Finite Difference Numerical Method Research Articles

Influence of V-shaped plasmonic nanostructures on beam propagation

The influence of V-shaped plasmonic nanostructures caused by focused ion beam milling on transmission properties are addressed in this paper. In order to evaluate the influence, a finite-difference and time-domain numerical method is employed for simulation of the plasmonic structures flanked with the corrugations with equal groove depth at the exit side of a metal film which is coated on a quartz substrate. Our simulation results have shown that a V-shaped subwavelength central slit/aperture is helpful for beam shaping. It plays a positive and dominant role in transmissivity. However, the V-shaped grooves cause a red shift of the peak wavelength and a broadening of the cut-off wavelength. This leads to the beam diverging in the region of the far field. In the case of the existence of both the V-shaped central slit and the V-shaped grooves, the influence of the grooves is less compared to that of the central slit, and can be ignored theoretically.

Lattice thermal conductivity of nanoscale AlN∕GaN∕AlN heterostructures: Effects of partial phonon spatial confinement

We report a detailed theoretical modeling of lattice thermal conductivity in nanoscale AlN∕GaN∕AlN heterostructures. Thermal conductivity in such heterostructures is derived based on the solution of the phonon Boltzmann equation in the relaxation-time approximation. Phonon dispersion relations are obtained in the elastic continuum approximation using the finite-difference numerical method. Quasi-two-dimensional (2D) phonon density of states is derived using the actual phonon dispersion. To investigate the effects of partial phonon spatial confinement, numerical simulations are performed for both the three-layer heterostructures and the single GaN thin films. The dependence of the thermal conductivity on the core or cladding layer thickness in an AlN∕GaN∕AlN heterostructure is also discussed. We have demonstrated that partial phonon confinement leads to a higher thermal conductivity in an AlN∕GaN∕AlN heterostructure than that in a single GaN thin film with the same total structure thickness. Such thermal conductivity can also be tuned higher or lower by adjusting the core or cladding layer thickness without changing the total thickness of the structure. Obtained results have quantitatively shown that it is possible to improve heat conduction in semiconductor nanostructures through phonon spectrum engineering, in agreement with other recent studies.

Numerical method of moments for solute transport in a nonstationary flow field conditioned on hydraulic conductivity and head measurements

The unconditional stochastic studies on groundwater flow and solute transport in a nonstationary conductivity field show that the standard deviations of the hydraulic head and solute flux are very large in comparison with their mean values (Zhang et al. in Water Resour Res 36:2107–2120, 2000; Wu et al. in J Hydrol 275:208–228, 2003; Hu et al. in Adv Water Resour 26:513–531, 2003). In this study, we develop a numerical method of moments conditioning on measurements of hydraulic conductivity and head to reduce the variances of the head and the solute flux. A Lagrangian perturbation method is applied to develop the framework for solute transport in a nonstationary flow field. Since analytically derived moments equations are too complicated to solve analytically, a numerical finite difference method is implemented to obtain the solutions. Instead of using an unconditional conductivity field as an input to calculate groundwater velocity, we combine a geostatistical method and a method of moment for flow to conditionally simulate the distributions of head and velocity based on the measurements of hydraulic conductivity and head at some points. The developed theory is applied in several case studies to investigate the influences of the measurements of hydraulic conductivity and/or the hydraulic head on the variances of the predictive head and the solute flux in nonstationary flow fields. The study results show that the conditional calculation will significantly reduce the head variance. Since the hydraulic head measurement points are treated as the interior boundary (Dirichlet boundary) conditions, conditioning on both the hydraulic conductivity and the head measurements is much better than conditioning only on conductivity measurements for reduction of head variance. However, for solute flux, variance reduction by the conditional study is not so significant.

Separation of Organic Compounds by Gradient Simulated Moving Bed Chromatography

A promising solution for the separation of organic compounds in pharmaceutical industry is the gradient simulated moving bed chromatography (SMB). The above mentioned process can be used for the production of high purity materials (isomers, optical isomers, biomolecules). We assumed isotherm, isochor equilibrium adsorption (competitive multicomponent Langmuir-adsorption equilibrium) in the mathematical model and neglected the effects of axial dispersion. In our present work we extended the mathematical model with solvent adsorption-desorption processes (acetone-dichloromethane eluent, steroid compounds, silicagel). The mathematical model was solved by finite differences numerical mathematical method using PC. The gradient SMB separations were carried out with a laboratory scale four column equipment in open loop system at 1:1:2:0 column configuration. The SMB equipment (L=25 cm, I.D.=1 cm) was planned and constructed for separation of a steroid mixture using YMC S-50 silica gel as adsorbent (specific surface=798.63 m g). The effect of switching time change (5.5 min, 9 min, 11.5 min, 22.5 min) on component separation was examined. During our measurements the amount of acetone in dichloromethane was 55 % v/v in the fresh eluent and pure dichloromethane in the feed. The process variables of the gradient SMB (product purity, yield, productivity, specific solvent consumption) are very favourable. Our conclusion is that the measured and the calculated data agree well and we have produced more than 99.9 %m/m purity and 90 % yield , productivity 1300-3000 g steroid kg adsorbent day at 0.1-0.3 m fresh eluent kg steroid eluent consumption.

English

Predicting post-mining water quality is a key task in managing mine drainage. The most scientifically-defensible approach would be to undertake thermodynamics-based modeling of coupled geochemical and hydrodynamic processes occurring in flooded and partly flooded underground workings. Most models currently in use make the fatal assumption of local equilibrium, or else do not include all important geochemical and geometrical components of an abandoned mine system. Hydrologic and geochemical data have been incorporated into a model (POllutant Sources & Sinks in Underground Mines) that models one-dimensional reactive transport along a mine roadway. POSSUM is an object-oriented program that models the evolution of mine water quality, accounting for contaminant sources and sinks, water level changes (due to rebound and seasonal fluctuations), reaction kinetics, variable flow regimes, infiltration through porous media, and inflow from secondary roadways. The main metals included in POSSUM's analysis are Fe, Al, Mn, Ni, and Zn. Additionally, carbonates, including limestone, ankerite and siderite, and silicates, including K-feldspar, anorthite, albite and muscovite are represented in POSSUM. Reversible sorption is also accounted for. Contaminant transport is solved using the NOAH 1-D program developed at University of Newcastle. POSSUM solves for concentrations in both the aqueous and solid phases using sequential iterations of finite difference numerical methods. Data sets from the National Coal Mining Museum in Wakefield and Coal Authority boreholes have been collected for comparison with model results. In contrast to pre-existing models, POSSUM couples kinetically controlled multi-component reactions with variable transport and will further the 'state of the art' by improving the defensibility of predictions of water quality changes over time.

Experimental and theoretical analysis of annular two-phase flow regimen in direct steam generation for a low-power system

This study aims to quantify and to model the temperature profile around an absorber tube of a parabolic trough concentrator with low fluid flow. This study was specifically developed for the solar power plant of the Engineering Institute, National University of Mexico. This work presents experimental results under saturated conditions and low pressures (1.5–3 bar) using water as the thermal and working fluid for direct steam generation (DSG). The control variable was feed flow. Solar irradiance was used as the restriction variable because all experimental tests should be developed under very specific values of this variable (for example, I > 700 W/m 2). The objective of this experiment was to study the thermal behavior of a temperature gradient around the absorber tube under steady-state conditions and with low flow. Additionally, a theoretical analysis was carried out by means of the homogeneous heat conduction equation in the cylindrical coordinate system using only two dimensions ( r, ϕ). The finite-difference numerical method was used with the purpose of proposing a solution and obtaining a temperature profile. The objective of this theoretical analysis was to complement the experimental tests carried out for direct steam generation (DSG) with annular two-phase flow patterns for low powers in parabolic trough concentrators with carbon steel receivers.

Electromagnetic energy in a dispersive metamaterial

An expression for the electromagnetic field energy density in a dispersive lossy left-handed metamaterial, consisting of an array of split-ring resonators and an array of wires, is derived. An electromagnetic field with general time dependence is considered. The outcome is compared with previously published results. In the absence of losses, agreement with the general result for the energy density in a dispersive material is obtained. The formulae are verified using the finite-difference time-domain numerical method. The applicability of two commonly used permeability models to the problem of calculating the energy stored in an array of split-ring resonators is discussed.

Concentration profiles and effective diffusion coefficients of sucrose and water in osmo-dehydrated apples

The spatial distribution of water and sugars in half-fresh apples dehydrated in sucrose solutions (30% and 50% w/w, 27 °C) for 2, 4 and 8 h, was determined. Each half was sliced as from the exposed surface. The density, water and sugar contents were determined for each piece. A mathematical model was fitted to the experimental data of the water and sucrose contents considering the overall flux and tissue shrinkage. A numerical method of finite differences permitted the calculation of the effective diffusion coefficients as a function of concentration, using material coordinates and integrating the two differential equations (for water and sucrose) simultaneously. The coefficients obtained were one or even two orders of magnitude lower than those for pure solutions and presented unusual concentration dependence. The behaviour of the apple tissue was also studied using light microscopy techniques to obtain images of the osmotically treated pieces (20%, 30% and 50% w/w sucrose solutions for 2, 4 and 8 h).

Oscillating viscoelastic flow in a curved duct—Exact analytical and numerical solution

The analytical solution of the equations of motion is presented for a fully developed laminar flow in a curved channel with circular walls, when an oscillating circumferential pressure gradient is imposed (finite gap oscillating Dean flow). The fluid studied, is a viscoelastic (Jeffrey) fluid. The validity of the solution is verified by a finite difference numerical method.

High-fidelity simulations of ion trajectories in miniature ion traps using the boundary-element method

In this paper we present numerical modeling results for endcap and linear ion traps, used for experiments at the National Physical Laboratory in the UK and Innsbruck University respectively. The secular frequencies for Strontium-88 and Calcium-40 ions were calculated from ion trajectories, simulated using boundary-element and finite-difference numerical methods. The results were compared against experimental measurements. Both numerical methods showed high accuracy with boundary-element method being more accurate. Such simulations can be useful tools for designing new traps and trap arrays. They can also be used for obtaining precise trapping parameters for desired ion control when no analytical approach is possible as well as for investigating the ion heating rates due to thermal electronic noise.

A Numerical Method of Moments for Solute Transport in Nonstationary Flow Fields

A Lagrangian perturbation approach has been applied to develop the method of moments for predicting mean and variance of solute flux through a three-dimensional nonstationary flow field. The flow nonstationarity may stem from medium nonstationarity, finite domain boundaries, and/or fluid pumping and injecting. The solute flux is described as a space–time process where time refers to the solute flux breakthrough and space refers to the transverse displacement distribution at the control plane. The analytically derived moment equations for solute transport in a nonstationary flow field are too complicated to solve analytically, a numerical finite difference method is implemented to obtain the solutions. This approach combines the stochastic model with the flexibility of the numerical method to boundary and initial conditions. This method is also compared with the numerical Monte Carlo method. The calculation results indicate the two methods match very well when the variance of log-conductivity is small, but the method of moment is more efficient in computation.

On uniformly accurate high-order Boussinesq difference equations for water waves

A new accurate finite-difference (AFD) numerical method is developed specifically for solving high-order Boussinesq (HOB) equations. The method solves the water-wave flow with much higher accuracy compared to the standard finite-difference (SFD) method for the same computer resources. It is first developed for linear water waves and then for the nonlinear problem. It is presented for a horizontal bottom, but can be used for variable depth as well. The method can be developed for other equations as long as they use Padé approximation, for example extensions of the parabolic equation for acoustic wave problems. Finally, the results of the new method and the SFD method are compared with the accurate solution for nonlinear progressive waves over a horizontal bottom that is found using the stream function theory. The agreement of the AFD to the accurate solution is found to be excellent compared to the SFD solution. Copyright © 2005 John Wiley & Sons, Ltd.

A fast and accurate numerical tool to model the modal properties of photonic-bandgap fibers

We describe a finite-difference numerical method that allows us to simulate the modes of air-core photonic-bandgap fibers (PBF) of any geometry in minutes on a standard PC. The modes' effective indices and fields are found by solving a vectorial transverse magnetic-field equation in a matrix form, which can be done quickly because this matrix is sparse and because we reduce its bandwidth by rearranging its elements. The Stanford Photonic-Bandgap Fiber code, which is based on this method, takes about 4 minutes to model 20 modes of a typical PBF on a PC. Other advantages include easy coding, faithful modeling of the abrupt discontinuities in the index profile, high accuracy, and applicability to waveguides of arbitrarily complex profile.

Schwarzschild tests of the Wahlquist-Estabrook-Buchman-Bardeen tetrad formulation for numerical relativity

A first order symmetric hyperbolic tetrad formulation of the Einstein equations developed by Estabrook and Wahlquist and put into a form suitable for numerical relativity by Buchman and Bardeen (the WEBB formulation) is adapted to explicit spherical symmetry and tested for accuracy and stability in the evolution of spherically symmetric black holes (the Schwarzschild geometry). The lapse and shift, which specify the evolution of the coordinates relative to the tetrad congruence, are reset at frequent time intervals to keep the constant-time hypersurfaces nearly orthogonal to the tetrad congruence and the spatial coordinate satisfying a kind of minimal rate of strain condition. By arranging through initial conditions that the constant-time hypersurfaces are asymptotically hyperbolic, we simplify the boundary value problem and improve stability of the evolution. Results are obtained for both tetrad gauges (``Nester'' and ``Lorentz'') of the WEBB formalism using finite difference numerical methods. We are able to obtain stable unconstrained evolution with the Nester gauge for certain initial conditions, but not with the Lorentz gauge.

Modeling Depth Filtration of Activated Sludge Effluent Using a Compressible Medium Filter

A new filter, using a compressible-filter medium, has been evaluated for the filtration of secondary effluent. The ability to adjust the properties of the filter medium by altering the degree of the medium compression is a significant departure from conventional depth-filtration technology. Unlike conventional filters, it is possible to optimize the performance of the compressible-medium filter (CMF) by adjusting the medium properties (i.e., collector size, porosity, and depth) to respond to the variations in influent quality. Because existing filter models cannot be used to predict the performance of the CMF, a new predictive model has been developed to describe the filtration performance of the CMF and the effect of medium-compression ratio. The model accounts for the fact that the properties of the filter medium change with time and depth. The model, developed for heterodisperse suspensions and variable influent total suspended solids concentrations, can be used to predict all possible phases of filtration (i.e., ripening, constant removal, and breakthrough). A hyperbolic-type, second-order, nonlinear, partial-differential equation was derived to model the CMF. The equation was solved using the finite-difference numerical method. The accuracy of the numerical method was tested by a sensitivity analysis and a convergence test. The model is first-order accurate with respect to medium depth and time. Field data were obtained for the filtration of settled secondary effluent using a CMF with a capacity of 1200 m3/d. Model predictions were compared with observed performance from filter runs conducted at medium-compression ratios between 15 and 40% and filtration rates from 410 to 820 L/m2 min. The difference between the observed and the predicted values was found to be within 0 to 15%.

High order finite difference numerical methods for time-dependent convection--dominated problems

We continue to investigate a family of fully discrete finite difference implicit methods already proposed for the numerical solution of one-dimensional hyperbolic systems of conservation laws. In t...

Diffusion of substrate and oxygen in aerobic granule

In this study, one-dimensional models were developed to simultaneously describe diffusion of substrate and dissolved oxygen in aerobic granule under various conditions in sequencing batch reactor (SBR), meanwhile the models were further applied to predict changes of the bulk substrate concentration in the course of SBR operation. The proposed models were solved numerically by the Numerical Finite Difference Methods without making any additional assumptions. Results showed that diffusions of substrate and dissolved oxygen in aerobic granule would be a dynamic process, and they were closely interrelated. For aerobic granules with a size smaller than 0.4 mm, no diffusion limitation of substrate and dissolved oxygen occurred. However, when aerobic granules grew to a size larger than 0.5 mm, dissolved oxygen became a major limiting factor of metabolic activity of aerobic granule over substrate. These imply that the optimal size of aerobic granule would be less than 0.5 mm in the sense of mass transfer. It was revealed that the substrate removal rate was inversely related to the size of aerobic granules, the substrate removal rate by aerobic granules with a size of 0.5 mm was almost three times of that by aerobic granules with a size of 1 mm.

Modeling of temperature distributions in canned tomato based dip during industrial pasteurization

A mathematical model for the time prediction of three-dimensional temperature distributions in the bottled tomato based dip (originally named “Ajvar”) during industrial pasteurization is presented. A model follows the pasteurization process through six zones of industrial pasteurizator with different operational conditions. The thermal diffusivity of tomato based dip is experimentally determined. The numerical method of finite differences is used to solve a three-dimensional heat conduction equation with variable boundary and initial conditions. The temperature predictions obtained by dynamic model enable considerable optimization of the described industrial process of canned food pasteurization.

PyNSol: Objects as Scaffolding

The author presents a design for a software framework in which object-orientation is used intensively during the construction of an executable but where there is no runtime presence of objects at all. Indeed, they can't be present because the final executable will be compiled and linked from Fortran 77 source. This counterintuitive strategy emerges from the problem domain and its constraints; it is presented here both for its intrinsic interest and for the new light it sheds on how object-orientation can be useful in scientific programming. The hypothesis presented here isn't that object-oriented programming is generally inappropriate - rather, the author proposes that the effective use of object techniques is a consequence of good design, not a precondition for it. PyNSol, the Pythonic Numeric Solver is an application development environment for finite difference and finite volume numerical methods. PyNSol is targeted to support researchers in the environmental sciences, but it could also be useful in other settings. PyNSol is discussed here primarily as an example of an alternative approach to scientific computing.

Beaming effect from increased-index photonic crystal waveguides

We study the beaming effect of light for the case of increased-index photonic crystal (PhC) waveguides, formed through the omission of low-dielectric media in the waveguide region. We employ the finite-difference time-domain numerical method for characterizing the beaming effect and determining the mechanisms of loss and the overall efficiency of the directional emission. We find that, while this type of PhC waveguides is capable of producing a highly collimated emission as was demonstrated experimentally, the inherent characteristics of the structure result in a restrictively low efficiency in the coupling of light into the collimated beam of light.