Physics-Based Simulation Homework: Multigrid Stable Fluids

This homework is developed by Kangbo Lyu (Tsinghua University).

Installation Guide

• Use git to clone the codebase:

git clone --recursive https://github.com/taodu-eecs/phys_sim_multigrid_stable_fluids.git

• Install Taichi Lang on your computer:

pip install --upgrade taichi

• Run the homework with the following command:

python fluid_hw.py

Programming Tasks

This programming assignment invites you to implement a few key functions in a real-time stable fluid program. The fluid domain is relatively simple: it is a 2D square full of fluid surrounded by free-slip, no-separation boundaries at the four static (zero velocity) edges of the square. In particular, you will implement two Poisson solvers: one based on Jacobi iterations and one based on multigrids. The code skeleton runs the script on GPUs (line 5 in fluid_hw.py):

ti.init(arch=ti.cuda, default_fp=ti.f32, fast_math=True, debug=False)

If you do not have access to a GPU, feel free to replace ti.cuda with ti.cpu. The choice of CPU/GPU architecture won't affect our grading of your homework.

Task 1 (8 points)

Implement the semi-Lagrangian advection algorithm split in three functions:

• Task 1.1 (3 points): Read the instructions in sample to implement bilinear interpolation in a scalar/vector field with the (clamped) coordinates rel_pos.
• Task 1.2 (3 points): Implement the Runge-Kutta 2 (RK2) time integration in backtrace. Note that this function computes the position backward in time.
• Task 1.3 (2 points): Use the two functions above to implement semi-Lagrangian advection in advect_velocity.

Task 2 (6 points)

Next, we invite you to implement the Poisson solver in stable fluids split in three functions:

• Task 2.1 (2 points): Implement the divergence function to compute the divergence of a given velocity field. We will need the divergence value in the right-hand side of the Poisson equation.
• Task 2.2 (2 points): Read the instructions in iteration and implement a modified Jacobi iteration in this function. It is also helpful to read the solve function that calls iteration.
• Task 2.3 (2 points): Use the two functions above as building blocks to finish the project function.

Running fluid_hw.py after finishing Tasks 1 and 2 should show an interactive stable fluid demo in a squared domain, thanks to the high-performance computational power offered by Taichi Lang. Read the GUI code in fluid_hw.py to figure out how to interact with this fluid demo. The frame rate of your demo depends on your hardware platform and CPU/GPU architecture.

The Poisson solver in stable fluids typically consumes a substantial portion of the whole computational time. In the next task, you will implement a few functions in PoissonMultiGridSolver class and compare the performance of the multi-grid preconditioned conjugate gradient (MGPCG) solver with the previous Jacobi solver.

Task 3 (6 points)

We highly encourage you to read the full PoissonMultiGridSolver class carefully before you start working on the three tasks below.

• Task 3.1 (2 points): Implement smooth, the smoothing operator used in the V-cycle scheme. It may be helpful to skim over functions calling smooth and get a sense of its inputs and outputs before your start coding;
• Task 3.2 (2 points): Implement prolongate, the prolongation/interpolation operator.
• Task 3.3 (2 points): Implement restrict, the restriction operator.
• Task 3.4 (0 points): Try to run fluid_hw.py with Jacobi and MGPCG Poisson solvers, respectively, and compare their frame rates on your computer. Feel free to adjust the solver's tolerance and the fluid domain's resolution and repeat the comparison. What is your feeling about the computational speed of these two solvers?

Implementing a correct multigrid can be tricky. If you struggle with Task 3, please feel free to swing by our office hours for help.

Submission Guide

Submit your fluid_hw.py file to Web Learning. It is your responsibility to double check that your submission file is correct. Email your fluid_hw.py to [email protected] or our Teaching Assistant if you miss the deadline but still have late days left. We do not accept submissions in any other formats.

Acknowledgments

The code skeleton is from the stable_fluid.py example in Taichi Lang. Feel free to read their implementation before you start working on this homework, but you should not copy and paste code from it. In fact, we have made substantial changes in fluid_hw.py so that directly copying the implementation in stable_fluid.py may not lead to a correct solution to our homework.