## 4 Numerical Results and Discussion  
In this part, we present the numerical results of the implementation of IGA with T-Splines and its numerical advantages to study mechanical structure issues. The LPG cylinder is subjected to a static load benchmark to check the integration and the adaptation of the IGA with T-Splines and to investigate the numerical convergence and robustness of the mechanical performance of the LPG cylinder in the case of energy storage issues. The T-Splines numerical results obtained are examined and verified with the FEM in Abaqus/Standard. Firstly, a comparative study of computational time between the various numerical methods utilized in this study was carried out. The obtained results demonstrate that the processing time (Table 2) of IGA with T-Splines is less than that of FEM in Abaqus/Standard. This affirms the results in studies by [PERSON] et al. [23] and [PERSON] et al. [25]. The reduced computation time of the IGA compared to the NURBS method and FEM is determined, and the T-Splines method uses less control points. And the benefits of polygonal and NURBS modeling approaches are combined in T-Splines, as well as the better numerical convergence of the results. \begin{table}
\begin{tabular}{c c} \hline
**Numerical Method** & **FEM in (Abaqus/Standard)** & **IGA with T-Splines** \\ \hline Time Calculation(s) & 24.81 & 13.22 \\ \hline \end{tabular}
\end{table} 
Table 2: A comparison of the time calculations for the FEM and the IGA with T-Splines methods, performed with Intel® Core TM i5-8250U CPU 1.60 GHz (4 CPUs). Figure 9: Loading conditions of the LPG cylinder. A displacement comparison was made for an LPG cylinder subjected to the force of an empty LPG cylinder during storage application, computed using the finite Element method (FEM) in Abaqus/Standard and the isogeometric analysis (IGA) T-Splines approach. As shown in Figure 10, the displacement distribution differs between the two methods, with the T-Splines approach yielding smoother results with fewer numerical artifacts. The comparison in Figure 11 confirms that the IGA T-Splines method provides slightly lower displacement values than the FEM while maintaining better numerical stability and accuracy. This difference is due to the inherent properties of T-Splines, which allow for a more flexible and continuous representation of 3D surfaces without the need for mesh refinement, unlike the FEM, which relies on structured meshing that may introduce local discretization errors (Figure 12). The T-Splines approach also enables local refinement without increasing the global computational cost, making it advantageous for complex geometries and large deformation problems. Similar conclusions were drawn by [PERSON] et al. (2019); [PERSON] et al. (2020); [PERSON] et al. (2020), reinforcing the idea that IGA-based T-Splines outperform the traditional FEM in terms of smoothness, accuracy, and computational efficiency. Figure 11: Load–displacement curves of the LPG cylinder. Figure 10: The LPG cylinder’s displacement magnitude (Um): (**a**) numerical result of Abaqus/Standard FEA, (**b**) numerical result of IGA with T-Splines. The obtained numerical results demonstrate that the IGA with T-Splines provides enhanced accuracy, particularly in capturing localized stress concentrations. The maximum Von Mises stress obtained using T-Splines is 225 MPa, whereas the result for the Abaqus/Standard FEM is 204 MPa. This enhanced precision is attributed to the ability of T-Splines to introduce local control points, refining the solution in critical zones without requiring a global mesh refinement. This feature is particularly beneficial for structural analyses where stress concentration areas are crucial for design validation. Furthermore, the structural integrity of the LPG cylinder is evaluated by varying the number of applied empty cylinders from 5 to 20.