Periodic Grid Research Articles

On radial basis approximation on periodic grids

A radial basis function approximation in n variables has the formwhere ø:ℝn → ℝ denotes the n-variate, spherically symmetric function associated with a prescribed radial basis function ø+:ℝ+ → ℝ, i.e. ø = ø+(‖ · ‖), the norm being Euclidean. The are real coefficients (often, approximants s above are considered where only finitely many λjs are non-zero), and is a fixed set of points in ℝn (of course, only the xj with non-zero coefficient λj affect s). Thus s is a linear combination of translates of a radially symmetric function which can be of global support, the simplest choice being , where c is a positive parameter. The latter is referred to as the multiquadric function and is usefull in applications.

A computational model of the cochlea using the immersed boundary method

In this work we describe a two-dimensional computational model of the cochlea (inner ear). The cochlea model is solved by modifying and extending Peskin's immersed boundary method, originally applied to solving a model of the heart. This method solves the time-dependent incompressible Navier-Stokes equations in the presence of immersed boundaries. The fluid equations are specified on a fixed Eulerian grid while the immersed boundaries are specified on a moving Lagrangian grid. The immersed boundaries exert forces locally on the fluid. These local forces are seen by the fluid as external forces that are added to the other forces, pressure and viscous, acting on the fluid. The modifications and extension of Peskin's method involve both the fluid solver and the calculation and transfer of immersed-boundary forces to the fluid. For the fluid, the Navier-Stokes equations are solved on a doubly periodic rectangular grid in a second-order accurate manner using a projection method developed by J. Bell, P. Colella, and H. Glaz (Lawrence Livermore National Laboratory Report UCRL-98225, 1988). The extension of the immersed-boundary forces from the moving grid to the fixed fluid grid and the restriction of the fluid velocities from the fixed fluid grid to the moving grid have been modified to be second-order accurate. The calculation of the immersed-boundary forces can be done either explicitly or implicitly or a combination of both. The cochlea is modelled as two fluid chambers separated by a flexible partition whose stiffness varies exponentially along its length. The stapes is represented by a moving piston and the bony outer walls are allowed to be either straight or tapered. A travelling wave propagates along the flexible partition under the influence of the moving piston, and the dependence of this wave on partition stiffness is studied. Also included are studies of transient signal analysis and comparisons of model results to experimental data and asymptotic results.

“Checker Board” Periodic Boundary Conditions in Molecular Dynamics Codes

Abstract Checker board periodic boundary conditions for molecular simulation are proposed. They represent a simple geometric transformation of the well known periodic boundary conditions formulae for variety of periodic and regular computational boxes (e.g. the truncated octahedron, rhombic dodecahedron, hexagonal prism, etc.) into a shifted parallelepiped periodic grid. As a result, the choice of an optional computational box shape is equivalent to the proper choice of the basic parallelepiped sides lengths. The checker board periodic boundary conditions have been applied in a molecular dynamics (MD) code based on the link-cell method (using Lennard-Jones interactions). Simulation efficiency is approximately the same as for the classical cubic periodic boundary transformations.

A Structural Model of Ceramic Deformation and Fracture∗

ABSTRACT A new structural model of deformation and fracture of ceramics is presented. The real material is simulated as a periodic grid of hexagonal elastic grains connected with elastic bondings. A method for deformation analysis of such a medium is introduced, and the effective modules of the material are calculated. The solution of the problem of wave propagation gives dispersion correlations. The deformation model introduced and the procedure of Fourier transforms yields a Green function for an infinite periodic grid of nondeformable hexagonal grains that are connected with elastic bondings. The fundamental solution is used to examine the strength of the medium with local defects and to compare it with the strength of a defectless material.

Numerical investigation of a plane periodic array of spiral strip elements

In this paper, we examine a direct numerical method which is suitable for the analysis of a periodic grid of spiral strip elements. The class of strip elements can be extended by representing the strip structure as a collection of elements of constant curvature. The method is also applicable to the investigation of slot analogs of these elements. The method is based on the derivation of the Fredholm integral equation of the first kind for the current through a spiral strip located in the spatial cell of a periodic grid with a transversal stratified-homogeneous filling. The self-regularization principle is applied for numerical solution of the integral equation [2]. As an example, we develop an algorithm and present the calculation results for an equiangular spiral strip in a phased antenna array. We consider an unbounded plane grid (Fig. 1) which for z > 0 is formed by a periodic system of regular spatial waveguides and for z _ 1, and k 0 = 2~/2 (2 is the working wavelength). Between the spiral arms there is a quasistatic region, so that the spiral input can be determined. The spiral is located at the boundary of the stratified-homogeneous medium with the parameters e(z), a(z), Po. The medium may contain a conducting screen as a boundary layer. The spiral elements coincide with the nodes of a rectangular grid. The case of an oblique grid is considered similarly. We examine two cases of array excitation. In the first case, the grid is excited by the field of a plane electromagnetic wave (E °, tI 0) incident at an angle (00, ~o0) in the direction of the wavevector k (see Fig. 1). This field induces currents of equal amplitude on the inputs of the spiral elements, whose phase varies linearly along the axes x and y. The phase shift between the currents of neighboring elements is related to the wave incidence angle by the relationships

Phase shifts in transmission line models of thin periodic metal grids

This paper describes simple formulas for the phase shifts associated with reflection and transmission of electromagnetic waves by a thin periodic metal grid on a plane boundary between two lossless dielectrics, for the case where the wavelengths on both sides of the boundary are greater than the grid period. The phase shifts are expressed in terms of the zero-order intensity reflectance and transmittance of the grid and are correct even when the grid has loss. They correspond exactly to the phase shifts that can be calculated using the well-known equivalent-circuit models for thin metallic grids but are applicable for any thin nondiffracting layer on a dielectric boundary, for normal incidence.

An n-gonal quasilattice in two dimensions as a dual lattice to a multiple periodic grid

An n-gonal quasilattice in two dimensions is constructed for the case of n=6m (m>or=2) as a dual to an m-tuple periodic grid Gm=G0 union rG0 union . . . union rm-1G0, where G0 is a periodic grid with hexagonal point symmetry and r the rotation operation of the plane by angle 2 pi /n. The author investigates the case where G0 is a honeycomb, triangular, Kagome or diced grid. The resulting quasilattice is not of 'Bravais' type unless G0 is the honeycomb grid. It is shown also that the same quasilattice is obtained with the projection method from the 2m-dimensional lattice L=L0*L0*. . .*L0, where L0 is a triangular, honeycomb, diced or Kagome lattice being dual to G0. In the projection method, the 2m-dimensional Euclidean space is divided into cells with a 2m-dimensional grid G which is dual to L. It is shown that Gm is identical to a two-dimensional section of G.

Equivalent thin film of a periodic metal grid

This paper establishes that any thin periodic metal grid on a plane boundary between two semi-infinite media, irradiated normally at any specified wavelength larger than its period, can be represented by an equivalent thin film. Explicit formulas for the parameters of this film are derived, and a numerical example of the equivalence is given.

Image transmission by a Bragg-Fresnel X-ray lens

A periodic grid image transmission by a Bragg-Fresnel lens (BFL) in a twice reduced scaled scheme has been carried out. The experiments have been performed in white synchrotron radiation in a hard spectral region (λ≈1 Å).

Time-asymptotics and the self-organization hypothesis for 2D Navier-Stokes equations

In this paper we study the long-time behaviour of solutions with a two-dimensional structure of the Navier-Stokes equations on a periodic grid. From a rigorous investigation of the decrease of the energy E and the enstrophy W, it follows that the Rayleigh quotient Q= W E is monotonically decreasing on solutions. This is shown to imply that the spatial structure of any solution tends to a critical point of Q, which is the structure of some planar vortex, and that all these structures are unstable except for the fundamental one which has the longest wavelength. The time dependence of the approach towards this self-organized state is investigated in some detail. In the spectral plane, the mean-squared wave-numbers of both the spectral energy- and enstrophy density are shown to decrease as a function of time. For the enstrophy this implies that the normal cascade does not match the discriminating effect of the viscous dissipation.

Non-periodic Packings of Icosahedral Polyhedra

A method for deriving packings of icosahedral polyhedra from periodic and non-periodic space grids based on icosahedral symmetry is presented. Corresponding to the three axes of symmetry, three classes of packings exist. For each class, an inter-transforming family of 64 (theoretically) possible packings is suggested. Their details remain to be worked out and examples are illustrated with packings of icosahedra.

Electrohydrodynamic Distortion Patterns in Wedged Homeotropic Samples of MBBA†

Abstract Homeotropically aligned nematic liquid crystal wedges up to 1[ddot] arc angle have been examined for the threshold of their low frequency field induced electrohydrodynamic instabilities. As the threshold value of the electric field was reached, a two dimensional array of domains resembling a honeycomb structure appeared at the thinner end of the wedge. A further increase in the applied field resulted in a reduction in the period of the honeycomb structure which eventually changed to a reasonably periodic one-dimensional grid structure resembling Williams domains. The grid lines were found to be aligned approximately along the direction perpendicular to the sample's refracting edge. Two sets of focussed images were found to be located on mutually inclined planes approximately parallel to the walls of the wedge. The phenomenon was field dependent and the variable thickness of the wedge caused the anisotropic index of refraction, and hence the associated focussing power of the cylindrical lens-array...

Dynamic magnetoconductivity of inversion electrons in periodic MOS-microstructures on Si

In laterally microstructured MOS-samples on Si(100) with an optically transparent periodic wire grid of submicron periodicity serving as gate, the quasistatic and dynamic conductivity of the inversion electrons are investigated at low temperatures and in high magnetic fields. At infrared frequencies we observe in addition to collective modes cyclotron resonance like modes that are shifted to higher frequencies if compared to cyclotron resonance in laterally homogeneous MOS-samples.

Filtration of spatial frequencies in a periodic square mesh grid

An experiment of image formation of a periodic square mesh grid under incoherent light shows the properties of optical filtering. At the same time, the periodic structures (reticular lines) of the grid are displayed.

Masked ion beam resist exposure using grid support stencil masks

Submicrometer resolution ion beam resist exposure using a new type of stencil mask is described. The mask uses a grid support structure in transmission areas to remove restrictions on pattern geometry and improve stability. The image of the grid is eliminated by rocking the incident angle of the ion beam during exposure, or by special processing of the resist. We describe the mask fabrication sequence and show examples of masks that employ a 320 nm period grid structure. Patterns with a minimum feature size of less than one-half micrometer have been replicated using this type of mask with both single and multilayer resists. In addition, experiments are described which used low energy ion beams to test the ultimate resolution of stencil masks without grid support. 40 nm wide lines were printed in PMMA at a mask-to-wafer gap of 25 μm. The applicability of such masks in high resolution proximity printing systems is discussed.

Tunable spectral filtering of a spatially incoherent source using an acousto-optic deflector

A tunable spectral filter working with spatially non coherent fields is described. Its principle is based on the chromatic filtering properties of a combination of three periodic grids, one of them being an acousto-optic deflector.

Simple formulas for transmission through periodic metal grids or plates

A simple closed-form approximate solution is given for the transmission coefficient of a normally incident electromagnetic plane wave through a screen made of periodic metal grids (inductive screen), or made of metal plates (the complementary capacitive screen). Explicit formulas are also presented for cascading screens and dielectric slabs. When compared with the exact solution, our approximate simple formulas show good numerical accuracy.

Electrohydrodynamic instabilities observed in a nematic phase under oblique boundary conditions

Electrohydrodynamic instabilities in the nematic phase of Merck ’’Phase V’’ with oblique boundary conditions were optically observed with a polarizing microscope in 25–100 μm ’’sandwich’’ cells. Oblique anchoring of the nematic was achieved by oblique evaporation of SiO on the plates. Two types of cells were used having the respective in-plane projection of the direction of evaporation on the two plates either parallel (p-type cells), or antiparallel (a-type cells). The low voltage dc instability observed for the p-type cells forms in an almost regular hexagonal pattern. By gradually increasing the voltage, the dc instability observed for the a-type cells forms at first as flows which originate at order disturbances created at imperfections in the SiO coating. Voltage increase causes these flows to detach themselves from the places of the imperfections and move solitarily. The moving flows are associated with what appears to be moving tilt inversion deformations (of splay-bend type) extending from the central part of the flow to some distance from it. When the voltage is further increased, a repeated process of replication of the flows, occurring on the associated tilt inversion deformations, leads to the creation of a periodic grid of moving flows. Other observed types of static and dynamic patterns under ac and dc excitation are reported, in particular: different types of cross rolls (ac conduction regime); variations of a pattern of what appears to be walls associated with flows, exhibiting an approximate wave number dependence on the electric field k∼E; a striped pattern associated with what appears to be twist walls and the propagating interference patterns associated with their oscillations; a toroidal flow (sometimes associated with closed inversion walls) which creates and caries along closed nematic threads (dc regime); a polygonal grid of turbulent flows (dc regime); a flow pattern correlated with the movement of the moving chevron pattern; a cellular fast turn-off pattern related to the chevron pattern. This cellular pattern appears at first as moving snakelike regions in the chevron pattern which are bordered by disclination lines. Some features of dark, spotlike figures appearing on the chevron pattern are described. Preliminary interpretations of some of the observations are offered.

Fast generation of three-dimensional computational boundary-conforming periodic grids of C-type

A fast computer program, GRID3C, has been developed to generate multilevel three-dimensional, C-type, periodic, boundary conforming grids for the calculation of realistic turbomachinery and propeller flow fields. The technique is based on two analytic functions that conformally map a cascade of semi-infinite slits to a cascade of doubly infinite strips on different Riemann sheets. Up to four consecutively refined three-dimensional grids can be automatically generated and permanently stored on four different computer tapes. Grid nonorthogonality is introduced by a separate coordinate shearing and stretching performed in each of three coordinate directions. The grids can be easily clustered closer to the blade surface, the trailing and leading edges and the hub or shroud regions by changing appropriate input parameters. Hub and duct (or outer free boundary) can have different axisymmetric shapes. A vortex sheet of arbitrary thickness emanating smoothly from the blade trailing edge is generated automatically by GRID3C. Blade cross-sectional shape, chord length, twist angle, sweep angle, and dihedral angle can vary in an arbitrary smooth fashion in the spanwise direction. Input coordinates must be Cartesian, while the output grid coordinates can be Cartesian or cylindrical.

Numerical Treatment of Unsteady Subsonic Flow Past an Oscillating Cascade

A numerical procedure for determining unsteady subsonic flow past a finite-deflection, oscillating cascade is described. Based on the assumption of small-amplitu de blade motions, the unsteady flow is governed by linear equations with variable coefficients which depend on the underlying steady flow. These equations are solved on a nonorthogonal, body-fitted, and periodic grid which facilitates the implementation of blade, wake, and cascade-periodicity conditions but disallows the use of standard difference approximations. Instead, difference approximations are based on an implicit least-squares development applicable on arbitrary grids. This permits flexibility in the choice of difference neighbors and the simultaneous approximations of differential equation and boundary conditions at boundary points, strategies which can significantly improve numerical accuracy. Sample results illustrating the effects of blade thickness, motion frequency, and inlet Mach number on the unsteady response of staggered cascades are presented. N unsteady aerodynamic analysis which accounts for the effects of mean flow deflection due to blade geometry and flow turning and is capable of providing efficient predictions of unsteady response coefficients is an important requirement for turbomachinery applications. A general aerodynamic model which fully includes the effects of nonuniform mean or steady flow on a small-disturbance unsteady flow has been formulated and described in Refs. 1 and 2. The steady flow is governed by the full potential equations, and the unsteady flow is governed by linear equations with variable coefficients which depend on the underlying steady flow. In contrast to classical linear theory, an unsteady aerodynamic formulation which includes the effects of nonuniform mean flow requires numerical solutions of the governing steady and unsteady boundary value problems. Numerical procedures for determining steady subsonic3 or transonic 4 potential flow through cascades are currently available. A solution procedure for the linear, variable-coefficient, unsteady problem is described in the present paper. Since the unsteady aerodynamic model treats realistic cascade configurations, the numerical approximation must take into account nonrectangular geometries. Generally, approximations to differential equations in nonrectangula r regions are developed in one of two ways. A mapping may be defined so that in mapped variables calculation points can be placed in a rectangular grid; the transformed equations are then approximated using rectangular difference expressions in the usual way. Alternatively, more general difference ap- proximations may be defined so that differential equations can be approximated on nonrectangula r meshes in physical variables. Thompson, Thames, and Mastin5 have defined global mappings for arbitrary two-dimensional bodies, but these mappings are inappropriate for cascades since the resulting meshes generally do not allow for cascade periodicity. Ives and Liutermoza4 have determined global mappings which transform infinite two-dimensional cascade regions into rectangular computational regions while allowing for cascade periodicity. However, the necessary mappings are not presently known for all cascades, particularly those with