Local Boundary Conditions Research Articles

Additional small-scale boundary effects on free vibration of carbon nanotubes and their macroscopic energy meaning

Abstract In this paper, additional small-scale boundary effects on the free vibration of single-walled carbon nanotubes under various boundary conditions are investigated using an analytical nonlocal Euler-beam model. From the numerical results, it seems that the various beam boundary conditions due to local and nonlocal elasticity are equivalent. For simplicity, local boundary conditions can still be used even for the free end. It is convenient to analyze the vibration of single-walled carbon nanotubes under various types of boundary conditions. In addition, Hamilton's energy function, which can be explicitly expressed by transverse displacement, has also been derived to explain the boundary equivalence of nonlocal and local beams.

Numerical solution of the nonlinear Schrödinger equation with wave operator on unbounded domains.

In this paper, we generalize the unified approach proposed in Zhang etal. [J. Zhang, Z. Xu, and X. Wu, Phys. Rev. E 78, 026709 (2008)] to design the nonlinear local absorbing boundary conditions (LABCs) for the nonlinear Schrödinger equation with wave operator on unbounded domains. In fact, based on the methodology underlying the unified approach, we first split the original equation into two parts-the linear equation and the nonlinear equation-then achieve a one-way operator to approximate the linear equation to make the wave outgoing, and finally combine the one-way operator with the nonlinear equation to achieve the nonlinear LABCs. The stability of the equation with the nonlinear LABCs is also analyzed by introducing some auxiliary variables, and some numerical examples are presented to verify the accuracy and effectiveness of our proposed method.

Analytical solutions for stream‐aquifer flow exchange under varying head asymmetry and river penetration: Comparison to numerical solutions and use in regional groundwater models

AbstractAn analytical approach is presented to characterize the local flow exchange conductance and boundary condition between a stream and a hydraulically connected aquifer. Due to the curvilinear nature of the flow pattern in the vicinity of the stream, with numerical procedures, very fine grids would have to be used in order to secure accurate results. This is a waste of numerical efforts when the solution can be obtained analytically. Here, we show that the local analytical procedure for the determination of the stream‐aquifer flow exchange (SAFE) coefficient can be integrated within a regional numerical groundwater model through an original boundary condition. This is a very practical result. With this new approach, it is quite simple to calculate accurately the discharges on the two sides of the river when the heads are different. This is an important capability for example when wells are present near a river. The results show also that the extent of penetration of the stream is a factor in the estimation of the discharges. At the very least, the results can be used to choose a grid size that will yield correct estimates within an a priori selected acceptable level of accuracy.

Large-scale seafloor waveforms on the flanks of insular volcanoes (Aeolian Archipelago, Italy), with inferences about their origin

High-resolution multibeam data in the central and eastern part of the Aeolian volcanic arc (Southern Tyrrhenian Sea) reveal widespread large-scale waveforms on the submarine flanks of these insular volcanoes. These features show high variability in wave parameters (wavelength of 60–1600 m and wave height of 4–200 m) and morphology (from sinuous to arcuate in plan-view, and from undulating to stepped, with very steep scarps, in cross-section), indicating that different processes, at variable spatial and temporal scale, are responsible for their genesis. By relating the morphological characters of the different waveforms and the local boundary conditions (morphological setting, regional slope gradients, sediment source and dynamics) where these features form, we were able to distinguish bedforms formed by sedimentary gravity flows from bedform-like features due to gravity instability processes. In detail, bedforms have been interpreted as coarse-grained sediment waves according to their wave parameters, geometry and texture. Bedforms are always found where the slope gradients markedly decrease to values < 4°–8°, promoting a hydraulic jump in the sedimentary gravity flow. Hence, we tested if the recognized coarse-grained sediment waves can be generated as cyclic steps by comparing their morphometric characteristics with the results of experimental evidence and numerical modeling proposed in literature. The results show that cyclic step is a reasonable interpretation for these features, and we discuss the effect of flow depth, flow concentration and discharge as governing parameters affecting their wavelength, wave height and shape. • Large bedforms are widespread on the submarine flanks of the Aeolian volcanoes. • These features are related to gravity-driven processes. • Bedforms are always found downslope of a decrease of slope gradients below 4°–8°. • Bedform variability is a useful proxy to reconstruct sedimentary processes. • Bedforms can be interpreted as cyclic steps.

Dynamic charge mapping in layered polymer films

This paper describes a hybrid algorithm for the solution of the drift-diffusion equations for bipolar charge transport in layered polymer films. Dynamic charge mapping is achieved using a source distribution technique for the solution of the Poisson equation and time-integration using a fourth order total variation diminishing Runge-Kutta scheme that satisfies the Courant-Friedrichs-Levy stability criterion. An integral equation formulation allow conducting and insulating boundaries and material interfaces to be represented by equivalent free and bound charge distributions that collectively satisfy all local and far-field boundary conditions. This hybrid technique solves the drift-diffusion equations that simulate charge injection, field-dependent mobility transport, recombination, and trapping/de-trapping in the bulk and at material and physical interfaces. The resulting charge map is the taxonomy of the different charge types and their abundance; presenting a dynamic view of the temporal and spatial distributions. Results are discussed for single-layer and multi-layered samples with physical and material interfaces. Conditions for creation of charge packets and electroluminescence due to recombination are computed and discussed.

Local absorbing boundary conditions for a linearized Korteweg-de Vries equation.

The aim of this paper is to construct highly accurate local absorbing boundary conditions for a linearized Korteweg-de Vries equation on unbounded domain. The local absorbing boundary conditions are derived by Padé approximation with high accuracy, and a sequence of auxiliary variables are utilized to avoid the high-order derivatives in the absorbing boundary conditions. Then the original problem on unbounded domain is replaced by an equivalent initial boundary value problem defined on a finite domain. The finite difference method is applied to solve the reduced problem on the finite computational domain. Finally, numerical results are presented to demonstrate the effectiveness and accuracy of the proposed method.

Acoustic eigenanalysis of 2D open cavity with Vekua approximations and the method of particular solutions

This paper discusses the efficient extraction of acoustic resonances in 2D open cavities using a meshless method, the method of particular solutions. A first order, local non-reflecting boundary condition is chosen to account for the opening and Fourier-Bessel functions are employed to approximate the pressure at the borders of the cavity and inside. A minimization problem is then solved for the complex frequency range of interest. For the investigated cavity, the minimum values obtained match those found in previously published studies or by other numerical methods. But, unlike the perfectly matched layer absorbing boundary conditions now usually employed, this approach is free from spurious eigenfrequencies. Moreover, the specific treatments of the geometric singularities allow this method to be particularly efficient in the presence of corner singularities.

A Note on Gluing Dirac Type Operators on a Mirror Quantum Two-Sphere

The goal of this paper is to introduce a class of operators, which we call quantum Dirac type operators on a noncommutative sphere, by a gluing construction from copies of noncommutative disks, subject to an appropriate local boundary condition. We show that the resulting operators have compact resolvents, and so they are elliptic operators.

2D simulations of transport dynamics during tokamak fuelling by supersonic molecular beam injection

Time-dependent transport of both plasma and neutrals is simulated during supersonic molecular beam injection (SMBI) yielding the evolution of edge plasma and neutral profiles. The SMBI model is included as a module, called trans-neut, within the original BOUT++ boundary plasma turbulence code. Results of calculations are reported for the realistic divertor geometry of the HL-2A tokamak. The model can also be used to study the effect of gas puffing. A seven-field fluid model couples plasma density, heat, and momentum transport equations together with neutral density and momentum transport equations for both molecules and atoms. Collisional interactions between molecules, atoms, and plasma include dissociation, ionization, recombination and charge-exchange effects. Sheath boundary conditions and particle recycling are applied at both the wall and the divertor plates. A localized boundary condition of constant molecular flux (product of density times speed) is applied at the outermost flux surface to model the SMBI. Steady state profiles with and without particle recycling are achieved before SMBI. During SMBI, the simulation shows that neutrals can penetrate several centimetres inside the last closed (magnetic) flux surface (LCFS). Along the SMBI path, plasma density increases while plasma temperature decreases. The molecule penetration depth depends on both the SMBI flux and the initial plasma density and temperature along its path. As the local plasma density increases substantially, molecule and atom penetration depths decrease due to their higher dissociation and ionization rates if the electron temperature does not drop too low. Dynamic poloidal spreading of the enhanced plasma density region is observed due to rapid ion flow along the magnetic field (parallel) driven by a parallel pressure asymmetry during SMBI. Profile relaxation in the radial and poloidal directions is simulated after SMBI termination, showing that the plasma returns to pre-SMBI conditions on a time scale of 60 ms.

Local artificial boundary conditions for Schrödinger and heat equations by using high-order azimuth derivatives on circular artificial boundary

The aim of the paper is to design high-order artificial boundary conditions for the Schrödinger equation on unbounded domains in parallel with a treatment of the heat equation. We first introduce a circular artificial boundary to divide the unbounded definition domain into a bounded computational domain and an unbounded exterior domain. On the exterior domain, the Laplace transformation in time and Fourier series in space are applied to achieve the relation of special functions. Then the rational functions are used to approximate the relation of the special functions. Applying the inverse Laplace transformation to a series of simple rational function, we finally obtain the corresponding high-order artificial boundary conditions, where a sequence of auxiliary variables are utilized to avoid the high-order derivatives in respect to time and space. Furthermore, the finite difference method is formulated to discretize the reduced initial–boundary value problem with high-order artificial boundary conditions on a bounded computational domain. Numerical experiments are presented to illustrate the performance of our method.

Finite element model updating of flexural structures based on modal parameters extracted from dynamic distributed macro-strain responses

This article presents a novel strategy for finite element model updating of flexural structures. The method is based on modal parameters extracted from dynamic distributed macro-strain responses. The objective function that comprised low-order modal macro-strain and frequency was established first, while local bending stiffness, density, and boundary conditions of structures can be selected as the design variables. Both numerical simulation and experiment were conducted to verify the effectiveness of the proposed method. The long-gauge macro-strain sensors chosen in this article are first addressed, and the fundamental sensing properties of the macro-strain sensors confirm great dynamic measurement capacity. Simulation and experimental results show that both the local parameter (bending stiffness) and the global parameters (mass density and rotational stiffness of support) can be well identified. Notably, the updated finite element model can predict local response modal macro-strain and global response (frequency and displacement mode) because the local information-sensitive index and global information-sensitive index are included in the objective function. Therefore, the proposed finite element model updating strategy constitutes a new alternative in performance assessment of flexural structures.

A Local Pressure Boundary Condition Spectral Collocation Scheme for the Three-Dimensional Navier–Stokes Equations

A spectral collocation scheme for the three-dimensional incompressible $$({\varvec{u}},p)$$ formulation of the Navier–Stokes equations, in domains $$\varOmega $$ with a non-periodic boundary condition, is described. The key feature is the high order approximation, by means of a local Hermite interpolant, of a Neumann boundary condition for use in the numerical solution of the pressure Poisson system. The time updates of the velocity $${\varvec{u}}$$ and pressure $$p$$ are decoupled as a result of treating the pressure gradient in the momentum equation explicitly in time. The pressure update is computed from a pressure Poisson equation. Extension of the overall methodology to the Boussinesq system is also described. The uncoupling of the pressure and velocity time updates results in a highly efficient scheme that is simple to implement and well suited for simulating moderate to high Reynolds and Rayleigh number flows. Accuracy checks are presented, along with simulations of the lid-driven cavity flow and a differentially heated cavity flow, to demonstrate the scheme produces accurate three-dimensional results at a reasonable computational cost.

Techno-economic Assessment of Recovery and Reuse of Low Temperature Heat (T

This article presents a techno-economic evaluation of various options for recovery and reuse of low temperature heat (LTH) with temperatures below 350°C. A process integration approach has been applied to illustrate the investment strategies for a typical European integrated steel plant towards positive energy and environmental effects. The modelling results indicate a CO2 reduction potential of 0.44-1.80% of the total CO2 emissions by recovering and reuse of LTH from the flue gas in various process units. The pay-back time depends on reuse pathways and waste heat temperature, varying from 0.5 to 7.6 years. The results are only valid for the simulated generic site. For the results implementation at real steel plants, local boundary conditions should be considered.

On the existence of fuzzy solutions for partial hyperbolic functional differential equations

AbstractIn this paper, we consider the boundary valued problems for fuzzy partial hyperbolic functional differential equations with local and integral boundary conditions. A new weighted metric is used to investigate the existence and uniqueness of fuzzy solutions for these problems in a complete fuzzy metric space. Our results are demonstrated in some numerical examples in which we use the same strategy as Buckley-Feuring to build fuzzy solutions from fuzzifying the deterministic solutions. Then by using the continuity of the Zadeh’s extension principle combining with numerical simulations for α−cuts of fuzzy solutions, we give some representations of the surfaces of fuzzy solutions.

Computation of the Schrödinger Equation in the Semiclassical Regime on an Unbounded Domain

The study of this paper is twofold: On the one hand, we generalize the high-order local absorbing boundary conditions (LABCs) proposed in [J. Zhang et al., Comm. Comput. Phys., 10 (2011), pp. 742--766] to compute the Schrödinger equation in the semiclassical regime on an unbounded domain. We analyze the stability of the equation with LABCs and the convergence of the Crank--Nicolson scheme that discretizes it and we conclude that when the rescaled Planck constant $\varepsilon$ gets small, the accuracy deteriorates and the requirements on time step and mesh size get tough. This leads to the second part of our study. We propose an asymptotic method based on the frozen Gaussian approximation. The absorbing boundary condition is dealt with by a simple strategy that all the effects of the Gaussian functions which contribute to the outgoing waves will be eliminated by stopping the Hamiltonian flow of their centers when they get out of the domain of interest. We present numerical examples in both one and two dimensions to verify the performance of the proposed numerical methods.

Integrated Approach to CO2 Capture: Fuel Gas Decarbonisation

In this presentation, the opportunities for pre-combustion capture of fuel gas at a world-scale manufacturing complex will be evaluated. The work focusses on decarbonising fuel gas at a central location in a manufacturing complex to create a hydrogen rich fuel gas pool ready for use in existing or slightly modified heating equipment and a high pressure CO2 stream ready for storage. Evaluations are based on current reforming and gasification technologies in combination with available gas treating line-ups, as well as new line-ups with emerging gas separation technologies, like high temperature membranes. Dependent on the local process boundary conditions (sweet or sour fuel gas) one or more technologies can be suitable. Current conclusion is that the concept of pre- combustion capture at a world scale manufacturing location is feasible with available technologies. It is advised to demonstrate at a scale which is sufficiently large to evaluate the performance and reliability of the individual technologies as well as the complexity of integration with the manufacturing. This presentation will reveal some of the technology aspects regarding the minimum scale for a demo regarding membrane technology. Several membrane based process schemes have been short listed for these fuel decarbonisation demonstration projects. Recovery of CO2 at high pressure and high purity (>90 mol%) as well as carbon conversions in excess of equilibrium limits may be achieved using various process schemes patented by Shell [1], [2].Successful evaluations of CRI/Criterion H2 separation membranes in membrane steam reformers and as high temperature membrane separators have been performed by CRI/Criterion [3]. These membranes can be produced with a hydrogen permeance in the range of 50-70 [Nm3/m2.h.bar0.5]. Both the hydrogen flux and the separation selectivity are stable at temperatures of 300 - 500 oC and large differential pressures (tested up to 68 bar). H2 purity of >99% has been demonstrated for long periods (over 1 year) in high temperature gas separations.

Membrane Fusion by X-Rays: From Model Membranes to Organelles

Understanding the physical mechanisms underlying membrane fusion requires a multi winged approach, involving model systems as well as biological membranes. We study fusion intermediates occurring in form of ordered passages or stalks connecting neighbouring bilayers in multilamellar model membrane stacks. The stalks exhibit long range crystalline order with rhombohedral symmetry in a fluid ‘host’ membrane stack, which is studied by high resolution x-ray diffraction under grazing incidence angles. Information on membrane curvature, and hydration interaction can be revealed by analyzing the quantitative electron density maps, collected for controlled environmental parameters and membrane composition [1]. Phase diagrams can be analyzed in view of stabilizing or destabilizing agents for stalk formation.While in these equilibrium phase, dehydration forces bring bilayers together favoring at some point the formation of stalks, it is specific membrane proteins and their interaction which set the local boundary conditions for membrane apposition in biological membrane fusion. In view of studying fusion in the presence of SNARE proteins, we have started a x-ray structural characterization of synaptic vesicles (SV) by small-angle x-ray scattering, and currently extent this work towards studies of SV dockled to and interaction with model bilayers [2].Finally we present a novel high resolution x-ray imaging scheme capable of yielding a magnified hologram of a freely suspended lipid membrane illuminated by highly divergent and coherent x-ray beams. We propose this setup to image fusion trajectories at high resolution in future experiments [3].[1] S. Aeffner et al., Proc. Natl. Ac. Sc. doi: 10.1073/pnas.1119442109 (2010)[2] S. Ghosh et al., Biophys. J. 2012 Biophysical Journal (102), 1394-1402, (2012).[3] A. Beerlink et al., Soft Matter 8, 4595-4601 (2012).

Numerical Solution of Blow-Up Problems for Nonlinear Wave Equations on Unbounded Domains

AbstractThe numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered. Applying the unified approach, which is based on the operator splitting method, we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation, and reduce the nonlinear problem on the unbounded spatial domain to an initial-boundary-value problem on a bounded domain. Then the finite difference method is used to solve the reduced problem on the bounded computational domain. Finally, a broad range of numerical examples are given to demonstrate the effectiveness and accuracy of our method, and some interesting propagation and behaviors of the blow-up problems for nonlinear wave equations are observed.

Interaction between an incompressible flow and elastic cantilevers of circular cross-section

The purpose of this work is to study the deformation of elastic cantilevers due to hydrodynamic forces by coupled fluid–structure interaction simulations. The cantilever is placed in a rectangular duct and the Reynolds number based on bulk velocity and cantilever diameter is 400. Reduced velocities in the range π/4 to 2π are studied, which covers both un-synchronised motion and the initial branch of synchronisation. The cantilever surface is represented by a virtual boundary method which replaces a solid object in flow by additional force distribution to satisfy local boundary condition. The flow field is solved using a Cartesian finite difference code and the deformation of the cylinder a finite element approach using one-dimensional beam elements is used.

STRESS–VELOCITY COMPLETE RADIATION BOUNDARY CONDITIONS

A new high-order local Absorbing Boundary Condition (ABC) has been recently proposed for use on an artificial boundary for time-dependent elastic waves in unbounded domains, in two dimensions. It is based on the stress–velocity formulation of the elastodynamics problem, and on the general Complete Radiation Boundary Condition (CRBC) approach, originally devised by Hagstrom and Warburton in 2009. The work presented here is a sequel to previous work that concentrated on the stability of the scheme; this is the first known high-order ABC for elastodynamics which is long-time stable. Stability was established both theoretically and numerically. The present paper focuses on the accuracy of the scheme. In particular, two accuracy-related issues are investigated. First, the reflection coefficients associated with the new CRBC for different types of incident and reflected elastic waves are analyzed. Second, various choices of computational parameters for the CRBC, and their effect on the accuracy, are discussed. These choices include the optimal coefficients proposed by Hagstrom and Warburton for the acoustic case, and a simplified formula for these coefficients. A finite difference discretization is employed in space and time. Numerical examples are used to experiment with the scheme and demonstrate the above-mentioned accuracy issues.