August 17, 2016

Back in the early nineties, the demoscene was in love with a simple fire effect, that generally looked something like this:

Fire demo by Iguana - screenshot by pouet.net

That doesn’t look much like real fire, but with some simple tweaks you can get something that looks good, runs on most graphics hardware and is easy to implement:

Try touching the fire, or clicking and dragging with your mouse. If you’re on a phone or tablet, you can rotate your device and see how the fire aligns itself correctly.

Even though the effect is easy to implement, fairly cheap to simulate and looks really good, I rarely see it in games. It doesn’t translate very well to three dimensions (since all you get is a flat texture), but it works great for 2D games. It seems like there should be tons of indie 2D games with awesome fire effects, flamethrowers, burning zombies, torches or whatever, but since I haven’t found any, here’s my attempt to document how you make a simple and awesome fire simulation.

## A very simple model of fire

The very simple model of fire used for the effect contains four parts: a heat source (generally something that is burning, i.e the top of a torch), convection (the fact that hot stuff tends to rise and cool stuff tends to sink), diffusion (heat will gradually spread out), and cooling.

The basic idea is to start with a grid of pixels, each pixel representing heat at that location. To get the fire started we plot heat (i.e pixels with maximum intensity) at the locations in the grid which have burning fuel.

To simulate diffusion, the grid of pixels is blurred. This simulates heat spreading to neighboring grid points. Convection is simulated by scrolling the grid upwards, and cooling by simply subtracting a constant from each pixel.

To render the fire, we map pixel intensities in the heat texture to a color ramp that goes from black, via red and orange to white, simulating black body radiation.

Example color ramp

Then we repeat the process, injecting heat, blurring and convecting and so on.

It is simple to combine the blurring and convection into a single step. In pseudocode:

sum = pixel(x+1, y) + pixel(x-1, y) + pixel(x, y+1) + pixel(x, y-1)
pixel(x, y - 1) = (sum / 4) - cooling_amount


This is easy to translate into WebGL, we store the heat in a texture, and use a pixel shader to blur and convect the heat. To inject heat where fuel is burning we render the burning parts to the heat texture with additive blending, modulated with a high frequency noise texture to simulate flicker. By translating the noise texture a random amount every flame, we get randomly flickering burning fuel.

Implementing this will give you something that looks a bit like this:

It sort of looks like fire, but the movement is strictly vertical, it doesn’t really move in the billowing, fluid-like way real fire does. Also the shape of the “flames” is very blobby and soft.

## Uneven cooling

The easiest way to dramatically improve the look of the fire is to introduce uneven cooling. Instead of simply using a constant cooling factor, we use a cooling map, a grid of pixels of the same size as our heat grid. You want the cooling texture to be fairly smooth and to contain a mix of large blobs and high frequency noise. Here’s a good cooling texture:

This particular texture was generated in processing.js by drawing circles of random sizes and intensities and blurring the image repeatedly.

The cooling grid/texture should tile and scroll upwards at approximately twice the speed of the heat grid. Varying the rate at which the cooling texture scrolls affects the look of the flames, a slower scroll speed gives a more billowing look, and a faster scroll speed makes the fire look more turbulent.

As far as I know, the first description of using a cooling texture instead of constant cooling was by Hugo Elias.

Uneven cooling improves the look of the fire dramatically, implementing it will make the fire simulation look something like this:

## Adding more fluid movement

In the current model, fire always rises upwards at the same speed, it never moves sideways or stops, and there are no swirls or turbulence. We sort of fake turbulence by scrolling the cooling map, but that doesn’t affect the direction the fire moves. To get the flames to move sideways and swirl, we must move the fire by a vector field instead, i.e. the direction we sample neighbors from when blurring and scrolling must vary across the grid.

A good candidate for computing this vector field is curl noise. Curl noise is fairly cheap to compute and is divergence free, i.e. the vector field generated won’t contain any sinks or “gutters”, areas where heat will get stuck. Being divergence free is equivalent to being incompressible when using the vector field to advect a fluid (i.e. fire in our case) and incompressibility is an important visual cue for fluids (although incompressibility probably is more visually important when simulating water).

The curl of a scalar valued potential field $F(x, y)$ is simply

The potential field $F$ is just noise, e.g. Perlin or Simplex noise. If we view the noise field as a height map, with high values indicating hills, and low values indicating valleys, then the gradient vector $(\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y})$ points along the slope of the height field. If we define the perpendicular product of a vector as $perp(x, y) = (y, -x)$, (i.e. the perpendicular product of a vector rotates it by 90 degrees clockwise), then the curl is simply the perpendicular product of the fields gradient vector. In other words, moving along the curl of the noise field is the same thing as moving along points of equal elevation, the contour lines of the field (think of if as moving along the contour lines of a topological map).

Now we just need to implement curl noise advection in WebGL. A simple approach is to evaluate a 3D noise function (we want the noise to vary both spatially and with time) per pixel and use the pixel shader’s partial derivative instructions to get the curl of that noise. GPUs process pixels in groups of four, so computing partial derivatives in the pixel shader can be done cheaply with the finite difference method.

float noise = noise(texcoord.s, texcoord.t, time);
vec2 curl   = vec2(dFdy(noise), -dFdx(noise));
vec2 offset = texcoord - upvector + (curl * scalingfactor);


This is simple but probably not optimal. Instead of computing the curl noise per pixel we can evaluate it at evenly spaced points and then simply interpolate the vectors to get a per pixel value. That way the curl noise can be computed in a vertex shader and we can simply render a tessellated quad to get an interpolated velocity vector per pixel. When we compute the curl in the vertex shader, we lose the ability to use cheap partial derivative instructions to get the derivatives. To fix this we can either use the finite difference method directly (but this of course requires multiple evaluations of the noise function) or we can use a form of noise where it is simple to directly get the derivatives, e.g. simplex noise.

The fire simulation on this page uses the simpler per pixel implementation. Since the fire shader does plenty of texture sampling, we have quite a lot of free arithmetic instructions to spend anyway, and it makes the code simpler.

## Other improvements

### Improved curl noise

Currently we use a single evaluation of the noise function at a medium frequency to compute our curl noise. This gives some nice swirling motion, but lacks higher frequency details. By summing several evaluations of noise function at different frequencies, we can add high frequency detail to the curl noise. This is generally referred to as fractal browninan motion and works like this:

Usually $s=2$ and $w=1/2$, so each term of the sum is called an octave of noise, because the frequency doubles with each term, just like a musical octave.

Of course, this requires multiple evaluations of the noise function and is therefore more computationally expensive, but it does add a lot of detail. There are many ways to mitigate this, perhaps computing high frequency detail per pixel, and low frequency detail per vertex.

Using a non uniform scaling of the curl noise to stretch it a bit vertically might also improve the shape of the flames, making them more elongated.

### Improved dynamic range

Since we are doing all computations on modern graphics hardware, there is really no need to use a standard 8-bit per channel RGBA texture when computing our simulation, 16-bit floating point makes much more sense. Fire is potentially very bright, so we want to represent values larger than 1.0. Rendering to floating point textures is not necessarily supported in WebGL (some mobile chipsets don’t support it), so another alternative is to encode a larger range than [0, 1] into several channels of an 8-bit per channel render target.

Tone mapping is straight forward since we already have a rendering step that maps heat to color, we just need some sort of exposure function to map values outside [0, 1] to our color ramp.

### Bloom

To really sell how hot and bright the fire is, we can add bloom. This is simply a matter of blurring and optionally downsampling the fire texture and blending the result on top of the rendered fire.

### Better looking fuel

The base of the flames (the burning fuel we add to the heat texture every simulation step) doesn’t really look like fire at all. They are just a flickering mess (which happens to produce pleasing flame like shapes), but they look nothing like fire. Maybe we can use animated noise to make the base of the flames more realistic, or we could just hide the flicker at the base by sampling the fire outside the base when we’re performing the final rendering step.

### Physics based calculations

The current fire simulation isn’t really based on physics, it’s just a bunch of hacks that look good. We could try to improve realism by using math more based on the actual physics. For example, the rate of cooling is actually proportional to the temperature to the fourth power, while our model just uses a randomly fluctuating cooling factor based on the cooling map, independent of the temperature.

Another inaccuracy is the speed of convection. When hot gas rises, the speed at which it does isn’t constant like in our model, but proportional to the temperature difference. Hotter air rises faster.

Of course, using physics based calculations might not improve the look of the fire at all. Since the simulation isn’t physics based to start with, we might be better off going with a full fluid simulation instead of making small changes to our fundamentally incorrect model.

## Get cracking

There’s no limit to all the stuff you could add: sparks or glowing pieces of fuel floating along the fire’s velocity field, smoke, motion blur for the flames, lightning the environment based on the fire texture, sound, a cellular automata based model of fuel consumption causing the fire to burn stuff away, a water simulation that causes the fire to extinguish, heat distortion and on and on and on.

So get started implementing an awesome fire effect for your game, I want to see flamethrowers, fireballs, fire brigade simulations and more!

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