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Assignment 5 — PBD Cloth

Due Wednesday, 4/9 at 23:59:59. You must work individually.

• Goals
• Setting up Eigen
• Starting Point
• Task 1: Setting up the Cloth
• Task 2: Time Stepper
• Task 3: Sphere Collision
• Task 4: Parameter Exploration
• Rubric
• What to Hand in

Goals

In this assignment, we’re going to simulate cloth with Position Based Dynamics.

Setting up Eigen

For linear algebra, we’ll be using a C++ library called Eigen. You’ll have to install this in the same way you installed glm in lab 0. You’ll need to create a new environment variable called EIGEN3_INCLUDE_DIR that points to the Eigen directory. Once you have it installed, you may want to look at this quick tutorial.

Starting Point

Please download the assignment and go over the code.

Your main task is to complete the function Cloth::step(). The skeleton code has a lot of functionalities that deal with setting up the scene and rendering the cloth. Study the code carefully to see what is happening.

• main()
  • The simulation thread is separate from the rendering thread. The thread is defined and started on line 301, right before the main thread enters the render loop. The space bar turns on/off the simulation, and the h key runs one simulation step.
• Scene
  • This class is responsible for holding the things in the scene. Currently, it holds a pointer to a cloth, and an empty vector of pointers to spheres.
  • To change the number of rows and columns of the cloth, modify Scene::load(). You should start with a 2 by 2 cloth.
• Particle
  • The particle class is used to represent both cloth nodes and spheres.
  • Member variables x and v represent the current position and velocity.
  • Member variable p is the previous position, used by the PBD stepper.
  • Member variables x0 and v0 represent the initial position and velocity, which are used for resetting the cloth.
  • Member variable r is the radius used for collisions. A cloth particle has a small radius that is noticeably larger than the thickness of the cloth.
  • Member variable m is the mass of the particle.
  • Member variable d is the damping coefficient.
  • Member variable fixed indicates whether this particle is fixed or free.
• Spring
  • This class is used to represent a spring between two particles.
  • Member variables p0 and p1 are references to the two particles.
  • Member variable L is the rest length of the spring.
  • Member variable alpha is the compliance constant of the spring.
• Cloth
  • Cloth::Cloth()
    • The particles and the springs are created. This is your first task, described below in “Setting up the Cloth.”
    • The vertex buffers for rendering the cloth are allocated.
  • Cloth::updatePosNor()
    • Whenever the simulation step changes the particle positions, we have to resend the position and normal data to the GPU. This function fills the position buffer from the particles and recomputes the normals using the cross product.
  • Cloth::draw()
    • Sends the updated position and normal buffers to the GPU and then draws the cloth mesh using triangle strips.
  • Cloth::step()
    • The most important function that you will implement later in this assignment.
    • The last argument (spheres) is used for collisions and can be ignored initially.

Task 1: Setting up the Cloth

First, add some code into Cloth::Cloth() to set up the particles and the springs. Start with rows = cols = 2.

• Particles
  • The particles should be created to form a grid with corners x00, x01, x10, and x11. With rows = cols = 2, there should be four particles.
  • The mass of each particle should be the total cloth mass (the input mass argument) divided by the number of particles.
• Springs
  • There are ‘x’ springs, ‘y’ springs, two sets of ‘shear’ (or diagonal) springs, and two sets of ‘bending’ springs.
  • Use the same alpha parameter for all spring types.

Task 2: Time Stepper

Apply the XPBD stepping scheme:

for each particle i
   fi = mi * g - di * vi
   vi += (h / mi) * fi
   pi = xi
   xi += h * vi
end
for each constraint j between x0 and x1
   [Cj, ∇Cj] = springConstraint(x0, x1)
   λj = -Cj / (w0 * |∇Cj0|^2 + w1 * |∇Cj1|^2 + αj/h^2)
   x0 += λj * w0 * ∇Cj0
   x1 += λj * w1 * ∇Cj1
end
for each particle i
   vi = (1 / h) * (xi - pi)
end

Keep in mind that some particles are marked as “fixed.” The positions and velocities of these particles should not be modified.

The spring constraint between particles \(\vec{x}_0\) and \(\vec{x}_1\) is \[ C = l - L, \] where \[ l = \| \Delta \vec{x} \|, \quad \Delta \vec{x} = \vec{x}_1 - \vec{x}_0. \] The gradients are \[ \nabla C_0 = -\frac{\Delta \vec{x}}{l}, \quad \nabla C_1 = \frac{\Delta \vec{x}}{l}. \] Note that for this constraint, \(\| \nabla C_i \| = 1\) because the norm of a unit vector is one. This means that the denominator for the calculation of \(\lambda\) can be simplified.

Task 3: Sphere Collision

If you uncomment line 47 in Scene::load(...), you’ll see a collision sphere moving, but it won’t be affecting the cloth yet. We need to write some collision code first.

Implement collisions in Cloth::step(), right before the final update for the velocities. You must check all the cloth particles against all the collision spheres (which are also particles) in the scene, which in this case is just a single sphere. For each particle, if there is a collision, snap the position of the cloth particle onto the surface of the sphere.

Task 4: Parameter Exploration

Try changing the compliance, damping, and time step parameters. Write a short paragraph in the README discussing the following:

• What is a reasonable range for the compliance parameter? What happens outside of this range?
• What is a reasonable range for the damping parameter? What happens outside of this range?
• What is a reasonable range for the time step parameter? What happens outside of this range?
• What is the highest resolution you can simulate on your computer?

Rubric

• [10] Task 1: Cloth setup
• [60] Task 2: Simulation without collisions
• [20] Task 3: Simulation with sphere collisions
• [5] Task 4: Parameter exploration paragraph
• [5] Coding style and general execution

Total: 100 points

What to hand in

Failing to follow these points may decrease your “general execution” score.

If you’re using Mac/Linux, make sure that your code compiles and runs by typing:

> mkdir build
> cd build
> cmake ..
> make
> ./A5 <SHADER DIR>

If you’re using Windows, make sure your code builds using the steps described in Lab 0.

• Put your name on the program window.
• Include an ASCII README that includes
  • Your name
  • The highest stage you’ve completed
  • Any bonus?
  • Anything else of note
• Make sure you don’t get any compiler warnings.
• Remove unnecessary debug printouts.
• Remove unnecessary debug code, including commented out code.
• Do not hand in:
  • The build directory
  • The executable
  • Old save files (*.~)
  • Object files (*.o)
  • Visual Studio files (.vs)
  • Git folder (.git)
• Create a single zip file of all the required files.
  • The filename of this zip file should be UIN.zip (e.g., 12345678.zip).
  • The zip file should extract a single top-level folder named UIN/ (e.g. 12345678/).
  • This top-level folder should contain your README, src/, CMakeLists.txt, etc.
  • Use the standard .zip format (not .gz, .7z, .rar, etc.).