Iterative Finite Difference Method Research Articles

Dislocation dynamics during the growth of silicon ribbon

The thermal viscoplastic stresses and the dislocation densities in silicon ribbon are computed for an axially changing thermal profile by using an iterative finite difference method. A material constitutive equation (Haasen–Sumino model) which involves an internal variable (mobile dislocation density) is used. The results are interpreted as showing that there is a maximum width of silicon ribbon that can be grown when viscoplasticity and dislocations are considered. This maximum width limitation does not exist if the material behavior is elastic.

Two directional coextrusion flow of two molten polymers in flat dies

AbstractA theoretical study has been carried out of the two directional isothermal flow of two molten polymers in a flat die. The viscous behavior of each fluid is described by a power‐law. Using an iterative finite difference method, a numerical program predicts the influence of geometrical and Theological parameters on the variations of the interface position. Theoretical values are in good agreement with experimental results obtained on a fish‐tail die. In the case of a coat hanger‐die, the mean value of the interface position is correct, but the general shape may be quite different.

Non‐isothermal flow of a molten polymer in a coat‐hanger die

AbstractThe non‐isothermal flow of a power‐law fluid in a coal‐hanger flat die is studied. Equilibrium and thermal equations are solved by using an iterative finite difference method. The pressure distribution, the flow lines, the residence time distribution, the temperature distribution, and the flow uniformity at the end of the die are obtained, with a stress on the effect of thermal regulation and of temperature dependence of the viscosity.

COMPARISON OF SOIL WATER INFILTRATION PROFILES OBTAINED EXPERIMENTALLY AND BY SOLUTION OF RICHARDSʼ EQUATION

We compared Hesperia and Columbia soil experimental infiltration data to calculated profiles obtained by solutions of Richards' equation requiring soil water diffusivity D and hydraulic conductivity K as functions of the water content θ. The calculations were made using a finite-difference, iterative method (FINDIT). Predicting wetting profiles by this method is somewhat more rigorous than some other solutions of Richards' equation. The method reduces calculations for infiltration to a two-term algebraic equation, partitions infiltration into matric and gravitational components, and gives an asymptotic relationship between the infiltration rate and the saturated conductivity as time approaches infinity. An array of solutions for the two indicated soils, obtained with and without the diffusion lip procedure, raises questions of the need for its use. A new method for estimating the Λ - θ area in horizontal infiltration analysis, using only diffusivity data, gives estimates to less than 3 percent of the final value. Integrated mean values of D and K were required for accurately predicting soil moisture wetting profiles over a range of time periods and θ divisions. Arithmetic and geometric means and nonaveraged values, particularly of D, produced unreliable wetting profiles.

Numerical solution of an electrically conducting viscous incompressible fluid

Abstract This paper investigates the flow pattern of an axisymmetric, electrically conducting viscous and incompressible fluid in a pipe, having a throat at its centre. An iterative finite difference method, based on Picards approximations for the spatial derivatives of the governing equations, is used to obtain the streamline patterns of the fluid flow field in presence of a magnetic field. An initial parabolic velocity profile is assumed at the entrance of the pipe. Numerical solutions which are computed for the values of Reynolds number lying between 50 and 2000 and the magnetic pressure number varying from 0 to 80 show the occurrence of a secondary flow in the downstream region and a recirculating flow in the upstream region of the pipe.

Modelling the Flow of Mass and Energy within a Snowpack for Hydrological Forecasting

This paper describes a deterministic, distributed snow-melt model which has been developed for the Systeme Hydrologique Européen (SHE), The model is based on partial differential equations describing the flow of mass and energy through the snow. These equations are solved by an implicit, iterative finite-difference method. The behaviour of the model is investigated using data from a sub-Arctic site in Scotland.

Modelling the Flow of Mass and Energy within a Snowpack for Hydrological Forecasting

This paper describes a deterministic, distributed snow-melt model which has been developed for the Systeme Hydrologique Européen (SHE), The model is based on partial differential equations describing the flow of mass and energy through the snow. These equations are solved by an implicit, iterative finite-difference method. The behaviour of the model is investigated using data from a sub-Arctic site in Scotland.

FINITE-DIFFERENCE SOLUTIONS OF THE INFILTRATION EQUATION

ABSTRACT An accurate, two-term solution of Richards' equation for one-dimensional, vertical infiltration was obtained by a finite difference, iterative method (FINDIT). Wetting distances and soil water content distributions closely resemble those obtained by Philip with his series solution, which is extremely accurate at short times but fails to converge for long times. Solutions appear to be equally accurate for all times using the proposed procedure. Hence, the proposed procedure does not require additional approximations or matching factors to link separate analyses for short or long infiltration times. Because the solution equation consists of only two terms, the inflow is logically partitioned into matric and gravitational components. The matric component is expressed in a constant sorptivity term S, conceptually identical to the sorptivity S of Philip. The gravitational component is time-dependent and increases to a maximum value equal to the hydraulic conductivity at the soil surface as time approaches infinity. After appropriate coefficients have been determined, both matric and gravitational components of cumulative infiltration may be expressed independently by a logarithmic relationship that avoids the use of D and K functions and iterative computer procedures.

An Accurate Slot-Flow Model for Non-Newtonian Fluid Flow Through Eccentric Annuli

AbstractThis paper describes a new model for obtaining analytical solutions to the problem of non-Newtonian fluid flow through eccentric annuli. A discussion on non-Newtonian rheology is presented, followed by the development and solution of applicable differential equations using the Ostwald de Waele power-law model and a nonrectangular slot.Results indicate that velocity values are reduced greatly in the reduced region of the eccentric annulus. This is important in directional drilling where the drillpipe tends to lie against the hole. Design of mud flow for cuttings transport on the basis of the nominal average velocity could lead to serious problems associated with cuttings buildup in the low-velocity region of the annulus. Other practical ap-plications of this work include the determination of velocity distribution in chemical processes involving fluid flow through eccentric annuli – e.g., heat exchangers and extruders – and more accurate velocity profiles inside journal bearings, particularly for small diameter ratios.The main advantage in the new approach is that iterative finite difference methods used by previous investigators are avoided. Previous work along present lines used a linearized model and resulted in velocity profiles of unacceptable accuracy. This study improves both the accuracy and the solution technique.

The Effects of Cold Gas Injection on a Confined Arc Column

This paper considers the interaction of a thermal argon arc plasma confined in a relatively long water-cooled cylindrical tube with a cold argon flow injected radially into the tube through a circumferential slit. An analytical model is established to predict the thermal, the fluiddynamic, as well as the electrical behavior, of such an arc assuming laminar flow and Local Thermodynamic Equilibrium (LTE) of the arc plasma. Numerical solutions for the field variables are obtained by solving simultaneously the mass, momentum, energy, and charge conservation equations by an iterative finite difference method. The results show that the arc column becomes constricted at the location of gas injection due to thermal and fluiddynamic effects associated with the injected cold flow. The arc responds to the constriction by an increase of the temperature in the core region which resists the penetration of the cold flow into the arc. This penetration remains relatively small even at high injection ratios because the arc temperature and, therefore, the resistance to flow penetration increases with increasing injection ratio. The enhanced Joule heating in the injection region leads to a minor thermal expansion of the base flow which can be observed upstream of the injection slit. This effect becomes less pronounced as the injection mass flow rate increases.

Modeling of the Anode Contraction Region of High Intensity Arcs

A self-consistent model for the anode contraction region of a high intensity dc arc is based on a wall-stabilized axisymmetric arc operated at atmospheric pressure with a plane cooled nonablating anode perpendicular to the arc axis. Arc constriction in front of the anode gives rise to an entrainment of cold gas from the vicinity of the anode leading to a more or less pronounced anode jet. The conservation equations for the anode region represent a set of highly nonlinear integro-differential equations which describe the temperature and the flow field in the arc. Numerical solutions of these equations are obtained by using an iterative finite-difference method. Results for a nitrogen arc at 250 A indicate that heat transfer close to the anode is dominated by the electron enthalpy transport. The cold gas approaches the arc fringes with velocities in the order of 1 m/s, and reaches velocities of up to 250 m/s in the hot core of the arc, indicating the existence of an anode jet which has been confirmed by experimental investigations.

Numerical solution of static and dynamic equations of cables

Abstract The mechanical equations of an extensible, perfectly flexible curvilinear material (cable) are formulated. The static problem can be solved either by a minimization technique or by an iterative finite difference method which also permits dealing with forces that are not derived from a potential. In the dynamic alternative the solution of the equations is obtained from finite difference discretization explicit in time - this can apply to different kinds of constitutive laws and to different kinds of boundary conditions. Application of these numerical methods to electric transmission lines subject to electromagnetic forces shows a good agreement between computation and experimental data.

A Method for solving non-linear boundary value problems describing the minority carrier distribution in the base of a semiconductor structure

A COMPUTATIONAL algorithm is given for solving non-linear boundary value problems it uses the familiar methods of quasi-linearization, pivotal condensation, and continuation of the solution with respect to a parameter. The algorithm is applied in the study of a mathematical model of the carrier distribution in the base of a semiconductor structure, described by a non-linear boundary value problem for an ordinary second-order differential equation. Most numerical methods for solving non-linear boundary value problems, such as Newton's method [1–3], the finite-difference iterative method [4], and the method of quasi-linearization [5], are based on the construction of iterative processes. The continuous analogues of single-step iterative processes, described in [6], have come to be widely used in the investigation of various physical problems [7]. In particular, the study of certain models of the charge transport process in the quasi-neutral zone of a semiconductor structure reduces to the solution of a non-linear boundary value problem for an ordinary second-order differential equation. Attempts to solve this boundary value problem [8,9,10] have so far been limited to certain approximations, whereas modem requirements and technological possibilities have made it necessary to solve the initial (non-approximated) problem. It is in this connection that the present paper describes a computational algorithm for solving non-linear boundary value problems, based on the method of quasi-linearization, a discrete version of the method of continuing the solution with respect to a parameter [11,12], and the pivotal condensation method [13]. This algorithm makes it possible to increase substantially the domain of convergence of the relevant iterative procedures. The algorithm is described in general terms in Section 1, while in Section 2 it is applied to the solution of a concrete problem.

Acoustic surface-wave scattering on a homogeneous three-quarter space

The full-wave solution to the problem of acoustic surface-wave scattering on a homogeneous isotropic three-quarter space of Poisson's ratio σ=0.245 has been obtained by a finite-difference iterative method. The amplitude coefficients for Rayleigh wave transmission and reflection at the 270° corner were found to be 0.28±0.02 and 0.09±0.01, respectively, while the corresponding phase shifts were 140±10° and −125±10°. All of these results, which are independent of wavelength, agree well with previously published experimental and theoretical work. About 90% of the incident energy is lost to body waves and the scattered energy-density pattern indicates that over half of this energy sweeps past the corner and propagates into the body of the medium in the form of bulk modes. Some of the incident energy tends to follow the free surface by swinging around the corner while the reflected wave appears to be launched mainly by the action of a virtual source located at the corner which is excited by the incoming wave and radiates into the three-quarter space. The accuracy of the finite-difference technique has also been verified by comparing the iterated results for Rayleigh wave propagation on a homogeneous isotropic semi-infinite half-space with the analytically exact results. This basic method may be extended to solve problems involving layered geometries.

A note on the stability of an iterative finite-difference method for hyperbolic systems

In this note, we find analytically the linear stability criteria for two finite-difference methods for hyperbolic systems in conservation-law form, presented recently by S. Abarbanel and G. Zwas and by S. Abarbanel and M. Goldberg.

A Note on the Stability of an Iterative Finite-Difference Method for Hyperbolic Systems

In this note, we find analytically the linear stability criteria for two finite-difference methods for hyperbolic systems in conservation-law form, presented recently by S. Abarbanel and G. Zwas and by S. Abarbanel and M. Goldberg.

Invariant imbedding, difference equations, and elliptic boundary value problems

This paper presents an invariant imbedding approach to the solution of coupled matrix-vector difference equations under two-point boundary conditions. It is shown that the finite difference approach to the solution of elliptic equations results in a special case of the coupled difference equations considered. For linear self-adjoint equations, the invariant imbedding approach yields stable initial value problems. Finally, the standard direct and iterative finite difference methods are derived and analyzed from the invariant imbedding equations.

The interaction region in the boundary layer of a shock tube

Velocity and temperature boundary layers developed on a plane wall by ideal shock-tube flow are considered for weak shock and expansion waves. Analytically, the boundary layer consists of three regions, bounded by (1) expansion-wave head, (2) diaphragm location, (3) contact discontinuity, (4) shock. The flow fields (1, 2) and (3, 4) are, essentially, known. In the interaction region (2, 3), these flow fields merge, the governing equations are ‘singular parabolic’ and admit boundary conditions usually associated with elliptic equations. It is convenient to replace the weak expansion wave in the main flow by a line discontinuity. A consistent linearization scheme can now be devised to obtain the solution in the three regions. In (2, 3), the resulting linear singular parabolic equations for the first-order solutions are solved successfully by an iterative finite difference method, normally applied to elliptic equations.

An iterative finite-difference method for hyperbolic systems

An iterative finite-difference scheme for initial value problems is presented. It is applied to the quasi-linear hyperbolic system representing the one-dimensional time dependent flow of a compressible polytropic gas. The emphasis in this research was on the handling of discontinuities, such as shock waves, and overcoming the post-shock oscillations resulting from nonlinear instabilities. The linear stability is investigated as well. The success of the method is indicated by the monotonic profiles which were obtained for almost all the cases tested.

An iterative finite difference method for the frequency domain solution of some dynamic countercurrent laminar flow transport problems with split boundary conditions

An iterative finite difference method for the frequency domain solution of some dynamic countercurrent laminar flow transport problems with split boundary conditions