Rectangular Boundary Research Articles

An exact solution to the plane orthotropic elasticity problem with rectangular boundaries under arbitrary edge forces

Mathieu’s approach to the fundamental problem of plane strain with rectangular boundaries was futher developed by Baker, Pavlovic and Tahan. The method consists in breaking up the full solution into eight basic problem types. In this paper, an exact solution to two-dimensional orthotropic elasticity problems with rectangular contours is obtained. Some convergent results are proved.

Decaying two-dimensional turbulence with rigid walls

Decaying two-dimensional Navier-Stokes turbulence (or turbulence for the corresponding electrostatic “guiding center” plasma) has usually been studied in spatially periodic boundary conditions. It is known to exhibit a variety of features which are different from the three-dimensional case. By means of spectral-method direct numerical computation, we show that circular no-slip boundaries can significantly alter the decay scenario in two dimensions. In particular, the wall functions as an important source of vorticity and vorticity gradient, and the decay of energy is enhanced by this effect, relative to the decay in periodic boundary conditions. In addition, the very slow decay of angular momentum in high Reynolds number cases constrains the evolution in a way that has no analogue for rectangular periodic boundary conditions.

Experimental evaluation of basic circuit components using buried microstrip lines for constructing high-density microwave integrated circuits

The buried microstrip line (BMSL) is a promising transmission line structure for realizing low crosstalk characteristics. The measured crosstalk characteristics of model BMSL's have shown good agreement with the estimated ones based on the rectangular boundary division method and the finite-difference time-domain (FDTD) method, indicating that these calculation methods are appropriate for estimating the performance of BMSL's. These results also confirm our expectation that BMSL's have extremely low crosstalk characteristics such as -100 dB in real circuits at high microwave frequencies. The basic circuit components to construct high-density microwave circuits using BMSL's, couplers, stub matching circuits, and gap-coupled resonance filter circuits, were fabricated and experimentally evaluated.

Configurational transition in a Fleming - Viot-type model and probabilistic interpretation of Laplacian eigenfunctions

We analyze and simulate a two dimensional Brownian multi-type particle system with death and branching (birth) depending on the position of particles of different types. The system is confined in the two dimensional box, whose boundaries act as the sink of Brownian particles. The branching rate matches the death rate so that the total number of particles is kept constant. In the case of m types of particles in the rectangular box of size a,b and elongated shape $a\gg b$ we observe that the stationary distribution of particles corresponds to the m-th Laplacian eigenfunction. For smaller elongations $a>b$ we find a configurational transition to a new limiting distribution. The ratio a/b for which the transition occurs is related to the value of the m-th eigenvalue of the Laplacian with rectangular boundaries.

Characterization of buried microstrip lines for constructing high-density microwave integrated circuits

This paper describes the characterization of a guided wave structure, buried microstrip line (BMSL), which is considered to be promising for constructing high-density microwave and millimeter-wave integrated circuits because of its high isolation characteristics. The BMSL includes a dielectric medium surrounded by ground conductor walls and a strip conductor on the top of the dielectric. The BMSL structure is characterized by the two methods, the rectangular boundary division (RBD) method and the finite-difference time-domain (FDTD) method. The RBD method is employed to obtain basic parameters of the BMSL such as characteristic impedances and coupling coefficients over a wide range of line sizes taking advantages of its high calculation efficiency. On the other hand, the FDTD method has been used for more detailed characterization such as the frequency performances of stub matching circuits. The FDTD method is also used to confirm the validity of the quasi-TEM wave approximation which the RBD is based on. The analysis results reveal that the BMSL structure possesses much lower coupling coefficients than a conventional microstrip line does, from -15 dB to -100 dB depending on their burial depths.

New infinite elements for a finite element analysis of 2D scattering problems

Two new formulations of infinite elements are proposed. These elements in combination with the conventional finite elements yield coupled finite element formulations for 2D scattering problems. The first formulation works for any shape of the finite element region, while the second one has been developed for a rectangular outer boundary. In comparison with the previous infinite elements, the present infinite elements are constructed in such a way that simple expressions have been obtained for their element matrices. As a result of this, no formula for numerical quadrature over the infinite region is necessary and the infinite elements can easily be incorporated into standard finite element codes. Two numerical examples are given.

Vector absorbing boundary conditions for nodal or mixed finite elements

We present 2D and 3D vector absorbing boundary conditions for unbounded microwave problems. Coupling with vector formulation leads to a non-symmetric system matrix. We show bow it is possible to symmetrise these conditions with the same degree of precision. Edge and corner conditions are also presented for rectangular boundaries.

Topographic distribution of inguinal lymph nodes metastasis: significance in determination of treatment margin for elective inguinal lymph nodes irradiation of low pelvic tumors

Purpose : To study the distribution of gross inguinal lymph node metastasis and, in particular, its correlation with major pelvic bony structures on a simulation film. Methods and Materials : Thirty-seven cases of low pelvic tumors having gross inguinal lymph node metastasis that were treated with radiation therapy between November 1987 and December 1992 were segregated for study. The patient's nodes were palpated and marked with lead wire before the simulation film was taken. The geometric center of the usually round or elliptical node on the film was assumed to be the origin of the previously uninfested node. A total of 84 such labeled nodes was obtained from these 37 cases. These centers were transferred to and mapped collectively on a new simulation film showing major pelvic bony structures of left hemipelvis and upper femur. Results : Distribution of gross inguinal lymph nodes was found confined to the following area, as related to major pelvic bony structure: laterally, just abutting the tangential line that passes through lateral border of the femoral head; medially: 3 cm away from the body's midline axis; superiorly: 1 cm below the line that joins both upper borders of the femoral head; inferiorly: 2.5 cm below the low borders of ischial tuberosity. According to this rectangular boundary, three nodes were out of field, nine nodes near the border less than 1 cm margin. This area adequately covered 86% (72 of 84) of the studied nodes. Conclusion : Distribution study is important in determining the treatment margin. In general, an additional 1–2 cm beyond the area described above is the recommended treatment margin for elective inguinal lymph nodes irradiation with high confidence level of coverage.

Spatial properties of retinal mosaics: an empirical evaluation of some existing measures.

Mosaics of neurons are usually quantified by methods based on nearest-neighbor distance (NND). The commonest indicator of regularity has been the ratio of the mean NND to the standard deviation, here termed the 'conformity ratio.' However, an accurate baseline value of this ratio for bounded random samples has never been determined; nor was its sampling distribution known, making it impossible to test its significance. Instead, significance was assessed from goodness-of-fit to a Rayleigh distribution, or from another ratio, that of the observed mean NND to an expected mean predicted by theory, termed the dispersion index. Neither approach allows for boundary effects that are common in experimental mosaics. Equally common are 'missing' neurons, whose effects on the statistics have not been studied. To address these deficiencies, random patterns and real neuronal mosaics were analyzed statistically. Ns independent random-point samples of size Np were generated for 13 Np values between 25 and 6400, where Ns x Np > or = 144,000. Samples were generated with rectangular boundaries of aspect ratio 1:1, 1:5, and 1:10 to examine the influence of sample geometry. NND distributions, conformity ratios, and dispersion indices were computed for the resulting 45,997 independent random patterns. From these, empirical sampling distributions and critical values were determined. NND distributions for small-to-medium, bounded, random populations were shown to differ significantly from Rayleigh distributions. Thus, goodness-of-fit tests are invalid for most experimental mosaics. Charts are presented from which the significance of conformity ratios or dispersion indices can be read directly. The conformity ratio reacts conservatively to extremes of sample geometry, and provides a useful and safe test. The dispersion index is nonconservative, making its use problematic. Tests based on the theoretical distribution of the dispersion index are unreliable for all but the largest samples. Random deletions were also made from 33 real retinal ganglion cell mosaics. The mean NND, conformity ratio, and dispersion index were determined for each original mosaic and 36 independent samples at each of nine sampling levels, retaining between 90% and 10% of the original population. An exclusion radius, based on a spatial autocorrelogram, was also calculated for each of these 10,725 mosaic samples. The mean NND was moderately insensitive to undersampling, rising smoothly. The exclusion radius was remarkably insensitive. The conformity ratio and dispersion index fell steeply, sometimes failing to reach significance while half of the cells still remained. For the same 33 original mosaics, linear regression showed the exclusion radius to be 62 +/- 3% of the mean NND.

GROMACS: A message-passing parallel molecular dynamics implementation

A parallel message-passing implementation of a molecular dynamics (MD) program that is useful for bio(macro)molecules in aqueous environment is described. The software has been developed for a custom-designed 32-processor ring GROMACS (GROningen MAchine for Chemical Simulation) with communication to and from left and right neighbours, but can run on any parallel system onto which a a ring of processors can be mapped and which supports PVM-like block send and receive calls. The GROMACS software consists of a preprocessor, a parallel MD and energy minimization program that can use an arbitrary number of processors (including one), an optional monitor, and several analysis tools. The programs are written in ANSI C and available by ftp (information: gromacs@chem.rug.nl). The functionality is based on the GROMOS (GROningen MOlecular Simulation) package (van Gunsteren and Berendsen, 1987; BIOMOS B.V., Nijenborgh 4, 9747 AG Groningen). Conversion programs between GROMOS and GROMACS formats are included. The MD program can handle rectangular periodic boundary conditions with temperature and pressure scaling. The interactions that can be handled without modification are variable non-bonded pair interactions with Coulomb and Lennard-Jones or Buckingham potentials, using a twin-range cut-off based on charge groups, and fixed bonded interactions of either harmonic or constraint type for bonds and bond angles and either periodic or cosine power series interactions for dihedral angles. Special forces can be added to groups of particles (for non-equilibrium dynamics or for position restraining) or between particles (for distance restraints). The parallelism is based on particle decomposition. Interprocessor communication is largely limited to position and force distribution over the ring once per time step.

Scalar Quantum Fields Confined by Rectangular Boundaries

A collection of local Casimir problems is studied in detail. These problems are generated by confining a scalar quantum field within boundaries consisting of planes, edges and corners. The boundaries include isolated infinite planes, rectangular edges and corners, parallel planes, the rectangular waveguide and the rectangular cavity. Calculations are done for massless and massive confined fields and for fields at zero and nonzero temperature. Each example has its finite global Casimir effect, which must be computable from the exact local description. The latter however contains known boundary divergences which seem to imply an infinite global Casimir effect. A physical picture and mathematically explicit method is presented for computing finite global Casimir effects from the divergent local description. Divergent boundary functions extending outward in space away from a given boundary ϱ𝔐 represent the distortion of the virtual particle sea of the confined field caused by ϱ𝔐. This distortion of the virtual sea is inseparable from the existence of ϱ𝔐, and may be regarded as the cost of construction of ϱ𝔐 out of nothing. Thus a given divergent boundary function is intrinsic to its ϱ𝔐 — where here is meant specifically the complete boundary function extending (if it does) all the way to infinity. Different boundaries cut off parts of each other's boundary functions, and this plays a vital role in the global Casimir effect as will be explained. It is shown in the examples considered how to compute the global Casimir effect by integration of the exact local description over the region within which the quantum field is confined. Rectangular boundary geometry makes calculations relatively simple; however the method can be applied to boundaries of arbitrary shape. At finite temperature the confined quantum field represents a thermal gas as well as a virtual particle sea. Boundaries also distort this gas of real particles in a way which can be computed. Exact local calculations are done for the rectangular boundaries considered. Beyond these considerations for free fields, systems with interacting fields present additional very interesting local Casimir phenomena. A model two-field interacting theory is discussed briefly for illustration.

Analysis of metallic waveguides with rectangular boundaries by using the finite-difference method and the simultaneous iteration with the Chebyshev acceleration

A numerical procedure based on the finite-difference method and simultaneous iteration of the power method in conjunction with the Chebyshev acceleration technique is utilized to analyze the metallic waveguides. Due to the efficiency of the present sparse matrix eigenproblem solver, lots of unknowns can be used in the domains of the waveguide cross-sections. Therefore, accurate cutoff wavenumbers or frequencies can be obtained by using the simple finite-difference method for the commonly used metallic waveguides such as the L-shaped, single-ridged, double-ridged, and rectangular coaxial waveguides. Some discrepancies with the numerical results in the recent literature are found and detailed discussions are provided to verify the correctness of the present results. >

Efficient method for the capacitance calculation of circularly symmetric via in multilayered media

The equivalent capacitance produced by a circularly symmetric via connecting two microstrip lines on different planes is efficiently calculated by using the rectangular boundary division method. The calculation is carried out considering effects of inhomogeneous media within the line structures as well as the ground conductor thickness on the capacitance produced by a microstrip via. Computational results for various via geometries are presented and compared with presently available data.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Large amplitude free vibration of thick shallow shells supported by shear diaphragms

The effect of thickness and curvature upon the large amplitude vibration of shallow shells is studied in this paper. For this purpose, the non-linear governing equations for thick shallow shells which have principle curvatures and rectangular planform are derived by Hamilton's principle using first order shear deformation theory. Applying Galerkin's procedure and eliminating variables except for transverse displacement, the governing equations are reduced to an elliptic ordinary differential equation in time. The period of vibration for the shell is calculated by integrating the equation using a Gauss-Legendre integration method. The present method is applied to a shallow shell which has a rectangular boundary supported by shear diaphragms.

Experimental results on buried microstrip lines for constructing high-density microwave integrated circuits

This letter describes the characterization and some experimental results of a buried microstrip line (BMSL), a guided-wave structure considered to be promising for constructing high-density microwave and millimeter-wave integrated circuits because of its high isolation characteristics. The BMSL structure is characterized practically by the rectangular boundary division (RED) method and rigorously by the finite difference time domain (FDTD) method. The analyzed results reveal that the BMSL structure possesses much lower coupling level than the conventional microstrip line does, from -15 to -100 dB depending on their burial depth. Experimental data show good agreement with numerical results.

Elastic stability of orthotropic plates using the conjugate load/displacement method

It is shown that the conjugate load/displacement method which has been used for the elastic stability analysis of isotropic plates can also be used for orthotropic plates. Results are presented for a range of plates of various rectangular configurations and boundary conditions, as well as for different values of the material modular ratios. A range of loadings is covered; in particular the case of concentrated loads is dealt with. For this loading case the conjugate load/displacement method has been found to be particularly useful.

Asymptotic boundary conditions for axisymmetric finite element electrostatic analysis

Asymptotic boundary conditions (ABC) for rectangular outer boundaries are formulated and used for the finite element computation of axisymmetric electrostatic fields. ABC-s for rectangular boundaries are more efficient than ABC-s with circular boundaries for problems with large aspect ratios. To illustrate this formulation, three axisymmetric electrostatic problems are solved. Potential distributions and electric field strengths are obtained. The results using ABC-s are compared with analytic solutions or solutions obtained by boundary truncation techniques.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Error analysis when solving Laplace's equation numerically by iteration

Numerical solution methods of Laplace's equation /spl Delta/V=0 when boundary values of potential V are specified abound, and many computer programs employing relaxation techniques, finite-element techniques, etc. have been discussed in the literature. The finite-mesh relaxation method of numerical iteration is discussed in many physics and electrical engineering texts, but little attention is given to the error analysis, which, moreover, is incorrect more often than not in these texts. The authors show that the error in the iterated solution can be found by a relatively simple analysis, and discuss its implications. The authors illustrate the problem by using a very simple PC program that solves the two-dimensional Laplace's equation with Dirichlet conditions on a rectangular boundary.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Three‐dimensional analytical solutions to the advection‐dispersion equation in arbitrary Cartesian coordinates

Analytical solutions for the advection‐dispersion equation (ADE) usually assume that boundary and initial conditions are orthogonal to the principal axes of the dispersion tensor. However, this is not always the case in field studies or modeling scenarios. Using the method of Green's functions, a generalized analytical solution of the three‐dimensional ADE set in an arbitrary Cartesian coordinate system (i.e., not orthogonal to the principal flow direction) is given for the solute resident concentration in a semi‐infinite porous medium with an arbitrary initial condition, surface boundary condition, and sink/source terms. Two particular solutions for a rectangular surface flux boundary condition and a buried parallelopiped, respectively, are derived from the general solution. The corresponding frequency domain solutions are also given which provide a more efficient method of computation for generating two‐and three‐dimensional grids via use of a two‐dimensional fast Fourier transform algorithm (as an alternative to the two‐dimensional numerical integration required to calculate the concentration using the real space solution). When the arbitrary coordinate system is orthogonal to the principal axes, it is shown that these particular solutions are the same as previously published results.

Rectangular boundary division method and its applications (invited review article)

AbstractThe rectangular boundary division method within the quasi‐TEM wave approximation is described as an analysis method to give basic parameters for designing complicated microwave transmission lines. Analysis procedures of some practical planar transmission lines are outlined and practical techniques for numerical computations with this method are discussed. © 1993 John Wiley &amp; Sons, Inc.