Gauss-Seidel Iterative Method Research Articles

Elastohydrodynamic lubrication of circular contacts at pure squeeze motion with micropolar lubricants

Purpose This paper aims to explore pure squeeze elastohydrodynamic lubrication (EHL) motion of circular contacts with micropolar lubricants under constant load. The proposed model can reasonably calculate the pressure distributions, film thicknesses and normal squeeze velocities during the pure squeeze process. Design/methodology/approach The transient modified Reynolds equation is derived in polar coordinates using micropolar fluids theory. The finite difference method and the Gauss–Seidel iteration method are used to solve the transient modified Reynolds equation, the elasticity deformation equation, load balance equation and lubricant rheology equations simultaneously. Findings The simulation results reveal that the effect of the micropolar lubricant is equivalent to enhancing the lubricant viscosity. As the film thickness is enlarged, the central pressure and film thickness for micropolar lubricants are larger than those of Newtonian fluids under the same load in the elastic deformation stage. The greater the coupling parameter (N), the greater the maximum central pressure. However, the smaller the characteristic length (L), the greater the maximum central pressure. The time needed to achieve maximum central pressure increases with increasing N and L. Originality/value A numerical method for general applications was developed to investigate the effects of the micropolar lubricants at pure squeeze EHL motion of circular contacts under constant load.

Numerical investigation of local defectiveness control of diblock copolymer patterns

We numerically investigate local defectiveness control of self-assembled diblock copolymer patterns through appropriate substrate design. We use a nonlocal Cahn-Hilliard (CH) equation for the phase separation dynamics of diblock copolymers. We discretize the nonlocal CH equation by an unconditionally stable finite difference scheme on a tapered trench design and, in particular, we use Dirichlet, Neumann, and periodic boundary conditions. The value at the Dirichlet boundary comes from an energy-minimizing equilibrium lamellar profile. We solve the resulting discrete equations using a Gauss-Seidel iterative method. We perform various numerical experiments such as effects of channel width, channel length, and angle on the phase separation dynamics. The simulation results are consistent with the previous experimental observations.

Periodically unsteady thermal response of metal-foam solid composite

With the development of materials and manufacturing techniques, using highly conductive metallic porous materials, such as metal foam, metal fiber felt, lattice-frame material, and so on, to enhance heat transfer becomes possible. As a highly-conductive multi-functional material, metal foam owns many good features including high porosity, light weight, large surface area, high strength, etc. To date, the thermal response characteristics of porous media with transient heat transfer have been investigated to some extent. For metal foam, the local thermal non-equilibrium (LTNE) effect can be found during the unsteady heat conduction process. However, for the unsteady heat conduction of the highly-conductive metal foam saturated with solid with a low thermal conductivity, the LTNE effect exists, but the thermal response characteristics of the metal foam and the infill solid with the periodically changing thermal boundary condition are rarely investigated. In this paper, based on the two equations model, the thermal response of the composite material which is made up of the metal foam and the saturated paraffin wax was numerically investigated with the periodic fluctuating temperature boundary condition, in which the temperature difference between metal foam and the infill paraffin wax was considered. The finite volume method (FVM) was used to discrete the coupled governing equations including the porous solid equation for metal foam and the infill solid equation for paraffin wax. The governing equations were solved by Gauss-Seidel iteration method. The present simulation method was validate by comparing with the analytical results under a constant temperature boundary. Then, the effects of various parameters on the thermal response characteristics were discussed. It was found that the LTNE effect exists in the metal foam and paraffin wax and thus the two equations model must be employed. When the ambient temperature periodically fluctuates with time, the temperature field inside the metal foam also reflects the cyclical fluctuations, and with the increasing of temperature fluctuation cycles, the LTNE effect increases firstly and then decreases. This means that there is a resonant time cycle corresponding to the most significant LTNE effect. When the fluctuant amplitude of ambient temperature is fixed, the temperature amplitudes of the metal foams increase with the increasing of temperature fluctuation cycles. The decay degrees of temperature amplitudes are different at different positions. The farther the position away from the heating surface, the more lagging the temperature amplitudes. The thermal diffusivity of the metal foam has a significant impact on the temperature amplitude and decay degree of temperature, while the porosity, pore density and nano-particles additive have less notable effect on them. For copper foam, pore density has a larger impact on LTNE phenomena, compared with porosity. When porosity and pore density are constant, the thermal diffusivity of metal foam has main impact on LTNE phenomena. Therefore, these factors should be considered synthetically so that the heat transfer rate of the composite is optimal. This paper reveals the temperature differences between porous solid phase and the infill solid phase during the metal foam heat conduction process, which shows direct scientific significance for the LTNE effect of an unsteady heat conduction in porous media.

A generalization of the Gauss–Seidel iteration method for solving absolute value equations

Based on the Gauss–Seidel splitting, we present a new matrix splitting iteration method, called generalized Gauss–Seidel (GGS) iteration method, for solving the large sparse absolute value equation (AVE) Ax−|x|=b where A∈Rn×n and b∈Rn and investigate its convergence properties. Moreover, by preconditioning AVE, a preconditioned variant of the GGS (PGGS) method is presented. Numerical experiments illustrate the efficiency of both GGS and PGGS iterations.

The solution of the linear complementarity problem by the matrix analogue of the accelerated overrelaxation iterative method

The Linear Complementarity Problem (LCP), with an H+źmatrix coefficient, is solved by using the new "(Projected) Matrix Analogue of the AOR (MAAOR)" iterative method; this new method constitutes an extension of the "Generalized AOR (GAOR)" iterative method. In this work two sets of convergence intervals of the parameters involved are determined by the theories of "Perron-Frobenius" and of "Regular Splittings". It is shown that the intervals in question are better than any similar convergence intervals found so far by similar iterative methods. A deeper analysis reveals that the "best" values of the parameters involved are those of the (projected) scalar Gauss-Seidel iterative method. A theoretical comparison of the "best" (projected) Gauss-Seidel and the "best" modulus-based splitting Gauss-Seidel method is in favor of the former method. A number of numerical examples support most of our theoretical findings.

A New Iterative Algorithm for Efficient 3-D Laguerre-Based FDTD Method

This letter presents a new iterative algorithm for efficient 3-D Laguerre-based finite-difference time-domain (FDTD) method. A new perturbation term and the Gauss–Seidel method are introduced in the algorithm. The theoretical analysis in the frequency domain shows that the splitting error introduced by the new perturbation term grows slower than that of the original one at the high frequency range. The Gauss–Seidel iterative method is used in the iterative procedure to speed up the convergence further. Numerical results show that the new iterative algorithm achieves a comparable accuracy with three iterations comparing to the original one with eight iterations, and the whole CPU time is reduced to about 34.5% of the original one.

ISOGEN: Interactive isotope generation and depletion code

ISOGEN is an interactive code for solving first order coupled linear differential equations with constant coefficients for a large number of isotopes, which are produced or depleted by the processes of radioactive decay or through neutron transmutation or fission. These coupled equations can be written in a matrix notation involving radioactive decay constants and transmutation coefficients, and the eigenvalues of thus formed matrix vary widely (several tens of orders), and hence no single method of solution is suitable for obtaining precise estimate of concentrations of isotopes. Therefore, different methods of solutions are followed, namely, matrix exponential method, Bateman series method, and Gauss–Seidel iteration method, as was followed in the ORIGEN-2 code. ISOGEN code is written in a modern computer language, VB.NET version 2013 for Windows operating system version 7, which enables one to provide many interactive features between the user and the program. The output results depend on the input neutron database employed and the time step involved in the calculations. The present program can display the information about the database files, and the user has to select one which suits the current need. The program prints the “WARNING” information if the time step is too large, which is decided based on the built-in convergence criterion. Other salient interactive features provided are (i) inspection of input data that goes into calculation, (ii) viewing of radioactive decay sequence of isotopes (daughters, precursors, photons emitted) in a graphical format, (iii) solution of parent and daughter products by direct Bateman series solution method, (iv) quick input method and context sensitive prompts for guiding the novice user, (v) view of output tables for any parameter of interest, and (vi) output file can be read to generate new information and can be viewed or printed since the program stores basic nuclide concentration unlike other batch jobs. The sample problems are chosen to serve two purposes, namely to validate the results of the code against problems where the analytical solution is possible, and the other is to demonstrate the use of particular solution method adopted for solving the problem. Besides spent fuels, results are validated for many of the useful deduced parameters of practical interest such as radioactivity, thermal power, alpha activity, neutron emission rate, and photon emission spectrum. These parameters are of utmost important in handling spent fuels, in waste disposal, in fuel management, in radiation shielding and many other areas of nuclear fuel cycle facilities.

JOR Iterative Method for the Modeling of MTLs Excited by EMP

In recent years, the distributed analytical representation and iterative technique (DARIT) based on the Jacobi iteration method (Jacobi-DARIT-field) and the Gauss-Seidel iteration method (Seidel-DARIT-field) were proposed to handle the transient electromagnetic field or the electromagnetic pulse (EMP) radiated field coupling to multiconductor transmission lines (MTLs). Compared to the Seidel-DARIT-field method, the Jacobi-DARIT-field method has a higher computational efficiency but a lower convergence rate. In order to improve its convergence rate, this letter extended the Jacobi-DARIT-field method with the Jacobi over relaxation (JOR) iteration algorithm and proposed a JOR-DARIT-field method. The performances of the JOR-DARIT-field method and the Jacobi-DARIT-field method were compared through the MTL modeling examples. It could be concluded that both methods have a high computational efficiency when computing the EMP coupling to MTLs. However, the JOR-DARIT-field method has a higher convergence rate than the Jacobi-DARIT-field method with a proper relaxation factor (RF).

Об одной вычислительной реализации блочного метода Гаусса-Зейделя для нормальных систем уравнений

Статья посвящена модификации блочного варианта метода Гаусса-Зейделя для нормальных систем уравнений, который является одним из достаточно эффективных методов решения, в общем случае переопределенных, систем линейных алгебраических уравнений большой размерности. Основным недостатком методов, основанных на нормальных системах уравнений, является тот факт, что число обусловленности нормальной системы равно квадрату числа обусловленности исходной задачи. Этот факт отрицательно влияет на скорость сходимости итерационных методов, основанных на нормальных системах уравнений. Для повышения скорости сходимости итерационных методов, основанных на нормальных системах уравнений, при решении плохо обусловленных задач в настоящее время используются различные варианты предобуславливателей, позволяющие снизить число обусловленности исходной системы уравнений. Однако универсального предобуславливателя для всех задач не существует. Одним из эффективных подходов, позволяющих повысить скорость сходимости итерационного метода Гаусса-Зейделя для нормальных систем уравнений, является использование его блочного варианта. Недостатком блочного метода Гаусса-Зейделя для нормальных систем является тот факт, что на каждой итерации необходимо вычислять псевдообратную матрицу. Известно, что нахождение псевдообратной матрицы является достаточно сложной вычислительной процедурой. В настоящей работе предлагается процедуру псевдообращения матрицы заменить на задачу решения нормальных систем уравнений методом Холецкого. Нормальные уравнения, возникающие на каждой итерации метода Гаусса-Зейделя, имеют сравнительно невысокую размерность по сравнению с исходной системой. Приводятся результаты вычислительных экспериментов, демонстрирующие эффективность предлагаемого подхода.

Second degree generalized gauss-Seidel iteration method for solving linear system of equations

The main objective of Automatic Generation Control (AGC) is to balance the total system generation against system load and losses so that the desired frequency and power interchange with neighbouring systems are maintained. Any mismatch between generation and demand causes the system frequency to deviate from its nominal value. Thus the high frequency deviation may lead to system collapse. Therefore, it is necessary to include the active controller as Superconducting Magnetic Energy Storage Units (SMES) to maintain the nominal system frequency. This paper presents the application of energy storage units in an interconnected two area hydropower system. The proposed work consist of two area interconnected hydro power system with Superconducting Magnetic Energy Storage units has been designed to improve the dynamic performance of the system and also Integral Square Error (ISE) technique is used to obtain the optimal integral gain settings. The simulation result shows that the Load Frequency Control in an interconnected hydro power system with SMES units is considerably improved the system dynamics such as peak overshoot, settling time and frequency oscillation as compared to that of the system without SMES units. The system simulation realized by using MATLAB software.Key words: Automatic Generation Control, Load Frequency Control, Energy Storage Units, SMES unit, System dynamics, Hydro Power System

A numerical model for simulating single-phase gas flow in anisotropic coal

Numerical simulation has become an important approach for investigating coalbed methane (CBM) recovery. This demand has enabled the development of numerical simulators and each simulator has its own features. This paper first modifies the mathematical model of single-phase gas flow in coal by incorporating the anisotropic permeability model presented in our previous work. A numerical model is then developed based on this modified mathematical model by using the finite difference method. The numerical model is solved by using the Gauss-Seidel iteration method. A simulator is developed with the C++ programing language to solve the iteration equation. The numerical model is validated with a set of published experimental data and a set of published production data modeled by commercial CBM simulators. The results show that the numerical model developed in this paper agrees well with both data sets. A series of numerical simulation is conducted to evaluated the effects of permeability anisotropy on dry CBM production by using the numerical model developed in this paper. The results show that if the equivalent permeability is equal to the isotropic permeability, the permeability anisotropy has few effects on gas rate but impacts the distributions of pore pressure and gas content significantly. These results may be meaningful for dry CBM production with multiple vertical wells.

On the Use of a Direct Radiative Transfer Equation Solver for Path Loss Calculation in Underwater Optical Wireless Channels

In this letter, we propose a fast numerical solution for the steady state radiative transfer equation in order to calculate the optical path loss of light propagation suffering from attenuation due to the absorption and scattering in various water types. We apply an optimal non-uniform method to discretize the angular space and an upwind type finite-difference method to discretize the spatial space. A Gauss-Seidel iterative method is then applied to solve the fully discretized system of linear equations. Finally, we extend the resulting radiance in two-dimensional to three-dimensional by the azimuthal symmetric assumption to compute the received optical power under the given receiver aperture and field of view. The accuracy and efficiency of the proposed scheme are validated by uniform RTE solver and Monte Carlo simulations.

Rigorous convergence analysis of alternating variable minimization with multiplier methods for quadratic programming problems with equality constraints

We discuss unique solvability of the equality-constraint quadratic programming problem, establish a class of preconditioned alternating variable minimization with multiplier (PAVMM) methods for iteratively computing its solution, and demonstrate asymptotic convergence property of these PAVMM methods. We also discuss an algebraic derivation of the PAVMM method by making use of matrix splitting, which reveals that the PAVMM method is actually a modified block Gauss–Seidel iteration method for solving the augmented Lagrangian linear system resulting from the weighted Lagrangian function with respect to the equality-constraint quadratic programming problem.

Second degree generalized gauss-Seidel iteration method for solving linear system of equations

In this paper, a second degree generalized Gauss –Seidel iteration (SDGGS) method for solving linear system of equations whose iterative matrix has real and complex eigenvalues are less than unity in magnitude is presented. Few numerical examples are considered to show the efficiency of the new method compared to first degree Gauss-Seidel (GS), first degree Generalized Gauss-Seidel (GGS) and Second degree Gauss-Seidel (SDGS) methods. It is observed that the spectral radius of the new Second degree Generalized Gauss-Seidel (SDGGS) method is less than the spectral radius of the methods GS, GGS and SDGS. By use of second degree iteration (SD) method, it is possible to accelerate the convergence of any iterative method.Key words: Gauss-Seidel method (GS); Generalized Gauss-Seidel method (GGS); Strictly Diagonally Dominant Matrix.

Modulus-based GSTS Iteration Method for Linear Complementarity Problems

Inthis paper, amodulus-basedgeneralizedskew-Hermitiantriangularsplit- ting (MGSTS) iteration method is present for solving a class of linear complemen- tarity problems with the system matrix either being an H+-matrix with non-positive off-diagonal entries or a symmetric positive definite matrix. The convergence of the MGSTS iteration method is studied in detail. By choosing differentparameters, a series of existing and new iterative methods arederived, including the modulus-based Jacobi (MJ) and the modulus-based Gauss-Seidel (MGS) iteration methods and so on. Exper- imental results are given to show the effectiveness and feasibility of the new method when it is employed for solving this class of linear complementarity problems.

Numerical Solution for Solving Space-Fractional Diffusion Equations using Half-sweep Gauss-seidel Iterative Method

The main purpose of this paper is to examine theeffectiveness of Half-Sweep Gauss-Seidel (HSGS) method forSpace-Fractional diffusion equations. The Caputo’s derivativeand implicit finite difference scheme will be used to discretizelinear space-fractional equation of the first order to constructsystem linear equation. The basic formulation and application ofthe HSGS iterative method are also presented. Two numericalexamples and comparison with other iterative methods showsthat the present method is effective. Based on computationalnumerical result, the solution obtained by proposed iterativemethod is in excellent agreement, it can be concluded that theproposed iterative method is superior to the Full-Sweep Gauss-Seidel (FSGS) iterative method

Self-adaptive Extrapolated Gauss-Seidel Iterative Methods

In this paper, we consider a self-adaptive extrapolated Gauss-Seidel method for solving the Hermitian positive definite linear systems. Based on optimization mod- els, self-adaptive optimal factor is given. Moreover, we prove the convergence of the self-adaptive extrapolated Gauss-Seidel method without any constraints on optimal factor. Finally, the numerical examples show that the self-adaptiveextrapolatedGauss- Seidel method is effective and practical in iteration number.

Effects of Surface Forces on Pure Squeeze Elastohydrodynamic Lubrication Motion of Circular Contacts with Coated Layer

The effects of surface forces (SF) on pure squeeze elastohydrodynamic lubrication (EHL) motion of circular contacts with coated layer are explored under constant load condition by using the finite element method (FEM) and the Gauss-Seidel iteration method. The difference between SFEHL model and EHL model is apparent as the film thickness is thinner than 5 nm. The oscillation phenomena in pressure and film thickness come mainly from the action of solvation forces. The effects of surface forces become significant as the film thickness becomes thinner.

Convergence on Gauss-Seidel iterative methods for linear systems with general H-matrices

It is well known that as a famous type of iterative methods in numerical linear algebra, Gauss-Seidel iterative methods are convergent for linear systems with strictly or irreducibly diagonally dominant matrices, invertible H−matrices (generalized strictly diagonally dominant matrices) and Hermitian positive definite matrices. But, the same is not necessarily true for linear systems with non-strictly diagonally dominant matrices and general H−matrices. This paper firstly proposes some necessary and sufficient conditions for convergence on Gauss-Seidel iterative methods to establish several new theoretical results on linear systems with nonstrictly diagonally dominant matrices and general H−matrices. Then, the convergence results on preconditioned Gauss-Seidel (PGS) iterative methods for general H−matrices are presented. Finally, some numerical examples are given to demonstrate the results obtained in this paper.

Virtual flight simulation of a dual rotor micro air vehicle

In this paper, a new computational method is developed based on computational fluid dynamics (CFD) coupled with rigid body dynamics (RBD) and flight control law in an in-house programmed source code. The CFD solver is established based on momentum source method, preconditioning method, lower–upper symmetric Gauss–Seidel iteration method, and moving overset grid method. Two-equation shear–stress transport k − ω turbulence model is employed to close the governing equations. Third-order Adams prediction-correction method is used to couple CFD and RBD in the inner iteration. The wing-rock motion of the delta wing is simulated to validate the capability of the computational method for virtual flight simulation. Finally, the developed computational method is employed to simulate the longitudinal virtual flight of a dual rotor micro air vehicle (MAV). Results show that the computational method can simulate the virtual flight of the dual rotor MAV.