# Application of a Coupled Eulerian-Lagrangian Approach to the Shape and Force of Scientific Balloons
## 2 CEL Modeling Method and Simulation Framework  
### Introduction to CEL Modeling Method  
The description of fluids can be categorized into Lagrangian and Eulerian approaches. In the Lagrangian description, the coordinate system follows the motion of the fluid, recording its physical quantities as it moves. In contrast, the Eulerian description is based on fixed points in space, recording the physical quantities of the fluid passing through these points. In finite element simulation software, analyses based on these two approaches are referred to as Lagrangian analysis and Eulerian analysis, respectively. In Lagrangian analysis, the mesh nodes are determined by the material, and the elements deform along with the material. This approach yields relatively accurate results for solids with small strains. In Eulerian analysis, the mesh nodes are fixed, and the elements do not deform. The material flows between elements, making Eulerian analysis well suited for fluid calculations. It is particularly effective for computations involving extreme deformations, such as fluid flow and soil motion. In these analyses, traditional Lagrangian analysis may lose accuracy as the mesh elements become highly distorted. Eulerian analysis can effectively handle issues such as liquid sloshing, gas flow, and permeation problems. For fluid-structure interaction problems, combining Eulerian and Lagrangian analyses can yield better results than using either method alone [17]. In previous computations of large deformation problems involving flexible materials in balloon membranes, methods such as the Arbitrary Lagrangian-Eulerian (ALE) method, Smoothed Particle Hydrodynamics (SPH), and Coupled Eulerian-Lagrangian (CEL) analyses have been commonly used [18; 19; 20]. The ALE method tracks element deformations and remeshes when the deformation becomes excessive. However, during the launching process of scientific balloons, the large deformations require frequent remeshing with short step intervals, resulting in high computational costs. Additionally, the ALE method has strict requirements for the mesh quality and is prone to errors when singularities occur in the mesh. The SPH method simulates the helium inside the balloon membrane using smoothed particles. However, since the helium is replaced by a finite number of particle groups, it cannot fully replicate the physical properties of helium, such as pressure gradients. Moreover, as the balloon's volume increases, the number of SPH particles required grows geometrically, leading to a sharp increase in computational effort. In contrast, the CEL method effectively simulates the physical environment of scientific balloons while keeping the modeling process relatively simple. For larger scientific balloons, scaling the mesh size proportionally can meet computational requirements without significantly increasing the computational load [21]. In Lagrangian analysis in Abaqus, the traditional finite element analysis method is used, where nodes and elements are attached to the material, and the material moves along with the deformation of the mesh, as shown in Figure 1(a). However, in Eulerian analysis, the nodes and elements are fixed in space, and the material can move freely through the nodes without causing mesh deformation, as shown in Figure 1(b). Eulerian meshes are usually simple rectangular elements whose volume extends beyond the boundaries of the material within the Eulerian domain, providing space for material movement and deformation. Thus, the Eulerian mesh is not fully occupied by material, leaving some empty space. Material that moves outside the Eulerian domain during the simulation is no longer included in the computation. Eulerian calculations are based on the volume-of-fluid method, which tracks material flow through the mesh by calculating the Eulerian volume fraction for each element. By definition, if a material completely fills an element, its volume fraction is 1; if no material is  
Figure 2: CEL method: schematic drawing. present in the element, its volume fraction is zero.