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DISORDER6 1Disruptive Publishers Themes and Pics 2. Disturbia 6Divinity II DKS 6DJCAZPA 3DOA5 Ultimate 1Doctor Who Gamer Pics and Themes 8Do. Don. Pachi Res. 1. Dollar Dash 2Domino Master 2Do. Nn. An. 94 7Donnie. T 1Donnyskillz 7DOOM 2Dopiest. Product 1Doritos 5Doritos Crash Course 2Doritos Crash Course 2 2Doritos Crash the Super Bowl 5Doritos Unlock Xbox 4Doritos Unlock Xbox II 9Double D Dodgeball 5Double Dragon 1. Double Dragon Neon 2Double. Cowboy Programming Practical Fluid MechanicsThis article originally appeared in two parts in Game Developer Magazine, March and April, 2. Fluid effects such as rising smoke and turbulent water flow are everywhere in nature, but are seldom implemented convincingly in computer games. The simulation of fluids which covers both liquids and gasses is computationally very expensive. It is also mentally very expensive, with even introductory papers on the subject relying on the reader having math skills at least at the undergraduate calculus level. In this article I will attempt to address both these problems from the perspective of a game programmer not necessarily conversant with vector calculus. Ill explain how certain fluid effects work without using advanced equations and without too much new terminology. I shall also describe one way of implementing the simulation of fluids in an efficient manner without the expensive iterative diffusion and projection steps found in other implementations. A working demonstration with source accompanies this article and can be downloaded from here, example output from this can be seen in figure 1. GRIDS OR PARTICLES There are several ways of simulating the motion of fluids. These generally divide into two common types of methods grids methods and particle methods. In grid methods, the fluid is represented by dividing up the space a fluid might occupy into individual cells, and storing how much of the fluid is in each cell. In particle methods the fluid is modeled as a large number of particles that each move around and react to collision with the environment and interacting with nearby particles. Here Im going to concentrate on simulating fluids with grids. It is simplest to discuss the grid methods with respect to a regular two dimensional grid, although the techniques apply equally well to three dimensions. At the simplest level, to simulate fluid in the space covered by a grid you need two grids, one to store the density of liquid or gas at each point, and another to store the velocity of the fluid. Figure 2 shows a representation of this, with each point having a velocity vector, and also containing a density value not shown. The actual implementation of these grids in CC is most efficiently done as one dimensional arrays. The amount of fluid in each cell is represented as a float. The velocity grid also referred to as a velocity field, or vector field could be represented as an array of 2. D vectors, but for coding simplicity it is best represented as two separate arrays of floats, one for x and one for y. In addition to these two grids we can have any number of other matching grids that store various attributes. Again each will be stored as matching array of floats, and can store things such as the temperature of the fluid at each point, or the color of the fluid whereby you can mix multiple fluids together. You can also store more esoteric quantities such as humidity, for if you were simulating steam or cloud formation. ADVECTIONThe fundamental operation in grid based fluid dynamics is advection. Advection is basically moving things around on the grid, but more specifically its moving the quantities stored in one array by the movement vectors stored in the velocity arrays. Its quite simple to understand what is going on here if you think of each point on the grid as being an individual particle, with some attribute the density and a velocity. You are probably familiar with the process of moving a particle by adding the velocity vector to the position vector. On the grid, however, the possible positions are fixed, so all we can do is move advect the quantity the density from one grid point to another. In addition to advecting the density value, we also need to advect all the other quantities associated with the point. This would obviously include additional attributes such as temperature and color, but also includes the velocity of the point itself. The process of moving a velocity field over itself is referred to as self advection. The grid does not represent a series of discreet quantities, density or otherwise, it actually represents inaccurately a smooth surface, with the grid points just being sampled points on that surface. Think of the points as being X,Y vertices of a 3. D surface, with the density field being the Z height. Thus you can pick any X and Y position on the mesh, and find the Z value at that point by interpolating between the closest four points. Similarly while advecting a value across the grid the destination point will not fall directly on a grid point, and you will have to interpolate your value into the four grid points closest to the target position. In figure 3, the point at P has a velocity V, which, after a time step of dt, will put it in position P P dtV. This point falls between the points A, B, C and D, and so a bit of P has to go into each of them. Generally dt will be significantly smaller than the width of a cell, so one of the points A,B,C or D will be P itself. Advecting the entire grid like this sufferers from various inaccuracies, particularly that quantities dissipate when moving in a non axis axis aligned direction. This inaccuracy can actually be turned to our advantage. STAMS ADVECTIONProgrammers looking into grid based fluid dynamics for the first time will most often come across the work of Jos Stam and Ron Fedkiw, particularly Stams paper Real Time Fluid Dynamics for Games, presented at the 2. Game Developer Conference. In this paper Stam presents a very short implementation of a grid based fluid simulator. In particular he describes implementing the advection step using what he terms a linear backtrace, which simply means instead of moving the point forward in space, we invert the velocity and find the source point in the opposite direction, essentially back in time. We then take the interpolated density value from that source which, again, will lay between four actual grid points, and then move this value into the point P. See figure 4. Stams approach produces visually pleasing results, yet suffers from a number of problems. Firstly the specific collection of techniques discussed may be covered by U. S. patent 6,2. 66,0. Stam notes, the approach of backtracing dates back to 1. Check with your lawyer if this is a concern to you. On a more practical note the advection alone as described by Stam simply does not work accurately unless the velocity field is smooth in a way termed mass conserving, or incompressible. Consider the case of a vector field where all the velocities are zero except for one. In this situation the velocity cannot move advect forward through the field, since there is nothing ahead of it to pull it forward, instead the velocity simply bleeds backwards. The resultant velocity field will terminate at the original point, and any quantities moving through this field will end up there. This problem is solved by adding a step to the algorithm termed projection, which is basically smoothes out the velocity by making it incompressible, thus allowing the backtracing advection to work perfectly, and making the paths formed by the velocity be swirly, as would be the case in real water. The problem with this approach is that projection is quite expensive, requiring 2. Another performance problem with Stams approach is that there is a diffusion step, which also involves 2.

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