A Direct Spectral Collocation Poisson Solver in Polar and Cylindrical Coordinates

In this paper, we present a direct spectral collocation method for the solution of the Poisson equation in polar and cylindrical coordinates. The solver is applied to the Poisson equations for several different domains including a part of a disk, an annulus, a unit disk, and a cylinder. Unlike other Poisson solvers for geometries such as unit disks and cylinders, no pole condition is involved for the present solver. The method is easy to implement, fast, and gives spectral accuracy. We also use the weighted interpolation technique and nonclassical collocation points to improve the convergence.     

A High-Order Discontinuous Galerkin Method for 2D Incompressible Flows

In this paper we introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments.     

A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows

We propose a new model and a solution method for two-phase compressible flows. The model involves six equations obtained from conservation principles applied to each phase, completed by a seventh equation for the evolution of the volume fraction. This equation is necessary to close the overall system. The model is valid for fluid mixtures, as well as for pure fluids. The system of partial differential equations is hyperbolic. Hyperbolicity is obtained because each phase is considered to be compressible. Two difficulties arise for the solution: one of the equations is written in non-conservative form; non-conservative terms exist in the momentum and energy equations. We propose robust and accurate discretisation of these terms. The method solves the same system at each mesh point with the same algorithm. It allows the simulation of interface problems between pure fluids as well as multiphase mixtures. Several test cases where fluids have compressible behavior are shown as well as some other test problems where one of the phases is incompressible. The method provides reliable results, is able to compute strong shock waves, and deals with complex equations of state.     

A Finite Volume Method for the Approximation of Diffusion Operators on Distorted Meshes

A new finite volume method is presented for discretizing general linear or nonlinear elliptic second-order partial-differential equations with mixed boundary conditions. The advantage of this method is that arbitrary distorted meshes can be used without the numerical results being altered. The resulting algorithm has more unknowns than standard methods like finite difference or finite element methods. However, the matrices that need to be inverted are positive definite, so the most powerful linear solvers can be applied. The method has been tested on a few elliptic and parabolic equations, either linear, as in the case of the standard heat diffusion equation, or nonlinear, as in the case of the radiation diffusion equation and the resistive diffusion equation with Hall term.     

A rotational approach to localized SPAMM 1-1 tagging

Magnetic resonance tagging usually relies on controlling the phase dispersion of the transverse magnetization component. Phase dispersion is, however, affected by the inherent phase of selective excitation pulses, thus limiting their combination with tagging sequences to the application of refocusable pulses, as in the localized spatial modulation of magnetization (L-SPAMM) technique. In this study, we examine the effect of selective excitation pulses on a L-SPAMM 1-1 sequence, showing that in the case of two identical pulses the phase component is canceled out, and thus preemphasis and refocus gradients are not needed, allowing us to take advantage of a constant gradient throughout the tagging sequence, and also that one might choose nonrefocusable maximum and minimum phase pulses.     

Particle Transport through Scattering Regions with Clear Layers and Inclusions

This paper introduces generalized diffusion models for the transport of particles in scattering media with nonscattering inclusions. Classical diffusion is known as a good approximation of transport only in scattering media. Based on asymptotic expansions and the coupling of transport and diffusion models, generalized diffusion equations with nonlocal interface conditions are proposed which offer a computationally cheap, yet accurate, alternative to solving the full phase-space transport equations. The paper shows which computational model should be used depending on the size and shape of the nonscattering inclusions in the simplified setting of two space dimensions. An important application is the treatment of clear layers in near-infrared (NIR) spectroscopy, an imaging technique based on the propagation of NIR photons in human tissues.     

Modeling Three-Dimensional Multiphase Flow Using a Level Contour Reconstruction Method for Front Tracking without Connectivity

Three-dimensional multiphase flow and flow with phase change are simulated using a simplified method of tracking and reconstructing the phase interface. The new level contour reconstruction technique presented here enables front tracking methods to naturally, automatically, and robustly model the merging and breakup of interfaces in three-dimensional flows. The method is designed so that the phase surface is treated as a collection of physically linked but not logically connected surface elements. Eliminating the need to bookkeep logical connections between neighboring surface elements greatly simplifies the Lagrangian tracking of interfaces, particularly for 3D flows exhibiting topology change. The motivation for this new method is the modeling of complex three-dimensional boiling flows where repeated merging and breakup are inherent features of the interface dynamics. Results of 3D film boiling simulations with multiple interacting bubbles are presented. The capabilities of the new interface reconstruction method are also tested in a variety of two-phase flows without phase change. Three-dimensional simulations of bubble merging and droplet collision, coalescence, and breakup demonstrate the new method's ability to easily handle topology change by film rupture or filamentary breakup. Validation tests are conducted for drop oscillation and bubble rise. The susceptibility of the numerical method to parasitic currents is also thoroughly assessed.     

Wave Simulation in Frozen Porous Media

We propose a numerical algorithm for simulation of wave propagation in frozen porous media, where the pore space is filled with ice and water. The model, based on a Biot-type three-phase theory, predicts three compressional waves and two shear waves and models the attenuation level observed in rocks. Attenuation is modeled with exponential relaxation functions which allow a differential formulation based on memory variables. The wavefield is obtained using a grid method based on the Fourier differential operator and a Runge–Kutta time-integration algorithm. Since the presence of slow quasistatic modes makes the differential equations stiff, a time-splitting integration algorithm is used to solve the stiff part analytically. The modeling is second-order accurate in the time discretization and has spectral accuracy in the calculation of the spatial derivatives.     

Completely Conservative and Oscillationless Semi-Lagrangian Schemes for Advection Transportation

In this paper, we present a new type of semi-Lagrangian scheme for advection transportation equation. The interpolation function is based on a cubic polynomial and is constructed under the constraints of conservation of cell-integrated average and the slope modification. The cell-integrated average is defined via the spatial integration of the interpolation function over a single grid cell and is advanced using a flux form. Nonoscillatory interpolation is constructed by choosing proper approximation to the cell-center values of the first derivative of the interpolation function, which appears to be a free parameter in the present formulation. The resulting scheme is exactly conservative regarding the cell average of the advected quantity and does not produce any spurious oscillation. Oscillationless solutions to linear transportation problems were obtained. Incorporated with an entropy-enforcing numerical flux, the presented schemes can accurately compute shocks and sonic rarefaction waves when applied to nonlinear problems.     

Simultaneous selective detection of multiple quantum spectra

A three-dimensional multiple-quantum NMR experiment that produces individual spectra of all quantum orders is described. The separation of different quantum orders is accomplished via Fourier transformation with respect to the phase of the first two pulses of a generic three-pulse multiple-quantum sequence. This dramatically reduces the time required to obtain several selectively detected spectra and enhances the sensitivity and digital resolution from that obtained using the original two-dimensional technique. The experiment is demonstrated on the protons of para-chlorotoluene dissolved in the nematic liquid crystal Merck ZLI-1132.     

NMR spectral quantitation by principal component analysis. III. A generalized procedure for determination of lineshape variations

We present a general procedure for automatic quantitation of a series of spectral peaks based on principal component analysis (PCA). PCA has been previously used for spectral quantitation of a single resonant peak of constant shape but variable amplitude. Here we extend this procedure to estimate all of the peak parameters: amplitude, position (frequency), phase and linewidth. The procedure consists of a series of iterative steps in which the estimates of position and phase from one stage of iteration are used to correct the spectra prior to the next stage. The process is convergent to a stable result, typically in less than 5 iterations. If desired, remaining linewidth variations can then be corrected. Correction of (typically) unwanted variations of these types is important not only for direct peak quantitation, but also as a preprocessing step for spectral data prior to application of pattern recognition/classification techniques. The procedure is demonstrated on simulated data and on a set of 992 (31)P NMR in vivo spectra taken from a kinetic study of rat muscle energetics. The proposed procedure is robust, makes very limited assumptions about the lineshape, and performs well with data of low signal-to-noise ratio.     

High-Order Nonreflecting Boundary Conditions without High-Order Derivatives

A wave problem in an unbounded domain is often treated numerically by truncating the infinite domain via an artificial boundary , imposing a so-called nonreflecting boundary condition (NRBC) on , and then solving the problem numerically in the finite domain bounded by . A general approach is devised here to construct high-order local NRBCs with a symmetric structure and with only low (first- or second-) order spatial and/or temporal derivatives. This enables the practical use of NRBCs of arbitrarily high order. In the case of time-harmonic waves with finite element discretization, the approach yields a symmetric C⁰ finite element formulation in which standard elements can be employed. The general methodology is presented for both the time-harmonic case (Helmholtz equation) and the time-dependent case (the wave equation) and is demonstrated numerically in the former case.     

A two-dimensional artifact from asynchronous decoupling

Many heteronuclear NMR experiments employ decoupling to collapse the heteronuclear multiplet, using decoupling schemes with a periodic phase modulation like WALTZ, MLEV, or GARP. Because of the periodic nature of these schemes, cycling sidebands are generated, whose intensity can be strongly reduced by decoupling asynchronously. We show that the most common implementation of asynchronous decoupling on modern spectrometers is such that the cycling sidebands are subjected to a periodic modulation. For multidimensional experiments, this results in ridges that can seriously compromise the quality of the spectrum. Based on our model, the artifact in a 2D [(1)H]-(15)N NOE equilibrium experiment is simulated and it is shown that the artifact can be prevented by using synchronous decoupling.     

Multigrid Solution of the Incompressible Navier–Stokes Equations for Density-Stratified Flow past Three-Dimensional Obstacles

The steady incompressible Navier–Stokes equations in three dimensions are solved for neutral and stably stratified flow past three-dimensional obstacles of increasing spanwise width. The continuous equations are approximated using a finite volume discretisation on staggered grids with a flux-limited monotonic scheme for the advective terms. The discrete equations which arise are solved using a nonlinear multigrid algorithm with up to four grid levels using the SIMPLE pressure correction method as smoother. When at its most effective the multigrid algorithm is demonstrated to yield convergence rates which are independent of the grid density. However, it is found that the asymptotic convergence rate depends on the choice of the limiter used for the advective terms of the density equation, and some commonly used schemes are investigated. The variation with obstacle width of the influence of the stratification on the flow field is described and the results of the three-dimensional computations are compared with those of the corresponding computation of flow over a two-dimensional obstacle (of effectively infinite width). Also given are the results of time-dependent computations for three-dimensional flows under conditions of strong static stability when lee-wave propagation is present and the multigrid algorithm is used to compute the flow at each time step.     

A Volume of Fluid Based Method for Fluid Flows with Phase Change

This paper presents a numerical method directed towards the simulation of flows with mass transfer due to changes of phase. We use a volume of fluid (VOF) based interface tracking method in conjunction with a mass transfer model and a model for surface tension. The bulk fluids are viscous, conducting, and incompressible. A one-dimensional test problem is developed with the feature that a thin thermal layer propagates with the moving phase interface. This test problem isolates the ability of a method to accurately calculate the thermal layers responsible for driving the mass transfer in boiling flows. The numerical method is tested on this problem and then is used in simulations of horizontal film boiling.     

Introducing Physical Relaxation Terms in Bloch Equations

The Bloch equation models the evolution of the state of electrons in matter described by a Hamiltonian. To model more physical phenomena we have to introduce phenomenological relaxation terms. The introduction of these terms has to conserve some positiveness properties. The aim of this paper is to review possible relaxation models and to provide insight into how to discretize them properly in view of numerical computations.     

A new method for variable temperature gradient shimming

Sample convection can severely attenuate the signals observed in pulsed field gradient spin--echo experiments such as those used for gradient shimming. A new class of pulse sequences is proposed, in which a double spin--echo refocuses the phase errors caused by sample convection, enabling gradient shimming to be performed reliably over a wide range of temperatures.     

Phase-Accuracy Comparisons and Improved Far-Field Estimates for 3-D Edge Elements on Tetrahedral Meshes

Edge-element methods have proved very effective for 3-D electromagnetic computations and are widely used on unstructured meshes. However, the accuracy of standard edge elements can be criticised because of their low order. This paper analyses discrete dispersion relations together with numerical propagation accuracy to determine the effect of tetrahedral shape on the phase accuracy of standard 3-D edge-element approximations in comparison to other methods. Scattering computations for the sphere obtained with edge elements are compared with results obtained with vertex elements, and a new formulation of the far-field integral approximations for use with edge elements is shown to give improved cross sections over conventional formulations.     

An Incompressible Three-Dimensional Multiphase Particle-in-Cell Model for Dense Particle Flows

A three-dimensional, incompressible, multiphase particle-in-cell method is presented for dense particle flows. The numerical technique solves the governing equations of the fluid phase using a continuum model and those of the particle phase using a Lagrangian model. Difficulties associated with calculating interparticle interactions for dense particle flows with volume fractions above 5% have been eliminated by mapping particle properties to an Eulerian grid and then mapping back computed stress tensors to particle positions. A subgrid particle, normal stress model for discrete particles which is robust and eliminates the need for an implicit calculation of the particle normal stress on the grid is presented. Interpolation operators and their properties are defined which provide compact support, are conservative, and provide fast solution for a large particle population. The solution scheme allows for distributions of types, sizes, and density of particles, with no numerical diffusion from the Lagrangian particle calculations. Particles are implicitly coupled to the fluid phase, and the fluid momentum and pressure equations are implicitly solved, which gives a robust solution.