On Finite Element Methods for 2nd order (semi–) periodic Eigenvalue Problems

We deal with a class of elliptic eigenvalue problems (EVPs) on a rectangle Ω ⊂ R^2 , with periodic or semi–periodic boundary conditions (BCs) on ∂Ω. First, for both types of EVPs, we pass to a proper variational formulation which is shown to fit into the general framework of abstract EVPs for symmetric, bounded, strongly coercive bilinear forms in Hilbert spaces, see, e.g., [13, §6.2]. Next, we consider finite element methods (FEMs) without and with numerical quadrature. The aim of the paper is to show that well–known error estimates, established for the finite element approximation of elliptic EVPs with classical BCs, hold for the present types of EVPs too. Some attention is also paid to the computational aspects of the resulting algebraic EVP. Finally, the analysis is illustrated by two non-trivial numerical examples, the exact eigenpairs of which can be determined.

Optimal Control of a Second Order Parabolic Heat Equation

In this paper, we are concerned with the optimal control boundary control of a second order parabolic heat equation. Using the results in [Evtushenko, 1997] and spatial central finite difference with diagonally implicit Runge-Kutta method (DIRK) is applied to solve the parabolic heat equation. The conjugate gradient method (CGM) is applied to solve the distributed control problem. Numerical results are reported.

Full-vectorial modeling of femtosecond pulses for laser inscription of photonic structures

During the last decade, microfabrication of photonic devices by means of intense femtosecond (fs) laser pulses has emerged as a novel technology. A common requirement for the production of these devices is that the refractive index modification pitch size should be smaller than the inscribing wavelength. This can be achieved by making use of the nonlinear propagation of intense fs laser pulses. Nonlinear propagation of intense fs laser pulses is an extremely complicated phenomenon featuring complex multiscale spatiotemporal dynamics of the laser pulses. We have utilized a principal approach based on finite difference time domain (FDTD) modeling of the full set of Maxwell's equations coupled to the conventional Drude model for generated plasma. Nonlinear effects are included, such as self-phase modulation and multiphoton absorption. Such an approach resolves most problems related to the inscription of subwavelength structures, when the paraxial approximation is not applicable to correctly describe the creation of and scattering on the structures. In a representative simulation of the inscription process, the signature of degenerate four wave mixing has been found. © 2012 Optical Society of America.

Development of a mathematical and computer model to assess water quality impacts of hazardous compounds from minor gasoline spills in the inter-tidal zone of the Miami River, Florida

The objective of this study was to develop a model to predict transport and fate of gasoline components of environmental concern in the Miami River by mathematically simulating the movement of dissolved benzene, toluene, xylene (BTX), and methyl-tertiary-butyl ether (MTBE) occurring from minor gasoline spills in the inter-tidal zone of the river. Computer codes were based on mathematical algorithms that acknowledge the role of advective and dispersive physical phenomena along the river and prevailing phase transformations of BTX and MTBE. Phase transformations included volatilization and settling. ^ The model used a finite-difference scheme of steady-state conditions, with a set of numerical equations that was solved by two numerical methods: Gauss-Seidel and Jacobi iterations. A numerical validation process was conducted by comparing the results from both methods with analytical and numerical reference solutions. Since similar trends were achieved after the numerical validation process, it was concluded that the computer codes algorithmically were correct. The Gauss-Seidel iteration yielded at a faster convergence rate than the Jacobi iteration. Hence, the mathematical code was selected to further develop the computer program and software. The model was then analyzed for its sensitivity. It was found that the model was very sensitive to wind speed but not to sediment settling velocity. ^ A computer software was developed with the model code embedded. The software was provided with two major user-friendly visualized forms, one to interface with the database files and the other to execute and present the graphical and tabulated results. For all predicted concentrations of BTX and MTBE, the maximum concentrations were over an order of magnitude lower than current drinking water standards. It should be pointed out, however, that smaller concentrations than the latter reported standards and values, although not harmful to humans, may be very harmful to organisms of the trophic levels of the Miami River ecosystem and associated waters. This computer model can be used for the rapid assessment and management of the effects of minor gasoline spills on inter-tidal riverine water quality. ^

Three dimensional circulation of water with variable wind in the Caspian Sea

This research is based on a numerical model for forecasting the three-dimensional behavior of (sea) water motion due to the effect of a variable wind velocity. The results obtained are then analyzed and compared with observation. This model is based on the equations that overcome the current and distribution of temperature by applying the method of finite difference with assuming Δx, Δy as constant and Δz, variable. The model is based on the momentum equation, continuity equation and thermodynamic energy equation and tension at the surface and middle layers and bottom stress. The horizontal and vertical eddy viscosity and thermal diffusivity coefficients we used in accordance with that of the Bennet on Outario Lake (1977). Considering the Caspian Sea dimension in numerical model the Coriolis parameter used with β effects and the approximation Boussines have been used. For the program controlling some simple experiment with boundary condition similar to that of the Caspian Sea have been done. For modeling the Caspian Sea the grid of the field was done as follows: At horizontal surface grid size is 10×10km extension and at vertical in 10 layers with varying thickness from surface to bed respectively as: 5, 10, 20, 3, 50, 100, 150, 200, 25, 500 and higher. The data of wind as velocity، direction and temperature of water related to 15th September 1995 at 6،12 and 18 o’clock were obtained from synoptic station at the Caspian Sea shore and the research marine of Haji Alief. The information concerning shore wind was measured and by the method of SPM (shore protection manual) was transferred to far shore winds through interpolation and by use of inverse square distance of position distribution of the wind velocity at the Caspian surface field was obtained. The model has been evaluated according to the reports and observations. Through studying the position of the current in different layers، the velocity in the cross section in the northern، southern and the middle layers، will be discussed. The results reveal the presence of the circulation cells in the three above mentioned areas. The circulation with depth is reduced too. The results obtained through the numerical solution of the temperature equation have been compared with the observation. The temperature change in different layers in cross section illustrates the relative accordance of the model mentioned.

Energy spectra for quantum wires and two-dimensional electron gases in magnetic fields with Rashba and Dresselhaus spin-orbit interactions

We introduce an analytical approximation scheme to diagonalize parabolically confined two-dimensional (2D) electron systems with both the Rashba and Dresselhaus spin-orbit interactions. The starting point of our perturbative expansion is a zeroth-order Hamiltonian for an electron confined in a quantum wire with an effective spin-orbit induced magnetic field along the wire, obtained by properly rotating the usual spin-orbit Hamiltonian. We find that the spin-orbit-related transverse coupling terms can be recast into two parts W and V, which couple crossing and noncrossing adjacent transverse modes, respectively. Interestingly, the zeroth-order Hamiltonian together with W can be solved exactly, as it maps onto the Jaynes-Cummings model of quantum optics. We treat the V coupling by performing a Schrieffer-Wolff transformation. This allows us to obtain an effective Hamiltonian to third order in the coupling strength k(R)l of V, which can be straightforwardly diagonalized via an additional unitary transformation. We also apply our approach to other types of effective parabolic confinement, e. g., 2D electrons in a perpendicular magnetic field. To demonstrate the usefulness of our approximate eigensolutions, we obtain analytical expressions for the nth Landau-level g(n) factors in the presence of both Rashba and Dresselhaus couplings. For small values of the bulk g factors, we find that spin-orbit effects cancel out entirely for particular values of the spin-orbit couplings. By solving simple transcendental equations we also obtain the band minima of a Rashba-coupled quantum wire as a function of an external magnetic field. These can be used to describe Shubnikov-de Haas oscillations. This procedure makes it easier to extract the strength of the spin-orbit interaction in these systems via proper fitting of the data.

Solution of a heterogeneous modeling of a horizontal-flow anaerobic immobilized biomass (HAIB) reactor by the sequencing method

A modeling study was completed to develop a methodology that combines the sequencing and finite difference methods for the simulation of a heterogeneous model of a tubular reactor applied in the treatment of wastewater. The system included a liquid phase (convection diffusion transport) and a solid phase (diffusion reaction) that was obtained by completing a mass balance in the reactor and in the particle, respectively. The model was solved using a pilot-scale horizontal-flow anaerobic immobilized biomass (HAIB) reactor to treat domestic sewage, with the concentration results compared with the experimental data. A comparison of the behavior of the liquid phase concentration profile and the experimental results indicated that both the numerical methods offer a good description of the behavior of the concentration along the reactor. The advantage of the sequencing method over the finite difference method is that it is easier to apply and requires less computational time to model the dynamic simulation of outlet response of HAIB.

Modelling grain boundary migration during geometric dynamic recrystallization

High-angle grain boundary migration is predicted during geometric dynamic recrystallization (GDRX) by two types of mathematical models. Both models consider the driving pressure due to curvature and a sinusoidal driving pressure owing to subgrain walls connected to the grain boundary. One model is based on the finite difference solution of a kinetic equation, and the other, on a numerical technique in which the boundary is subdivided into linear segments. The models show that an initially flat boundary becomes serrated, with the peak and valley migrating into both adjacent grains, as observed during GDRX. When the sinusoidal driving pressure amplitude is smaller than 2 pi, the boundary stops migrating, reaching an equilibrium shape. Otherwise, when the amplitude is larger than 2 pi, equilibrium is never reached and the boundary migrates indefinitely, which would cause the protrusions of two serrated parallel boundaries to impinge on each other, creating smaller equiaxed grains.

a-SiC : H anti-resonant layer ARROW waveguides

Over the last decades, anti-resonant reflecting optical waveguides (ARROW) have been used in different integrated optics applications. In this type of waveguide, light confinement is partially achieved through an anti-resonant reflection. In this work, the simulation, fabrication and characterization of ARROW waveguides using dielectric films deposited by a plasma-enhanced chemical vapor deposition (PECVD) technique, at low temperatures(similar to 300 degrees C), are presented. Silicon oxynitride (SiO(x)N(y)) films were used as core and second cladding layers and amorphous hydrogenated silicon carbide(a-SiC:H) films as first cladding layer. Furthermore, numerical simulations were performed using homemade routines based on two computational methods: the transfer matrix method (TMM) for the determination of the optimum thickness of the Fabry-Perot layers; and the non-uniform finite difference method (NU-FDM) for 2D design and determination of the maximum width that yields single-mode operation. The utilization of a silicon carbide anti-resonant layer resulted in low optical attenuations, which is due to the high refractive index difference between the core and this layer. Finally, for comparison purposes, optical waveguides using titanium oxide (TiO(2)) as the first ARROW layer were also fabricated and characterized.

Computationally efficient solution techniques for adsorption problems involving steep gradients in bidisperse particles

A piecewise uniform fitted mesh method turns out to be sufficient for the solution of a surprisingly wide variety of singularly perturbed problems involving steep gradients. The technique is applied to a model of adsorption in bidisperse solids for which two fitted mesh techniques, a fitted-mesh finite difference method (FMFDM) and fitted mesh collocation method (FMCM) are presented. A combination (FMCMD) of FMCM and the DASSL integration package is found to be most effective in solving the problems. Numerical solutions (FMFDM and FMCMD) were found to match the analytical solution when the adsorption isotherm is linear, even under conditions involving steep gradients for which global collocation fails. In particular, FMCMD is highly efficient for macropore diffusion control or micropore diffusion control. These techniques are simple and there is no limit on the range of the parameters. The techniques can be applied to a variety of adsorption and desorption problems in bidisperse solids with non-linear isotherm and for arbitrary particle geometry.

Towards realistic simulations of lava dome growth using the level set method

The level set method has been implemented in a computational volcanology context. New techniques are presented to solve the advection equation and the reinitialisation equation. These techniques are based upon an algorithm developed in the finite difference context, but are modified to take advantage of the robustness of the finite element method. The resulting algorithm is tested on a well documented Rayleigh–Taylor instability benchmark [19], and on an axisymmetric problem where the analytical solution is known. Finally, the algorithm is applied to a basic study of lava dome growth.

Flow and axial dispersion simulation for traveling axisymmetric Taylor vortices

Numerical experiments using a finite difference method were carried out to determine the motion of axisymmetric Taylor vortices for narrow-gap Taylor vortex flow. When a pressure gradient is imposed on the flow the vortices are observed to move with an axial speed of 1.16 +/- 0.005 times the mean axial flow velocity. The method of Brenner was used to calculate the long-time axial spread of material in the flow. For flows where there is no pressure gradient, the axial dispersion scales with the square root of the molecular diffusion, in agreement with the results of Rosen-bluth et al. for high Peclet number dispersion in spatially periodic flows with a roll structure. When a pressure gradient is imposed the dispersion increases by an amount approximately equal to 6.5 x 10(-4) (W) over bar(2)d(2)/D-m, where (W) over bar is the average axial velocity in the annulus, analogous to Taylor dispersion for laminar flow in an empty tube.

A two-dimensional analytical solution of groundwater responses to tidal loading in an estuary and ocean

Previous studies on tidal dynamics of coastal aquifers have focussed on the inland propagation of oceanic tides in the cross-shore direction, a configuration that is essentially one-dimensional. Aquifers at natural coasts can also be influenced by tidal waves in nearby estuaries, resulting in a more complex behaviour of head fluctuations in the aquifers. We present an analytical solution to the two-dimensional depth-averaged groundwater flow equation for a semi-infinite aquifer subject to oscillating head conditions at the boundaries. The solution describes the tidal dynamics of a coastal aquifer that is adjacent to a cross-shore estuary. Both the effects of oceanic and estuarine tides on the aquifer are included in the solution. The analytical prediction of the head fluctuations is verified by comparison with numerical solutions computed using a standard finite-difference method. An essential feature of the present analytical solution is the interaction between the cross- and along-shore tidal waves in the aquifer area near the estuary's entry. As the distance from the estuary or coastline increases, the wave interaction is weakened and the aquifer response is reduced, respectively, to the one-dimensional solution for oceanic tides or the solution of Sun (Sun H. A two-dimensional analytical solution of groundwater response to tidal loading in an estuary, Water Resour Res 1997;33:1429-35) for two-dimensional non-interacting tidal waves. (C) 2000 Elsevier Science Ltd. All rights reserved.

Numerical modelling of double diffusion driven reactive flow transport in deformable fluid-saturated porous media with particular consideration of temperature-dependent chemical reaction rates

Numerical methods ave used to solve double diffusion driven reactive flow transport problems in deformable fluid-saturated porous media. in particular, thp temperature dependent reaction rate in the non-equilibrium chemical reactions is considered. A general numerical solution method, which is a combination of the finite difference method in FLAG and the finite element method in FIDAP, to solve the fully coupled problem involving material deformation, pore-fluid flow, heat transfer and species transport/chemical reactions in deformable fluid-saturated porous media has been developed The coupled problem is divided into two subproblems which are solved interactively until the convergence requirement is met. Owing to the approximate nature of the numerical method, if is essential to justify the numerical solutions through some kind of theoretical analysis. This has been highlighted in this paper The related numerical results, which are justified by the theoretical analysis, have demonstrated that the proposed solution method is useful for and applicable to a wide range of fully coupled problems in the field of science and engineering.