Boundary Problem Research Articles

Global well‐posedness for the one‐phase Muskat problem

AbstractThe free boundary problem for a two‐dimensional fluid permeating a porous medium is studied. This is known as the one‐phase Muskat problem and is mathematically equivalent to the vertical Hele‐Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global‐in‐time Lipschitz solution in the strong sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.

Invadopodia Formation in Cancer Cell: The Mathematical and Computational Modelling Based on Free Boundary Problem

We present a mathematical model of an individual cell to expand the simulation of invadopodia formation to a three-dimensional (3D) domain for a more realistic complexity. Simulating invadopodia replication in order for it to be biologically relevant is important since it helps us to understand cancer invasion and metastasis better as well as giving some insight into investigating ways to stop the spread of this fatal disease. Invadopodia formation is formulated using the Stefan problem approach, where the free boundary is characterised by the Stefan free boundary condition, in which the boundary membrane is not known in advance. Level set method is proposed to indicate the behaviour of the cell interface and the motion of the plasma membrane. An enthalpy method (phase-transition problem) is used to describe the cell membrane diffusion. In addition to this, we were able to improve the simulation outcome, giving it a more realistic complexity by using a different simulation technique and domain as well as a different data set. Singularities and instabilities were eliminated. The results that were achieved have the potential to be helpful for novel approaches or to be extended to other methods in the development of a more accurate numerical simulation.

Discriminative feature constraints via supervised contrastive learning for few-shot forest tree species classification using airborne hyperspectral images

In scenarios where sample collection is limited, studying few-shot learning algorithms such as prototypical networks (P-Net) is a keynote topic for supervised multiple tree species classification. In a previous study, we improved the P-Net by combining the feature enhancement algorithm based on the convolutional block attention module and several popular data augmentation methods in the computer vision domain, the classification accuracy can be significantly increased, and the degree of model overfitting can be reduced. However, there is a clear boundary between the data augmentations and the feature enhancement algorithm, which is manifested in that data augmentations are only used to enrich the diversity of the learned samples, but cannot directly affect the construction of the objective function, thus limiting the ability of data augmentations. In fact, in the supervised contrastive learning research, data augmentation methods are often used to generate positive samples of an anchor image to construct the objective function, i.e. supervised contrastive loss. The core idea for solving such a boundary problem is to use contrastive learning to make the anchor image close to its positive samples and the negative samples away from each other. Inspired by this, we introduced supervised contrastive learning in the P-Net, i.e., SCL-P-Net, which takes the discriminative feature representations as the constraints of the prototype clustering algorithm. In SCL-P-Net, data augmentation methods can not only extend the sample distribution, but also be used to construct the supervised contrastive loss directly. The study involves four airborne hyperspectral image datasets related to tree species classification, including the GFF-A and GFF-B datasets collected from Gaofeng Forest Farm in Nanning City, Guangxi Province, South China, the Xiongan dataset from Matiwan Village in Xiongan New Area, Hebei Province, North China, and the Tea Farm dataset from Fanglu Tea Farm in Changzhou City, Jiangsu Province, East China. The highest overall accuracy (OA) for the four datasets is 99.23% for GFF-A, 98.39% for GFF-B, 99.30% for Xiongan, and 99.54% for Tea Farm. It is concluded that the proposed SCL-P-Net classification framework can achieve multiple tree species classification with high-precision. Without changing the basic classification framework of P-Net, the introduction of supervised contrastive learning makes the combination of the data augmentations and the feature enhancement algorithm and plays a positive role in improving the distinguishability between samples.

Global existence and asymptotic stability of the free boundary problem of the primitive equations with heat insulation

In this paper, we deal with the free boundary value problem of the three-dimensional full primitive equations for the large-scale ocean with heat insulation in a bounded domain. The domain is periodic in the horizontal direction and finite in the vertical direction, and its upper boundary is a free surface driven by the fluid and bottom boundary is fixed. We show that the unique strong solution of the free boundary value problem for the three-dimensional full primitive equations exists globally in time and converges to the equilibrium state with exponential decay rate, when the initial data is a small perturbation of the steady state in regular Sobolev space.

Integrative taxonomy and revision of the Neotropical Biomphalaria tenagophila (Mollusca: Planorbidae) complex taxa

ABSTRACT Morphological and molecular studies of the taxa of the Biomphalaria tenagophila complex were conducted on specimens from Argentina, Brazil, Paraguay, Peru, Uruguay and Venezuela. The morphological analysis included aspects of the shell, mantle, excretory organs and reproductive system. DNA barcode, ABGD and mPTP were used to group specimens into Operational Taxonomic Units (OTUs) and in attempt to solve species boundary delimitation problems within the Biomphalaria tenagophila complex. Population analyzes were also performed based on the cytochrome C oxidase I (COI) and 16S genes. The seminal vesicle, extension of the oviduct in relation to the distal ovispermioduct, vaginal pouch, aspect of the penial complex, aspect of the prepuce diaphragm, diaphragm papilla, extension of the penial sheath in relation to the prepuce and prepuce dimensions are essential for the determination of the taxa of the Biomphalaria tenagophila complex. An updated molecular phylogeny for Biomphalaria based on COI, 16S and ITS2 genes is presented and recover Biomphalaria tenagophila complex as monophyletic. The morphological, population and phylogenetic data observed in an integrated way define taxonomic questions, such as the proposition of a new status for B. guaibensis, which integrates into the Biomphalaria tenagophila complex. http://www.zoobank.org/urn:lsid:zoobank.org:pub:478145E5-D4A5-4128-AA80-DB6A69164E1E

A free boundary problem for the p-Laplacian with nonlinear boundary conditions

We study a nonlinear generalization of a free boundary problem that arises in the context of thermal insulation. We consider two open sets Omega subseteq A, and we search for an optimal A in order to minimize a nonlinear energy functional, whose minimizers u satisfy the following conditions: Delta _p u=0 inside A{setminus }Omega , u=1 in Omega , and a nonlinear Robin-like boundary (p, q)-condition on the free boundary partial A. We study the variational formulation of the problem in {{,textrm{SBV},}}, and we prove that, under suitable conditions on the exponents p and q, a minimizer exists and its jump set satisfies uniform density estimates.

Physics-informed neural networks for mesh deformation with exact boundary enforcement

In this work, we have applied physics-informed neural networks (PINN) for solving mesh deformation problems. We used the collocation PINN method to capture the new positions of the vertex nodes while preserving the connectivity information. We use linear elasticity equations for mesh deformation. To prevent vertex collisions or edge overlap, the mesh movement in this work is conducted in steps with relatively small movements. For moving boundary problems, the exact position of the boundary is essential for having an accurate solution. However, PINNs are frequently unable to satisfy Dirichlet boundary conditions exactly. To overcome this issue, we have used hard boundary condition enforcement to automatically satisfy Dirichlet boundary conditions. Specifically, we first trained a PINN with soft boundary conditions to obtain a particular solution. Then, this solution was tuned with exact boundary positions and a proper distance function by using a new PINN considering only the equation residual. To assess the accuracy of our approach, we used the classical translation and rotation tests and compared them with a proper mesh quality metric considering the change in the element area and shape. The results show the accuracy of this approach is comparable with that of finite element solutions. We also solved different moving boundary problems, resembling commonly used fluid–structure interaction problems. This work provides insight into using PINN for mesh-deformation problems without needing a discretization scheme with reasonable accuracy.

GIPC-GAN: an end-to-end gradient and intensity joint proportional constraint generative adversarial network for multi-focus image fusion

As for the problems of boundary blurring and information loss in the multi-focus image fusion method based on the generative decision maps, this paper proposes a new gradient-intensity joint proportional constraint generative adversarial network for multi-focus image fusion, with the name of GIPC-GAN. First, a set of labeled multi-focus image datasets using the deep region competition algorithm on a public dataset is constructed. It can train the network and generate fused images in an end-to-end manner, while avoiding boundary errors caused by artificially constructed decision maps. Second, the most meaningful information in the multi-focus image fusion task is defined as the target intensity and detail gradient, and a jointly constrained loss function based on intensity and gradient proportional maintenance is proposed. Constrained by a specific loss function to force the generated image to retain the information of target intensity, global texture and local texture of the source image as much as possible and maintain the structural consistency between the fused image and the source image. Third, we introduce GAN into the network, and establish an adversarial game between the generator and the discriminator, so that the intensity structure and texture gradient retained by the fused image are kept in a balance, and the detailed information of the fused image is further enhanced. Last but not least, experiments are conducted on two multi-focus public datasets and a multi-source multi-focus image sequence dataset and compared with other 7 state-of-the-art algorithms. The experimental results show that the images fused by the GIPC-GAN model are superior to other comparison algorithms in both subjective performance and objective measurement, and basically meet the requirements of real-time image fusion in terms of running efficiency and mode parameters quantity.

Editorial: Moving boundary problems in multi-physics coupling processes

Editorial: Moving boundary problems in multi-physics coupling processes

Some global topological properties of a free boundary problem appearing in a two dimensional controlled ruin problem

AbstractIn this paper a two-dimensional Brownian motion (modeling the endowment of two companies), absorbed at the boundary of the positive quadrant, with controlled drift, is considered. The volatilities of the Brownian motions are different. We control the drifts of these processes and allow that both drifts add up to the maximal value of one. Our target is to choose the strategy in a way, s.t. the probability that both companies survive is maximized. It turns out that the state space of the problem is divided into two sets. In one set the first company gets the full drift, and in the other set the second one. We describe some topological properties of these sets and their separating curve.

Dynamics of nonlocal and local SIR diffusive epidemic model with free boundaries

This paper is concerned with nonlocal and local diffusive SIR epidemic model with free boundaries including convolution, which is natural extension of reaction diffusion systems with free boundary problems and local diffusions. The existence of unique global solution for this model is considered. Dichotomy of the spreading and vanishing is established. A spreading barrier line is found to determine whether the spreading of disease will fail finally. The spreading of disease will fail when it cannot spread across the spreading barrier line l∗, while it will be successful when it transcends over this barrier line. The results show that if the basic reproduction number R0<1, the spreading of disease will fail eventually, and if R0>1+d2μ2+α, the spreading of disease will get success finally. We also find that the spreading coefficients play important role in the spreading achievement. When 1+βθμ1<R0<1+d2μ2+α, the spreading coefficient decides whether the spreading of disease will be successful. It is shown that the spreading will be successful when the spreading coefficient is relatively big, while the spreading will fail if the spreading coefficient is relatively small.

Boundary value problems of integrodifferential equations under boundary conditions taking into account physical nonlinearity

When solving integrodifferential equations under boundary conditions, taking into account physical nonlinearity, a broad class of boundary-value problems of oscillations arises associated with various boundary conditions at the edges of a flat element. When taking into account non-stationary external influences, the main parameters is the frequency of natural vibrations of a flat component, taking into account temperature, prestressing, and other factors. The study of such problems, taking into account complicating factors, reduces to solving rather complex problems. The difficulty of solving these problems is due to both the type of equations and the variety. We analyze the results of previous works on the boundary problems of vibrations of plane elements. Possible boundary conditions at the edges of a flat element and the necessary initial conditions for solving particular problems of self-oscillation and forced vibrations, and other problems are considered. The set of equations, boundaries, and initial conditions make it possible to formulate and solve various boundary value problems of vibrations for a flat element. The oscillation equations for a flat element in the form of a plate given in this paper contain viscoelastic operators that describe the viscous behavior of the materials of a flat component. In studying oscillations and wave processes, it is advisable to take the kernels of viscoelastic operators regularly, since only such operators describe instantaneous elasticity and then viscous flow.

Application of the FEM Method to Modeling and Analysis of the Cold Thread Rolling Process-Part 1: Conditions for Ensuring a Plane State of Deformations.

The article concerns the application of the FEM method for the prediction of stress and deformation states in a workpiece during the thread rolling process (TR). The analysis covered a new kinematic variant of the TR process in which the basket of the head rotates and is torque-driven, while the workpiece is stationary and the head with the rollers moves axially relative to the workpiece. The TR process was considered as a geometrical and physical non-linear initial and boundary problem with non-linear, moving, and variable in time and space boundary conditions. The boundary conditions in the contact areas of the tool with the workpiece were unknown. An updated Lagrange (UL) description was used to describe the physical phenomena at a typical incremental step. The states of strain and strain rate were described by non-linear relationships without linearization. New discrete systems of motion and deformation equations of the object in the TR were introduced, which take into account the change in the stiffness of the system during the TR process. This equation was solved by the central differences method (explicit). The material parameters were estimated during tensile tests to determine the characteristics of the C45 steel, and a new semi-empirical method was used to determine the relationship yield stress, effective true strain, and effective true strain rate in the thread rolling process. A modified Cowper-Symonds material model was also used to model the displacement process of the wedge on an elastic/visco-plastic body reflecting the TR process. A non-linear dependency of material hardening module depending on strain and strain rate was introduced. To confirm the plane state of deformation and spatial state of stress, an experimental investigation was carried out. The computer models were validated, and a good convergence of the results was obtained. Applications in the ANSYS/LS-Dyna program were developed to simulate the TR process. The developed applications enable a comprehensive time analysis of the states of displacement, strain, and stress occurring in an object consisting of a workpiece (shaft) and a tool (roller) for the case of a plane strain state and a spatial stress state. Exemplary results of numerical analyzes are presented to explain the influence of the friction coefficient on the condition of the thread quality, and the state of deformations and stresses were shown.

Analysis and Numerical Results of p-Laplacian Effect on Steklov Neumann Problems

This paper explains how to obtain the existence and uniqueness of weak solutions for the weighted p-Laplacian boundary problem of Steklov and Neumann, relying on Browder’s theorem under conditions of the monotonous function f. The paper also discusses the behavior of the problems through the numerical results. Additionally, to illustrate the effectiveness of our approach, examples of both linear and nonlinear equations are presented.

The Unsettled Agreement of the 2019 Re-Demarcation of the Indonesia-Malaysia Boundary Line’s Impact

Referring to the principle of uti possidetis juris, the establishment of the boundary line on Sebatik Island was based on a previous agreement between the Netherlands and the British. It was stated in the 1915 Agreement between the Netherlands and the British, which declared that the boundary line follows a parallel line of 4º10' North Latitude. Unfortunately, The Boundary Agreement of 1915 does not mention the datum and reference ellipsoid used to determine the abovementioned 4° 10' North Latitude. The absence of a datum and reference ellipsoid causes disputes in determining the precise location of boundary pillars during the field reconstruction procedure. A joint survey between Indonesia and Malaysia in 1983 found that most of the boundary pillars on Sebatik Island were not precisely located along the 4° 10' North Latitude line and based on the results of the 10th meeting of the Joint Working Group on the Outstanding Boundary Problems in 2018, Indonesia and Malaysia agreed to re-demarcate the boundary, which was carried out in 2019. This study aims to determine the impact of the 2019 re-demarcation of the boundary line between Indonesia and Malaysia on Sebatik Island. The methods applied in this research are in-depth interviews, observation, and document studies. The results found that the 2019 re-demarcation resulted in exchanging areas between Indonesia and Malaysia. This area exchange generates uncertainty about land ownership by citizens in border zones.

Construction of effective computational algorithms for solving free boundary problems

An efficient method for numerically solving the one-dimensional Stefan problem is proposed herein. A computational algorithm for solving free boundary problems has been developed. This provides a means of solving problems with an arbitrary and variable number of phases, both in terms of thermal conductivity and diffusion. The solution algorithm is based on the application of the finite element method. The calculations are performed according to a homogeneous scheme. This makes the method universal and renders it possible to be referred to the class of shock-capturing methods. Accurate tracking of the position of the boundaries is carried out, in the same manner as in the methods with edge detection, which makes it possible to solve problems with high accuracy, inherent in methods of this type.

Numerical solution of linear pseudo-parabolic equation with time delay using three layer difference method

In this paper, we consider the one-dimensional initial–boundary problem for a pseudo-parabolic equation with a time delay in the second spatial derivative. To solve this problem numerically, we construct a higher-order difference method and obtain the error estimate for its solution. Based on the method of energy estimates the fully discrete scheme is shown to be convergent of order four in space and of order two in time. Given numerical results illustrate the convergence and effectiveness of the numerical method.

A symmetry group based supervised learning method for solving partial differential equations

Deep neural network provides an effective way to solve partial differential equations (PDEs) where physics-informed neural network (PINN) is a typical representative of which the outputs are constrained to approximate the solution of PDEs. However, the PINN method still fails to learn some solutions of particular properties and the reasons of failure are still unclear. In this paper, we employ both the loss landscape of the final optimization parameters of network and the imbalance of back-propagation gradient during the training to analyze why the PINN method fails to learn large amplitude solution and high-frequency solution of the Klein–Gordon equation. Consequently, we find that the PDE-based soft constraints in the loss function take a major responsibility for the failures of the PINN method in predicting such types of solutions. To remedy the pathologies we remove the PDE-based soft constraints in the loss function and calibrate a supervised learning model for predicting such types of particular solutions for a class of PDEs which cannot be learned by the PINN method and the two weight-related improved PINN methods. Specifically, if the governing PDEs under study admit a Lie symmetry group and the unique solution of the corresponding initial boundary problem is obtained via the symmetry group, we utilize the symmetry group to generate labeled data in the interior domain of PDEs from the discrete points on the known initial and boundary conditions. Then, with the labeled data we leverage the supervised learning model to predict the large amplitude solution and the high-frequency solution of the Klein–Gordon equation on a rectangular domain by means of a space–time translation group and a singular solution of the heat equation on a moving boundary domain with a physical symmetry group. Numerical experiments confirm the good performances of the proposed supervised learning method.

MKIoU loss: toward accurate oriented object detection in aerial images

Oriented bounding box regression is crucial for oriented object detection. However, regression-based methods often suffer from boundary problems and the inconsistency between loss and evaluation metrics. A modulated Kalman intersection over union (IoU) loss of approximate SkewIoU is proposed, named MKIoU. To avoid boundary problems, we convert the oriented bounding box to Gaussian distribution then use the Kalman filter to approximate the intersection area. However, there exists significant difference between the calculated and actual intersection areas. Thus, we propose a modulation factor to adjust the sensitivity of angle deviation and width-height offset to loss variation, making the loss more consistent with the evaluation metric. Furthermore, the Gaussian modeling method avoids the boundary problem but causes the angle confusion of square objects simultaneously. Thus, the Gaussian angle loss (GA loss) is presented to solve this problem by adding a corrected loss for square targets. The proposed GA loss can be easily extended to other Gaussian-based methods. Experiments on three publicly available aerial image datasets, DOTA, UCAS-AOD, and HRSC2016, show the effectiveness of the proposed method.

DPCTN: Dual path context-aware transformer network for medical image segmentation

Accurate segmentation of lesions in medical images is a key step to assist clinicians in diagnosis and analysis. Most studies combine the Transformer model with CNN at a single scale or use the highest-level feature tensor extracted by CNN as input to Transformer without fully exploiting Transformer’s potential. In addition, for the problems of structural boundary blurring, heterogeneous textures, etc., in medical images, most existing methods pay attention to using contour information to solve this problem but simply fuse the contour information and ignore the potential relationship between the regions and the contours. We propose the DPCTN network based on the traditional encoding–decoding structure, consisting of the CNN, Transformer dual backbone networks and parallel attention mechanisms, to achieve accurate segmentation in medical image lesions. Local and global multiscale feature information is extracted by CNN and Transformer. The Transformer block of channel cross fusion can implement multiscale information fusion of the high-level local features and reduce the impact of the redundant information. The dual backbone feature fusion module effectively couples the local and global high-level feature information. The decoder refines and enriches the boundary and regional features, layer by layer, to achieve effective supervision of the boundary and region. Considering the possible dimension collapse in the attention mechanism, a novel three branch transposed self-attention module is designed to reduce the information loss caused by feature pooling. To verify the effectiveness of our proposed method, subjective and objective comparative experiments and ablation experiments were performed on four medical segmentation tasks, polyps, skin lesions, glands and breast tumors. A large number of experimental results show that our method is superior to the current state-of-the-art method, reduces the standard deviation and is more robust. Source code is released at https://github.com/sd-spf/DPCTN.