What can we do with ODEs? It turns out the ODEs are an integral part of the cloth simulation process.

For simulation purposes cloth is often modeled as a quad mesh whose vertices are connected by stiff springs. In general cloth resists stretching, this property is called extensibility. In order to simulate this, the springs must be made stiffer and stiffer in the model, and as stiffness increases numerical methods for simulating the springs in cloth become intractable [1].

Another approach builds on this by adding constrained dynamics to enforce inextensibility of cloth. Constraints include edge length preservation in the cloth. This approach incorporates two topics from scientific computing, ordinary differential equation solvers and root finding methods. The general process can be described as follows [2]:

1) Start with some initial position of the cloth x0 and the world model.
2) Solve all constraints simultaneously.
3) Apply collision response.
4) Implicitly integrate the springs.

While there is some somewhat fancier math involved, solving the implicit constraint dynamics in step 2 boils down to finding the zeros of an equations map calculated points to points which preserve the length of edges of the cloth. As we know from class, root finding can be performed by numerous methods and in this case Newton’s is generally used.
Additionally, the cloth simulation is performed in discrete time steps. In this setup in order to perform the implicit integration of the springs and ODE solver is needed. An equation of the form below must be solved.

x’’ = (M-1) * f(x, x’)

As is popular in many areas of engineering and simulation, RK4 is the method of choice due to its speed to stability ratio.

The result of each of these steps is life-like cloth simulation with realistic folds and creases that maintains its length without stretching unnecessarily.

[1] Kwang-Jin Choi and Hyeong-Seok Ko. Stable but responsive cloth. In John Hughes, editor, SIGGRAPH 2002 Conference Proceedings, Annual Conference Series, pages 604–611. ACM Press/ACM SIGGRAPH, 2002.
[2] Goldenthal, R., Harmon, D., Fattal, R., Bercovier, M., and Grinspun, E. 2007. Efficient simulation of inextensible cloth. ACM Trans. Graph. 26, 3 (Jul. 2007), 49.