[ModernMech]

If you are financially independently wealthy, then you can just apply to the school and pay the full-rate tuition. Usually the way admission works is that you indicate a faculty advisor with whom you want to work. They will act as your mentor. I don't know of a situation where you would work alone, nor would you want to if you wanted to get the most out of your time there. So you would want to talk to that actual person you want to work with, and have a conversation about the fact that you can fund your own research. I imagine the funding process would have to go through the University. They get a cut of research funding, so they would want to funnel your funds through the same system set up for private grants, of which there are plenty.

So yeah, I don't see why you couldn't do this. You would just set up a private research grant with a faculty member and fund yourself with it.