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### 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 droplets tougher
- Gravity
- Interfacial , surface tension
- Air resistance
- Accurate rendering
- Two Problems : Flow and Rendering

Work Presented

- Kaneda93 : Water Droplets on Glass Plate
- Kaneda96 : Droplets on a Curved Surface

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”

Simulation of Flow

- Surface divided into a nxn lattice
- An affinity for water fi,jassigned to each lattice point
- Depends on surface properties, impurities etc.

Simulation of Flow

- 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?

Critical Mass

- Each lattice point associated with a mass mi,j
- Droplet moves if

msc() is the static critical weight when inclination angle is

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

w1(,k) is the inclination weighting function

Speed of water

- Depends in wetness of new direction and inclination
- Acceleration ai,j() can be dry acceleration ad() or wet acceleration aw(),

aw() > ad()

t is time from generation/collision

Mass of Moving Water

- Mass of remaining water m’i,jdepends on fi,j, mmin, mmax
- Mass of water at new position

Merging

- Conservation of mass and momentum

Stopping

- With no water ahead, decelerates and stops when mass is less than dynamic critical mass mdc()
- Speed during deceleration given by

Shape of Droplet

- Shape of static droplet on glass plate determined by interfacial, surface tensions (de Gennes 85)
- Dynamic droplets complicated – different contact angles at head & tail, several contact angles to minimize energy

Shape of Droplet

- Approximated by a sphere
- Radius of sphere dependent on mass

Rendering

- Background textures generated for environment mapping
- Front face at higher resolution

Rendering

- Tracing reflected/refracted rays
- Most rays either reflected or refracted : follow one principle direction

Rendering

- Determining pixel color using the background texture at the intersection point

Curved Surfaces

- Modeled as Bezier Patches
- Discretized to a quad mesh – approximated by a plane
- A more generic method that reduces discretization artifacts

Critical Mass

- Movement Test : Sum of external (gravity, wind) forces greater than affinity force

Fi,j is the static critical force

s is degree of affinity coefficient

Direction of Motion

- “Roulette” of 8 adjacent meshes used
- Probability assigned to each mesh
- Using probabilities allows to simulate random ‘meandering’ of water

Direction of Motion

- Weight of each direction determined by
- Newton’s Laws of Motion
- Degree of affinity for water
- Wetness of neighboring meshes

Newton’s Laws

- Compute principle velocity vector VP from Newton\'s mechanics.

Solve to find position Pwhere droplet crosses boundary and the velocity VPat crossing point

Newton’s Laws

- Probabilities distributed between 2 nearest neighboring directions of VP (l, l+1)

Affinity

- Probability distributed to directions whose angle from VPis less than 90 deg

ckis degree of affinity in neighboring mesh

u(x) is unit step function

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) is a function to check wetness

Determining direction

- All 3 probabilities weighted and summed up

1 controls regularity, 2 controls meandering

- Direction depends on generated random number r

p is the desired direction of movement

Speed of Travel

- Project acceleration in direction of principle vector
- Internal forces depend on affinity

Mass of moving drop

- “Wetting” : Mass of remaining water

h(c) is a function for remaining water based on affinity

- “Merging” : Follow conservation of mass, momentum

Algorithm

- 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

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

References

- 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 Wwater 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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