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Expand Up @@ -16,7 +16,175 @@ By [Yuanming Hu (MIT CSAIL)](http://taichi.graphics/me/), [Yu Fang (Tsinghua Uni
<img src="https://github.com/yuanming-hu/public_files/raw/master/graphics/mls-mpm-cpic/bunny.gif"> <img src="https://github.com/yuanming-hu/public_files/raw/master/graphics/mls-mpm-cpic/robot_forward.gif">
<img src="https://github.com/yuanming-hu/public_files/raw/master/graphics/mls-mpm-cpic/banana.gif"> <img src="https://github.com/yuanming-hu/public_files/raw/master/graphics/mls-mpm-cpic/cheese.gif">

## Installation

## 88-Line Version [[Download](https://github.com/yuanming-hu/taichi_mpm/releases/download/SIGGRAPH2018/mls-mpm88.zip)]

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

<img src="https://github.com/yuanming-hu/public_files/raw/master/graphics/mls-mpm-cpic/mls-mpm88-lowres.gif">
<img src="https://github.com/yuanming-hu/public_files/raw/master/graphics/mls-mpm-cpic/mls-mpm88-highres.gif">


``` C++
//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 [http://mpm.graphics 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)canvas.circle(p.x).radius(2).color(p.c);//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 mls-mpm88.zip (Link: http://bit.ly/mls-mpm88)
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
./mls-mpm

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

* 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: Coming soon. If you don't want to wait, just install XQuartz and follow
the Linux instructions.

** FAQ:
Q1: What does "1e-4_f" mean?
A1: The same as 1e-4f.

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
https://color.adobe.com/Copy-of-Copy-of-Core-color-theme-11449181/

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.


For more questions, please email yuanming _at_ mit.edu
or visit https://github.com/yuanming-hu/taichi_mpm/issues.

Last Update: Oct 30, 2018
Version 1.3

----------------------------------------------------------------------------- */
```
## 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](https://github.com/yuanming-hu/taichi_mpm#88-line-version-download))
### Linux
Install [`taichi`](https://github.com/yuanming-hu/taichi) [[Instructions](https://taichi.readthedocs.io/en/latest/installation.html)].
Expand Down Expand Up @@ -182,169 +350,6 @@ Syntax:
- There might be some artifact due to the effect of gravity. You can reduce that artifact by increasing `update_frequency`.
- Example: `source_sampling.py`, `source_sampling_2d.py`

## 88-Line Version [[Download](https://github.com/yuanming-hu/taichi_mpm/releases/download/SIGGRAPH2018/mls-mpm88.zip)]
Supports Linux and Windows. Tested on Ubuntu 16.04, Arch Linux, MinGW, VS2017.

No need to install `taichi` or `taichi_mpm` - see the end of code for instructions.

``` C++
//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 [http://mpm.graphics 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)canvas.circle(p.x).radius(2).color(p.c);//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 mls-mpm88.zip (Link: http://bit.ly/mls-mpm88)
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
./mls-mpm

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

* 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: Coming soon. If you don't want to wait, just install XQuartz and follow
the Linux instructions.

** FAQ:
Q1: What does "1e-4_f" mean?
A1: The same as 1e-4f.

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
https://color.adobe.com/Copy-of-Copy-of-Core-color-theme-11449181/

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.


For more questions, please email yuanming _at_ mit.edu
or visit https://github.com/yuanming-hu/taichi_mpm/issues.

Last Update: Oct 30, 2018
Version 1.3

----------------------------------------------------------------------------- */
```

## Mathematical Comparisons with Traditional MPM
<img src="/data/images/comparisons.jpg" with="1000">
Expand Down

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