IANAME (Mech. Engineer) - but I'd bet it was mostly known & boring physics (rates of thermal conduction, coefficients of thermal expansion, etc.), experience with the material being used (stress-strain curves, crack propagation, and such), and some differential equations that weren't all that hard to get approximate solutions to. A lot of things like this are "If you could get an "A" on the Mech. Engin. 407 final, then you could pretty much figure it all out."