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14 pages, 3665 KiB  
Article
An Irregular Pupil Localization Network Driven by ResNet Architecture
by Genjian Yang, Wenbai Chen, Peiliang Wu, Jianping Gou and Xintong Meng
Mathematics 2024, 12(17), 2703; https://doi.org/10.3390/math12172703 - 30 Aug 2024
Viewed by 519
Abstract
The precise and robust localization of pupils is crucial for advancing medical diagnostics and enhancing user experience. Currently, the predominant method for determining the center of the pupil relies on the principles of multi-view geometry, necessitating the simultaneous operation of multiple sensors at [...] Read more.
The precise and robust localization of pupils is crucial for advancing medical diagnostics and enhancing user experience. Currently, the predominant method for determining the center of the pupil relies on the principles of multi-view geometry, necessitating the simultaneous operation of multiple sensors at different angles. This study introduces a single-stage pupil localization network named ResDenseDilateNet, which is aimed at utilizing a single sensor for pupil localization and ensuring accuracy and stability across various application environments. Our network utilizes near-infrared (NIR) imaging to ensure high-quality image output, meeting the demands of most current applications. A unique technical highlight is the seamless integration of the efficient characteristics of the Deep Residual Network (ResNet) with the Dense Dilated Convolutions Merging Module (DDCM), which substantially enhances the network’s performance in precisely capturing pupil features, providing a deep and accurate understanding and extraction of pupil details. This innovative combination strategy greatly improves the system’s ability to handle the complexity and subtleties of pupil detection, as well as its adaptability to dynamic pupil changes and environmental factors. Furthermore, we have proposed an innovative loss function, the Contour Centering Loss, which is specifically designed for irregular or partially occluded pupil scenarios. This method innovatively calculates the pupil center point, significantly enhancing the accuracy of pupil localization and robustness of the model in dealing with varied pupil morphologies and partial occlusions. The technology presented in this study not only significantly improves the precision of pupil localization but also exhibits exceptional adaptability and robustness in dealing with complex scenarios, diverse pupil shapes, and occlusions, laying a solid foundation for the future development and application of pupil localization technology. Full article
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Figure 1
<p>Diagram of the ResDenseDilateNet architecture.</p>
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<p>Principle Diagram of Dilated Convolution. (<b>a</b>) The outcome of a 1-dilated convolution; (<b>b</b>) The effect of a 2-dilated convolution; (<b>c</b>) The impact of a 4-dilated convolution.</p>
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<p>Schematic diagram illustrating the principle of the DDCM.</p>
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<p>Diagram illustrating the principles of Contour Centering Loss. The blue lines represent tangents, and the red lines represent normals.</p>
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<p>Examples of images from the dataset.</p>
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<p>Schematic diagram of the human eye data collection process.</p>
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<p>Comparative analysis of ablation experiments on the RPE.</p>
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14 pages, 7290 KiB  
Article
Optimizing Trapezoidal Labyrinth Weir Design for Enhanced Scour Mitigation in Straight Channels
by Ahmed H. Shehata, Tahani F. Youssef, Hamada A. Hamada, Ibrahim M. M. and Abeer Samy
Water 2024, 16(17), 2443; https://doi.org/10.3390/w16172443 - 29 Aug 2024
Viewed by 751
Abstract
Designing hydraulic structures requires careful consideration of local scouring downstream. This study investigated the performance of trapezoidal labyrinth weirs in controlling flow and mitigating scour in straight channels through physical model experiments. Sixty configurations were examined, using weir apex angles of 20°, 45°, [...] Read more.
Designing hydraulic structures requires careful consideration of local scouring downstream. This study investigated the performance of trapezoidal labyrinth weirs in controlling flow and mitigating scour in straight channels through physical model experiments. Sixty configurations were examined, using weir apex angles of 20°, 45°, 60°, and 80°, heights of 30 cm, 35 cm, and 40 cm, and flow rates of 50–200 L/s. A linear weir served as a reference. The results showed that the 60° apex angle consistently outperformed other configurations, reducing scour depth by up to 41% and scour length by up to 50% compared to the linear weir. It also decreased deposition depth by 40% and length by 50%. Lowering weir height from 40 cm to 30 cm led to reductions of 35% in scour depth and 40% in scour length at low discharges. These improvements remained significant even at higher flow rates, with a 29% reduction in scour depth and 25% in scour length at 200 L/s. This study provides evidence-based recommendations for optimizing labyrinth weir designs to define the relationship between hydraulic efficiency and erosion control. It offers valuable insights into weir geometry, flow conditions, and the resulting scour and deposition patterns. These findings contribute to the optimization of labyrinth weir designs to minimize downstream bed configurations. The tests were conducted under limited flow conditions. Full article
(This article belongs to the Section Hydraulics and Hydrodynamics)
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Figure 1

Figure 1
<p>General layout of the physical model (20.5 m long, 2 m width, and 0.85 m depth), (<b>a</b>) an upper view of the physical model, (<b>b</b>) a side view of the model showing its longitude, (<b>c</b>) a view of the upstream entrance, and (<b>d</b>) a view of the downstream outlet.</p>
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<p>Schematic diagram for the weir model, and experimental setup: (<b>a</b>) side view, (<b>b</b>) plan.</p>
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<p>The schematic diagram for the trapezoidal labyrinth weir with four different apex angles (20°, 45°, 60°, and 80°)—“the red line shows the borders of one cycle of the weir from the schematic diagram to the natural view in the flume”.</p>
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<p>Grain size distribution of the bed material downstream of the solid apron.</p>
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<p>The calibrated point gauges to measure (<b>a</b>) the upstream water level, (<b>b</b>) the developed bed configurations, and (<b>c</b>) the tailgate water level.</p>
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<p>Scour developed downstream of the apron: (<b>a</b>) before the test run, (<b>b</b>) after the test run, and (<b>c</b>) a side view of the resulting scour and deposition.</p>
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<p>The relation between flow discharge and relative scour/deposition parameters (D<sub>s</sub>/Y<sub>t</sub>), (L<sub>s</sub>/Y<sub>t</sub>), (D<sub>d</sub>/Y<sub>t</sub>), and (L<sub>d</sub>/Y<sub>t</sub>) for the linear case and the other apex angles at different weir heights: (<b>a</b>) P = 40 cm, (<b>b</b>) P = 35 cm, and (<b>c</b>) P = 30 cm.</p>
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<p>The relation between flow discharge and relative scour/deposition parameters (D<sub>s</sub>/Y<sub>t</sub>), (L<sub>s</sub>/Y<sub>t</sub>), (D<sub>d</sub>/Y<sub>t</sub>), and (L<sub>d</sub>/Y<sub>t</sub>) for the linear case and the other apex angles at different weir heights: (<b>a</b>) P = 40 cm, (<b>b</b>) P = 35 cm, and (<b>c</b>) P = 30 cm.</p>
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<p>Relationship between passing discharge and scour/deposition parameters (D<sub>s</sub>/y<sub>t</sub>), (L<sub>s</sub>/y<sub>t</sub>), (D<sub>d</sub>/y<sub>t</sub>), and (L<sub>d</sub>/y<sub>t</sub>), for a weir apex angle of 60° and different weir heights (P = 40 cm, 35 cm, and 30 cm).</p>
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20 pages, 14212 KiB  
Article
ReLoki: A Light-Weight Relative Localization System Based on UWB Antenna Arrays
by Joseph Prince Mathew and Cameron Nowzari
Sensors 2024, 24(16), 5407; https://doi.org/10.3390/s24165407 - 21 Aug 2024
Viewed by 680
Abstract
Ultra Wide-Band (UWB) sensing has gained popularity in relative localization applications. Many localization solutions rely on using Time of Flight (ToF) sensing based on a beacon–tag system, which requires four or more beacons in the environment for 3D localization. A lesser researched option [...] Read more.
Ultra Wide-Band (UWB) sensing has gained popularity in relative localization applications. Many localization solutions rely on using Time of Flight (ToF) sensing based on a beacon–tag system, which requires four or more beacons in the environment for 3D localization. A lesser researched option is using Angle of Arrival (AoA) readings obtained from UWB antenna pairs to perform relative localization. In this paper, we present a UWB platform called ReLoki that can be used for ranging and AoA-based relative localization in 3D. To enable AoA, ReLoki utilizes the geometry of antenna arrays. In this paper, we present a system design for localization estimates using a Regular Tetrahedral Array (RTA), Regular Orthogonal Array (ROA), and Uniform Square Array (USA). The use of a multi-antenna array enables fully onboard infrastructure-free relative localization between participating ReLoki modules. We also present studies demonstrating sub-50cm localization errors in indoor experiments, achieving performance close to current ToF-based systems, while offering the advantage of not relying on static infrastructure. Full article
(This article belongs to the Section Navigation and Positioning)
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Figure 1
<p>Illustration of the relative localization problem. On the left, we show ReLoki attached to an existing motion platform and capable of relative localization based on fully onboard sensing. Here, the RX agent senses the relative positions <math display="inline"><semantics> <msub> <mi>q</mi> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </semantics></math> of the TX agents w.r.t its body frame whenever a message is received from <span class="html-italic">j</span>. On the right, we show the scenario where ReLoki can act as a mobile beacon. All beacons are capable of localizing a transmitting agent in 3D and adding more beacons will improve estimates.</p>
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<p>Illustration of the 4-antenna configurations that can be used with ReLoki. Here, we show the ROA, where the antennas are placed orthogonal w.r.t the central antenna, the RTA, where the antennas are placed at the vertices of a regular tetrahedron, and the USA, where the antennas are placed as a square on the same plane.</p>
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<p>Illustration of angle of incidence for RTA, ROA, and USA Antennas. The angle of incidence measured is used for bearing estimates based on the specific geometry of the antenna array.</p>
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<p>Angle of incidence measured for the redundant pairs. Here, the measured value is the average of 20 readings. The plot shows the saturation of the angle of incidence measured over <math display="inline"><semantics> <mrow> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> in one pair and under <math display="inline"><semantics> <mrow> <mo>−</mo> <msup> <mn>60</mn> <mo>∘</mo> </msup> </mrow> </semantics></math> in the other.</p>
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<p>Timing diagram showing the different phases of transmissions. Message Transfer phase is shown in red, TWR Ranging phase in blue, and AoA Blink phase in green.</p>
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<p>Single-antenna design for ReLoki. (<b>a</b>) Finished PCB antenna along with the copper plane showing the circular patch antenna and the ground plane. (<b>b</b>) Return loss for the designed antenna showing less than <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>10</mn> <mspace width="3.33333pt"/> <mi>dB</mi> </mrow> </semantics></math> return loss in almost all the UWB band for Ch. 1, 2, and 3. (<b>c</b>) Center frequency and the bandwidth of the UWB bands supported by proposed antenna and DW1000.</p>
Full article ">Figure 6 Cont.
<p>Single-antenna design for ReLoki. (<b>a</b>) Finished PCB antenna along with the copper plane showing the circular patch antenna and the ground plane. (<b>b</b>) Return loss for the designed antenna showing less than <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>10</mn> <mspace width="3.33333pt"/> <mi>dB</mi> </mrow> </semantics></math> return loss in almost all the UWB band for Ch. 1, 2, and 3. (<b>c</b>) Center frequency and the bandwidth of the UWB bands supported by proposed antenna and DW1000.</p>
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<p>ReLoki controller design. (<b>a</b>) ReLoki hardware block diagram showing the components. Here, we start with the host <span class="html-italic">i</span> initiating a communication request. ReLoki connects to host <span class="html-italic">i</span> and transmits the data. The information is transferred to the receiving ReLoki where it is then combined with the estimated localization data. Finally, the data are sent to the receiving host <span class="html-italic">j</span>. (<b>b</b>) ReLoki PCB design showcasing the different components mentioned in the block diagram.</p>
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<p>ReLoki experimental setup for covariance measurement. On the left, we have the pan and tilt mechanism and on the right we have the test setup for the <math display="inline"><semantics> <mrow> <mn>1.5</mn> <mspace width="3.33333pt"/> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> range from source.</p>
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<p>Covariance maps for RTA and ROA antennas. In the top left we have RTA and in the top right we have the ROA array. A darker color means lower error. On the bottom, we show the comparison of RTA antenna array to the ROA antenna array. Here, green boxes represent lower errors for RTA and red represents lower errors for ROA. Yellow a represents comparable performance (combined azimuth and elevation difference within <math display="inline"><semantics> <mrow> <msup> <mn>10</mn> <mo>∘</mo> </msup> </mrow> </semantics></math>) between both.</p>
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<p>Covariance map for USA antenna. On the top, we have the covariance maps, with darker colors showing lower errors in localization and lighter colors showing higher errors in localization. On the bottom, we show the average of measured vs actual values for azimuth and elevation for 50 readings at a given pan–tilt pair.</p>
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<p>Localization experiment with RTA antenna on ReLoki. On the left, we have the composite of overlayed frames from the video captured during the experiment. Agent 1 is executing a rectangular motion and Agent 2 is executing a straight-line motion. On the right, we have the output from ReLoki as seen by Agent 3 as well as the Opti-Track data captured. We show both the raw estimation data, in a lighter color, and filtered data using a low-pass filter in a darker color.</p>
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<p>ReLoki beacon test. On the top, we show the experimental setup. Two beacons are placed <math display="inline"><semantics> <mrow> <mn>8</mn> <mspace width="3.33333pt"/> <mi mathvariant="normal">m</mi> </mrow> </semantics></math> apart and the human operator moves the tag in a hour glass pattern. On the bottom, we show plots of the localization data along with the captured MoCAP data. Here, we show localization data with only one beacon active on the right side and both beacons active on the left. The unused beacon is marked with an “X”. We show the localization errors in both cases.</p>
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20 pages, 15846 KiB  
Article
Modelling the Control of Groundwater on the Development of Colliery Spoil Tip Failures in Wales
by Lingfeng He, John Coggan, Patrick Foster, Tikondane Phiri and Matthew Eyre
Land 2024, 13(8), 1311; https://doi.org/10.3390/land13081311 - 19 Aug 2024
Viewed by 678
Abstract
Legacy colliery spoil tip failures pose a significant hazard that can result in harm to persons or damage to property and infrastructure. In this research, the 2020 Wattstown tip landslide caused by heavy rainfall was examined to investigate the likely mechanisms and developmental [...] Read more.
Legacy colliery spoil tip failures pose a significant hazard that can result in harm to persons or damage to property and infrastructure. In this research, the 2020 Wattstown tip landslide caused by heavy rainfall was examined to investigate the likely mechanisms and developmental factors contributing to colliery spoil tip failures in Welsh coalfields. To achieve this, an integrated method was proposed through the combination of remote sensing mapping, stability chart analysis, 2D limit equilibrium (LE) modelling, and 3D finite difference method (FDM) analysis. Various water table geometries were incorporated into these models to ascertain the specific groundwater condition that triggered the occurrence of the 2020 landslide. In addition, sensitivity analyses were carried out to assess the influence of the colliery spoil properties (i.e., cohesion, friction angle, and porosity) on the slope stability analysis. The results indicate that the landslide was characterised by a shallow rotational failure mode and spatially constrained by the critical water table and an underlying geological interface. In addition, the results also imply that the landslide was triggered by the rise of water table associated with heavy rainfall. Through sensitivity analysis, it was found that the properties of the colliery spoil played an important role in confining the extent of the landslide and controlling the process of its development. The findings underscore the detrimental effects of increased pore pressures, induced by heavy rainfall, on the stability of colliery tips, highlighting the urgent needs for local government to enhance water management strategies for this region. Full article
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Figure 1
<p>Study area—Wattstown tip which is south of Wattstown in the county borough of Rhondda Cynon Taf, Wales, and displayed on ESRI world imagery.</p>
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<p>Images of Wattstown standard tip. (<b>a</b>) A Google satellite image prior to the landslide (05/2020), (<b>b</b>) a Google satellite image after the landslide (07/2021), (<b>c</b>) delineation of the landslide boundary.</p>
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<p>Daily rainfall located at the study area from November 2020 to December 2020 (Met Office Hadley Centre, 2023).</p>
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<p>Remote-sensing mapping of the Wattstown standard tip prior to the 2020 landslide occurrence. (<b>a</b>) Hillshade map, (<b>b</b>) aspect map, and (<b>c</b>) slope angle map.</p>
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<p>Methodology of this research, including the input part to collect data for landslide analysis, the methods part of different methods used for landslide analysis, the output part of the final results obtained using these methods, and the validation process using post-landslide satellite images.</p>
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<p>Slope models for numerical analysis. (<b>a</b>) A 3D model and 9 monitoring points, (<b>b</b>) a satellite image showing the position of the 9 points, (<b>c</b>) a 2D model constructed along an N–S profile in the 3D model.</p>
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<p>A representative Hoek and Bray circular failure chart to estimate the FS of a soil slope where the surface water is 8×H behind the toe of the slope.</p>
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<p>FS values of the slope angle of 30° (grey line), 35° (red line), and 40° (blue line) in response to different groundwater conditions.</p>
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<p>Results of 2D LE analysis based on the simplified Bishop method, showing FS estimation corresponding to the slip surface. (<b>a</b>) Dry slope, (<b>b</b>) partially saturated slope (regime_1).</p>
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<p>FDM modelling result showing the total displacement of a dry slope (regime_1). (<b>a</b>) overview of the modelling result, (<b>b</b>) an N–S cross section of the total displacement contour.</p>
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<p>The curves of total displacement at 9 monitoring points when the slope is in a dry condition.</p>
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<p>FDM modelling result of a partially saturated slope (regime_1). (<b>a</b>) Contour of slope displacement and modelled displacement vectors of the unstable zone, (<b>b</b>) an N–S cross section of the slope displacement contour, (<b>c</b>) close-up image of the N–S cross section showing the geometry of the unstable zone and monitoring points P1, P6, and P8 on the slope.</p>
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<p>The curves of total displacement at 9 monitoring points when the slope is in the regime_1 water condition.</p>
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<p>The curves of total displacement at 9 monitoring points. (<b>a</b>) c = 0, (<b>b</b>) c = 20 kPa.</p>
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<p>Results of sensitivity analysis associated with the cohesion of colliery spoil. (<b>a</b>) c = 0, (<b>b</b>) c = 20 kPa, (<b>c</b>) N–S cross section of the cohesionless modelling result, (<b>d</b>) N–S cross section of the 20 kPa modelling result.</p>
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<p>The curves of total displacement at 9 monitoring points. (<b>a</b>) Φ = 32°, (<b>b</b>) Φ = 42°.</p>
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<p>Results of sensitivity analysis associated with the friction angle of colliery spoil. (<b>a</b>) Φ = 32°, (<b>b</b>) Φ = 42°, (<b>c</b>) N–S cross section of the 32° friction angle modelling result, (<b>d</b>) N–S cross section of the 42° friction angle modelling result.</p>
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<p>The curves of total displacement at 9 monitoring points. (<b>a</b>) φ = 10%, (<b>b</b>) φ = 30%.</p>
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<p>Results of sensitivity analysis associated with the porosity of colliery spoil. (<b>a</b>) φ = 10%, (<b>b</b>) φ = 30%, (<b>c</b>) N–S cross section of the 10% porosity modelling result, (<b>d</b>) N–S cross section of the 30% porosity modelling result.</p>
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<p>UKCP18 daily precipitation projection of a 5 km grid located in the study area from 1 July 2024 to 31 December 2028.</p>
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18 pages, 3123 KiB  
Article
A Method to Evaluate Forchheimer Resistance Coefficients for Permeable Screens and Air Louvers Modelled as a Porous Medium
by Yuriy Marykovskiy, Giulia Pomaranzi, Paolo Schito and Alberto Zasso
Fluids 2024, 9(7), 147; https://doi.org/10.3390/fluids9070147 - 22 Jun 2024
Viewed by 771
Abstract
Porous medium models are commonly used in Computational Fluid Dynamics (CFD) to simulate flow through permeable screens of various types. However, the setup of these models is often limited to replicating a pressure drop in cases where fluid inflow is orthogonal to the [...] Read more.
Porous medium models are commonly used in Computational Fluid Dynamics (CFD) to simulate flow through permeable screens of various types. However, the setup of these models is often limited to replicating a pressure drop in cases where fluid inflow is orthogonal to the screen. In this work, a porous medium formulation that employs a non-diagonal Forchheimer tensor is presented. This formulation is capable of reproducing both the pressure drop and flow deflection under varying inflow angles for complex screen geometries. A general method to determine the porous model coefficients valid for both diagonal and non-diagonal Forchheimer tensors is proposed. The coefficients are calculated using a nonlinear least-squares optimisation based on an analytical solution of a special case of the Navier–Stokes equations. The applicability of the proposed method is evaluated in four different scenarios supplemented by local CFD simulations of permeable screens: wire mesh, perforated screens, air louvers, and expanded mesh panels. The practical application of this method is demonstrated in the modelling of windbreaks and permeable double-skin facades, which typically employ the aforementioned types of porous screens. Full article
(This article belongs to the Section Mathematical and Computational Fluid Mechanics)
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Figure 1
<p>Permeable screen types on building facades: perforated screens, louvers, expanded mesh (left to right).</p>
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<p>Computational domain and coordinate systems.</p>
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<p>Resolved portions of the permeable screens.</p>
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<p>Wired mesh: analytical, FVM porous vs. resolved geometry CFD solutions in the Case 1 flow.</p>
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<p>Perforated panels: analytical, FVM porous vs. resolved geometry CFD solutions in the Case 1 flow; (<b>a</b>) 0 mm thickness, (<b>b</b>) 2 mm thickness, (<b>c</b>) 4 mm thickness.</p>
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<p>Louvers: analytical, FVM porous vs. resolved geometry CFD solutions in the Case 1 flow.</p>
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<p>Expanded Mesh: analytical, FVM porous vs. resolved geometry CFD solutions for Fx and Fy in the Case 1 flow.</p>
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<p>Expanded mesh: analytical, FVM porous vs. resolved CFD solutions for Fz in the Case 2 flow.</p>
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17 pages, 1309 KiB  
Article
Timelike Constant Axis Ruled Surface Family in Minkowski 3-Space
by Areej A. Almoneef and Rashad A. Abdel-Baky
Symmetry 2024, 16(6), 677; https://doi.org/10.3390/sym16060677 - 31 May 2024
Viewed by 437
Abstract
A timelike (TL) constant axis ruled surface in E13 (Minkowski 3-space), as determined by its ruling, forms a constant dual angle with its Disteli-axis (striction axis or curvature axis). In this article, we employ the symmetry through point geometry [...] Read more.
A timelike (TL) constant axis ruled surface in E13 (Minkowski 3-space), as determined by its ruling, forms a constant dual angle with its Disteli-axis (striction axis or curvature axis). In this article, we employ the symmetry through point geometry of Lorentzian dual curves and the line geometry of TL ruled surfaces. This produces the capability to expound a set of curvature functions that specify the local configurations of TL ruled surfaces. Then, we gain some new constant axis ruled surfaces in Lorentzian line space and their geometrical illustrations. Further, we also earn several organizations among a TL constant axis ruled surface and its striction curve. Full article
(This article belongs to the Section Mathematics)
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<p>The hyperbolic and Lorentzian <math display="inline"><semantics> <mi mathvariant="script">DU</mi> </semantics></math> spheres.</p>
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<p><math display="inline"><semantics> <mrow> <mover accent="true"> <mi mathvariant="bold">e</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo form="prefix">cosh</mo> <mover accent="true"> <mi>ω</mi> <mo>^</mo> </mover> <msub> <mover accent="true"> <mi mathvariant="bold">z</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <mo form="prefix">sinh</mo> <mover accent="true"> <mi>ω</mi> <mo>^</mo> </mover> <msub> <mover accent="true"> <mi mathvariant="bold">z</mi> <mo>^</mo> </mover> <mn>3</mn> </msub> </mrow> </semantics></math>.</p>
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<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> cylindroid.</p>
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<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> Archimedes helicoid with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Γ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mi>μ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p>
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<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> Archimedes helicoid with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Γ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mi>μ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p>
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<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> Archimedes helicoid with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Γ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mi>μ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mi>ϰ</mi> </mrow> </semantics></math>.</p>
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<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> Archimedes helicoid with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Γ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mi>μ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mi>ϰ</mi> </mrow> </semantics></math>.</p>
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<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> tangential surface with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </semantics></math></p>
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<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> tangential surface with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </semantics></math></p>
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<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> binormal surface with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Γ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </semantics></math></p>
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<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> binormal surface with <math display="inline"><semantics> <mrow> <msup> <mi>ω</mi> <mo>*</mo> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="normal">Γ</mi> <mo>(</mo> <mi>ϰ</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </semantics></math></p>
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<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> cone.</p>
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<p><math display="inline"><semantics> <mi mathvariant="script">TL</mi> </semantics></math> cylinder with <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> </mrow> </semantics></math></p>
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23 pages, 11342 KiB  
Article
Geometric Implications of Photodiode Arrays on Received Power Distribution in Mobile Underwater Optical Wireless Communication
by Tharuka Govinda Waduge, Boon-Chong Seet and Kay Vopel
Sensors 2024, 24(11), 3490; https://doi.org/10.3390/s24113490 - 28 May 2024
Cited by 1 | Viewed by 926
Abstract
Underwater optical wireless communication (UOWC) has gained interest in recent years with the introduction of autonomous and remotely operated mobile systems in blue economic ventures such as offshore food production and energy generation. Here, we devised a model for estimating the received power [...] Read more.
Underwater optical wireless communication (UOWC) has gained interest in recent years with the introduction of autonomous and remotely operated mobile systems in blue economic ventures such as offshore food production and energy generation. Here, we devised a model for estimating the received power distribution of diffused line-of-sight mobile optical links, accommodating irregular intensity distributions beyond the beam-spread angle of the emitter. We then used this model to conduct a spatial analysis investigating the parametric influence of the placement, orientation, and angular spread of photodiodes in array-based receivers on the mobile UOWC links in different Jerlov seawater types. It revealed that flat arrays were best for links where strict alignment could be maintained, whereas curved arrays performed better spatially but were not always optimal. Furthermore, utilizing two or more spectrally distinct wavelengths and more bandwidth-efficient modulation may be preferred for received-signal intensity-based localization and improving link range in clearer oceans, respectively. Considering the geometric implications of the array of receiver photodiodes for mobile UOWCs, we recommend the use of dynamically shape-shifting array geometries. Full article
(This article belongs to the Special Issue Underwater Vision Sensing System)
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Figure 1

Figure 1
<p>(<b>a</b>) The polar distribution of the emitter’s optical intensity is non-uniform, especially for diffused line-of-sight optical links; the received power is contingent on multiple factors. (<b>b</b>) a miniaturized visual of a typical transmitter–receiver alignment expected for a mobile UOWC. Here, <math display="inline"><semantics> <mrow> <mi>θ</mi> </mrow> </semantics></math> is the beam divergence angle and <math display="inline"><semantics> <mrow> <mi>d</mi> </mrow> </semantics></math> is the separation between the transmitter and the receiver.</p>
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<p>Spectral attenuation variation of Jerlov water types [<a href="#B25-sensors-24-03490" class="html-bibr">25</a>].</p>
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<p>Relative intensity distribution over annuli concentric about the LED center axis.</p>
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<p>Array configuration and naming convention. Array center axis is orthogonal to the tangent to curve at the middle PD, with coordinate <math display="inline"><semantics> <mrow> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>c</mi> </mrow> </msub> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>y</mi> </mrow> <mrow> <mi>c</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math>. C-type and D-type arrays observe the light from the outside and inside of the curve, respectively. Orange circles and purple squares are representative of the PD placements of the 3PD and 5PD arrays, respectively. Each PD is equidistantly placed along the curve starting from the array center.</p>
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<p>Each array is moved within the positive quadrant of a cartesian coordinate plane with the origin centered at the location of the light source. The results are generated for horizontal and vertical displacements.</p>
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<p><math display="inline"><semantics> <mrow> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mrow> </semantics></math> for 5PD arrays at: (<b>a</b>–<b>c</b>) <span class="html-italic">β</span> = 90°; (<b>d</b>–<b>f</b>) <span class="html-italic">β</span> = 60°; (<b>g</b>–<b>i</b>) <span class="html-italic">β</span> = 0°; (<b>j</b>–<b>l</b>) <span class="html-italic">β</span> = −60°; (<b>m</b>–<b>o</b>) <span class="html-italic">β</span> = −90° for Jerlov I; <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>0.022</mn> </mrow> </semantics></math> m<sup>−1</sup>. Red and green lines are 10- and 0-dB contours, respectively.</p>
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<p><math display="inline"><semantics> <mrow> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mrow> </semantics></math> for 5PD arrays at: (<b>a</b>–<b>c</b>) <span class="html-italic">β</span> = 90°; (<b>d</b>–<b>f</b>) <span class="html-italic">β</span> = 60°; (<b>g</b>–<b>i</b>) <span class="html-italic">β</span> = 0°; (<b>j</b>–<b>l</b>) <span class="html-italic">β</span> = −60°; (<b>m</b>–<b>o</b>) <span class="html-italic">β</span> = −90° for Jerlov III; <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>1.045</mn> </mrow> </semantics></math> m<sup>−1</sup>. Red and green lines show 10- and 0-dB contours, respectively.</p>
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<p><math display="inline"><semantics> <mrow> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mrow> </semantics></math> for 5PD arrays at: (<b>a</b>–<b>c</b>) <span class="html-italic">β</span> = 90°; (<b>d</b>–<b>f</b>) <span class="html-italic">β</span> = 60°; (<b>g</b>–<b>i</b>) <span class="html-italic">β</span> = 0°; (<b>j</b>–<b>l</b>) <span class="html-italic">β</span> = −60°; (<b>m</b>–<b>o</b>) <span class="html-italic">β</span> = −90° for Jerlov 3C; <math display="inline"><semantics> <mrow> <mi>c</mi> <mo>=</mo> <mn>1.142</mn> </mrow> </semantics></math> m<sup>−1</sup>. Red and green lines show 10- and 0-dB contours, respectively.</p>
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<p><math display="inline"><semantics> <mrow> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mrow> </semantics></math> for respective 3PD arrays at <span class="html-italic">β</span> = 60° rotation in: (<b>a</b>–<b>c</b>) Jerlov I (<span class="html-italic">λ</span> = 450 nm); (<b>d</b>–<b>f</b>) Jerlov III (<span class="html-italic">λ</span> = 550 nm) and (<b>g</b>–<b>i</b>) Jerlov 3C (<span class="html-italic">λ</span> = 550 nm). Red and green lines show 10- and 0-dB contours, respectively.</p>
Full article ">Figure 10
<p>BER analysis for respective 5PD arrays in: (<b>a</b>–<b>c</b>) Jerlov I; (<b>d</b>–<b>f</b>) Jerlov III; and (<b>g</b>–<b>i</b>) Jerlov 3C. A wavelength of 450 nm was used for Jerlov I and 550 nm was used for Jerlov III and 3C, respectively, due to them being the least attenuated light wavelengths in the respective water types, as per <a href="#sensors-24-03490-t003" class="html-table">Table 3</a>. Jerlov I was modeled for a 15 m vertical displacement, and Jerlov III and 3C at a 1 m vertical displacement.</p>
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<p><math display="inline"><semantics> <mrow> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mrow> </semantics></math> for 5PD arrays of: (<b>a</b>,<b>b</b>) type F; (<b>c</b>,<b>d</b>) type C25; and (<b>e</b>,<b>f</b>) type C55; at <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mo>{</mo> </mrow> </semantics></math>−60°, 0°, 60°} respectively, for Jerlov I, III, 1C, and 3C, for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mo>{</mo> </mrow> </semantics></math>450, 500, 550} nm.</p>
Full article ">Figure 11 Cont.
<p><math display="inline"><semantics> <mrow> <msub> <mrow> <mi>P</mi> </mrow> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> </mrow> </msub> </mrow> </msub> </mrow> </semantics></math> for 5PD arrays of: (<b>a</b>,<b>b</b>) type F; (<b>c</b>,<b>d</b>) type C25; and (<b>e</b>,<b>f</b>) type C55; at <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mo>{</mo> </mrow> </semantics></math>−60°, 0°, 60°} respectively, for Jerlov I, III, 1C, and 3C, for <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mo>{</mo> </mrow> </semantics></math>450, 500, 550} nm.</p>
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25 pages, 6814 KiB  
Article
Study on the Seismic Response of a Water-Conveyance Tunnel Considering Non-Uniform Longitudinal Subsurface Geometry and Obliquely Incident SV-Waves
by Erlei Yao, Yu Rao, Meishan Liu, Zhifang Liu and Ang Cao
Appl. Sci. 2024, 14(11), 4398; https://doi.org/10.3390/app14114398 - 22 May 2024
Cited by 1 | Viewed by 608
Abstract
The longitudinal seismic response characteristics of a shallow-buried water-conveyance tunnel under non-uniform longitudinal subsurface geometry and obliquely incident SV-waves was studied using the numerical method, where the effect of the non-uniform longitudinal subsurface geometry due to the existence of a local one-sided rock [...] Read more.
The longitudinal seismic response characteristics of a shallow-buried water-conveyance tunnel under non-uniform longitudinal subsurface geometry and obliquely incident SV-waves was studied using the numerical method, where the effect of the non-uniform longitudinal subsurface geometry due to the existence of a local one-sided rock mountain on the seismic response of the tunnel was focused on. Correspondingly, a large-scale three-dimensional (3D) finite-element model was established, where different incidence angles and incidence directions of the SV-wave were taken into consideration. Also, the non-linearity of soil and rock and the damage plastic of the concrete lining were incorporated. In addition, the wave field of the site and the acceleration response as well as damage of the tunnel were observed. The results revealed the following: (i) a local inclined subsurface geometry may focus an obliquely incident wave due to refraction or total reflection; (ii) a tunnel in a site adjacent to a rock mountain may exhibit a higher acceleration response than a tunnel in a homogeneous plain site; and (iii) damage in the tunnel in the site adjacent to a rock mountain may appear in multiple positions, and the effect of the incidence angle on the mode of compressive deformation and damage of the lining is of significance. Full article
(This article belongs to the Special Issue Earthquake Engineering and Seismic Risk)
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Figure 1

Figure 1
<p>The spatial location relationship between the tunnel and the mountain for the Kunming interval of the Dianzhong Water Diversion Project.</p>
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<p>Schematic diagram for the local one-sided mountain–plain site tunnel model: (<b>a</b>) left side incidence; and (<b>b</b>) right side incidence.</p>
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<p>Dimensions of the one-sided mountain tunnel model.</p>
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<p>Mesh of the one-sided mountain tunnel model.</p>
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<p>Simulated ground motions: (<b>a</b>) acceleration time–history; (<b>b</b>) velocity time–history; (<b>c</b>) displacement time–history; and (<b>d</b>) the PSD of the simulated ground motion.</p>
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<p>Wave propagation diagram for SV-waves: (<b>a</b>) left-side incidence; and (<b>b</b>) right-side incidence.</p>
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<p>Displacement contour under incident SV–wave for (<b>a</b>) plain site and left-side incidence at t = 0.27 s; (<b>b</b>) plain–mountain site and left–side incidence at t = 0.27 s; (<b>c</b>) plain site and right–side incidence at t = 0.36 s; and (<b>d</b>) plain–mountain site and right-side incidence at t = 0.36 s. (Unit: m).</p>
Full article ">Figure 7 Cont.
<p>Displacement contour under incident SV–wave for (<b>a</b>) plain site and left-side incidence at t = 0.27 s; (<b>b</b>) plain–mountain site and left–side incidence at t = 0.27 s; (<b>c</b>) plain site and right–side incidence at t = 0.36 s; and (<b>d</b>) plain–mountain site and right-side incidence at t = 0.36 s. (Unit: m).</p>
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<p>Seismic response at the arch vault of the tunnel: (<b>a</b>) acceleration response in the horizontal direction; and (<b>b</b>) acceleration response in the vertical direction.</p>
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<p>Seismic response at the arch bottom of the tunnel: (<b>a</b>) acceleration response in the horizontal direction; and (<b>b</b>) acceleration response in the vertical direction.</p>
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<p>Seismic response at the left arch of the tunnel: (<b>a</b>) acceleration response in the horizontal direction; and (<b>b</b>) acceleration response in the vertical direction.</p>
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<p>Seismic response at the right arch of the tunnel: (<b>a</b>) acceleration response in the horizontal direction; and (<b>b</b>) acceleration response in the vertical direction.</p>
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<p>Diagram of the wave field for (<b>a</b>) left-side-incident SV-wave and (<b>b</b>) right-side-incident SV-wave with an incidence angle = 11.31° in the plain–mountain site.</p>
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<p>Diagram of the wave field for (<b>a</b>) left-side-incident SV-wave and (<b>b</b>) right-side-incident SV-wave with an incidence angle = 33.69° in the plain–mountain site.</p>
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<p>Tensile damage of tunnel lining under right-side-incident SV-wave: (<b>a</b>) incidence angle = 33.69° and t = 2.25 s; (<b>b</b>) incidence angle = 33.69° and t = 20 s; (<b>c</b>) incidence angle = 26.57° and t = 2.31 s; (<b>d</b>) incidence angle = 26.57° and t = 20 s; (<b>e</b>) incidence angle = 18.43° and t = 2.26 s; (<b>f</b>) incidence angle = 18.43° and t = 20 s; (<b>g</b>) incidence angle = 11.31° and t = 2.21 s; and (<b>h</b>) incidence angle = 11.31° and t = 20 s.</p>
Full article ">Figure 15
<p>Tunnel lining deformation at the time when damage initially shows up for right-side-incident SV-wave with the following incidence angles: (<b>a</b>) 33.69°; (<b>b</b>) 26.57°; (<b>c</b>) 18.43°; and (<b>d</b>) 11.31° (scaling factor = 300; unit: m).</p>
Full article ">Figure 16
<p>Tunnel lining deformation at t = 6.0 s for right-side-incident SV-wave with the following incidence angles: (<b>a</b>) 33.69°; (<b>b</b>) 26.57°; (<b>c</b>) 18.43°; and (<b>d</b>) 11.31° (scaling factor = 300; unit: m).</p>
Full article ">Figure 17
<p>Tensile damage of tunnel lining under left-side-incident SV-wave: (<b>a</b>) incidence angle = 33.69° and t = 2.25 s; (<b>b</b>) incidence angle = 33.69° and t = 20 s; (<b>c</b>) incidence angle = 26.57° and t = 2.31 s; (<b>d</b>) incidence angle = 26.57° and t = 20 s; (<b>e</b>) incidence angle = 18.43° and t = 2.26 s; (<b>f</b>) incidence angle = 18.43° and t = 20 s; (<b>g</b>) incidence angle = 11.31° and t = 2.21 s; and (<b>h</b>) incidence angle = 11.31° and t = 20 s.</p>
Full article ">Figure 18
<p>Tunnel lining deformation at the time when damage initially shows up for left-side-incident SV-wave with the following incidence angles: (<b>a</b>) 33.69°; (<b>b</b>) 26.57°; (<b>c</b>) 18.43°; and (<b>d</b>) 11.31° (scaling factor = 300; unit: m).</p>
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16 pages, 4660 KiB  
Article
Elastic Critical Buckling Coefficients for Skew Plates of Steel Structures under Biaxial Normal Stress
by Kazuya Mitsui, Kikuo Ikarashi and Keiichiro Sada
Buildings 2024, 14(4), 901; https://doi.org/10.3390/buildings14040901 - 26 Mar 2024
Viewed by 833
Abstract
In steel structures, skew thin steel plates serve as panel zones in structures spanning large spaces (e.g., warehouses and gymnasiums). Considerable research has been conducted on the shear buckling of panels due to seismic loads acting on a structure. Conversely, under snow or [...] Read more.
In steel structures, skew thin steel plates serve as panel zones in structures spanning large spaces (e.g., warehouses and gymnasiums). Considerable research has been conducted on the shear buckling of panels due to seismic loads acting on a structure. Conversely, under snow or wind loads, the panel zone may experience compressive and tensile stresses simultaneously from two directions. Considering the economic preference for thin steel plates, evaluating the elastic critical local buckling stresses in the panel zone under biaxial normal stress may provide essential information to structural engineers. In this study, an elastic buckling analysis based on the energy method is performed to clarify the impact of panel geometry and boundary conditions on the elastic local buckling stresses of skew panel zones. As confirmed from the results, the local buckling stresses calculated using the energy method were consistent with those determined using finite element analysis. The findings indicate that a skew angle of up to 30° marginally affects the elastic buckling stress under uniaxial stress. Consequently, engineer-friendly design formulas were developed based on these findings. Comparisons with previous research demonstrated that the buckling loads reported were generally higher than those determined by finite element analysis. The study also established the correlation of the buckling stresses under biaxial stresses, which implied that the skew angle posed minimal influence on buckling stress for skew plates under biaxial stress. Additionally, a method for evaluating this correlation was presented. Engineers can utilize the provided design equations to more efficiently and accurately calculate buckling loads, facilitating a safer and more economical design of structures with skew plates. Full article
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Figure 1

Figure 1
<p>Stress state on panel zones varying with multiple loads.</p>
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<p>Skew plates under uniform biaxial normal stresses and its geometry: (<b>a</b>) Panel zone in moment frame; (<b>b</b>) Theoretical analysis model.</p>
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<p>Overview of FE model for skew plates under normal stress.</p>
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<p>Convergence study for mesh size.</p>
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<p>Graphical comparisons between present theoretical and FEA results and buckling modes under uniaxial compression where <span class="html-italic">θ</span> = 30°: (<b>a</b>) Under stress parallel to longitudinal direction; (<b>b</b>) Under stress perpendicular to longitudinal direction.</p>
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<p>Effect of geometries on the local buckling load: (<b>a</b>) Under stress parallel to longitudinal direction; (<b>b</b>) Under stress perpendicular to longitudinal direction.</p>
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<p>Effect of geometries on the local buckling modes with simply supported condition: (<b>a</b>) Under stress parallel to longitudinal direction; (<b>b</b>) Under stress perpendicular to longitudinal direction.</p>
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<p>Comparison between FEA results and calculations based on the proposed design equations: (<b>a</b>) Stress parallel to longitudinal direction; (<b>b</b>) Stress perpendicular to longitudinal direction.</p>
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<p>Correlation curve of buckling coefficient for simply supported condition: (<b>a</b>) <span class="html-italic">a</span>/<span class="html-italic">b</span> = 1.0; (<b>b</b>) <span class="html-italic">a</span>/<span class="html-italic">b</span> = 3.0; (<b>c</b>) <span class="html-italic">a</span>/<span class="html-italic">b</span> = 5.0.</p>
Full article ">Figure 10
<p>Correlation curve of buckling coefficient for clamped-supported condition: (<b>a</b>) <span class="html-italic">a</span>/<span class="html-italic">b</span> = 1.0; (<b>b</b>) <span class="html-italic">a</span>/<span class="html-italic">b</span> = 3.0; (<b>c</b>) <span class="html-italic">a</span>/<span class="html-italic">b</span> = 5.0.</p>
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32 pages, 8588 KiB  
Article
Heat Convection in a Channel-Opened Cavity with Two Heated Sources and Baffle
by Farhan Lafta Rashid, Asseel M. Rasheed Al-Gaheeshi, Hayder I. Mohammed and Arman Ameen
Energies 2024, 17(5), 1209; https://doi.org/10.3390/en17051209 - 3 Mar 2024
Cited by 2 | Viewed by 1011
Abstract
This study employs COMSOL software v 5.6 to investigate a novel approach to heat transfer via mixed convection in an open hollow structure with an unheated 90° baffle elbow. Two 20 W heat sources are strategically positioned on the cavity’s bottom and right-angled [...] Read more.
This study employs COMSOL software v 5.6 to investigate a novel approach to heat transfer via mixed convection in an open hollow structure with an unheated 90° baffle elbow. Two 20 W heat sources are strategically positioned on the cavity’s bottom and right-angled wall for this research. Notably, the orientation of the baffle perpendicular to the airflow is used to direct external, unrestricted flow into the square cavity. The research investigates a range of air velocities (0.1, 0.5, 1.0, and 1.5 m/s) and the intricate interaction between input air velocity, dual heated sources, and the presence of a right-angle baffle on critical thermodynamic variables, such as temperature distribution, isotherms, pressure variation, velocity profile, air density, and both local and mean Nusselt numbers. Validation of the applicable computational method is achieved by comparing it to two previous studies. Significant findings from numerical simulations indicate that the highest velocity profile is in the centre of the channel (2.3–2.68 m/s at an inflow velocity of 1.5 m/s), while the lowest profile is observed along the channel wall, with a notable disruption near the inlet caused by increased shear forces. The cavity neck temperature ranges from 380 to 640 K, with inflow air velocities varying from 0.1 to 1.5 m/s (Re is 812 to 12,182), respectively. In addition, the pressure fluctuates at the channel-cavity junction, decreasing steadily along the channel length and reaching a maximum at the intake, where the cavity neck pressure varies from 0.01 to 2.5 Pa with inflow air velocities changing from 0.1 to 1.5 m/s, respectively. The mean Nusselt number exhibits an upward trend as air velocity upon entry increases. The mean Nusselt number reaches up to 1500 when the entry air velocity reaches 1.5 m/s. Due to recirculation patterns, the presence of the 90° unheated baffle produces a remarkable cooling effect. The study establishes a direct correlation between input air velocity and internal temperature distribution, indicating that as air velocity increases, heat dissipation improves. This research advances our understanding of convective heat transfer phenomena in complex geometries and provides insights for optimising thermal management strategies for a variety of engineering applications. Full article
(This article belongs to the Special Issue New Challenges in Heat Transfer Enhancement)
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<p>Geometric passage flow diagram (all dimensions are measured in units of meters (m)).</p>
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<p>Mesh employed in this case study; (<b>a</b>) overview of the model and (<b>b</b>) a close up of the region of interest.</p>
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<p>Velocity contours comparison with the study by Salhi et al. (2020) [<a href="#B10-energies-17-01209" class="html-bibr">10</a>].</p>
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<p>Isotherm curve at various inlet velocities.</p>
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<p>Isotherm curve at various inlet velocities.</p>
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<p>Different inlet velocities’ contours of the 2D passage temperature.</p>
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<p>Different inlet velocities’ contours of the 2D passage temperature.</p>
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<p>Two-dimensional inlet velocity contours.</p>
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<p>Two-dimensional inlet velocity contours.</p>
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<p>Contours of the passage pressure streamline at various inlet velocities.</p>
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<p>Velocities distribution at various inlet velocities when the <span class="html-italic">y</span>-axis of the passage flow is positive.</p>
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<p>Pressure distribution along the <span class="html-italic">y</span>-axis for a range of inflow velocities.</p>
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<p>Pressure distribution along the <span class="html-italic">y</span>-axis for a range of inflow velocities.</p>
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<p>Upper side of the vertical position passage flow temperature distribution for different inlet flows.</p>
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<p>Upper side of the vertical position passage flow temperature distribution for different inlet flows.</p>
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<p>Cavity velocity profile along the lower side vertical position for various inflow speeds.</p>
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<p>Cavity velocity profile along the lower side vertical position for various inflow speeds.</p>
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<p>Cavity pressure distribution along the lower side vertical position for a range of inflow velocities.</p>
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<p>Cavity pressure distribution along the lower side vertical position for a range of inflow velocities.</p>
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<p>Cavity temperature distribution along the lower side vertical position for a range of input speeds.</p>
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<p>Cavity temperature distribution along the lower side vertical position for a range of input speeds.</p>
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<p>The local Nusselt number in the cavity along the lower side vertical position for a range of inflow velocities.</p>
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<p>The local velocity profile in the hollow neck at different intake speeds.</p>
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<p>Cavity neck pressure at various inflow velocities.</p>
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<p>Local temperature in cavity neck at various velocities.</p>
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<p>Cavity neck Nusselt number for different inflow velocities.</p>
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<p>Profile of velocity in outlet passage flow at various inlet velocities.</p>
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<p>Profile of Temperature in outlet passage flow at various inlet velocities.</p>
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<p>Profile of Density in outlet passage flow at various inlet velocities.</p>
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13 pages, 2862 KiB  
Article
Self-Shielding of X-ray Emission from Ultrafast Laser Processing Due to Geometrical Changes of the Interaction Zone
by Julian Holland, Christian Hagenlocher, Rudolf Weber and Thomas Graf
Materials 2024, 17(5), 1109; https://doi.org/10.3390/ma17051109 - 28 Feb 2024
Viewed by 828
Abstract
Materials processing with ultrashort laser pulses is one of the most important approaches when it comes to machining with very high accuracy. High pulse repetition rates and high average laser power can be used to attain high productivity. By tightly focusing the laser [...] Read more.
Materials processing with ultrashort laser pulses is one of the most important approaches when it comes to machining with very high accuracy. High pulse repetition rates and high average laser power can be used to attain high productivity. By tightly focusing the laser beam, the irradiances on the workpiece can exceed 1013 W/cm2, and thus cause usually unwanted X-ray emission. Pulsed laser processing of micro holes exhibits two typical features: a gradual increase in the irradiated surface within the hole and, with this, a decrease in the local irradiance. This and the shielding by the surrounding material diminishes the amount of ionizing radiation emitted from the process; therefore, both effects lead to a reduction in the potential X-ray exposure of an operator or any nearby person. The present study was performed to quantify this self-shielding of the X-ray emission from laser-drilled micro holes. Percussion drilling in standard air atmosphere was investigated using a laser with a wavelength of 800 nm a pulse duration of 1 ps, a repetition rate of 1 kHz, and with irradiances of up to 1.1·1014 W/cm. The X-ray emission was measured by means of a spectrometer. In addition to the experimental results, we present a model to predict the expected X-ray emission at different angles to the surface. These calculations are based on raytracing simulations to obtain the local irradiance, from which the local X-ray emission inside the holes can be calculated. It was found that the X-ray exposure measured in the surroundings strongly depends on the geometry of the hole and the measuring direction, as predicted by the theoretical model. Full article
(This article belongs to the Section Manufacturing Processes and Systems)
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<p>Model geometry of the percussion-drilled hole with a qualitative sketch of the distribution of the irradiance <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics></math>. As a consequence of multiple reflections inside of the hole, as indicated by the red rays (<b>right</b>), the distribution of the irradiance <math display="inline"><semantics> <mrow> <mi>J</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics></math> exhibits a maximum near the tip of the hole (<b>left</b>).</p>
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<p>Geometrical parameters used to model the spectral power of the X-ray emission detectable at a given point outside of a percussion-drilled conical hole.</p>
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<p>Experimental setup. (<b>a</b>) The beam of a Ti:Sapphire laser with a wavelength of 800 nm and a pulse repetition rate of 1 kHz was focused on a sample of stainless steel by means of an F-theta lens, leading to a peak irradiance of 1.1·10<sup>14</sup> W/cm<sup>2</sup> on the initially plane surface. The spectrometer was placed at a distance of 35 cm from the processing region. (<b>b</b>) A total of 20 holes were processed for each measurement. All holes were repeatedly processed with the same number <span class="html-italic">N<sub>i</sub></span> of pulses, and the X-ray emission was measured separately for each processing step <span class="html-italic">i</span>. Hence, every measurement corresponds to the emission from a specific depth <span class="html-italic">z<sub>h</sub></span> of the drilled holes.</p>
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<p>Yield of the detected X-ray emission from percussion-drilled holes with increasing aspect ratio. The detector was placed at a distance of 35 cm from the processing zone. Each data point corresponds to the average over the 20 holes drilled with the same parameters.</p>
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<p>Calculated X-ray emission reaching a detector placed in 35 cm from the drilled holes in the direction <math display="inline"><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math>. (<b>a</b>) Results according to Equation (14). (<b>b</b>) Results from Equation (14) with further details, including an unshielded emission from the surface next to the hole (green dotted line) following Equation (15).</p>
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<p>Comparison of measurements and model calculation. Depending on the detection angle, the yield of X-ray emission first increases (<b>b</b>,<b>c</b>) before dropping to a constant value, which is identical for all three detection angles. For small detection angles (<b>a</b>) there is no increase at the beginning visible, because the discussed shielding effect starts immediately.</p>
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18 pages, 16502 KiB  
Article
Intricacies of Opening Geometry Detection in Terrestrial Laser Scanning: An Analysis Using Point Cloud Data from BLK360
by Jinman Jung, Taesik Kim, Hong Min, Seongmin Kim and Young-Hoon Jung
Remote Sens. 2024, 16(5), 759; https://doi.org/10.3390/rs16050759 - 21 Feb 2024
Cited by 2 | Viewed by 1005
Abstract
This study investigates the use of terrestrial laser scanning (TLS) in urban excavation sites, focusing on enhancing ground deformation detection by precisely identifying opening geometries, such as gaps between pavement blocks. The accuracy of TLS data, affected by equipment specifications, environmental conditions, and [...] Read more.
This study investigates the use of terrestrial laser scanning (TLS) in urban excavation sites, focusing on enhancing ground deformation detection by precisely identifying opening geometries, such as gaps between pavement blocks. The accuracy of TLS data, affected by equipment specifications, environmental conditions, and scanning geometry, is closely examined, especially with regard to the detection of openings between blocks. The experimental setup, employing the BLK360 scanner, aimed to mimic real-world paving situations with varied opening widths, allowing an in-depth analysis of how factors related to scan geometry, such as incidence angles and opening orientations, influence detection capabilities. Our examination of various factors and detection levels reveals the importance of the opening width and orientation in identifying block openings. We discovered the crucial role of the opening width, where larger openings facilitate detection in 2D cross-sections. The overall density of the point cloud was more significant than localized variations. Among geometric factors, the orientation of the local object geometry was more impactful than the incidence angle. Increasing the number of laser beam points within an opening did not necessarily improve detection, but beams crossing the secondary edge were vital. Our findings highlight that larger openings and greater overall point cloud densities markedly improve detection levels, whereas the orientation of local geometry is more critical than the incidence angle. The study also discusses the limitations of using a single BLK360 scanner and the subtle effects of scanning geometry on data accuracy, providing a thorough understanding of the factors that influence TLS data accuracy and reliability in monitoring urban excavations. Full article
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Graphical abstract

Graphical abstract
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<p>(<b>a</b>) Supporting wall and safety fence standing at an excavation site; (<b>b</b>) block pavement for pedestrians adjacent to the safety fence outside the excavation site; (<b>c</b>) example of misaligned blocks adjacent to the safety fence; (<b>d</b>) example of severe distortion in the block pavement.</p>
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<p>(<b>a</b>) Test specimens with the opening width of (<b>c</b>) 2 mm, (<b>d</b>) 5 mm, (<b>e</b>) 10 mm, (<b>f</b>) 15 mm, and (<b>g</b>) 20 mm. (<b>b</b>) The point densities of Zones A and B were evaluated separately.</p>
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<p>Configuration of the specimen and the scanner at (<b>a</b>) the incidence angle, <math display="inline"><semantics> <mrow> <mi>θ</mi> </mrow> </semantics></math>, and the orientation of the opening geometry, <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math>. The scanner and the specimens were configured at the incidence angles of (<b>b</b>) 0° and (<b>c</b>) 60°. The TLS was flipped downward to scan the specimen on the ground.</p>
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<p>The configuration of the scanner and the opening geometry specimen (<b>a</b>) in the cross-section view and (<b>b</b>) in the plan view.</p>
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<p>Structured overview of the methodology to process the point cloud datasets.</p>
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<p>Planar distribution of point density: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>90</mn> <mo>°</mo> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>°</mo> </mrow> </semantics></math>, and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>45</mn> <mo>°</mo> </mrow> </semantics></math>. The block opening is positioned at a vertical orientation in the figure.</p>
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<p>Planar distribution of point density: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>90</mn> <mo>°</mo> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>°</mo> </mrow> </semantics></math>, and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>45</mn> <mo>°</mo> </mrow> </semantics></math>. The block opening is positioned at a vertical orientation in the figure.</p>
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<p>Planar distribution of point density: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>90</mn> <mo>°</mo> </mrow> </semantics></math>, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0</mn> <mo>°</mo> </mrow> </semantics></math>, and (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>45</mn> <mo>°</mo> </mrow> </semantics></math>. The block opening is positioned at a vertical orientation in the figure.</p>
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<p>Variation in the detection levels for the different incidence angles and set widths: (<b>a</b>) at <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> = 90<math display="inline"><semantics> <mrow> <mo>°</mo> </mrow> </semantics></math>; (<b>b</b>) at <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> = 0<math display="inline"><semantics> <mrow> <mo>°</mo> </mrow> </semantics></math>; (<b>c</b>) at <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> = 45<math display="inline"><semantics> <mrow> <mo>°</mo> </mrow> </semantics></math>.</p>
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<p>Variation in the detection levels for the different point cloud densities: (<b>a</b>) at <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> = 90<math display="inline"><semantics> <mrow> <mo>°</mo> </mrow> </semantics></math>; (<b>b</b>) at <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> = 0<math display="inline"><semantics> <mrow> <mo>°</mo> </mrow> </semantics></math>; (<b>c</b>) at <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> = 45<math display="inline"><semantics> <mrow> <mo>°</mo> </mrow> </semantics></math>.</p>
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<p>Variation in the detection levels for the different numbers of laser beams passing: (<b>a</b>) at <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> = 90<math display="inline"><semantics> <mrow> <mo>°</mo> </mrow> </semantics></math>; (<b>b</b>) at <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> = 0<math display="inline"><semantics> <mrow> <mo>°</mo> </mrow> </semantics></math>; (<b>c</b>) at <math display="inline"><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> = 45<math display="inline"><semantics> <mrow> <mo>°</mo> </mrow> </semantics></math>.</p>
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<p>Correlation matrix detailing the Pearson’s correlation values between the detection level and eight parameters. “DL” in the chart denotes the detection level.</p>
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<p>Ranked lists of nine different metrics.</p>
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<p>Ranked lists of nine different metrics.</p>
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<p>Borda count of the influencing parameters.</p>
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20 pages, 14402 KiB  
Article
Refined Coseismic Slip Model and Surface Deformation of the 2021 Maduo Earthquake: Implications for Sensitivity of Rupture Behaviors to Geometric Complexity
by Xiaoli Liu, Debeier Deng, Zhige Jia, Jing Liu-Zeng, Xinyu Mo, Yu Huang, Qiaozhe Ruan and Juntao Liu
Remote Sens. 2024, 16(4), 713; https://doi.org/10.3390/rs16040713 - 18 Feb 2024
Viewed by 1168
Abstract
Geometric complexities of a fault system have a significant impact on the rupture behavior of the fault. The 2021 Mw7.4 Maduo earthquake occurred on a multi-segmented complex sinistral fault in the interior of the Bayan-Har block in the northern Tibetan Plateau. Here, we [...] Read more.
Geometric complexities of a fault system have a significant impact on the rupture behavior of the fault. The 2021 Mw7.4 Maduo earthquake occurred on a multi-segmented complex sinistral fault in the interior of the Bayan-Har block in the northern Tibetan Plateau. Here, we integrate centimeter-resolution surface rupture zones and Sentinel-2 optical displacement fields to accurately determine the geometric parameters of the causative fault in detail. An adaptive quadtree down-sampling method for interferograms was employed to enhance the reliability of the coseismic slip model inversion for interferograms. The optimal coseismic slip model indicated a complex non-planar structure with varying strike and dip angles. The largest slip of ~6 m, at a depth of ~7 km, occurred near a 6 km-wide stepover (a geometric complexity area) to the east of the epicenter, which occurred at the transition zone from sub-shear to super-shear rupture suggested by seismological studies. Optical and SAR displacement fields consistently indicated the local minimization of effective normal stress on releasing stepovers, which facilitated rupture through them. Moreover, connecting intermediate structures contributes to maintaining the rupture propagation through wide stepovers and may even facilitate the transition from subshear to supershear. Our study provides more evidence of the reactivation of a branched fault at the western end during the mainshock, which was previously under-appreciated. Furthermore, we found that a strong asymmetry in slip depth, stress drop, and rupture velocity east and west of the epicenter was coupled with variations in geometric and structural characteristics of fault segments along the strike. Our findings highlight the sensitivity of rupture behaviors to small-scale details of fault geometry. Full article
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<p>Topographic and tectonic setting map of the Tibetan plateau. Gray dashed lines and thick black and thin lines indicate block suture zones and main and secondary active faults, respectively [<a href="#B8-remotesensing-16-00713" class="html-bibr">8</a>]. Blue/white and red/white beach balls denote focal mechanisms of historical and Maduo earthquakes (M ≥ 7) in the Bayan-Har block reported in the Global CMT (GCMT) and USGS catalog from 1 January 1966 to 1 June 2021, respectively. The light blue and yellow boxes depict footprints of SAR and optical images used in this study, respectively.</p>
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<p>Surface rupture map of the 2020 Madoi earthquake, with relocated aftershocks 8 days after the main earthquake [<a href="#B14-remotesensing-16-00713" class="html-bibr">14</a>]. Inverted triangles with numbers indicate the projected position of the maximum slip of every asperity on the surface. Diamonds with numbers indicate possible projected positions of the west branch on the surface.</p>
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<p>Full-resolution phase gradient and optical deformation fields. (<b>a</b>) EW- and (<b>b</b>) NS-deformation maps, (<b>c</b>) descending range and (<b>d</b>) azimuth phase gradient maps, (<b>e</b>) EW-trending stepover, (<b>f</b>) Optical displacement, and (<b>g</b>,<b>h</b>) linear features from the range and azimuth phase gradients. The yellow star indicates the CENC epicenter. W20 in Panel (<b>a</b>) indicates 20 km west of the CENC epicenter. Black hollow squares and rectangles mark sites and tail bifurcations, respectively. Black and blue arrows mark the locations of bends and stepovers along the main fault, respectively. Numbers in drop-shaped signs show stepover width.</p>
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<p>Coseismic LoS direction deformation and three-dimensional displacement fields for the 2021 Maduo earthquake. (<b>a</b>) Ascending, (<b>b</b>) Descending, (<b>c</b>) EW-, (<b>d</b>) NS-, and (<b>e</b>) Vertical-component displacement fields. The yellow star indicates the epicenter. Blue solid and dashed lines represent the main and branch faults, respectively, divided into 8 segments. The caption is as for <a href="#remotesensing-16-00713-f003" class="html-fig">Figure 3</a>.</p>
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<p>The downsampled results used the traditional variance method (<b>a</b>,<b>b</b>) and the adaptive threshold methods (<b>c</b>,<b>d</b>), respectively, for the ascending and descending interferograms.</p>
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<p>The preferred coseismic slip model of the 2021 Maduo earthquake. Coseismic slip distribution for the total (<b>a</b>), strike- (<b>b</b>), and dip-component (<b>c</b>) slip on the seismogenic fault. Pink bars indicate the maximum slip on the fault plane along the strike. The yellow star denotes the earthquake hypocenter. Red beach balls represent the focal mechanism solution of every asperity. Captions are as in <a href="#remotesensing-16-00713-f003" class="html-fig">Figure 3</a>.</p>
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<p>Distribution of released seismic moment and slip with depth. Distribution of the total (<b>a</b>), strike- (<b>b</b>), and dip-direction (<b>c</b>) slip-along depth for every asperity. Distribution of the total (<b>d</b>), strike- (<b>e</b>), and dip-direction (<b>f</b>) released seismic moment along depth for every asperity.</p>
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<p>Simplified surface rupture zone (<b>a</b>) and seismogenic fault models used in the coseismic slip distribution inversion (<b>b</b>–<b>g</b>) of the 2021 Maduo earthquake [<a href="#B9-remotesensing-16-00713" class="html-bibr">9</a>,<a href="#B10-remotesensing-16-00713" class="html-bibr">10</a>,<a href="#B12-remotesensing-16-00713" class="html-bibr">12</a>,<a href="#B17-remotesensing-16-00713" class="html-bibr">17</a>,<a href="#B18-remotesensing-16-00713" class="html-bibr">18</a>]. The numbers show the rupture speeds of the western and eastern sections of the epicenter, respectively. The caption is as for <a href="#remotesensing-16-00713-f002" class="html-fig">Figure 2</a>.</p>
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<p>Stress drop and depth of the peak slip of every asperity for the optimal slip distribution model. The symbol size indicates the peak slip of every asperity.</p>
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13 pages, 389 KiB  
Review
Gravitational Light Bending in Weyl Gravity and Schwarzschild–de Sitter Spacetime
by Joseph Sultana
Symmetry 2024, 16(1), 101; https://doi.org/10.3390/sym16010101 - 14 Jan 2024
Cited by 2 | Viewed by 1192
Abstract
The topic of gravitational lensing in the Mannheim–Kazanas solution of Weyl conformal gravity and the Schwarzschild–de Sitter solution in general relativity has featured in numerous publications. These two solutions represent a spherical massive object (lens) embedded in a cosmological background. In both cases, [...] Read more.
The topic of gravitational lensing in the Mannheim–Kazanas solution of Weyl conformal gravity and the Schwarzschild–de Sitter solution in general relativity has featured in numerous publications. These two solutions represent a spherical massive object (lens) embedded in a cosmological background. In both cases, the interest lies in the possible effect of the background non-asymptotically flat spacetime on the geometry of the local light curves, particularly the observed deflection angle of light near the massive object. The main discussion involves possible contributions to the bending angle formula from the cosmological constant Λ in the Schwarzschild–de Sitter solution and the linear term γr in the Mannheim–Kazanas metric. These effects from the background geometry, and whether they are significant enough to be important for gravitational lensing, seem to depend on the methodology used to calculate the bending angle. In this paper, we review these techniques and comment on some of the obtained results, particularly those cases that contain unphysical terms in the bending angle formula. Full article
(This article belongs to the Section Physics)
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<p>The deflected light trajectory with the one-sided bending angle <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mo>∞</mo> </msub> </semantics></math>. (adapted from [<a href="#B50-symmetry-16-00101" class="html-bibr">50</a>]).</p>
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<p>The deflected photon trajectory with the one-sided bending angle given by <math display="inline"><semantics> <mrow> <mi>ϵ</mi> <mo>=</mo> <mi>ψ</mi> <mo>−</mo> <mi>ϕ</mi> </mrow> </semantics></math> (adapted from [<a href="#B3-symmetry-16-00101" class="html-bibr">3</a>]).</p>
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<p>The observer R, source S, and lens L, with <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Ψ</mi> <mi>R</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Ψ</mi> <mi>S</mi> </msub> </semantics></math> being the angles between the light ray and the corresponding radial directions at these positions. The coordinate angular separation between the source and observer is <math display="inline"><semantics> <mrow> <msub> <mi>ϕ</mi> <mrow> <mi>R</mi> <mi>S</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ϕ</mi> <mi>R</mi> </msub> <mo>−</mo> <msub> <mi>ϕ</mi> <mi>S</mi> </msub> </mrow> </semantics></math>. (adapted from [<a href="#B40-symmetry-16-00101" class="html-bibr">40</a>]).</p>
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12 pages, 1280 KiB  
Article
A Thermodynamic Comparison of Nanotip and Nanoblade Geometries for Ultrafast Laser Field Emission via the Finite Element Method
by Joshua Mann and James Rosenzweig
Physics 2024, 6(1), 1-12; https://doi.org/10.3390/physics6010001 - 19 Dec 2023
Viewed by 1243
Abstract
Strong laser field emission from metals is a growing area of study, owing to its applications in high-brightness cathodes and potentially as a high harmonic generation source. Nanopatterned plasmonic cathodes localize and enhance incident laser fields, reducing the spot size and increasing the [...] Read more.
Strong laser field emission from metals is a growing area of study, owing to its applications in high-brightness cathodes and potentially as a high harmonic generation source. Nanopatterned plasmonic cathodes localize and enhance incident laser fields, reducing the spot size and increasing the current density. Experiments have demonstrated that the nanoblade structure outperforms nanotips in the peak fields achieved before damage is inflicted. With more intense surface fields come brighter emissions, and thus investigating the thermomechanical properties of these structures is crucial in their characterization. We study, using the finite element method, the electron and lattice temperatures for varying geometries, as well as the opening angles, peak surface fields, and apex radii of curvature. While we underestimate the energy deposited into the lattice here, a comparison of the geometries is still helpful for understanding why one structure performs better than the other. We find that the opening angle—not the structure dimensionality—is what primarily determines the thermal performance of these structures. Full article
(This article belongs to the Section Applied Physics)
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Figure 1

Figure 1
<p>Cross-section of the nanoblade and nanotip geometries. We represent the nanoblade using the right half of this 2-dimensional (2D) representation and the tip by revolving that 2D half about the vertical axis by 90 deg. We denote with A the apex of the structure and T the tangent point. The apex has a radius of curvature of <span class="html-italic">R</span> and an opening angle of <math display="inline"><semantics> <mi>θ</mi> </semantics></math>.</p>
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<p>Time plots for the electron (solid) and lattice (dot-dashed) temperatures at the apex as well as the laser envelope (dotted) for the nanoblade structure with <span class="html-italic">R</span> = 20 nm, the peak surface field, <span class="html-italic">E</span><sub>0</sub> = 80 V/nm, and <math display="inline"><semantics> <mrow> <mi>θ</mi> </mrow> </semantics></math> = 30 deg. (I) The laser heats up the electron distribution, and the apex comes to near-equilibrium. (II) The electronic distribution disperses energy throughout the bulk of the structure and begins to deposit energy into the lattice. (III) The electronic distribution has cooled to below the now-heated lattice temperature and begins to aid in cooling.</p>
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<p>Spacetime plots for the electron (<b>left</b>) and lattice (<b>right</b>) temperatures for the nanoblade structure with <span class="html-italic">R</span> = 20 nm, <span class="html-italic">E</span><sub>0</sub> = 80 V/nm and <math display="inline"><semantics> <mrow> <mi>θ</mi> </mrow> </semantics></math> = 30 deg. The curves in <a href="#physics-06-00001-f002" class="html-fig">Figure 2</a> are these data evaluated at the apex at the maximal <span class="html-italic">y</span> value.</p>
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<p>The lattice temperatures at the apex over time for structures with a 3 deg opening angle (blade: dot-dashed; tip: solid and circles) and for a 30 deg opening angle (blade: solid and crosses; tip: dashed). The peak field strength here is 80 V/nm. The structures corresponding most closely to real structures are the 3-degree tip and 30-degree blade. Note that the opening angle appears to have the strongest impact on peak achieved temperatures, followed by the dimensionality.</p>
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<p>The peak lattice temperatures achieved as a function of the peak applied field. The 3-degree tip (solid and circles) reaches a higher temperature than the 30-degree blade (solid and crosses). The 30-degree tip (dashed) outperforms the other structures here, and the fictional 3-degree blade (dot-dashed) is the only structure here which heats to the melting point of gold (solid).</p>
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<p>The peak lattice temperatures as a function of the opening angle <math display="inline"><semantics> <mi>θ</mi> </semantics></math> for the blade (crosses) and tip (circles). Even for the same small opening angles, the blade becomes hotter than the tip. This further indicates that the opening angle strongly impacts thermal dissipation.</p>
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<p>The peak lattice temperatures as a function of the apex radius <span class="html-italic">R</span> for the 30 deg blade (crosses) with <span class="html-italic">E</span><sub>0</sub> = 80 V/nm. Unfortunately, we could not perform the same sweep for the tips. The peak temperature increases as the apex size increases.</p>
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<p>Comparison of tip and blade structures when doubling the height of the center of the apex circle. This moves the 300 K heat bath farther away from the apex, slightly increasing the peak achieved temperatures.</p>
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<p>Comparison of the tip structure with no refinement (grid size of <math display="inline"><semantics> <mrow> <mn>0.5</mn> <mi>R</mi> </mrow> </semantics></math> throughout) and with refinement (<math display="inline"><semantics> <mrow> <mn>0.5</mn> <mo>×</mo> <mi>R</mi> </mrow> </semantics></math> from the apex, <math display="inline"><semantics> <mrow> <mn>0.25</mn> <mo>×</mo> <mi>R</mi> </mrow> </semantics></math> at the tangent point and <math display="inline"><semantics> <mrow> <mn>0.125</mn> <mi>R</mi> </mrow> </semantics></math> at the apex). Surprisingly, this refinement shows little effect on the results, only being off by up to about 10 K.</p>
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<p>The peak lattice temperatures achieved as a function of the peak applied field, using the low-temperature electron-phonon coupling constant <span class="html-italic">g</span> = 0.2 × 10<sup>17</sup> Wm<sup>−3</sup>K<sup>−1</sup>. The conclusions drawn are unchanged. The 3-degree tip (solid and circles) reaches a higher temperature than the 30-degree blade (solid and crosses). The 30-degree tip (dashed) outperforms the other structures here, and the fictional 3-degree blade (dot-dashed) represents the worst performer.</p>
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