# Dynamics of a Floating Object

Project report for CPSC 533B: Algorithmic Animation
Ken Deeter

 Solid rendering Wireframe rendering

## 1. Introduction

The simple goal of this project was to build a dynamics simulator that models the interaction of solid objects with water (or other liquids). Although there has been much work in the areas of simulating rigid body dynamics, as well as water, there has not been a significant amount studying the interaction of the two. Only the most complex of commercial particle system simulators provide support for modeling the complicated interaction between solid objects, with non-solid fluids.

## 2. Relevant Work

Much effort past has gone into modeling many different kinds of liquids. Approaches range from computational fluid dynamics, procedurally generated liquid surfaces, and approximations or simplifications of the Navier-Stokes equation. Study of this topic dates back to several reports such as [FOURNIER] and [PEACHY].

A recent paper by Foster and Fedikw [FOSTER], describe a system which can handle moving solid objects, but does not model the forces that a fluid exerts on that object.

As for the physics of floating objects, various levels of understanding have been around since the time of Archimedes. The ideas referenced in this project (in the domain of physics and mechanics) can be found in any standard physics textbook or related web reference.

## 3. Approach and Implementation

In order to make the project feasible in the limited timeframe, complexity of the model was reduced down to its basics as much as possible. This simple design assumes a height-field representation of a surface of a deep body of water, with directional sinewaves simulating water waves (with controllable amplitude, frequency, and direction). Only one type of object, a simple cube, is modeled. Although the size of the cube can be controlled, no other aspects of the shape can be modified. The implementation, however, does not make simulation of other shapes impossible. The cube was chosen because it simplifies many of the calculations required in modeling its interaction with a liquid. Both vertical and lateral forces are modeled, but rotation of the cube is not taken into account, resulting in a cube that is always "upright."(see future work)

To model the motion of solid objects in water, three main forces must be computed.
1. Buoyancy Force: According to Archimedes' principle, the net vertical force on a submerged object is equal to the weight of the water displaced.
2. Drag Force: As an object moves through a liquid, a force proportional to the velocity of the object, viscosity of the fluid, and cross-sectional area of the object, exists that resists the motion.
3. Gravity: As in most real-world simulations, the gravity exerts a constant downward force proportional to an object's mass

### Calculating the Buoyancy Force

A true complete calculation of a buoyancy force on an object involves integrating the pressure exerted by the surrounding water over the object's surface. Because static pressure at a point in water depends on depth, a net upward force results from the fact that the surfaces on the shallower end of an object experience less pressure than the deeper surfaces.

For completely submerged objects, this net force is roughly equal to the weight of the fluid displaced by the object, or in other words, the volume of the object times the fluid's density.

For partially submerged objects, a slight approximation is necessary to avoid calculating the surface integral. In my implementation, I treat the partial submersion case in exactly the same manner as the fully submerged case, except that the "weight of the fluid displaced" is only calculated for the partial volume of the object that is actually below the water's surface.

To account for the fact that the partially submerged portion of the object may not necessariy correspond to the "lower" portion of the object (which would imply that the resulting buoyancy force would always be pointing "up"), the force is applied in a direction obtained from the water surface's normal vector. This simple approximation allows for somewhat realistic lateral movement of an object at the water's surface, as if it is being pushed around by a wave.

### Calculating the Submerged Volume of the Cube

The most general method of calculating the volume of water displaced by an object, is to divide its volume into many smaller sub-volumes, and determe the depth of a point representing each sub-volume. If the "height" coordinate (in world-space) of that point is lower than the height of the water's surface at that point, then we can make an approximation and treat the entire sub-volume as being submerged. Summing up all such submerged sub-volumes will yield an approximation of the total partial submerged volume.

In my implementation, I use a simpler method to approximate the submerged volume of the cube. Because the cube is always upright, the amount of submersion can be approximated by evaluating the height of the water surface at each of vertically oriented edges of the cube. Specifically, the approximation is computed as follows:
1. Compute the depth of each of the four lower corners of the cube
2. If a corner is found to be above the water of the surface, then assign its corresponding vertical edge a submerged height value of 0. If it's depth is greater than the length of one of the edges of the cube, then assign it a height value equal to the single edge length. Otherwise, assign the edge a height value equal to the depth of the corner.
3. Average the four height values, and multiply the average by the area of the cube's base.
This approximation only works for relatively small cubes and relatively low frequency water surfaces. In effect, it samples four surface locations (for corners of a square), and approximates the surface height across the square patch to be the average of the heights at each of the four locations.

### Calculating the Drag Force

The drag force on an object moving through a liquid is proportional to the product of the square of its velocity, the density of the fluid, and the cross-sectional area of the object. A constant is also introduced, which depends on the shape of the object.

In this implementation, the three values mentioned above are incorporated into the calculation of the drag force. The density of water is fixed at 1000 kilograms per cubic meter and the cube's cross-sectional area is approximated by the area of one of the cube's faces.

### Object-water collision

When an object collides into a water surface, a great portion of its kinetic energy is transferred to the water, resulting in splashing and waves. To simulate this loss of energy, a hard-coded percentage of the cube's velocity is deducted when a collision with the water's surface is detected. The percentage of determined empirically. A value of approximately 90% appears to produce reasonable results.

### Modeling Waves

Originally, I intended to use a method based on [HINSINGER] to model the water waves. Although this model produced more realistic results, it has one major problem: it is difficult, given an (x,y) coordinate in the surface plane of the water, to determine the height of the water at exactly that point. This is due to the fact that Hinsinger et al.'s method distorts the (x,y) location of a point on the surface to model the realistic elliptical motion actual water molecules.

In calculating buoyancy forces, it is necessary to often query the height field for a water surface height value at an arbitrary position (x,y) in the plane. Because of this requirement, I chose instead, to use simple sinusoidal functions to model each wave, as they only modify the height of each position on the surface, and not the lateral position.

The tradeoff in realism is not show-stopping, especially for small waves, and the added simplicity makes the computation much more efficient. One idea that was retained from the implementation, is that the water surface's normal vectors are always computed analytically, providing for better results in both rendering and dynamics computation.

### Integration

The midpoint method (Heun's method) is used to acheive numerical stability. The default maximum time increment for the integration is 100 milliseconds.

## 4. Results

Click here (2.2MB) to view a 15fps 24 second recording of the system in action. Note that, the actualy system runs at full framerate (30fps+) on a pentium4 1.8ghz. The lower framerate was required since framebuffer recording happens in realtime. (NOTE: this file was encoded using the Divx4 codec)

The cube object modeled in this video has a mass of 200 kg, has 1 meter length on each edge. As the equivalent weight of water for such a volume is 1000kg, the object is fairly buoyant, and can be seen being tossed into the air by passing waves.

In a second file (1.0MB), the same object is shown being affected by a very large slow moving wave. (The slight change in framerate is due to a rendering glitch)

## 5. Future Work

The work presented in this report can be extended in many straightforward ways:
• Support for a variety of shapes. One need only solve the submerged partial volume computation problem for each new shape.
• Support for multiple solid objects and interactions among them.
• Support for modeling rotation, torque, and angular velocity. A method similar to the previously described integration method can be used to compute a net torque on the floating object.
• Model effects of object on water. As of now, the system only models the forces and effects of the water on the solid object. To be complete, the system also needs to acount for perturbations to the water's surface caused by the object's presence, including ripples and splashes.
• Better simulation of water. Although a more realistic simulation of water waves was originally intended, a simpler method was chosen for computational simplicity. In theory, however, any model of a body of water's surface can be used for this type of simulation.

## References

kdeeter cs ubc ca
Last modified: Fri Apr 25 03:03:59 PDT 2003