A variational principle for ascent shapes of large scientific balloons

Abstract

Ascent shapes of large scientific balloons possess a number of well-defined features, including ( F 1) a spherical top, ( F 2) a lobe pattern surrounding the gas bubble, ( F 3) internally folded material within the gas bubble, and ( F 4) flat wing-like structures in the lower portion of the balloon. The standard Σ-shape model for a balloon assumes an axisymmetric solution and is inadequate for handling shapes with these features. The energy E of a balloon shape S is modeled as the sum of the gravitational potentials of the lifting gas and the balloon fabric. Our approach is to minimize E[ S] over a class of shapes that satisfy certain constraints and are invariant under the dihedral group D k . A solution computed in this fashion is called an Energy Minimizing Shape or EM-shape. EM-shapes are found to possess features ( F 1)–( F 4). In this paper, we introduce a structure called a pseudo-gore into our mathematical model. Numerical solutions of EM-shapes with square-shaped and hexagaonal symmetries are included.