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 subspace time-domain algorithm for automated NMR spectral normalization

Recently, two methods have been proposed for quantitatively comparing NMR spectra of control and treated samples, in order to examine the possible occurring variations in cell metabolism and/or structure in response to numerous physical, chemical, and biological agents. These methods are the maximum superposition normalization algorithm (MaSNAl) and the minimum rank normalization algorithm (MiRaNAl). In this paper a new subspace-based time-domain normalization algorithm, denoted by SuTdNAl (subspace time-domain normalization algorithm), is presented. By the determination of the intersection of the column spaces of two Hankel matrices, the common signal poles and further on the components having proportionally varying amplitudes are detected. The method has the advantage that it is computationally less intensive than the MaSNAl and the MiRaNAl. Furthermore, no approximate estimate of the normalization factor is required. The algorithm was tested by Monte Carlo simulations on a set of simulation signals. It was shown that the SuTdNAl has a statistical performance similar to that of the MiRaNAl, which itself is an improvement over the MaSNAl. Furthermore, two samples of known contents are compared with the MiRaNAl, the SuTdNAl, and an older method using a standard. Finally, the SuTdNAl is tested on a realistic simulation example derived from an in vitro measurement on cells.     

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.     

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 New Unconditionally Stable Algorithm for Steady-State Fluid Simulation of High Density Plasma Discharge

A new unconditionally stable algorithm for steady-state fluid simulation of high density plasma discharge is suggested. The physical origin of restriction on simulation time step is discussed and a new method to overcome it is explained. To compare the new method with previous other methods, a one-dimensional fluid simulation of inductively coupled plasma discharge is performed.     

An Efficient Algorithm for Truncating Spatial Domain in Modeling Light Scattering by Finite-Difference Technique

The finite-difference time domain technique is one of the most robust and accurate numerical methods for the solution of light scattering by small particles with arbitrary composition and geometry. In practice, this method requires that the spatial domain for the computation of near-field be truncated. An absorbing boundary condition must be imposed in conjunction with this truncation. The performance of this boundary condition is essential to the stability of numerical computations and the reliability of results. In the present study, a new boundary condition, referred to as the mixed T algorithm, has been developed, which is a generalization of the transmitting boundary condition originally developed by Liao and co-workers. The present algorithm does not require spatial interpolation for wave values at interior grid points. In addition, it produces two minima of spurious reflections at small and large incident angles, allowing efficient absorption of the scattered waves at the boundary for large incident angles. When the third-order mixed T algorithm is used, the reflection coefficient of the boundary is less than 1% for incident angles from 0° to about 70°. We find that the numerical instability associated with the transmitting boundary condition is caused by the location-dependent amplitude of outgoing waves in the vicinity of the boundary. For this reason, the mixed T algorithm is stabilized by consistently introducing diffusive coefficients into the boundary equation. When the stabilized algorithm is applied, the near-field within the truncated domain can be computed by using single-precision arithmetic without overflows for more than 10⁵steps in the time-marching iteration. Finally, the new absorbing boundary condition is validated by carrying out numerical experiments involving the propagation of a TM wave excited by a sinusoidal point source, simultaneous simulation of the wave propagation in small and large domains, and the scattering of a TM wave by an infinite circular cylinder.     

A new time-domain frequency-selective quantification algorithm

In this paper a new time-domain frequency-selective quantification algorithm is presented. Frequency-selective quantification refers to a method that analyzes spectral components in a selected frequency region, ignoring all the other components outside. The algorithm, referred to as MeFreS (Metropolis Frequency-Selective), is based on rank minimization of an opportune Hankel matrix. The minimization procedure is satisfied by the down-hill simplex method, implemented with the simulated annealing method. MeFreS does not use any preprocessing step or filter to suppress nuisance peaks, but the signal model function is directly fitted. In this manner, neither inherent signal distortions nor estimation biases to be corrected occur. The algorithm was tested with Monte Carlo simulations. A comparison with VARPRO and AMARESw algorithms was carried out. Finally, two samples of known content from NMR data were quantified.     

Moving Mesh Methods in Multiple Dimensions Based on Harmonic Maps

In practice, there are three types of adaptive methods using the finite element approach, namely the h-method, p-method, and r-method. In the h-method, the overall method contains two parts, a solution algorithm and a mesh selection algorithm. These two parts are independent of each other in the sense that the change of the PDEs will affect the first part only. However, in some of the existing versions of the r-method (also known as the moving mesh method), these two parts are strongly associated with each other and as a result any change of the PDEs will result in the rewriting of the whole code. In this work, we will propose a moving mesh method which also contains two parts, a solution algorithm and a mesh-redistribution algorithm. Our efforts are to keep the advantages of the r-method (e.g., keep the number of nodes unchanged) and of the h-method (e.g., the two parts in the code are independent). A framework for adaptive meshes based on the Hamilton–Schoen–Yau theory was proposed by Dvinsky. In this work, we will extend Dvinsky's method to provide an efficient solver for the mesh-redistribution algorithm. The key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure. Each iteration step is to move the mesh closer to the harmonic map. This procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration. The numerical schemes are applied to a number of test problems in two dimensions. It is observed that the mesh-redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi-dimensional computations.     

Solution of Time-Dependent Diffusion Equations with Variable Coefficients Using Multiwavelets

A new numerical algorithm is developed for the solution of time-dependent differential equations of diffusion type. It allows for an accurate and efficient treatment of multidimensional problems with variable coefficients, nonlinearities, and general boundary conditions. For space discretization we use the multiwavelet bases introduced by Alpert (1993,SIAM J. Math. Anal.24, 246–262), and then applied to the representation of differential operators and functions of operators presented by Alpert, Beylkin, and Vozovoi (Representation of operators in the multiwavelet basis, in preparation). An important advantage of multiwavelet basis functions is the fact that they are supported only on non-overlapping subdomains. Thus multiwavelet bases are attractive for solving problems in finite (non periodic) domains. Boundary conditions are imposed with a penalty technique of Hesthaven and Gottlieb (1996,SIAM J. Sci. Comput., 579–612) which can be used to impose rather general boundary conditions. The penalty approach was extended to a procedure for ensuring the continuity of the solution and its first derivative across interior boundaries between neighboring subdomains while time stepping the solution of a time dependent problem. This penalty procedure on the interfaces allows for a simplification and sparsification of the representation of differential operators by discarding the elements responsible for interactions between neighboring subdomains. Consequently the matrices representing the differential operators (on the finest scale) have block-diagonal structure. For a fixed order of multiwavelets (i.e., a fixed number of vanishing moments) the computational complexity of the present algorithm is proportional to the number of subdomains. The time discretization method of Beylkin, Keiser, and Vozovoi (1998, PAM Report 347) is used in view of its favorable stability properties. Numerical results are presented for evolution equations with variable coefficients in one and two dimensions.     

Improved Lanczos algorithms for blackbox MRS data quantitation

Magnetic resonance spectroscopy (MRS) has been shown to be a potentially important medical diagnostic tool. The success of MRS depends on the quantitative data analysis, i.e., the interpretation of the signal in terms of relevant physical parameters, such as frequencies, decay constants, and amplitudes. A variety of time-domain algorithms to extract parameters have been developed. On the one hand, there are so-called blackbox methods. Minimal user interaction and limited incorporation of prior knowledge are inherent to this type of method. On the other hand, interactive methods exist that are iterative, require user involvement, and allow inclusion of prior knowledge. We focus on blackbox methods. The computationally most intensive part of these blackbox methods is the computation of the singular value decomposition (SVD) of a Hankel matrix. Our goal is to reduce the needed computational time without affecting the accuracy of the parameters of interest. To this end, algorithms based on the Lanczos method are suitable because the main computation at each step, a matrix-vector product, can be efficiently performed by means of the fast Fourier transform exploiting the structure of the involved matrix. We compare the performance in terms of accuracy and efficiency of four algorithms: the classical SVD algorithm based on the QR decomposition, the Lanczos algorithm, the Lanczos algorithm with partial reorthogonalization, and the implicitly restarted Lanczos algorithm. Extensive simulation studies show that the latter two algorithms perform best.     

Physics-Based Preconditioning and the Newton–Krylov Method for Non-equilibrium Radiation Diffusion

An algorithm is presented for the solution of the time dependent reaction-diffusion systems which arise in non-equilibrium radiation diffusion applications. This system of nonlinear equations is solved by coupling three numerical methods, Jacobian-free Newton–Krylov, operator splitting, and multigrid linear solvers. An inexact Newton's method is used to solve the system of nonlinear equations. Since building the Jacobian matrix for problems of interest can be challenging, we employ a Jacobian–free implementation of Newton's method, where the action of the Jacobian matrix on a vector is approximated by a first order Taylor series expansion. Preconditioned generalized minimal residual (PGMRES) is the Krylov method used to solve the linear systems that come from the iterations of Newton's method. The preconditioner in this solution method is constructed using a physics-based divide and conquer approach, often referred to as operator splitting. This solution procedure inverts the scalar elliptic systems that make up the preconditioner using simple multigrid methods. The preconditioner also addresses the strong coupling between equations with local 2×2 block solves. The intra-cell coupling is applied after the inter-cell coupling has already been addressed by the elliptic solves. Results are presented using this solution procedure that demonstrate its efficiency while incurring minimal memory requirements.     

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.     

Nonlinear Landau Damping in Spherically Symmetric Vlasov Poisson Systems

The Vlasov Poisson system is a partial differential equation widely used to describe collisionless plasma. It is formulated in a six-dimensional phase space, this prohibits a numerical solution on a complete phase space grid. In some applications, however, spherical symmetry is given, which introduces singularities into the Vlasov Poisson equation. We focus on such problems and propose a stable algorithm using accommodating boundaries. At first, the method is tested in the linear regime, where analytical solutions are available. Thereafter it is applied to large disturbances from equilibrium.     

Two-Level Hierarchical FEM Method for Modeling Passive Microwave Devices

In recent years multigrid methods have been proven to be very efficient for solving large systems of linear equations resulting from the discretization of positive definite differential equations by either the finite difference method or theh-version of the finite element method. In this paper an iterative method of the multiple level type is proposed for solving systems of algebraic equations which arise from thep-version of the finite element analysis applied to indefinite problems. A two-levelV-cycle algorithm has been implemented and studied with a Gauss–Seidel iterative scheme used as a smoother. The convergence of the method has been investigated, and numerical results for a number of numerical examples are presented.     

Tagging of the Magnetization with the Transition Zones of 360° Rotations Generated by a Tandem of Two Adiabatic DANTE Inversion Sequences

A new technique is presented for generating myocardial tagging using the signal intensity minima of the transition zones between the bands of 0° and 360° rotations, induced by a tandem of two adiabatic delays alternating with nutations for tailored excitation (DANTE) inversion sequences. With this approach, the underlying matrix corresponds to magnetization that has experienced 0° or 360° rotations. The DANTE sequences were implemented from adiabatic parent pulses for insensitivity of the underlying matrix to B₁ inhomogeneity. The performance of the proposed tagging technique is demonstrated theoretically with computer simulations and experimentally on phantom and on the canine heart, using a surface coil for both RF transmission and signal reception. The simulations and the experimental data demonstrated uniform grid contrast and sharp tagging profiles over a twofold variation of the B₁ field magnitude.     

Analysis of protein/ligand interactions with NMR diffusion measurements: the importance of eliminating the protein background

Pulsed-field gradient nuclear magnetic resonance (PFG-NMR) is a well-established method for the determination of translational diffusion coefficients. Recently, this method has found applicability in the combinatorial arena with the introduction of affinity NMR for characterizing protein/ligand interactions. Although affinity NMR has been reported to be an effective method for the identification of active compounds in a complex mixture, there are limitations of this method. We have developed a simple mathematical model to predict optimum concentration ratios of the ligand and protein in order to observe maximum changes in the ligand diffusion coefficient upon protein binding. The ligand/protein systems of L-tryptophan and ibuprofen binding to human serum albumin were chosen to demonstrate the usefulness of this model. However, even when the conditions of the mathematical model are satisfied, the spectral background arising from the protein in proton-detected experiments can be problematic. To this end, we have employed spectral subtraction of the protein spectrum to yield ligand diffusion coefficients that are in agreement with those predicted by simulation.     

Determination of the chemical shielding tensor orientation from two or one of the three conventional rotations of a single crystal

Chemical shift anisotropy (CSA) is an immensely useful interaction to study the structure, dynamics, and function of a wide variety of chemical and biological molecules. Traditionally the only unambiguous way to determine both the principal values and the orientation of the principal axes of the CSA tensor has been to follow the chemical shift frequency changes as a crystal of known structure is rotated relative to the direction of the external magnetic field. This classic method employs rotations about three mutually orthogonal axes of a single crystal. It is shown here that just two, or one, of the above rotations suffice to determine the CSA tensor orientation by borrowing, the easy to obtain, principal values of CSA from an independent source. Methods for using two rotation patterns or even a single rotation pattern are described and illustrated with known chemical shielding tensors.     

PELDOR at S- and X-band frequencies and the separation of exchange coupling from dipolar coupling

A pulsed electron double resonance (PELDOR) setup working at S-band frequencies is introduced and its performance compared with an X-band setup. Furthermore, to verify experimentally that it is possible to disentangle the dipolar coupling nu(Dip) from the exchange coupling J by PELDOR we synthesized and investigated four bisnitroxide radicals. They exhibit in pairs the same distances r(AB) between the nitroxide moieties but only one of each pair possesses a non-zero J. The experimental values for r(AB) match the ones from molecular modeling very well for the molecules without exchange coupling. For one bisnitroxide it was possible to separate nu(Dip) from J and to ascertain the magnitude and sign of J to +11 MHz (antiferromagnetic spin-spin coupling).     

The Lattice Solid Model to Simulate the Physics of Rocks and Earthquakes: Incorporation of Friction

The particle-based lattice solid model developed to study the physics of rocks and the nonlinear dynamics of earthquakes is refined by incorporating intrinsic friction between particles. The model provides a means for studying the causes of seismic wave attenuation, as well as frictional heat generation, fault zone evolution, and localisation phenomena. A modified velocity–Verlat scheme that allows friction to be precisely modelled is developed. This is a difficult computational problem given that a discontinuity must be accurately simulated by the numerical approach (i.e., the transition from static to dynamical frictional behaviour). This is achieved using a half time step integration scheme. At each half time step, a nonlinear system is solved to compute the static frictional forces and states of touching particle-pairs. Improved efficiency is achieved by adaptively adjusting the time step increment, depending on the particle velocities in the system. The total energy is calculated and verified to remain constant to a high precision during simulations. Numerical experiments show that the model can be applied to the study of earthquake dynamics, the stick–slip instability, heat generation, and fault zone evolution. Such experiments may lead to a conclusive resolution of the heat flow paradox and improved understanding of earthquake precursory phenomena and dynamics.