Diffusion generated motion for grain growth in two and three dimensions

An efficient algorithm for accurately simulating curvature flow for large networks of curves in two dimensions and surfaces in three dimensions on uniform grids is proposed. This motion arises in the technologically important problem of simulating grain boundary motion in polycrystalline materials. In this formulation grain boundaries are zero-level sets of signed distance functions. Curvature motion is achieved by first diffusing locally maintained signed distance functions followed by a reinitialization step. A technique is devised to allow a single signed distance function to represent a large subset of spatially separated grains. Hundreds of thousands of grains can be simulated using a small number of signed distance functions (in this work, 32 in two dimensions and 64 in three dimensions are more than sufficient) using modest computational hardware.     

Efficient mesh motion using radial basis functions with data reduction algorithms

Mesh motion using radial basis functions has been demonstrated previously by the authors to produce high quality meshes suitable for use within unsteady and aeroelastic CFD codes. In the aeroelastic case the structural mesh may be used as the set of control points governing the deformation, which is efficient since the structural mesh is usually small. However, as a stand alone mesh motion tool, where the surface mesh points control the motion, radial basis functions may be restricted by the size of the surface mesh, as an update of a single volume point depends on all surface points. In this paper a method is presented that allows an arbitrary deformation to be represented to within a desired tolerance by using a significantly reduced set of surface points intelligently identified in a fashion that minimises the error in the interpolated surface. This method may be used on much larger cases and is successfully demonstrated here for a 10⁶ cell mesh, where the initial solve phase cost reduces by a factor of eight with the new scheme and the mesh update by a factor of 55. It has also been shown that the number of surface points required to represent the surface is only geometry dependent (i.e. grid size independent), and so this reduction factor actually increases for larger meshes.     

Diffusion generated T1 and T2 contrast

In MR images of porous organic samples (such as roots or wood) in water media, the sample is often surrounded by a bright ring, with a corresponding decreased T1 value in T1 maps. When the medium is removed, or contrast agents are added, the ring disappears, indicating that the signal does not originate in the outer layers of the sample, but from the medium itself. It can be shown that this "bright ring effect" is only observed when the medium experiences a reduction in T1 when permeating the sample. In order to investigate this effect, a computer model was used to simulate the diffusion of magnetisation between regions that exhibit different relaxation constants. Using this model, the origin of the signal increase was found to be an inflow effect, as diffusion transports relaxed magnetisation from the boundary regions of the sample into the surrounding medium. In the case of the "bright ring" around the plants described above, a mixing of short T1 values from within the sample and long T1 values within the medium occurs, yielding a "transition region" between the two values. There, a signal increase can be observed at T1 weighted images, compared to the signal from the medium beyond this transition region. The width of the transition region is on the order of magnitude of the diffusion displacement that is calculated from the T1 value as diffusion time. In addition to causing the bright ring around the plant samples, this diffusion effect also limits the resolution of the relaxation time maps. This effect is not limited to T1 relaxation but also applies to T2 relaxation. However, at high B0 field strengths such as those used in this study (11.7 T), a T2 effect is not usually observed due to the considerably shorter T2 times in plants (about 50 ms, compared to T1 times of higher than 1 s). Because the diffusion length during this T2 relaxation is short with respect to the resolution of the imaging experiments, no T2 ring effect is seen.     

Kähler metrics generated by functions of the time-like distance in the flat Kähler–Lorentz space

We prove that every Kähler metric, whose potential is a function of the time-like distance in the flat Kähler–Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kähler manifolds with the above-mentioned metrics. New examples of Sasakian space forms are obtained as real hypersurfaces of a Kähler space form with special invariant distribution. We introduce three types of even dimensional rotational hypersurfaces in flat spaces and endow them with locally conformal Kähler structures. We prove that these rotational hypersurfaces carry Kähler metrics of quasi-constant holomorphic sectional curvatures satisfying some conditions, corresponding to the type of the hypersurfaces. The meridians of those rotational hypersurfaces, whose Kähler metrics are Bochner–Kähler (especially of constant holomorphic sectional curvatures), are also described.     

Diffusion in different models of active Brownian motion

Active Brownian particles (ABP) have served as phenomenological models of self-propelled motion in biology. We study the effective diffusion coefficient of two one-dimensional ABP models (simplified depot model and Rayleigh-Helmholtz model) differing in their nonlinear friction functions. Depending on the choice of the friction function the diffusion coefficient does or does not attain a minimum as a function of noise intensity. We furthermore discuss the case of an additional bias breaking the left-right symmetry of the system. We show that this bias induces a drift and that it generally reduces the diffusion coefficient. For a finite range of values of the bias, both models can exhibit a maximum in the diffusion coefficient vs. noise intensity.     

Holodiagrams using Mahalanobis distance

We propose the generalization of the Holodiagram concept that has demonstrated many useful properties in optics, to include other metric measurement, namely the Mahalanobis distance, thus involving several other optical possible uses. So, besides being useful in the study of decision and classification problems it can be used in geometric solutions involving Fermat's principle in a rather wide range of situations.     

Diffusion and collective motion of rotlets in 2D space

We investigate the collective motion of rotlets that are placed in a single plane. Due to the hydrodynamic interactions,the particles move through the two-dimensional(2D) plane and we analyze these diffusive motions. By analyzing the scaling of the values, we predict that the diffusion coefficient scales with φ~(0.5), the average velocity with φ, and relaxation time of the velocity autocorrelation function with φ~(-1.5), where φ is the area fraction of the particles. In this paper, we find that the predicted scaling could be seen only when the initial particle position is homogeneous. The particle collective motions are different by starting the simulation from random initial positions, and the diffusion coefficient is the largest at a minimum volume fraction of our parameter range, φ = ~(0.05). The deviations based on two initial positions can be explained by the frequency of the collision events. The particles collide during their movements and the inter-particle distances gradually increase. When the area fraction is large, the particles will result in relatively homogeneous configurations regardless of the initial positions because of many collision events. When the area fraction is small(φ 0.25), on the other hand, two initial positions would fall into different local solutions because the rare collision events would not modify the inter-particle distances drastically. By starting from the homogeneous initial positions, the particles show the maximum diffusion coefficient at φ≈ 0.20. The diffusion coefficient starts to decrease from this area fraction because the particles start to collide and hinder each other from a critical fraction ～23%. We believe our current work contributes to a basic understanding of the collective motion of rotating units.     

Diffusion measurements using oscillating gradients

A novel method for the measurement of self-diffusion coefficients employing oscillating gradients is presented. The method used has advantages over conventional techniques and will allow measurements to be made at very short diffusion times (<1 ms) and should prove particularly useful for shortT ₂ materials.     

Computer generated holograms using personal computers

Holography or the method of wave-front reconstruction to produce an image was introduced by D. Gabor in 1948. Normally a laser is necessary to produce holograms in the laboratory. This paper introduces the usage of the high resolution display of the IBM-PC and a graphics printer to generate holograms. The advantages of using this method are that holograms of real or fictitious objects can be made, since all objects can be represented by a collection of points and one does not need an elaborate set up to make the holograms. This method is also useful for making Fourier transform holograms, image plane holograms, spatial filters, optical elements, etc.^1,2     

Dark-state polaritons using spontaneously generated coherence

A three-level atomic system in V-configuration (interacting with a single mode laser field) with parallel transition dipole moments exhibiting spontaneously generated coherence due to quantum interference of decaying channels is considered here for the purpose of storing light pulses. This system is equivalent to (with some restrictions) another three-level system in which ground state is coupled with one of the upper states but the upper states are coupled through a DC field and hence can be used to store electromagnetic pulse using the concept of dark-state-polariton.     

Optical Mellin transforms using computer generated holograms

The scale invariant correlation possible with Mellin transforms has made this operation extremely important in pattern recognition for image and signal processing. This operation is realized by a geometrical transformation $\text{?(x, y) → ?(}\text{exp}\text{ξ,}\text{exp}\text{η)}$ followed by an optical Fourier transform. The use of computer generated hologram masks to effect the ξ = ln x, η = ln y transformation are discussed.     

Pitch extraction using a generated decision function

A pitch extraction method is considered that employs a decision function whose maximum is greater than the possible spurious maxima of the analyzed signal on the pitch period. Smoothing by a wavelet window is proposed as a preprocessing procedure for the telephone signal. Experiments have shown that the proposed pitch extraction method is advantageous in almost all situations except for the case of the undistorted signal, where the LLK method proved to be the best one.     

Diffusion equation for the quasi-incoherent motion of Frenkel excitons in molecular crystals

In a recent paper a master equation describing the quasi-incoherent motion of Frenkel excitons in molecular crystals has been derived within the Haken-Strobl model for the coupled coherent and incoherent motion of Frenkel excitons. Starting from this master equation and using only the translational symmetry of the crystal lattice, for crystals with one molecule in the unit cell a diffusion equation is derived. For crystals with several molecules in the unit cell instead of a diffusion equation one obtains a set of diffusion-like equations. These equations are solved explicitly for the case of two molecules in the unit cell and asymptotic expressions are discussed. It is shown that this asymptotic behaviour is again described by a diffusion equation.     

Structural and flow properties of binary media generated by fractional Brownian motion models

In the present paper, the structural and flow properties of binary media generated by two-dimensional lattices that follow fractional Brownian motion statistics are studied. A modification of the midpoint displacement and random addition method is employed in order to generate multicell binary media with sizes that are considerably larger than the correlation length of the medium. Several structural properties, such as the autocorrelation function, the surface area, and the percolation threshold, are studied for different values of porosity and degree of correlation. In addition, transport properties are investigated in the above media, by solving numerically the momentum and continuity equations, to determine the absolute permeability of the medium in directions parallel and normal to the fractional Brownian motion (fBm) plane. It is found that multicell fBm porous media possess very interesting structural properties that are functions of the Hurst exponent and porosity, and are independent of the lattice size, in contrast to the traditional single-cell fBm media. In addition, they exhibit stronger structural correlation, lower specific surface area, higher percolation threshold, and lower permeabilities than those of the corresponding single-cell porous media.     

Self-consistent wave functions,center-of-mass motion and high-energy particle-nucleus scattering

Elastic electron and hadron scattering by ⁴He and ¹⁶O is calculated using for the latter nucleus Brueckner-Hartree-Fock (BHF) wave functions obtained in the harmonic oscillator representation. The effects of the center-of-mass motion spuriosity are estimated.