# Application of a Coupled Eulerian-Lagrangian Approach to the Shape and Force of Scientific Balloons
## 3 Quasi-Static Shape Solving During the Ascent Process  
Compared to the scenario where the balloon is fully inflated at its maximum altitude, the ascent of the balloon is a dynamic process primarily governed by the buoyancy force. During ascent, as the altitude increases, the atmospheric pressure decreases, causing the volume of the gas-filled portion (the bubble) to gradually expand, and the balloon membrane gradually unfolds. As shown in Figure 5a, the actual photographed shape of the balloon during ascent is spherical. In reality, this unfolding process is non-axisymmetric. Some of the membrane sections unfold, while others pile up circumferentially, and some are sandwiched between the unfolded sections. Additionally, the dynamic loading from airflows causes varying force states on each membrane at the same horizontal section. This unevenness is inevitable overall but random for each individual membrane section. However, from the perspective of the overall change in the balloon's shape, there is still a regular pattern: the head of the balloon remains in a positive pressure state, with an outwardly convex shape. Along the meridian, the negative pressure gradually decreases to zero, followed by a negative pressure zone at the lower part of the bubble, causing the balloon to adopt an inwardly concave shape. Between the positive and negative pressure zones, there is a point where the internal and external pressure difference is zero, referred to as the zero-pressure point. As the balloon ascends, this zero-pressure point gradually moves downward, and, when the balloon reaches its maximum altitude, the zero-pressure point coincides with the bottom of the balloon [19, 23]. This section uses the CEL method in Abaqus/Explicit to simulate the ascent and continuous expansion of the balloon in order to verify the feasibility of the method in solving the fluid-structure coupling problem during the balloon's launch and in other  
Figure 5: (**a**) The actual captured balloon shape during the ascent process. (**b**) The calculated meridional profile of the balloon during ascent. operational conditions. A meridian equation with sigma = 0.1 was selected, and it was scaled proportionally to a meridian equation with a balloon height of 20 m. The balloon design parameters are shown in Table 1.  
In previous designs, balloon designers established differential control equations for the balloon shape and used numerical integration methods to solve the ascent balloon shape, as shown in Figure 5b [24]. This section aims to verify the feasibility of the simulation setup by comparing the balloon ascent shape from the meridian equation with the results obtained from CEL simulations. First, the sigma = 0.1 meridian equation is scaled proportionally to a height of 20 m. Then, rotating shell elements are used to create the zero-pressure balloon part. A 35 m \(\times\) 35 m \(\times\) 40 m Eulerian domain is used to enclose the balloon. Meshing is performed for both the balloon membrane and the Eulerian domain. The zero-pressure balloon is set as M3D4R elements, with a sweep meshing strategy, while the Eulerian domain uses EC3D8R mesh elements. The material properties of the balloon membrane, air, and helium are separately defined. We employ the volume fraction tool to assign materials to the Eulerian domain inside and outside the balloon membrane, setting the exterior to air and the interior to helium. We configure the dynamic explicit analysis step and apply mass scaling to accelerate the computation. As shown in Figure 6a, the bottom point of the zero-pressure balloon is fully fixed. Constraints are applied to the balloon nodes in the XY and YZ planes, allowing movement only in the Y-direction within these planes. According to the standard atmospheric formula, we know that  
\[P_{air}=