# Application of a Coupled Eulerian-Lagrangian Approach to the Shape and Force of Scientific Balloons
## 1 Introduction
The process of a zero-pressure balloon from launch to ascent and full launching involves the interaction of the external atmosphere with the balloon membrane, the folding and unfolding of the membrane, and changes in the internal flow field, making it a relatively complex physical process. In the 1940s, researchers at the University of Minnesota proposed that, as a balloon ascends and gradually inflates, the membrane unfolds circumferentially in an axisymmetric manner, maintaining the principle of a natural shape. Once the balloon reaches level flight, the membrane becomes fully unfolded circumferentially. Based on this idea, they first proposed the cylindrical balloon shape. Shortly after, they introduced a more efficient semi-cylindrical shape [8]. For cylindrical balloons, excess membrane material accumulates at both ends, where the circumferential stress is naturally zero. They deduced that, if this excess membrane was removed while maintaining the naturally formed balloon shape, the circumferential stress at both ends would still remain zero. Inspired by this, they eventually derived equations to describe the natural shapes of balloons, laying the theoretical foundation for natural-shaped balloons. In the early 1960s, [PERSON] at the National Center for Atmospheric Research (NCAR) used digital computers to solve the differential equations proposed by the University of Minnesota that described the balloon shape. Through numerical calculations, he systematically obtained precise dimensionless numerical results under various conditions, such as different loads, inflation levels, and altitudes, describing the natural shape of the balloon. His notable contribution was providing dimensionless numerical results regarding balloon shapes for different payloads and flight altitudes [9,10]. [PERSON] (1984) demonstrated that extremely high strain rates can occur for a brief moment during the instant of balloon launch. This condition may potentially cause the failure of the thin film material; however, a more detailed model of the shear state development, accounting for wave reflections from the edges of the balloon cap, is required before reaching any definitive conclusions. Even with careful measures taken by operational personnel and a smooth launch process, the balloon's thin film is subjected to a violently hostile loading environment, particularly in the vicinity of the load tape. This loading poses a significant risk of damaging the thin film. Although preliminary in nature, this research provides valuable insights into the mechanics of the dynamic launch process [11]. Prof. [PERSON] of George Washington University not only used the shooting method to obtain the two-dimensional, fully expanded balloon generatrix but also assumed that the folded membrane partially disappeared. By introducing a change in the membrane surface density to keep the total mass of the membrane unchanged, the two-dimensional partially expanded generatrix was obtained by the multi-stage shooting method [12,13]. According to the folding and wrinkling characteristics of a three-dimensional shape, [PERSON] further proposed the energy minimization method, which divided the energy of the balloon into membrane strain energy, hydrostatic potential energy, and gravity potential energy. Then, [PERSON] established the variational equation for the shape change function and solved the continuous variational equations by using triangular discrete elements [14,15]. Figure 1: NASA’s 1.14 million cubic meter zero-pressure balloon. Other researchers have attempted to use the finite element method to perform the mechanical analysis of balloons. In the field of fluid-structure interaction, [PERSON] [16] used the Smoothed Particle Hydrodynamics (SPH) method and Arbitrary Lagrangian-Eulerian (ALE) method in the simulation software LS-DYNA R11.2 to model the fluid-structure interaction within the balloon membrane.