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Large values and gradients of stress and strain, triggering concentrated stress and strain, arise in angular areas of a structure. The strain action, leading to the finite loss of contact between structural elements, also triggers concentrated stress. The loss of contact reaches an irregular point and a line on the boundary. The theoretical analysis of the stress–strain state (SSS) of areas with angular cutouts in the boundary under the action of discontinuous strain is reduced to the study of singular solutions to the homogeneous problem of elasticity theory with power-related features. The calculation of stress concentration coefficients in the domain of a singular solution to the elastic problem makes no sense. It is experimentally proven that the area located near the vertex of an angular cutout in the boundary features substantial strain and rotations, and it corresponds to higher values of the first and second derivatives of displacements along the radius in cases of sufficiently small radii in the neighborhood of an irregular boundary point. As far as these areas are concerned, it is necessary to consider the plane problem of the elasticity theory, taking into account the geometric nonlinearity under the action of strain, to analyze the effect of relationships between strain orders, rotations, and strain on the form of the equation of equilibrium. The purpose of this work is to analyze the effect of relationships between strain orders, rotations, and strain on the form of the equilibrium equation in the polar system of coordinates for a V-shaped area under the action of temperature-induced strain, taking into account geometric non-linearity and physical linearity.

In corner areas of structures, high stress values and gradients occur, and lead to stress concentrations. Infinite stress and deformations are determined by a solution of the linear elasticity theory problem in the area with a wedge-shape boundary notch. Infinite solutions of the elasticity problem occur under impact of forced deformations, when a surge of the deformation value reaches beyond the area boundary. Relative values of stress concentrations for corner area zones make no more sense. At finite displacements, high deformation and stress values occur in the corner zones of the area. For a linear statement of the elasticity theory problem, at minor deflections, not only first-order, but also second-order derivatives of the displacements function are significant. To account for finite deformations of such corner zones of the area, correct formulations of elasticity problems are required. Study objective: influence determination of the infinitesimal order of the deformation on the appearance of equilibrium equations of an area with induced (temperature) deformations. This allows for the analysis of the influence of linear, shear deformations, and of the swing on the solution of the elasticity problem with induced deformations.

Density peaks clustering (DPC) algorithm provides an efficient method to quickly find cluster centers with decision graphs. In recent years, due to its unique parameters, no iteration, and good robustness, it has been widely studied and applied. However, it also has some shortcomings, such as no adaptability, inadaptability to high-dimensional data and accuracy is easily affected. For reducing the higher time complexity of DPC, we introduce the polar coordinates to DPC (PC-DPC). Firstly, obtain the distance from every point to third-party point and the cosine value of the angle formed with the third-party vector, and reorder the points by distances and cosine values. Then, select other points in the adjacent sequence number of each point to calculate distances, and build a sparse distance matrix. Finally, the sparse distance matrix is used as the input of DPC to obtain clustering results. Theoretical analysis and experiments show that, compared with DPC and other algorithms, PC-DPC greatly reduces running time of DPC while maintaining clustering precision.

Application of higher order finite element to the analysis of thin solid circular plates on elastic foundation using the modified Vlasov model in polar coordinate system is attempted. A Matlab code has been written for present formulation that can handle any type of loading and boundary conditions. Validation of the code is carried out after the convergence study. The results are compared to those obtained by other researchers. It is concluded that the present formulation behaves extremely well for thin solid circular plate resting on elastic foundations with good convergence rate and accuracy. The present plate bending element resting on elastic foundation using a higher order finite element is simple and requires less computational efforts, resources and time.

Airborne LiDAR (ALS) and terrestrial LiDAR (TLS) data integration provides complementary perspectives for acquiring detailed 3D forest information. However, challenges in registration arise due to feature instability, low overlap, and differences in cross-platform point cloud density. To address these issues, this study proposes an automatic point cloud registration method based on the consistency of the single-tree position distribution in multi-species and complex forest scenes. In this method, single-tree positions are extracted as feature points using the Stepwise Multi-Form Fitting (SMF) technique. A novel feature point matching method is proposed by constructing a polar coordinate system, which achieves fast horizontal registration. Then, the Z-axis translation is determined through the integration of Cloth Simulation Filtering (CSF) and grid-based methods. Finally, the Iterative Closest Point (ICP) algorithm is employed to perform fine registration. The experimental results demonstrate that the method achieves high registration accuracy across four forest plots of varying complexity, with root-mean-square errors of 0.0423 m, 0.0348 m, 0.0313 m, and 0.0531 m. The registration accuracy is significantly improved compared to existing methods, and the time efficiency is enhanced by an average of 90%. This method offers robust and accurate registration performance in complex and diverse forest environments.

Defect pattern analysis of wafer bin maps (WBMs) is an important means of identifying process problems. Recently, automated analysis methods using machine learning or deep learning have been studied as alternatives to manual classification by engineers. In this paper, we propose a method to improve the feature extraction performance of defect patterns by transforming the polar coordinate system instead of the existing WBM image input. To reduce the variability of the location representation, defect patterns in the Cartesian coordinate system, where the location of the distributed defect die is not constant, were converted to a polar coordinate system. The CNN classifier, which uses polar coordinate transformed input, achieved a classification accuracy of 91.3%, which is 4.8% better than the existing WBM image-based CNN classifier. Additionally, a tree-structured classifier model that sequentially connects binary classifiers achieved a classification accuracy of 94%. The method proposed in this paper is also applicable to the defect pattern classification of WBMs consisting of different die sizes than the training data. Finally, the paper proposes an automated pattern classification method that uses individual classifiers to learn defect types and then applies ensemble techniques for multiple defect pattern classification. This method is expected to reduce labor, time, and cost and enable objective labeling instead of relying on subjective judgments of engineers.

Modulation format identification (MFI) is one of the most critical functions embedded in digital coherent receivers in elastic optical networks (EONs). In view of inherent amplitude and phase characteristics of received signals, different modulation formats exhibit a set of notable features in the polar coordinate system, based on which an MFI scheme incorporating the Gaussian weighted k-nearest neighbors (KNN) algorithm was proposed to identify polarization division multiplexed (PDM)-QPSK/-16QAM/-32QAM/-64QAM/-128QAM signals. The performance of the proposed scheme was numerically verified in 28GBaud coherent optical communication systems. The numerical simulation results show that, to achieve 100% correct identification rates for all of the five modulation formats, the required minimum optical signal-to-noise ratios (OSNRs) were less than their relevant thresholds corresponding to the 20% forward error correction (FEC). The tolerable ranges of the residual chromatic dispersion (CD) for QPSK, 16QAM, 32QAM, 64QAM, and 128QAM were −1920 ps/nm~1920 ps/nm, −720 ps/nm~360 ps/nm, −1200 ps/nm~1680 ps/nm, −600 ps/nm~360 ps/nm, and −600 ps/nm~480 ps/nm, respectively. Meanwhile, the results demonstrate the maximum tolerable differential-group delay (DGD) for the QPSK, 16QAM, 32QAM, 64QAM, and 128QAM signals were 34 ps, 16 ps, 20 ps, 6 ps, and 1.2 ps, respectively. In addition, the simulated results also show that the proposed MFI scheme is robust against the fiber nonlinearities, even if the launch power is increased to 4 dBm.

To obtain the distribution rules of pore water pressure for twin shallow circular tunnels, the shallow circular tunnels were regarded as fully saturated, homogeneous, isotropic structures in a semi-infinite space. Coordinate transformations were utilized to obtain the water pressure control equations in a bipolar coordinate system. Through the Schwarz alternating method, the calculation method for the pore water pressure around a single tunnel was generalized to twin tunnels, and a new method for determining the pore water pressure around twin shallow circular tunnels in a semi-infinite space was proposed for the first time by using the bipolar coordinate system method and the Schwarz alternating method. Solutions for the pore water pressure were obtained via multiple iterations. The calculation results obtained via the proposed method were compared with the other theoretical calculation results and numerical simulation results, and we further utilized the proposed method to analyse the influencing factors of the pore water pressure. The results show that the maximum error between the theoretical calculation results and the numerical simulation results was only 2.15%; thus, the rationality of this new method was effectively verified. The relative error between the results obtained by the proposed method and the results obtained by the existing method was only 1.0%, and both very well matched the numerical simulation results; thus, the high accuracy of the proposed method was confirmed. The number of tunnels and the centre distance between the twin tunnels are the main influencing factors of the pore water pressure. According to their influences, the distribution pattern of the pore water pressure around the twin shallow circular tunnels can be preliminarily determined. Therefore, the method proposed in this study provides a theoretical guidance for the development of design and construction programs for water-rich tunnel engineering.

This study sought to examine the impact of dynamic mathematics software, GeoGebra, on university mathematics students' understanding of polar coordinates. A quasi-experimental design with an equivalent group of 42 participants in each group was used. A simple random sampling approach was used to select the 84 participants from the population and a purposive sampling technique was used to assign the participants to the various groups. A t-test analysis was conducted on the pre-test and post-test activities. It was revealed that students who were taught polar coordinates with the aid of GeoGebra performed better than those who were taught with the conventional approach. Analysis of students in the experimental group's responses to the questionnaire showed that they demonstrated positive attitudes and perceptions concerning the use of GeoGebra software in the learning of polar coordinates. It was concluded that GeoGebra is more effective in improving university mathematics students' understanding of polar coordinates.

We introduce a new nonstationary spatial covariance model for analyzing geostatistical point-referenced data that contain point sources (i.e., known locations that impact the outcome). Our model is based on viewing the spatial domain on the polar coordinate scale, with the point source representing the reference location. As a result, we incorporate distances from the point source and angles of the separation vector with respect to the point source into the covariance model definition in order to describe complex correlation patterns that may be induced by the point source. We apply the new model and several competing options to analyze the impact of a hog lot on house sales prices in Cedar Falls, Iowa. We find that the new model offers improved model fit and predictive ability through Watanabe–Akaike information criterion and cross-validation, respectively. Additionally, we design a simulation study to determine the impact that mean misspecification has on each model’s ability to produce quality predictions. Overall, the new model is shown to consistently outperform the competitors and is useful even when the point source has no impact on the outcome.