Successive Over-relaxation Technique Research Articles

Solving the Small Disturbance Equation and Numerical Method Optimization in Subsonic and Supersonic Flows

This paper addresses the solution of the Small Disturbance Equation (SDE) under subsonic and supersonic flow conditions using Successive Over-Relaxation (SOR) techniques. The study implements and validates numerical methods tailored to the unique dynamics of these regimes by setting precise boundary conditions, adjusting computational stencils, and managing the Mach number ( M∞ ) across the solution domain. A detailed comparison of streamline patterns between the regimes illustrates the efficacy of the applied numerical strategies. In the subsonic domain, the flow demonstrates uniform and smooth characteristics, conducive to standard SOR techniques, as evidenced by the rapid decline in residual errors, confirming the method’s efficiency for accurate solutions. Conversely, the supersonic regime presents increased complexity where standard finite difference methods encounter notable challenges, necessitating more sophisticated approaches to capture the intricate flow behaviors effectively. Conversely, in the supersonic regime, where the flow behavior exhibits more complex characteristics, the standard finite difference method faces challenges.

Numerical Solution to Poisson’s Equation for Estimating Electrostatic Properties Resulting from an Axially Symmetric Gaussian Charge Density Distribution: Charge in Free Space

Poisson’s equation frequently emerges in many fields, yet its exact solution is rarely feasible, making the numerical approach notably valuable. This study aims to provide a tutorial-level guide to numerically solving Poisson’s equation, focusing on estimating the electrostatic field and potential resulting from an axially symmetric Gaussian charge distribution. The Finite Difference Method is utilized to discretize the desired spatial domain into a grid of points and approximate the derivatives using finite difference approximations. The resulting system of linear equations is then tackled using the Successive Over-Relaxation technique. Our results suggest that the potential obtained from the direct integration of the distance-weighted charge density is a reasonable choice for Dirichlet boundary conditions. We examine a scenario involving a charge in free space; the numerical electrostatic potential is estimated to be within a tolerable error range compared to the exact solution.

ELASTOHYDRODYNAMIC EFFECTS IN LUBRICATION OF JOURNAL BEARINGS UNDER TURBULENT CONDITIONS

The current paper is aimed to investigate the deformability effects of the bearing surface on the lubrication performance of cylindrical journal bearing with finite length under turbulent conditions. The bush is coated with a thin resilient layer with a smooth surface, the radial distortions of which are of the same order of magnitude as the film thickness. The elastohydrodynamic problem is studied under elasticity conditions in accordance with the plain strain hypothesis (Winkler model). The modified Reynolds equation is carried out on the base of the turbulent lubrication model of Hirs (bulk flow model). The partial differential equation is solved numerically by successive over-relaxation technique on a finite difference grid, considering different values of coating materials, lubricant viscosity, and journal speed. Results for the performance characteristics of the bearing are given for several selected values of Reynolds number and elasticity parameters of the soft liner on the bush.

Magnetohydrodynamic thermo-bioconvective flow of 3D rotating Williamson nanofluid with Arrhenius activation energy in a Darcy–Forchheimer medium over an exponentially stretching Surface

This paper aims to analyze the heat and mass transfer characteristics of the 3D thermo-bioconvective flow of rotating Williamson nanofluid over an exponentially stretching vertical sheet. The flow distribution considers the influence of nonlinear thermal radiation, Hall and ion-slip currents, joule heating, coupled effects of Dufour and Soret diffusion, internal heat generation, Arrhenius activation energy, Brownian, and thermophoresis diffusion in a Darcy–Forchheimer porous medium. The modeled partial differential equations (PDE) in terms of continuity, momentum, energy, concentration, and density of motile microorganisms are reformed into ordinary differential equations (ODE) forms employing similarity variables. The resulting equations are solved numerically using the spectral local linearization method (SLLM). The successive over-relaxation technique is applied to accelerate and enhance the convergence of the method. The obtained results are graphically and tabularly represented and numerically explored for the effects of pertinent parameters on the velocity, temperature, nanoparticle concentration, and motile microorganism concentration profiles. The rotation and mixed convection parameters dropped the primary velocity curve, while the Williamson and heat source parameters uplifted the microbes density distributions. The temperature field rises with an increase in radiation and microbes Brownian motion and declines with an increase in Hall and ion-slip numbers. Furthermore, with the increase in activation energy from 0.5 to 2 and the mixed convection parameter from 0.6 to 1.3, we observe a respective 3.93% rise and 8.30% fall in mass transfer rates. The current investigation has broad applications, including MHD accelerators, biotechnology, the formation of elastic sheets, the production of polymer solutions, and the synthesis of nanoparticles through microbial processes.

Buoyancy-driven flow of magnetohydrodynamic hybrid nanofluids in an open cavity with permeable horizontal walls

The buoyancy-driven flow of hybrid nanofluids in an open cavity is examined in the presence of a magnetic field with an angle to the horizontal axis. The right end of the cavity is open, and the left wall is supposed to be heated. Moreover, the horizontal walls are deemed to be permeable and maintained at ambient temperature. The successive over-relaxation (SOR) technique is applied with the finite-difference method to solve the dimensionless equations. A comparison is executed with formerly published results which provide a good agreement. The grid refinement test has also been carried out to increase the accuracy. Flow and temperature profiles are investigated for the Rayleigh number ( Ra = 104, 105, 106), Reynolds number (Re = 5, 8, 10, 20, 100), Hartmann number ( Ha = 0, 5, 10), concentration of nanoparticles ( φ1 or φ2 = 0.0, 0.05, 0.1), angle of inclination of the magnetic field ( γ = 0°, 30°, 45°), aspect ratio ( A = 1, 2, 3, 4), and suction/blow parameter ( S = −1, 0, 1). Results have been elucidated based on streamlines, isotherms, local Nusselt number ( Nu), and average Nusselt number ( Nuavg). It is noted that the changes in the variables significantly affect streamlines and isotherms. The heat transfer for Cu-Fe3O4/water hybrid nanofluid is higher about 9.98% compared to Fe3O4/water nanofluid and 26.41% compared to water. Furthermore, all other parameters noticeably augment both Nu and Nuavg.

Numerical Analysis of Fluid Flow Problems Using Spectral Relaxation Method (SRM)

The paper presents four examples arising from mathematical models in fluid flow. Examples 1 and 2 illustrate the implementation of the spectral relaxation method (SRM) on problems involving ordinary differential equations. Examples 3 and 4 illustrate the application of the SRM on partial differential equations. The SRM is accurate and robust if it is used together with the successive overrelaxation (SOR) technique. The method is easy to implement and requires less computational time than similar methods that can be used to solve similar problems. The method converges after a few iterations and is stable. The method can be used as an alternative method to solve problems arising in fluid flow.

Generalized Second-Order Value Iteration in Markov Decision Processes

Value iteration is a fixed point iteration technique utilized to obtain the optimal value function and policy in a discounted reward Markov decision process (MDP). Here, a contraction operator is constructed and applied repeatedly to arrive at the optimal solution. Value iteration is a first-order method and, therefore, it may take a large number of iterations to converge to the optimal solution. Successive relaxation is a popular technique that can be applied to solve a fixed point equation. It has been shown in the literature that under a special structure of the MDP, successive overrelaxation technique computes the optimal value function faster than standard value iteration. In this article, we propose a second-order value iteration procedure that is obtained by applying the Newton–Raphson method to the successive relaxation value iteration scheme. We prove the global convergence of our algorithm to the optimal solution asymptotically and show the second-order convergence. Through experiments, we demonstrate the effectiveness of our proposed approach.

Numerical computation of magnetized bioconvection nanofluid flow with temperature-dependent viscosity and Arrhenius kinetic

The current study focused on the impact of activation energy and magnetic field on immiscible steady nanofluid moving over an elastic stretched surface containing motile gyrotactic microorganisms. We use the Reynolds exponential model to develop the mathematical modeling since the viscosity of the fluid is temperature-dependent. Under the presence of uniformly disseminated nanoparticles, the base fluid is electrically conductive. The Buongiorno model, which incorporates thermophoretic forces and Brownian motion, has been used. Using the spectral local linearization approach (SLLM), the looming nonlinear, coupled differential equations are numerically addressed. Because it is based on a smooth Uni-variant linearization of non-linear functions, the SLLM method is simple to construct and use. The numerical performance of SLLM is even better since it grows a collection of equations that are solved sequentially by operating the findings from one equation into the next. The successive over-relaxation technique was used to accelerate and enhance the convergence of the SLLM system. The reliability of the SLLM will be verified through the use of well-known techniques and comparisons with other data We have illustrated and analyzed in detail the graphical behavior of all emergent variables for temperature, velocity, and concentration distributions, and also the Nusselt number, Sherwood number, skin friction, and density number of motile microorganisms. It is significant to claim that the spectral local linearization approach has been demonstrated to be extremely stable and adaptive for the solution of non-linear differential equations.

Impact of Swimming Gyrotactic Microorganisms and Viscous Dissipation on Nanoparticles Flow through a Permeable Medium: A Numerical Assessment

In this paper, heat and mass transportation flow of swimming gyrotactic microorganisms (microbes) and solid nanoparticles under the viscous dissipation effect is investigated. The flow model PDEs are renovated with ordinary ones using suitable boundary layer approximations. The system governing the flow model dimensionless equations as well as boundary conditions is numerically treated with the SOR (successive over relaxation) technique. The flow, heat, and mass transport characteristics are examined against the prime parameters. A comparison is examined to be in a good agreement with the earlier results. It is found here that flow and thermal characteristics of the problem are substantially affected by the porous medium. The outcomes evidently point out that porous medium causes an enhancement in the skin friction and density of the motile microbes. Further, the rate of heat transport is devaluated at the surface of sheet due to viscous dissipation and elevated due to suction.

Total solar spectrum energy converter with integrated photovoltaics, thermoelectrics, and thermal energy storage: System modeling and design

A total-spectrum-utilizing integrated photovoltaic (PV), thermoelectric (TEG), and thermal energy storage fluid (TES) solar energy converter (PV-TEG-TES) with novel device architecture is proposed, and its performance is modeled to demonstrate its viability and optimize its system-level design. By incorporating a top-layer GaAs PV, ultraviolet and visible light is absorbed and converted directly to electricity, while infrared radiation passes through the PV. This infrared radiation is then captured by the TEG and TES layers to supply additional electricity or to be stored as thermal energy. In addition, the TEG converts to electricity otherwise wasted heat from thermalization of photons within the PV layer, and the TES effectively cools the PV, keeping it within adequate operating temperatures. According to analysis predicting upper bound performance, this PV-TEG-TES device can outperform the electrical conversion efficiency of a single GaAs solar cell module by 1.9 percentage points (from 24.1% for PV alone to 26.0% for the combined device, a relative 7.9% improvement), and can convert and store a total of 79.3% of incident solar energy under a concentration ratio of 673 suns per direct solar spectrum. An analytical framework using a successive overrelaxation technique for solutions to the nonlinearly coupled radiative, thermal, fluid, and electrical phenomena in such a PV-TEG-TES device is also discussed, offering a starting point for practical device parameters.

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.

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.

A heterogeneous parallel Red–Black SOR technique and the numerical study on SIMPLE

A basic heterogeneous parallel Red–Black successive over-relaxation (SOR) implement, the mono-color floating-point scheme, was developed on graphics processing units (GPU) with OpenCL platform. Designed in fine granularity, compact data structure, and stencil function, a concise mapping relationship was created to implicitly describe the complex rules for searching neighbor elements, which could avoid low utilization of GPU in the traditional scheme of Red–Black SOR. The new mono-color floating-point scheme was applied to build fast Semi-Implicit Method for Pressure Linked Equations (SIMPLE) solver with OpenCL and OpenMP on the heterogeneous parallel computing device. Compared with SIMPLE solver in the traditional Red–Black SOR scheme, the new scheme can achieve 1.7 to 1.8 faster accelerative performance on the same GPU. And this scheme can eliminate the complex searching module in mono-color logical scheme and behave better than the mono-color logical scheme by 20–30% acceleration. Numerical cases in double precision showed that SIMPLE solver on GPU with new scheme of Red–Black SOR could save up to 92% computing time compared with the serial solver on CPU.

Stationary probability densities of generalized Maxwell-type viscoelastic systems under combined harmonic and Gaussian white noise excitations

Viscoelasticity has been widely studied over the past few decades as a combination of the viscous effects of energy dissipation and the elastic effects of energy storage. As an application, viscoelastic dampers are commonly used to suppress the dynamic behaviour of structures. However, in an actual physical environment, it is difficult to explore the stationary probability densities of random responses for viscoelastic systems. For this purpose, we present a numerical method to investigate the dynamic behaviour of a generalized Maxwell-type viscoelastic system under harmonic and Gaussian white noise excitations. Using approximate equivalent and stochastic averaging, we establish an averaged Ito differential equation for the amplitude of the system. For primary external resonance, we solve the reduced Fokker–Planck–Kolmogorov equation by using the successive over-relaxation technique combined with a finite difference method. Through Monte Carlo simulations, we verify the applicability, accuracy and efficiency of the proposed methodology and demonstrate that viscoelasticity has a significant impact on the dynamic behaviour of the viscoelastic systems. This work reveals the remarkable influences of viscoelasticity, excitation intensities, linear and nonlinear damping and linear and nonlinear stiffness on the probability density functions of system responses, which can guide the selection of materials, stiffnesses and structures in viscoelastic damper design.

Thermo-solute natural convection with heat and mass lines in a uniformly heated and soluted rectangular enclosure for low Prandtl number fluids

Thermo-solute natural convection with heat and mass lines in a uniformly heated and soluted rectangular enclosure for low Prandtl number fluids is studied. The left and right walls are maintained at constant temperature and solute while the bottom and top walls are adiabatic and non-diffusive. The finite difference method, together with the successive over-relaxation (SOR) technique, is used to solve the flow governing equations after converted into the vorticity-stream function form. Heat and mass lines visualization techniques are used for the better visualization of energy and solute distribution in the enclosure. The results obtained in this study are compared with experimental and numerical those from literature and found to be in good agreement. The influence of Rayleigh number Ra=103,104,105, Prandtl number Pr=0.015,0.025,0.71, Lewis number Le=1,2,5, and Buoyancy ratio N=−1,0,1 on streamlines, isotherms, isosolutes, heatlines, masslines, total heat and mass transfer are examined. The parameters mentioned above considerably influence the total heat and solute transfer rates; moreover, the heat and mass lines play significant roles in understanding the distribution of energy and solute transfers in the enclosure.

Large-margin Distribution Machine-based regression

This paper presents an efficient and robust Large-margin Distribution Machine formulation for regression. The proposed model is termed as ‘Large-margin Distribution Machine-based Regression’ (LDMR) model, and it is in the spirit of Large-margin Distribution Machine (LDM) (Zhang and Zhou, in: Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, ACM, 2014) classification model. The LDM model optimizes the margin distribution instead of minimizing a single-point margin as is done in the traditional SVM. The optimization problem of the LDMR model has been mathematically derived from the optimization problem of the LDM model using an interesting result of Bi and Bennett (Neurocomputing 55(1):79–108, 2003). The resulting LDMR formulation attempts to minimize the $$\epsilon$$ -insensitive loss function and the quadratic loss function simultaneously. Further, the successive over-relaxation technique (Mangasarian and Musicant, IEEE Trans Neural Netw 10(5):1032−1037, 1999) has also been applied to speed up the training procedure of the proposed LDMR model. The experimental results on artificial datasets, UCI datasets and time-series financial datasets show that the proposed LDMR model owns better generalization ability than other existing models and is less sensitive to the presence of outliers.

Accelerated low rank matrix approximate algorithms for matrix completion

In this paper, we propose two accelerated algorithms for the low-rank approximate method in Wang et al. (0000) for matrix completion. The main idea is to use the successive over-relaxation technique. Based on the successive over-relaxation method for the feasible matrices or projection matrices, the low-rank matrix approximate method is modified and accelerated. Meanwhile, we discuss the convergence of the over-relaxation algorithm for the feasible matrix. Finally, the numerical experiments show them to be effective.

A novel projection nonparallel support vector machine for pattern classification

In this paper, we propose a novel nonparallel classifier termed as projection nonparallel support vector machine (PNPSVM), which is fully different from the existing nonparallel classifiers. The new classifier needs two steps to obtain the optimal proximal hyperplanes. The first step is to obtain two projection directions, which could achieve maximum class separability by minimizing the within-class distance and maximizing the between-class distance simultaneously, to be treated as the normal vectors of the optimal proximal hyperplanes, and the second step is to determine the specific locations of the optimal proximal hyperplanes based on an appropriate central sample. Furthermore, the improved successive overrelaxation (SOR) algorithm is applied to solve our PNPSVM. The incomparable advantages of this paper can be summarized as follows: (1) implementing the structural risk minimization principle in the primal problems; (2) utilizing the potential structural information of data by considering both the tightness between the similar patterns and the discrepancy between the dissimilar pairs; (3) the kernel trick can be applied directly since the dual problems have the similar elegant formulation as that of standard support vector machine; (4) SOR technique is introduced to solve our optimization problems. The comprehensive experimental results on an artificial dataset and twenty-four UCI datasets demonstrate the effectiveness of our method in classification accuracy.

Analysis and comparison of steady-state performance characteristics of two-axial groove and multilobe hydrodynamic bearings lubricated with non-Newtonian fluids

Rotating machines are one of the most important and widely used machineries in the modern engineering world. They are also the required to run at high rotating speed. The plain journal bearing is mostly replaced by some other bearing, as it does not suit the stability requirement of high-speed machines and precision machine tools. In order to improve the stability of a circular bearing, many researchers tried to change its geometrical configuration. In view of this, an attempt has been made in the present work to study the performance characteristics of four different bearing configurations using non-Newtonian lubricant and comparison of the results has been made with plain journal bearing. To calculate the steady-state performance characteristics, nondimensional form of the Reynolds equation using non-Newtonian lubricant has been written and solved for the steady-state nondimensional pressure distribution in a finite grid using finite difference method with successive over-relaxation technique satisfying appropriate boundary condition. From the study, it can be concluded that three-lobe bearing have maximum load-carrying capacity and minimum friction variable, whereas two-lobe bearing have maximum flow coefficient.

Improved preconditioned conjugate gradient algorithm and application in 3D inversion of gravity-gradiometry data

With the continuous development of full tensor gradiometer (FTG) measurement techniques, three-dimensional (3D) inversion of FTG data is becoming increasingly used in oil and gas exploration. In the fast processing and interpretation of large-scale high-precision data, the use of the graphics processing unit process unit (GPU) and preconditioning methods are very important in the data inversion. In this paper, an improved preconditioned conjugate gradient algorithm is proposed by combining the symmetric successive over-relaxation (SSOR) technique and the incomplete Choleksy decomposition conjugate gradient algorithm (ICCG). Since preparing the preconditioner requires extra time, a parallel implement based on GPU is proposed. The improved method is then applied in the inversion of noisecontaminated synthetic data to prove its adaptability in the inversion of 3D FTG data. Results show that the parallel SSOR-ICCG algorithm based on NVIDIA Tesla C2050 GPU achieves a speedup of approximately 25 times that of a serial program using a 2.0 GHz Central Processing Unit (CPU). Real airborne gravity-gradiometry data from Vinton salt dome (southwest Louisiana, USA) are also considered. Good results are obtained, which verifies the efficiency and feasibility of the proposed parallel method in fast inversion of 3D FTG data.