## 4 Discussion  
The computational performance of the proposed \(p\)-refinement-based numerical modeling approach was validated by analyzing the deflection of a cylindrical shell subjected to internal pressure. The results demonstrated that even with a coarse mesh, the solution started to converge from the fourth polynomial order. A comparison with the LW element [25] showed a high level of agreement, confirming that the proposed method maintained computational efficiency while achieving a high accuracy. The minor discrepancies observed in references [32; 33] are attributed to the fact that these studies did not consider shear deformation effects. The analysis of cracked pipes without patch reinforcement showed that the computed nondimensional SIFs exhibited a high degree of agreement with previous studies, with relative errors within 10%. Furthermore, applying the VCCT revealed that the choice of \(\Delta a/a\) significantly influenced the accuracy of SIFs, and values within the range of \(0.015\leq\Delta a/a\leq 0.05\) provided stable results. Figure 16: Variations in nondimensional SIFs with varying adhesive thickness. Figure 15: Variations in nondimensional SIFs with patch thickness. To evaluate the computational efficiency of the proposed model, a comparison was conducted using the same coarse element mesh as in the LW model, focusing on DOFs. The results demonstrated that the proposed model required only one-third of the DOFs, while still producing comparable accuracy. This finding highlights the advantage of p-refinement-based modeling over traditional finite element approaches, particularly for large-scale fracture mechanics problems. This study also investigated axially cracked pipes reinforced with composite patches, confirming that patch reinforcement effectively reduces SIF values. Among the patch materials considered, boron-epoxy patches yielded lower SIF values than graphite-epoxy patches, which was attributed to the higher stiffness of boron-epoxy. As the stiffness of the reinforced region increased, crack resistance improved, and the difference in SIF values between the two materials became more pronounced with increasing crack length. This result suggests that for longer cracks, a stiffer patch material enhances the reinforcement effect. The effect of patch size on SIF values was also examined, revealing that increasing the patch size led to a decrease in SIF values, thereby improving crack resistance. However, beyond a certain threshold, the reduction in SIF values stabilized, indicating that increasing the patch size indefinitely does not necessarily maximize reinforcement effectiveness. A similar trend was observed with patch thickness, where an increase resulted in a continuous decrease in SIF values, demonstrating enhanced reinforcement efficiency. However, beyond a certain thickness, the rate of reduction in SIF values diminished, suggesting that while thicker patches contribute to structural improvement, excessive thickness does not provide significant additional benefits. The influence of adhesive thickness on SIF values was relatively minor compared to patch thickness. In general, increasing the adhesive thickness did not lead to substantial changes in SIF values. However, for shorter cracks, a thicker adhesive layer resulted in slightly lower SIF values. This suggests that adhesive thickness can provide some reinforcement benefits for smaller cracks, but its overall influence remains limited compared to patch material, size, and thickness. These findings emphasize the importance of selecting optimal patch properties to maximize reinforcement effectiveness. While increasing patch size and thickness reduces SIF values, achieving a balance between these parameters is crucial to optimize the reinforcement efficiency and minimize unnecessary material usage. Despite the excellent performance of the proposed model, several limitations exist.