Local Boundary Conditions Research Articles

Classical echoes of quantum boundary conditions

Abstract We consider a non-relativistic particle in a one-dimensional box with all possible quantum boundary conditions that make the kinetic-energy operator self-adjoint. We determine the Wigner functions of the corresponding eigenfunctions and analyze in detail their classical limit, governed by their behavior in the high-energy regime. We show that the quantum boundary conditions split into two classes: all local and regular boundary conditions collapse to the same classical boundary condition, while a dependence on singular non-local boundary conditions persists in the classical limit.

Analysis and Implementation of Nonlocal Governing Operators with Local Boundary Conditions on a General Interval

Analysis and Implementation of Nonlocal Governing Operators with Local Boundary Conditions on a General Interval

Construction of Nonlocal Governing Operators with Local Boundary Conditions on a General Interval

Construction of Nonlocal Governing Operators with Local Boundary Conditions on a General Interval

Adaptive local boundary conditions to improve deformable image registration

Objective. In medical imaging, it is often crucial to accurately assess and correct movement during image-guided therapy. Deformable image registration (DIR) consists in estimating the required spatial transformation to align a moving image with a fixed one. However, it is acknowledged that for DIR methods, boundary conditions applied to the solution are critical in preventing mis-registration. This poses an issue particularly when areas of interest are located near the image border. Despite the extensive research on registration techniques, relatively few have addressed the issue of boundary conditions in the context of medical DIR. Our aim is a step towards customizing boundary conditions to suit the diverse registration tasks at hand. Approach. We analyze the behavior of two typical global boundary conditions: homogeneous Dirichlet and homogeneous Neumann. We propose a generic, locally adaptive, Robin-type condition enabling to balance between Dirichlet and Neumann boundary conditions, depending on incoming/outgoing flow fields on the image boundaries. The proposed framework is entirely automatized through the determination of a reduced set of hyperparameters optimized via energy minimization. Main results. The proposed approach was tested on a mono-modal computed tomography (CT) thorax registration task and an abdominal CT-to-MRI registration task. For the first task, we observed a relative improvement in terms of target registration error of up to 12% (mean 4%), compared to homogeneous Dirichlet and homogeneous Neumann. For the second task, the automatic framework provides results close to the best achievable. Significance. This study underscores the importance of tailoring the registration problem at the image boundaries. In this research, we introduce a novel method to adapt the boundary conditions on a voxel-by-voxel basis, yielding optimized results in two distinct tasks: mono-modal CT thorax registration and abdominal CT-to-MRI registration. The proposed framework enables optimized boundary conditions in image registration without prior assumptions regarding the images or the motion.

Comparative study between observed and numerical downscaled data of surface air temperature

Surface Air temperature, being one of the key parameters of climate, plays an important role in occurrence severe environmental issues like droughts, floods, rate of evaporation etc. In this study, an endeavor has been made to investigate the experiments available in Educational Global Climate Model (EdGCM) at Sylhet Agricultural University for recognizing best suited experiment to predict the changing pattern of surface air temperature at Sylhet region. Downscaling is a cutting-edge procedure to produce high-goals atmosphere change expectation from coarse to fine goals of worldwide atmosphere model outcomes and a conspicuous device for estimating future atmosphere situation at the local scale for some information alarm zone like Bangladesh. Maps are generated from EdGCM simulation run to collect data and depict global temperature variations. Transform software have been connected to downscale the outcomes of temperature data by Doubled_CO2, Global_Warming_01, Ice_Age_21kya, IPCC_A1F1_CO2, Modern_PredictedSST and Modern_SpecifiedSST. The trend analysis was performed using the Mann-Kendall Trend test and Sen-slope estimator. Temperature changes are significant for observed results, Doubled_CO2 and IPCC_A1F1_CO2 results where p-value is less than alpha 0.05. Doubled_CO2 results are more suited for dry season and IPCC_A1F1_CO2 provides better results for monsoon. The investigation could have been unquestionably progressively broad in the event that more situations of EdGCM information were utilized, and local boundary conditions were used.

Investigation of the Influence of Including or Omitting the Oxide Layer on the Result of Identifying the Local Boundary Condition during Water Spray Cooling

In the case of products made of steel, the presence of an oxide layer, which is formed during the steel production process as a result of high temperature, has a significant impact on the process of heat removal from the surface of the cooled material. For this reason, it is necessary to take into account the presence of the oxide layer in mathematical and numerical models used to simulate the distribution of the temperature field in cooled steel products. These models are based on the boundary conditions identified for given production conditions. This paper presents a comparison of the results of the identification of the boundary condition during water spray cooling of Armco iron with the use of the inverse solution. Numerical calculations were carried out using two models of heat conduction. In the first model, the presence of an oxide layer with different thermophysical properties than the base material (Armco iron) was taken into account. The second model assumed no oxide layer on the cooled Armco iron surface. It was found that the inverse solution obtained in the case of the heat conduction model taking into account the thickness of the oxide layer is correct in time and as a function of temperature. Thus, the boundary condition model obtained as a function of temperature is universal. However, this model requires an additional layer of oxides with different thermophysical properties than the base material to be included in the finite element model (FEM). Based on the conducted uncertainty tests of the inverse solution, it was found that the results of the determined boundary condition in the absence of the oxide layer on the cooled surface are subject to an error higher than 10% in comparison to the maximum reference value of the heat transfer coefficient.

Inverse Coefficient Problem for the 2D Wave Equation with Initial and Nonlocal Boundary Conditions

In this paper, we consider direct and inverse problems for the two-dimensional wave equation. The direct problem is an initial boundary value problem for this equation with nonlocal boundary conditions. In the inverse problem, it is required to find the time-variable coefficient at the lower term of the equation. The classical solution of the direct problem is presented in the form of a biorthogonal series in eigenvalues and associated functions, and the uniqueness and stability of this solution are proven. For solution to the inverse problem, theorems of existence in local, uniqueness in global, and an estimate of conditional stability are obtained. The problems of determining the right-hand sides and variable coefficients at the lower terms from initial boundary value problems for second-order linear partial differential equations with local boundary conditions have been studied by many authors. Since the nonlinearity is convolutional, the unique solvability theorems in them are proven in a global sense. In the works, the method of separation of variables is used to find the classical solution of the direct problem in the form of a biorthogonal series in terms of eigenfunctions and associated functions. The nonlocal integral condition is used as the overdetermination condition with respect to the solution of the direct problem. The direct problem reduces to equivalent integral equations of the Fourier method. To establish integral inequalities, the generalized Gronwall-Bellman inequality is used. We obtain an a priori estimate of the solution in terms of an unknown coefficient, that are useful for studying the inverse problem.

Numerical study of liquid metal magnetoconvection and heat transfer in an electrically conductive square duct

Mixed convection of an electrically conductive fluid in a square duct with imposed transverse magnetic field is studied using Large Eddy Simulation (LES) paradigms. The duct walls are electrically conductive, with the wall conductivity parameter cw ranging from 0 to 0.5. The Reynolds number is Re=5602 and the Prandtl number is Pr=0.0238. The focus of the study is on flows at Hartmann numbers Ha⩽125, Richardson numbers Ri⩽10 and two different thermal boundary conditions are considered: four wall uniform heat fluxes and one-sided heating (fixed wall temperature). The results show that the transition from laminar to turbulent flow depends not only on the ratio Ri/Ha, but also on cw and on the local thermal boundary conditions. In the turbulent regime with one-sided heating, the turbulent heat fluxes play an important role in the total heat transfer, in contrast with the typical behaviours of liquid metals. Moreover, the turbulent and thermal structures are highly dependent on the thermal boundary conditions, which completely alter the flow structure. It is also found that at cw≥0.01 the turbulent heat fluxes decrease.

Nonlocal Problems with Local Boundary Conditions I: Function Spaces and Variational Principles

Nonlocal Problems with Local Boundary Conditions I: Function Spaces and Variational Principles

Revealing molecular insights into surface charge and local viscosity in electroosmotic flows

The limitations of the continuum theory in predicting osmotic response at the nanoscale stem from its lack of molecular-level insight into local fluid properties and the interfacial structure of fluid and electrolyte solutions. To overcome this challenge, our study integrates molecular dynamics (MD) simulation with the continuum framework to explore how surface charge and various hydrodynamic properties impact electroosmotic flow (EOF). The failure of continuum theories to account for molecular interactions and geometric boundaries leads to significant disparities between MD simulations and continuum predictions, influenced by local fluid properties and the electric field. Emphasizing the importance of incorporating appropriate local hydrodynamic properties and atomic interface boundary conditions, our findings bridge the gap between MD simulations and continuum EOF predictions. Our computational results and theoretical model, considering surface charge, atomic interface boundaries, and dynamic structure-based hydrodynamic properties, provide crucial insights and guidance for EOF investigations.

Complex phases in quantum mechanics

Schrödinger's equation is a local differential equation and boundary conditions are required to determine the solution uniquely. Depending on the choice of boundary conditions, a given Hamiltonian may describe several different physically observable phases, each exhibiting its own characteristic global symmetry.

Stress-Equivalent Spalart–Allmaras Wall Model with Local Boundary Conditions for Reynolds-Averaged Navier–Stokes

In this paper, we present a wall model based on a modified version of the Spalart–Allmaras turbulence model. Unlike typical wall models, this method avoids the need to query the interior solution by utilizing solution information solely at the boundary, making it well-suited for unstructured grids and mesh adaptation. Furthermore, this wall-modeling method is formulated to lessen the near-wall grid resolution requirements by, below the log layer, making the eddy viscosity approach a constant, nonzero value and the velocity varying approximately linearly with distance from the wall. Using output-based mesh adaptation, this new wall-modeling method is compared to standard Reynolds-Averaged Navier-Stokes (RANS-SA). It is shown that this wall model results in accurate predictions of quantities of interest such as aerodynamic coefficients, surface pressure and temperature, skin friction, and heat transfer compared with RANS-SA, while requiring substantially less near-wall grid.

Well posedness of linear parabolic partial differential equations posed on a star-shaped network with local time Kirchhoff's boundary condition at the vertex

The main purpose of this work is to provide an existence and uniqueness result for the solution of a linear parabolic system posed on a star-shaped network, which presents a new type of Kirchhoff's boundary transmission condition at the junction. This new type of Kirchhoff's condition - that we decide to call here local-time Kirchhoff's condition - induces a dynamical behavior with respect to an external variable that may be interpreted as a local time parameter, designed to drive the system only at the singular point of the network. The seeds of this study point towards a theoretical inquiry of a particular generalization of Walsh's random spider motions, whose spinning measures would select the available directions according to the local time of the motion at the junction of the network.

An Efficient Local Artificial Boundary Condition for Infinite Long Finite Element Euler–Bernoulli Beam

To solve the wave propagation problems of the Euler–Bernoulli beam in an unbounded domain effectively and efficiently, a new local artificial boundary condition technology is proposed. It replaces the residual right-hand side of the truncated discrete equation with an equivalent linear algebraic system. First, the equivalent Schrodinger equation is discussed. Its artificial boundary condition is obtained by first rationalizing the Dirichlet-to-Neumann condition in the frequency domain with a Pade approximation and then inverse transforming each Pade term back into the time domain by introducing auxiliary degrees of freedom. Frequency shifting is employed such that it performs better near a prescribed frequency. Then, the artificial boundary condition of the finite element Euler–Bernoulli beam is obtained by simple algebraic manipulations on that of the corresponding Schrodinger equation. This method only makes local changes to the original truncated discrete dynamic system and thus is very efficient and easy to use. The accuracy of the proposed method can be improved by using more Pade terms and a proper shift frequency. The numerical example shows, with only a few additional degrees of freedom, the proposed artificial boundary condition effectively eliminates the spurious reflection. The idea of the proposed method can also be used in other dispersive wave systems.

Existence and Uniqueness of Solutions for Fractional-Differential Equation with Boundary Condition Using Nonlinear Multi-Fractional Derivatives

In this article the existence as well as the uniqueness (EU) of the solutions for nonlinear multiorder fractional-differential equations (FDE) with local boundary conditions and fractional derivatives of different orders (Caputo and Riemann–Liouville) are covered. The existence result is derived from Krasnoselskii’s fixed point theorem and its uniqueness is shown using the Banach contraction mapping principle. To illustrate the reliability of the results, two examples are given.

Analytical approach to solving linear diffusion–advection–reaction equations with local and nonlocal boundary conditions

Initial–boundary value problems for a linear diffusion–advection–reaction equation are considered, with general nonhomogeneous linear boundary conditions and general linear nonlocal boundary conditions. Analytical solutions are obtained using an embedding method. The solutions are expressed in terms of time‐varying functions that satisfy coupled linear Volterra integral equations of the first kind. A boundary element method is applied to numerically solve the integral equations. Three examples are given to demonstrate the accuracy of the numerical solutions when compared with the analytical solutions. The embedding method is applicable to problems with bounded and unbounded spatial domains.

General Local Reactive Boundary Condition for Dissolution and Precipitation Using the Lattice Boltzmann Method

AbstractA general and local reactive boundary condition (RBC) for studying first‐order equilibrium reactions using the lattice Boltzmann method is presented. Its main characteristics are accurate reproduction of wall diffusion, invariance to the wall and grid orientation, and absence of nonphysical artifacts. The scheme is successfully tested for different benchmark cases considering diffusion, advection, and reactions of fluids at solid‐liquid interfaces. Unlike other comparable RBCs from the literature, the novel scheme is valid for a large range of Péclet and Damköhler numbers, and shows realistic pattern formation during precipitation. In addition, quantitative results are in good accordance with analytical solutions and values from literature. Combining the new RBC with the rest fraction method, Péclet‐Reynolds ratios of up to 1,000 can be achieved. Overall, the novel RBC accurately models first‐order reactions, is applicable for complex geometries, and allows efficiently simulating dissolution and precipitation phenomena in fluids at the pore scale.

Spatial stress analysis in an elastic parallelepiped

Abstract The direct integration method is extended onto the 3D analysis of an elastic rectangular parallelepiped subject to arbitrary force loadings on its sides. By making use of the equilibrium equations, the integral-form expressions are derived for the stress-tensor components through the introduced Vihak functions. These expressions were efficiently used to reduce the original sets of the local boundary conditions to the equivalent sets of the integral conditions for the Vihak functions. In such a manner, the original problems are managed to be reduced to the auxiliary boundary value problems for the governing integro-differential equations with accompanying integral conditions for the Vihak functions. For solving the auxiliary problems for the key functions, special semi-analytical algorithms are suggested in engaging a specific approach for the separation of variables by making use of the complete systems of orthogonal eigen- and associated functions. This allows for determining the Vihak key functions and, consequently, the stress-tensor components in the form of explicit analytical dependencies on the applied force loadings. The solution is quite beneficial for both theoretical and practical implementations. It was shown by the numerical evidence that the solutions are efficient for the analysis of stress fields in the entire domain including edges and corners.

A graph model to describe the network connectivity of trabecular plates and rods.

Introduction: The trabecular network is perceived as a collection of interconnected plate- (P) and rod-like (R) elements. Previous research has highlighted how these elements and their connectivity influence the mechanical properties of bone, yet further work is required to elucidate better the deeply interconnected nature of the trabecular network with distinct element formations conducting forces per their mechanical boundary conditions. Within this network, forces act through elements: a rod or plate with force applied to one end will transmit this force to a component connected to the other end, defining the boundary conditions for the loading of each element. To that end, this study has two aims: First, to investigate the connectivity of individually segmented elements of trabecular bone with respect to their local boundary conditions as defined by the surrounding trabecular network and linking them directly to the bone's overall mechanical response during loading using a mathematical graph model of the plate and rod (PR) Network. Second, we use this model to quantify side artifacts, a known artifact when testing an excised specimen of trabecular bone, where vertical trabeculae lose their load-bearing capacity due to a loss of connectivity, ultimately resulting in a change of the trabecular network topology. Resuts: Connected elements derived from our model predicted apparent elastic modulus by fitting a linear regression (R 2 = 0.81). In comparison, prediction using conventional bone volume fraction results in a lower accuracy (R 2 = 0.72), demonstrating the ability of the PR Network to estimate compressive elastic modulus independent of specimen size or loading boundary condition. Discussion: PR Network models are a novel approach to describing connectivity within the trabecular network and incorporating mechanical boundary conditions within the morphological analysis, thus enabling the study of intrinsic material properties of trabecular bone. Ultimately, PR Network models may be an early predictor or provide further insights into osteo-degenerative diseases.

Error estimate and superconvergence of a high-accuracy difference scheme for 2D heat equation with nonlocal boundary conditions

<p>In this work, we initially construct an implicit Euler difference scheme for a two-dimensional heat problem, incorporating both local and nonlocal boundary conditions. Subsequently, we harness the power of the discrete Fourier transform and develop an innovative transformation technique to rigorously demonstrate that our scheme attains the asymptotic optimal error estimate in the maximum norm. Furthermore, we derive a series of approximation formulas for the partial derivatives of the solution along the two spatial dimensions, meticulously proving that each of these formulations possesses superconvergence properties. Lastly, to validate our theoretical findings, we present two comprehensive numerical experiments, showcasing the efficiency and accuracy of our approach.</p>