Overrelaxation Method Research Articles

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.

Optimized Accelerated Over-Relaxation Method for Robust Signal Detection: A Metaheuristic Approach

Massive MIMO technology is recognized as a key enabler for beyond 5G (B5G) and next-generation wireless networks. By utilizing large-scale antenna arrays at the base station (BS), it significantly improves both system capacity and energy efficiency. Despite these advantages, the deployment of a high number of antennas at the BS presents considerable challenges, particularly in the design of signal detectors that can operate with low computational complexity. While the minimum mean square error (MMSE) detector offers optimal performance in these large-scale systems, it suffers from the computational burden that makes its practical implementation challenging. To mitigate this, various iterative methods and their improved versions have been introduced. However, these iterative methods often converge slowly and are less accurate. To address these challenges, this study introduces an improved variant of traditional accelerated over-relaxation (AOR), called optimized AOR (OAOR). AOR is an over-relaxation method, and its performance is highly dependent on its relaxation parameters. To find the optimal parameters, we have developed an innovative approach that integrates a nature-inspired meta-heuristic algorithm known as Particle Swarm Optimization (PSO). Specifically, we introduce a novel variant of PSO that improves upon basic PSO by enhancing the cognitive coefficients to optimize the relaxation parameters for OAOR. These key modifications to the standard PSO improve its ability to explore various solutions efficiently and help to find the optimal parameters more quickly for signal detection. It facilitates the OAOR with faster convergence towards the optimal solution by reducing the error rate, resulting in high detection accuracy and simultaneously decreasing computational complexity from O(K3) to O(K2) making it suitable for modern wireless communication systems. We conduct extensive simulations across various configurations of massive MIMO systems. The results indicate that our proposed method achieves better performance compared to existing techniques. This improvement is particularly evident in terms of both computational complexity and error rate.

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.

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

Analysis of turbulent cavitation effects on water-lubricated bearing in single screw compressors

PurposeTo guide the stable radius clearance choice of water-lubricated bearings for single screw compressors, this paper aims to analyze the effects of turbulence and cavitation on bearing performance under two conditions of specified external load and radius clearance.Design/methodology/approachA modified Reynolds equation considering turbulence and cavitation is adopted, based on the Jakobsson–Floberg–Olsson boundary condition, Ng–Pan model and turbulent factors. The equation is solved using the finite difference method and successive over-relaxation method to investigate the bearing performance.FindingsThe turbulent effect can increase the hydrodynamic pressure and cavitation. In addition, the turbulent effect can lead to an increase in the equilibrium radius clearance. The turbulent region exhibits a higher load capacity and cavitation rate. However, the increased cavitation negatively impacts the frictional coefficient and end flow rate. The impact of turbulence increases as the radius clearance decreases. As the rotating speed increases, the turbulence effect has a greater impact on the bearing characteristics.Originality/valueThe research can provide theoretical support for the design of water-lubricated journal bearings used in high-speed water-lubricated single screw compressors.Peer reviewThe peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-01-2024-0029/

A comparative study on the acceleration techniques for solving finite difference discretization poisson’s equation in the PIC/MCC Method

A wide variety of plasma phenomena have been investigated during the past decades using the particle-in-cell/Monte Carlo collisions (PIC/MCC) method. As an important component of the PIC/MCC method, solving Poisson’s equation is crucial for the accuracy and efficiency of calculations. Different acceleration techniques for solving finite difference discretization Poisson’s equation are investigated and compared, including direct method, iterative method, multigrid (MG) method, parallel computing and inherited initial value. The charge density distribution with a known analytical solution is used to validate the algorithm and code. The optimal relaxation factor for the successive over-relaxation (SOR) method in 2D Poisson’s equation with unequal grid node numbers in different dimensions is derived, which is only related to the dimension with the largest grid number. Although there will be a ‘more optimal’ relaxation factor deviated from in some simulation cases, selecting the optimal relaxation factor derived always leads to a not slow solving speed. However, when SOR is used in MG for smoothing, the optimal relaxation factor will shift to 0.5–1.2 from the theoretical optimal value derived with the increase of MG levels. By comparing the convergence order under different relaxation factors and MG levels, the suitable MG level is proposed as log2[min(N x , N y )]−2. Combining the optimal SOR relaxation factor, MG, parallel computing and inherited initial values, the computational cost may decrease by 5 orders of magnitude than that by the simple Gaussian elimination (GE). Based on the optimal acceleration techniques mentioned above, a benchmark simulation case electron cyclotron drift instability (ECDI) in magnetized plasmas was run to further validate the developed PIC/MCC code. The distributions of electric field in the x-direction, electron density and electron temperature are all consistent with the literatures. This paper provides a reference for the acceleration strategy selection for solving Poisson’s equation quickly in plasma simulations.

NUMERICAL STUDY FOR A THREE DIMENSIONAL LAMINAR NATURAL CONVECTION HEAT TRANSFER FROM AN ISOTHERMAL HEATED HORIZONTAL AND INCLINED SQUARE PLATE AND WITH A CIRCULAR HOLE

A theoretical study for a three-dimensional natural convection heat transfer from an isothermal horizontal , vertical and inclined heated square flat plates (with and without circular hole) has been done in the present work. The study involved the numerical solution of the transient Navier-Stokes and energy equations by using finite deference method (F.D.M.). The complete Navier-Stokes equation are transformed and expressed in terms of vorticity-vector potential. The Energy and Vorticity equations were solved by using an Alternating Direction Implicit (ADI) method because they are transient equations of parabolic portion, and the Vector potential is solved by using an equations Successive Over-Relaxation (S.O.R) method because it is from elliptic portion. The numerical solution is capable of calculating the Vector potential, three components of Vorticity and temperature field of the calculation domain. The numerical results were obtained in rang of Grashof number (103≤Gr≤5x104 ) with Prandtl number of (0.72) for square flat plate and the other consist a circle hole with ratio 0.6 and 0.8 diameter of the hole to main square side length. The numerical results showed that the main process of heat transfer is conduction for Grashof number less than 103 and convection for Grashof number larger than 103 and the results of local Nusselt number show fairly large dependence on inclination angle. For horizontal plate facing upward and downward, average Nusselt number is proportional to one-fifth power of Rayleigh number, and there is a significant difference in heat transfer rates between the upward and downward cases. For horizontal plate with circle hole facing upward for Grashof number 104 , the effect of core portion caused a limited increment in the heat transfer rate, where as for the facing downward case, the effect was larger and the maximum value of heat transfer rates is be for square flat plate with circle hole by ratio 0.6 for all inclination angles. With the increase of Grashof number to 5x104 heat transfer rates decrease except the square horizontal flat plate with circle hole by ratio 0.6 . The average Nusselt number increases with the increase of inclination of plates facing upward to reach to the higher average Nusselt number at vertical position then decrease with increase of inclination of plates. And the maximum value of average Nusselt number is depended on the ratio of diameter of the hole to main square side length, showed that the maximum temperature gradient occurs at the external edge of the horizontal plate (with and without circle hole) facing upward and at the lower external edge in inclined case. The numerical results was made through comparison with a previous numerical and experimental work, the agreement was good.

Triangular intuitionistic fuzzy linear system of equations with applications: an analytical approach

This study extended an existing semi-analytical technique, the Homotopy Perturbation Method, to the Block Homotopy Modified Perturbation Method by solving two n × n crisp triangular intuitionistic fuzzy (TIF) systems of linear equations. In the original system, the coefficient matrix is considered as real crisp, while the unknown variable vector and right hand side vector are regarded as triangular intuitionistic fuzzy numbers. The Block Homotopy Modified Perturbation Method is found to be efficient and practical to solve n × n TIF linear systems as it only requires the non-singularity of the n × n TIF linear system's coefficient matrix, whereas the point Homotopy Perturbation Method and other classical numerical iterative methods typically require non-zero diagonal entries in the coefficient matrix. A set of theorems relevant to this study are presented and demonstrated. We solve an engineering application, i.e. a current flow circuit problem that is represented in terms of a triangular intuitionistic fuzzy environment, using the suggested method. The unknown current is then obtained as a triangle intuitionistic fuzzy number. The proposed semi-analytic method is used to solve some numerical test problems in order to validate their performance and efficiency in comparison to other existing techniques. The numerical results of the example are displayed on graphs with different degrees of uncertainty. The efficiency and accuracy of the proposed method are further demonstrated by comparisons to block Jacobi, Adomain Decomposition method, Successive Over-Relaxation method and the classical Gauss-Seidel numerical method.

Substitutional Based Gauss-Seidel Method For Solving System of Linear Algebraic Equations

In this research paper a new modification of Gauss-Seidel method has been presented for solving the system of linear algebraic equations. The systems of linear algebraic equations have an important role in the field of science and engineering. This modification has been developed by using the procedure of Gauss-Seidel method and the concept of substitution techniques. Developed modification of Gauss-Seidel method is a fast convergent as compared to Gauss Jacobi’s method, Gauss-Seidel method and successive over-relaxation (SOR) method. It works on the diagonally dominant as well as positive definite symmetric systems of linear algebraic equations. Its solution has been compared with the Gauss Jacobi’s method, Gauss-Seidel method and Successive over-Relaxation method by taking different systems of linear algebraic equations and found that, it was reducing to the number of iterations and errors in each problem.

Research on 2D SDE Method of Fluid Field Effect above Airfoil at Transonic Speed

In this research, a Python-centric approach is proposed for simulating fluid dynamics over transonic airfoils. The study employs the finite difference method to solve the small disturbance equation (SDE), incorporating Newman boundary conditions. Iterative computations are adeptly managed using the successive over-relaxation (SOR) method. These methodologies, as validated through rigorous code testing, enable precise calculations of velocity distributions and vivid visualizations of fluid velocity streamlines. For this investigation, the airfoil is conceptualized as a minor obstruction and is abstracted into a trigonometric function. A pivotal aspect of this study is the exploration of the interplay between the Mach number and the 'shark angle'—a critical determinant in aerodynamic performance. Simulation outcomes reveal an inverse relationship between the Mach number and the airfoil's shark angle; a surge in the Mach number corresponds to a reduction in the shark angle. These findings, encapsulated in flow field diagrams, offer invaluable insights for the strategic design and optimization of airfoils, aiming to maximize lift while concurrently curtailing drag, thereby enhancing overall aerodynamic efficiency.

Various Flow in a Slender Tube: Mach Number and Domain Shape Influenced by Boundary Obstacles Via Small Disturbance Equations

With the exponential advancements in computer hardware and software in recent years, computational fluid dynamics (CFD) has emerged as a highly efficient alternative to traditional fluid mechanics studies. Its prowess in accurately predicting and analyzing flow mechanics has led to widespread adoption across diverse engineering disciplines. This research delves into a numerical analysis of flow conditions utilizing the small disturbance equation (SDE). The study bridges the gap between mathematical formulations and computational language, exemplified through Python code, by harnessing the successive over-relaxation (SOR) method, Neumann boundary conditions (BC), and the ghost point technique. The research underscores its predictions' precision in subsonic and supersonic scenarios by meticulously evaluating the resultant errors. This accuracy is further corroborated by examining overarching flow patterns and Mach number distributions. However, the program's outputs indicate a discernible lack of precision in the transonic case, with errors surpassing acceptable thresholds. This discrepancy is hypothesized to stem from potential mischaracterizations of points at the inflow and outflow boundaries. This study, thus, offers invaluable insights while also highlighting areas necessitating further refinement.

Static performance analysis of gas foil thrust bearings under inclined thrust disc considering the slip flow effect

As one of the important components in air compressors for fuel cell vehicles, the static performance of the gas foil thrust bearings greatly affects the operating stability of the rotor system. In actual operation, thrust disc tilt and the slip flow effect often occur, which will change the characteristics of bearings. Therefore, this paper focuses on the static performance of gas foil thrust bearings under tilting conditions with account of the slip flow effect. Firstly, the gas film thickness equation is established when the thrust disc is inclined, and the Reynolds equation is modified by the boundary slip conditions. Secondly, the load capacity and friction torque of bearings are solved by the finite difference method and successive over-relaxation method. Finally, the sensitivity of rigid and elastic bearings to tilt angles is analyzed by comparing the static characteristics under tilting conditions. The results show that the load capacity will be overestimated without considering the slip flow effect, and the tilting of the thrust disc increases both the load capacity and friction torque. Furthermore, compared with rigid bearings, elastic foil bearings have better adaptability to tilting conditions, which is conducive to the stable operation of the rotor system.

Enhancing Autonomous Guided Vehicles with Red-Black TOR Iterative Method

To address an autonomous guided vehicle problem, this article presents extended variants of the established block over-relaxation method known as the Block Modified Two-Parameter Over-relaxation (B-MTOR) method. The main challenge in handling autonomous-driven vehicles is to offer an efficient and reliable path-planning algorithm equipped with collision-free feature. This work intends to solve the path navigation with obstacle avoidance problem explicitly by using a numerical approach, where the mobile robot must project a route to outperform the efficiency of its travel from any initial position to the target location in the designated area. The solution builds on the potential field technique that uses Laplace’s equation to restrict the formation of potential functions across operating mobile robot regions. The existing block over-relaxation method and its variants evaluate the computation by obtaining four Laplacian potentials per computation in groups. These groups can also be viewed as groups of two points and single points if they’re close to the boundary. The proposed B-MTOR technique employs red-black ordering with four different weighted parameters. By carefully choosing the optimal parameter values, the suggested B-MTOR improved the computational execution of the algorithm. In red-black ordering, the computational molecules of red and black nodes are symmetrical. When the computation of red nodes is performed, the updated values of their four neighbouring black nodes are applied, and conversely. The performance of the newly proposed B-MTOR method is compared against the existing methods in terms of computational complexity and execution time. The simulation findings reveal that the red-black variants are superior to their corresponding regular variants, with the B-MTOR approach giving the best performance. The experiment also shows that, by applying a finite difference method, the mobile robot is capable of producing a collision-free path from any start to a given target point. In addition, the findings also verified that numerical techniques could provide an accelerated solution and have generated a smoother path than earlier work on the same issue.

Mathematical model of matter transfer in a helical magnetic field using boundary conditions at infinity

The paper presents a mathematical model of plasma transfer in an open magnetic trap using the condition of zero plasma concentration at infinity. New experimental data obtained at the SMOLA trap at the Budker Institute of Nuclear Physics SB RAS were used. Plasma confinement in the plant is carried out by transmitting a pulse from a magnetic field with helical symmetry to a rotating plasma. The mathematical model is based on a stationary plasma transfer equation in an axially symmetric formulation. The stationary equation of the transfer of matter contains second spatial derivatives. The optimal template for the approximation of the mixed derivative based on the test problem is selected. The numerical implementation of the model by the establishment method and the method of successive over-relaxation is compared.

A new matrix splitting generalized iteration method for linear complementarity problems

The linear complementarity problems (LCPs) can be encountered in various scientific computing, management science, and operations research. In this study, we introduce and analyze a new generalized accelerated overrelaxation (NGAOR) method for solving LCPs, in which one special case reduces to a new generalized successive overrelaxation (NGSOR) method. Moreover, we prove the convergence of the proposed methods when the system matrix is an H-matrix (irreducible or strictly diagonally dominant matrix). Numerical results for several experiments are present to show the effectiveness and efficiency of the proposed methods.AMS classification: 65F10, 90C33

Symmetrized Versions of the Seidel and Successive OverRelaxation Methods for Solving Two-Dimensional Difference Problems of Elliptic Type

Introduction. This article is devoted to the consideration of options for symmetrization of two-layer implicit iterative methods for solving grid equations that arise when approximating boundary value problems for two-dimensional elliptic equations. These equations are included in the formulation of many problems of hydrodynamics, hydrobiology of aquatic systems, etc. Grid equations for these problems are characterized by a large number of unknowns — from 106 to 1010, which leads to poor conditionality of the corresponding system of algebraic equations and, as a consequence, to a significant increase in the number of iterations, necessary to achieve the specified accuracy. The article discusses a method for reducing the number of iterations for relatively simple methods for solving grid equations, based on the procedure of symmetrized traversal of the grid region.Materials and Methods. The methods for solving grid equations discussed in the article are based on the procedure of symmetrized traversal along the rows (or columns) of the grid area.Results. Numerical experiments have been performed for a model problem — the Dirichlet difference problem for the Poisson equation, which demonstrate a reduction in the number of iterations compared to the basic algorithms of these methods.Discussion and Conclusions. This work has practical significance. The developed software allows it to be used to solve specific physical problems, including as an element of a software package.

Accelerated Relaxation Two-Sweep Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

In this paper, by applying the accelerated technique and relaxation two-sweep strategy to the modulus-based matrix splitting iteration method, we establish the accelerated relaxation two-sweep modulus-based matrix splitting method. The proposed method contains the accelerated modulus-based matrix splitting and the generalized accelerated modulus-based matrix splitting methods presented recently as special cases. Compared with some existing methods, the proposed method can use more information during each iteration, which may lead to higher computational efficiency. We investigate the convergence properties of the new method, and prove that it is convergent under certain assumptions. Also, for the accelerated relaxation two-sweep modulus-based accelerated overrelaxation method, we analyze the convergence regions of the parameters in detail. Numerical examples are offered to show that the proposed method is efficient and performs better than the generalized accelerated modulus-based matrix splitting one in actual implementation.

On GSOR, the Generalized Successive Overrelaxation Method for Double Saddle-Point Problems

On GSOR, the Generalized Successive Overrelaxation Method for Double Saddle-Point Problems