Successive Overrelaxation Research Articles

Overview of Precoding Techniques for Massive MIMO

Massive multiple-input multiple-output (MIMO) is playing a crucial role in the fifth generation (5G) and beyond 5G (B5G) communication systems. Unfortunately, the complexity of massive MIMO systems is tremendously increased when a large number of antennas and radio frequency chains (RF) are utilized. Therefore, a plethora of research efforts has been conducted to find the optimal precoding algorithm with lowest complexity. The main aim of this paper is to provide insights on such precoding algorithms to a generalist of wireless communications. The added value of this paper is that the classification of massive MIMO precoding algorithms is provided with easily distinguishable classes of precoding solutions. This paper covers linear precoding algorithms starting with precoders based on approximate matrix inversion methods such as the truncated polynomial expansion (TPE), the Neumann series approximation (NSA), the Newton iteration (NI), and the Chebyshev iteration (CI) algorithms. The paper also presents the fixed-point iteration-based linear precoding algorithms such as the Gauss-Seidel (GS) algorithm, the successive over relaxation (SOR) algorithm, the conjugate gradient (CG) algorithm, and the Jacobi iteration (JI) algorithm. In addition, the paper reviews the direct matrix decomposition based linear precoding algorithms such as the QR decomposition and Cholesky decomposition (CD). The non-linear precoders are also presented which include the dirty-paper coding (DPC), Tomlinson-Harashima (TH), vector perturbation (VP), and lattice reduction aided (LR) algorithms. Due to the necessity to deal with a high consuming power by the base station (BS) with a large number of antennas in massive MIMO systems, a special subsection is included to describe the characteristics of the peak-to-average power ratio precoding (PAPR) algorithms such as the constant envelope (CE) algorithm, approximate message passing (AMP), and quantized precoding (QP) algorithms. This paper also reviews the machine learning role in precoding techniques. Although many precoding techniques are essentially proposed for a small-scale MIMO, they have been exploited in massive MIMO networks. Therefore, this paper presents the application of small-scale MIMO precoding techniques for massive MIMO. This paper demonstrates the precoding schemes in promising multiple antenna technologies such as the cell-free massive MIMO (CF-M-MIMO), beamspace massive MIMO, and intelligent reflecting surfaces (IRSs). In-depth discussion on the pros and cons, performance-complexity profile, and implementation solidity is provided. This paper also provides a discussion on the channel estimation and energy efficiency. This paper also presents potential future directions in massive MIMO precoding algorithms.

Low Complexity Linear Detectors for Massive MIMO: A Comparative Study

Massive multiple-input multiple-output (M-MIMO) is a significant pillar in fifth generation (5G) networks where a large number of antennas is deployed. It provides massive advantages to modern communication systems in data rate, spectral efficiency, number of users serviced simultaneously, energy efficiency, and quality of service (QoS). However, it requires advanced signal processing for data detection. The growing MIMO size leads to complicated scenarios, which makes the detector design a knotty problem. The problem is also becoming more complicated when high-order modulation schemes are exploited and more users are multiplexed. Therefore, it is not practical to employ the maximum likelihood (ML) detector despite the excellent performance. Linear detectors are alternative solutions and relatively simple. Unfortunately, they still need an exact matrix inversion computation, which bears to a significant high complexity. Therefore, several iterative methods are utilized to approximate or evade the matrix inversion rather than computing it. This paper studies the pros and cons of iterative matrix inversion methods where the number of computations and bit-error-rate (BER) are considered to compare between the methods. The comparison is conducted in several scenarios such as different ratio between the number of base station (BS) antennas and user terminal (UT) antennas (β), the number of iterations (n), and the relaxation parameter (ω). This paper also studies the impact of ω in the performance of Richardson (RI) and the successive over-relaxation (SOR) methods. Numerical results show that the conjugate gradient (CG) and optimized coordinate descent (OCD) methods exhibit the lowest complexity with an acceptable performance. In addition, the Gauss-Seidel (GS) method outperforms all other detectors with a trivial complexity increment. It is also noticed that the performance is not improved with every iteration. It is also shown that ω has a great impact and a significant role in achieving a satisfactory performance in both RI and SOR based detectors. From implementation point of view, detectors based on RI, OCD, and CG methods have achieved the highest hardware efficiency (HE) while Jacobi (JA) based detector has obtained the lowest HE. Recent research advances of detection methods are also presented in the open research direction with a potential impact of linear detection methods in initialization and pre-processing.

Effect of Surface Texture on the Steady-state Characteristics of Journal Bearing and Its Comparisons

The journal bearing has been an important mechanical part of the mechanical system, and over the decade's researchers have been investigating how to analyse and enhance its performance. In order to improve the bearing performance, the surface texture and texture shape have the potential to improve its performance efficiently. Therefore, the present paper investigates the effect of different types of texture such as elliptical, cylindrical for the steady-state characteristics of journal bearing. It has been found that the incorporation of texture on the inner side surface of the bearing has a significant influence on the steady-state characteristics of the bearing. The resultsare obtained in terms of non-dimensional Sommerfeld number (S), attitude angle (ф), friction variable (μ), and flow coefficient (q_z), by using the finite difference method and Gauss-Seidel method. The successive over-relaxation (SOR) technique has been applied to satisfy the boundary conditions. The obtained reluts show the improvement in tribological performance of the bearing characteristics through incorporation of texture. In addition, a comparative analysis of textured bearings i.e., elliptical, cylindrical is also done withplain journal bearing. It has been investigated that elliptical surfacestexturehas superior performance in contrast to cylindrical and plain textures.

Chebyshev Periodical Successive Over-Relaxation for Accelerating Fixed-Point Iterations

A novel method which is called the Chebyshev inertial iteration for accelerating the convergence speed of fixed-point iterations is presented. The Chebyshev inertial iteration can be regarded as a valiant of the successive over relaxation or Krasnosel'ski\v{\i}-Mann iteration utilizing the inverse of roots of a Chebyshev polynomial as iteration dependent inertial factors. One of the most notable features of the proposed method is that it can be applied to nonlinear fixed-point iterations in addition to linear fixed-point iterations. Linearization around the fixed point is the key for the analysis on the local convergence rate of the proposed method. The proposed method appears effective in particular for accelerating the proximal gradient methods such as ISTA. It is also proved that the proposed method can successfully accelerate almost any fixed-point iterations if all the eigenvalues of the Jacobian at the fixed point are real.

An optimized AOR iterative method for solving absolute value equations

In recent years, the AOR iterative method has been proposed for solving absolute value equations. This method has two parameters ? and ?. In this paper, we intend to find the optimal parameters of this method to improve convergence rate by suitable optimization techniques. Meanwhile, the convergence of the optimized AOR iterative method is discussed. It is both theoretically and experimentally demonstrated efficiency of the optimized AOR iterative method in contrast with the AOR and SOR methods.

Robot pathfinding with obstacle avoidance capabilities in a static indoor environment via TOR iterative method using harmonic potentials

Mobile robots are always in a state where they have to find a collision-free path in their environment from start to the target point. This study tries to solve the problem of mobile robot iteratively by using a numerical technique. It is based on potential field technique that was modelled using the Laplace’s equation to restrain the creation of a potential functions across regions in the mobile robot’s configuration space. The gradient formed by the potential field is then used to generate a path for the robot to advance through. The present paper proposes a Two-Parameter Over-Relaxation (TOR) iterative method that is used to solve Laplace’s equation for obtaining the potential field that is then utilized for finding path of the robot, thus solving the robot pathfinding problem. The experiment indicates that it is capable of producing a smooth path between the starting and target points through the use of a finite-difference technique. Furthermore, the simulation results show that this numerical approach executes quicker and provides a smoother trail than to the previous works, that includes Successive Over-Relaxation (SOR) and Accelerated Over-Relaxation (AOR) methods.

A Fractional Drift Diffusion Model for Organic Semiconductor Devices

Because charge carriers of many organic semiconductors (OSCs) exhibit fractional drift diffusion (Fr-DD) transport properties, the need to develop a Fr-DD model solver becomes more apparent. However, the current research on solving the governing equations of the Fr-DD model is practically nonexistent. In this paper, an iterative solver with high precision is developed to solve both the transient and steady-state Fr-DD model for organic semiconductor devices. The Fr-DD model is composed of two fractional-order carriers (i.e., electrons and holes) continuity equations coupled with Poisson’s equation. By treating the current density as constants within each pair of consecutive grid nodes, a linear Caputo’s fractional-order ordinary differential equation (FrODE) can be produced, and its analytic solution gives an approximation to the carrier concentration. The convergence of the solver is guaranteed by implementing a successive over-relaxation (SOR) mechanism on each loop of Gummel’s iteration. Based on our derivations, it can be shown that the Scharfetter–Gummel discretization method is essentially a special case of our scheme. In addition, the consistency and convergence of the two core algorithms are proved, with three numerical examples designed to demonstrate the accuracy and computational performance of this solver. Finally, we validate the Fr-DD model for a steady-state organic field effect transistor (OFET) by fitting the simulated transconductance and output curves to the experimental data.

DOUBLE-DIFFUSIVE CONVECTION IN A DISSIPATIVE ELECTRICALLY CONDUCTING NANOFLUID UNDER ORTHOGONAL ELECTRIC AND MAGNETIC FIELDS: A NUMERICAL STUDY

Two-dimensional double-diffusive convective flow in a duct is studied numerically. The duct is filled with electrically conducting nanofluid and subjected to mutually orthogonal static electrical and magnetic fields. The one-phase Tiwari-Das model is employed to simulate nanoscale effects. The study is conducted for four different electroconductive nanofluids using water as a base fluid. The left and right plates of the enclosure are kept at different constant temperatures and concentrations. The top and bottom faces are insulated and impermeable to heat and mass transfer respectively. The transport equations describe the velocity, temperature and nanoparticle concentration fields. These coupled differential Navier-Stokes equations are nonlinear, and therefore discretized via a robust Finite Difference Method (FDM). The reduced difference equations are solved by incorporating the Successive-Over-Relaxation (SOR) method. The results are shown graphically for various governing parameters. The skin friction, Nusselt and Sherwood numbers for the impact of selected electromagnetic, nanoscale and thermophysical parameters are computed. The study is relevant to thermal power technologies, bioelectromagnetic therapy and nuclear engineering heat transfer control.

Lubrication performance of magnetorheological fluid-lubricated rubber stern bearing test ring

The purpose of this paper is to propose magnetorheological fluid-lubricated rubber stern bearing test ring, to study the effects of temperature, magnetic field strength and eccentricity ratio on lubrication performance. The Reynolds equation considering effects of eccentricity ratio, temperature and magnetic field strength is adopted and solved by finite difference method and successive over-relaxation method to calculate pressure distribution and friction coefficient with real bearing shapes and boundary conditions. Bearing capacity can be improved and friction coefficient can be reduced by decreasing temperature, increasing magnetic field strength and increasing eccentricity ratio. The paper’s results may provide important design theoretical and experimental guidelines support for rubber stern bearing in this kind of magnetorheological fluids-lubricated.

Optimal Complex Relaxation Parameters in Multigrid for Complex-Shifted Linear Systems

We derive optimal complex relaxation parameters minimizing smoothing factors associated with multigrid using red-black successive overrelaxation or damped Jacobi smoothing applied to a class of lin...

Heat transfer enhancement with nanofluid in an open enclosure due to discrete heaters mounted on sidewalls and a heated inner block

Purpose The purpose of this study is to explore the heat transfer enhancement in copper–water nanofluid flowing in a diagonally vented rectangular enclosure with four discrete heaters mounted centrally on the sidewalls and a square-shaped embedded heated block in the influence of a static magnetic field. Design/methodology/approach Four discrete heaters are mounted centrally on each sidewall of the rectangular enclosure that embraces a heated square block. A static transverse magnetic field is acting on the vertical walls. The Navier–Stokes equations of motion and the energy equation are modified by incorporating Lorentz force and basic physical properties of nanofluid. The derived momentum and energy equations are tackled numerically using the successive over-relaxation technique associating with the Gauss–Seidel iteration technique. The effects of physical parameters connected to dynamics of flow and heat convection are explored from streamlines and isotherms graphs and discussed numerically in terms of Nusselt number. Findings The effect of the embedded heated square block size and its location in the enclosure, nanoparticles volume fraction and the intensity of the magnetic field on flow and heat transfer are computed. Compared with the case when no heated block is embedded in the enclosure, in free convection at Ra = 106, the average local Nusselt number on the wall-mounted heaters is attenuated by 8.25%, 11.24% and 12.75% when the enclosure embraced a heated square block of side length 10% of H, 20% of H and 30% of H, respectively. An increase in Hartmann number suppresses the heat convection. Research limitations/implications The enhancement in the convective heat is greater when the buoyancy effect dominates the viscous effects. Placing the embedded heated block near the inlet vent, the lower temperature zone has reduced while the embedded heated block is at the central location of the enclosure, the high-temperature zone has expanded. The external magnetic field can be used as a non-invasive controlling device. Practical implications The numerically simulated results for heat convection of water-based copper nanofluid agreed qualitatively with the existing experimental results. Social implications The models could be used in designing a target-oriented heat exchanger. Originality/value The paper includes a comparative study for three locations of the embedded heated square. The optimal results for the centrally located heated block are also performed for three different sizes of the embedded block. The numerically simulated results are compared with the published numerical and experimental studies.

Effects of Using a Memory Stalled Core for Handling MPI Communication Overlapping in the SOR Solver on SX-ACE and SX-Aurora TSUBASA

Modern high-performance computing (HPC) systems consist of a large number of nodes featuring multi-core processors. Many computational fluid dynamics (CFD) codes utilize a Message Passing Interface (MPI) to exploit the potential of such systems. In general, the MPI communication costs increase as the number of MPI processes increases. In this paper, we discuss performance of the code in which a core is used as a dedicated communication core when the core cannot contribute to the performance improvement due to memory-bandwidth limitations. By using the dedicated communication core, the communication operations are overlapped with computation operations, thus enabling highly efficient computation by exploiting the limited memory bandwidth and idle cores. The performance evaluation shows that this code can hide the MPI communication times of 90% on the supercomputer SX-ACE system and 80% on the supercomputer SX-Aurora TSUBASA system, and the performance of the successive over-relaxation (SOR) method is improved by 32% on SX-ACE and 20% on SX-Aurora TSUBASA.

Comparative Analysis of Data Detection Techniques for 5G Massive MIMO Systems

Massive multiple-input multiple-output (MIMO) is a backbone technology in the fifth-generation (5G) and beyond 5G (B5G) networks. It enhances performance gain, energy efficiency, and spectral efficiency. Unfortunately, a massive number of antennas need sophisticated processing to detect the transmitted signal. Although a detector based on the maximum likelihood (ML) is optimal, it incurs a high computational complexity, and hence, it is not hardware-friendly. In addition, the conventional linear detectors, such as the minimum mean square error (MMSE), include a matrix inversion, which causes a high computational complexity. As an alternative solution, approximate message passing (AMP) algorithm is proposed for data detection in massive MIMO uplink (UL) detectors. Although the AMP algorithm is converging extremely fast, the convergence is not guaranteed. A good initialization influences the convergence rate and affects the performance substantially together and the complexity. In this paper, we exploit several free-matrix-inversion methods, namely, the successive over-relaxation (SOR), the Gauss–Seidel (GS), and the Jacobi (JA), to initialize the AMP-based massive MIMO UL detector. In other words, hybrid detectors are proposed based on AMP, JA, SOR, and GS with an efficient initialization. Numerical results show that proposed detectors achieve a significant performance enhancement and a large reduction in the computational complexity.

Dynamics of fluid flow through Soret-Dufour impacts subject to upward and downward motion of rotating disk

Flow features are encountered through numerical way using central finite differences approximations and Successive over Relaxation (SOR) method. Fluid flow is generated due to disk rotation, which moves in vertical direction both up and down. Similarity variables are accomplished in such a way that similarity equations can be limited to those communicated in viscous pumping analysis of traditional Von Karman problem for non-vertical motion of rotating disk. The significance of Soret and Dufour effects is also incorporated in fluid model. Additionally, the occurrence of convective boundary conditions associated to heat and mass are of physical concern in existing study. Graphical descriptions and tabular structures of numerical results are presented. The impacts of wall injection and suction are reviewed. It is judged that the uphill motion of rotating disk contributes to similar observations as in scenario of injection, while the downward direction signifies the wall suction like effects.

NATURAL CONVECTION IN A MICROPOLAR NANOFLUID FILLED OPEN RECTANGULAR ENCLOSURE EMBEDDED WITH A HEATED OBJECT OF DIFFERENT GEOMETRIES

A numerical investigation has been performed to analyze heat convection through a micropolar nanofluid in an open rectangular enclosure. It embedded the inner heated object of different geometries and finite heat source length on the bottom wall with uniform heat flux. The remaining portion of the wall containing heat source and all the remaining walls are assumed as adiabatic except the top wall. Successive over-relaxation (SOR) method coupled with Gauss-Seidel iteration technique are employed in order to numerically tackled the nonlinear model momentum and energy equations. The effect of the calibrated parameters such as Rayleigh number, length of the wall heat source, the geometry of the inner block, vortex viscosity parameter, the type of nanoparticles, and the concentration of nanoparticles on the flow and thermal performance is studied. The computed results show that increasing in the Rayleigh number and the concentration of nanoparticles have a positive effect on Nusselt number whereas increasing in wall heat source length and vortex viscosity attenuates the Nusselt number. Also, the geometry of the inner block has effect in the flow pattern and the temperature distribution.

Fast Iterative Techniques for Polarized Radiative Transfer in Spherically Symmetric Moving Media

Abstract For a more precise modeling of polarized spectral lines formed in extended and expanding stellar atmospheres, the solution of the radiative transfer equation for the Stokes vectors must be obtained in a spherical geometry rather than in a planar geometry. In this paper, we present the modern iterative techniques based on operator perturbation to solve the spherically symmetric polarized radiative transfer equation with velocity fields. We consider scattering on a two-level atom and account for partial frequency redistribution. An accurate numerical solution to such problems requires the use of spatial grids with higher resolution. Consequently, Jacobi-based methods lead to slower convergence rate. The convergence rate can be improved by a factor of 2 or more when fast iterative schemes based on Gauss–Seidel (GS) and successive overrelaxation (SOR) methods are used over the Jacobi-based method. Here we present the Jacobi, GS, and SOR iterative techniques for solving the abovementioned problem, and discuss their convergence behavior.

Numerical Evaluation of Quarter-Sweep KSOR Method to Solve Time-Fractional Parabolic Equations

We study the performance of the combination of quarter-sweep iteration concept with the Kaudd Successive Over-Relaxation (KSOR) iterative method in solving the discretized one-dimensional time-fractional parabolic equation. We called the mixed of these two concepts as QSKSOR. The time-fractional derivative in Grunwald sense, together with the implicit finite difference scheme was used to discretized the tested problems to form the quarter-sweep implicit finite difference approximation equations in the sense of Grunwald type. This approximation equation of half-sweep will then generate a linear system. Next, we used the proposed QSKSOR iterative method to the generated linear systems before comparing the effectiveness between the other family of KSOR method, FSKSOR and HSKSOR with respect to the full- and half-sweep cases respectively. To do so, three examples are included. The results of this study show the superiority of the QSKSOR iterative method in terms of iteration numbers and execution time in comparison to the other two methods.

Unsteady 2D and 3D Navier-Stokes Solver with Application of Multigrid Scheme to Pressure Poisson Fractional Step on Arbitrary Unstructured Grids in Various Applications with Emphasis on Ship Motion

A 3D unsteady computer solver is presented to compute incompressible Navier-Stokes equations combined with the volume of fraction (VOF) method on an arbitrary unstructured domain. This is done to simulate fluid flows in various applications, especially around a marine vessel. The Navier-Stokes solver is based on the fractional steps method coupled with a finite volume scheme and collocated grids by which velocity components and pressure fields are defined at the center of the control volume. However, the fluxes are defined at the midpoint on their corresponding cell faces. On the other hand, the CICSAM (Compressive Interface Capturing Scheme for Arbitrary Meshes) scheme is applied to capture the free surface. In the presented fractional step method, the pressure Poisson equation suffers from poor convergence rate by simple iterative methods like Successive Overrelaxation (SOR), especially in simulating complex geometrics like a ship with appendages. Therefore, to accelerate the convergence rate, an agglomeration multigrid method is applied on arbitrary moving mesh for solving pressure Poisson equation with two well-known cycles, V and W. In order to maintain accuracy, the geometry details should not change in grid coarsening procedure. Therefore, the boundary faces are assumed to be fixed in all grids level. This assumption requires nonstandard cells in coarsening procedures. To investigate the performance of the applied algorithm, various flows including one and two-phase flows are studied in two and three dimensions. It is found that the multigrid method can speed up the convergence rate of fractional step twofold. In most cases (not all), W cycle displays better performance. It is also concluded that the efficiency of the cycle depends on the number of meshes and complexity of the problem and this is mainly due to the data transferring between grids. Therefore, the type of cycle should be selected judiciously and carefully, while considering the mesh size and flow properties.

Numerical simulation of natural convection of dusty nanofluids within a curve-shaped enclosure

The current research expounds the numerical simulation of a two phase dusty nanofluid flow in a curve-shaped enclosure surrounded by three straight walls and a curved wall. The flow is generated in the enclosure because of the buoyancy force due to the heated lower curved wall. An apt variable transform that transfigures the curve-shaped domain into a rectangular domain is introduced in this study. The governing equations for the nanofluid phase and the dusty phase in transformed coordinates are solved by incorporating a finite difference technique with a successive over-relaxation method. The requisite results to analyze the flow and heat transfer are blazoned for physically significant parameters, namely, the nanoparticles’ volume fraction, the Rayleigh number, the amplitude, the density ratio, and the dusty parameter. The results assert that the flow strength can be decreased slightly by increasing the nanoparticles’ volume fraction and the amplitude but can be increased notably by incrementing the Rayleigh number. Besides, heat transfer can be enhanced about 4.33% at the lower wall and about 5.13% at the upper wall by incrementing the nanoparticles’ volume fraction 4% for the Rayleigh number 105. Moreover, the increment in the Rayleigh number and decrement in the density ratio and the dusty parameter can augment the heat transfer rate.

Thermophoretic particles deposition in time-dependent magneto flow over oscillatory spinning disk

Thermophoresis particles deposition is a fundamental procedure of micro-particles transmission through temperature gradient and has major significance in aerosol and electronic technologies. To observe the variation of mass deposition, the phenomenon of thermophoresis is encountered in time-dependent electrically conducting flow of viscous fluid by rotating disk which also oscillates periodically. The swirl flow finds significance in hydraulic pumps and conical diffusers. The fluid model initially based upon the system that comprises on the partial differential equations (PDEs) and then re-framed into the ordinary differential equations (ODEs) by similar transformations. To obtain the numerical results in graphical and tabular formats, we first apply the finite difference scheme on governing equations and then implemented successive over relaxation (SOR) method for faster convergence. Statistical approach is also executed to discuss the Pearson’s correlation and probable error for the emerging physical parameters. Three-dimensional flow graphs, contour plots and scatter plots of shear stresses are also drawn. The oscillatory behavior of radial curves is reduced by increasing the magnetic parameter. Thermophoretic co-efficient and relative temperature difference parameters demonstrated the opposite impact on oscillatory concentration field. A moderate positive correlation is noticed between swirl parameter and radial shear stress.