Rectangular Area Research Articles

Ultra-wide-angle peripheral refraction using a laser-scanning instrument

We compared the peripheral refractive measurements of a recently proposed laser-scanning instrument with an established peripheral refractor. Two-dimensional refractive maps were obtained using both instruments for 18 young subjects with differing values of central refraction. The comparison shows a strong correlation between devices in the overlapping measurement area, with the new device extending the range of the explored retinal area to a 100-degree-diameter circular patch, compared to the 60°x35° rectangular area of the older peripheral refractor. Larger refractive maps exhibit trends that cannot be easily predicted from narrower scans. These results demonstrate that the new instrument can be a useful tool for assessing wide-angle peripheral optical data in the human eye.

Control-Oriented Free-Boundary Equilibrium Solver for Tokamaks

A free-boundary equilibrium solver for an axisymmetric tokamak geometry was developed based on the finite difference method and Picard iteration in a rectangular computational area. The solver can run either in forward mode, where external coil currents are prescribed until the converged magnetic flux function ψ(R,Z) map is achieved, or in inverse mode, where the desired plasma boundary, with or without an X-point, is prescribed to determine the required coil currents. The equilibrium solutions are made consistent with prescribed plasma parameters, such as the total plasma current, poloidal beta, or safety factor at a specified flux surface. To verify the mathematical correctness and accuracy of the solver, the solution obtained using this numerical solver was compared with that from an analytic fixed-boundary equilibrium solver based on the EAST geometry. Additionally, the proposed solver was benchmarked against another numerical solver based on the finite-element and Newton-iteration methods in a triangular-based mesh. Finally, the proposed solver was compared with equilibrium reconstruction results in DIII-D experiments.

TSAE-UNet: A Novel Network for Multi-Scene and Multi-Temporal Water Body Detection Based on Spatiotemporal Feature Extraction

The application of remote sensing technology in water body detection has become increasingly widespread, offering significant value for environmental monitoring, hydrological research, and disaster early warning. However, the existing methods face challenges in multi-scene and multi-temporal water body detection, including the diverse variations in water body shapes and sizes that complicate detection; the complexity of land cover types, which easily leads to false positives and missed detections; the high cost of acquiring high-resolution images, limiting long-term applications; and the lack of effective handling of multi-temporal data, making it difficult to capture the dynamic changes in water bodies. To address these challenges, this study proposes a novel network for multi-scene and multi-temporal water body detection based on spatiotemporal feature extraction, named TSAE-UNet. TSAE-UNet integrates convolutional neural networks (CNN), depthwise separable convolutions, ConvLSTM, and attention mechanisms, significantly improving the accuracy and robustness of water body detection by capturing multi-scale features and establishing long-term dependencies. The Otsu method was employed to quickly process Sentinel-1A and Sentinel-2 images, generating a high-quality training dataset. In the first experiment, five rectangular areas of approximately 37.5 km2 each were selected to validate the water body detection performance of the TSAE-UNet model across different scenes. The second experiment focused on Jining City, Shandong Province, China, analyzing the monthly water body changes from 2020 to 2022 and the quarterly changes in 2022. The experimental results demonstrate that TSAE-UNet excels in multi-scene and long-term water body detection, achieving a precision of 0.989, a recall of 0.983, an F1 score of 0.986, and an IoU of 0.974, significantly outperforming FCN, PSPNet, DeepLabV3+, ADCNN, and MECNet.

Excellent Canopy Structure in Soybeans Can Improve Their Photosynthetic Performance and Increase Yield

In the maize-soybean intercropping system, varying degrees of maize leaf shading are an important factor that reduces the uniformity of light penetration within the soybean canopy, altering the soybean canopy structure. Quantitative analysis of the relationship between the soybean canopy structure and canopy photosynthesis helps with breeding shade-tolerant soybean varieties for intercropping systems. This study examined the canopy structure and photosynthesis of intercropped soybeans during the shading stress period (28 days before the corn harvest), the high light adaptation period (15 days after the corn harvest), and the recovery period (35 and 55 days after the corn harvest), using a field high-throughput phenotyping platform and a plant gas exchange testing system (CAPTS). Additionally, indoor shading experiments were conducted for validation. The results indicate that shade-tolerant soybean varieties (STV varieties) have significantly higher yields than shade-sensitive soybean varieties (SSV varieties). This is attributable to the STV varieties having a larger top area, lateral width, and lateral external rectangular area. Compared to the SSV varieties, the four top areas of the STV varieties are, on average, 52.09%, 72.05%, and 61.37% higher during the shading stress, high light adaptation, and recovery periods, respectively. Furthermore, the average maximum growth rates (GRs) for the side mean width (SMW) and side rectangle area (SRA) of the STV varieties are 62.92% and 22.13% in the field, and 83.36% and 55.53% in the indoor environment, respectively. This results in a lower canopy overlap in STV varieties, leading to a more uniform light distribution within the canopy, which is reflected in higher photosynthetic rates (Pn), apparent quantum efficiency, and whole-leaf photosynthetic potential (WLPP) for the STV varieties, thereby enhancing their adaptability to shading stress. Above-ground dry matter accumulation was higher in STV varieties, with more assimilates stored in the source and sink, promoting assimilate accumulation in the grains. These results provide new insights into how the superior canopy structure and photosynthesis of shade-tolerant soybean varieties contribute to increased yield.

A radioactive source-seeking method based on angle constraint and particle diffusion

The deployment of mobile robots to conduct search for radioactive sources plays a crucial role in preventing radiation pollution and safeguarding public health. And it is a significant challenge to accurately locate radioactive sources in unknown environments. Herein, a particle filter based on mobile robot search method is proposed to localize radiation sources. By applying an angle constraint technique, the complexity of this algorithm is reduced, the high accuracy of source estimation is maintained, and the range of search is expanded. Additionally, a particle diffusion technology is applied to address particle loss that occurs during filtering. Furthermore, an adaptive gradient descent strategy is adopted to enhance the search efficiency. Experimental results demonstrate that this method achieves a success rate of over 95 % within a rectangular area (e.g., 40m by 40m), outperforming traditional methods, and it could perform effectively in dual radioactive source search tasks.

In vivo video microscopy of the rupturing process of thin blood vessels to clarify the mechanism of bruising caused by blunt impact: an animal study

BackgroundThe thresholds of mechanical inputs for bruising caused by blunt impact are important in the fields of machine safety and forensics. However, reliable data on these thresholds remain inadequate owing to a lack of in vivo experiments, which are crucial for investigating the occurrence of bruising. Since experiments involving live human participants are limited owing to ethical concerns, finite-element method (FEM) simulations of the bruising mechanism should be used to compensate for the lack of experimental data by estimating the thresholds under various conditions, which requires clarifying the mechanism of formation of actual bruises. Therefore, this study aimed to visualize the mechanism underlying the formation of bruises caused by blunt impact to enable FEM simulations to estimate the thresholds of mechanical inputs for bruising.MethodsIn vivo microscopy of a transparent glass catfish subjected to blunt contact with an indenter was performed. The fish were anesthetized by immersing them in buffered MS-222 (75–100 mg/L) and then fixed on a subject tray. The indenter, made of transparent acrylic and having a rectangular contact area with dimensions of 1.0 mm × 1.5 mm, was loaded onto the lateral side of the caudal region of the fish. Blood vessels and surrounding tissues were examined through the transparent indenter using a microscope equipped with a video camera. The contact force was measured using a force-sensing table.ResultsOne of the processes of rupturing thin blood vessels, which are an essential component of the bruising mechanism, was observed and recorded as a movie. The soft tissue surrounding the thin blood vessel extended in a plane perpendicular to the compressive contact force. Subsequently, the thin blood vessel was pulled into a straight configuration. Next, it was stretched in the axial direction and finally ruptured.ConclusionThe results obtained indicate that the extension of the surrounding tissue in the direction perpendicular to the contact force as well as the extension of the thin blood vessels are important factors in the bruising mechanism, which must be reproduced by FEM simulation to estimate the thresholds.

Elastic wave control in reticulated plates using Schwarz primitive cells.

In this work, the Schwarz primitive unit cell is used as the building block of different types of metastructures for steering and focusing elastic vibrations. The emergence of a Bragg-type bandgap when constructing a two-dimensional plate from such unit cells is experimentally validated. It is demonstrated that increasing both mass and porosity of the Schwarz primitive leads to a decrease in the frequency of the out-of-plane propagating wave targeted in this study. By arranging these modified Schwarz primitive unit cells in constant and graded layouts, two-dimensional plates with an embedded metabarrier and a metalens are numerically designed. The metabarrier protects an interior area of the plate from the propagating waves on a wide frequency band (approx. 1.4-3.4 kHz). Equally, the refractive index profile necessary for gradient index lenses is obtained via a progressive variation of the added mass or, alternatively, the porosity of the unit cell over a rectangular area. For the first time, bending of the out-of-plane mode towards the focusing point is practically validated in a challenging mesoscale experiment requiring the assembly of different three-dimensional printed sections of the plate. The increased porosity design is advantageous not only in terms of overall lightweight, but also towards additive manufacturing as it requires less material.This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

Pragmatic structure optimization: Achieving optimal crosstalk delay and gate oxide reliability of randomly mixed CNT bundle interconnects

This study explores the potential of randomly mixed carbon nanotube bundle (RMCB) as a viable on-chip interconnect. Achieving high-quality carbon nanotubes (CNTs) with uniform diameters is challenging for the current framework of enhanced fabrication techniques. The Stoyan and Yaskov technique is employed to optimize CNT arrangement within a specified rectangular area. This method accounts for statistical variation in CNT diameters, offering a more realistic and fabrication-focused approach to designing CNT bundle interconnects. Eight such practical RMCB structures (RMCB-50 to RMCB-350) are selected using this technique, each characterized by distinct CNT counts and variable diameters. Comprehensive average crosstalk-delay and reliability assessments are conducted by comparing different CNT bundle interconnects with the best-optimized RMCB (O-RMCB) interconnect, placed on various dielectric substrates such as SiO2, SiC, BN. The study unequivocally indicates that O-RMCB produces highly favorable results and stands as the most suitable future solution for VLSI circuits. Additionally, the thickness optimization of O-RMCB interconnect is explored, yielding in improvements in both performance and reliability compared to other well-known CNT bundled interconnects.

Improved ant colony optimization algorithm based on islands type for mobile robot path planning

According to the characteristics of ant colony optimization (ACO) algorithm in mobile robot path planning, such as local optimal solution, slow convergence speed, low search efficiency, and propensity to produce numerous deadlock ants, an improved ACO algorithm based on island type (insular ACO (INACO)) is introduced. In this algorithm, firstly, several islands are established between the starting and the ending position of environment, serving as intermediate nodes for the paths searched by the ants. This greatly reduces the number of deadlock ants. Additionally, rectangular areas are initialized with non-uniform pheromone levels between adjacent islands, while other areas are set to a constant minimum pheromone value. This prevents blind searches during the initial stages of path planning. Furthermore, an adaptive volatilization coefficient is introduced into the global pheromone update rules to balance the convergence and global search ability. Finally, optimal parameter combinations of INACO are determined by simulation. INACO algorithm is simulated in various grid maps and compared with other improved ACO algorithms. Results demonstrate its superior global optimal search capability and rapid convergence speed. The average iterative times is 2.7 in a 20 × 20 grid environment and 4.7 in a 30 × 30 grid environment. Notably, INACO produces very few lost ants, with mean values of 50.1 and 95.2 in 20 × 20 and 30 × 30 grid environments, respectively.

Soft pneumatic actuators for pushing fingers into extension

BackgroundCompliant pneumatic actuators possess many characteristics that are desirable for wearable robotic systems. These actuators can be lightweight, integrated with clothing, and accommodate uncontrolled degrees of freedom. These attributes are especially desirable for hand exoskeletons, where the soft actuator can conform to the highly variable digit shape. In particular, locating the pneumatic actuator on the palmar side of the digit may have benefits for assisting finger extension and resisting unwanted finger flexion, but this configuration requires suppleness to allow digit flexion while retaining sufficient stiffness to assist extension.MethodsTo meet these needs, we designed an actuator consisting of a hollow chamber long enough to span the joints of each digit while sufficiently narrow not to inhibit finger adduction. We explored the geometrical design parameter space for this chamber in terms of shape, dimensions, and wall thickness. After fabricating an elastomer-based prototype for each actuator design, we measured active extension force and passive resistance to bending for each chamber using a mechanical jig. We also created a finite element model for each chamber to enable estimation of the impact of chamber deformation, caused by joint rotation, on airflow through the chamber. Finally, we created a prototype hand exoskeleton with the chamber parameters yielding the best outcomes.ResultsA rectangular cross-sectional area was preferable to a semi-obround shape for the chamber; wall thickness also impacted performance. Extension joint torque reached 0.33 N-m at a low chamber pressure of 48.3 kPa. The finite element model confirmed that airflow for the rectangular chamber remained high despite deformation resulting from joint rotation. The hand exoskeleton created with the rectangular chambers enabled rapid movement, with a cycle time of 1.1 s for voluntary flexion followed by actuated extension.ConclusionsThe developed soft actuators are feasible for use in promoting finger extension from the palmar side of the hand. This placement utilizes pushing rather than pulling for digit extension, which is more comfortable and safer. The small chamber volumes allow rapid filling and evacuation to facilitate relatively high frequency finger movements.

Fracture energy of birch in tension perpendicular to grain: experimental evaluation and comparative numerical simulations

The present work has experimentally determined the specific fracture energy of the hardwood species silver birch (Betula pendula), which in recent times has caught increased attention for utilization in structural applications. The single-edge-notched beam loaded in three-point-bending was utilized for evaluating the fracture energy with the work-of-fracture method. In addition to birch, Norway spruce (Picea abies) was utilized as a reference material. The effect of two different geometries of the fracture area for each species was evaluated—one triangular and one rectangular fracture area. It should be noted that the geometry of the fracture area did influence the evaluated fracture energy, and this influence was not consistent between species. This was likely in part due to manufacturing difficulties with the triangular fracture area. In addition to the experimental testing, a numerical 2d-model including linear strain-softening behavior was used for comparative simulations. The numerical 2d-models showed reasonable agreement with the experimental results regarding the global load vs. displacement response, despite their relative simple nature. The specific fracture energy for the spruce specimens was evaluated to 221 J/m2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {m}^2$$\\end{document} and for the birch specimens to 656 J/m2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {m}^2$$\\end{document}. Consequently, the present work implies a marked increase in specific fracture energy for birch, compared to spruce. This increase in specific fracture energy could potentially have a large influence on the failure behavior of birch when used in structural applications which is something that needs to be considered in future work.

Numerical simulation of laser shot peening and fatigue performance test of AA2524 thin plate

In this study, laser shot peening (LSP) is applied to 2524 aluminum alloy (AA2524). The effects of LSP parameters (the range and the position of LSPed area and LSP times) on the residual stress distribution of the specimen are analyzed using ABAQUS software. The fatigue crack growth (FCG) experiment is conducted, and the FCG of both unpeened and LSPed specimens is simulated using the extended finite element method (XFEM). Then the Paris equation is used to predict specimens’ fatigue life. The results suggest that the compressive residual stress (CRS) gain effect increases as the width perpendicular to the crack tip increases in rectangular LSPed areas of the same size. Furthermore, when comparing specimens of LSPed area situated 0 and 3 mm from the fracture tip, it is observed that the specimen of LSPed area located 5 mm away exhibits the slowest rate of crack propagation.

Optimal fare and headway for a demand adaptive paired-line hybrid transit system in a rectangular area with elastic demand

Demand adaptive paired-line hybrid transit systems that integrate fixed- and flex-route transit have emerged in the last decade and attracted increasing attention because of their potential to improve accessibility for passengers. To facilitate the operation of such a hybrid transit system, this study develops a model to determine the optimal fare and headways associated with fixed- and flex-route transit along a rectangular corridor. Compared with existing literature, the novelty of this study lies in designing the fare structure while simultaneously considering demand elasticity and passenger behaviour. A continuous approximation modelling approach is employed to derive the agency’s and travellers’ cost components. Using these, a nonlinear programming optimisation model is formulated to minimise the total user cost subject to the agency’s nonnegative revenue constraints and passengers’ route choice behaviour, which is characterised as a path-size logit model. Numerical experiments are performed using a stylised network to examine the properties of the model, in which the solution is obtained by combining a brute force method that enumerates headway and fare combinations and an iterative method that determines equilibrated passenger choices. The results show that as potential demand density increases, fare and headway fluctuate and drop, while the percentage of passengers choosing to ride on flex-route transit increases. In addition, there may be an optimal maximum offset distance, which is defined as the width of the corridor to be covered by the transit system, when the potential demand density is low, leading to minimum user cost and maximum travel demand within the service area.

Crustal thickness variations beneath Egypt through gravity inversion and forward modeling: linking surface thermal anomalies and Moho topography

This study aims to quantify the topography of the Moho boundary, the lower crust and uppermost mantle contact of Egypt, in order to estimate the crustal thickness variation and its link to the distribution of thermal anomalies under Egypt. This is accomplished by modeling satellite gravity, supported by the passive seismic constraints throughout Egypt. However, when estimating the thickness of the crust in Egypt using just seismic data, substantial uncertainty and deviation are produced due to the sparsely dispersed stations. Integrating seismic and gravity data minimizes uncertainty and improves estimate accuracy. The investigation is broken down into four stages, the first involving utilizing the Sentinel-3B satellite to create land surface temperature maps. The subsequent steps consist of gravity and seismic data adjustments, inversion and forward modeling. We used seismically restricted nonlinear inversion to look at Goco06s satellite gravity data to model the Moho’s topographic surface. The data gathered from deep seismic refraction and receiver functions adjusted the analyzed data. The inversion process relies on the adapted Bott's approach and Tikhonov regularization, using the assumption of the sphericity of the Earth planet. Reference values for depth of Moho and density contrast were set at 35 km and 500 kg/m3, respectively. The average statistical difference for Moho depth between gravity-based model and seismic data is − 0.10 km. Through forward gravity modeling, five gravity profiles were chosen and interpreted in 2.5D models. The results indicated that the Moho depth in the south varies from 35 to 39 km and decreases in the north and the Mediterranean. In upper Egypt, the highest Moho depth is 39 km. The depth varies beneath the Sinai Peninsula as it is about 35 km in its south, reaches 30 km in the northern portion, and ranges along the Red Sea’s Rift Margin from 29 to 32 km. Moreover, the final model shows the relation between Moho coincides with the surface temperature anomalies approved by satellite images and hot springs. The model reveals a correlation between Moho discontinuity and surface temperature anomalies, revealing the highest geothermal potential in a rectangular area in central Egypt, between latitudes 25°N and 30°N, based on satellite imagery and hot springs distribution.

Clustering-based interactive image segmentation

Objectives. The task of color image segmentation without the use of preliminary training is considered. It arises, for example, when it is necessary to perform image segmentation with semantic and color properties unknown in advance immediately after their acquisition, or when the set of images intended for segmentation is too small, as well as when performing preliminary "exploratory" analysis of images. In such cases, powerful neural network and other segmentation tools that require deep learning can not be used.Methods. An algorithm for interactive image segmentation is proposed, based on the analysis of the colors of areas selected interactively. First, in interactive mode, the image areas belonging to the objects are selected very approximately, and then regions belonging to the background are chosen. In the next step, the set of colors of the selected object areas and the set of colors of the selected background areas are clustered separately by one of the clustering algorithms, for example, k-means, fuzzy c-means, or the multi-level clustering algorithm proposed by the author. After this, non-informative elements are removed from the set of cluster centers describing the objects and the set of clusters presenting the background. The modified sets of object and background cluster centers are used for image segmentation.Results . The constructed algorithm allows selection of the required objects in color images if the colors of the objects and the background are different. Interactive selection of object areas and background areas does not require accuracy or much effort and usually takes several tens of seconds. For selection, rectangular areas that lie entirely inside the object images, and rectangular areas that belong completely to the background can be used. Below an example of interactive regions selection and color image segmentation is shown.Conclusion. The experiments performed showed the effectiveness of the proposed approach to segmenting color images. It can be used in cases where the semantic and color properties of images are not known in advance, and in cases where the use of more powerful deep learning methods, including neural networks, is too expensive or impossible.

What Prevents Us From Learning Virtual Reality Space: A Study of Indoor Corridor Angle and Obstacle Transparency

People sometimes become disorientated and feel unnatural in virtual reality (VR) spaces created via devices such as head-mounted displays. The main problem is that multiple structures make it difficult to learn the VR space, particularly indoors. These structures segment the space and form boundaries for sightlines and activity areas. How obstacles in segmented space affect human behavior have been less discussed in studies related to navigation. This study sought to clarify the mechanism for the effect of spatial segmentation effect on human spatial cognition, in the context of indoor navigation. This study was conducted in a VR scenario with an omni-directional treadmill; a total of 76 college students participated, and 52 of them were included in the analysis. The study found that the cognitive distortion from spatial segmentation primarily operates by activity area rather than visual obstruction, and it can be inferred that the basis of human abstraction of spatial information may perhaps be the activity area. Regardless of obstacle transparency, non-rectangular spatial segmentation deteriorates navigational performance. The results suggest that, future metaverse communities should consider designing rectangular segmented activity areas to improve navigation performance and reduce disorientation.

Research on survey line arrangement of a multibeam bathymetric system in ocean mapping

Submarine topography measurement technology is an important technology for exploring marine resources. Multi-beam bathymetry systems have high efficiency and important application value in underwater resource surveys, marine delimitation, and other fields. They are currently one of the most widely used measurement technologies. This study aims to optimize multi-beam bathymetry technology and apply it to seabed topography measurement, achieving a transition from point measurement to line measurement and minimizing measurement paths. A mathematical model was established for the coverage width and overlap rate between adjacent bands of multi-beam bathymetry at a given water depth based on planar geometric modeling and three-dimensional geometric modeling. The model was applied to a rectangular sea area with a length of 2 nautical miles (3704 meters) from north to south and a width of 4 nautical miles (7408 meters) from east to west. The optimal survey line planning was obtained as follows: all survey lines are in the north-south direction, with a total of 34 survey lines and a total length of 68 nautical miles. This model has good solution results for optimizing the layout of survey lines under different seabed conditions and can adjust important parameters of the system according to different conditions. The model has broad practical significance and can provide a reference for ocean-sounding work and other fields with high accuracy.

Fluid–structure interaction of a marine rudder at incidence in the wake of a propeller

The structural response of a rudder in the wake of a marine propeller is investigated by one-way fluid–structure interaction approach. The unsteady pressure field gathered by detached eddy simulations is provided to a structural solver for the computation of deformations and stresses of the rudder. The study compares the structural response of the rudder at neutral and two equal and opposite rotations, which are representative of design conditions in straight motion and maneuvering conditions that are experienced under the action of the autopilot for course control or weak maneuvering. The analysis sheds light on the different structural behavior at the two opposite rotation angles, caused by asymmetrical variation along the span of the rudder of the angle of attack induced by the propeller slipstream, by considering the different role played by the tip and hub vortical systems. The test case consists of a rudder with a rectangular plane area and National Advisory Committee for Aeronautics 0015 sectional profile located past the E779A propeller. The propeller operates at low loading conditions, and the rudder is set at incidence δ=0∘,±4∘. The study shows that the response of the rudder is driven by flap and torsion and is asymmetric for the two and opposite rotations. The mean deformation and vibratory response are magnified for δ=−4∘ by at least 70% and 20% for the lateral and edgewise deflections, respectively, with respect to the opposite rudder incidence. In general, the excitation generated by the tip vortex is stronger than that of the hub vortex. In the most critical condition, at δ=−4∘, the excitation associated with the tip vortex is nearly double that of the hub vortex.

Calculation of Plates of Different Shapes Using Generalized Equations of the Finite Difference Method

Statement of the problem. The article reviewed nonaxisymmetric tasks for calculating slabs in shape of a circle sector, wedge and a slab formed by a connection of rectangular and circular areas. The solution is based on generalized equations of the finite differences method. The algorithm allows us to take into account finite discontinuities of the desired function, the right side of the original differential equations, as well as discontinuities of derivatives of these functions without accumulation of the grid and using points outside the contour. It resolves oneself into a system of differential algebraic equations generated for field and contour points. In this paper the initial differential equations and their numerical approximation are provided. Results. Calculations of a semicircle and a wedge under different boundary conditions are presented, as well as an uncut slab in the form of a semicircle with an annular support and a composite slab are consi-dered. The solution of the above problems has been carried out on the minimum possible computational grid, comparison with the available solutions has been provided, and static checks of the calculation have been carried out. Conclusions. Along with the most common numerical method for calculating buildings and structures, it is possible and quite effective to use these equations. It is proved by means of some examples that such an approximation has a number of advantages: the results allow us to talk about the sufficient accuracy of the solution with a small number of partitions; fundamental simplicity and universality of the method.

Minimal number of sensors for 3D coverage

This paper presents some theoretical results on the smaller number Nk(a, b, c) of sensors to achieve k coverage for the 3D rectangular area [0, a] × [0, b] × [0, c]. The first properties outline some theoretical results for the numbers Nk(a, b, c), including symmetry, subadditivity, and monotony on each variable. We use then these results to establish some lower and upper bounds for Nk(a, b, c). The main contribution proposes a result concerning the minimal density of sensors to achieve k-coverage.