation Scale (α

str

) 0.375 ·ϑ

Compression Variation Scale (α

cmp

) 0.25 ·ϑ

Structural Variation Scale (α

noise

) 2.25 ·ϑ

Figure 11: Parameters of our animation system.

5 CONCLUSIONS AND FUTURE

WORK

We simulate a model of jellyﬁsh numerically, which

accounts for the elastic forces of the organism as it

contracts its muscles, as well as the reaction of the

sea water that surrounds the organism. Simulation al-

lows us to model the direct interaction of the jellyﬁsh

and its environment. We simulate only a 2D vertical

slice of the jellyﬁsh and exploit the axial symmetry

of the organism. We also concentrate on the resonant

gait of an adult jellyﬁsh, in which it contracts at its

submerged natural resonant frequency.

We represent the jellyﬁsh’s ﬂesh and subumbrellar

muscles as a mass-spring system (Terzopoulos, Platt,

Barr & Fleischer, 1987) that consists of a combina-

tion of linear and angular springs. We shorten the rest

lengths of subumbrellar springs to simulate the con-

traction of muscles within the umbrella based on an

artistically chosen periodic function. We account for

sea water with a semi-Lagrangian ﬂuid solver (Stam,

1999) in conjunction with the immersed boundary

method (Peskin, 2002).

We generate a higher resolution model from our

coarse simulations by threading a cubic spline inter-

polant around the simulated point-masses. We then

extrapolate to a 3D surface by deﬁning discs that go

through either side of the jellyﬁsh’s umbrella, and add

several forms of variation to the resulting 3D surface.

These variations are partly based on observations of

real jellyﬁsh, and partly artistic.

Much work could still be done with respect to jel-

lyﬁsh animation. We would like to remove the as-

sumption of axial symmetry, possibly by simulating

multiple 2D slices. Also, jellyﬁsh are capable of other

locomotion modes besides the common resonant gait,

and simulating these would increase the range of mo-

tion of our virtual jellyﬁsh.

INTERACTIVE JELLYFISH ANIMATION USING SIMULATION

247

We still know little about exactly how jellyﬁsh

contract their muscles to achieve jet propulsion. Van

de Panne and Fiume (1993) designed a means of

experimenting with the control processes for sim-

ple creatures using sensor-actuator networks. We

could perhaps leverage this kind of control explo-

ration within the context of jellyﬁsh.

In our work, we have not discussed how some

species of jellyﬁsh are able to reorient themselves.

Megill (2002) states that there are sparse muscle ﬁbres

in the bell mesoglea that are normal to the umbrella’s

surface. These ﬁbres can change the symmetry of the

umbrella and thus effect a course change for the or-

ganism as it swims, though the exact process is not

well understood. So far, we were not able to repro-

duce this phenomenon within our model.

We could improve our rendering of the jellyﬁsh by

considering translucence and bioluminescence, and

by improving our noise model. Also, our rendering

model currently lacks the venous structure visible in

some species of jellyﬁsh.

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