It seems that the type of algorithms differ considerably depending on whether the problem is:
Quasistatic elastic or
In the quasistatic elastic case, a simple approach is the following:
As the first part of each timestep, the displacement field $u$ is computed.
Since the displacement $u$ is now known at each node of the mesh, the nodes can be ...
I would suggest looking into the papers of Mavriplis, Diskin and Nishikawa. Mavriplis and one of his post-docs (Reza-Ahrabi) worked on implicit block ILU preconditioners and other smoothing strategies for Finite Element CFD Codes. Mavriplis himself does a lot of work on Finite Volume FAS (including working a great deal on general agglomeration strategies) ...
If movement is removed there, deformation of the ring under variable load remains. This can be calculated in the model of an elastic body with the addition of Rayleigh damping. The animation shows the deformation of a rubber ring calculated using FEM and Mathematica 12.
Whether any mesh updating (i.e. re-meshing) is required as part of the
solution of the nonlinear equations depends very much on the specific
problem you are trying to solve (geometry, loading, material properties).
Many problems of this type can be solved with no re-meshing.
For example, if you take a block of rubber, create a uniform mesh of
What you find is indeed correct. It is known that positivity is lost if very small time steps are chosen, see
This loss of positivity happens even for semi-discrete scheme.
The analysis for 1-d case is given in section 6 of this paper.
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 ...
Basis functions on quad elements must be constructed by mapping to a reference element using a bilinear map. Then the element-wise approximations will be continuous across common faces. This is because the restriction of the solution to any face will be a polynomial of degree 1 in one variable, so it is completely determined by the two nodal values on that ...