esa-sceva/satcom-chunk-collection huggingface dataset:
## 5 Results and Discussions
### BMP for the GCC Region
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Electricity Generation (GWh)**} & \multicolumn{2}{c}{**Freshwater Production (10\({}^{6}\) m\({}^{3}\))**} \\ \cline{2-5}  & **Electricity** & **Current** & **Percent Increase** & **Freshwater** & **Percent Increase** \\  & **From CH\({}_{4}\)** & **Renewables** & **from CH\({}_{4}\)** & **From CH\({}_{4}\)** & **from CH\({}_{4}\)** \\ \hline UAE & 172–250 & 1315 & 13–19\% & 31–45 & 4–6\% \\ Bahrain & 0 & 9 & 0\% & 0 & \\ KSA & 812–1177 & 219 & 373–53\% & 148–214 & 5–7\% \\ Oman & 27–40 & 16 & 175–276\% & 4–8 & 5–8\% \\ Qatar & 61–88 & 123 & 50–72\% & 11–16 & 4–6\% \\ Kuwait & 50–105 & 88 & 88–120\% & 13–19 & 4–6\% \\ Total & 1152–1665 & 1770 & & 150–302 & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Electrical power and desalinated water production from Biomethane. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **CH\({}_{4}\) (mL/g)** & **CO\({}_{2}\) (mL/g)** & **Biogas (mL/g)** & **CO\({}_{2}\) (g/g)** \\ \hline COD & 144.33 & - & - & - \\ C\({}_{7}\)H\({}_{15}\)O\({}_{3}\)N & 100.2 & 39.3 & 139.5 & 0.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Biogas production per gram of COD and sludge (substrate) at standard conditions at 0 \({}^{\circ}\)C and 1.0 atm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Whesterar Influent** & **COD Loading** & **CH\({}_{4}\) (10\({}^{3}\) m\({}^{3}\))** & **CH\({}_{4}\) (10\({}^{3}\) m\({}^{3}\))** \\  & **2019 (10\({}^{6}\) m\({}^{3}\)/Year)** & **10\({}^{3}\) Tonne** & **C\({}_{7}\)H\({}_{15}\)O\({}_{14}\)N** & **COD** \\ \hline UAE & 771 & 348 & 34,870 & 50,227 \\ Bahrain & 154 & - & - & - \\ KSA & 3083 & 1642 & 164,528 & 236,991 \\ Oman & 95 & 56 & 5611 & 8083 \\ Qatar & 278 & 123 & 12,325 & 17,753 \\ Kuwait & 334 & 147 & 14,729 & 21,217 \\ Total & 4715 & 2316 & 232,063 & 334,268 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Biomethane estimates for GCC states. wind and solar-based power production. Unlike the variable solar and wind-based energy production sources, biomethane can be directly utilized in gas turbines that can be dispatched in a timely matter to facilitate the integration of a high share of intermittent renewables. One key advantage of biomethane is that it can exploit existing natural gas infrastructures, such pipelines, storage and turbines.
## Abbreviations  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} \(x\) & The original dataset \\ \(\mu\) & The mean of the dataset \\ \(\sigma\) & The standard deviation of the dataset \\ \(z\) & The formation pressure, MPa \\ \(v\prime\) & The processed value \\ \(v\) & A value from the original dataset \\ _median_ & The median of the dataset \\ \(IQR\) & The interquartile range of the dataset \\ \(g_{i}^{(t)}\) & The prediction of the \(i\)-th instance at the \(t\)-th iteration \\ \(f_{k}(x_{i})\) & The prediction of the \(k\)-th tree for instance \(x_{i}\) \\ \(w_{q(x_{i})}\) & The leaf weight assigned to the \(i\)-th instance when it reaches a leaf in the tree \\ \(q(x_{i})\) & The function mapping an instance to the corresponding leaf in the tree \\ \(Obj(\Theta)\) & The objective is a sum of a loss term (how well the model predicts) and a regularization term (to keep the model simple) \\ \(l\left(y_{i},g_{i}^{(t)}\right)\) & The loss function that measures the difference between the true label \(y_{i}\) and the predicted \\  & label \(\hat{y}_{i}^{(t)}\) for the \(i\)-th instance at the \(t\)-th iteration \\ \(g_{i}\) & The first-order gradient statistics of the loss function with respect to the prediction \(\hat{y}^{(t-1)}\) from the previous iteration \\ \(h_{i}\) & The second-order gradient statistics of the loss function with respect to the prediction \\ \(\hat{y}^{(t-1)}\) from the previous iteration \\ \(\Omega(f_{k})\) & Regularization term for the \(k\)-th tree \\ \(y_{k}\) & The \(k\)-th real production, \(\mathrm{m}^{3}/\mathrm{d}\) \\ \(\hat{y}_{k}\) & The \(k\)-th predicted production, \(\mathrm{m}^{3}/\mathrm{d}\) \\ \(\overline{y}\) & The mean real production of all wells, \(\mathrm{m}^{3}/\mathrm{d}\) \\ \(\overline{y}\) & The mean predicted production of all wells, \(\mathrm{m}^{3}/\mathrm{d}\) \\ \(\sigma_{l}\) & The standard deviation of pipelines' real production, \(\mathrm{m}^{3}/\mathrm{d}\) \\ \(\sigma_{p}\) & The standard deviation of pipelines' predicted production, \(\mathrm{m}^{3}/\mathrm{d}\) \\ \(n\) & The total count of specimens \\ \(Q_{j}\) & The gas production of a sampling point after segmenting, \(\mathrm{m}^{3}/\mathrm{d}\) \\ \(Q\) & The total gas production, \(\mathrm{m}^{3}/\mathrm{d}\) \\ \(K\) & Permeability, md \\ \(S\) & Oil saturation, \% \\ \(\phi\) & Porosity, \% \\ \(a\) & Tortuosity factor \\ \(b\) & The coefficient related to lithological characteristics \\ \(c\) & The cementation exponent of the rock \\ \(d\) & The saturation exponent \\ \(R_{t}\) & The total fluid-saturated rock resistivity \\ \(R_{w}\) & The resistivity of the fluid itself (\(w\) meaning water or an aqueous solution containing dissolved salts with ions bearing electricity in solution) \\ \end{tabular}  
Figure A1: The technical approach of post-fracture production prediction.
# Optimization of Number of GCPs and Placement Strategy for UAV-Based Orthophoto Production
###### Abstract  
Unmanned aerial vehicles (UAVs) have been employed to perform aerial surveys in many industries owing to their versatility, relatively low cost, and efficiency. Ground control points (GCPs) are used for georeferencing to ensure orthophoto geolocation/positioning accuracy. In this study, we investigate the impact of the number and distribution of GCPs on the accuracy of orthophoto production based on images acquired by UAVs. A test site was selected based on regulatory requirements, and several scenarios were developed considering the specifications of the UAVs used in this study. The locations of GCPs were varied to obtain the results. Based on the results obtained for different numbers of GCPs per unit area and distribution of GCPs, it is shown that UAV-based platforms can be more extensively utilized in a range of applications. The findings of this study will significantly impact the development process of GCP automation algorithms and enable a more cost-effective approach when determining target sites for UAV-based orthophoto production.  
ground control points; unmanned aerial photogrammetry; unmanned aerial vehicle; orthophoto; digital elevation model +
Footnote †: journal: Medical Imaging
## 6 Route Discovery by TW
### TW Algorithms  
TW algorithm uses two nested loops, so each loop has a time complexity of \(O(n)\times O(n)=O(n^{2})\). If \(H\) is searching depth of network and \(H=K\) time complexity its will be \(O(Kn^{2})\). It is much suitable for a highly dense network as compared to other route search algorithm, i.e., AT [11]. AT algorithm has high time complexity because it needs to search one to one user in network. However, this is not feasible for trust assessment in high-density VIoT. The time complexity in computing each user trustworthiness will be \(O((n-1)n^{k-1})=O(n^{k})\). Here, \(k\) shows the total searching depth in the network.  
Algorithm 1 starts from line 3, defining a minimum number of hops. Line 4, iteratively search the number of hops on the network. In TW, we used two nestle loops in which line 5-14 updates the indirect opinion \(\Omega_{ij}\), and lines 7-12 compute all opinions from \(w_{sj}\
eq u\).  
\[\Theta(\triangle(\Omega_{i_{1}}^{(K-1)},w_{1j}),..,\triangle(\Omega_{i_{n-1}}^ {(K-1)},w_{n-1j},\triangle(\Omega_{i_{n}}^{(K-1)},w_{nj}).\]  
Line 13 uses \(\Omega_{ij}\) to updates corresponding elements in individual opinion vector. After updating All possible opinions at line 14, TW search next level.  
Figure 4: Matrix formation and initialization in TW route discovery.  
```
REQUIRE: A directed graph G with a truster \(i\) and the maximum searching level \(H\) ENSURE: \(i^{\prime}s\) opinion on \(J\) where \(i\
eq j\)
1. Initialize \(OM\) and \(OV_{i}^{(1)}\) based on \(G\)
2. \(k\gets 1\)
3. While\(k<H\)do
4. \(k\gets k+1\)
5. for all columns \(C_{j}\in OM\) s.t. \(j\
eq i\)do
6. \(\Omega_{ij}^{k}\gets U\)
7. for all direct opinion \(w_{sj}\in C_{j}\) s.t. \(w_{sj}\
eq u\)do
8. \(\Omega_{is}^{(k-1)}\gets y_{i}^{(k-1)}[s]\)
9. if\(\Omega_{is}^{(k-1)}\
eq u\)then
10. \(\Omega_{ij}^{k}\leftarrow\Theta\left(\Omega_{ij^{\prime}}^{K}\Delta(\Omega_{ ij}^{K-1},w_{sj})\right)\)
11. end if
12. endfor
13. \(y_{i}^{(k)}[j]\leftarrow\Omega_{ij}^{(K)}\)
14. endfor
15. endwhile
16. return\(y_{i}^{k}\)
```  
**Algorithm 1**Route Discovery by TW  
The pseudo-code for user authentication is described in Algorithm 2. Here, we set the threshold with 0.5 for user authentication and make decisions to update \(\Omega_{ij}^{k}\) according to the rules given in Table 2.  
```
Exchange opinion about opinion matrix \(i\) with all neighbors of individual vector \(j\) using the TW information exchange protocol.;
1. /* Verify the trustworthiness of \(i-j^{*}\)/
2. /* \(\Omega_{ij}^{k}\
eq u\) and judge the next step using conditions set in Table 1 */
3. if\(\Omega_{ij,j}^{k}\geq 0.5\)
4. trust \(i-j\) and forward RREQ/RREP
5. elseif\(\Omega_{ij,d}^{k}\geq 0.5\)
6. distrust \(i-j\) for expiry time
7. elseif\(\Omega_{ij,u}^{k}\geq 0.5\) request and verify digital certificate
8. else
9. /* the confidence about trustworthiness is decreased*/
10. endif
```  
**Algorithm 2**Authentication from user \(i\) to \(j\) in \(k\)-hop by TW
## 3 Results and Discussion
### Contact Angle Test  
The contact angle represents the intersection formed between a liquid droplet and the solid mineral substrate, offers crucial information about the surface's hydrophobic characteristics and its behavior during flotation processes. In general, a higher contact angle suggests increased hydrophobicity and superior flotation performance, whereas a lower value reflects weaker hydrophobic tendencies and reduced flotation capacity [30; 31; 32; 33]. Figure 6 illustrates the variation in the contact angle of molybden surface under different  
Figure 5: Flotation behavior of chalcopyrite and molybdenite as affected by **(a)** STG concentration at natural pH; **(b)** Cu\({}^{2+}\) concentration at pH 8 with 1.0 \(\times\) 10\({}^{-3}\) mol/L STG; **(c)** reagent conditions; and **(d)** Cu\({}^{2+}\) concentration in an artificial mixed ore system at pH 8 with 1.0 \(\times\) 10\({}^{-3}\) mol/L STG.  
conditions. Pristine molybdenum exhibited an initial contact angle of 86\({}^{\circ}\). Only a slight change in the contact angle occurred when treated with either Cu\({}^{2+}\) or STG alone, with values of 87\({}^{\circ}\) and 84\({}^{\circ}\), respectively, remaining close to the initial contact angle. However, when molybdenum was in the STG system with Cu\({}^{2+}\), the contact angle decreased from 86\({}^{\circ}\) to 42\({}^{\circ}\). This suggested that neither Cu\({}^{2+}\) nor STG, when applied individually, substantially altered the floatability of molybdenum, whereas in the STG system with Cu\({}^{2+}\), the floatability of molybdenum was significantly inhibited.  
Further analysis revealed that the contact angle of molybdenum after kerosene treatment was 108\({}^{\circ}\), whereas after the individual applications of Cu\({}^{2+}\) or STG followed by kerosene treatment, the contact angles were 110\({}^{\circ}\) and 107\({}^{\circ}\), respectively, indicating that either reagent alone had minimal impact on the hydrophobicity imparted by kerosene. In contrast, when kerosene was added to molybdenum in the STG system with Cu\({}^{2+}\), the contact angle decreased to 55\({}^{\circ}\), suggesting that Cu\({}^{2+}\) in the STG system significantly weakened the hydrophobic effect induced by kerosene. These findings aligned with the flotation experiment results, implying that while Cu\({}^{2+}\) or STG alone has limited influence on surface hydrophobicity, their combined presence exerts a synergistic inhibitory effect, markedly reducing the hydrophobicity of the molybden surface.
# Reliable Visual Exploration System with Fault Tolerance Structure
## 6 Experiments
### Evaluation of Active Controller  
Both simulations in Gazebo and experiments in the real world are conducted with the active controller, and a differential driven platform is used. In addition, demonstrations are shown to evaluate the active controller. In addition, all the experiments and simulations in this section are performed without any visual localization.  
In simulation, an indoor environment is set-up for ground texture generation. In addition, some obstacles are placed to test the ability of obstacle avoidance. With respect to the mobile platform, the TurtleBot model is utilized, and a monocular camera is fixed on it in the heading direction. The simulation result of obstacle avoidance is shown in Figure 9. In addition, given enough time (60 s), the process of exploration is shown in Figure 10.  
For experiments in the real world, instead of the ideal illumination in the simulation environment, the illumination changes greatly outdoors, as well as the inconsistency of the ground texture. Due to the illumination invariance of the CNN based image segmentation method, the mobile platform is driven properly and avoids the obstacles successfully. The environment of experiment is shown in Figure 11. To evaluate the robustness of the controller, the experiment is also run on rainy nights (Figure 12), whose illumination is much different from that of Figure 11. Some shortcuts of the experiment are shown in Figure 13, and the demonstration can be seen in the attached video \"Demonstration of Active Controller\".  
Figure 9: Trajectory of obstacle avoidance in simulation. The blue line with arrow is the trajectory, and the colored points in the trajectory are the places where the avoidances of correspondent colored obstacles happen.  
Figure 11: Real world experiment in the sunny morning. The array is the robot trajectory, where the red dotted line in the upper figure is the place where obstacle avoidance happens.  
Figure 12: Environment of the experiment on rainy nights, for evaluation of illumination invariance.  
Figure 10: The results of covering in simulation. The trajectory of the robot is represented by a set of red arrows. The green point is the start point of exploration and the blue one is the end point.
# MIMO LS-SVR-Based Multi-Point Vibration Response Prediction in the Frequency Domain
## 2 MIMO LS-SVR-Based Vibration Response Prediction under Unknown Uncorrelated Loads
### Theoretical Comparison of Response Prediction Methods  
The computation of the condition number of transfer function matrix \(\mathbf{H}(\omega)\) is given by  
\[\text{{cond}}(\mathbf{H}(\omega))=\
ormalsize\mathbf{H}(\omega)^{\text{T}} \mathbf{H}(\omega)\
ormalsize\bullet\
ormalsize\left\lVert\left(\mathbf{H}( \omega)^{\text{T}}\mathbf{H}(\omega)\right)^{-1}\right\rVert \tag{10}\]  
For nearby resonance frequencies, the condition number of the transfer function matrix \(\mathbf{H}(\omega)\) is very large. Moreover, there exists an ill-conditioned part in the least squares generalized inverse of the transfer function matrix [36].  
The least squares generalized inverse is needed in the transfer function method and the transmissibility matrix method. The least squares generalized inverse is also needed in the multiple linear regression methods. Because of the existence of an ill-conditioned part in the least squares generalized inverse, the transmissibility matrix and multiple linear regression are not robust to the measurement error, particularly for nearby resonance frequencies.  
However, because there are no ill-conditioned parts or a need to use the least squares generalized inverse to calculate the transfer function matrix, using the measured multi-point vibration response to predict the unmeasured multi-point vibration response in the frequency domain using MIMO LS-SVR methods is robust for measurement noise. Simultaneously, the MIMO neural network [42] also has the above advantages, but the neural network requires many samples and hyperparameters to improve the prediction performance, and it is easy to overfit. It is not easy for the MIMO neural network to achieve a good prediction in small sample experiments. Therefore, MIMO LS-SVR is more suitable for response prediction in this study.  
The comparison of the different methods are shown in Table 1.  
Figure 2: Procedure for building and applying the multi-input multi-output least squares support vector regression (MIMO LS-SVR)-based multi-point vibration response prediction model in the frequency domain under unknown uncorrelated multi-source loads.
## 4 Numerical Simulation and Discussion
### Numerical Results  
The multi-field response of the study area was obtained by using the transient finite element solver in COMSOL Multiphysics(r). The calculation is divided into two stages: A stationary state before impoundment and a transient state after impoundment. The pre-project exploration data are used to calibrate the model in the stationary state. Next, the transient state of the study area is calculated with the initial multi-field distribution, which is obtained from the calibrated stationary state results. Two specific sections shown in Figure 13 are used to represent the simulation result in the following chapters.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Weathered Basalt**} & \multicolumn{4}{c}{**Weathered Limestone**} \\ \hline \multirow{2}{*}{**Buried Depth (m)**} & \multicolumn{2}{c}{**Hydraulic Conductivity (m/d)**} & \multirow{2}{*}{**Buried Depth (m)**} & \multicolumn{2}{c}{**Hydraulic Conductivity (m/d)**} \\ \cline{2-5}  & **Horizontal** & **Vertical** & & **Horizontal** & **Vertical** \\ \hline
0–40 & 2.80–1.40 & 0.84–0.35 & 0–30 & 30.0–10.0 & 20.0–15.0 \\
40–90 & 1.62–0.60 & 0.55–0.15 & 30–80 & 20.0–5.0 & 10.0–2.0 \\
90–190 & 0.54–0.10 & 0.17–0.02 & 80–120 & 10.0–1.0 & 5.0–0.51 \\ \textgreater{}190 & 0.06–0.02 & 0.015–0.003 & 120–300 & 3.6–0.5 & 2.0–0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Hydraulic conductivity of the weathered zone in the river valley.  
Figure 13: 2D sections used for displaying the simulation result (B–B1: Transversal section, C–C1: Longitudinal section).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & **Dam** & **Grouted Curtain** & **Shale** & **Limestone** & **Basalt** & **Weathered Zone** \\ \hline Density (kg/m\({}^{3}\)) & \(2.40\times 10^{3}\) & \(2.20\times 10^{3}\) & \(2.60\times 10^{3}\) & \(2.60\times 10^{3}\) & \(2.65\times 10^{3}\) & \(2.60\times 10^{3}\) \\ \hline Thermal conductivity (W/(m\text{-}K)) & 1.80 & 2.10 & 2.72 & 2.90 & 4.65 & 4.65 \\ \hline Specific heat capacity (R/(kg\text{-}K)) & 0.88 & 0.85 & 0.86 & 0.86 & 0.86 & 0.86 \\ \hline Thermal expansion (I/K) & \(7.5\times 10^{-6}\) & \(8.0\times 10^{-6}\) & \(8.0\times 10^{-6}\) & \(9.5\times 10^{-6}\) & \(8.0\times 10^{-6}\) & \(8.5\times 10^{-6}\) \\ \hline Elastic modulus (GPa) & 43.0 & 15.0 & 10.0 & 20.0 & 25.0 & 15.0 \\ \hline Possion’s ratio & 0.17 & 0.20 & 0.22 & 0.30 & 0.20 & 0.20 \\ \hline Porosity & 0.040 & 0.10 & 0.25 & 0.20 & 0.050 & 0.20 \\ \hline Biot’s coefficient & - & 0.80 & 0.85 & 0.95 & 0.80 & 0.95 \\ \hline Hydraulic conductivity (m/d) & - & \(1.0\times 10^{-4}\) & \(2.0\times 10^{-2}\) & \(5.0\times 10^{-2}\) & \(2.0\times 10^{-2}\) & Table 4.
# Green Synthesis Method and Application of NaP
## 4 Conclusions  
NaP zeolite with good physical properties can be synthesized by adding a small amount of NaOH and deionized water to the discarded CGCS. Adjust the molar ratio to SiO\({}_{2}\):Al\({}_{2}\)O\({}_{3}\):Na\({}_{2}\)O:H\({}_{2}\)O = 5.2:1.0:5.0:100. This process does not require additional silicon source aluminum source and template agent, and does not require high temperature calcination. After aging for 24 h and crystallization at 105 \({}^{\circ}\)C for 12 h, NaP zeolite with high crystallinity and regular morphology and a specific surface area of 161.06 m\({}^{2}\)/g can be obtained. The process has the characteristics of simple operation, low energy consumption, low cost, small secondary pollution, and easy industrialization. It is a resource utilization method of CGCS, and the water used in the process can be treated and recycled, which can basically achieve \"zero discharge\". The synthetic NaP zeolite has a good removal effect on ammonia nitrogen in industrial wastewater, the best removal rate is 92.67%, and the adsorption amount is 4.63 mg/g. Although the adsorption amount is not large, it has a large application prospect because of its simple preparation and low cost. Na\({}^{+}\) plays an important role in the formation of NaP zeolite crystals, and at the same time it can ion-exchange with NH\({}_{4}\)\({}^{+}\), which is beneficial to the removal of ammonia nitrogen in wastewater.  
Our work proposes a new way to solve the problems of large-scale accumulation, difficulty of utilization, huge environmental pressure, and high transportation cost of CGCS. The environmental functional materials prepared by CGCS can be applied to wastewater treatment in the industrial production process, which reduces the cost of the enterprise, realizes the resource utilization of waste, and achieves the purpose of \"treating waste by waste\".  
Conceptualization, Y.M.; Investigation, W.J. and L.L.; Methodology, W.J.; Resources, W.J. and Y.M.; Supervision, Y.S. and Y.L.; Visualization, S.Z. and N.F.; Writing - original draft, P.Z. and X.Z.; Writing - review & editing, W.J. and S.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by The National Science Foundation of China (No. 21663019) and The East-West Cooperation Project of Ningxia Key R & D Plan (2019 BFH02014).  
The authors declare that there are no conflicts of interest.  
Figure 9: Effect of contact time on ammonia nitrogen removal rate (adsorbent dosage: 20 g/L; pH = 7).
# Fast Simulation of Laser Heating Processes on Thin Metal Plates with FFT Using CPU/GPU Hardware
## 2 Literature Review
### Conclusions of the Literature Review  
Current analytic methods for simulation of the laser heating/cutting problem already provide fast solutions to the problem in the frequency domain. However, such methods perform brute-force evaluation of the Fourier transforms, whose computation complexity for a 2D plate is \(\mathcal{O}(M^{2}N^{2})\). As a consequence, these applications quickly become computationally expensive as more resolution of the plate is required.  
To overcome this problem, this manuscript presents four different schemes that cast the existing brute-force solutions into equivalent DST and DFT problems. Mathematical proof for the validity of each scheme is presented and algorithms that make use of FFT libraries are introduced, reducing the computational complexity of the problem from \(\mathcal{O}(M^{2}N^{2})\) (squared) to \(\mathcal{O}(MN\log(MN))\) (logarithmic). These algorithms are implemented both in CPU and GPU architectures. Numerical validation against the brute-force approach results in a measured absolute error that is below \(10^{-10}\) K along the 2D plate. The results show significant computation time improvements to such brute-force simulations (i.e., References [22; 23; 24; 25]), reducing the measured computation times from 1 s to 0.01 s (100\(\times\) faster) for a \(1024\times 1024\) rectangular plate, and enabling simulations for larger plate discretization sizes (up to \(4096\times 4096\)).  
This manuscript extends the work presented in Reference [3]. In this previous work, two of the four presented schemes are briefly introduced. The research presented in this paper presents two additional FFT schemes, and provides further details of the four schemes (with added illustrations), to make easier the understanding of the algorithms. Furthermore, new simulations have been executed and an application case of the algorithms being implemented into an interactive simulator is presented.