Cloth Simulation and Collision

Triangular FEM

We often simulate cloth with triangle surfaces instead of a tetrahedron. Therefore, we cannot totally rely on the volumetric approach (while we do introduce "thickness" to adapt the mechanics).

Finite Elements

Given the reference space in 3D, and the reference position as \(\mathbf X = (X, Y, Z)^T \in \mathbb R^3\), just as tetrahedron case, we represent the finite elements at point \(\mathbf X\) given by the 3 corners of the triangle \(T=(\mathbf X_0, \mathbf X_1, \mathbf X_2)\)

where \(\phi_i\) is the shape function, and as before, we apply the constraint \(\phi_0 + \phi_1 + \phi_2 = 1\Rightarrow \phi_0 = 1- \phi_1 - \phi_2\).

However, note that \(\mathbf X\) may not lie exactly on the surface of the triangle. In this case, we want to find the function for the nearest point of \(\mathbf X\) on triangle, i.e. \(proj_{T}(\mathbf X)\). In this case, we can turn the question of solving the shape function to an optimization problem. i.e.

And the solution to this optimization problem is \begin{align*}

T & = \begin{pmatrix}\Delta \mathbf X_1 &\Delta \mathbf X_2\end{pmatrix}\ T^TT\begin{pmatrix}\phi_1 \\phi_2\end{pmatrix} &= T^T \Delta \mathbf X\ \phi_0 &= 1 - \phi_1 - \phi_2 \end{align*}

where \(\Delta \mathbf X = \mathbf X - \mathbf X_0\)

Note that this shape function is the barycentric coordinate for \(proj_T(\mathbf X)\), and for points off the triangle

where \(N\) is the normal of \(T\)

Deformation

Note that we introduce \(N_T\), while the world space have different normal on the deformed triangle \(\mathbf n\), therefore, we can simply have the deformation as

Generalized Coordinates

As before, let the generalized coordinates and generalized velocity for the triangle as

Kinetic Energy

Let the cloth domain be \(\Omega\), since the cloth is thin, we can just integrate over the triangle surface, and then multiple the thickness to get the volumetric kinetic energy, i.e.

where

and just as before, we can assemble the total kinetic energy for cloth as

Potential Energy

As before, we derive the potential energy from the deformation gradient.

Note that the deformation can be written as

Therefore, the deformation gradient is simply

Note that deformation gradient in this case is constant.

Then, we can apply the same potential energy function as in volumetric approach with constant thickness over the triangle area.

Energy by Principal Stretches

From volumetric approach, we have the strain represented by Right Cauchy Green Deformation

Apply an eigendecomposition on the deformation, we have

where \((V\Delta\mathbf X)\) represents the rotation, and \(\Lambda\) is the squared singular values of \(F\). Therefore, via Principal Stretches, we have

where \(\Sigma(F) = (\sigma_0(F), \sigma_1(F), \sigma_2(F))\) is the diagonal of singular value matrix from SVD of \(F\).

With this, we can filter out rotations and translations, which are unwanted as they should not change the potential energy.

Then, we want our cloth to go back to undeformed shape when external forces are removed, therefore, we introduce the quadratic energy model Co-Rotational Linear Elasticity

Note that $\mathbf D\varphi(1, 1, 1) = 0\Rightarrow $ minimum at \((1, 1, 1)\), which is the identity transformation, this energy is minimized at undeformed shape.

Since \(F\) is constant across the triangle, now we simply have the potential energy for the triangle be

and then assemble the total potential energy as

The Lagrangian

The Lagrangian and EL equation is still the same, since we only change the same of the tetrahedron.

Time Integration

Usually we use linearly implicit integrator for cloth due to high membrane stiffness. Therefore, we still need the generalized force and generalized stiffness.

Note that the Hessian of \(\varphi\) will be extremely messy, therefore, instead of using the full Hessian to do the Newton's method, we can use an approximation of the Hessian and Quasi-Newton Method. In this case, we will have

Then, the problem is left with \(\frac{\partial^2 \varphi}{\partial F^2}\), which involves with the SVD.

Derivative of SVD

For a SVD \(A = U\Sigma V^T\), we can parameterized by \(A\) as \(U(A)\Sigma(A)V(A)^T\), hence the derivative w.r.t to entries of \(A\) can be written as

Therefore, we can compute \(\frac{\partial^2 \varphi}{\partial F^2}\) for each entry as

where

Vectorized Deformation Gradient

As before, we need to vectorize the deformation gradient

Then, the gradient is

then, note that the normal can be written as

so that the gradient is

where \([\cdot]_\times\) is the skew-symmetric matrix

Collision with Sphere

Given the query point \(\mathbf x\) and a sphere of center and radius \((\mathbf c, r)\) , we need to first detect the collision and then respond.

For the detection, simply check

For the response, we don't want the cloth to move into the sphere, hence we can just filter normal component of velocity. i.e.

where the filtering component coefficient \(\alpha\) satisfies

In addition, we want the velocity change be minimal and if velocity if away from the sphere, then we do nothing. Therefore, we have