UnityTerrainErosionGPU

Hydraulic and thermal erosion implement in Unity using compute shaders.

This is an example implementation of hydraulic and thermal erosion with shallow water equations. My initial motivation was to implement a game mechanic like in the From Dust game.

Disclaimer

This project is still in progress. Hydraulic erosion requires a bit more parameter tweaking and revisiting the actual implementation. But overall it works.

Demo

Just run Main Scene and use the mouse to draw modify terrain/water.

How it works

Explanation is still in progress...

Data / notation

We first need to list all the data we will operate with during the simulation. Since we are using a grid-based simulation we will need some per-cell information. Assuming we have a grid of size <img src="https://tex.s2cms.ru/svg/W%20%5Ctimes%20H" alt="W \times H" /> and current cell coordinates are <img src="https://tex.s2cms.ru/svg/(i%2C%20j)" alt="(i, j)" />.

• Simulation parameters
  • <img src="https://tex.s2cms.ru/svg/W" alt="W" /> - grid width
  • <img src="https://tex.s2cms.ru/svg/H" alt="H" /> - grid height
  • <img src="https://tex.s2cms.ru/svg/%5CDelta%20t" alt="\Delta t" /> - simulation time delta
  • <img src="https://tex.s2cms.ru/svg/g" alt="g" /> - gravity
  • <img src="https://tex.s2cms.ru/svg/l_x" alt="l_x" /> - cell size along <img src="https://tex.s2cms.ru/svg/x" alt="x" /> axis
  • <img src="https://tex.s2cms.ru/svg/l_y" alt="l_y" /> - cell size along <img src="https://tex.s2cms.ru/svg/y" alt="y" /> axis
  • <img src="https://tex.s2cms.ru/svg/l" alt="l" /> - pipe length
  • <img src="https://tex.s2cms.ru/svg/A" alt="A" /> - pip cross-section area
  • <img src="https://tex.s2cms.ru/svg/K_e" alt="K_e" /> - evaporation rate
  • <img src="https://tex.s2cms.ru/svg/K_r" alt="K_r" /> - rain rate
  • Hydraulic erosion parameters
    • <img src="https://tex.s2cms.ru/svg/K_c" alt="K_c" /> - sediment capacity
    • <img src="https://tex.s2cms.ru/svg/K_s" alt="K_s" /> - soil suspension rate
    • <img src="https://tex.s2cms.ru/svg/K_d" alt="K_d" /> - soil deposition rate
    • <img src="https://tex.s2cms.ru/svg/l_%7Bmax%7D" alt="l_{max}" /> - erosion depth limit
  • Thermal erosion parameters
    • <img src="https://tex.s2cms.ru/svg/K_t" alt="K_t" /> - thermal erosion rate
    • <img src="https://tex.s2cms.ru/svg/K%5Es_%5Calpha" alt="K^s_\alpha" /> - talus angle scale
    • <img src="https://tex.s2cms.ru/svg/K%5Eb_%5Calpha" alt="K^b_\alpha" /> - talus angle bias
• Grid-values
  • <img src="https://tex.s2cms.ru/svg/h%5E%7Bterrain%7D_%7Bi%2Cj%7D" alt="h^{terrain}_{i,j}" /> - height of terrain in cell <img src="https://tex.s2cms.ru/svg/(i%2C%20j)" alt="(i, j)" />, must be positive
  • <img src="https://tex.s2cms.ru/svg/h%5E%7Bwater%7D_%7Bi%2Cj%7D" alt="h^{water}_{i,j}" /> - height of water in cell <img src="https://tex.s2cms.ru/svg/(i%2C%20j)" alt="(i, j)" />, must be positive
  • <img src="https://tex.s2cms.ru/svg/s_%7Bi%2Cj%7D" alt="s_{i,j}" /> - suspended sediment amount in cell <img src="https://tex.s2cms.ru/svg/(i%2C%20j)" alt="(i, j)" />
  • <img src="https://tex.s2cms.ru/svg/r_%7Bi%2Cj%7D" alt="r_{i,j}" /> - terrain hardness in cell <img src="https://tex.s2cms.ru/svg/(i%2C%20j)" alt="(i, j)" />
  • <img src="https://tex.s2cms.ru/svg/f%5E%7Bleft%7D_%7Bi%2Cj%7D" alt="f^{left}_{i,j}" />, <img src="https://tex.s2cms.ru/svg/f%5E%7Bright%7D_%7Bi%2Cj%7D" alt="f^{right}_{i,j}" />, <img src="https://tex.s2cms.ru/svg/f%5E%7Bup%7D_%7Bi%2Cj%7D" alt="f^{up}_{i,j}" />, <img src="https://tex.s2cms.ru/svg/f%5E%7Bdown%7D_%7Bi%2Cj%7D" alt="f^{down}_{i,j}" /> - water flow (flux) in each direction in cell <img src="https://tex.s2cms.ru/svg/(i%2C%20j)" alt="(i, j)" />
  • <img src="https://tex.s2cms.ru/svg/p%5E%7Bleft%7D_%7Bi%2Cj%7D" alt="p^{left}_{i,j}" />, <img src="https://tex.s2cms.ru/svg/p%5E%7Bright%7D_%7Bi%2Cj%7D" alt="p^{right}_{i,j}" />, <img src="https://tex.s2cms.ru/svg/p%5E%7Bup%7D_%7Bi%2Cj%7D" alt="p^{up}_{i,j}" />, <img src="https://tex.s2cms.ru/svg/p%5E%7Bdown%7D_%7Bi%2Cj%7D" alt="p^{down}_{i,j}" /> - terrain mass flow (flux) in each direction in cell <img src="https://tex.s2cms.ru/svg/(i%2C%20j)" alt="(i, j)" />
  • <img src="https://tex.s2cms.ru/svg/v%5Ex_%7Bi%2Cj%7D" alt="v^x_{i,j}" /> - water velocity in <img src="https://tex.s2cms.ru/svg/x" alt="x" /> direction in cell <img src="https://tex.s2cms.ru/svg/(i%2C%20j)" alt="(i, j)" />
  • <img src="https://tex.s2cms.ru/svg/v%5Ey_%7Bi%2Cj%7D" alt="v^y_{i,j}" /> - water velocity in <img src="https://tex.s2cms.ru/svg/y" alt="y" /> direction in cell <img src="https://tex.s2cms.ru/svg/(i%2C%20j)" alt="(i, j)" />

Step 1. Water sources

We need to increase the water amount from water sources (e.g. rain). Right now the brush-drawing is omitted for clarity, but it happens in this step.

<img src="https://tex.s2cms.ru/svg/h%5E%7Bwater%7D_%7Bi%2Cj%7D%20%3A%3D%20h%5E%7Bwater%7D_%7Bi%2Cj%7D%20%2B%20%5CDelta%20t%20K_e" alt="h^{water}_{i,j} := h^{water}_{i,j} + \Delta t K_e" />

Step 2. Water flow computation

Water flow is proportional to the height difference in each cell because of pressure emerging from the height of water. Thus, to compute flow in each direction of cell we need to calculate the height differences:

<img src="https://tex.s2cms.ru/svg/%5CDelta%20h%5E%7Bleft%7D_%7Bi%2Cj%7D%20%3A%3D%20(h%5E%7Bterrain%7D_%7Bi%2Cj%7D%20%2B%20h%5E%7Bwater%7D_%7Bi%2Cj%7D)%20-%20(h%5E%7Bterrain%7D_%7Bi-1%2Cj%7D%20%2B%20h%5E%7Bwater%7D_%7Bi-1%2Cj%7D)" alt="\Delta h^{left}_{i,j} := (h^{terrain}_{i,j} + h^{water}_{i,j}) - (h^{terrain}_{i-1,j} + h^{water}_{i-1,j})" />

And repeat for other directions. Then we need to compute the outgoing amount of water from the current cell to the neighbors (in each direction) which is proportional to the volume diefference - height difference <img src="https://tex.s2cms.ru/svg/%5CDelta%20h%5E%7Bleft%7D_%7Bi%2Cj%7D" alt="\Delta h^{left}_{i,j}" /> multiplied by cell area (<img src="https://tex.s2cms.ru/svg/l_x%20%5Ctimes%20l_y" alt="l_x \times l_y" />):

<img src="https://tex.s2cms.ru/svg/f%5E%7Bleft%7D_%7Bi%2Cj%7D%20%3A%3D%20k%20%5Ccdot%20max%5CBig(0%2Cf%5E%7Bleft%7D_%7Bi%2Cj%7D%20%2B%20%5Cfrac%7B%5CDelta%20t%20g%20l_x%20l_y%20h%5E%7Bleft%7D_%7Bi%2Cj%7D%7D%7Bl%7D%20%20%5CBig)" alt="f^{left}_{i,j} := k \cdot max\Big(0,f^{left}_{i,j} + \frac{\Delta t g l_x l_y h^{left}_{i,j}}{l} \Big)" />

and also repeat the computation for each direction. <img src="https://tex.s2cms.ru/svg/k" alt="k" /> - is a special scaling factor to prevent situations when total outflow in 4 directions is higher than water volume in cell (since each direction is computed independently). Thus <img src="https://tex.s2cms.ru/svg/k" alt="k" /> is defined to split the scale the outflow accordingly - cell water volume divided by total outflow per single step:

<img src="https://tex.s2cms.ru/svg/k%20%3D%20min%5CBig(1%2C%20%5Cfrac%7Bh%5E%7Bwater%7D_%7Bi%2Cj%7D%20l_x%20l_y%7D%7B%5CDelta%20t%20%5Cbig%5Bf%5E%7Bleft%7D_%7Bi%2Cj%7D%20%2B%20f%5E%7Bright%7D_%7Bi%2Cj%7D%20%2B%20f%5E%7Bup%7D_%7Bi%2Cj%7D%20%2Bf%5E%7Bdown%7D_%7Bi%2Cj%7D%20%5Cbig%5D%7D%5CBig)" alt="k = min\Big(1, \frac{h^{water}_{i,j} l_x l_y}{\Delta t \big[f^{left}_{i,j} + f^{right}_{i,j} + f^{up}_{i,j} +f^{down}_{i,j} \big]}\Big)" />

If outflow will exceed the total volume, the fraction will be less than 1 thus flow will be reduced.

Boundaries

If we need the water to bounce off the walls we need to disable outgouing flow at the boundaries:

<img src="https://tex.s2cms.ru/svg/f%5E%7Bleft%7D_%7B0%2Cj%7D%20%3A%3D%200" alt="f^{left}_{0,j} := 0" /> <img src="https://tex.s2cms.ru/svg/f%5E%7Bright%7D_%7BW-1%2Cj%7D%20%3A%3D%200" alt="f^{right}_{W-1,j} := 0" /> <img src="https://tex.s2cms.ru/svg/f%5E%7Bdown%7D_%7Bi%2C0%7D%20%3A%3D%200" alt="f^{down}_{i,0} := 0" /> <img src="https://tex.s2cms.ru/svg/f%5E%7Bup%7D_%7Bi%2CH-1%7D%20%3A%3D%200" alt="f^{up}_{i,H-1} := 0" />

At the end of this step, we got all the outgouing flow computed at each cell so we now need to use this flow information to adjust the height of the water.

Step 3. Appling the water flow

The water height in each should increase by the total amount of incoming flow and decrease by the amount of total outgoing flow.

<img src="https://tex.s2cms.ru/svg/f%5E%7Bout%7D%20%3D%20f%5E%7Bleft%7D_%7Bi%2Cj%7D%20%2B%20f%5E%7Bright%7D_%7Bi%2Cj%7D%20%2B%20f%5E%7Bup%7D_%7Bi%2Cj%7D%20%2Bf%5E%7Bdown%7D_%7Bi%2Cj%7D" alt="f^{out} = f^{left}_{i,j} + f^{right}_{i,j} + f^{up}_{i,j} +f^{down}_{i,j}" />

outgoing flow is the same as at the previous step.

<img src="https://tex.s2cms.ru/svg/f%5E%7Bin%7D%20%3D%20f%5E%7Bright%7D_%7Bi-1%2Cj%7D%20%2B%20f%5E%7Bleft%7D_%7Bi%2B1%2Cj%7D%2B%20f%5E%7Bup%7D_%7Bi%2Cj-1%7D%20%2B%20f%5E%7Bdown%7D_%7Bi%2Cj%2B1%7D" alt="f^{in} = f^{right}_{i-1,j} + f^{left}_{i+1,j}+ f^{up}_{i,j-1} + f^{down}_{i,j+1}" />

incoming flow is the outgoing flow from neighbor cells in opposite directions.

The total volume change of the column is:

<img src="https://tex.s2cms.ru/svg/%5CDelta%20V%20%3D%5CDelta%20t%20(%20f%5E%7Bin%7D%20-%20f%5E%7Bout%7D%20)" alt="\Delta V =\Delta t ( f^{in} - f^{out} )" />

Finally apply the volume change to the water column, since we store the height and not the volume we need to divide by cell area:

<img src="https://tex.s2cms.ru/svg/h%5E%7Bwater%7D_%7Bi%2Cj%7D%20%3A%3D%20h%5E%7Bwater%7D_%7Bi%2Cj%7D%20%2B%20%5Cfrac%7B%5CDelta%20V%7D%7Bl_x%20l_y%7D" alt="h^{water}_{i,j} := h^{water}_{i,j} + \frac{\Delta V}{l_x l_y}" />

And that's it for water flow. It is also called shallow water equations using pipe model. At the end of this step, the water can slide down the terrain, create vertical waves and so on. But to apply erosion we will need to do more stuff.

Step 3.5. Compute water velocity

In further computation we will need the information about the water velocity