Gauss-Seidel Iterative Procedure Research Articles

Falling clouds of particles with finite inertia in viscous flows

When sedimenting in a viscous fluid under gravity, a cloud of particles undergoes a complex shape evolution due to the hydrodynamic interactions. In this work, Lagrange particle dynamic simulation, which combines the Oseen solution for flow around a particle and a Gauss–Seidel iterative procedure, is adopted to investigate the effects of the particle inertia and the hydrodynamic interactions on the cloud's sedimentation behavior. It is found that, with a small Stokes number (St), the cloud evolves into a torus and then breaks up into secondary clouds. In contrast, the cloud with a finite Stokes number becomes compact in the horizontal direction and is elongated along the vertical direction. The critical St value that separates the breakup mode and the vertical elongation mode is around 0.2. The cloud response time (t̂r) and the maximum settling velocity (V̂max) are measured at different Stokes numbers, particle Reynolds numbers, and particle volume fractions. A linear relationship, t̂r=aSt, is found between t̂r and the Stokes number and the correlation between V̂max and St can be well described by an exponential function V̂max=b1exp−b2St+b3. At last, the chaotic dynamics of the sedimentation system are discussed. A small difference between the initial configurations diverges exponentially. The sedimentation system containing particles with larger inertia has a lower divergence rate.

Transient numerical analysis of thermophoresis and particle dynamics in a nanofluid – pool boiling conditions

This paper attempts to apply two different equations of thermophoresis (applicable to micro-fluids) to nanofluids. Three significant parameters namely thermophoretic diffusion coefficient, thermophoretic velocity and thermophoretic mobility are used to analyse thermophoresis in a nanofluid. Both experiments and CFD simulations (ANSYS Fluent) are performed for water-alumina nanofluid under transient pool boiling conditions. A hollow stainless steel/glass chamber with a brass heater is used to boil water-alumina nanofluid under sub-cooled conditions. Three different heater surface temperatures (60, 70 and 80°C) are investigated to realize thermophoresis phenomena of the nanofluid. Random variation of temperature measurements in experiments is justified using an uncertainty analysis. Although, volume fraction of nanoparticles (diameter 46 nm) dispersed in the fluid is only 2% by volume, they are featured to exist in every frame of the numerical model for better understanding. 2-D transient implicit pressure based Navier-Stokes solver with SIMPLE (Semi-implicit Method for Pressure Linked Equation) pressure correction technique is used to discretize the numerical grid. Gauss-Seidel iterative procedure is used to solve the discretized equations. Simulation results are validated using a sequence of investigation (thermal property variation of water-alumina nanofluid with respect to pure water under steady-state/transient conditions, thumb rule from literature and experiments). A thin line of differentiation is found to exist between viscosity induced particle migration and thermophoresis induced particle migration for the nanofluid. Implications of equations and parameters used for estimating thermophoresis of nanofluid are discussed.

A Finite Difference Method and Effective Modification of Gradient Descent Optimization Algorithm for MHD Fluid Flow over a Linearly Stretching Surface

Present contribution is concerned with the construction and application of a numerical method for the fluid flow problem over a linearly stretching surface with the modification of standard Gradient descent Algorithm to solve the resulted difference equation. The flow problem is constructed using continuity, and Navier Stoke equations and these PDEs are further converted into boundary value problem by applying suitable similarity transformations. A central finite difference method is proposed that gives third-order accuracy using three grid points. The stability conditions of the present proposed method using a Gauss-Seidel iterative procedure is found using VonNeumann stability criteria and order of the finite difference method is proved by applying the Taylor series on the discretised equation. The comparison of the presently modified optimisation algorithm with the Gauss-Seidel iterative method and standard Newton’s method in optimisation is also made. It can be concluded that the presently modified optimisation Algorithm takes a few iterations to converge with a small value of the parameter contained in it compared with the standard descent algorithm that may take millions of iterations to converge. The present modification of the steepest descent method converges faster than Gauss-Seidel method and standard steepest descent method, and it may also overcome the deficiency of singular hessian arise in Newton’s method for some of the cases that may arise in optimisation problem(s).

An Efficient Laguerre-Based FDTD Iterative Algorithm in 3D Cylindrical Coordinate System

Here an efficient Laguerre-based finite-difference time-domain iterative algorithm is proposed. Different from the previously developed iterative procedure used in the efficient FDTD algorithm, a new perturbation term combined with the Gauss–Seidel iterative procedure is introduced to form the new Laguerre-based FDTD algorithm in the 3D cylindrical coordinate system. Such a treatment scheme can reduce the splitting error to a low level and obtain a good convergence; in other words, it can improve the efficiency and accuracy than other algorithms. To verify the performance of the proposed algorithm, two scattering numerical examples are given. The computation results show that the proposed algorithm can be better than the ADI-FDTD algorithm in terms of efficiency and accuracy. Meanwhile, the proposed algorithm is extremely useful for the problems with fine structures in the 3D cylindrical coordinate system.

Numerical modelling of thermophoresis in water-alumina nanofluid under pool boiling conditions

The Soret-effect or thermo-diffusion in liquids has been a subject of long-standing consideration for research. Numerical modelling of thermophoresis in water-alumina nanofluid subjected to sub-cooled pool boiling conditions is attempted in this paper. Pool boiling experiments are conducted using water-alumina nanofluid for different heater input conditions (50 °C–90 °C). Computational fluid dynamics (CFD) code ANSYS Fluent is used further to analyse thermophoresis in water-alumina nanofluid. 2-D steady pressure-based implicit solver with Gauss-Seidel iterative procedure is used to discretize and numerically solve the mathematical model. Pressure-correction in the numerical procedure is done using SIMPLE (Semi-implicit Method for Pressure Linked Equation) technique. Flow boiling models for micro-sized particles are revised to outfit pool boiling conditions of nanofluid. Results of temperature from the numerical model were found to have a worthy-match with measured temperature values of the experiments. Uncertainty analysis is carried out for the random variation of temperature measurements in experiments. Thermophoretic parameters, Peclet numbers (rotational/translational) and time-scale equations for diffusion are used to analyse minute particle motion due to advection/diffusion nano-scale heat transfer. Results of thermophoretic velocity reveal a diminishing effect of particle motion near heater surface with amplified heater inputs. Results of Peclet numbers (translational/rotational) reveal the significant role of very small nano particle motion on heat diffusion in nanofluids. Combined results of Peclet number and time-scale equations reveal the strengths of advection and diffusion mechanisms in the nano-scale heat transfer process.

A New Approach and Solution Technique to Solve Time Fractional Nonlinear Reaction-Diffusion Equations

A new application of the hybrid generalized differential transform and finite difference method is proposed by solving time fractional nonlinear reaction-diffusion equations. This method is a combination of the multi-time-stepping temporal generalized differential transform and the spatial finite difference methods. The procedure first converts the time-evolutionary equations into Poisson equations which are then solved using the central difference method. The temporal differential transform method as used in the paper takes care of stability and the finite difference method on the resulting equation results in a system of diagonally dominant linear algebraic equations. The Gauss-Seidel iterative procedure then used to solve the linear system thus has assured convergence. To have optimized convergence rate, numerical experiments were done by using a combination of factors involving multi-time-stepping, spatial step size, and degree of the polynomial fit in time. It is shown that the hybrid technique is reliable, accurate, and easy to apply.

Electromagnetic Scattering From Electrically Large Arbitrarily-Shaped Conductors Using the Method of Moments and a New Null-Field Generation Technique

In this work, a new numerical procedure is developed to apply the well-known method of moments (MoM) formulation to electrically large conducting bodies of arbitrary shape. The numerical procedure involves developing a combination of subdomain-entire domain basis functions which result in a sparse moment matrix as opposed to a full matrix in the traditional method. Moreover, the zeros in the MoM matrix are precisely at the same locations where one would have encountered the most significant values. The solution of the new matrix may be obtained using the simple Gauss-Seidel iterative procedure with only two or three iterations. All the traditional advantages of the MoM procedure are retained including the solution for multiple incident fields. Several numerical results are presented to illustrate the validity of the new approach.

Numerical modeling of field-activated combustion synthesis process of the B4C system

Abstract A field-activation combustion synthesis process of the 4B + C reactive system was numerically simulated to investigate the effect of external field and porosity on the combustion reaction by an implicit difference method and a Gauss–Seidel iteration procedure. The new features of the model include a consideration of the melting of each constituent of the reactants and product and the inclusion of considerations involving porosity and dilution. The results show that the self-sustaining reactions are not possible until field-activated temperature is more than 1100 K, which agree with the theoretic calculation. As the reactant porosity values are decreased from 60% to 20%, the combustion velocity first increases because of an increase in the thermal conductivity. The combustion velocity, after reaching a maximum, decreases with a further decrease in the porosity because of the high value of the thermal conductivity of the reactants.

Fixed-point convergence of modular, steady-state heat transfer models coupling multiple scales and phenomena for melt–crystal growth

A block Gauss–Seidel iteration procedure is employed to couple two independent heat transfer codes representing different scales and phenomena in a melt–crystal growth system via shared domain boundaries. In general, such strategies are attractive because of their simplicity for solving problems that may be represented by linking, via shared boundary conditions, existing codes using a modular design. However, these approaches often meet with limited success due to convergence difficulties associated with the overall model. The mathematical framework of fixed-point iterations is employed to assess convergence behaviour for steady-state, non-linear heat transfer models. Analytical forms describe the convergence of one-dimensional problems and provide insight to the behaviour of more complicated, two-dimensional models. Certain implementations that are physically reasonable result in algorithms that will never converge, demonstrating that notions based on physical intuition may not be useful for predicting algorithm performance. A mixing of flux and temperature information along the shared model boundary leads to successful iterative strategies under many conditions. Copyright © 2006 John Wiley & Sons, Ltd.

Simulation Studies on Multimode Heat Transfer from a Square-Shaped Electronic Device with Multiple Discrete Heat Sources

ABSTRACT The results of a numerical study of the problem of multimode heat transfer from a square-shaped electronic device provided with three identical flush-mounted discrete heat sources are presented here. Air, a radiatively nonparticipating fluid, is taken to be the cooling medium. The heat generated in the discrete heat sources is first conducted through the device, before ultimately being dissipated by convection and surface radiation. The governing partial differential equations for temperature distribution are converted into algebraic form using a finite-volume based finite difference method, and the resulting algebraic equations are subsequently solved using Gauss-Seidel iterative procedure. A grid size of 151 × 91 is used for discretizing the computational domain. The effects of all relevant parameters, including volumetric heat generation, thermal conductivity, convection heat transfer coefficient, and surface emissivity, on various important results, such as the local temperature distribution, the peak temperature of the device, and the relative contributions of convection and surface radiation to heat dissipation from the device, are studied in sufficient detail. The exclusive effect of surface radiation on pertinent results of the present problem is also brought out.

DEVELOPMENT OF A DYNAMIC MODEL FOR INVESTIGATING THE INTERACTION BETWEEN RURAL TRANSPORT AND DEVELOPMENT: A CASE OF SOUTHEAST SULAWESI, INDONESIA

A lack of tool in understanding the dynamic relation between rural transport intervention and development level has contributed to the ignorance of policy makers to develop a pro-poor investment. A dynamic model for an interaction between rural transportation, i.e. accessibility and mobility, and development was proposed with a simultaneous equation block using regression analysis and a Gauss-Seidel iteration procedure. Data were obtained from three districts in Southeast Sulawesi (Tinanggea, Poleang Timur, and Watubangga Districts), using 360 HH and 984 respondents. The equation sets were tested against 5 provinces in Indonesia to acquire nation-wide data on the link between rural transport and development. The result shows that rural transport has a significant impact on rural development indicators with a diminishing return principle. The role of rural transport intervention disappears as the mobility increase. This conclusion of the research can be use as a base for future investment prioritization and budget allocation.

A solution to mean delay in the ∑ Mc/ Gck/1 cyclic priority queue with cycle ( k) and class ( c) dependent feedback and service times

In this paper we develop an original solution to mean delay in a general ∑ M c / G ck /1 cyclic priority queue with class ( c) and cycle ( k)-dependent service time and feedback. Each class has its priority assigned. There may be one or more classes with the same priority. This model is suitable for performance analysis of round robin processor sharing policies. The analysis may be used to examine the effects of quantum sizes in round robin scheduling used in operating systems. Each customer, upon entering the system, requires a number of service cycles before it leaves the system. Each service cycle is characterized by its service-time distribution, and the probability of having at least one more cycle before the customer leaves the system. These characteristics of cycles may be different for different cycles of a service process. The solution is represented as a system of linear equations. It may be efficiently solved using the Gauss–Seidel iterative procedure. A general solution is developed of which, the two special cases are a non-priority (single-priority) and a one-class-per-priority non-preemptive queues. Computational complexity of the numerical procedure is between computational complexity of the two special cases, O( C 3 K 3) and O( CK 3), respectively. C stands for the number of classes, and K stands for the maximum number of cycles being numerically considered.

An implicit numerical scheme for the simulation of internal viscous flow on unstructured grids

The development of a cell-centered unstructured grid solution for the two-dimensional Navier-Stokes equations is described. A finite-volume approach is used to discretize the conservation law form of the compressible flow equations written in terms of primitive variables. Temporal preconditioning is employed so that low Mach number flows can be solved economically. The equations are marched in time using either an implicit Gauss-Seidel iterative procedure or a solver based on a conjugate gradient-like method. A four color scheme is employed to vectorize the block Gauss-Seidel relaxation procedure. This increases the memory requirements minimally and decreases the computer time spent solving the resulting system of equations substantially. A factor of 7.6 speedup in the matrix solver is typical for the viscous equations. The grids are generated based on the method of Delaunay triangulation. Numerical results are obtained for inviscid flow over a bump in a channel at subsonic and transonic conditions for validation with structured solvers. Viscous results are computed for developing flow in a channel, a symmetric sudden expansion, periodic tandem cylinders in a cross-flow, and a fourport valve. Comparisons are made with available results obtained by other investigators.

On general zero-skew clock net construction

We propose a simulated annealing based zero-skew clock net construction algorithm that works in any routing spaces, from Manhattan to Euclidean, with the added flexibility of optimizing either the wire length or the propagation delay. We first devise an O(log n) tree grafting perturbation function to construct a zero-skew clock tree under the Elmore delay model. This tree grafting scheme is able to explore the entire solution space asymptotically. A Gauss-Seidel iteration procedure is then applied to optimize the Steiner point positions. Experimental results have shown that our algorithm can achieve substantial delay reduction and encouraging wire length minimization compared to previous works.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Cone- and wedge-gated field emitter diode and microtriode modeling

Two kinds of vertical-type gated field emission diodes (GFED) and field emission microtriodes (FEMT) are modeled and their performances compared, one based on cone emitters (CE) and the other based on wedge emitters (WE). The emitters have parabolic shape. The CE and WE structure basal areas are chosen to be the same, allowing extension of the results to GFED and FEMT arrays. The planar Fowler-Nordheim (FN) current density-electric field J( E) relationship is assumed to be valid. The current I is obtained by integration of J over the emitter's surface. No “field enhancement” and “area” factors are used. The two-dimensional Laplace equation for the electric potential is solved numerically using a Gauss-Seidel iterative procedure, having as special features: appropriate coordinate systems; a unique lattice for both CE and WE modeling; lattice steps in both directions in geometrical progression; emitters described by lattice points lying on it. The GFED model parameters are: emitter curvature radius R, height h and work function φ, together with the gated anode circular aperture radius r, height H and gate voltage V g. Besides these parameters, the FEMT model includes also as parameters the anode height D and the anode voltage V a. GFED- and FEMT-obtained simulation results refer to the effect of model parameters on the emission current. FEMT modeling results also include transconductance, base resistance, gain, capacitance and cut-off frequency. Several device design suggestions are drawn.

VIPRE-02—A Two-Fluid Thermal-Hydraulics Code for Reactor Core and Vessel Analysis: Mathematical Modeling and Solution Methods

The VIPRE-02 code is a thermal-hydraulic analysis code designed to model steady-state conditions and operational transients in light water reactor cores and vessels. It uses a two-fluid representation of two-phase flow that solves conservation equations for mass, momentum, and energy for each phase. The code uses a subchannel formulation of the conservation equations but also contains an optional three-dimensional (r-θ coordinates) representation of the lower plenum for vessel modeling. The six-equation formulation is solved implicitly, by a modified Gauss-Seidel iteration procedure, and has no time step size limitation for stability. Models for phase interaction based on flow regime mapping are provided that use empirical models and correlations for heat and mass transfer at the interface and vapor generation. In addition, the code contains as an option a dynamic flow regime model, which uses an interfacial area transport equation to determine the phase interaction terms.

Flow and heat transfer in a micropolar fluid past a flat plate with suction and heat sources

The laminar boundary layer flow of a micropolar fluid past a flat plate subject to uniform suction has been examined. The heat transfer study has been made in the presence of both constant as well as temperature dependent heat sources. The governing equations of momentum, first stress momentum and energy have been solved using numerical integration and Gauss-Seidel iterative procedure. For a particular value of suction parameter λ, as compared to the Newtonian fluid, the velocity decreases on increasing the value of micropolar parameter R. The skin friction and the Nusselt number have been calculated and presented in tables.

Surface impedance-BEM technique for nonlinear TM-eddy current problems

The impedance boundary condition (IBC) approximation is used to formulate the problem as an exterior one, in other words, in terms of only the exterior fundamental solution. The first order boundary element technique has been used to represent the boundary integral equation numerically. The Gauss–Seidel iterative procedure was applied. The results obtained with the present approach are in good agreement with those obtained experimentally and theoretically by other authors.

Steady forced flow of a micropolar fluid against a rotating disc

The steady forced flow of an incompressible micropolar fluid against a rotating disc is considered. The flow due to a rotating disc in an infinite fluid which is at rest and the axisymmetric stagnation flow on a flat plate are particular cases of this present problem. The equations of motion are solved numerically using the Gauss-Seidel iterative procedure and the Simpson's rule. The results are given in tabular form and compared with the known results for a Newtonian fluid.

Application of integral equations to the magnetic field analysis of rotating electrical machines

An integral equation (IE) method is applied to the magnetostatic field analysis of salient-pole synchronous machines. A brief description of the method and verification with a known analytical solution are given. The method requires discretization only in the steel regions and does not necessitate any special treatment of the boundary conditions. For a two dimensional solution, the machine cross section is discretized into small triangular elements over which the magnetization can effectively be assumed constant. The resulting nonlinear algebraic equations are solved using the Gauss-Seidel iterative procedure. Application of Ampere's circuital law around the current carrying regions, combined with overrelaxation, is used to speed convergence. An open circuit analysis is performed on a synchronous motor, where, because of the field symmetry, only one half pole pitch need be discretized. The resulting open-circuit voltage characteristic shows very good agreement with the test curve.