A key feature of any aircraft is its fuel system. Fuel is stored in large tanks (typically within the wing of the aircraft) which are approximately rectangular, but have detailed internal structure. A typical tank is illustrated in Figure 1, in which we can see the internal stringers and other small details. Figure 1: An example of a typical fuel tank showing the internal structures including corners and stringers. A key aspect of aircraft safety is an accurate measurement of the amount of fuel in the tank. To determine this, the height h of the fuel at certain points within the tank is measured and the fuel volume V is determined from this. However the relationship between height and volume depends crucially on the pitch angle, α, and roll angle, β, of the tank. (This is clear to all of us when we drive a car and see that the indicated fuel level in the tank depends upon whether we are going up or down hill at the time.) At present Airbus uses lookup tables to link height to volume for a set of pitch and roll angles. As the fuel tanks become more and more intricate, the huge amount of data required becomes an issue. The problem posed to the Study Group was whether the lookup tables can be replaced by suitable functions. (Note that we only considered the static problem, which assumed that the fuel was in equilibrium. The much harder problem of determining the volume of a sloshing fuel was not considered, but might make a very interesting problem for a future Study Group.) To solve this problem we require a combination of numerical analysis and geometry. Numerical methods are needed to determine suitable approximations to the h-V function. However, generating such approximations turns out to be very subtle, because of the way that the fuel interacts with the tank geometry. This leads to h-V curves which are nonsmooth and have to be approximated very carefully. An understanding of the way that this nonsmoothness arises and how the resulting curves can be approximated forms the basis of this Study Group report
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