When vehicle designs, spacecraft trajectorys and flight profiles are
designed, do the designers use a method of trial and error, then
interpolation to calculate the optimal specifications? Or do they use
numerical intergration to create one huge bunch of equations that when
computed produce the optimal specifications?

For instance, if NASA wanted to build a new vehicle, to determine how much
fuel it needs to carry and what flight profile it needs to fly to get to
orbit, what would they do? Any references on the process of design would be
useful.

I've been working on my own equations for doing this stuff, but I'm just
wondering if it's better to go with simulations and trial and error instead
of lots of equations. Thanks,

David