Neumann Type Boundary Research Articles

A convergence result for the ergodic problem for Hamilton–Jacobi equations with Neumann-type boundary conditions

We consider the ergodic (or additive eigenvalue) problem for the Neumann-type boundary-value problem for Hamilton–Jacobi equations and the corresponding discounted problems. Denoting by uλ the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family to a solution of the ergodic problem as λ → 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting by-product, we introduce Mather measures associated with Hamilton–Jacobi equations with the Neumann-type boundary conditions. These results are variants of the main results in a recent paper by Davini et al., who study the same convergence problem on smooth compact manifolds without boundary.

The numerical solution of advection–diffusion problems using new cubic trigonometric B-splines approach

A new cubic trigonometric B-spline collocation approach is developed for the numerical solution of the advection–diffusion equation with Dirichlet and Neumann's type boundary conditions. The approach is based on the usual finite difference scheme to discretize the time derivative while a cubic trigonometric B-spline is utilized as an interpolation function in the space dimension with the help of θ-weighted scheme. The present scheme stabilizes the oscillations that are normally displayed by the approximate solution of the transient advective–diffusive equation in the locality of sharp gradients produced by transient loads and boundary layers. The scheme is shown to be stable and the accuracy of the scheme is tested by application to various test problems. The proposed approach is numerically verified to second order and shown to work for the Péclet number ≤ 5.

Nesting an incompressible-flow code within a compressible-flow code: A two-dimensional study

We consider numerical algorithms appropriate for one- and two-way coupling between meso-scale and micro-scale fluid-dynamics codes for wind energy computing. At the meso-scale is a numerical weather-prediction code, which is typically based on the compressible-flow Euler equations. At the micro-scale, surrounding one or more wind turbines, is a computational fluid dynamics code, which is typically based on the incompressible-flow Navier–Stokes equations. When calculating short-duration flow around wind turbines, one-way coupling is sufficient, where the meso-scale computational model drives the micro-scale model. However, in long-duration simulations involving large wind farms, the influence of the wind farm on the meso-scale weather may no longer be insignificant and two-way coupling is warranted. In this study, we focus on a simple two-dimensional system, for which our goal is to devise one- and two-way coupling algorithms that can effectively transport a vortex propagating in laminar flow from one domain to the other. Two coupling schemes and their numerical implementation are described: partial-boundary coupling and projection coupling. In the former, the micro-scale-domain boundary is decomposed, based on the meso-scale solution, into sections corresponding to inflow and outflow. The micro-scale model has Dirichlet- and Neumann-type boundary conditions on these sections, respectively. In projection coupling, the meso-scale solution is projected onto the incompressible-flow solution space in the micro-scale domain, from which Dirichlet-type boundary conditions are derived. In these simulations, the uncoupled meso-scale solution is taken as the reference, and the best coupling method is that which produces solutions that deviate the least from the reference. In one-way coupling, under a simple two-dimensional laminar-flow test case, partial-boundary coupling was more effective than projection coupling. However, in two-way coupling, projection coupling was the best performer.

Coupled lattice Boltzmann method for simulating electrokinetic flows: A localized scheme for the Nernst–Plank model

We present a coupled lattice Boltzmann method (LBM) to solve a set of model equations for electrokinetic flows in micro-/nano-channels. The model consists of the Poisson equation for the electrical potential, the Nernst–Planck equation for the ion concentration, and the Navier–Stokes equation for the flows of the electrolyte solution. In the proposed LBM, the electrochemical migration and the convection of the electrolyte solution contributing to the ion flux are incorporated into the collision operator, which maintains the locality of the algorithm inherent to the original LBM. Furthermore, the Neumann-type boundary condition at the solid/liquid interface is then correctly imposed. In order to validate the present LBM, we consider an electro-osmotic flow in a slit between two charged infinite parallel plates, and the results of LBM computation are compared to the analytical solutions. Good agreement is obtained in the parameter range considered herein, including the case in which the nonlinearity of the Poisson equation due to the large potential variation manifests itself. We also apply the method to a two-dimensional problem of a finite-length microchannel with an entry and an exit. The steady state, as well as the transient behavior, of the electro-osmotic flow induced in the microchannel is investigated. It is shown that, although no external pressure difference is imposed, the presence of the entry and exit results in the occurrence of the local pressure gradient that causes a flow resistance reducing the magnitude of the electro-osmotic flow.

Computational homogenization in dissipative electro-mechanics of functional materials

The homogenization of the complex microscopic response of functional materials with electro-mechanical coupling is a highly demanding research topic and of utmost importance for technological design. Smart materials are characterized by microstructural properties, which can be changed by external stress and electric field stimuli, and hence find use as the active components in sensors and actuators. In this context, an important challenge is to combine phase field descriptions of microscopic electric domain evolution with homogenization techniques, in order to define the macroscopic overall response. This paper offers a new approach to computational homogenization in dissipative electro-mechanics based on a rigorous exploitation of rate-type and incremental variational principles, governing the scale bridging between micro- and macroscales. It provides a canonical insight in the structure of the coupled electro-mechanical problems, where all field equations appear as the Euler–Lagrange equations of variational principles. A challenge is to link the gradient-type continuum description of the microstructure, governed by a phase field of micro-electric polarization, to a local electro-mechanical macro-continuum. This is achieved by the definition of boundary conditions on the surface of the microstructure, which include the phase field. To this end, Dirichlet-, Neumann-type and periodic boundary constraints are derived from a generalization of the Hill-Mandel macro-homogeneity condition towards the coupled electro-mechanical problem. We outline in detail techniques for the numerical implementation of these boundary conditions by means of Lagrange multiplier and penalty methods. Two- and three-dimensional numerical simulations treat electric domain wall motions in microstructures and their hysteretic homogenized overall response for macroscopic strain- and electric-field- driven volume elements.

SCATTERED FIELD COMPUTATION WITH AN EXTENDED FETI-DPEM2 METHOD

Due to the increasing number of applications in engineering design and optimization, more and more attention has been paid to full-wave simulations based on computational electromagnetics. In particular, the flnite-element method (FEM) is well suited for problems involving inhomogeneous and arbitrary shaped objects. Unfortunately, solving large-scale electromagnetic problems with FEM may be time consuming. A numerical scheme, called the dual-primal flnite element tearing and interconnecting method (FETI-DPEM2), distinguishes itself through the partioning on the computation domain into non-overlapping subdomains where incomplete solutions of the electrical fleld are evaluated independently. Next, all the subdomains are \glued together using a modifled Robin-type transmission condition along each common internal interface, apart from the corner points where a simple Neumann-type boundary condition is imposed. We propose an extension of the FETI-DPEM2 method where we impose a Robin type boundary conditions at each interface point, even at the corner points. We have implemented this Extended FETI- DPEM2 method in a bidimensional conflguration while computing the fleld scattered by a set of heterogeneous, eventually anistropic, scatterers. The results presented here will assert the e-ciency of the proposed method with respect to the classical FETI-DPEM2 method, whatever the mesh partition is arbitrary deflned.

Thrust Bump Air Foil Bearings with Variable Axial Load: Theoretical Predictions and Experiments

Thrust air foil bearings are critical components in high-efficiency turbomachinery, such as two-stage compressors subjected to large and irregular axial forces. In this article, a model of thrust bump foil bearings that predicts deflection with variable axial load is developed assuming no tilting effect of the thrust collar. To predict the air clearance, deflection of the elastic foundation was used in the air film height equation. Combined Dirichlet and Neumann-type boundary conditions were used for static load performance predictions. To verify the theoretical model, tests were performed with three different thrust air foil bearings with outer radii of 45, 50, and 55 mm. The rotating speed ranged from 10,000 to 25,000 rpm. From the test results, the model using nonlinear stiffness was in better agreement with the experimental results than the model using linear stiffness.

Saturation of front propagation in a reaction diffusion process describing plasma damage in porous low-kmaterials

We propose a three-component reaction-diffusion system yielding an asymptotic logarithmic time dependence for a moving interface. This is naturally related to a Stefan problem for which both one-sided Dirichlet-type and von Neumann-type boundary conditions are considered. We integrate the dependence of the interface motion on diffusion and reaction parameters and we observe a change from transport behavior and interface motion $~{t}^{1/2}$ to logarithmic behavior $~\mathrm{ln}t$ as a function of time. We apply our theoretical findings to the propagation of carbon depletion in porous dielectrics exposed to a low temperature plasma. This diffusion saturation is reached after about one minute in typical experimental situations of plasma damage in microelectronic fabrication. We predict the general dependencies on porosity and reaction rates.

Inverse problems for Bessel-type differential equations on noncompact graphs using spectral data

Bessel-type differential operators on noncompact star-type graphs under generalized Neumann-type boundary conditions are studied. We establish properties of the spectral characteristics and then we investigate the inverse problem of recovering the operator from the so-called spectral data. For this inverse problem, we prove a uniqueness theorem and propose a procedure for constructing the solution using the method of spectral mappings.

Dynamical gauge fields in holographic superconductors

Holographic superconductors have been studied so far in the absence of dynamical electromagnetic fields, namely in the limit in which they coincide with holographic superfluids. It is possible, however, to introduce dynamical gauge fields if a Neumann-type boundary condition is imposed on the AdS-boundary. In 3 + 1 dimensions, the dual theory is a 2 + 1 dimensional CFT whose spectrum contains a massless gauge field, signaling the emergence of a gauge symmetry. We study the impact of a dynamical gauge field in vortex configurations where it is known to significantly affect the energetics and phase transitions.

Polynomial spline approach for solving second-order boundary-value problems with Neumann conditions

In this paper, a new difference scheme based on quartic splines is derived for solving linear and nonlinear second-order ordinary differential equations subject to Neumann-type boundary conditions. The scheme can achieve sixth order accuracy at the interior nodal points and fourth order accuracy at and near the boundary, which is superior to the well-known Numerov’s scheme with the accuracy being fourth order. Convergence analysis of the present method for linear cases is discussed. Finally, numerical results for both linear and nonlinear cases are given to illustrate the efficiency of our method.

Perturbation of essential spectra of exterior elliptic problems

For a second-order symmetric strongly elliptic differential operator on an exterior domain in ℝ n , it is known from the works of Birman and Solomiak that a change in the boundary condition from the Dirichlet condition to an elliptic Neumann or Robin condition leaves the essential spectrum unchanged, in such a way that the spectrum of the difference between the inverses satisfies a Weyl-type asymptotic formula. We show that one can increase, but not diminish, the essential spectrum by imposition of other Neumann-type nonelliptic boundary conditions. The results are extended to 2m-order operators, where it is shown that for any selfadjoint realization defined by an elliptic normal boundary condition (other than the Dirichlet condition), one can augment the essential spectrum at will by adding a suitable operator to the mapping from free Dirichlet data to Neumann data. We also show here an extension of the spectral asymptotics formula for the difference between inverses of elliptic problems. The proofs rely on Kreĭn-type formulae for differences between inverses, and cutoff techniques, combined with results on singular Green operators and their spectral asymptotics.

An implicit Galerkin finite element Runge–Kutta algorithm for shock-structure investigations

This paper introduces an implicit high-order Galerkin finite element Runge–Kutta algorithm for efficient computational investigations of shock structures. The algorithm induces no spatial-discretization artificial diffusion, relies on cubic and higher-degree elements for an accurate resolution of the steep shock gradients, uses an implicit time integration for swift convergence to steady states, and employs original Neumann-type outlet boundary conditions in the form of generalized Rankine–Hugoniot conditions on normal stress and balance of heat flux and deviatoric-stress work per unit time. The formulation automatically calculates the spatial extent of the shock and employs the single non-dimensional (0, 1) computational domain for the determination of any shock structure. Since it is implicit, the algorithm rapidly generates steady shock structures, in at most 150 time steps for any upstream Mach number considered in this study. The finite element discretization is shown to be asymptotically convergent under progressive grid refinements, in respect of both the H 0 and H 1 error norms, with an H 0 accuracy order as high as 6 and reduction of the discretization error to the round-off-error threshold of 1 × 10 −9 with just 420 computational cells and 5th-degree elements. For upstream Mach numbers in the range 1.05 ⩽ M ⩽ 10.0, the computational results satisfy the Rankine–Hugoniot conditions and reflect independently published Navier–Stokes results.

Emergent gauge fields in holographic superconductors

Holographic superconductors have been studied so far in the absence of dynamical electromagnetic fields, namely in the limit in which they coincide with holographic superfluids. It is possible, however, to introduce dynamical gauge fields if a Neumann-type boundary condition is imposed on the AdS-boundary. In 3+1 dimensions, the dual theory is a 2+1 dimensional CFT whose spectrum contains a massless gauge field, signaling the emergence of a gauge symmetry. We study the impact of a dynamical gauge field in vortex configurations where it is known to significantly affect the energetics and phase transitions. We calculate the critical magnetic fields H_c1 and H_c2, obtaining that holographic superconductors are of Type II (H_c1 < H_c2). We extend the study to 4+1 dimensions where the gauge field does not appear as an emergent phenomena, but can be introduced, by a proper renormalization, as an external dynamical field. We also compare our predictions with those arising from a Ginzburg-Landau theory and identify the generic properties of Abrikosov vortices in holographic models.

Multiple-relaxation-time lattice Boltzmann model for the convection and anisotropic diffusion equation

A lattice Boltzmann model with a multiple-relaxation-time (MRT) collision operator for the convection–diffusion equation is presented. The model uses seven discrete velocities in three dimensions (D3Q7 model). The off-diagonal components of the relaxation-time matrix, which originate from the rotation of the principal axes, enable us to take into account full anisotropy of diffusion. An asymptotic analysis of the model equation with boundary rules for the Dirichlet and Neumann-type (specified flux) conditions is carried out to show that the model is first- and second-order accurate in time and space, respectively. The results of the analysis are verified by several numerical examples. It is also shown numerically that the error of the MRT model is less sensitive to the variation of the relaxation-time coefficients than that of the classical BGK model. In addition, an alternative treatment for the Neumann-type boundary condition that improves the accuracy on a curved boundary is presented along with a numerical example of a spherical boundary.

Diffusion Wave Model for Simulating Storm-Water Runoff on Highway Pavement Surfaces at Superelevation Transition

On a curved section of highway, the cross slope of the road is often designed to be superelevated to balance the centrifugal force and gravity applied on vehicles. The accumulation of storm-water runoff (sheet flow) near superelevation transitions may significantly increase due to the extended flow path and converging flow lines. A two-dimensional finite-volume-based diffusion wave model is developed to simulate the sheet flow on these geometrically complex surfaces. Both Dirichlet- and Neumann-type boundary conditions are developed for open boundaries based on kinematic wave theory. Results show that the distribution of sheet flow is closely related to the cross slope, longitudinal slope, rainfall intensity, and the width of the road. The analysis of sheet flow characteristics on superelevation transition areas suggests that the optimal longitudinal slope in the range of 0.3–0.4% minimizes the depth of storm-water runoff on the road surface.

A boundary-value problem for a non-linear equation with a fractional derivative

We consider a problem on a half-line with a Neumann-type boundary condition. We prove the global existence of solutions and find the leading term of their large-time asymptotic expansion. The case of large initial data and critical non-linearity is considered in the case when the non-linear term of the equation decays in time as rapidly as the linear terms.

Cavity phenomena in mesas of cuprate high-Tcsuperconductors under voltage bias

Modeling a single crystal of cuprate high-${T}_{c}$ superconductor, such as ${\text{Bi}}_{2}{\text{Sr}}_{2}{\text{CaCu}}_{2}{\text{O}}_{8+\ensuremath{\delta}}$, as a stack of intrinsic Josephson junctions, we formulate explicitly the cavity phenomenon of plasma oscillations and electromagnetic (EM) waves in mesas of cylindrical and annular shapes. The phase differences of the junctions are governed by the inductively coupled sine-Gordon equations, with the Neumann-type boundary condition for sample thickness much smaller than the EM wavelength, which renders the superconductor single crystal a cavity. Biasing a dc voltage in the $c$ direction, a state with $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}$ kinks in the superconductivity phase difference piled up alternatively along the $c$ axis is stabilized. The $\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}$ phase kinks provide interlock between superconductivity phases in adjacent junctions, taking the advantage of huge inductive couplings inherent in the cuprate superconductors, which establishes the coherence across the whole system of more than $\ensuremath{\sim}600$ junctions. They also permit a strong coupling between the lateral cavity mode of the transverse Josephson plasma and the $c$-axis bias, and enhance the plasma oscillation significantly at the cavity modes which radiates EM waves in the terahertz band when the lateral size of mesa is set to tens of micrometers. It is discussed that the cavity mode realized in a very recent experiment using a cylindrical mesa can be explained by the present theory. In order to overcome the heating effect, we propose to use annular geometry. The dependence of frequency on the radius ratio is analyzed, which reveals that the shape tailor is quite promising for improving the present technique of terahertz excitation. The annular geometry may be developed as a waveguide resonator, mimicking the fiber lasers for visible lights.

The traction initial-boundary value problem for bending of thermoelastic plates with cracks

The initial-boundary value problem for bending of a thermoelastic plate weakened by a crack, with Neumann-type boundary conditions along the edges of the crack, is studied, and its unique solvability in spaces of distributions is proved by means of a combination of the Laplace transformation and variational methods.

An asymptotic model for the formation and evolution of air gaps in vertical continuous casting

The formation of an air gap at the mould–metal interface in vertical continuous casting has long been known to have a detrimental effect on the efficiency of the process, and has therefore attracted attempts at mathematical modelling. While almost all current efforts consist of complex three-dimensional numerical simulations of the phenomenon, this paper considers instead an asymptotic model that captures the essential characteristics. The model is thermomechanical and is derived for a geometry, where the generalized plane strain approximation is appropriate. Although two-way coupling between the thermal and mechanical problems is accounted for, it is found that the problems decouple at leading order anyway, and that the thickness of the air gap does not depend on the constitutive relation used for describing the inelastic strains. Furthermore, a criterion for the onset of air-gap formation is derived in terms of the process operating parameters. Mathematically, we obtain a moving boundary problem for a parabolic partial differential equation with a degenerate initial condition and a non-standard Neumann-type boundary condition. Sample computations are performed using parameters for the continuous casting of the copper, and the results, qualitative trends and possible extensions are discussed.