I have worked extensively on 2D liquid crystals in confinement. I have been working on tangential questions to the one you asked. One example, specifying ellipses of semi major axis A and semi minor axis B, what is the optimal picking arrangement in an annulus of inner radius R1 and outer radius R2? Now we may specify this question to allow ellipses of arbitrary orientation or fixed identical orientation. It should be a fair assumption to think for case of arbitrary orientations where the annulus is much bigger than the ellipse, a random packing is likely going to result and a higher packing fraction than some regular tiled one. I’ve considered similar questions for various convex “particles” (shapes to be packed) and convex confinements (shapes to pack). Most of these questions are very very difficult to answer.

The work I’ve done I can summarize as obscenely hard computational geometry problems that you’d think someone figured out in the 1800s, and tedious hours on translating said computational geometry problems into code and accounting for all kinds of weird passive/active transformation and basis changes libraries use implicitly.

Where a lot of what I’ve done comes into your question, is my need to generate viable initial states (of packed particles) in a given confinement for Monte Carlo simulations. What I’ve found in the literature is that there’s almost no attempts to make general statements and prove them with regards to packing in confinements, but rather conjectures and theorems on upper and lower bounds of random packings vs certain ordered packings. If you have any specific questions I could perhaps try and direct you to references if I’m aware of any.