esa-sceva/satcom-chunk-collection huggingface dataset:
# Iterative Oblique Decision Trees Deliver Explainable RL Models
## 4 Results
### Explainability for Oblique Decision Trees in RL  
Explainability is significantly more difficult to achieve for agents operating in environments with multiple time steps than in those with single time steps (by \"single time step\" we mean that a decision follows directly in response to a single input record, e.g., classification or regression). This is because the RL agent has to execute a possibly long sequence of action decisions before collecting the final return (also known as the credit assignment problem [30; 31]).  
Given their simple, transparent nature, DTs offer interesting options for explainability in such environments with multiple time steps, which are not immediately applicable to opaque DRL models. This section will illustrate steps towards explainability in RL through shallow DTs.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Environment** & **dim \(D\)** & **Algorithm** & \(t_{run}\) [s] & \(t_{O\!PCT}\) [s] & \(n_{s}\) \\ \hline \multirow{3}{*}{Acrobot-v1} & \multirow{3}{*}{6} & EPS & 124.82 \(\pm\) 1.96 & 11.71 \(\pm\) 0.95 & 30,000 \\  & & BB & 477.70 \(\pm\) 7.89 & 47.56 \(\pm\) 2.92 & 30,000 \\  & & ITER & 1189.14 \(\pm\) 28.70 & 10.76 \(\pm\) 1.02 & 30,000 \\ \hline \multirow{3}{*}{CartPole-v1} & \multirow{3}{*}{4} & EPS & 187.39 \(\pm\) 3.64 & 18.06 \(\pm\) 2.29 & 30,000 \\  & & BB & 202.65 \(\pm\) 1.26 & 19.11 \(\pm\) 1.13 & 30,000 \\  & & ITER & 2040.16 \(\pm\) 21.46 & 18.50 \(\pm\) 1.64 & 30,000 \\ \hline \multirow{3}{*}{MountainCar-v0} & \multirow{3}{*}{2} & EPS & 85.39 \(\pm\) 1.50 & 7.93 \(\pm\) 0.57 & 30,000 \\  & & BB & 104.92 \(\pm\) 2.05 & 10.27 \(\pm\) 0.54 & 30,000 \\ \cline{1-1}  & & ITER & 849.90 \(\pm\) 19.79 & 7.68 \(\pm\) 0.61 & 30,000 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Computation times of our algorithms for some environments (we selected those with different observation space dimensions \(D\)). Shown are the averaged results from 10 runs. \(t_{run}\): average time elapsed for one run covering all depths \(d=1,\ldots,10\). \(t_{O\!PCT}\): average time elapsed to train and evaluate 10 OPCTs. \(n_{s}\): total number of samples used. Note how the bad performance of BB in the Acrobot environment leads to longer episodes and therefore longer evaluation times \(t_{O\!PCT}\).
## 2 Discrete-Time Volterra Series Model for TIADC
### Discrete-Time Volterra Series Model for a Four-Channel TIADC  
The four-channel TIADC system contains four ADCs that sample the input analog signal alternately under the trigger of the interleaved time signal; thus, four interleaved infinite unit sampling sequences can be used as the model for the trigger signal when modeling the clock of the four-channel TIADC system. The sampling time interval of two adjacent ADC channels is \(T_{s}\), so the expression of the unit sampling sequence of the \(m\)th ADC is:  
\[S_{m}(n)=\sum_{k=-\infty}^{\infty}\delta(n-mT_{S}-4 kT_{S}) \tag{2}\]  
When modeling a four-channel TIADC, it should be noted that the time interval between two adjacent samples of the same ADC is \(4T_{s}\), and the discrete-time Volterra series expression for the four-channel TIADC can be derived as:\[y(n) =\sum\limits_{m=1}^{4}V_{m}[x(n)]*S_{m}(n)\] \[=\sum\limits_{m=1}^{4}\left\{\begin{array}{c}\left[\begin{array}[ ]{c}h_{m,0}+\sum\limits_{k_{1}=0}^{2}h_{m,1}(k_{1})x(4n-k_{1})\\ +\sum\limits_{k_{1}=0}^{2}\sum\limits_{k_{2}=0}^{2}h_{m,2}(k_{1},k_{2})x(4n-k _{1})x(4n-k_{2})\\ \ast\sum\limits_{k=-\infty}^{\infty}\delta(n-mT_{S}-4 kT_{S})\end{array}\right] \end{array}\right\} \tag{3}\]  
The TIADC system block diagram is shown in Figure 3.  
Figure 3: Three-order Volterra series model of the TIADC.  
Figure 2: Three-order Volterra series model of a single ADC.  
For computer programs, infinite sequences cannot be realized; thus, for infinite unit sampling sequences, the Poisson summation formula can be used to convert infinite sequences to the finite complex number for computation. The Poisson summation formula is as follows:  
\[\sum\limits_{n=-\infty}^{+\infty}f(t+nT_{S})=\sum\limits_{K=-\infty}^{+\infty} \frac{1}{T_{S}}F(\frac{K}{T_{S}})e^{2\pi i\frac{K}{T_{S}}t} \tag{4}\]  
where \(F(t)\) denotes the Fourier transform of \(f(t)\), so that the unit sampling sequence can also be expressed as:  
\[\begin{split} S_{m}(n)&{=\sum\limits_{k=-\infty}^ {+\infty}\delta[nT_{S}-mT_{S}-4 kT_{S}]}\\ &=\frac{1}{4T_{S}}\sum\limits_{k=0}^{3}e^{j\pi k(n-m)/2}\end{split} \tag{5}\]  
Then the unit sampling sequence becomes a finite sequence. However, due to the introduction of complex numbers into the calculation, the computational complexity in the field programmable gate array (FPGA) used to calibrate the system is very high. Substituting Equation (5) into Equation (3), the TIADC expression becomes:  
\[\begin{split} y(n)&{=\sum\limits_{m=1}^{4}V_{m}[x( n)]\;*S_{m}(n)}\\ &=\sum\limits_{m=1}^{4}\left\{\begin{array}{c}\left[\begin{array} []{c}h_{m,0}+\sum\limits_{k_{1}=0}^{2}h_{m,1}(k_{1})x(4n-k_{1})\\ +\sum\limits_{k_{1}=0}^{2}\sum\limits_{k_{2}=0}^{2}h_{m,2}(k_{1},k_{2})x(4n-k_ {1})x(4n-k_{2})\\ \ast\frac{1}{4T_{S}}\sum\limits_{k=0}^{3}e^{j\pi k(n-m)/2}\end{array}\right] \end{array}\right\}\\ \end{split} \tag{6}\]
## 3 Results and Discussion
### Extraction of Schottky Parameters
#### 3.3.1 Method 1: \(ln(I)\) vs. \(V\)  
Equations (2) and (3) when combined with consideration of \(V_{d}=V-R_{s}I>3\ kT/q\) will give the following equation:  
\[I\ =\ AA^{\star\star}T^{2}exp\left(-\frac{q\phi_{B}}{kT}\right)exp\left(\frac{q \left(V-RsI\right)}{nkT}\right). \tag{4}\]  
Taking the natural logarithm of Equation (4) will result in  
\[ln\ (I)\ =\ ln\ (I_{s})+\frac{qV_{d}}{nkT}\ =\ ln\ (I_{s})\ +\ \frac{q\left(V-R_{s}I \right)}{nkT}. \tag{5}\]  
A graph of \(ln(I)\) vs. \(V\) can then be plotted as shown in Figure 6 and least square fitting is performed in the linear region to determine \(\phi_{B}\) from the y-intercept and \(n\) from the slope. For this study, the linear region of the curve is defined from 0 to 0.5 V. Above 0.5 V, the \(Rs\) becomes dominant and deviates from the linearity.  
Figure 6: \(ln\ (I)\) vs. \(V\) plot using method 1.
# Security Performance Analysis of Full-Duplex UAV Assisted Relay System Based on SWIPT Technology
## Appendix A Connection Probability (CP) Derivation Procedure:  
Assuming that \(X=S_{au}\), \(Y=S_{ub}\),  
\[P_{c} \triangleq P_{r}\{C_{M}>R_{t}\}=P_{r}\{\gamma_{AB}>\delta_{t}\}\] \[=P_{r}\bigg{\{}\frac{\eta\beta(1-\beta)P_{a}L_{au}XY}{\eta\beta(1 -\beta)P_{a}L_{au}YS_{a}+\eta\beta(1-\beta+\xi)L_{au}YN_{0}+\big{(}P_{a}S_{b}+ N_{0}\big{)}(1-\beta)}>\delta_{t}\bigg{\}} \tag{11}\]  
assuming that  
\[A=\frac{(1-\beta)P_{u}S_{u}+N_{0}(1-\beta+\xi)\delta_{t}}{(1-\beta)P_{a}L_{au}} \tag{12}\]  
\[B=\frac{[P_{b}S_{b}(1-\beta)+N_{0}(1-\beta)]\delta_{t}}{\eta\beta(1-\beta)P_{a }L_{au}L_{ub}} \tag{13}\]  
Since the nodes A and U are Rice fading between them, the calculations are based on the PDF and CDF of the Rice distribution, which leads to Equations (A4).  
\[\begin{split} P_{c}&=P_{r}\Big{\{}X>A+\frac{B}{Y} \Big{\}}=1-E_{Y}\Big{\{}F_{X|Y}\Big{(}A+\frac{B}{Y}\Big{)}\Big{\}}\\ &=\int_{0}^{\infty}Q\bigg{(}\sqrt{2K_{au}},\sqrt{2(1+K_{au}) \Big{(}A+\frac{B}{y}\Big{)}}\bigg{)}\cdot(1+K_{ub})\cdot e^{-K_{ub}}\\ &\cdot exp(-(1+K_{ub})y)\cdot I_{0}\Big{(}2\sqrt{K_{ub}(K_{ub}+1) y}\Big{)}dy\end{split} \tag{14}\]  
where, \(Q(\cdot,\cdot)\) is a first-order Marcu-Q function and \(I_{0}(\cdot)\) is a first-order modified Bessel function of order 0.  
\[I_{0}(y)=\sum_{r=0}^{R}\frac{\Gamma(R+r)R^{r-2r}}{\Gamma^{2}(r+1)\Gamma(R-r+1) }\Big{(}\frac{y}{2}\Big{)}^{2r} \tag{15}\]  
\[Q(x,y)=\sum_{d=0}^{D}\sum_{u=0}^{d}\frac{\Gamma(D+d)D^{1-2d}x^{2d}y^{2u}}{ \Gamma(D-d+1)d\cdot u!\cdot 2^{d+u}}\cdot e^{-\frac{x^{2}+y^{2}}{2}} \tag{16}\]  
Substituting Equations (A5) and (A6) into Equation (A4) the simplification leads to:  
\[\begin{split}& P_{c}\triangleq(1+K_{ub})\cdot\exp[-K_{au}-K_{ub}-A(1+ K_{au})]\cdot\\ &\sum_{d=0}^{D}\sum_{u=0}^{d}\sum_{s=0}^{u}\sum_{r=0}^{\infty} \frac{\Gamma(D+d)D^{1-2d}(\Gamma R+r)R^{1-2r}R_{u}^{2u}(1+K_{au})^{r}K_{ub}^{ r}(1+K_{ub})^{r}}{\Gamma(D-d+1)\Gamma(d+1)\Gamma(d+1)^{2}(r+1)\Gamma(R-r+1) \Gamma(R-r+1)\Gamma(u-s+1)\Gamma(s+1)}\\ & A^{s}\cdot B^{u-s}\cdot\int_{0}^{\infty}exp\Big{[}-(1+K_{au}) \frac{B}{y}-(1+K_{ub})\,y)\Big{]}\cdot y^{s+r-u}dy\end{split} \tag{17}\]  
According to the formula equation (Equation (3.471.9) [24]), the integral of Equation (17) is calculated, and the final closed expression for the connection probability is obtained after complex calculations, which is represented by Equation (16).
## 4 Conclusions  
1. Sasobit modifiers can enhance the resistance to deformation, high-temperature stability, and anti-aging properties of bitumen; however, they can reduce its low-temperature ductility. After mixing with XT-W3 and Evotherm modifiers, there is a reduction in its deformation resistance and high-temperature stability and improvement in its aging resistance and low-temperature ductility.
2. Brookfield rotational viscosity tests indicated that incorporating Sasobit, XT-W3, and Evotherm modifiers can effectively reduce the high-temperature viscosity of bitumen and improve its construction and ease.
3. The dynamic shear rheology test of Sasobit, XT-W3, and Evotherm-modified bitumen indicated that the elastic component in the bitumen increases after incorporating modifiers, improving the high-temperature rutting resistance of bitumen with Sasobit-modified bitumen having the strongest high-temperature rutting resistance.
4. Incorporating Sasobit, XT-W3, and Evotherm modifiers into bitumen mixtures can enhance the compressive properties, high-temperature rutting resistance, water stability, and fatigue resistance of bitumen mixtures. However, Evotherm modifiers reduce the deformation resistance of bitumen mixtures. XT-W3 and Evotherm modifiers can effectively improve the low-temperature crack resistance, but the Sasobit modifier has a negative impact.
5. The compaction performance of the modified bitumen mixtures demonstrated that the gross bulk density of the bitumen mixture Marshall specimens and the compaction degree increased linearly with the increase in compaction temperature and exponentially as a function of the increase in the number of compactions. If the same compaction degree is achieved, the number of compaction times and temperatures required for the Sasobit-modified, XT-W3-modified, and Evotherm-modified bitumen mixtures are reduced by about 10 \({}^{\circ}\)C-30 \({}^{\circ}\)C compared with the virgin and SBS-modified bitumen mixtures.  
Figure 10: Relationship between compaction degree and number of compactions.  
**Author Contributions:** Conceptualization, writing--review and editing, R.L.; investigation, writing--original draft, N.S.; methodology, J.Y.; supervision, B.L. All authors have read and agreed to the published version of the manuscript.  
**Funding:** 1. The authors would like to acknowledge the financial support provided by the National Natural Science Foundation of China (No.51708513). 2. This research was funded by the Open Subject of Coal-based Eco-Fine Chemicals Engineering Laboratory of Henan Province (No. B202001).  
**Data Availability Statement:** The data is unavailable due to privacy restrictions.  
**Conflicts of Interest:** The authors declare no conflict of interest.
## 3 Results and Discussion
### Event 6 (2 October 2022)  
The event of 2 October 2022 began at 19:53 UTC, reached its peak intensity at 20:25 UTC, and ended at 20:34 UTC. The genesis of this event can be traced back to the coordinates N16 W50, pinpointing the origin within the solar region AR13110. The impact of the solar flare on each of the satellites for all stations can be observed in the wavelet results. While some satellites show disturbances in the wavelets independent of the solar flare's impact, the maximum values of the wavelets occur precisely when the solar flare happens (see Figure 11).  
The wavelet outcomes for this event revealed subtle alterations during the solar flare. Despite their subtlety, it was feasible to discern the moment when the flare impacted the ionosphere and distinguish between the results of a calm day and a day affected by the flare-induced disturbances (Figure 11).  
Figure 10: Visualization of Total Electron Content Wavelet Analysis from GNSS Signals for the Event on 10 May 2022.  
Conducting the methodology's implementation on days characterized by tranquil Total Electron Content (TEC) conditions, unaffected by storms or phenomena that atypically disrupt the TEC, is pivotal. This approach allows for the comparison of results between normal days and days influenced by solar flares, as seen in prior events. However, by conducting an aggregation of wavelet results for each station, a coherent and distinctive pattern emerges, revealing the solar flare's consistent impact on the total electron content within the ionosphere. This visual representation is showcased in Figure 12, where the average wavelet trends per station are depicted. This illustration unmistakably underscores that all events exhibit a discernible surge in wavelet values across all the stations employed in the analysis.  
Figure 11: Visualization of Total Electron Content Wavelet Analysis from GNSS Signals for the Event on 2 October 2022.
## 2 Data Acquisition and Preprocessing
### Evaluation Indicators  
In the evaluation of the regression model in this study, R\({}^{2}\), RMSE, and MAE are three commonly used indicators employed, each reflecting the predictive performance of the model from different perspectives [29]. R\({}^{2}\) measures the model's ability to explain the variation in the data; the closer the value is to 1, the better the model's fitting effect. RMSE measures the model's error by calculating the square root of the difference between the predicted and actual values; the smaller the RMSE, the more accurate the model's prediction. MAE calculates the average of the absolute differences between each predicted and actual value, providing a direct reflection of the model's prediction accuracy and being less affected by extreme values compared with RMSE. This study combines the use of these three indicators to comprehensively assess the predictive ability and accuracy of the PCHE microchannel thermal hydraulic performance prediction model.  
\[R^{2}=1-\frac{\sum_{i=1}^{n}(y_{i}-\hat{y_{i}})^{2}}{\sum_{i=1}^{n}(y_{i}-\overline {y_{i}})^{2}} \tag{2}\]  
\[\text{RMSE}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(\hat{y_{i}}-y_{i})^{2}} \tag{3}\]  
\[\text{MAE}=\frac{1}{n}\sum_{i=1}^{n}|\hat{y_{i}}-y_{i}| \tag{4}\]
# Study on Asymmetric Support of Anchor Cable with C-Shaped Tube in Inclined Coal Seam Roadway
## 3 Roadway Deformation Law in Inclined Coal Seam
### Asymmetric Deformation Law of the Initial Support System
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Distance from** & \multicolumn{2}{c}{**The Side Is 2.4 m High**} & \multicolumn{2}{c}{**The Side Is 1.2 m High**} \\ \cline{2-4}
**Roadway Surface** & **Right Displacement/cm** & **Left Displacement/cm** & **Right Displacement/cm** & **Left Displacement/cm** \\ \hline
0 & \(-\)7.83 & 6.2 & \(-\)7.13 & 6.59 \\
0.6 & \(-\)6.85 & 4.94 & \(-\)6.33 & 5.09 \\
1.2 & \(-\)5.26 & 3.67 & \(-\)5.49 & 3.27 \\
1.8 & \(-\)3.11 & 2.57 & \(-\)3.48 & 1.98 \\
2.4 & \(-\)2.5 & 2.57 & \(-\)2.81 & 1.98 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Horizontal displacements on both sides of the displacement monitoring points under the initial support scheme. It is revealed that the inclined coal seam's roadway deformation is considerable, and the plastic zone tends to develop along the coal seam. The two sides clearly exhibit asymmetric deformation characteristics, and when the separation from the roadway surface increases, the degree of asymmetry initially rises and then falls. Design of Anchor Cable with C-Shaped Tube Supporting Reinforcement Based on Asymmetric Deformation of the Roadway
## 3 Temperature Prediction Model for Subgrade
### Evaluation Metrics of Models  
This study employs three metrics to evaluate model performance: Root Mean Square Error (_RMSE_), Mean Absolute Error (_MAE_), and Coefficient of Determination (\(R^{2}\)). _RMSE_, where smaller values indicate better performance, is the most widely used error assessment method for point predictions and is particularly sensitive to larger deviations between actual and predicted values. _MAE_, also favoring smaller values, effectively prevents  
Figure 8: Comparison of MSE values with different numbers of neurons in the hidden layer.  
errors from canceling each other out. \(R^{2}\), with larger values indicating better performance, measures the degree of match between the predicted and actual values. The specific equations for these metrics are as follows:  
\[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}[\hat{y}(i)-y(i)]^{2}}\quad MAE=\frac{1}{n} \sum_{i=1}^{n}|\hat{y}(i)-y(i)|\quad R^{2}=1-\frac{\sum\limits_{i=1}^{n}[\hat{y }(i)-y(i)]^{2}}{\sum\limits_{i=1}^{n}[\hat{y}(i)-\overline{y}]^{2}} \tag{4}\]  
where \(\hat{y}(i)\) is the predicted value, \(y(i)\) is the actual value, \(\overline{y}\) represents the average of the actual values, and \(n\) is the total number of samples.
# Investigating the Influence of Feature Sources for Malicious Website Detection
## 3 Preliminary Experiments
### Expanding [PERSON]'s Dataset
#### 3.3.3 Content-Based Features  
Any source-code behind the webpages serves as the potential input for content-based features. This can be HTML, JavaScript, CSS, PHP, or any other code. Example features include: _total_ and _unique URL count_ in the code, _total JavaScript function count_, and _malicious function count_. Studies that have successfully utilized these features tackle detecting malicious web links alongside their attack types [23] and detecting the presence of malicious JavaScript code [24]. The _suspicious JavaScript function count_ feature involved counting the number of functions associated with potentially risky JavaScript code [25].
## 2 Research Status and Significance at Home and Abroad
### Describe the Problem  
A typical urban logistics network includes urban distribution centers, urban transit centers, and end-of-line demand points close to customers. The city distribution center is responsible for the inflow and outflow of goods (or parcels) in the city. The transit center connects the distribution center and the terminal demand point and is responsible for transit processing. The terminal demand point is generally located in the community, school, business district, and other locations, responsible for the delivery and collection of parcels in the area.  
When standard parcels and oversized parcels are delivered together, according to the functional differences, transit centers can be divided into three categories: oversized transit centers, standard transit centers and hybrid transit centers. In practice, most of the point-to-point direct distribution mode is adopted between the distribution center and the transit center, and the circular distribution mode is adopted between the transit center and the terminal demand point. From the transit center to the end demand point, standard parcels and bulky parcels are independently distributed and collected, the solid line between the terminal demand points in Figure 1 represents the standard parcel travel route, and the dotted line between the end demand points represents the oversized parcel travel route. When the terminal demand point has both standard parcel and bulky parcel delivery or collection demand, it needs to be accessed twice separately, as shown in point A and point B in Figure 1. Urban logistics enterprises face the following decision-making problems when implementing multi-type parcel collaborative distribution:  
1. Threshold setting for bulky parcels and standard parcels;
2. The number and location of three types of transit centers: large-size, standard, and mixed;
3. The coverage of transit centers of different functional types, that is, the allocation of terminal demand points.  
Figure 1: Urban logistics network under the coordinated distribution for standard and large items.
# Increasing Photovoltaic Systems Efficiency Through the Implementation of Statistical Methods
## 5 Discussion
For example, during Romania's cold and cloudy months, photovoltaic energy production can drop below 10%, even when using tracking systems, as only diffuse light is available. Trackers require more space between rows to avoid shadowing as they move, reducing land-use efficiency. Considering all the mentioned factors, the conclusion was that adding extra production capacity to a photovoltaic system (more panels) was more cost-effective than increasing efficiency with trackers. The value of the STEM method lies in its flexibility to incorporate various technical and economic parameters. For instance, S1 represents a value directly proportional to the quality of the product, while S2 denotes a value inversely proportional to the product's quality.
## 2 Materials and Methods
### Generation of ANN Training Data
#### 2.1.2 Modelling and Meshing of the Quarter Models  
In this study, quarter pipes with a rectangular shape idealization of corrosion defects were modelled using AutoCAD. Quarter models reduce computation time, while rectangular defect idealization allows for a safer, lower bound failure pressure prediction without compromising accuracy. The pipes were modelled with end caps for even distributions of axial compressive stress, and the full length of the model was set to 2000 mm to eliminate end-cap influence [2,4,8,12]. Figure 3 illustrates an example of the quarter model used during FEA.  
Prior to finite element simulations, the quarter models were meshed using ANSYS 16.1 Structural Product of Mechanical ANSYS Parametric Design Language (APDL), referred to as ANSYS. Hexahedral SOLID185 (linear order) and tetrahedral SOLID186 (quadratic order) elements were used to mesh the pipe body and end cap, respectively [2,4,8,30,31]. These elements were used to represent the solid structure, while shell elements were used to represent the outer surface of a structure [32].  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Input Parameters** & **Values** \\ \hline Outer diameter of pipe, \(D\) (mm) & 300 \\ \hline Length of pipe, \(L\) (mm) & 2000 \\ \hline Wall thickness, \(t\) (mm) & 10 \\ \hline Normalised defect width, \(w/t\) & 10 \\ \hline Normalised effective defect depth, \({\left({d}/{t}\right)_{e}}\) & 0.00–0.80 \\ \hline Normalised effective defect length, \({\left({l}/{D}\right)_{e}}\) & 0.00–2.95 \\ \hline Normalised circumferential defect spacing, \({s_{c}}/{\sqrt{Dt}}\) & 0.00–3.00 \\ \hline Normalised longitudinal compressive stress, \({\sigma_{c}}/{\sigma_{y}}\) & 0.00–0.60 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Geometric parameters of the corroded pipe models.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Properties**} & \multicolumn{2}{c}{**Pipe Body**} & \multirow{2}{*}{**Pipe End Cap**} \\ \cline{2-3} \cline{4-4}  & **API 5L X52** & **API 5L X65** & **API 5L X80** \\ \hline Modulus of elasticity, \(E\) & & 210.0 GPa & 200.0 TPa \\ \hline Poisson’s ratio, \(v\) & & 0.3 & 0.3 \\ \hline Yield strength, \({\sigma_{y}}\) & 359.0 MPa & 464.0 MPa & 531.0 MPa & - \\ \hline True ultimate tensile strength, \({\sigma_{UITS}^{*}}\) & 612.0 MPa & 629.0 MPa & 718.2 MPa & - \\ \hline \hline \end{tabular}
\end{table}
Table 3: Material properties of the pipe grades used in this study.  
With a total of three layers and mesh size of 2 mm in length and depth, the mesh settings at the defect region are in line with the recommendations by the British Standards Institution (BSI) [33]. Prior to finalizing the mesh settings, a convergence test was carried out to optimise the number of elements to ensure minimum computation time without compromising accuracy. Moving away from the defect region, a mesh bias with an aspect ratio of 0.5 was applied to the elements with a total of 80 divisions. The details and results of the convergence test are presented in Table 5.