Gauss-Seidel Relaxation Research Articles

Double-Diffusive MHD Viscous Fluid Flow in a Porous Medium in the Presence of Cattaneo-Christov Theories

The Cattaneo-Christov model will be used to examine the significance of heat generation, viscous dissipation, and thermal radiation on a double-diffusive MHD flow in this study. In this study, it was discovered that heat and mass transfer can be affected by nonlinear buoyancy significance. The flow direction was subjected to a uniform magnetic field. A set of partial differential equations governs the current design (PDEs). In order to simplify these equations, they are converted into ordinary differential equations (ODEs). In order to numerically solve the nonlinear ODEs, the spectral relaxation method (SRM) is utilized. In order to decouple and linearize the equation sets, the SRM employs the Gauss-Seidel relaxation method. Geothermal power generation and underground storage systems are just a few examples where this research could be put to use. When compared to previous findings, the current outcomes were discovered to be closely related. Owing to an increase in Lorentz force, the imposed magnetic field slows down fluid motion. Viscosity dissipation and heat generation all contribute to the formation of an ever-thicker thermal boundary layer. When the Cattaneo-Christov models are used, the thermal and concentration boundary layers get a lot thicker.

Predetermination of Performance Parameters of 3-phase Induction Motor using Numerical Technique Tools

The Circle Diagram was used in past for the performance evaluation of 3-phase induction motor under different load conditions. But, now it is not preferred due to availability of programs for detailed analysis. The per-phase equivalent circuit is also not that suitable to estimate the dynamic performance. In this paper, a method is suggested to determine the performance using dq-model. The objective of this paper is to describe modelling in terms of dq variables. The analysis under steady state condition is done using numerical technique tools. The numerical tools such as Gauss-Seidel and Successive Relaxation methods are used and the results are compared with that obtained from the experiment conducted in the laboratory. The comparison of results confirms the validity as well as the accuracy of the proposed methods, as yet another approach to predetermine the performance.

Spectral relaxation approach and velocity slip stagnation point flow of inclined magnetized cross-nanofluid with a quadratic multiple regression model

In this paper, a comprehensive analysis of the second-order velocity slip phenomenon and stagnation point flow of cross nanofluid with a spectral relaxation approach over the geometry of porous medium is presented. Several other features are considered: mixed convection, passive control of nanoparticles, joules and viscous heating effect, thermophoresis, and Brownian motion. The velocity of the cross-nanofluid is judged by placing an inclined magnetic field. A set of partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) using transform variables, and further ODEs are solved with the spectral relaxation approach, which is based on Gauss–Seidel relaxation method for computing numerical results. The numerical scheme using a physical quantity, the local Nusselt number, is made by quadratic multiple regression model and for the numerical solution of ODEs through Bvp4c. Comparison of the numerical values with quadratic multiple regression model and the Bvp4c technique is tabulated. This numerical outcome suggests that a large perturbation in the Nusselt Number is found when an increment in the thermophoretic parameter is given. Eckert number strengthens the thermophoretic and Brownian motion.

Effects of surface topography on cavitation evolution of liquid film lubricated mechanical seals

PurposeThe purpose of this paper is to investigate the effects of surface topography, including surface roughness, waviness and taper, on the cavitation of liquid film lubricated mechanical seals (LFL-MS).Design/methodology/approachA universal governing equation considering cavitation is established, and an equivalent relative density is defined to characterize the cavitation degree. The equation is discretized by the finite volume method and solved by the Gauss–Seidel relaxation scheme.FindingsResults indicate that both radial length and a circumferential width of the cavitation zone and cavitation degree are affected significantly by the waviness amplitude and taper, but the effect of surface roughness is limited.Originality/valueEffect mechanism of surface topography on the cavitation of LFL-MS is investigated and cavitation degree is reflected by an equivalent relative density. The results further help to comprehensively explore the cavitation mechanism.

Convergence of the Resistive Coupling-Based Waveform Relaxation Method for Chains of Identical and Symmetric Circuits

The convergence of the waveform relaxation (WR) method is demonstrated for a class of circuits: Chains of identical and symmetrical passive subcircuits. The WR algorithm uses resistive coupling to implement the iteration. Every part is modeled as a symmetric and reciprocal linear two-port network. The iteration matrices of the WR operator are constructed for the Gauss-Jacobi and Gauss-Seidel relaxations in the Fourier domain. An upperbound estimate of the spectral radius of the WR operator is presented. It demonstrates the convergence of the method independently of the number of cascaded parts in the chain and the coupling resistance.

Effects of operating conditions on cavitation induction of spiral groove liquid-film seal (SG-LFS)

PurposeThe purpose of this paper is to study the effects of operating conditions including process coefficient, lubricant viscosity and cavitation pressure on the cavitation of spiral groove liquid-film seal (SG-LFS).Design/methodology/approachA mathematical model of SG-LFS is established based on the JFO boundary and a relative density is introduced. The universal governing equation after a coordinate transformation is discretized by the FVM method and solved by the Gauss-Seidel relaxation scheme.FindingsThe results indicate that the two-dimensional size of cavitation and cavitation degree are affected significantly by the process coefficient and lubricant viscosity but the effect of cavitation pressure can be ignored.Originality/valueThe effect mechanisms of operating conditions on the cavitation of SG-LFS are studied by the JFO boundary and cavitation degree characterized by a relative density. The results presented are helpful to perfect and deeply understand the cavitation mechanism of liquid-film seal.Peer reviewThe peer review history for this article is available at:https://publons.com/publon/10.1108/ILT-03-2020-0083/

Coupling multi-body dynamics and fluid dynamics to model lubricated spherical joints

A new approach of coupling multibody dynamics and fluid dynamics is developed to model hydrodynamic lubrication of spherical clearance joints with thin fluid film and relative multidirectional motion. The model accounts for dynamics motion of articulating components as well as both squeeze- and wedge-film actions of the synovial fluid. Multibody dynamics methodology is employed to derive the motion equations and Reynolds equation governs the fluid dynamics. The finite difference method is utilized to discretize the governing equation of lubricant and the multi-grid method augments computational efficiency to acquire outcomes employing a Gauss–Seidel relaxation scheme. Fluid–structure interaction is incorporated into the methodology using a partitioned formulation embedded in a high-order Runge–Kutta time integrators for integrating the nonlinear equations of the coupled system over time of interest. A demonstrative example of total hip arthroplasty is considered and the developed model is assessed against outcomes available in the literature. The effect of initial conditions on the pressure, film thickness and dynamics of the lubricated spherical joint is analyzed and discussed. It is illustrated that maximum fluid pressure is undergone by the hip implant at the first walking cycle of movement due to an unstable state, which is strongly dependent upon the initial condition. Finally, the approach presented in this research work is a robust dynamic model to study hydrodynamic lubrication of spherical joints.

Modelling and numerical simulation of hygrothermal transfer through a building wall for locations subjected to outdoor conditions in Sub-Saharan Africa

In the building sector, the improvement of the heat and mass insulation of the walls can reduce the values of the global energy demand and the CO2 emission. A coupled one-dimensional heat and mass transfer model through a building wall was proposed in this study. The governing equations integrate the influences of Dufour and Soret effects and are numerically resolved using FORTRAN 90 and finite difference method in Crank-Nicolson scheme. Gauss-Seidel relaxation iteration method is used in order to generate the numerical solutions. The accuracy of the model was evaluated with experimental data on wooden wall and concrete wall from literature. A very satisfactory agreement is noted between the numerical predictions, experimental observations and physical phenomenon obtained. The model was further used to simulate the hygrothermal transfer through five tropical woods used as building walls in sub-Saharan African region after integration of outdoor conditions. These five woods have densities ranging from 430 to 800 kg/m3 and the conditions of the interior air were fixed to maintain a comfortable state for the occupants. Twelve towns located within latitudes 4.33°S to 12.37°N in Sub – Saharan Africa were used to discuss the results given by the model. At the end, influences of anatomical direction, tropical wood types, wall thickness, latitudes of the locations and solar radiation on evolutions of temperature and moisture content in the wall are studied. Our results showed that these parameters have significant effects on temperature and moisture content propagation within the wall. Using a wooden wall of 5 cm thickness, it is possible to reduce significantly the effect of the exterior heat.

Numerical Analysis of Flow Around Bluff Bodies with 4th and 2nd order compact formulations

In the present study, fourth-order compact formulation has been improved for Navier-Stokes (N-S) equations, which is expressed for two-dimensional, steady, incompressible flow problems. N-S equation system has been expressed with Stream Function-Vorticity Approach using Finite Difference Method (FDM) from the numerical methods. In order to test the functionality and applicability of the improved numerical formulation, a sample for submerged bluff bodies, flow problem around cylinder with square cross-section was chosen as a benchmark problem. As a result of applying improved numerical formulation with Gauss-Seidel Relaxation Method was used for this benchmark problem. The benchmark problem was also solved with second-order accuracy and obtained numerical results were compared with fourth-order accuracy numerical results. With the same Reynolds Number and the same free-stream velocity values, fourth-order numerical results are more convergent than second-order numerical results. Furthermore, in the flow field for considered benchmark problem, separation bubble length that consisted in the wake region is increased proportionally depending on the alteration of the Reynolds Number values.

Effects of surface topography on hydrodynamic performance of liquid film seals considering cavitation

PurposeThe purpose of this paper is to investigate the effects of surface topography, including surface roughness, circumferential waviness and radial taper, on hydrodynamic performance of liquid film seals considering cavitation.Design/methodology/approachA mathematical model of liquid film seals with surface topography was established based on the mass-conservative algorithm. Liquid film governing equation was discretized by the finite control volume method and solved by the Gauss–Seidel relaxation iterative algorithm, and the hydrodynamic performance parameters of liquid film seals were obtained considering surface roughness, circumferential waviness and radial taper separately.FindingsThe results indicate that the values of load-carrying capacity and frication torque are affected by the surface topography in varying degrees, but the effect is limited.Originality/valueThe results presented in the study are expected to aid in determining the optimum value of structural parameters for the optimum seal performance because of the realistic model which considers both surface topography and cavitation.

New method for initial density reconstruction

A theoretically interesting and practically important question in cosmology is the reconstruction of the initial density distribution provided a late-time density field. This is a long-standing question with a revived interest recently, especially in the context of optimally extracting the baryonic acoustic oscillation (BAO) signals from observed galaxy distributions. We present a new efficient method to carry out this reconstruction, which is based on numerical solutions to the nonlinear partial differential equation that governs the mapping between the initial Lagrangian and final Eulerian coordinates of particles in evolved density fields. This is motivated by numerical simulations of the quartic Galileon gravity model, which has similar equations that can be solved effectively by multigrid Gauss-Seidel relaxation. The method is based on mass conservation, and does not assume any specific cosmological model. Our test shows that it has a performance comparable to that of state-of-the-art algorithms which were very recently put forward in the literature, with the reconstructed density field over $\sim80\%$ ($50\%$) correlated with the initial condition at $k\lesssim0.6h/{\rm Mpc}$ ($1.0h/{\rm Mpc}$). With an example, we demonstrate that this method can significantly improve the accuracy of BAO reconstruction.

Effects of high-frequency damping on iterative convergence of implicit viscous solver

This paper discusses effects of high-frequency damping on iterative convergence of an implicit defect-correction solver for viscous problems. The study targets a finite-volume discretization with a one parameter family of damped viscous schemes. The parameter α controls high-frequency damping: zero damping with α=0, and larger damping for larger α(>0). Convergence rates are predicted for a model diffusion equation by a Fourier analysis over a practical range of α. It is shown that the convergence rate attains its minimum at α=1 on regular quadrilateral grids, and deteriorates for larger values of α. A similar behavior is observed for regular triangular grids. In both quadrilateral and triangular grids, the solver is predicted to diverge for α smaller than approximately 0.5. Numerical results are shown for the diffusion equation and the Navier–Stokes equations on regular and irregular grids. The study suggests that α=1 and 4/3 are suitable values for robust and efficient computations, and α=4/3 is recommended for the diffusion equation, which achieves higher-order accuracy on regular quadrilateral grids. Finally, a Jacobian-Free Newton–Krylov solver with the implicit solver (a low-order Jacobian approximately inverted by a multi-color Gauss–Seidel relaxation scheme) used as a variable preconditioner is recommended for practical computations, which provides robust and efficient convergence for a wide range of α.

Multigrid cell-centered techniques for high-order incompressible flow numerical solutions

A multigrid pressure correction scheme suitable for high order discretizations of the incompressible Navier–Stokes equations is developed and demonstrated. The pressure correction equation is discretized with fourth-order compact finite-difference approximations. Iterative methods based on multigrid techniques accelerate the most demanding part of the overall solution algorithm, which is the numerical solution of the arised large and sparse linear system. Geometrical multigrid methods, using partial semicoarsenig strategy and zebra line Gauss–Seidel relaxation, are employed to efficiently approximate the solution of the resulting algebraic linear system. Effects of various multigrid components on the pressure correction procedure are evaluated and new high-order transfer operators are developed for the case of cell-centered grids. Their convergence rates are also compared with commonly used intergrid transfer operators. Furthermore, numerically comparisons between different multigrid cycle approaches, such as V-, W- and F-cycle, are presented. The performance tests demonstrate that the new pressure correction approach significantly reduces the computational effort compared to single-grid algorithms. Furthermore, it is shown that the overall high order accuracy of the numerical method is retained in space and time with increasing Reynolds number.

Visualization of heat transfer in material for varians of boundary value with Relaxation Iteration Gauss-Seidel method

The research was aimed to know the effect of initial boundary value to the heat propagation rate pattern using iterations over Gauss-Seidel relaxation method and to analyze the exact value of each node descritization profile of test material. This study was an analytical study to fine analytical an numerical solution. Result from this study is that the pattern of variation of the boundary or initial conditions of a material with regard conductivity value remains at steady state the exact value of the smallest are in the same iteration value. The indicates that the value of the thermal equilibrium tend to be at the same iteration. Result from study showed that the pattern of initial boundary values that causes steady state of heat propagation of test material that has smallest exact similar to the iteration value.

Fast solution of elliptic partial differential equations using linear combinations of plane waves.

Given an arbitrary elliptic partial differential equation (PDE), a procedure for obtaining its solution is proposed based on the method of Ritz: the solution is written as a linear combination of plane waves and the coefficients are obtained by variational minimization. The PDE to be solved is cast as a system of linear equations Ax=b, where the matrix A is not sparse, which prevents the straightforward application of standard iterative methods in order to solve it. This sparseness problem can be circumvented by means of a recursive bisection approach based on the fast Fourier transform, which makes it possible to implement fast versions of some stationary iterative methods (such as Gauss-Seidel) consuming O(NlogN) memory and executing an iteration in O(Nlog(2)N) time, N being the number of plane waves used. In a similar way, fast versions of Krylov subspace methods and multigrid methods can also be implemented. These procedures are tested on Poisson's equation expressed in adaptive coordinates. It is found that the best results are obtained with the GMRES method using a multigrid preconditioner with Gauss-Seidel relaxation steps.

求解Oseen流的交替线松弛多重网格方法

The 1st-order upwind discretization form of the Oseen flow was obtained through the Godunov-type flux-difference splitting approach based on the Riemann solver. The convergence analysis of 2 kinds of cycling algorithms, i.e., the V-cycle and the W-cycle in the multigrid method for the solution of the discretized equations, was performed. Furthermore, the smooth properties of the collective symmetrical alternating-line Gauss-Seidel relaxation was investigated by means of the local Fourier analysis. The numerical results show that the collective symmetrical alternating-line Gauss-Seidel relaxation has sound smooth properties, and the convergence of the W-cycle algorithm is better than that of the V-cycle one in the multigrid method for the solution of the Oseen flow with different Reynolds numbers.

Globally consistent correspondence of multiple feature sets using proximal Gauss–Seidel relaxation

Feature correspondence between two or more images is a fundamental problem towards many computer vision applications. The case of correspondence between two images has been intensively studied, however, few works so far have been concerned with multi-image correspondence. In this paper, we address the problem of establishing a globally consistent correspondence among multiple (more than two) feature sets given the pairwise feature affinity information. Our main contribution is to propose a novel optimization framework for solving this problem based on the so-called Proximal Gauss–Seidel Relaxation (PGSR). The proposed method is distinguished from previous works mainly in three aspects: (1) it is more robust to noise and outliers; (2) its solution is based on convex relaxation and the principled PGSR method, which in general has convergence guarantee; (3) the scale of the problem in our method is linear with respect to the number of feature sets, making it computationally practical to be used in real-world applications. Experimental results both synthetic and real image datasets have demonstrated the effectiveness and superiority of the proposed method.

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell’s Equations

We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell’s equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their “immediate” neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiments.

A Multigrid Solver based on Distributive Smoother and Residual Overweighting for Oseen Problems

AbstractAn efficient multigrid solver for the Oseen problems discretized by Marker and Cell (MAC) scheme on staggered grid is developed in this paper. Least squares commutator distributive Gauss-Seidel (LSC-DGS) relaxation is generalized and developed for Oseen problems. Residual overweighting technique is applied to further improve the performance of the solver and a defect correction method is suggested to improve the accuracy of the discretization. Some numerical results are presented to demonstrate the efficiency and robustness of the proposed solver.

Multigrid Method for Solution of 3D Helmholtz Equation Based on HOC Schemes

A higher order compact difference (HOC) scheme with uniform mesh sizes in different coordinate directions is employed to discretize a two- and three-dimensional Helmholtz equation. In case of two dimension, the stencil is of 9 points while in three-dimensional case, the scheme has 27 points and has fourth- to fifth-order accuracy. Multigrid method using Gauss-Seidel relaxation is designed to solve the resulting sparse linear systems. Numerical experiments were conducted to test the accuracy of the sixth-order compact difference scheme with Multigrid method and to compare it with the standard second-order finite-difference scheme and fourth-order compact difference scheme. Performance of the scheme is tested through numerical examples. Accuracy and efficiency of the new scheme are established by using the errors normsl2.