esa-sceva/satcom-chunk-collection huggingface dataset:
## 4 Dummy Generation Scheme
### Historical Location Sub-Dataset Generation Algorithm  
In our scheme, the mobile terminal is used to implement the process selecting dummy locations and query content. So, it needs to store the historical locations dataset and basic service set. Considering its lesser storage, the server must send a smaller historical location sub-dataset \(G_{sub}\) to the mobile terminal. Here, we only consider the \(G_{sub}\), since the basic service set itself is a smaller dataset and can store on the mobile terminal. So, we design an algorithm to generate the \(G_{sub}\)-Algorithm 2 shows the formal description of it. First, Algorithm 2 determines whether the historical locations within the range of \(Area\) are distributed in \(s\) different second-level grids. If so, they meet the requirement and return \(G_{sub}\), which are the historical locations in the given range. Otherwise, the range of the \(Area\) is extended by a degree of \(10^{-3}\) and we get a new \(Area\) which is expressed as  
Figure 6: Algorithm framework.  
\(Area_{1}=(s,lo_{min}-10^{-3},lo_{max}+10^{-3},la_{min}-10^{-3},la_{max}+10^{-3})\). Repeat the above process until the given historical location range can contain different second-level grids. In general, iterating once will meet the requirement.  
```
Input:\(L_{0}\), \(Area\), \(G\) Output:\(G_{sub}\)
1\(G_{sub}\leftarrow\emptyset\);
2 Count the \(n_{A}\) in \(Area\); // \(n_{A}\) is the number of the second-level grids in \(Area\);
3 while\(n_{A}<s\)do
4 let \(lo_{min}\gets lo_{min}-10^{-3},lo_{max}\gets lo_{max}+10^{-3}\), \(la_{min}\gets la_{min}-10^{-3}\), \(la_{max}\gets la_{max}+10^{-3}\);
5 let \(Area\leftarrow\{s,lo_{min},lo_{max},la_{min},la_{max}\}\);
6 Count the \(n_{A}\) in \(Area\);
7 Update \(G_{sub}\) by adding all locations in \(G\) that are located in the specified range of \(Area\) into \(G_{sub}\);
8 Return\(G_{sub}\)
```  
**Algorithm 2**Historical location sub-dataset generation.
## 4 Result and Discussion
### Microstructural Analysis
#### 4.3.2 EDS Analysis of Samples  
EDS was used to determine the types and contents of elements in samples by analyzing the wavelength and intensity of characteristic X-rays emitted by the samples. Figure 16 shows the energy distribution spectrograms of samples obtained by means of EDS, and each peak in Figure 16 represents an element. It should be pointed out that the carbon content in the control sample was considered as zero by the device due to its small content in the selected area.  
Rubber contains many carbon elements and a small amount of aluminum, magnesium, potassium, etc. Table 7 lists the contents of major elements (excluding gold element) in each sample obtained by means of EDS. Table 7 shows that the carbon content was significantly increased after rubber was incorporated, and the greater the rubber particles, the higher the content of carbon element. For example, when the particle size was 3-5 mm, the carbon content (31.95%) of the sample with 20% rubber was increased by 96.62% as compared with the carbon content (16.25%) of the sample with 10% rubber. Keeping the rubber content unchanged, as the particle size decreased, the content of carbon element increased. For instance, when the rubber content was 10%, the carbon content (25.68%) of the sample with 1-3 mm rubber particle size was 58.03% higher than the carbon content (16.25%) of the sample with 3-5 mm rubber particle size.  
Figure 15: Micrographs of samples obtained by SEM: (**a**) Rubber content 0% (control group), (**b**) rubber content 10% (rubber particle size 3–5 mm), (**c**) rubber content 30% (rubber particle size 3–5 mm), (**d**) rubber content 10% (rubber particle size 1–3 mm), (**e**) rubber content 30% (rubber particle size 1–3 mm), 1–3 mm), (**f**) rubber content 10% (rubber particle size 40 mesh), and (**g**) rubber content 30% (rubber particle size 40 mesh).  
_Appl. Sci._**2019**, \(9\), 2943  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Rubber** & **Specimen** & **Rubber** & \multicolumn{6}{c}{**Element (\%)**} \\ \cline{4-10}
**Particle Size** & **No** & **Content (\%)** & **C** & **O** & **Ca** & **Si** & **Al** & **Mg** & **K** \\ \hline Control & C\({}_{\text{J}}\) & 0 & 0 & 45.31 & 30.01 & 24.68 & 0 & 0 & 0 \\
3–5 mm & C\({}_{\text{s-10}}\) & 10 & 16.25 & 46.29 & 32.12 & 3.64 & 1.70 & 0 & 0 \\  & C\({}_{\text{a-20}}\) & 20 & 31.95 & 35.06 & 27.37 & 5.62 & 0 & 0 & 0 \\
1–3 mm & C\({}_{\text{b-10}}\) & 10 & 25.68 & 33.70 & 23.21 & 10.74 & 2.59 & 0 & 3.95 \\  & C\({}_{\text{b-20}}\) & 20 & 42.10 & 31.85 & 17.28 & 6.67 & 2.10 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Content of major elements in the samples.  
Figure 16: Energy distribution spectrograms of samples: (**a**) Rubber content 0% (control group), (**b**) rubber content 10% (rubber particle size 3–5 mm), (**c**) rubber content 20% (rubber particle size 3–5 mm), (**d**) rubber content 10% (rubber particle size 1–3 mm), and (**e**) rubber content 20% (rubber particle size 1–3 mm).
## 5 Aircraft Parameter Optimization Method
The results are displayed in Table 5. The lift coefficient needs to increase by 5%, while the drag coefficient will increase by 2.5% as a result; the inlet area should increase by 25.93%, while the mass will increase by 8.64%, and the reference area will increase by 12.91%; the specific impulse needs to increase by 10%. It can be seen that the variation of the lift coefficient and impulse reach their boundaries, while the variation of the inlet area and reference area are within the feasible range. If the constraints are different, the optimization results will change as well. After the aircraft parameters are adjusted, the optimal steady-state cruise point estimated by the neural network model is located at (\(Ma4.99\), 25.95 km). Under the optimized aircraft parameters, the position of the accurate optimal cruise point and contour of fuel consumption are displayed in Figure 15. It can be seen that the actual optimal cruise point after adjustment is located at (\(Ma4.99\), 25.92 km), which basically meets the goal of adjusting the optimal cruise point from (\(Ma4.95\), 24.95 km) to (\(Ma5\), 26 km). In addition, the adjustment adopts a plan with the lowest fuel consumption at 0.3642 kg/km with the constraints satisfied. Therefore, the method of optimizing the aircraft parameters based on the neural network model has been demonstrated to be effective. Figure 15: Optimal point and contour of fuel consumption after adjustment. \begin{table}
\begin{tabular}{c c c c c c} \hline \(X_{D}\) & \(X_{I}\) & \(X_{L}\) & \(X_{m}\) & \(X_{S}\) & \(X_{A}\) \\ \hline
1.025 & 1.1 & 1.05 & 1.0864 & 1.1291 & 1.2593 \\ \hline \end{tabular}
\end{table} 
Table 5: Optimized results of aircraft parameters. So far, not only the optimal cruise point is adjusted to the target position but also the fuel consumption after adjustment is the lowest. However, it is difficult to greatly adjust the optimal cruise Mach number under the current constraints. If the technical bottleneck of the ramjet can be broken through and an engine with a larger size and significantly increased specific impulse can be obtained, it will be of great benefit to adjust the optimal cruise Mach number in a large range.
## 4 Analysis and Discussion
### Analysis of PUF-Based Authentication and Its Applications
The most popular encryption method used in a hash function is primarily used because of its simplicity and low computational power, and the advanced encryption key (AES) is also a popular technique. A range of platforms are used in order to evaluate the performance of the protocols. Simulation software, field-programmable gate arrays (FPGAs), and Arduino are popular choices to evaluate the performance of the protocols. One of the most important characteristics that must be studied further by the research community is \"Secure store key\". These keys are vulnerable to attacks and hacking, especially those which are stored in memory. Although secure keys have been designed [11], the developed keys are still weak because the method functions are not strong enough for decryption. We believe the authors used traditional methods, which mean that they are always displayed and thus vulnerable to machine learning attacks. Another major concern for users is trust in the data regarding their origin and location. This is especially true for IoT devices, which are typically low-cost. A lightweight protocol for data provenance in the IoT was given [74]. PUFs are used to give physical security and uniquely identify IoT devices. Channel impairments and characteristics are used to uniquely identify the communication link between the server and the IoT device. Next, the key agreement between two constrained IoT devices is essential to establish trust between the devices, and PUFs can play a role in this regard as well. The existing key
## 2 Materials and Methods
### Synthesis of Mn\({}^{2+}\)-Doped ZnS Quantum Dots Entrapped in a Sol-Gel Matrix  
Water-soluble Mn\({}^{2+}\)-doped ZnS quantum dots were synthesized in our laboratory following a synthetic route previously described by [PERSON] et al. [17]. In short, 50 mL of 0.02 M L-cysteine, 5 mL of 0.1 M ZnSO\({}_{4}\), and 0.15 mL of 0.1 M MnCl\({}_{2}\) were mixed in a beaker, adjusting the pH to 11 with 1 M NaOH. This mixture was then placed in a three-necked flask under stirring, deoxygenated with argon for 30 min, and 5 mL of 0.1 M Na\({}_{2}\)S was added to allow for nucleation of the nanoparticles. After 20 min, the solution was aged under an air atmosphere at 50 \({}^{\circ}\)C for 2 h to improve the crystallinity. To purify the obtained nanoparticles, they were precipitated with ethanol via centrifuging for 5 min at 5000 rpm. The QDs were then dried under vacuum and stored under an argon atmosphere as a water-soluble brown powder.  
After that, the obtained quantum dots were immobilized in a sol-gel matrix by adding them during the polymerization process [18]. For this purpose, 1 mL of Mn:ZnS QDs dissolved in a phosphate buffer at pH 7.4 was mixed with 2.5 mL ethanol in a small Teflon cup under constant stirring. Then, the reagents for the formation of a sol-gel were added: 750 \(\upmu\)L APTES, followed by 100 \(\upmu\)L EDC, 50 \(\upmu\)L 1 M NaOH, and finally, 750 \(\upmu\)L TMOS. APTES and TMOS were the necessary reagents to form the sol-gel structure, NaOH was employed as the catalyst, and EDC was employed to covalently bond the amino groups of the surface ligands of the QDs to the polymer structure. Once the polymerization process had taken place, the mixture was left to dry at 40 \({}^{\circ}\)C for two weeks.  
The obtained dry gel was ground using an agate mortar and the particles generated were separated by particle size using 80, 120, 200, and 300 \(\upmu\)m sieves. The 120 \(\upmu\)m fraction was selected for further experiments as a compromise: small particle sizes provide a more intense signal but requires higher pressures. The bigger particle sizes provide the better pressure stability although lower signal.
# Intelligent Transmission Control Scheme for 5G mmWave Networks Employing Hybrid Beamforming
## 4 Proposed Work
### Accurate Channel Estimation for mmWave Systems
#### 4.2.1 Sparse Signal Recovery Using LAMP Network
The row support \(\hat{\Omega}\) is employed to filter out the valuable rows of \(H_{row}^{e}\) and then is input into the second LAMP network as \(H_{row}^{e}\) (\(\hat{\Omega}\), \(\cdot\)), the power level of a single row \(H_{row}^{e}\) (\(i\), \(:\)) in the matrix. Input: Overall matrix \(Y\) for measurements  
1: Step 1: Give back the predicted row-sparse matrix \(H_{row}^{e}\) by applying the LAMP  
network created using \(A_{1}\) to \(Y\)  
2: Row Compression (COMP): \(H_{row}^{e}\) row support \(\hat{\Omega}\) is returned. 3: Step 2: Give back the expected angular cascaded channel \(\overset{\sim}{H_{c}}^{e}\) on the support \(\hat{\Omega}\) from \(H_{row}^{e}\)(\(\hat{\Omega}\), \(:\)) with the LAMP network created from \(A_{2}\). 4: Step 3: \(A_{B}\) (\(:,\hat{\Omega}\)) \(\overset{\sim}{H_{c}}^{e}\)(\(\hat{\Omega}\), \(:\))\(A_{S}^{H}\)  
5: Cascaded channel \(\overset{\sim}{H_{c}}^{e}\) Approximation is given. **Algorithm 2** LAMP network with compression in rows in two stages (RCTS-LAMP network). The size will be significant if the spatial frequency matches. The bandwidth of that particular line is near the actual spatial frequency \(\varphi_{G}^{L_{c}}\). To efficiently indicate power, we utilize the \(l_{2}\)-norm of a row \(\|H_{row}^{e}(i,\cdot)\|\). Due to Herow being a \(L_{G}\) row-sparse matrix; the \(l_{2}\)-norm distribution is skewed to the right, with the majority of rows having small \(l_{2}\)-norms and only a few rows having large \(l_{2}\)-norms, creating a long right tail in the distribution. Additionally, because the small norms primarily come from the residual noise in the rows, the average of the norms will be near the level of noise, particularly at higher grid resolution \(K_{b}\). Thus, utilizing the average value as the fluctuating threshold will lead to the exclusion of rows with norms lower than or near the noise level. Given that the excluded rows may introduce noise that could affect the cascaded channel's information, we can safely ignore them in the second stage without sacrificing performance. This situation is shown as follows:  
\[\hat{\Omega}=\left\{i:\|H_{row}^{e}(i,\cdot)\|>\sum_{H=1}^{K_{b}}\|H_{row}^{e }(H,\cdot)\|/K_{b}\right\} \tag{18}\]  
where \(\sum_{j=1}^{K_{b}}\|H_{row}^{e}(H,\cdot)\|/K_{b}\) denotes the changing threshold calculated by averaging all the \(l_{2}\) norms. Consequently, only \(\overset{\sim}{H_{c}}^{e}\) (\(\hat{\Omega}\), \(:\)) is restored in the second LAMP network, disregarding the remaining \(K_{b}\) - \(|\hat{\Omega}|\) rows of \(\overset{\sim}{H_{c}}\) to reduce computation and suppress noise. In the end, Algorithm 2's Step 4 regains the time domain channel \(\overset{\sim}{H_{c}}^{e}\) from the angular domain using dictionary matrices \(A_{B}\)(\(\hat{\Omega}\), \(:\)), and \(A_{S}\).
## 3 Overview of Multihop CA-IoT Architectures and Testbeds
#### Theoretical Analysis of Multihop CA-IoT Design and Operation
For our MCA-IoT network, \(P_{n-hops}\) with \(P_{Rx-elc}\) can be expressed as  
\[E_{ij}=\frac{(P_{Tx}+P_{Tx-{\it{dec}}}+P_{Rx-{\it{dec}}})}{L-bits} \tag{1}\]4  
From Equations (11) and (12), it can be concluded that multihop routing is more energy-efficient than single-hop routing only if the receiver's power consumption obeys Equation (13). The \(P_{Rx-elc}\) values from the transceiver platforms are approximately the same as the \(P_{1-hop}\), which makes them unsuitable for multihop CA-IoT networks. \[E_{ij}=\frac{(P_{Tx}+P_{Tx-{\it{dec}}}+P_{Rx-{\it{dec}}})}{L-bits} \tag{1}\]5 Equidistantmultihop transmission framework. However, recent communication standards such as LoRa, SIGFOX, BLE, and NB-IoT can support multihop routing since the \(P_{Rx-elec}\) values are significantly less than the following:  
\[E_{ij}=\frac{(P_{Tx}+P_{Tx-{\it{dec}}}+P_{Rx-{\it{dec}}})}{L-bits} \tag{1}\]6  
From the above expressions, energy-efficient multihop routing in large-scale CA-IoT networks can be achieved under the following conditions:  
1. The ratio of multihop distances between the MN, CH, RCH, and edge RCH must be unity along the path of a distant CH (Equation (8)). 2. To ensure the desired energy savings in multihop routing, \(\alpha\geq 2\) must be met, and the condition in Equation (13) must be satisfied by the transceiver module. Specifically, \(P_{Rx-elec}\) of the RCH must be sufficiently small compared to \(P_{Tx}\) in single-hop routing, as observed in LoRa, SIGFOX, and BLE systems. 3. To further minimize \(P_{Rx-elec}\) due to extended durations in multihop routing, a novel routing technique integrating the timeslot assignments of a distant CH into the RCH's TDMA schedule for its MNs per duty cycle, as proposed in [17], is adopted.
## 2 Design of the Solar-Powered ECG Device
### Hardware Design  
#### 2.1.1 Solar Energy Harvesting  
According to the manufacturer's datasheet, we synthesize operating voltages and current consumption of the three main components, which are PIC16 LF19168 [18], ADS1293 [17], and RN4020 [19] in Table 1.  
Following Table 1, the main components in the design can operate from 1.8 V but to ensure the stability of the design, the operating voltage of the main parts is 1.9 V. The block diagram is described in Figure 2. The power supply module for the entire design includes BQ25570 [23] energy harvesting integrated circuit, solar cell (2.4 V, 250 mW), and 0.47 F - 5.5 V supercapacitor. The BQ25570 [23] is configured with an output voltage of 1.9 V to be compatible with all other modules in this system and to optimize the lowest power consumption level. The voltage used to charge is 4.2 V with this voltage supercapacitor, and it can store more power than the configuration at 1.9 V voltage and is easily compatible with some rechargeable batteries operating at voltage 3.0 V in cases where you want to use rechargeable batteries for supercapacitors to create design flexibility.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Components** & **Operating Voltage (V)** & **Operating Current (\(\mu\)A)** \\ \hline \multirow{4}{*}{PIC16 LF19168} & \multirow{4}{*}{1.8–3.6} & Sleep Mode 50 Na @ 1.8 V \\  & & Active Mode 8 \(\mu\)A/32 kHz @ 1.8 V \\  & & Active Mode 32 \(\mu\)A/MHz @ 1.8 V \\  & & Second OSC 500 nA @ 32 kHz \\ \hline \multirow{4}{*}{RN4020} & \multirow{4}{*}{1.8–3.6} & Dormant \textless{} 900 nA @ 3 V \\  & & Deep Sleep \textless{} 5.0 \(\mu\)A @ 3 V \\ \cline{1-1}  & & Idle \textless{} 1.5 mA @ 3 V \\ \cline{1-1}  & & TX/ RX Active 16 mA @ 3 V \\ \cline{1-1}  & & TX Power (dBm) - 19.1 with 14.0 mA @ 3.3 V \\ \hline \multirow{2}{*}{ADS1293} & 1.65–3.6 (Digital Module) & Duty-Cycle mode: 120 \(\mu\)A \\  & 2.7–5.5 (Analog Module) & Normal mode: 415 \(\mu\)A \\ \hline \hline \end{tabular}
\end{table}
Table 1: Voltage and current consumption of three devices PIC16 LF19186, ADS1293, and RN4020.  
Figure 1: The electrocardiogram (ECG) monitoring system.  
In this design, we used supercapacitors to store energy from solar cells. Under normal working conditions, solar cells provide power to the device and charge the supercapacitor. When there is not enough solar energy to supply the main circuit, the energy will be discharged from the supercapacitor to supply the system. The charging time for supercapacitors is tested and shown in Table 2 with different electrical values and voltage applied to the main circuit board [23].  
Calculation of the energy stored in a capacitor is as follows:  
\[E=\frac{1}{2}CV^{2} \tag{1}\]  
where \(E\) is energy stored in capacitor (J), \(V\) signifies capacitor voltage (V), and \(C\) is capacitance of capacitor (F). In this design, when \(V\) = 4.2 V and \(C\) = 0.47 F, \(E\) = 4.1454 (J) = 1.25 (mWh).
# Microcontroller Unit-Based Wireless Sensor Network Nodes: A Review
## 3 MCU-Based Hardware Platforms for Wireless Sensor Nodes
### Environmental Monitoring and Smart Farming
#### 3.2.7 Wireless Multi-Sensor Node  
The multi-sensor node [26] was designed to cater for monitoring environmental parameters. The node was developed around the STM32F103 ZET6 MCU, which is based on the ARM(r) Cortex(r)-M3 core operating at a 72 MHz frequency. It contains up to 512 kB Flash memory and up to 64 kB SRAM. The authors utilized the wireless fidelity (Wi-Fi) to link a gateway with the sensor motes at 2.4 GHz. The ESP8266 transceiver module is exploited for communication. The node accommodates UV light (GT-302), raindrop, noise and SM300D2 seven-in-one (temperature, humidity, PM2.5, PM10, formaldehyde concentration, CO\({}_{2}\) concentration and TVOC concentration) sensors to measure environmental parameters.  
The node is powered by a battery and a solar panel connected to a solar tracking system. Regarding the power consumption, the mote consumes around 0.65 W during sensing. The current consumption by the Wi-Fi during the transmission mode is approximately 100 mA. The sensor node consumes around 0.8 W of power in the transmission mode. During the sleep state, the current consumption is about 50 mA due to the STM32 core plus 10 mAbecause of other standby circuits. Approximately 0.3 W power is consumed by the mode in the sleep state.
## 2 Rigid-Body Model of an Earthquake Excited Tower-Like Structure  
A fundamental non-linear model of a tower-like structure is shown in Figure 1. We regard the tower-like structure as an inverted pendulum with a non-linear spring at its hinge support, which is excited in both vertical and horizontal directions. We denote the total mass of the structure by \(m\) and its radius of inertia by \(i\). The base accelerations are \(a_{\rm H}\) and \(a_{\rm V}\). The mass center is located at a distance \(L\) from the support. We take a non-linear cubic restoring moment for the spring: \(M=-k_{\rm L}\varphi+k_{\rm N}\varphi^{3}\). The horizontal and vertical support reactions are called \(F_{\rm H}\) and \(F_{\rm V}\). The unknown degree-of-freedom (DOF) in this 1 DOF model is the angle \(\varphi\) counted from the vertical direction. The relation of balance of moment of momentum about the center of mass reads  
\[m^{2}\ddot{\varphi}=F_{\rm V}L\sin\varphi+F_{\rm H}L\cos\varphi-k_{\rm L} \varphi+k_{\rm N}\varphi^{3}. \tag{1}\]  
Derivatives with respect to time \(t\) are indicated by superimposed dots. Balance of linear momentum may be written as  
\[ \begin{split}& m\Big{(}a_{\rm H}-L\ddot{\varphi}\cos\varphi+L \dot{\varphi}^{2}\sin\varphi\Big{)}=F_{\rm H},\\ & m\Big{(}(a_{\rm V}+g)-L\ddot{\varphi}\sin\varphi-L\dot{\varphi }^{2}\cos\varphi\Big{)}=F_{\rm V}.\end{split}  \tag{2}\]  
We may use the latter relation in order to eliminate the support reactions from balance of moments of momentum Equation (1) and end up with the following non-linear second-order ordinary differential Equation (ODE):  
\[m\Big{(}i^{2}+L^{2}\Big{)}\ddot{\varphi}=m((a_{\rm V}+g)L\sin\varphi+a_{\rm H }L\cos\varphi)-k_{\rm L}\varphi+k_{\rm N}\varphi^{3}. \tag{3}\]  
Figure 1: Earthquake excited tower-like structure: rigid-body model. This relation can be regarded as balance of moment of momentum about the moving support, see ([23], Chapter 7).With the frequency \(\Omega_{0}\) of the freely hanging linear pendulum when \(\pi-\varphi\ll 1\), we obtain  
\[\ddot{\varphi}=\Omega_{0}^{2}\bigg{[}\bigg{(}1+\frac{a_{\rm V}}{g}\bigg{)}\sin \varphi+\frac{a_{\rm H}}{g}\cos\varphi-\frac{1}{mgL}\Big{(}k_{\rm L}\varphi-k_ {\rm N}\varphi^{3}\Big{)}\bigg{]},\ \Omega_{0}=\sqrt{\frac{gL}{i^{2}+L^{2}}}. \tag{4}\]  
For moderately large rotations, this non-linear ODE may be simplified by truncating the Taylor expansions for trigonometric terms as follows:  
\[\sin\varphi=\varphi-\frac{1}{6}\varphi^{3}+...,\quad\cos\varphi=1-\frac{1}{2 }\varphi^{2}+.... \tag{5}\]  
The corresponding cubic non-linear ODE reads  
\[\ddot{\varphi}=\Omega_{0}^{2}\bigg{[}\frac{a_{\rm H}}{g}+\bigg{(}\frac{a_{\rm V }}{g}+1-\frac{k_{\rm L}}{mgL}\bigg{)}\varphi-\frac{1}{2}\frac{a_{\rm H}}{g} \varphi^{2}+\bigg{(}\frac{k_{\rm H}}{mgL}-\frac{1}{6}\bigg{(}\frac{a_{\rm V}} {g}+1\bigg{)}\bigg{)}\varphi^{3}\bigg{]}. \tag{6}\]  
This equation will be treated as benchmark problem subsequently.