## 3 Results
### Axially Cracked Pipes with Composite Patches  
Figure 9 depicts a steel pipe subjected to internal pressure (\(q\) = 1 MPa) with a patch repair, represented in cylindrical coordinates, where \(x\), \(\theta\), and \(r\) denote the axial, circumferential, and thickness directions, respectively. The parameters \(L\) and \(2a\) represent the axial length and crack length of the pipe, respectively, while \(t_{a}\) and \(t_{p}\) denote the thicknesses of the adhesive and patch material. Graphite-epoxy and boron-epoxy fiber composites are widely used to reinforce cracked structural components. The material properties and geometric parameters of the steel pipe are the same as those in the previous example without patch reinforcement. The material properties of the boron-epoxy, graphite-epoxy, and adhesive film are given in Table 3.  
Figure 10 shows the finite element mesh of the proposed model for patch-reinforced pipes. The cracked pipe is discretized using a 7 \(\times\) 6 mesh, while both the patch and adhesive layers are discretized using a 5 \(\times\) 4 element mesh. In the thickness direction, a single layer of elements is used for the pipe, adhesive, and patch materials. As a result, a total of 82 elements are used. The previously proposed LW model [25] can also be modeled using the same mesh configuration, as shown in Figure 10. The \(p\)-refinement LW element was validated by comparing it with the ANSYS R14.0 results [25] using \(h\)-refinement, confirming an error margin of less than 3%. While the crack orientation in the current model differs from that in [25], the consistency of the applied methodology ensures the reliability of the proposed approach. Additionally, since this study focuses on emphasizing computational efficiency compared to the conventional \(p\)-version LW element, the results \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Types**} & \multicolumn{6}{c}{**Crack Length \(a\) (mm)**} \\ \cline{2-7}  & **3** & **6** & **9** & **12** & **15** & **18** \\ \hline Present & \multirow{2}{*}{1.0638} & \multirow{2}{*}{1.2547} & \multirow{2}{*}{1.4952} & \multirow{2}{*}{1.7681} & \multirow{2}{*}{2.0253} & \multirow{2}{*}{2.3095} \\ analysis & & & & & & \\ Reference [25] & & & & & & \\
[25] & & & & & & \\ Reference [38] & & & & & & \\ Reference [39] & & & & & & \\
[39] & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of nondimensional SIFs in unpatched pipes. Figure 8: Dependence of computed nondimensional SIFs on element size at the crack tip. from [25] are used to compare and analyze the simulation outcomes before and after patch reinforcement. Both the proposed model and the LW model contain 82 elements, but they differ in the number of degrees of freedom (DOFs). Figure 11 compares the number of DOFs of the proposed model with the LW model at different \(p\)-levels. At the same \(p\)-level, the proposed model requires approximately one-third of the DOFs compared to the LW model. Figure 12 presents the variation in the nondimensional SIFs as a function of the crack length, comparing the unreinforced case and the boron-epoxy patch reinforcement (\(\theta_{p}=45^{\circ}\), \(l_{p}=35\) mm, \(t_{p}=0.25\) mm, \(t_{a}=0.05\) mm). At \(p\)-level = 6, the proposed model requires 11,055 DOFs, whereas the LW model requires 33,042 DOFs. Despite using only one-third of the DOFs compared to the LW model, the computational results show minimal differences between the two models. Even with the same comparatively coarse mesh, the proposed model demonstrates greater computational efficiency. Furthermore, the analysis results indicate that as the crack length increases, the difference in SIFs between the unreinforced and reinforced cases becomes more pronounced. This confirms that the effectiveness of patch reinforcement becomes more significant as the crack length increases. Figure 13 compares the \(F\) values as a function of crack length for graphite-epoxy and boron-epoxy. Overall, the boron-epoxy patch, with its higher stiffness, leads to lower \(F\) values.