A Hyperbolic Geometric Flowfor Evolving Films and Foams

1,2Sadashige Ishida, 2Masafumi Yamamoto, 3Ryoichi Ando, and 2Toshiya Hachisuka

1:Nikon Corporation, 2:The University of Tokyo, 3:National Institute of Informatics

Overview

Reformulation of soap film dynamics as geometric flow
• Surface-driven simulation of soap film dynamics•Related Work

Bubble Animation
Grid-based

Zheng et al. [2006]

Kim et al. [2007]

Particle-based

Hong et al. [2008]

Busaryev et al. [2012]

Surface-Driven Soap Films

Durikovic [2001]

Zhu et al. [2014]

Da et al. [2015]Soap Films in Physics

Dynamic Fluids with Three Layers
external airliquid film
internal air

Simulation
Ideally, computable via
the Navier-Stokes equationsDu
1µ= rp +u+Dt⇢⇢+µr⇥ + g⇢

Challenges
500 to 800 nm

Thickness of films is extremely thin
Super-high resolution grid is necessary
Volumetric computation is too expensive

Soap Films in Mathematics

Geometric Property
Soap films evolve to reduce their surface area,
while preserving inner volumes.

…

Steady State

Area-Minimized

Steady States and Plateau’s Laws
Area-minimized shapes with volume constraints

Steady States and Plateau’s Laws
Area-minimized shapes with volume constraints

arccos( 1/2) = 120

arccos( 1/3) ⇡ 109

Plateau’s Problem
Mathematical formulation of
the steady statesZd✏✏|Su ⇥ Sv |dudv = 0d✏ ✏=0 U
General solutions are not found yet.

Solving Plateau’s Problem

Solving Plateau’s Problem
A common approach:
Evolve surface under Mean Curvature Flow.

Mean Curvature Flow (MCF)dx=dt

H(x, t)n(x, t)mean curvaturesurface normal

Property of MCF

Single open surface

Local minimum of
the area functional

Property of MCF

leads films to the steady states• no oscillation•Hyperbolic Mean Curvature Flow (HMCF)
2d x=2dt
constantH(x, t)n(x, t)mean curvaturesurface normal

Property of HMCF

Single open surface

Local minimum ofthe area functionalWe integrate this geometric view
into the film dynamics.

HMCF seems very good for soap film dynamics.
leads films to the steady states• with oscillation•HMCF seems very good for soap film dynamics,but …

Issues

Mean curvature is undefined on non-manifold junctions.
• HMCF does not preserve inner volume.
•Non-manifold junctions

Evolution under HMCF

Our Solution•

Use variational derivative of the area functional,
instead of mean curvature normal@AHn !
@x•

Volume preservation for multiple regions

Model Overview

Initial state

Intermediate state
Next state@A
Evolution by
Volume preservation@xHyperbolic Mean Curvature Flow

2d x
=2dt

H(x, t)n(x, t)Volume Preserving Hyperbolic
Geometric Flow for Multiple Surfaces
2d x
=2dt

@A+@xpnpressure difference across films

Volume Preserving Hyperbolic
Geometric Flow for Multiple Surfaces
2d x
@A
=+pn
2dt@x
surface tension force

air pressure from both sides

The Algorithm

Algorithm Overview

2d x
=2dt

@A+@x

pn

Algorithm Overview

Initial state

Intermediate state
Next state@A
Evolution by
Volume preservation
@x

2d x
=
2
dt

@A+@x

pn

@A
Evolution by @x

Initial state

Intermediate state@A
Evolution by
@x

2d x
=
2dt

@A+@x

pn

Variational Derivative of Area

@A@x
Gradient of the surface area on each point
Moving x toward@Aminimizes the area.
@xDefined Everywhere on Films

manifold point

non-manifold junction

Natural Extension of Hn@A
Common properties of @x andHn

Negative direction: minimizes the local area
• Magnitude: difference from the area-minimized configuration•@A
Indeed,= Hn on manifold points@xAfter the First Step

Intermediate stateAfter the First Step

Intermediate state
Volume preservation
Enclosed volumes may decrease at this point.

Next stateAfter the First Step

Intermediate state

Next state

Volume preservation
Need to resolve the volume change for all the regions.

Volume Preservation

Intermediate state

Next state

Volume preservation

2d x
=2dt

@A+@x

pnMüller’s Volume Preservation [2006]

Move each point toward the normal direction.
The correction amount d(t) is spatially constant.

Extension to Multiple Regions

The correction amount d(t, x) for each pointis related to the pressure difference.

Pressure and Volume Preservation

Performing volume preservation

Computing the pressure difference term
Assumptions• incompressibility• constant pressure per regionpnAfter the Second Step

Initial state

Intermediate state
Next state@A
Evolution byVolume preservation@xConnection to Physics

The Navier-Stokes equations
DuH (x)1=n + rpDt⇢⇢

Our geometric flow
2d x
@A=+pn2dt@x

Assumptions• infinitesimal thickness• constant pressure per region

Results

Computational Timings

ABCD

Plateau’s Laws

: 120
A: 109BCDE

Convergence of Surface Area

- - - Analytical solution

Volume Control
Achieved simply by changing the initial inner volumes.

External Force
Not well handled in previous work
Da et al. 2015 : Velocity is determined by circulation
Cannot directly add external force as accelerationOurs: Velocity is determined by acceleration
Straightforward to add external force

External Force

動画貼るComparison to Real Footage

experiment [Pucci et al. 2015]

our simulation

Conclusion

Efficient simulation method for soap film dynamics
Mathematical contributions:•
Volume preserving hyperbolic geometric flow for multiple surfaces•
Numerical solver for Plateau’s problem, even with presence of externalforce

Conclusion

Efficient simulation method for soap film dynamics
Mathematical contributions:•
Volume preserving hyperbolic geometric flow for multiple surfaces•
Numerical solver for Plateau’s problem, even with presence of externalforce

Conclusion

Efficient simulation method for soap film dynamics
Mathematical contributions:•
Volume preserving hyperbolic geometric flow for multiple surfaces•
Numerical solver for Plateau’s problem, even with presence of externalforce

Acknowledgements

Sigurd Ofstad, for discussion and experiments• Hisanari Otsu, for customization of Mitsuba renderer• Christopher Batty, Fang Da, and Raymond Yun Fei,on insightful discussions on the previous work
• Jamorn Sriwasansak, for the advice on the Thai language• Nikon Corporation, for the support by grants•

Source Code is Available

https://github.com/sdsgisd/HGF