Integral Boundary Research Articles

A simple meshfree method based on Trefftz attitude for 2D and 3D elasticity problems

A simple meshfree method is developed for the solution of 2D and 3D elasticity problems in potentially heterogeneous media. The rationale of the method follows that of Trefftz approaches to apply the Partial Differential Equation (PDE) and the boundary conditions in separate steps, while the basis functions can satisfy the PDE. The solution domain is discretized by a regular nodal grid including DOFs as the displacement components. The boundary is also discretized by some boundary points independent from the nodes, making the method applicable for arbitrarily shaped domains without imposing irregularity to the nodal grid. Each node corresponds to a cloud that contains some of its adjacent nodes as well. The overlap of the clouds integrates the displacement and stress components throughout the domain. The governing elasticity PDEs in heterogeneous media have non-constant coefficients, preventing Trefftz techniques to be applicable. The present method, based on equilibrated basis functions, satisfies the PDE in weighted residual approach to extract some bases capable of its approximately satisfaction. The weighting may remove the boundary integrals, so the boundary conditions are simply collocated. The integrals are composed by combination of 1D predefined ones, then removing the numerical quadrature from the solution procedure.

Kernel-free boundary integral method for two-phase Stokes equations with discontinuous viscosity on staggered grids

A discontinuous viscosity coefficient makes the jump conditions of the velocity and normal stress coupled together, which brings great challenges to some commonly used numerical methods to obtain accurate solutions. To overcome the difficulties, a kernel free boundary integral (KFBI) method combined with a modified marker-and-cell (MAC) scheme is developed to solve the two-phase Stokes problems with discontinuous viscosity. The main idea is to reformulate the two-phase Stokes problem into a single-fluid Stokes problem by using boundary integral equations and then evaluate the boundary integrals indirectly through a Cartesian grid-based method. Since the jump conditions of the single-fluid Stokes problems can be easily decoupled, the modified MAC scheme is adopted here and the existing fast solver can be applicable for the resulting linear saddle system. The computed numerical solutions are second order accurate in discrete ℓ2-norm for velocity and pressure as well as the gradient of velocity, and also second order accurate in maximum norm for both velocity and its gradient, even in the case of high contrast viscosity coefficient, which is demonstrated in numerical tests. In addition, moving interface driven by the surface tension is also provided to validate the efficiency of the KFBI method.

An analytical study of curved frequency selective surfaces for shielding applications

AbstractThis paper demonstrates an analytical study of different nonplanar frequency selective surfaces (FSSs) for X‐band (8–12 GHz) electromagnetic (EM) shielding applications. A well‐known metallic square loop unit cell is used for the analysis, which is arranged in a periodic pattern on a three‐dimensional (3‐D) printer compatible dielectric substrate. The geometry is wrapped on various kinds of curved surfaces (cylindrical, paraboloid, and hemispherical surfaces) and investigated using the finite element boundary integral (FEBI) technique. The structure exhibits a band stop characteristic with a fractional bandwidth of 62.74% (6.68–12.66 GHz, having transmission coefficient below −10 dB), resonating around 9.53 GHz under different angles of incident EM wave. Finally, the 3‐D printing method is used to fabricate the prototypes for performing laboratory measurements. The simulated and measured results follow each other closely, thereby establishing the proposed technique in realizing nonplanar FSSs.

Buckling analysis of thin plates using direct interpolation boundary element method

The present study employs the direct integration boundary element method (DIBEM) to analyze the buckling behavior of thin plates. Unlike conventional approaches, this method eliminates the need for domain discretization or specialized solutions. Furthermore, it constitutes a continuation from the previous research in Doval et al. (2013), which employed the radial integration method for the same investigation.To showcase the efficacy of the proposed approach, it is utilized in the stability analysis of both perforated and non-perforated plates. The findings are then compared against analytical responses, in the case of non-perforated plates, and finite element method results, for perforated plates.Moreover, the use of boundary elements reduces the size of the eigenvalue problem involved in stability analysis, as compared to finite elements, provided that only the domain and a few internal points are needed for discretization.Overall, this paper presents a novel and effective approach to the analysis of buckling plates.

Computation of scattering poles using boundary integrals

Scattering resonances have important applications in mathematics, physics and engineering. They can be viewed as the poles of the meromorphic extension of the scattering operator. In this paper, we consider the computation of the scattering poles for sound soft obstacles. The scattering problem is formulated using boundary integrals, which is then discretized by the Nyström method. The discrete scattering poles are computed using the contour integral method. The proposed method is highly accurate, free of spurious modes, and can be extended to treat other obstacles, e.g., sound hard obstacles. Numerical examples are presented to validate the effectiveness and accuracy. The current paper is among the very few computational studies of scattering poles for obstacles.

An efficient numerical approach for solving singularly perturbed parabolic differential equations with large negative shift and integral boundary condition

In this study, we consider singularly perturbed large negative shift parabolic reaction–diffusion with integral boundary condition. The continuous solution's properties are discussed. On a non-uniform Shishkin mesh, the spatial derivative is discretized using the tension spline method, and the temporal derivative is discretized using the Crank–Nicolson method. In order to handle the integral boundary condition, Simpson's rule is used. After conducting an error analysis, it was determined that the method was uniformly convergent. To support the theoretical findings, numerical examples are taken into account and solved for different values of the perturbation parameter and mesh sizes. It is proved to be uniformly convergent to almost second order.

Analysis of JS-contractions with applications to fractional boundary value problems

In this article, we modify JS-contractions by weakening the conditions on the function θ, where theta : ( 0,infty ) rightarrow ( 1,infty ) is a strictly increasing function. We prove fixed-point results for obtained contractions. Some examples are given to validate the results and modifications. We use our main theorem to establish the existence results for the solutions of the Atangana–Baleanu–Caputo fractional boundary value problem with integral boundary conditions. We also present a new definition of θ-Ulam stability and find the stability of our fractional boundary value problem.

Exact solution of a non-Hermitian -symmetric spin chain

We construct the exact solution of a non-Hermitian -symmetric isotropic Heisenberg spin chain with integrable boundary fields. We find that the system exhibits two types of phases named A and B. In the B type phase, the symmetry remains unbroken and it comprises of eigenstates with only real energies, whereas the A type phase exhibits both symmetry broken and unbroken sectors, comprising of eigenstates with only complex and real energies respectively. The -symmetry broken sector comprises of pairs of eigenstates whose energies are complex conjugates of each other. The existence of two sectors in the A type phase is associated with the exponentially localized bound states at the edges with complex energies which are described by boundary strings. We find that both A and B type phases can be further divided into sub-phases which exhibit different ground states. We also compute the bound state wavefunction in one magnon sector and find that as the imaginary value of the boundary parameter is increased, the exponentially localized wavefunction broadens thereby protruding more into the bulk, which indicates that exponentially localized bound states may not be stabilized for large imaginary values of the boundary parameter.

A Non-Local Non-Homogeneous Fractional Timoshenko System with Frictional and Viscoelastic Damping Terms

We are devoted to the study of a non-local non-homogeneous time fractional Timoshenko system with frictional and viscoelastic damping terms. We are concerned with the well-posedness of the given problem. The approach relies on some functional analysis tools, operator theory, a priori estimates and density arguments. This work can be considered as a contribution to the development of energy inequality methods, the so-called a priori estimate method inspired from functional analyses and used to prove the well-posedness of mixed problems with integral boundary conditions.

On the existence of a positive solution to a boundary value problem for one nonlinear functional-differential equation of the second order

This article considers a boundary value problem for one non-linear second-order functional differential equation on the segment [0, 1] with an integral boundary condition at one of the ends of the segment. Using the well-known Go-Krasnoselsky theorem, sufficient conditions for the existence of at least one positive solution to the problem under consideration are established. A non-trivial example is given, illustrating the fulfillment of the conditions for the unique solvability of the problem posed.

Special finite elements with arbitrarily-shaped holes for heat conduction analysis of porous materials

Special finite elements are developed which adequately represent the state of heat conduction in the region around an arbitrarily-shaped hole. This is achieved with two assumed temperature fields. The frame field is defined along the element boundary while the intra-element field is independently approximated by using truncated T-complete solutions as trial functions for capturing hole effects. The geometrical irregularity of holes contribute to the difficulty in seeking T-complete solutions. By means of the charge simulation method and numerical conformal mapping technique, the systematic construction of T-complete solutions is shown which satisfy a priori the governing differential equation. Based on the modified variational principle, the two independent temperature fields are linked together at the element level, ensuring conformity across inter-element boundaries. The resultant element stiffness equation involves boundary integrals only, which provides a great flexibility in mesh generation with polygons. Numerical examples for the problems with a single, dual, and multiple holes are given to illustrate the efficiency and accuracy of the proposed method.

Hybrid differential inclusion with nonlocal, infinite-point or Riemann-Stieltjes integral boundary conditions

Hybrid differential inclusion with nonlocal, infinite-point or Riemann-Stieltjes integral boundary conditions

Insights on using the boundary integral SPH formulations to calculate Laplacians with Dirichlet boundaries

Handling boundary conditions in Smooth Particle Hydrodynamics (SPH) has been known as a challenging problem for a long time, especially when particle approximation is implemented for truncated kernels on the boundary. This work develops a novel particle consistent gradient formulation to impose the Dirichlet boundary condition on a Laplacian with boundary integrals. Boundary integral formulation allows the accurate imposition of boundary conditions without creating virtual particles to extend boundaries, as used in conventional SPH modeling practices. Here, boundary integral formulations are presented for Laplacian operators and the imposition of Dirichlet boundary conditions on arbitrary boundary shapes. We also explore using this particle consistent first-order derivative twice to evaluate the entire Laplacian. Further, the convergence characteristics of different Laplacian formulations with boundary integrals are discussed. Finally, numerical experiments solving explicit thermal and fluid problems are presented to demonstrate the robustness of various Laplacian operators for practical use.

Damping of three-dimensional waves on coating films dragged by moving substrates

Paints and coatings often feature interfacial defects due to disturbances during the deposition process, which, if they persist until solidification, worsens the product quality. In this article, we investigate the stability of a thin liquid film dragged by a vertical substrate moving against gravity, a fundamental flow configuration in various coating processes. The receptivity of the liquid film to three-dimensional disturbances is analyzed with Direct Numerical Simulations (DNS) and an in-house Integral Boundary Layer (IBL) film model. The latter was used for linear stability analysis and nonlinear wave propagation analysis. The numerical implementation of the IBL film model combines a finite volume formulation with a pseudo-spectral approach for the capillary terms that allows one to investigate non-periodic surface tension-dominated flows. The numerical model was successfully validated with DNS computations. The combination of these numerical tools allows one to describe the mechanisms of capillary and nonlinear damping and identify the instability threshold of the coating processes. The results show that transverse modulations can be beneficial for damping two-dimensional waves within the range of operational conditions considered in this study, which are relevant to air-knife and slot-die coating.

Elimination of aberrantly specified cell clones is independent of interfacial Myosin II accumulation.

Spatial organization within an organ is essential and needs to be maintained during development. This is largely implemented via compartment boundaries that serve as barriers between distinct cell types. Biased accumulation of junctional non-muscle Myosin II along the interface between differently fated groups of cells contributes to boundary integrity and maintains its shape via increased tension. Here, using the Drosophila wing imaginal disc, we tested whether interfacial tension driven by accumulation of Myosin is responsible for the elimination of aberrantly specified cells that would otherwise compromise compartment organization. To this end, we genetically reduced Myosin II levels in three different patterns: in both wild-type and misspecified cells, only in misspecified cells, and specifically at the interface between wild-type and aberrantly specified cells. We found that the recognition and elimination of aberrantly specified cells do not strictly rely on tensile forces driven by interfacial Myosin cables. Moreover, apical constriction of misspecified cells and their separation from wild-type neighbours occurred even when Myosin levels were greatly reduced. Thus, we conclude that the forces that drive elimination of aberrantly specified cells are largely independent of Myosin II accumulation.

Salient object detection algorithm based on diversity features and global guidance information

Aiming at the problems of traditional salient object detection methods such as fuzzy boundary and insufficient information integrity, a salient object detection network composed of feature diversity enhancement module, global information guidance module and feature fusion module is proposed. Firstly, asymmetric convolution, cavity convolution and common convolution are spliced to form a feature diversity enhancement module to extract different types of spatial features corresponding to each feature layer. Secondly, the global information guidance module transmits the information captured by the coordinate attention mechanism to each feature layer through the global guidance stream, so as to learn the semantic relationship between different feature layers and alleviate the dilution effect; Finally, the feature fusion module receives the high-level features output from the previous layer, the low-level features of the corresponding layer and the global context information generated by the global information guidance module, and the cascade feature diversity enhancement module gradually integrates the multi-level features to refine the saliency feature map. Comparative experiments on five public data sets show that this method can achieve the highest values of 0.959 and 0.030 in F-measure and MAE. Compared with other seven advanced algorithms, it has better detection performance.

APPLICATION OF CALCULATION METHODS FOR SOLVING APPLIED PROBLEMS OF HEAT AND MASS EXCHANGE IN COMPLEX SYSTEMS

The accuracy of the calculation and optimization of the objective function and its parameters when solving applied engineering problems depends on the accuracy of the formulation of the main optimization problem, calculation and applied optimization problems, as well as the accuracy of computational methods for their implementation. An increase in the considered features of applied optimization problems will complicate the setting and methods of implementing boundary value problems. Thus, for the implementation of modernized boundary value problems, it will be necessary to apply several computational methods that will create a computational structure. The main condition for constructing a physically based boundary value problem is to find and justify the conditions for the existence of a unique solution. To increase the efficiency of the use of methods of calculation and optimization of technical parameters, it is necessary to increase the number of considered features of calculation and applied optimization mathematical models for heat and mass transfer in technical systems. Along with the construction of boundary value problems, it is important to define and justify the conditions for the existence of a single solution.
 The research article deals with some aspects of solving applied problems of heat and mass transfer in technical systems. Nonlocal boundary value problems for inhomogeneous and homogeneous pseudodifferential equations in partial derivatives with integral boundary conditions are considered, methods of solving a nonlocal inhomogeneous boundary value problem are proposed, and the correctness conditions of this problem in the class of infinitely differentiable generalized functions of power growth are defined and proven. Proved conditions for the existence of a correct problem for pseudo-differential equations with an integral boundary condition. The research of this article should be applied for controlling possible risks when solving applied problems in technical systems, biotechnology and veterinary medicine.

Uniformly Convergent Numerical Scheme for Solving Singularly Perturbed Parabolic Convection-Diffusion Equations with Integral Boundary Condition

Uniformly Convergent Numerical Scheme for Solving Singularly Perturbed Parabolic Convection-Diffusion Equations with Integral Boundary Condition

Existence Results of Langevin Equations with Caputo–Hadamard Fractional Operator

In this manuscript, we deal with a nonlinear Langevin fractional differential equation that involves the Caputo–Hadamard and Caputo fractional operators, with nonperiodic and nonlocal integral boundary conditions. The results presented in this study establish the existence, uniqueness, and Hyers–Ulam (HU) stability of the solution to the proposed equation. We achieved our main result by using the Banach contraction mapping principle and Krasonoselskii’s fixed point theorem. Furthermore, we introduce an application to demonstrate the validity of the results of our findings.

A non-uniform Haar wavelet method for a singularly perturbed convection–diffusion type problem with integral boundary condition on an exponentially graded mesh

A non-uniform Haar wavelet method for a singularly perturbed convection–diffusion type problem with integral boundary condition on an exponentially graded mesh