Originally authored by Ivan Lee on 30 Jan. 2005

Droplet Modeling

The first approximation for a droplet model was originally an ellipsoid whose size was proportional to the mass of water droplet. Depending on the surface angle, the ellipsoid was then rotated to represent the gravitational pull on the droplet. This is quite primitive and was used mainly just to represent the droplet in a liquid flow simulation. [Djurcilov]

Later, [Kazufumi et al. 1993] created a model by taking the mass of the droplet and the contact angle to calculate the radius of the droplet. This equation is:

where the contact angle is defined as theta in the following diagram:

The contact angle can be solved for using the formula:

where is the surface tension of the surface, is the surface tension of the liquid, and is the interfacial tension between the liquid and the surface. However, the model does not take into account a dynamic droplet in which a droplet has more mass in the advancing portion than the receding one and thus has different contact angles between the two. This method instead provides a perfectly symmetric sphere intersected with a plane.

Using a different technique, [Yu et al. 1999] used a metaball representation to account for the contact angles. This model also accounts for the forces of friction and gravity such that the droplet deforms and is not symmetric when lying on an angled plane. The metaball isosurfaces are calculated using Wyvill's degree six polynomial to remove the additional computation time of calculating a square root in the cubic polynomial.

Merging adjacent metaballs using Yu et al.'s model is simply achieved by using a threshold value. When metaballs are close enough, the function values for each metaball is summed and set to some constant value. The following diagram shows the areas of influence for the droplets without gravity ( and ) which will not be merged since they do not exceed a threshold, while the deformed droplets and will be merged.

For additional control, the drops are also deformed using a series of control points. These are spread evenly across the droplet and equations are provided in [Yu et al. 1999] for calculating the new positions of surface points using the given deformed control points. This seems to be used more for user-controlled modeling than for following any physical models applied to these control points. Some results of using these control points can be seen in the following picture:

To account for friction and the gravitational field, [Yu et al. 1999] splits up the model of a droplet into a u-v-w coordinate system as shown in the following picture:

Here, u is coming out of the diagram, v is vertical, and w points to the left. The magnitudes of these vectors reflect the amount of distortion provided by gravity. Using these definitions, the force of friction is

where theta is the angle between w and the plate as labeled in the diagram above and R is the radius of the undeformed metaball. Additionally, given the ratio T defined as

we finally have a vector field, with the following equations showing the adjustments that need to be made to u, v, and w

This vector field is applied to the blending function for the metaball which will consequently be applied to each surface point of the metaball. The paper does mention a few pieces of future work to be done. First, the model is under the assumption that the droplet is stationary on the surface, ignoring the cases when the droplet is moving down the surface. This paper did not address the issue of making streaks as the droplet moves across the surface, however this is addressed by other papers. Additionally, the issue of unwanted blending is evident in some of the pictures provided by this paper. As droplets near each other, they have a tendency to melt into each other rather than pop once contact is made. Another drawback of this procedure is that it is an isosurface and cannot be rendered as quickly as other models.

[Murta and Miller 1999] also used isosurfaces with the metaball method with each particle having its own radius of influence. In addition to the forces discussed earlier, the paper also discusses sliding forces and its main concentration is the splitting and remerging of droplets after colliding with a surface. Once a droplet strikes a surface, it is split into between four and six child particles. This provides a more macroscopic view of droplet interaction but not as elaborate as a large body of water.

The actual modeling of this process uses the familiar technique of potential fields, but it also adds the idea of families. Families classify particles into different areas such that situations such as oil and water mixing can be simulated using this same general model. As multiple particles merge together, the isosurface value for the sum of the potential fields must be set to attempt to maintain a given volume. However, this value can be quite costly to calculate, as the paper outlines a method that voxelizes the cluster with an octree and sums up the interior and boundary voxels to estimate the volume.

Thus, the main drawback of this method is its cost. Most other papers' simulations provide quick results (less than 10 seconds, usually sub-second) and most of the computation is placed in the rendering of the droplets. [Murta and Miller 1999] however requires a large amount of collision detection and isosurface calculations that make this model a bit unreasonable in its raw form.

[Fournier et al. 1998] approached the problem differently by first declaring a set of constraints to be satisfied:

a. The tendency to have constant volume (a liquid is incompressible)

b. The tendency to have the smallest contact surface with air (capillarity forces)

c. The tendency to have the largest contact surface with the underlying surface

d. The tendency to be deformed by external forces (this paper only concerns itself with uniform force fields)

These constraints compete with each other to get the droplet into a shape satisfying all constraints or creating the best compromise. The paper explicitly states it does not follow any valid physical model, only the intuition behind these constraints. To satisfy the constraints, the model is based on a series of springs and masses laid out in a series of longitudinal and transverse springs, as shown in the following picture.

The model consists of M radial slices, N levels, and an apex point AN+1 Additionally, the guide point is the point on the surface to which the droplet is attached. This results in NM + 1 point masses, and the notation used is that point refers to point mass of level i and slice j. In the final rendering stages, the paper uses these points to form the control points of a B-spline patch.

The spring rest lengths are adjusted to intuitively follow the other constraints. For constraint b, all longitudinal springs are set to have zero rest length so that the droplet prefers to shrink together as tight as possible. The transverse springs are assigned rest lengths and stiffness values based on its height in the model so that the droplet maintains some structure:

where represents the rest length of the transverse springs of level , and is the spring stiffness of the springs in level . For constraint , the springs connecting each basis point to the guide point are set to have a large rest length, thus trying to stretch out the model. Constraint is trivially satisfied by taking any forces applied to the droplet and applying them to each mass.

Satisfying constraint a is a little more complicated. Instead of actually calculating the spline-interpolated shape�s volume, the volume of the model is expressed as:

where is the area of the basis level, is the distance between the th level and the basis level, and is a constant parameter depending on and . The value of remains relatively constant, thus the only part the affects the volume is the weighted sum of the heights. To preserve the volume, solco connections are introduced (short for sum-of-length-constant).

A solco connection has a series of L links connecting point-masses together with a certain stiffness ks, viscosity zs, and rest sum-of-lengths sol0. These are used along with the defined value sol:

which is the weighted sum of the lengths of these segments. Using these defined values results in a force to be applied on a point-mass as follows:

To use solco connections, each point mass is connected to the underlying surface using a solco connection. This effectively assures that the weighted sum of the heights of the levels maintains a certain rest height. Repulsive and attractive forces are exerted using the above equation to maintain a larger or smaller volume, respectively. Some results from this paper are shown below:

One limitation of this model is that the springs are forever fixed in this arrangement. This implies that two droplets cannot be animated to merge together, but this is reasoned as tolerable since the merging process is nearly instantaneous for this paper�s purposes, and thus is not an important issue. This also implies that droplets cannot split, however the same argument applies.

Also, due to the nature of the model, the transition between a free falling droplet and a droplet on a surface is not modeled appropriately. This is because the spring setup shown above has the level 1 springs attached to the planar surface with a large stiffness so that it is free to move but not by very much. This transition can possibly be solved by detection collisions between the point-masses and the surface.

Another assumption made by this model is that the underlying surface is planar due to the relative size of the droplet and the surface, though it does not seem that an extension to curved surfaces would be much more work. The triangles used for the surface are usually only big enough to fit a few droplets at once, thus this is a reasonable assumption to make and an extension to curved surfaces may not have much benefit.

Lastly, this method seems to require the use of several parameters that need to be experimentally tweaked. There is no physical parameter to adjust the mass of a droplet. In fact, the paper mentions the way to do this is to adjust the sol value to obtain heavier and lighter droplets. This can be justified to say that this model is mainly to find the equilibrium shape of the droplet rather than a dynamic model. Perhaps the dynamic aspects of droplet flow can be incorporated into maintaining constraint d.

References

[Djurcilov] Djurcilov, S. Animation of water dripping on geometry shapes and glass planes.

[Fournier et al. 1998] Fournier, P., Habibi, A., Poulin P. Simulating the flow of liquid droplets. Proc. Graphics Interface, 1998.

[Kazufumi et al. 1993] Kaneda K., Kagawa T., Yamashita, H. Animation of water droplets on a glass plate. Proc. Computer Animation '93, pp. 177-189 (1993).

[Murta and Miller 1999] Murta, A., Miller, J. Modelling and rendering liquids in motion. WSCG Proceedings, 1999.

[Yu et al. 1999] Yu, Y., Jung, H., Cho, H. A new water droplet model using metaball in the gravitational field, Computers & Graphics, 23(2):213-222, March, 1999.