Two-and Three-dimensional Cases Research Articles

Simulation of viscous flows by highly accurate aeroacoustic schemes on regular grids

Highly accurate finite-difference schemes that are widely used in computational aeroacoustics and referred to as dispersion-relation-preserving (DRP) schemes are adapted for the simulation of viscous flows. The main result consists in the construction and verification of numerical boundary conditions on a solid body, artificial boundaries, and on their interface. Calculations are performed for several test problems, including dissipation of the Lamb–Oseen vortex and decay of the Taylor–Green vortex in two-and three-dimensional cases and in infinite and semi-infinite (open-outlet) plane channels.

An Edge-Based Anisotropic Mesh Refinement Algorithm and Its Application to Interface Problems

Based on an error estimate in terms of element edge vectors on arbitrary unstructured simplex meshes, we propose a new edge-based anisotropic mesh refinement algorithm. As the mesh adaptation indicator, the error estimate involves only the gradient of error rather than higher order derivatives. The preferred refinement edge is chosen to reduce the maximal term in the error estimate. The algorithm is implemented in both twoand three-dimensional cases, and applied to the singular function interpolation and the elliptic interface problem. The numerical results demonstrate that the convergence order obtained by using the proposed anisotropic mesh refinement algorithm can be higher than that given by the isotropic one. AMS subject classifications: 65N22, 65N50, 65N55

Transition of Multidimensional Jumplike Processes from Anomalous Diffusion to Linear Diffusion

We consider multidimensional “quasi-anomalous” random-walk processes having linear-diffusion asymptotic representations at large times and obeying anomalous-diffusion laws at intermediate times (but which are also sufficiently large compared with microscopic time scales). The transition of a jumplike process from anomalous diffusion to linear diffusion is demonstrated. We use numerical computation to confirm the validity of the analytic calculations for the two-and three-dimensional cases.

Simulation of Flows and Acoustic Field Around Moving Body by ALE Formulation in Finite Difference Lattice Boltzmann Method

Flowfield and acoustic-field around moving bodies are simulated by the Arbitrary Lagrangian Eulerian formulation in the finite difference lattice Boltzmann method. The effect of the ALE is checked by comparing flow about a square cylinder in ALE formulation and that in the fixed coordinates, and both agree very well. Matching procedure between the moving grid and fixed grid is also considered. The proposed method in which the both grids are connected through buffer region is shown to be superior to moving overlapped grid. Dipole-like emissions of sound wave from harmonically vibrating bodies in two-and three-dimensional cases are simulated. Sound wave emitted from rapidly rotating elliptic cylinder is also successfully simulated.

Jumplike Variation of the Contact Area between Randomly Rough Surfaces

A numerical model of the contact between two solid surfaces with statistically random roughness has been considered. As the two surfaces are brought in contact and pressed against each other, the real contact area initially increases in proportion to the normal force, with a proportionality factor dependent on the spectral density of the surface profile. As the pressing force grows further, the contact area exhibits two sequential jumps. This behavior is universal, being manifested for various spectral densities of the surface roughness in both two-and three-dimensional cases. The physical reasons for the observed features and their role in the mechanics of contact between soft materials (rubber, biological tissues) are discussed. The effect can be used for creating new adhesive systems capable of exhibiting strong adhesion upon application of a critical pressing force.

Influence of the size distribution of granules and of their attractive interaction on the percolation threshold in granulated alloys

Numerical simulation was applied to study the influence of the size distribution of granules and the interaction between them on the percolation threshold in granulated metal-insulator alloys. An alloy model was considered in which metal granules have two characteristic sizes, l and L (with L>l), and the size distribution of granules of greater size L having an average value of approximately L0 is described by a normal distribution with a standard deviation d, by a step function with a halfwidth d, or by a delta function. A model with attraction between granules and mechanisms of trapping of an additional granule by an already developed cluster with a characteristic trapping range R was also considered. The percolation threshold significantly grows with the ratio L0/l and with R for both two-and three-dimensional cases and tends to flattening at large L/l or R. The calculated results make it possible to explain the high percolation threshold observed for the majority of granulated alloys.

One-, two-, and three-dimensional ising model in the static fluctuation approximation

Viewed as a prototype for strongly interacting many-body systems, the spin-1/2n-dimensional Ising model (n = 1, 2, 3) is studied within the so-calledstatic fluctuation approximation (SFA). The underlying physical picture is that the local fieldoperator σfz withquadratic fluctuations is replaced with its mean value [(σfz)2 ≃ 〈(σfz)2〉]. This means that the true quantum mechanical spectrum of the operator σfz is replaced with a distribution; along with the calculation of its mean value, we take into accountself-consistently the moments of this distribution. It is shown that this sole approximation is sufficient for deducing the equilibrium correlation functions and the main thermodynamic characteristics of the system. Special new features of this study include an analysis of the two-dimensional modelwithout periodic boundary conditions, and the demonstration that the phase-transition scenario is quite sensitive to the boundary conditions in the two-and three-dimensional cases. In passing, new boundary problems in mathematical physics are emphasized.

Parasupersymmetry and truncated supersymmetry in quantum mechanics

We describe how a class of quantum mechanical systems with parasupersymmetries is obtained by truncating supersymmetric models. A detailed analysis of the one-dimensional systems is given using this point of view. We identify the mechanism responsible for the appearance of the negative energy eigenstates of parasupersymmetric hamiltonians and find new models for a spin-1 particle in exactly solvable potentials. The two-and three-dimensional cases are also discussed.

Axisymmetric fundamental solutions for the equations of heat conduction in the case of cylindrical anisotropy of a medium

Numerical methods of solving the boundary-value problems for the equations of mathematical physics based on the application of fundamental solutions, i.e., solutions describing the reaction of infinite space or an infinite plane to a concentrated action, are currently in widespread favor. Among these methods we can include the direct and indirect methods of boundary integral equations [i], as well as the method of sources in which the solution of the boundary-value problem is constructed by superposition of concentrated actions in space, above some surface encompassing the area under investigation [2]. For the equations of steady and nonsteady heat conduction in an isotropic medium such solutions are well established (see [i] and the references cited there) both for the twoand three-dimensional cases, as well as for the case of the axisymmetric problem. The plane and three-dimensional equations of heat conduction for a rectilinear anisotropic medium can be reduced to the isotropic case. We know of three-dimensional fundamental solutions for the equations of elasticity theory in the case of a medium with rectilinear anisotropy [3] and for a rectilinear anisotropic hereditary (or memory) elastic medium [4, 5].

The reliability of wormlike polysaccharide chain dimensions estimated from electron micrographs

Electron micrographs of alginate, xylinan, xanthan, and scleroglucan were prepared by vacuum-drying aqueous glycerol-containing solutions, and then heavy-metal, low-angle rotary replicated. Quantitative methods for excluding streamlining effects and deformation artifacts were developed and applied to the digitized polymer contours prior to analysis of stiffness. The apparent macromolecular dimensionalities were not obtainable on the basis of the change in the scaling coefficient alpha relating the rms end-to-end distance and the contour length, mean value of r2(1/2) approximately L alpha, for chains subject to the excluded volume effect in two and three dimensions. Using a two-dimensional model, the persistence length of these molecules was estimated to be (9 +/- 1) nm (alginate), (25 +/- 4) nm (xylinan), (30 +/- 4) nm (single-stranded xanthan), (68 +/- 7) nm (double-stranded xanthan), and (80 +/- 10) nm (scleroglucan). Monte Carlo calculations for wormlike chains close to an interacting surface or confined to the region between two surfaces showed that (1) strongly adsorbed molecules are essentially two-dimensional and (2) molecules restricted to the space between two surfaces separated by a distance less than 20% of the persistence length are two-dimensional in their directional correlation. The somewhat low estimates of the persistence lengths obtained from the electron micrographs compared with those reported from solution measurements can be accounted for by the adoption of a strictly two-dimensional model in the analysis, whereas the absorbed polymers are most likely intermediate between the two-and three-dimensional cases. The model calculations and the analysis of the electron micrographs suggest that stiffness parameters are obtainable from the electron micrographs when the proper theoretical description are used in the analysis.

Influence functions for the unsteady surface element method

The unsteady surface element methods provide analytical and numerical procedures for the efficient solution of certain transient heat conduction problems. The methods are based on Duhamel's theorem which requires influence functions. Because the influence functions implicitly include the effects of boundary conditions and geometries, many different ones are needed. Sometimes the most difficult influence functions to evaluate are those at the heated surface and at the smallest dimensionless times. This paper provides some early time influence functions for some basic one-dimensional geometries as well as for some twoand three-dimensional cases for various heated regions on the surface of a semi-infinite body. A given early time solution has the advantage of being valid for a variety of boundary conditions distant from the surface. Some large-time influence functions are also given. An example is given that illustrates that a combination of a short-time solution and a large-time solution can sometimes adequately cover the complete time domain.

Statistical approach to the description of spherulite patterns. Two-and three-dimensional cases

A new statistical approach was applied for the description of spherulite patterns for athermal and thermal type of primary nucleation. The two-and three-dimensional cases were treated in the same way.

The construction of effective methods for electromagnetic modelling

Summary. This paper deals with the further development of finite-difference methods for electromagnetic field modelling in two-and three-dimensional cases. The main feature of the approach suggested here is the application of generalized asymptotic boundary conditions valid with the accuracy (1/ρN), where ρ is the distance from the heterogeneities. The finite-difference approximation of problems under solution is made using the balance method, which results in 5-point difference schemes in the 2-D case and 7-point difference schemes in the 3-D case. To solve the linear system of difference equations the successive over-relaxation (SOR) method is used, the relaxation factor being chosen during the iteration procedure. In view of the vectorial character of the problem for the 3-D case, a successive blocked over-relaxation method (SBOR) is applied. The model's validity is based on the comparison of the fields accounted at the ground surface with those computed by the integral transformation of excessive currents, determined in the heterogeneity region using the finite-difference scheme.

Remanence and non-exponential relaxation in an Ising chain with random bonds

We study the time-dependent properties of an Ising chain with random bonds Gaussianly distributed. We obtain some exact as well as Monte Carlo (MC) results. Remanence, as well as a seemingly logarithmic long-time decay of the magnetization and of the energy towards their values is observed at low temperatures by means of MC simulation. The remanent values of the magnetization ($\frac{1}{3}$, starting with all spins up) and of the energy are derived. A simple and explicit physical picture of the mechanism behind remanence and the logarithmic relaxation emerges. The value of $q(t)={〈〈{\ensuremath{\sigma}}_{i}(0){\ensuremath{\sigma}}_{i}(t)〉〉}_{J}$ is obtained via the MC technique; it also seems to relax logarithmically for low temperatures. In contrast with the two-and three-dimensional cases, it is shown how any MC calculation can start immediately from an equilibrium state in this model, a very convenient feature for very-low-temperature MC computations. We show that if $H$ is switched on at $t=0$, then ${[\frac{\ensuremath{\partial}m(H,t)}{\ensuremath{\partial}H}]}_{H=0}=(\frac{1}{\mathrm{kT}})[1\ensuremath{-}q(t)]$ holds exactly in any number of dimensions, where $m$ is the magnetization per spin. A time-dependent susceptibility $\ensuremath{\chi}(H,t)$ is defined and shown to vanish (as $H\ensuremath{\rightarrow}0$) for low enough temperatures and finite $t$. The MC results for $\ensuremath{\chi}$ are in accord with this result and, if graphed versus $T$, show a hump at a time-dependent temperature. Finally, we compare the exact specific heat, $C$, for this model with the results obtained by MC simulation of calorimetric measurements. Thus, a simple explicit case is exhibited of the difficulties which may arise in measurements of $C$ in spin-glasses due to long-time effects.

非圧縮完全流体中で飛行方向に非一様運動を行なう薄翼の理論

There are few papers for nonuniform in-plane motions of wings, practical applications of which may concern lead-lag motion of helicopter blades and motions of horizontal stabilizer in T-Tail flutter. In the present formulation Fouriertransform is used, which makes development of the theory straighttorward, universal, and routine. Drag problems (on zero-lift symmetric wings) and lift problems (on zero-thickness wings) are treated for both two-and three-dimensional cases. Drag problems show a drag due to the instant acceleration, while lift problems yield and upwash dependent on the whole history of th wing motions.

Dynamics of the Ising Model near the Critical Point. I

A simple dynamical model of interacting Ising spins is discussed. Each spin is assumed to flip spontaneously with a transition probability which depends on the temperature and the configuration of surrounding spins, but its functional form is assumed to be the simplest. The frequency-wave number dependent susceptibility χ( q , ω) is given exactly in the one-dimensional case. In two-and three-dimensional cases the model is treated in the molecular field and the generalized approximations.

Discussion of a paper by Mahdi S. Hantush, ‘On the validity of the Dupuit-Forchheimer well-discharge formula’

This paper by Hantush [1962] is particularly interesting in its presentation of a new proof of the Dupuit-Forchheimer formulas for the steady discharge of a gravity well that is fed from a concentric equipotential surface, and the corresponding two-dimensional case of flow through a bank with vertical faces. The analytic correctness of the Dupuit formula was first brought to my attention in a paper by Huard de la Marre [1956], who indicated that a proof had been obtained by Polubarinova-Kochina [1952] and suggested that a simpler approach would be the use of one of Green's identities. He did not give a detailed proof, and exhaustive enquiries failed to produce a copy of Polubarinova-Kochina's book. Apparently neither of these works were widely known, so when challenged in Geotechnique on my use of the discharge formula, I published in English a detailed proof of the two-and three-dimensional cases [Chapman, 19576] and subsequently extended this to cover the case in which capillary effects are not negligible [Chapman, 1960].

Some Multi-Dimensional Birth and Death Processes and Their Applications in Population Genetics

In this paper the change in number of the various genotypes in a population is formulated as a multi-dimensional birth and death process. Two classes of populations are considered, namely, bisexual diploid populations and asexual haploid populations. Various forms of the birth functions for the two classes of populations are deduced from biological considerations. Four multi-dimensional generalizations of classical one-dimensional cases are presented, and the generalized linear birth and death process is introduced. The generating function of the generalized linear birth and death process is found in some twoand three-dimensional cases, and the mean vector of the process for an arbitrary finite dimension and arbitrary parameters is also studied.

NOTES ON CONTINUOUS STOCHASTIC PHENOMENA

The study of stochastic processes has naturally led to the consideration of stochastic phenomena which are distributed in space of two or more dimensions. Such investigations are, for instance, of practical interest in connexion with problems concerning the distribution of soil fertility over a field or the relations between the velocities at different points in a turbulent fluid. A review of such work with many references has recently been given by Ghosh (1949) (see also Matern, 1947). In the present note I consider two problems arising in the twoand three-dimensional cases.