Arbitrary Geometrical Shape Research Articles

Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects

The recent development and extension of the method of moments technique for analyzing electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects is presented based on the combined field integral equations. The surfaces of the homogeneous three-dimensional arbitrary geometrical shapes are modeled using surface triangular patches, similar to the case of arbitrary shaped conducting objects. Further, the development and extensions required to treat efficiently three-dimensional lossy dielectric objects are reported. Numerical results and their comparisons are also presented for two canonical dielectric scatterers-a sphere and a finite circular cylinder.

Circumscribing a convex polygon by a polygon of fewer sides with minimal area addition

An approach and an algorithm are introduced for circumscribing a given n-sided convex polygon P n by an m-sided polygon P m , ( m < n), so that the added area is minimal. This algorithm constitutes one building block in an algorithm for efficient nesting of arbitrary geometric shapes in a given rectangular board. Flame cutting of steel sheets and laser cutting of textiles are two industrial situations where this problem is of great importance. The approach follows a top-down stepwise refinement and reduction of the original problem into simpler subproblems, the solutions of all of which permit the solution of the original problem. It is first shown that the optimal circumscription of P n by P m may be obtained by ( n − m) iterative single side reductions. The solution of the single side reduction problem is then characterized, and an algorithm which is based on the triangle rotating side problem is proposed. This last problem is concerned with passing the third side of a triangle through a given point that lies within the area bounded by the two other given sides so that the triangle area is minimized. On the way to proving the optimality of the algorithms for the original problem and its subproblems, new concepts and theorems are introduced. The algorithm was tested on a very large number of polygons with varying numbers of sides and shapes, which were circumscribed by hexagons. The average efficiency—defined as the ratio of the area of P n to that of P m —was 96%. As n increases, efficiency reduces and approaches asymptotically the maximum achievable efficiency for circumscribing a circle (“infinite” sided polygon) by a regular hexagon: 90.69%. With n = 50 the average efficiency was 91.8%.

A finite-element program for modeling transient phenomena in GaAs MESFET's

A two-dimensional finite-element program has been developed for analyzing transient and steady-state characteristics of GaAs devices with arbitrary geometric boundary shapes. The program code consists of two separate programs, GRID and FET, which are discussed in some detail. GRID serves to generate a nonuniform mesh, while FET computes a self-consistent solution of Poisson's and the current continuity equations. A GaAs FET with a trapezoidal recessed gate structure has been studied to demonstrate the capabilities of the program to analyze odd shapes. Current-voltage characteristics were computed for the recessed gate and a planar device. The results from both FET structures are compared and analyzed. In particular the effect of the recessed gate on the field distribution in the device is discussed. Large signal transient behaviors of both devices were examined, and it was found that both device structures produce similar results in steady-state and transient conditions.

Patched coordinate systems

Abstract : Numerical grid generation has been an excellent tool for producing curvilinear body-fitted coordinate systems. Curvilinear coordinate systems or grids are commonly used in the solution of partial differential equations in domains surrounding arbitrary geometrical boundary shapes. Body-fitted grids are particularly advantageous in the treatment of surface boundary conditions, and usually yield a degree of simplicity in the logic required to solve the hosted partial differential equations. In practice, numerical grid generation usually involves transformation of the physical domain of interest into a geometrically simple domain, such as a rectangular block or assembly of blocks. The solution of grid generation equations in the simple domain products the coordinates of a corresponding grid in the physical domain, subject to a variety of grid control procedures aimed at producing favorable grid characteristics. This process is usually straightforward when the topology of the physical domain is simple enough to allow transformation to a single rectangular domain. But when dealing with geometrically and topologically complex domains such as surround an aircraft configuration, the total issue of grid generation becomes more complex. The domain in general cannot be mapped into a single block. The configuration surface geometry itself may be nonanalytic, and these features will be manifest in any grid surrounding such complex boundary shapes.

Quasi-Parabolic Technique for Spatial Marching Navier-Stokes Computations

ACOMPUTATIONAL technique is presented for obtaining flowfield solutions to a parabolic form of the Navier-Stokes equations. The point of departure is the general interpolants method (GIM), which provides a discretization for partial differential equations on arbitrary threedimensional geometries. The new scheme, termed quasiparabolic (QP), treats the parabolized equations but with time-like terms appended. Addition of these extra terms, which are relaxed by iteration, avoids many of the singularities inherent in classical parabolic Navier-Stokes methods. Solutions are presented for viscous flows in boundary layers and free shear layers as computed with the GIM/QP scheme. Contents The full three-dimensional Navier-Stokes equations are elliptic in character and require either time-dependent or relaxation/iteration schemes to integrate the complete spatial flowfield simultaneously. This can require large amounts of computer storage and relatively long run times. For situations in which a region of the flow is inviscid and entirely supersonic, a spatial hyperbolic marching algorithm would be efficient. There are also many viscous problems of interest in which parabolic marching solutions are acceptable. Certain assumptions must be made in using a spatial marching technique. There must exist a dominant flow direction in which to march, and there can be no flow back upstream, i.e., no recirculation in the streamwise direction. Stress terms are not allowed to act on the cross planes, i.e., there can be no second-order terms (diffusion, viscosity) in the marching coordinate. The downstream pressure field also must not be allowed to propagate upstream. Reference 1 contains a review of the approaches to the solution of the parabolic Navier-Stokes (PNS) equations. The paper of Lin and Rubin2 also discusses many of the approaches and difficulties involved in PNS solutions. The intent of this research is to provide a parabolic spatial marching technique which can be readily incorporated into the general interpolants method (GIM). The GIM code provides a discretization of partial differential equations on arbitrary three-dimensional geometries. The GIM treatment of arbitrary geometric shapes in three dimensions and the method of solution of the full Navier-Stokes equations is described fully in Refs. 1,3, and 4 and will not be repeated

Preliminary computations of magnetic distributions in ion-implanted garnet films

Magnetic domain structures in a thin ion-implanted garnet film are calculated through numerical integration of Landau-Lifshitz-Gilbert equation. The fundamental interactions due to anisotropy (including uniaxial growth anisotropy, stress induced in-plane anisotropy, cubic crystalline anisotropy), exchange, and demagnetization are included. Arbitrary geometric shapes of ion-implanted regions can be accommodated. Results show that magnetic domain structures can be affected strongly by the cubic ansiotropy, the geometric shapes of implanted regions, and the in-plane magnetic field.

Potential Distribution and Multi-Terminal DC Resistance Computations for LSI Technology

Computer time and storage requirements are the two main considerations in the design of a packaging analysis software tool for the problem of calculating the electric potential distribution in arbitrary geometrical shapes. The FEM (Finite Element Method) is the accepted approach for solving such problems. A new formulation for the linear triangular element is presented which is used to derive a very simple and computationally inexpensive linear rectangular element equation interrelating only the geometrical centers of the elements. The result is a much sparser assembly matrix with a maximum of five non-zero entries per equation compared with the usual nine of the FEM formulation. In addition, a method to obtain the minimum bandwidth of the matrix is given for the efficient and static use of external storage, permitting the solution of any size problem. The methods are applicable to multi-plane, multi-terminal configurations for the production of equivalent-resistance networks and for the calculation of the potential distribution throughout the configurations.

Aerodynamics and Noise of Coaxial Jets

The objective of this investigation was to develop a unified prediction method for estimating the aerodynamic and noise characteristics of jets issuing from nozzles of arbitrary geometric shapes. The method has been developed and demonstrated for dual-flow coaxial jets. An extension of Reichardt's theory is utilized to predict the mean velocity, temperature, and axial turbulence intensity distributions throughout the jet plume. The generic noise intensity spectrum is synthesized by a slice-of-jet approach, wherein each axial location in the plume contributes to the sound generation in one dominant frequency band. The propagation of the source spectrum to the far field is modeled by means of convected quadrupoles embedded in a parallel slug flow. Extensive predictions of the aeroacoustic characteristics of coaxial jets were made and compared with experiment. The agreement between theory and experiment is quite good, except at high frequencies and shallow angles to the jet axis, where refraction is overestimated. A major conclusion drawn from these results is that the noise reduction attained by a coaxial jet conies primarily from a reduction in turbulence intensity.

Simultaneous brownian coagulation and diffusive deposition of monodisperse particles

Concept of the “wall-loss constant” used for the description of particle deposition on the container walls is criticised. Kinetics of simultaneous Brownian coagulation and diffusive deposition of monodisperse particles in a stagnant fluid contained in the spherical vessel are investigated. Assuming that the rate of coagulation and deposition are additive the differential equation is obtained whose solution describes the decay with time of the total particle concentration in the initial stage of the coagulation-deposition process. The results obtained are used for another checking of the concept of the “wall-loss constant”. It is concluded that even in this most simple case this concept is unrealistic. Furthermore the results obtained support the old idea that neglecting the wall losses may be one of the main reasons of the discrepancy between the experimental and theoretical values of the coagulation constant. Possible application of the results obtained in the theory and operation of static diffusion batteries is briefly discussed. The developed approach can be applied to containers of arbitrary geometrical shapes.

Theory of diffusive deposition of particles in a sphere and in a cylinder at small fourier numbers

Abstract The general theory of the diffusive deposition of particles in a sphere and in a cylinder is critically reviewed and further developed. Smoluchowski's probable residence time of the particle is recalculated and compared with the introduced and calculated half-life of the diffusive deposition. Applying the expression for the average absolute value of the Brownian displacement of the particle, the theory of diffusive deposition of particles at small Fourier numbers is developed. Theory is checked by comparison with the exact values of the deposition rate on the external surface of an infinitely long cylinder. As expected, the deviations of approximate from exact values increase with increasing Fourier number. The developed method is very simple and can be applied to containers and bodies of arbitrary geometrical shape. Possibilities of application of the developed theory are briefly discussed.

Simultaneous translational and rotational Brownian movement of particles of arbitrary shape

The coupling of the translational and rotational Brownian motion of a particle due to its arbitrary geometric shape is studied. The generalized kinetic equation for the Brownian particle distribution function is derived from the Liouville equation. The correlation functions for random force and torque are calculated from hydrodynamic fluctuation theory. It is found that the coupling effect plays an appreciable role in the diffusivities of asymmetric particles. For the special case of a sphere where there is no coupling, the autocorrelation function for the random torque in rotational Brownian motion is given in closed from. Its asymptotic from at large times is proportional to (time)−5/2. As an example of an asymmetric particle, the diffusivity tensor is worked out for two disks connected by a rigid rod.

Elastic plate analysis by Euler's variational method

A finite difference method is presented for analysis of elastic plates based on the small displacement theory. The proposed procedure is useful in computer programming, and it permits solution for isotropic and orthotropic plates arbitrary geometrical shape as well as variable thickness, under arbitrary loading. An illustrative example is also given.

The theory of micromixing for unsteady state flow reactors

The classical Danckwerts—Zwietering treatment of segregation has been generalized to unsteady state flow reactors of arbitrary geometric shape. Basic equations which can be used to predict two limits of the extent of chemical reaction are developed from the population balance equation. It is shown that the residence time distribution uniquely determines the extreme of micromixing and the distribution can be determined from a series of standard tracer experiments. Nauman's treatment is confirmed but the restriction to stirred tank systems is removed in this study. Examples are given to illustrate the use of the theory. The segragation effect on a second order reaction is found to be small for both the isothermal and adiabatically cases. The effect on a biological growth process is very significant.

A Theoretical Analysis of the Plain-Knitted Structure

A theoretical analysis was made of the relaxed plain-knitted structure. The analysis differs from previous ones in that no arbitrary geometrical loop shape is assumed and no empirical methods are used. The loop configuration is de rived from consideration of the reaction forces and couples acting within the structure, their magnitude being determined by the yarn displacement necessary for loop interlocking. A minimum-energy structure is shown to exist; this structure is fairly independent of fabric tightness and yarn properties. The actual energy minimum is fairly shallow, so that in practice some difficulty could be expected in obtaining a perfectly relaxed fabric; this would lead to a corresponding spread in experimental values of plain-knitted fabric shape and dimensions. The numerical values of fabric dimensions obtained from the analysis show good agreement with the experimental data reported by other workers.

Exact Solution for the Scattering of Electromagnetic Waves from Bodies of Arbitrary Shape. III. Obstacles with Arbitrary Electromagnetic Properties

The analytical technique developed earlier for the scattering of electromagnetic wave fields by perfect conductors is generalized to the case where the scattering obstacle is not only of arbitrary geometrical shape, but where both the scattering obstacle and the exterior environment have arbitrary, though homogeneous, electromagnetic properties. As before, the solution obtained is analytically exact and thus equally valid in the near and far zones, as well as over the entire frequency range. The special cases of the first-order solution and of an incident plane wave are considered in detail. The form of the solution is particularly well suited for methodical numerical evaluation.

Distribution of the density of intense radiation in an absorbing specimen

An expression for computing the spatial density of light energy in an absorbing specimen of arbitrary geometrical shape is obtained with allowance for clearing of the material under the action of powerful exciting radiation. Numerical computations are carried out and the condition for uniform energy distribution in a cylindrical rod is found.

On the diffusion of ions in membranes

A simple model of a membrane is used to obtain relations among five measurable quantities: the inward and outward ionic fluxes, the internal and external ionic concentrations, and the difference of electrical potential across the membrane. The Goldman equation is generalized to arbitrary geometrical shapes.