TL,DR: either the good old Galerkin finite element method, or mesh-free / particle methods.
There are a few things to unpack here.
First, the simulation you show includes contact between an elastic body (the ring) and a hard boundary, so the constraints are non-holonomic.
Contact problems are much more challenging than, say, elastic deformation under gravity, which only has holonomic constraints.
In the animation you showed above, the only thing that appears to change are the positions of the mesh nodes.
This is a reasonable approach to take when the deformations from the reference configuration of the material are small, which is exactly when the elasticity equations are linear. For very large deformations, the transformed mesh in the body configuration could become tangled and as you might imagine this is bad news.
There are a few ways around this.
First, you could change the internal workings of the simulation by periodic remeshing.
When the triangles become too degenerate, you can flip edges to restore mesh quality.
This approach can eliminate the possibility of mesh tangling but it's hard to code up right.
Second, you could change the physics problem that you're simulating in the first place.
If the deformations are that large, using a linearized system of equations is no longer a good way to describe the problem.
Nonlinear elasticity, for example see Neo-Hookean solids, includes terms that go to infinity when the Jacobian of the reference-to-body transformation becomes singular.
This prevents weird deformations at a mathematical level but it also makes the numerical solution of the problem more challenging.
Both of these ideas are very much in the mindset of Galerkin finite element methods on simplicial meshes.
A much larger departure from that way of thinking is to use mesh-free methods.
This includes things like smoothed particle hydrodynamics, the material point method, etc.
These are often more expensive computationally because you need to do neighbor queries on every timestep and this is hard to make cache-friendly.
But they can adapt much better to very deformed or moving geometries.