I am an engineer who is trying to get a deeper understanding of FEM. I have been using the Zienkiewicz texts as my bible. It touches on the issue of incompressibility but I need a more intuitive way to think about what is happening here.
My current understanding
A 1 field displacement-based formulation has convergence issues due to the incompressibility leading to an ill-conditioned matrix. I am still not sure how this arises and whether this hold for all material types. A crude way of resolving this would be to employ higher order elements, which is computationally prohibitive, demanding the need for low order work arounds. Enter two and three field formulations...
Explicitly having additional variables can serve as constraints on the system can help with the ill condition matrix at the expense of solving for more variables. So my understanding is that an issue arises when we are interpolating values in a disconnected manner. I guess that this lack of a connection is due to the overly simple continuity equation that arises from incompressibility. This causes the locking phenomenon which leads to spurious results. I know I have gaps in that flow of logic. I am also confused as to why this resolves the convergence issues. Wouldn't the matrix still be prone to ill-conditioning?
This locking requires a mixed formulation where the new variables require a higher order than the displacement field to be stable. So we need elements that satisfy the LBB or inf-sup condition. I do not have an intuitive feel for what is going on here. What would happen if the displacement field had a higher order than these extra variables?
I have tried looking at the original LBB paper and my eyes glazed over. Hopefully, there is a less formal way of describing what's going on here.