Successive Overrelaxation 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.

Advanced Numerical Techniques for Supersonic Flow Analysis: From Euler Equations to Practical Applications in Aerospace and Turbomachinery Engineering

This paper explores advanced numerical techniques for analyzing supersonic flow, with a particular focus on the application of Euler equations in aerospace and turbomachinery engineering. Theoretical foundations such as boundary conditions, error evaluation, grid partitioning, and the Successive Over-Relaxation (SOR) method are thoroughly examined. Rigorous numerical simulations are conducted to investigate the final velocity field distribution and convergence characteristics under various conditions. The research underscores the precision and efficiency of these numerical methods, highlighting their practical applications in the aerodynamic design of aircraft and potential use in turbomachinery design. The findings contribute significantly to the advancement of computational fluid dynamics in supersonic flow regimes, providing valuable insights for engineers and researchers engaged in this specialized area. The study not only enhances understanding of complex flow dynamics but also supports the development of more effective engineering solutions in high-speed applications.

Explicit Group Iterative Method with Semi-Approximate Implicit Approach for Solving One-Dimensional Burgers’ Equation

This paper is considering a semi-approximate approach in investigating the Burgers’ problem. The process of the discretization of Burgers’ problem has taken place which it starts with the second-order finite difference in the discretize process together with the linearization part by using the semi-approximate implicit scheme in a way to achieve the approximation equation, thus generating the corresponding linear system equations. Besides, the Gauss-Seidel (GS), Successive Over-Relaxation (SOR) and explicit group (EG) iterative method has combined together with the SOR iterative method, namely as 4-point EGSOR (4EGSOR) has been introduced in this study for resolving the linear system. To assess the proficiency of the suggested methods on the approximation equation, the numerical test has been conducted by considering three parameters, which are computational time, iteration number and maximum absolute error. The findings indicate that the 4EGSOR method outperforms both SOR and GS iterative methods.

Effect of lubricant inertia on textured journal bearing implementing mass conserving (JFO) boundary conditions

Purpose Cavitation plays a significant role in the performance of textured journal bearings. Furthermore, because of the usage of low-viscosity lubricants and the high working speed of machines, it is pertinent to consider the lubricant inertia while analyzing the operating characteristics of bearings. This paper aims to investigate the influence of lubricant inertia in the case of a spherically textured journal bearing, considering both protrusion and dimple texturing and implementing the mass-conserving (JFO) boundary conditions. Design/methodology/approach A novel modified Reynolds equation has been developed to accommodate the effects of lubricant inertia and cavitation. The cavitation is treated by using mass-conserving (Jakobsson−Floberg−Olsson [JFO]) boundary conditions. The governing equation is solved by the Gauss−Seidel method with successive over-relaxation. To enhance computational efficiency and expedite the convergence process, the progressive mesh densification (PMD) method has been integrated into the solution process. Findings The current study indicates that the JFO boundary conditions result in higher load-carrying capacity and lesser friction variables for heavily loaded bearings, whereas the flow coefficient is reduced due to the application of JFO boundary conditions. The lubricant inertia effect enhances the flow coefficients for lightly loaded and protrusion-textured bearings. Originality/value It is crucial to understand the combined effects of lubricant inertia and cavitation for the effective design of textured journal bearings. The findings from this work will help in designing textured journal bearings more effectively and accurately, particularly when low-viscosity oil is used. Peer review The peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-07-2024-0276/

The SOR and AOR Methods with Stepwise Optimized Values of Parameters for the Iterative Solutions of Linear Systems

We modify the successive overrelaxation (SOR) method and accelerated overrelaxation (AOR) method for solving linear equations systems. The optimal value of the acceleration parameter is determined, using the maximal reduction method of the residual vector’s length, or equivalently an orthogonality condition. Rather than the constant value, the MAOR method endows a step-by-step varying acceleration parameter to possess the property of absolute convergence and the orthogonality of consecutive residual vector. In SOR, the relaxation parameter is also optimized by using the orthogonality condition. Numerical examples ensure that the MSOR and MAOR iterative schemes converge faster than the original SOR and AOR iterative schemes. They are easily implemented with low computational cost, and without needing of a detailed spectral analysis to determine the optimal values of parameters has a great advantage.

Influence of surface texture on pocket pairs lubrication performance of cylindrical roller bearings

PurposeThis paper aims to reveal the influence of the grooved texture parameters on the lubrication performance of circular pocket-roller pairs in cylindrical roller bearings.Design/methodology/approachIn this paper, the thermal elastohydrodynamic lubrication mathematical model of the grooved texture circular pocket-roller pair was established, the finite difference method and successive over-relaxation method were used to solve the model, the influence of texture quantity, texture depth and texture area ratio on circumferential bearing capacity, friction coefficient, maximum temperature rise, stiffness and damping of the circular pocket-roller pairs were analyzed.FindingsThe results show that texture quantity, texture depth and texture area ratio significantly influence the static and dynamic characteristics of circular pocket-roller pairs. The suitable surface groove texture parameters can dramatically improve the circumferential bearing capacity, reduce the friction coefficient, inhibit the maximum temperature rise and increase the stiffness and damping of the circular pocket-roller pairs.Originality/valueThe research in this paper can provide a theoretical basis for the optimization design of pockets in cylindrical roller bearings to reduce friction and vibration.

Solution of the plane thermoelasticity problems based on a new set of equations

The interaction of thermal and mechanical fields in modern technologies for the synthesis and processing of materials attracts more and more attention. This is due to the need to predict the behavior and properties of new materials and products depending on the conditions of their creation. In the present work, attention is focused on the comparison of the solutions to the thermoelasticity plane problems with and without accounting for the temperature dependence of the material properties. It is assumed that an external heat source traveling over the surface with a specific velocity and following a given trajectory is responsible for a significant temperature increase. It may correspond to a laser source, for example. The new way of problem formulation in stresses is used. The temperature dependence of the characteristics leads to nonlinear thermoelasticity equations when the equilibrium equations, the requirement of strain compatibility, and the Duhamel-Neuman relation for the plane stress state case are combined. To determine the temperature field, a nonlinear thermal conductivity equation is used. To solve the problem numerically, the use of an implicit finite-difference scheme and coordinate splitting for the thermal conductivity equation and the finite-difference method of successive over-relaxation with Chebyshev acceleration for the equations for stresses was made. As a result, the difference in the solution of linear and nonlinear problems is demonstrated for the considered cases. Using the example of a mixture of titanium and aluminum powders, numerical modeling demonstrated that some stress components can reach values up to 3.5 times higher than those obtained in the case of temperature-independent material properties when there is a 500-degree local temperature increase.

Generalized and novel iterative scheme for best approximate solution of large and sparse augmented linear systems

This article presents a generalized novel iterative technique for obtaining the approximate solution of systems of linear equations(ABBT0)(xy)=(pq). Parameters involved in the scheme play a vital role to obtain the rapid and best approximate solution. The technique is applied to some important problems and the results are presented in detail. The article compares the effectiveness of the new presented iterative scheme with modified SOR-like method, generalized SOR and SOR-like method for solving large systems of linear equations. The convergence criteria for the iterative scheme are also analysed and it can be observed in the article that the theoretical derivation is supported by numerical calculations and graphical exhibition. Overall, the article provides a new and effective method with a detailed analysis of its performance and convergence criteria for solving systems of linear equations of the form presented. The newly derived technique has potential applications in various fields of science and engineering where systems of linear equations arise and need to be solved.

A novel iterative model updating for jointed structures using nonlinear FRFs

A novel iterative model updating method is developed to identify the local nonlinear joint properties using the frequency response functions (FRFs) of assembled structures. In this paper, the least-squares fitting method was used to transform the sensitivity matrix into a square iteration matrix to match the dimensions of the objective functions and nonlinear joint model parameters. Leveraging the Newton’s iteration, the adaptive successive over-relaxation (A-SOR) was used to ensure iteration convergence while the multi-scale parameter adjusting (MPA) strategy was developed to degrade the ill-condition of the iteration matrix. Two updating examples of phenomenological equivalence models were applied to demonstrate the effectiveness of the proposed method. The nonlinear FRFs of a lap-type bolted joint beam system with Iwan model were simulated as the objective functions to identify the local nonlinear joint properties, as well as experimental investigations of a metal rubber isolation system with a high-order polynomial model. The proposed method was validated by the good agreement of the comparison results, and it indicated a better model updating performance with a much smaller condition number of the iteration matrix.

Position-Based Nonlinear Gauss-Seidel for Quasistatic Hyperelasticity

Position based dynamics [Müller et al. 2007] is a powerful technique for simulating a variety of materials. Its primary strength is its robustness when run with limited computational budget. Even though PBD is based on the projection of static constraints, it does not work well for quasistatic problems. This is particularly relevant since the efficient creation of large data sets of plausible, but not necessarily accurate elastic equilibria is of increasing importance with the emergence of quasistatic neural networks [Bailey et al. 2018; Chentanez et al. 2020; Jin et al. 2022; Luo et al. 2020]. Recent work [Macklin et al. 2016] has shown that PBD can be related to the Gauss-Seidel approximation of a Lagrange multiplier formulation of backward Euler time stepping, where each constraint is solved/projected independently of the others in an iterative fashion. We show that a position-based, rather than constraint-based nonlinear Gauss-Seidel approach resolves a number of issues with PBD, particularly in the quasistatic setting. Our approach retains the essential PBD feature of stable behavior with constrained computational budgets, but also allows for convergent behavior with expanded budgets. We demonstrate the efficacy of our method on a variety of representative hyperelastic problems and show that both successive over relaxation (SOR), Chebyshev and multiresolution-based acceleration can be easily applied.

Enhance Stability of Successive Over-Relaxation Method and Orthogonalized Symmetry Successive Over-Relaxation in a Larger Range of Relaxation Parameter

Enhance Stability of Successive Over-Relaxation Method and Orthogonalized Symmetry Successive Over-Relaxation in a Larger Range of Relaxation Parameter

High-precision meshfree methods for solving 3D radiation diffusion problem in irregular spaces

Radiation diffusion is a significant phenomenon in inertial confinement fusion, geology and so on. Solving nonlinear radiation diffusion equation in irregular spaces is challenging. In this paper, the direct linearisation algorithm and the successive permutation iteration algorithm are constructed for solving 3D radiation diffusion equation, which are full-implicit discretization on time. The high-precision meshfree methods are provided by the Kansa's non-symmetric collocation method with the compactly supported radial basis function. Ultimately, the accuracy and efficiency of the presented algorithms are verified through the comparison of the fixed point, Newton and SOR methods in 3D numerical experiments. These novel meshfree methods not only avoid the complexity of grid generation, but also provide high-precision numerical solution for nonlinear radiation diffusion equation.

Numerical Optimization Analysis of Floating Ring Seal Performance Based on Surface Texture

Much research and practical experience have shown that the utilization of textures has an enhancing effect on the performance of dynamic seals and the dynamic pressure lubrication of gas bearings. In order to optimize the performance of floating ring seals, this study systematically analyzes the effects of different texture shapes and their parameters. The Reynolds equation of the gas is solved by the successive over-relaxation (SOR) iteration method. The pressure and thickness distributions of the seal gas film are solved to derive the floating force, end leakage, friction, and the ratio of buoyancy to leakage within the seal. The effects of various texture shapes, including square, 2:1 rectangle, triangle, hexagon, and circle, as well as their parameters, such as texture depth, angle, and area share, on the sealing performance are discussed. Results show that the texture can increase the air film buoyancy and reduce friction, but it also increases the leakage by a small amount. Square textures and rectangular textures are relatively effective. The deeper the depth of the texture within a certain range, the better the overall performance of the floating ring seal. As the texture area percentage increases, leakage tends to increase and friction tends to decrease. A fractal roughness model is developed, the effect of surface roughness on sealing performance is briefly discussed, and finally the effect of surface texture with roughness is analyzed. Some texture parameters that can significantly optimize the sealing performance are obtained. Rectangular textures with certain parameters enhance the buoyancy of the air film by 81.2%, which is the most significant enhancement effect. This rectangular texture reduces friction by 25.8% but increases leakage by 79.5%. The triangular textures increase buoyancy by 28.02% and leakage increases by only 10.08% when the rotation speed is 15,000 r/min. The results show that texture with appropriate roughness significantly optimizes the performance of the floating ring seal.

Computation of Green’s Function in a Strongly Heterogeneous Medium Using the Lippmann–Schwinger Equation: A Generalized Successive Over-Relaxion plus Preconditioning Scheme

The computation of Green’s function is a basic and time-consuming task in realizing seismic imaging using integral operators because the function is the kernel of the integral operators and because every image point functions as the source point of Green’s function. If the perturbation theory is used, the problem of the computation of Green’s function can be transformed into one of solving the Lippmann–Schwinger (L–S) equation. However, if the velocity model under consideration has large scale and strong heterogeneity, solving the L–S equation may become difficult because only numerical or successive approximate (iterative) methods can be used in this case. In the literature, one of these methods is the generalized successive over-relaxation (GSOR) iterative method, which can effectively solve the L–S equation and obtain the desired convergent iterative series. However, the GSOR iterative method may encounter slow convergence when calculating the high-frequency Green’s function. In this paper, we propose a new scheme that utilizes the GSOR iterative with a precondition to solve the complex wavenumber L–S equation in a slightly attenuated medium. The complex wavenumber with imaginary components localizes the energy of the background Green’s function and reduces its singularity by enabling exponential decay. Introduction of the preconditioning operator can further improve the convergence speed of the GSOR iterative series. Then, we provide a preconditioned generalized successive over-relaxation (pre-GSOR) iterative format. Our numerical results show that if an appropriate damping factor and a proper preconditioning operator are selected, the method presented here outperforms the GSOR iterative for the real wavenumber L–S equation in terms of the convergence speed, accuracy, and adaptation to high frequencies.

Structured Linear Systems and Their Iterative Solutions Through Fuzzy Poisson's Equation

A realistic version of the modified successive overrelaxation (MSOR) with four relaxation parameters is introduced (MMSOR) with application to a representative matrix partition. The one-dimensional Poisson’s equation with fuzzy boundary values is the standard source problem for our treatment (it is sufficient to introduce all the concepts in a simple form). The finite difference method with RedBlack (RB)-Labelling of the grid points is used to introduce a fuzzy algebraic system with characterized fuzzy weak solutions (corresponding to black grid points). We introduce the algorithmic structure and the implementation of MMSOR on the de-fuzzified linear system. The choice of relaxation parameters is based on the minimum Spectral Radius (SR) of the iteration matrices. A comparison with SOR (one relaxation parameter) and MSOR (two relaxation parameters) is considered, and a relation between the three methods is revealed. Assuming the same accuracy, the experimental results showed that the MMSOR runs faster than the SOR and the MSOR methods.

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.

Optimal Combination of the Splitting–Linearizing Method to SSOR and SAOR for Solving the System of Nonlinear Equations

The symmetric successive overrelaxation (SSOR) and symmetric accelerated overrelaxation (SAOR) are conventional iterative methods for solving linear equations. In this paper, novel approaches are presented by combining a splitting–linearizing method with SSOR and SAOR for solving a system of nonlinear equations. The nonlinear terms are decomposed at two sides through a splitting parameter, which are linearized around the values at the previous step, obtaining a linear equation system at each iteration step. The optimal values of parameters are determined to minimize the reciprocal of the maximal projection, which are sought in preferred ranges using the golden section search algorithm. Numerical tests assess the performance of the developed methods, namely, the optimal splitting symmetric successive over-relaxation (OSSSOR), and the optimal splitting symmetric accelerated over-relaxation (OSSAOR). The chief advantages of the proposed methods are that they do not need to compute the inverse matrix at each iteration step, and the computed orders of convergence by OSSSOR and OSSAOR are between 1.5 and 5.61; they, without needing the inner iterations loop, converge very fast with saving CPU time to find the true solution with a high accuracy.

Design of herringbone grooved thrust bearing for locomotive turbocharger rotor

The herringbone texture exhibited excellent tribological performance to minimize friction and wear. However, the application of this texture in the development of grooved thrust bearings is limited. Therefore, in this study, an attempt was made to design an oil-lubricated herringbone grooved thrust bearing for high-speed locomotive turbochargers. The designed bearing accommodates the axial load generated due to the pressure difference between the turbine and compressor wheel. The bearing design starts with applying Newton’s second law to predict the thrust load acting on the locomotive turbocharger rotor. The thrust load is calculated analytically and is found to be 4.54 kN for a design rotor speed of 1,00,000 rpm. Further, the herringbone grooved thrust bearing has been modeled numerically using non-linear Reynolds equation. The modified Reynolds equation is discretized using the finite volume method (FVM) and solved by successive over-relaxation (SOR) methodology to determine the static characteristics over the bearing surface. The developed HGTB is found to have a suitable load-carrying capacity of 4.6 kN, frictional torque of 0.25 N.m, and power loss of 2.98 kW. Further, a parametric analysis has been carried out to study the influence of design parameters such as the number of grooves, helix angle, angular groove width, groove depth, and speed on load-carrying capacity, frictional torque, and power loss.

On SMSNSSOR iteration method for solving complex symmetric linear systems

In this paper, we extend SMSNS [Pourbagher M, Salkuyeh DK. On the solution of a class of complex symmetric linear systems. Appl Math Lett. 2018;76:14–20.] iteration method for solving a class of complex symmetric system of linear equations. We propose a successive-overrelaxation (SOR) acceleration scheme for SMSNS (SMSNSSOR), discuss the convergence conditions of it and give the optimal parameters which make the fast convergence. Numerical results demonstrate that SMSNSSOR iteration method is feasible and effective for solving complex symmetric systems, and performs better than some other usually used iteration methods.

CFD analysis of paraffin-based hybrid (Co–Au) and trihybrid (Co–Au–ZrO2) nanofluid flow through a porous medium

Abstract Ternary hybrid nanofluids possess improved thermal characteristics, enhanced stability, better physical strength, and multi-functionality as compared to hybrid or usual nanofluids. The aim of the ongoing study is to explore the novel thermal attributes of hybrid and trihybrid nanofluids through a porous medium. Whereas the nano-composition of cobalt (Co), gold (Au), and zirconium oxide (ZrO2) make amalgamation in the paraffin (Pfin) which is a base fluid. This nano-composition of the proposed nanoparticles, specifically, subject to the base fluid Pfin has not been interpreted before. The analysis not only covers the features of trihybrid nanofluids (Co–Au–ZrO2–Pfin) but it also describes the characteristics of hybrid (Co–Au–Pfin) as well as pure nanofluids (Co–Pfin). An efficient numerical algorithm is developed for which the numerical simulations are carried out. The approximations are performed in MATLAB software using “Successive under Relaxation (SUR)” technique. A comparison, under certain limiting conditions, with the established results appraises the efficiency of the numerical code. The outcomes evidently designate that temperature raises with the change in thermal radiation and volume fraction of gold and zirconium oxide in either case of pure, hybrid, or ternary nanofluids. The concentration ϕ 3 {\phi }_{3} of ZrO2 has a significant impact on Nusselt number rather than the concentration ϕ 1 {\phi }_{1} of cobalt and ϕ 2 {\phi }_{2} of gold. It has been comparatively noticed that the ternary nanofluids (Co–Au–ZrO2–Pfin) portray embellished and improvised thermal characteristics as compared to the other two cases.