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Due to the tedious and expensive nature of annotating data for medical image segmentation tasks, semi-supervised learning (SSL) methods utilizing a small amount of labeled data have gained widespread attention. However, most existing methods overlook the importance of boundary regions in multi-class tasks and are sensitive to incorrect pseudo-label. In this paper, a novel Dual-Cycled Boundary Refine Network (DCBR-Net) is presented, consisting of two simple yet slightly different segmentation networks and a Boundary Residual Refine (BRR) module. First, a new boundary refine method for the semi-supervised medical image segmentation field is designed, that is a BRR module with a residual architecture. This module is capable of enhancing the representation of boundaries while ensuring the quality of inner regions. Besides, a dual-cycled pseudo-label scheme is designed to train the unlabeled data. Through providing diverse outputs and achieving double supervision for the result, the issue of ineffective guidance caused by model consistency after multiple iterations is alleviated. Furthermore, a novel dynamic loss function is developed based on the prediction disagreement between different models, which can suppress the influence of incorrect pseudo-labels. Extensive experiments conducted on four public medical datasets demonstrate that our network can achieve competitive results, especially with higher reliability in boundary regions. On the ACDC, LA, and Fundus datasets, with only a 20% labeled ratio, our network achieves DSC scores of 90.53%, 90.40%, and 89.19%, respectively, which are comparable to fully supervised performance.

Agricultural greenhouses (AGs) are an important component of modern facility agriculture, and accurately mapping and dynamically monitoring their distribution are necessary for agricultural scientific management and planning. Semantic segmentation can be adopted for AG extraction from remote sensing images. However, the feature maps obtained by traditional deep convolutional neural network (DCNN)-based segmentation algorithms blur spatial details and insufficient attention is usually paid to contextual representation. Meanwhile, the maintenance of the original morphological characteristics, especially the boundaries, is still a challenge for precise identification of AGs. To alleviate these problems, this paper proposes a novel network called high-resolution boundary refined network (HBRNet). In this method, we design a new backbone with multiple paths based on HRNetV2 aiming to preserve high spatial resolution and improve feature extraction capability, in which the Pyramid Cross Channel Attention (PCCA) module is embedded to residual blocks to strengthen the interaction of multiscale information. Moreover, the Spatial Enhancement (SE) module is employed to integrate the contextual information of different scales. In addition, we introduce the Spatial Gradient Variation (SGV) unit in the Boundary Refined (BR) module to couple the segmentation task and boundary learning task, so that they can share latent high-level semantics and interact with each other, and combine this with the joint loss to refine the boundary. In our study, GaoFen-2 remote sensing images in Shouguang City, Shandong Province, China are selected to make the AG dataset. The experimental results show that HBRNet demonstrates a significant improvement in segmentation performance up to an IoU score of 94.89%, implying that this approach has advantages and potential for precise identification of AGs.

A new asymptotic approximation of the n th order is proposed for use in calculations of radiation propagation in optically thick media without scattering; the asymptotic approximation is simpler and more accurate than the well-known diffusion approximation. It is shown that for optically thick media the asymptotic solution of the kinetic equation of radiation propagation without scattering is an asymptotic expansion of the exact integral solution of this kinetic equation. A rigorous derivation of the diffusion approximation equation is obtained. Refined boundary conditions that are important for practical application in calculations of radiation propagation are derived.

Galin’s solution for the problem of biaxial tension of a plate with a hole completely covered by the plastic region appears to be a pearl recognized by the world scientific community. This solution serves as a test for all sorts of approximate approaches to solving elastoplastic problems, including the semi-analytical iterative method being developed by the author, focused on solving more complex problems such as the Kirsch problem in the elastoplastic formulation. The proposed iterative approach for a semi-analytical solution involves an explicit analytical expression for stresses in the plastic region and an iterative numerical solution in the elastic region with a refined border. The paper shows the convergence of the results based on the iterative procedure for the elastoplastic region boundary approaching its analytical position, which follows from the analytical solution of Galin’s elastoplastic problem. Consideration has also been given to obtaining results on the determination of the boundary between the elastic and plastic regions using a competing approximate perturbation method. The advantage of the proposed method lays in not limited modifications in parameters due to the requirement for small differences while formulating a problem from the axisymmetric case as seen in the perturbation method.

A method with optimal (up to logarithmic terms) complexity for solving elliptic problems is proposed. The method relies on interior regularity, but the solution may have globally low regularity due to rough boundary data or geometries. Elliptic regularity results, high order approximation results, and an efficient preconditioner are presented. The method is utilized to realize, with linear-logarithmic complexity, an accurate and data-sparse approximation to the associated elliptic Poincare--Steklov operators. Further applications include the treatment of exterior boundary value problems and the solution of problems in the framework of domain decomposition methods.

The boundary concentrated FEM, a variant of the hp-version of the finite element method, is proposed for the numerical treatment of elliptic boundary value problems. It is particularly suited for equations with smooth coefficients and non-smooth boundary conditions. In the two-dimensional case it is shown that the Cholesky factorization of the resulting stiffness matrix requires O(Nlog4N) units of storage and can be computed with O(Nlog8N) work, where N denotes the problem size. Numerical results confirm theoretical estimates.

In this article, isogeometric analysis (IGA) based on the modified nonlocal couple stress theory (MNCST) is introduced to study bending and free vibration characteristics of functionally graded (FG) nanoplates placed on an elastic foundation (EF). The MNCST is a combination of nonlocal elasticity theory and modified couple stress theory to capture the small-size effects most accurately, hence this theory considers both softening and stiffening effects on responses of FG nanoplates. A higher order refined plate theory is adapted, because it satisfies parabolic distributions of transverse shear stresses across the nanoplate thickness and equals zero at the top and bottom surfaces without requiring shear correction factors. The governing equations are obtained using Hamilton's principle from which deduce the equations determining the natural frequency and displacement of the FG nanoplates. Several comparison studies are conducted to verify the proposed model with other results in the literature. Furthermore, the influence of nonlocal parameters, material length parameters, boundary conditions, material volume exponent on the bending, and free vibration response of FG nanoplates are fully studied.

A second order accurate (in time) numerical scheme is proposed and analyzed for the Poisson–Nernst–Planck equation (PNP) system, reformulated as a non-constant mobility H - 1 gradient flow in the Energetic Variational Approach (EnVarA). The centered finite difference is taken as the spatial discretization. Meanwhile, the highly nonlinear and singular nature of the logarithmic energy potentials has always been the essential difficulty to design a second order accurate scheme in time, while preserving the variational energetic structures. The mobility function is updated with a second order accurate extrapolation formula, for the sake of unique solvability. A modified Crank–Nicolson scheme is used to approximate the logarithmic term, so that its inner product with the discrete temporal derivative exactly gives the corresponding nonlinear energy difference; henceforth the energy stability is ensured for the logarithmic part. In addition, nonlinear artificial regularization terms are added in the numerical scheme, so that the positivity-preserving property could be theoretically proved, with the help of the singularity associated with the logarithmic function. Furthermore, an optimal rate convergence analysis is provided in this paper, in which the higher order asymptotic expansion for the numerical solution, the rough error estimate and refined error estimate techniques have to be included to accomplish such an analysis. This work combines the following theoretical properties for a second order accurate numerical scheme for the PNP system: (i) second order accuracy in both time and space, (ii) unique solvability and positivity, (iii) energy stability, and (iv) optimal rate convergence. A few numerical results are also presented.

In this article, a mathematical analysis of thermoelastic skin tissue is presented based on a refined dual-phase-lag (DPL) thermal conduction theory that considers accounting for the effect of multiple time derivatives. The thin skin tissue is regarded as having mechanically clamped surfaces that are one-dimensional. Additionally, the skin tissue undergoes ramp-type heating on its outer surface, whereas its inner surface keeps the assessed temperature from vanishing. Some of the previous generalized thermoelasticity theories were obtained from the proposed model. The distributions of temperature, displacement, dilatation, and stress are attained by applying the Laplace transform and its numerical reversal approaches. The outcomes are explicitly illustrated to examine the significant influences on the distributions of the field variables. The refined DPL bioheat conduction model in this study predicts temperature, and the findings revealed that the model is located among the existing generalized thermoelastic theories. These findings offer a more thorough understanding of how skin tissue behaves when exposed to a particular boundary condition temperature distribution.

In this article, we present a mathematical model of thermoelastic skin tissue based on a refined Lord–Shulman heat conduction theory. A small thickness of skin tissue is considered to be one-dimensional with mechanical clamped surfaces. In addition, the skin tissue’s outer surface is subjected to ramp-type heating while its inner surface is adiabatic. A simple Lord–Shulman theory, as well as the classical coupled thermoelasticity, are also applied in this article. Laplace transform techniques and their inversions are calculated to return to the time domain. Numerical outcomes are represented graphically to discuss the significant impacts on the temperature, dilatation, displacement, and stress distributions. Such results provide a more comprehensive and better insight for understanding the behavior of skin tissue during the temperature distribution of a specific boundary condition.