Transcript Water Droplet Simulation Avneesh Sud Problem • Realistic Animation of individual water droplets on surfaces – Lot of work on ‘large’ water bodies (Napier-Stokes, Stam) •

Water Droplet Simulation
Avneesh Sud
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Problem
• Realistic Animation of individual water droplets
on surfaces
– Lot of work on ‘large’ water bodies (Napier-Stokes,
Stam)
• Water droplets tougher
–
–
–
–
Gravity
Interfacial , surface tension
Air resistance
Accurate rendering
• Two Problems : Flow and Rendering
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Work Presented
• Kaneda93 : Water Droplets on Glass Plate
• Kaneda96 : Droplets on a Curved Surface
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Flow of Water Droplets
• Physics studied by Janosi89
• Droplet runs down glass plate if mass
exceeds critical mass
• Route of stream determined by impurities
and presence of other droplets
• Traces of water left behind on surface :
“Wetting”
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Simulation of Flow
• Surface divided into a nxn lattice
• An affinity for water fi,j assigned to each lattice
point
– Depends on surface properties, impurities etc.
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Simulation of Flow
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•
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•
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When does droplet move?
Which direction does it move in?
What speed does it move with?
How much moves/left behind?
When does it stop?
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Critical Mass
• Each lattice point associated with a mass
mi,j
• Droplet moves if
mi , j  m ( )
s
c
msc() is the static critical weight when inclination angle is 
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Direction of Motion
• From (i,j) to (i-1,j+1), (i,j+1) or (i+1,j+1)
• If water exists in one of the 3 directions, then
moves to that direction
– If exists in more than 1 direction, then to (i,j+1)
– If not exist in (i,j+1), then to direction with larger mass
• If not exist in any direction, then tendency to
move in direction (i+k,j+1) is
di k , j 1  w1 (, k ) fi k , j 1
w1(,k) is the inclination weighting function
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Speed of water
• Depends in wetness of new direction and
inclination
v  v0  ai k , j 1 ( )t
• Acceleration ai,j() can be dry acceleration
ad() or wet acceleration aw(),
aw() > ad()
t is time from generation/collision
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Mass of Moving Water
• Mass of remaining water m’i,j depends on
fi,j, mmin, mmax

m
(if mi , j  f i , j mmin )
 i, j
m  mmin

mi, j  w2 f i , j (mi , j  f i , j mmin )  f i , j mmin (if f i , j mmin  mi , j  f i , j mmin  max
)
w2

mmax  mmin
f m
(
if
m

f
m

)
i, j
i , j min
 i , j max
w
2

• Mass of water at new position
mik , j 1  mi k , j 1  mi , j  mi, j
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Merging
• Conservation of mass and momentum
m1v1  m2v2
v0 
m1  m2
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Stopping
• With no water ahead, decelerates and stops
when mass is less than dynamic critical
mass mdc()
• Speed during deceleration given by
v  v0  ad ( )t
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Shape of Droplet
• Shape of static droplet on glass plate determined
by interfacial, surface tensions (de Gennes 85)
 SL   S   L cos  0
• Dynamic droplets complicated – different contact
angles at head & tail, several contact angles to
minimize energy
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Shape of Droplet
• Approximated by a sphere
( x  x0 ) 2  ( y  y0 ) 2  ( z  z0 ) 2  r 2
z  r cos  z0
• Radius of sphere dependent on mass


3m

r  
3
  (2  cos   3 cos ) 
1
3
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Rendering
• Background textures generated for environment mapping
– Front face at higher resolution
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Rendering
• Tracing reflected/refracted rays
– Most rays either reflected or refracted : follow one principle
direction
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Rendering
• Determining pixel color using the
background texture at the intersection point
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Results (Flow)
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Results (Animation)
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Curved Surfaces
• Modeled as Bezier Patches
• Discretized to a quad mesh – approximated by a plane
• A more generic method that reduces discretization artifacts
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Critical Mass
• Movement Test : Sum of external (gravity,
wind) forces greater than affinity force
f i ,extj  Fi , j
f i ,extj  f g ( i , j )  f w( i , j )
Fi , j   s ci , j
Fi,j is the static critical force
s is degree of affinity coefficient
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Direction of Motion
• “Roulette” of 8 adjacent meshes used
• Probability assigned to each mesh
• Using probabilities allows to simulate
random ‘meandering’ of water
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Direction of Motion
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Direction of Motion
• Weight of each direction determined by
– Newton’s Laws of Motion
– Degree of affinity for water
– Wetness of neighboring meshes
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In the beginning …
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Newton’s Laws
• Compute principle velocity vector VP from
Newton's mechanics.
f i ,extj  mi , j ai , j (t )
f
ext
i, j
 f g (i , j )  f w(i , j )
Solve to find position P where droplet crosses
boundary and the velocity VP at crossing point
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Newton’s Laws
• Probabilities distributed between 2 nearest
neighboring directions of VP (l, l+1)


 1
*
d
(k  l , l  1)

k  VP
Dk   Dsum

(otherwise)
0
Dsum  dl*  VP  dl*1  VP

 

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Affinity
• Probability distributed to directions whose
angle from VP is less than 90 deg

ck
Ak 
u d k*  VP
Asum
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

Asum   cl u dl*  VP

l 1
ck is degree of affinity in neighboring mesh
u(x) is unit step function
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Wetness
• Probability distributed to directions whose angle
from principle vector is less than 90 deg
– Prevents a droplet from suddenly changing direction
back to a wet spot

g (d k* )
Wk 
u d k*  VP
Wsum
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

Wsum   g (d l* ) u d l*  VP

l 1
g(d*k) is a function to check wetness
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Determining direction
• All 3 probabilities weighted and summed up
Rk 
1 VP Dk   2 Ak  Wk
Rsum
1 controls regularity, 2 controls meandering
• Direction depends on generated random
p 1
p
number r
R
k 1
k
 r   Rk
k 1
p is the desired direction of movement
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Speed of Travel
• Project acceleration in direction of principle
vector
• Internal forces depend on affinity
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Mass of moving drop
• “Wetting” : Mass of remaining water
mi, j  hci , j mi , j
h(c) is a function for remaining water based on
affinity
• “Merging” : Follow conservation of mass,
momentum
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Algorithm
1.
2.
3.
4.
5.
6.
7.
8.
9.
Make a discretized surface model, specify degree of affinity for each
mesh element
Put new droplets at mesh points, specify weight and initial speed
Initialize timekeepers for all droplets to accumulate time taken for a
droplet to travel from one mesh to another
If droplet moves, goto step (5) else repeat (4) for all droplets
Determine mesh droplet will visit next
Calculate time required for droplet to move to next mesh
Move the droplet to next mesh based on timekeeper value
Repeat steps (4)-(7) for all droplets till end of frame
Repeat (2)-(8) for duration of animation
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Rendering
• Similar to glass plate –efficient bezier
intersection tests used
• Higher quality method based on metaballs
– Efficient intersection between ray and metaball
using bezier clipping
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Results
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Results
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References
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Janosi IM, Horvath VK Dynamics of Water Droplets on a Window
Pane. Physical Review 40(9): 5232-5237 (1989)
De gennes PG. Wetting: Statucs and Dynamics. Rev. Mod. Phys.
57(3): 827-863 (1985)
Kaneda K, Kagawa T, Yamashita H. Animation of Water Droplets on
a Glass Plate. Proc. Computer Animation’93: 177-189 (1993)
Kaneda K, Zuyama Y, Yamashita H, Nishita T. Animation of Water
Droplet Flow on Curved Surfaces. Proc. PACIFIC GRAPHICS '96:
50-65 (1996).
Yu YJ, Jung HY, Cho HG. A New Rendering Techniques for Water
Droplet using Metaball in the Gravitation Force. WCGS'98 (1998)
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