MPM Fluid Simulation

A C++ implementation of the Material Point Method for multi-layer fluid simulation, inspired by Grant Kot's Liquid Layers.

Build

# Clone / enter project
cd mpm_fluid

# Configure (downloads Polyscope + Eigen automatically via CMake FetchContent)
cmake -B build -DCMAKE_BUILD_TYPE=Release

# Compile
cmake --build build -j$(nproc)

# Run
./build/mpm_fluid

Requirements: CMake ≥ 3.18, a C++17 compiler (GCC 9+, Clang 10+, MSVC 2019+), and OpenGL 3.3+ (any modern desktop GPU).

First build takes ~3-5 minutes — Polyscope is fetched and compiled from source.

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Project phases

Phase	What you implement	Key concepts
1	Boilerplate + colored particle layers	Polyscope setup, particle struct
2	Particle & grid data structures	B-spline weight functions, P-G indexing
3	P2G + G2P transfer loop	APIC, mass/momentum rasterization
4	Constitutive model	Deformation gradient F, stress tensor, equation of state
5	Multi-material + boundaries	Per-particle material tag, sticky walls
6	Polish + interaction	Mouse forces, surface tension, UI sliders

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File structure

mpm_fluid/
├── CMakeLists.txt          # Build config; fetches Polyscope + Eigen
└── src/
    ├── types.h             # Particle, GridCell, MaterialType, SimParams
    ├── simulation.h        # Simulation class declaration
    ├── simulation.cpp      # initialize(), registerPolyscope(), updatePolyscope()
    └── main.cpp            # Polyscope init, render loop, ImGui UI panel

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Key MPM resources

• Jiang et al. 2016 — "The Material Point Method for Simulating Continuum Materials" SIGGRAPH course notes (free PDF, search "MPM course SIGGRAPH 2016")
• Hu et al. 2018 — "A Moving Least Squares Material Point Method with Displacement Discontinuity and Two-Way Rigid Body Coupling" (APIC/MLS-MPM)
• Polyscope docs — https://polyscope.run/
• Grant Kot's source — study it after you've implemented your own version!

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Phase 2

The MPM timestep loop looks like this:

for each timestep:
  1. Reset grid  (zero mass and momentum on every node)
  2. P2G         (each particle spreads its mass + momentum to nearby grid nodes
                  using B-spline weights; also compute volume if first step)
  3. Grid forces (compute stress → nodal forces from constitutive model)
  4. Grid update (F = ma on each node; apply gravity; enforce boundaries)
  5. G2P         (each particle reads velocity from nearby grid nodes;
                  update particle velocity, APIC affine matrix C, then position)

The B-spline weight for a particle at xp affecting grid node i is:

w = N((xp - xi) / dx)
where N(x) = 0.75 - x^2          for |x| < 0.5
             0.5*(1.5 - |x|)^2   for 0.5 <= |x| < 1.5
             0                   otherwise

This is the quadratic B-spline — it affects a 3×3 stencil of grid nodes per particle.