As an engineer, I've used lots of FEA to solve problems, but if you can't put part of your solution space where it can be approximated by a closed form solution

As an engineer, I've used lots of FEA to solve problems, but if you can't put part of your solution space where it can be approximated by a closed form solution (and there are many more than those shown, which are amenable to solution to appropriate methods), you're going to have a hard time building a trusted model. There's a reason we still have wind tunnels. The most interesting parts of good FEA (where you've shown your model and reality match on measurable components), is that you can see hidden and unmeasurable variables, which may be design limiting.