Is there an analytic solution to this problem or do I need to compute a discrete approximation using a relaxation procedure - or something similar?
I want to find the shape of a roughly spherical shell, which, when submerged in a liquid medium of density d, will experience no bending moments at any point on its surface. At depths much larger than the maximum dimension of the shell, the ideal shape approaches a sphere, but I am interested in the case of a relatively shallow submersion for an underwater habitat at atmospheric pressure.
Another way to describe this shape is that it is the same as the shape assumed by a small drop of liquid on a non-wetting surface - except for the flat portion against that surface. To better understand the dynamics of the situation, consider a pool of mercury on a flat, level piece of glass. If you put this in a chamber with a viewport and increase the air pressure in the chamber, you will see that the mercury will assume a shape that takes up less volume until the sum of the gravitational potential of the mercury and the potential energy of the compressed gas is at a minimum. The mercury will take up less volume to do this and thereby assume a more nearly spherical shape.
I read a paper on this a long time ago, but I haven't been able to find it again.