2006/11/01 by Frank Baginski, Baginski, Frank, Michael Barg +3
Engineering · Mathematics · #49J45 #Aerospace Engineering and Energy Systems #Analysis of PDEs (math.AP) #Control and Dynamics of Mobile Robots #FOS: Mathematics #Structural Analysis and Optimization #math.AP #msc:49J45
paper · pdf · doi:10.48550/arxiv.math/0611035
39 pages, 9 figures
arxiv created 2006/11/01 · openalex publication_date 2006/11/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we establish rigorous existence theorems for a mathematical model of a thin inflated wrinkled membrane that is subjected to a shape dependent hydrostatic pressure load. We are motivated by the problem of determining the equilibrium shape of a strained high altitude large scientific balloon. This problem has a number of unique features. The balloon is very thin (30 micron), especially when compared with its diameter (over 100 meters). Unlike a standard membrane, the balloon is unable to support compressive stresses and will wrinkle or form folds of excess material. Our approach can be adapted to a wide variety of inflatable membranes, but we will focus on two types of high altitude balloons, a zero-pressure natural shape balloon and a super-pressure pumpkin shaped balloon. We outline the shape finding process for these two classes of balloon designs, formulate the problem of a strained balloon in an appropriate Sobolev space setting, establish rigorous existence theorems using direct methods in the calculus of variations, and present numerical studies to complement our theoretical results.