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High-Performance MLS-MPM Solver with Cutting and Coupling (CPIC) (MIT License)

A Moving Least Squares Material Point Method with Displacement Discontinuity and Two-Way Rigid Body Coupling, ACM Transactions on Graphics (SIGGRAPH 2018).

By Yuanming Hu (MIT CSAIL), Yu Fang (Tsinghua University), Ziheng Ge (University of Science and Technology of China), Ziyin Qu (University of Pennsylvania), Yixin Zhu (UCLA), Andre Pradhana (University of Pennsylvania), Chenfanfu Jiang (University of Pennsylvania).

Welcome to join the Discussion Forum.


[Introduction & Demo Video] [Paper] [Supplemental Document] [88-Line MLS-MPM]

[SIGGRAPH 2018 Fast Forward] [PDF Slides] [PDF Slides with Notes]

88-Line Version (MIT License) [Download C++ & Javascript versions]

Update Nov 2021: with the new Taichi programming language, you can run MLS-MPM on GPU with Python 3 after pip install taichi

Supports Linux, OS X and Windows. Tested on Ubuntu 16.04, Ubuntu 18.04, Arch Linux, MinGW, VS2017, OS X 10.11~10.14. No need to install taichi or taichi_mpm - see the end of code for instructions.

//88-Line 2D Moving Least Squares Material Point Method (MLS-MPM)[with comments]
//#define TC_IMAGE_IO   // Uncomment this line for image exporting functionality
#include "taichi.h"    // Note: You DO NOT have to install taichi or taichi_mpm.
using namespace taichi;// You only need [taichi.h] - see below for instructions.
const int n = 80 /*grid resolution (cells)*/, window_size = 800;
const real dt = 1e-4_f, frame_dt = 1e-3_f, dx = 1.0_f / n, inv_dx = 1.0_f / dx;
auto particle_mass = 1.0_f, vol = 1.0_f;
auto hardening = 10.0_f, E = 1e4_f, nu = 0.2_f;
real mu_0 = E / (2 * (1 + nu)), lambda_0 = E * nu / ((1+nu) * (1 - 2 * nu));
using Vec = Vector2; using Mat = Matrix2; bool plastic = true;
struct Particle { Vec x, v; Mat F, C; real Jp; int c/*color*/;
  Particle(Vec x, int c, Vec v=Vec(0)) : x(x), v(v), F(1), C(0), Jp(1), c(c){}};
std::vector<Particle> particles;
Vector3 grid[n + 1][n + 1];          // velocity + mass, node_res = cell_res + 1

void advance(real dt) {
  std::memset(grid, 0, sizeof(grid));                              // Reset grid
  for (auto &p : particles) {                                             // P2G
    Vector2i base_coord=(p.x*inv_dx-Vec(0.5_f)).cast<int>();//element-wise floor
    Vec fx = p.x * inv_dx - base_coord.cast<real>();
    // Quadratic kernels  [   Eqn. 123, with x=fx, fx-1,fx-2]
    Vec w[3]{Vec(0.5) * sqr(Vec(1.5) - fx), Vec(0.75) - sqr(fx - Vec(1.0)),
             Vec(0.5) * sqr(fx - Vec(0.5))};
    auto e = std::exp(hardening * (1.0_f - p.Jp)), mu=mu_0*e, lambda=lambda_0*e;
    real J = determinant(p.F);         //                         Current volume
    Mat r, s; polar_decomp(p.F, r, s); //Polar decomp. for fixed corotated model
    auto stress =                           // Cauchy stress times dt and inv_dx
        -4*inv_dx*inv_dx*dt*vol*(2*mu*(p.F-r) * transposed(p.F)+lambda*(J-1)*J);
    auto affine = stress+particle_mass*p.C;
    for (int i = 0; i < 3; i++) for (int j = 0; j < 3; j++) { // Scatter to grid
        auto dpos = (Vec(i, j) - fx) * dx;
        Vector3 mv(p.v * particle_mass, particle_mass); //translational momentum
        grid[base_coord.x + i][base_coord.y + j] +=
            w[i].x*w[j].y * (mv + Vector3(affine*dpos, 0));
  for(int i = 0; i <= n; i++) for(int j = 0; j <= n; j++) { //For all grid nodes
      auto &g = grid[i][j];
      if (g[2] > 0) {                                // No need for epsilon here
        g /= g[2];                                   //        Normalize by mass
        g += dt * Vector3(0, -200, 0);               //                  Gravity
        real boundary=0.05,x=(real)i/n,y=real(j)/n; //boundary thick.,node coord
        if (x < boundary||x > 1-boundary||y > 1-boundary) g=Vector3(0); //Sticky
        if (y < boundary) g[1] = std::max(0.0_f, g[1]);             //"Separate"
  for (auto &p : particles) {                                // Grid to particle
    Vector2i base_coord=(p.x*inv_dx-Vec(0.5_f)).cast<int>();//element-wise floor
    Vec fx = p.x * inv_dx - base_coord.cast<real>();
    Vec w[3]{Vec(0.5) * sqr(Vec(1.5) - fx), Vec(0.75) - sqr(fx - Vec(1.0)),
             Vec(0.5) * sqr(fx - Vec(0.5))};
    p.C = Mat(0); p.v = Vec(0);
    for (int i = 0; i < 3; i++) for (int j = 0; j < 3; j++) {
        auto dpos = (Vec(i, j) - fx),
            grid_v = Vec(grid[base_coord.x + i][base_coord.y + j]);
        auto weight = w[i].x * w[j].y;
        p.v += weight * grid_v;                                      // Velocity
        p.C += 4 * inv_dx * Mat::outer_product(weight * grid_v, dpos); // APIC C
    p.x += dt * p.v;                                                // Advection
    auto F = (Mat(1) + dt * p.C) * p.F;                      // MLS-MPM F-update
    Mat svd_u, sig, svd_v; svd(F, svd_u, sig, svd_v);
    for (int i = 0; i < 2 * int(plastic); i++)                // Snow Plasticity
      sig[i][i] = clamp(sig[i][i], 1.0_f - 2.5e-2_f, 1.0_f + 7.5e-3_f);
    real oldJ = determinant(F); F = svd_u * sig * transposed(svd_v);
    real Jp_new = clamp(p.Jp * oldJ / determinant(F), 0.6_f, 20.0_f);
    p.Jp = Jp_new; p.F = F;
void add_object(Vec center, int c) {   // Seed particles with position and color
  for (int i = 0; i < 500; i++)  // Randomly sample 1000 particles in the square
    particles.push_back(Particle((Vec::rand()*2.0_f-Vec(1))*0.08_f + center, c));
int main() {
  GUI gui("Real-time 2D MLS-MPM", window_size, window_size);
  add_object(Vec(0.55,0.45), 0xED553B); add_object(Vec(0.45,0.65), 0xF2B134);
  add_object(Vec(0.55,0.85), 0x068587); auto &canvas = gui.get_canvas();int f=0;
  for (int i = 0;; i++) {                              //              Main Loop
    advance(dt);                                       //     Advance simulation
    if (i % int(frame_dt / dt) == 0) {                 //        Visualize frame
      canvas.clear(0x112F41);                          //       Clear background
      canvas.rect(Vec(0.04), Vec(0.96)).radius(2).color(0x4FB99F).close();// Box
      for(auto p:particles);//Particles
      gui.update();                                              // Update image
      // canvas.img.write_as_image(fmt::format("tmp/{:05d}.png", f++));
} //----------------------------------------------------------------------------

/* -----------------------------------------------------------------------------
** Reference: A Moving Least Squares Material Point Method with Displacement
              Discontinuity and Two-Way Rigid Body Coupling (SIGGRAPH 2018)

  By Yuanming Hu (who also wrote this 88-line version), Yu Fang, Ziheng Ge,
           Ziyin Qu, Yixin Zhu, Andre Pradhana, Chenfanfu Jiang

** Build Instructions:

Step 1: Download and unzip (Link:
        Now you should have "mls-mpm88.cpp" and "taichi.h".

Step 2: Compile and run

* Linux:
    g++ mls-mpm88.cpp -std=c++14 -g -lX11 -lpthread -O3 -o mls-mpm

* Windows (MinGW):
    g++ mls-mpm88.cpp -std=c++14 -lgdi32 -lpthread -O3 -o mls-mpm

* Windows (Visual Studio 2017+):
  - Create an "Empty Project"
  - Use taichi.h as the only header, and mls-mpm88.cpp as the only source
  - Change configuration to "Release" and "x64"
  - Press F5 to compile and run

* OS X:
    g++ mls-mpm88.cpp -std=c++14 -framework Cocoa -lpthread -O3 -o mls-mpm

** FAQ:
Q1: What does "1e-4_f" mean?
A1: The same as 1e-4f, a float precision real number.

Q2: What is "real"?
A2: real = float in this file.

Q3: What are the hex numbers like 0xED553B?
A3: They are RGB color values.
    The color scheme is borrowed from

Q4: How can I get higher-quality?
A4: Change n to 320; Change dt to 1e-5; Change E to 2e4;
    Change particle per cube from 500 to 8000 (Ln 72).
    After the change the whole animation takes ~3 minutes on my computer.

Q5: How to record the animation?
A5: Uncomment Ln 2 and 85 and create a folder named "tmp".
    The frames will be saved to "tmp/XXXXX.png".

    To get a video, you can use ffmpeg. If you already have taichi installed,
    you can simply go to the "tmp" folder and execute

      ti video 60

    where 60 stands for 60 FPS. A file named "video.mp4" is what you want.

Q6: How was taichi.h generated?
A6: Please check out my #include <taichi> talk:
    and the generation script:
    You can regenerate it using `ti amal`, if you have taichi installed.

Questions go to yuanming _at_

                                                      Last Update: March 6, 2019
                                                      Version 1.5

----------------------------------------------------------------------------- */

Installing the High-Performance 3D Solver

(This is for installing the high-performance 2D/3D solver including MLS-MPM and CPIC. If you want to play with the 88-line MLS-MPM, you don't have to install anything - see here)

Linux and OSX

Install taichi (legacy branch). Then, in command line

ti install mpm

and it will install this taichi package automatically.


Support coming later.

Run demos

Every script under the folder scripts/mls-cpic is executable with python3.

Visualize the results

  • Outputs are in taichi/outputs/mpm/;
  • Install Houdini Apprentice (which is free);
  • Create a File node in Houdini to visualize the bgeo (particles), obj (3D meshes), poly (2D polygons) files.

Python 3 API


(You only need to specify res in most cases. The default parameters generally work well.) All parameters:

  • res: (Vector<dim, int>) grid resolution. The length of this vector also specifies the dimensionality of the simulation.
  • base_delta_t : (real, default: 1e-4) delta t
  • delta_x: (real, default: 1.0 / res[0])
  • particle_collision (bool, default: False): push particles inside level sets out (turn off when you are using sticky level sets)
  • pushing_force: (real, default: 20000.0) If things do not separate, use this. Typical value: 20000.0.
  • gravity (Vector<dim>, default: (0, -10, 0) for 3D, (0, -10) for 2D)
  • frame_dt: (real, default: 0.01) You can set to 1 / 24 for real frame rate.
  • num_threads: (int, default: -1) Number of threads to use. -1 means maximum threads.
  • num_frames: (int, default: 1000) Number of frames to simulate.
  • penalty: (real, default: 0) Penetration penalty. Typical values are 1e3 ~ 1e4.
  • optimized: (bool, default: True) Turn on optimization or not. Turning it off if you need to benchmark the less optimized transfers.
  • task_id: (string, default: taichi will use the current file name)
  • rigid_body_levelset_collision: (bool, default: False) Collide rigid body with level set? (Useful for wine & glass.)
  • rpic_damping: (real, default: 0) RPIC damping value should be between 0 and 1 (inclusive).
  • apic_damping: (real, default: 0) APIC damping value should be between 0 and 1 (inclusive).
  • warn_particle_deletion: (bool, default: False) Log warning when particles get deleted
  • verbose_bgeo: (bool, default: false) If true, output particle attributes other than position.
  • reorder_interval: (int, default: 1000) If bigger than error, sort particle storage in memory every reorder_interval substeps.
  • clean_boundary: (int, default: 1000) If bigger than error, sort particle storage in memory every reorder_interval substeps.
  • ...


  • type: rigid, snow, jelly, sand. For non-rigid type, see Particle Attributes
  • color: (Vector<3, real>)
  • pd : (bool, default: True) Is poisson disk sampling or not? Doesn't support type: rigid, sdf
  • pd_periodic : (bool, default: True) Is poisson disk periodic or not? Doesn't support 2D
  • pd_source : (bool, default: False) Is poisson disk sampling from source or not (need to defineframe_update)? Doesn't support 2D
  • For type rigid:
  • rotation_axis : (Vector<3, real>, default: (0, 0, 0)) Let the object rotate along with only this axis. Useful for fans or wheels.
  • codimensional : (bool, must be explicitly specified) Is thin shell or not?
  • restitution: (real, default: 0.0) Coefficient of restitution
  • friction: (real, default: 0.0) Coefficient of friction
  • density: (real, default: 40 for thin shell, 400 for non-thin shell) Volume/area density.
  • scale: (Vector<3, real>, default: (1, 1, 1)) rescale the object, bigger or smaller
  • initial_position: (Vector<dim, real>, must be explicitly specified)
  • initial_velocity: (Vector<dim, real>, default: (0, 0, 0))
  • scripted_position: (function(real) => Vector<3, real>)
  • initial_rotation: (Vector<1 (2D) or 3 (3D), real>, default: (0, 0, 0)) Euler angles
  • initial_angular_velocity: (Vector<1 (2D) or 3 (3D), real>, default: (0, 0, 0))
  • scripted_rotation: (function(real) => Vector<1 (2D) or 3 (3D), real>) Takes time, returns Euler angles
  • (Translational/Rotational) static objects are also considered as scripted, but with a fixed scripting function i.e. tc.constant_function13(tc.Vector(0, 0, 0))
  • linear_damping: (real, default: 0) damping of linear velocity. Typical value: 1
  • angular_damping: (real, default: 0) damping of angular velocity. Typical value: 1
  • ...

Particle Attributes

  • jelly
    • E: (real, default: 1e5) Young's modulus
    • nu: (real, default: 0.3) Poisson's ratio
  • snow
    • hardeing (real, default: 10) Hardening coefficient
    • mu_0 (real, default: 58333.3) Lame parameter
    • lambda_0 (real, default: 38888.9) Lame parameter
    • theta_c (real, default: 2.5e-2) Critical compression
    • theta_s (real, default: 7.5e-3) Critical stretch
  • sand
    • mu_0 (real, default: 136038) Lame parameter
    • lambda_0 (real, default: 204057) Lame parameter
    • friction_angle (real, default: 30)
    • cohesion (real, default: 0)
    • beta (real, default: 1)
  • water
    • k: (real, default: 1e5) Bulk modulus
    • gamma: (real, default: 7)
  • von_mises
    • youngs_modulus: (real, default: 5e3) Young's modulus (for elasticity)
    • poisson_ratio: (real, default: 0.4) Poisson's ratio (for elasticity, usually no need to change)
    • yield_stress: (real, default:1.0) Radius of yield surface (for plasticity)
  • ...

Script Examples

  • Scripted motion:
  • Rigid-ground collison:
  • When you're making an rotating wheel example, e.g. and the wheel is not turning in the right direction, you can try reverse_vertices=True.
  • ...


  • Matrices in taichi are column major. E.g. A[3][1] is the element at row 2 and column 4.
  • All indices, unless explicitly specified, are 0-based.
  • Use real, in most cases, instead of float or double.
  • Float point constants should be suffixed with _f, so that it will have type real, instead of float or double. Example: 1.5_f (float or double depending on build precision) instead of 1.5 (always double) or 1.5f (always float)
  • Always pull taichi (the main lib, master branch) after updating taichi_mpm.
  • When a particles moves too close to the boundary (4-8 dx) it will be deleted.
  • Whenever you can any compile/linking problem:
    • Make sure taichi is up-to-date
    • Invoke CMake so that all no source files will be detected
    • Rebuild
  • ...

Friction Coefficient

  • Separate: positive values, 0.4 means coeff of friction 0.4
  • Sticky: -1
  • Slip: -2
  • Slip with friction: -2.4 means coeff of friction 0.4 with slip



    object1 = mpm.add_particles(...)
    object2 = mpm.add_particles(...)

    mpm.add_articulation(type='motor', obj0=object1, obj1=object2, axis=(0, 0, 1), power=0.05)
  • Rotation: enforce two objects to have the same rotation.

    • type: rotation
    • obj0, obj1: two objects
    • Use case: blabe and wheel in water wheel examples
  • Distance: enforce two points on two different object to have constant distance

    • type: distance
    • obj0, obj1: two objects
    • offset0, offset1: (Vector<dim, real>, default: (0, 0, 0)) offset of two points to the center of mass to each object, in world space
    • distance (real, default: initial distance between two poitns) target distance
    • penalty (real, default: 1e5) corrective penalty
    • Use case: hammer in crashing_castle examples
  • Motor: enforce object to rotate along an axis on another object, and apply torque

    • type: motor
    • obj0: the wheel object
    • obj1: the body object
    • axis: (Vector<dim, real>) the rotation axis in world space
    • power (real, default: 0) torque applied per second
    • Use case: wheels for cars, and legs for the robot
    • Example:
  • Stepper: enforce object to rotate along an axis on another object at a fixed angular velocity

    • type: motor
    • obj0: the wheel object
    • obj1: the body object
    • axis: (Vector<dim, real>) the rotation axis in world space
    • angular_velocity (real)
    • Use case: Fixed-rotation-speed wheels for cars, and legs for the robot

Source Sampling

  • If you want to source particles continuously from a object, please set pd_source = True in add_particles
  • initial_velocity should be a non-zero vector
  • Remember to also set delta_t=frame_dt in add_particles, which enables the frequency of sampling to be consistent with its initial velocity
  • There might be some artifact due to the effect of gravity. You can reduce that artifact by increasing update_frequency.
  • Example:,

Mathematical Comparisons with Traditional MPM



Please cite our paper if you use this code for your research:

  title={A Moving Least Squares Material Point Method with Displacement Discontinuity and Two-Way Rigid Body Coupling},
  author={Hu, Yuanming and Fang, Yu and Ge, Ziheng and Qu, Ziyin and Zhu, Yixin and Pradhana, Andre and Jiang, Chenfanfu},
  journal={ACM Transactions on Graphics (TOG)},


High-performance moving least squares material point method (MLS-MPM) solver. (ACM Transactions on Graphics, SIGGRAPH 2018)







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