# Application of a Coupled Eulerian-Lagrangian Approach to the Shape and Force of Scientific Balloons
## 2 CEL Modeling Method and Simulation Framework
### CEL Fluid-Structure Coupling Simulation Framework  
The process of establishing the finite element model is shown in Figure 3. First, we create models for the zero-pressure balloon and the Eulerian domain and mesh them separately. Next, we assign material properties to the balloon membrane, helium, and air and set up a dynamic explicit analysis step. Then, we use the volume fraction tool to distinguish the regions of air and helium within the Eulerian domain and assign predefined fields. We establish general contact between the Eulerian domain and the zero-pressure balloon to enable their interaction. Finally, we apply loads and constraints and submit the simulation for computation. In the Coupled Eulerian-Lagrangian (CEL) method in Abaqus, fluid-structure interaction is achieved through the General Contact functionality. General Contact uses a penalty function algorithm, which efficiently handles complex geometries, nonlinear behaviors, and multiphysics coupling problems. At each time step, it automatically checks whether the slave contact surface nodes penetrate the master contact surface. If penetration occurs,  
Figure 3: Simulation analysis flow chart. a contact force proportional to the penetration depth and contact stiffness is introduced, as described by the following formula:  
\[F=k\cdot\delta \tag{1}\]  
where \(F\) represents the contact force, \(k\) is the contact stiffness, and \(\delta\) is the penetration depth. This contact force (penalty function value) is equivalent to placing a normal spring between the contact surfaces to separate them, as shown in Figure 4. These springs keep the master and slave contact surfaces apart, preventing overlap. This method avoids severe mesh deformation and minimizes noise. Since equal and opposite interaction forces are applied on the master and slave surfaces, momentum conservation is maintained accurately. This contact approach is highly adaptive, checking for penetration at every time step. When the mesh undergoes significant deformation, reducing the time step size appropriately can mitigate inaccuracies in the results caused by abrupt changes in the contact force. The General Contact method utilizes the principle of the penalty function to ensure physical constraints between contact surfaces and prevent penetration. At each time step, General Contact detects the penetration state between the master and slave contact surfaces to determine whether penalty forces need to be applied. The penalty function exerts elastic forces in the contact region to separate the contact surfaces, avoiding non-physical intersections or overlaps. This approach enables General Contact to handle not only solid-solid interactions but also interactions between solids and fluids. Particularly under conditions of large deformations and complex flow fields, the penalty function ensures the stability of the contact state. Abaqus/Explicit employs the central difference method to explicitly integrate the equations of motion over time, using an incremental kinematic condition to compute the kinematic condition for the next increment. This explicit dynamic method can handle cases with localized large deformations and discontinuities at certain points. To ensure the quality of the calculation, a minimum time increment (\(\Delta t\)) must be introduced to maintain stability. This increment is determined by the smallest element length (\(L_{c}\)), Young's modulus (\(E\)), and density (\(\rho\)) of the divided mesh, as expressed in the following formula:  
\[\Delta t=\frac{L_{c}}{\sqrt{E/\rho}} \tag{2}\]  
From this formula, it can be seen that the minimum time increment is related to the smallest element length. This means that, if a mesh becomes distorted and the edge length of the distorted element is smaller than that of the regular mesh, the system will select the shortest edge of the distorted mesh as the calculation parameter to satisfy the stability conditions. As a result, the minimum time increment becomes smaller than expected,  
Figure 4: Penalty function of the spring model.