Successive Overrelaxation Iteration Research Articles

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.

Comprehensive Analysis of Internal Flow Dynamics in Pipeline Systems

This research delves into the intricate flow dynamics within a gas pipeline, leveraging the small perturbation equations to elucidate the flow scenario. Utilizing the Python programming language, the study derives the numerical solutions for the governing equations of the pipeline's flow field. This is achieved through the successive over-relaxation (SOR) iterative method in tandem with the Neumann boundary conditions. The flow dynamics are subsequently visualized and analyzed using streamlined diagrams and Mach number cloud representations, facilitating the identification of stress concentration points on the pipeline's inner wall. Such insights significantly affect the pipeline's optimal design and strategic reinforcement. Beyond its immediate application, the methodology presented herein offers a robust framework for addressing diverse fluid mechanics simulation challenges. Its versatility extends to realms such as automotive contour design, pipeline leakage simulations, and engine compressor configurations. When confronted with the need to dissect fluid flow and stress dynamics, professionals can harness the Python-based SOR iteration code to derive governing fluid equations, enabling a graphical interpretation through streamlined and Mach number cloud diagrams. This study, thus, underscores a pivotal toolset for fluid mechanics challenges across various engineering domains.

New structure-preserving algorithms of Gauss-Seidel and successive over-relaxation iteration methods for quaternion linear systems

New structure-preserving algorithms of Gauss-Seidel and successive over-relaxation iteration methods for quaternion linear systems

Thermo-Hydrodynamic Lubrication Analysis of Slipper Pair Considering Wear Profile

The profile of sealing land is a sensitive factor affecting the thermo-hydrodynamic lubrication characteristics of the slipper pair. In this paper, the non-uniform wear of the running surface under the slipper was presented and defined as the boundary condition. Based on the finite volume method and the successive over-relaxation iteration method, a discrete numerical model coupled with the temperature, pressure, and thickness of the oil film was constructed. The Newton and sequential circulation methods were used to solve the coupling equations. The influence of the wear profile on the film thickness, sliding attitude, and leakage were discussed. The analyzed results show that the control of the wear on the outer side of sealing land and the contour vertex position, and the avoidance of the wear on the inner side of sealing land could improve the thermo-hydrodynamic lubrication performance of the slipper pair.

A Rotated Similarity Reduction Approach with Half-Sweep Successive Over-Relaxation Iteration for Solving Two-Dimensional Unsteady Convection-Diffusion Problems

A Rotated Similarity Reduction Approach with Half-Sweep Successive Over-Relaxation Iteration for Solving Two-Dimensional Unsteady Convection-Diffusion Problems

Effect of the phase-transition fluid reaction heat on wellbore temperature in self-propping phase-transition fracturing technology

The key to the success of self-propping phase-transition fracturing (SPF) technology using two immiscible fluids to generate proppants in-situ in a reservoir lies in accurate calculations of temperature distribution. As the reaction heat of phase-transition fluid (PF) significantly affects wellbore temperature, reaction kinetic parameters were fitted by experimental data based on the Arrhenius equation, and the transient temperature model considering the reaction heat is established based on the first law of thermodynamics. This model is discretized by the finite difference method and solved by the successive over-relaxation iteration method. The results show that the reaction heat effect on wellbore temperature cannot be ignored. A temperature value and a phase transition time at the well bottom are the largest in the whole wellbore, so the phase transition ratio at the well bottom is the largest. Moreover, since the PF with incomplete phase transition in a wellbore is easier to enter fractures and prop fracture fronts, it is recommended to inject a pre-pad fracturing fluid before injecting PF to reduce wellbore temperature and prevent premature phase transition in the wellbore. These findings can help reveal the action mechanisms of different injection methods and parameters in a heat transfer process, which is of great significance for the theoretical research and field implementation of SPF technology.

Progressive Iterative Approximation of Non-Uniform Cubic B-Spline Curves and Surfaces via Successive Over-Relaxation Iteration

Geometric iteration (GI) is one of the most efficient curve- or surface-fitting techniques in recent years, which is famous for its remarkable geometric significance. In essence, GI can be thought of as the sum of iterative methods for solving systems of linear equations, such as progressive iterative approximation (PIA) which relies on the theory of Richardson iteration. Thus, when the curve- or surface-fitting error is at a desired level, we want to have as few iterations as possible to improve efficiency when dealing with large data sets. Based on the idea of successive over-relaxation (SOR) iteration, we formulate a faster PIA curve and surface interpolation scheme using classical non-uniform cubic B-splines, named SOR-PIA. The genetic algorithm is utilized to estimate the best approximate relaxation factor of SOR-PIA. Similar to standard PIA, SOR-PIA can also be regarded as a process in which the control points move in one direction, but it can greatly reduce the number of iterations in the iterative process with the same fitting accuracy. By comparing with the standard PIA and WPIA algorithms, the effectiveness of the SOR-PIA iterative interpolation algorithm can be verified.

A numerical technique to solve a problem of the fluid motion in a straight plane rigid duct with two axisymmetric rectangular constrictions. An alternative approach

A numerical technique is devised to solve a problem of the steady laminar fluid motion in a straight plane hard-walled duct with two local axisymmetric rectangular constrictions. It uses the stream function, the vorticity and the pressure as the basic variables, has the second order of accuracy in the spatial and the first order of accuracy in the temporal coordinates, provides high stability of a solution, and needs significantly less computational time to obtain a result compared to appropriate techniques available in a scientific literature. The technique consists in: a) introducing the stream function and the vorticity, and subsequent transiting from the non-dimensional governing relations for the fluid velocity and the pressure to the corresponding non-dimensional relations for the stream function, the vorticity and the pressure; b) deriving their discrete counterparts in the nodes of the chosen space-time computational grid; c) integrating the systems of linear algebraic equations obtained after making the discretization. The discretization is based on applying appropriate differencing schemes to the terms of the equations for the basic variables. These are the two-point temporal onward difference for the unsteady term of the vorticity equation, as well as the two-point backward differences (for its convective term) and the five-point approximations (for its diffusive term and for the Poisson’s equations for the stream function and the pressure) in the axial and cross-flow coordinates. As for the velocity components, the appropriate central differences are applied to discretize their expressions. The above-mentioned systems of linear algebraic equations for the stream function and the pressure are integrated by the iterative successive over-relaxation method. The algebraic relation for the vorticity does not need application of any method to be solved, because it is a computational scheme to find this quantity based on the known magnitudes computed at the previous instant of time.

Numerical Modeling of Fracture Height Propagation in Multilayer Formations Considering the Plastic Zone and Induced Stress.

Hydraulic fracturing and acid fracturing are very effective stimulation technologies and are widely used in unconventional reservoir development. Fracture height, as an essential parameter to describe the geometric size of a fracture, is not only the input parameter of two-dimensional fracturing models but also the output parameter of three-dimensional fracturing models. Accurate prediction of fracture height growth can effectively avoid some risks. For example, petroleum reservoirs produce a large amount of formation water because wrong fracture height prediction leads to the connection between the oil or gas reservoir and the water layer. Although some fracture height prediction models were developed, few models considered the effects of the plastic zone, induced stress, and heterogeneous multilayer formation and its interaction. Therefore, considering the influence of many factors, an improved fracture-equilibrium-height model was developed in this study. The successive over-relaxation iteration method and the displacement discontinuity method were used to solve the model. We investigated the effects of the geological and engineering factors on fracture height growth by using the model, and some important conclusions were obtained. The higher the fracture height, the larger the plastic zone size, and the more obvious its influence on fracture height propagation. High overlying or underlying in situ stress and fracture toughness and low fluid density played a positive role in limiting the growth of the fracture height. Induced stress caused by fracture 1 could not only inhibit the height growth of fracture 2 but also promote its growth. The model established in this paper could be coupled to a fracturing simulator to provide a more reliable fracture height prediction.

Modulus-based Successive Overrelaxation Iteration Method for Pricing American Options with the Two-asset Black–Scholes and Heston's Models Based on Finite Volume Discretization

In this paper we introduce a new numerical method for the linear complementarity problems (LCPs) arising from two-asset Black–Scholes and Heston's stochastic volatility American options pricing. Based on barycenter dual mesh, a class of finite volume method (FVM) is proposed for the spatial discretization, coupled with the backward Euler and Crank–Nicolson schemes are employed for time stepping of the partial differential equations (PDEs). Then, for the resulting time-dependent LCPs are solved by using an efficient modulus-based successive overrelaxation (MSOR) iteration method. Numerical experiments are carried out to verify the efficiency and usefulness of the proposed method.

MN-PGSOR Method for Solving Nonlinear Systems with Block Two-by-Two Complex Symmetric Jacobian Matrices

For solving the large sparse linear systems with 2 × 2 block structure, the generalized successive overrelaxation (GSOR) iteration method is an efficient iteration method. Based on the GSOR method, the PGSOR method introduces a preconditioned matrix with a new parameter for the coefficient matrix which can enhance the efficiency. To solve the nonlinear systems in which the Jacobian matrices are complex and symmetric with the block two-by-two form, we try to use the PGSOR method as an inner iteration, with the help of the modified Newton method as an efficient outer iteration method. This new method is called the modified Newton-PGSOR (MN-PGSOR) method. Local convergence properties of the MN-PGSOR are analyzed under the Hölder condition. Finally, we give the comparison of our new method with some previous methods in the numerical results. The MN-PGSOR method is superior in both iteration steps and computing time.

Numerical Analysis Based on Oil Film Characteristics of Sliding Bearing

Taking the finite-length sliding bearing as the research object, the Reynolds equation is solved by the finite difference method and the successive over-relaxation iteration method to obtain the oil film pressure. It is concluded that the dimensionless oil film characteristics are related to bearing eccentricity and length diameter ratio. In order to study the oil film characteristics of sliding bearing, the bearing eccentricity and length diameter ratio are taken as the research objects, and the oil film thickness, load component, attitude angle and end discharge flow are solved by numerical analysis. It is concluded that with the increase of eccentricity, the bearing capacity of oil film increases, the attitude angle decreases and the end discharge flow increases with the case of a certain length diameter ratio.

On the modulus-based successive overrelaxation iteration method for horizontal linear complementarity problems arising from hydrodynamic lubrication

In this paper, for the horizontal linear complementarity problems arising from hydrodynamic lubrication, the convergence of modulus-based successive overrelaxation iteration method is presented, which extends the existing results given in the recent work of Mezzadri F. and Galligani E. published on Math. Comput. Simulat. Numerical examples demonstrate the significance of the proposed theorems.

Modified Newton-GSOR method for solving complex nonlinear systems with symmetric Jacobian matrices

In this paper, a new method for solving complex nonlinear systems with symmetric Jacobian matrices is proposed—modified Newton generalized successive overrelaxation (MN-GSOR) iteration method. The new MN-GSOR method helps us to solve nonlinear systems using the GSOR method as an inner iterative approximation for solving the Newton equations. Next, we use the Holder continuous condition instead of the Lipschitz assumption to analyze and prove the convergence properties of the MN-GSOR method. Numerical experiments are conducted to show the superiority and effectiveness of MN-GSOR method, and comparisons are made to see the advantages over some other recently proposed methods. Furthermore, when the imaginary part of the Jacobian matrix is symmetric but non-definite, some recently proposed methods are not applicable, but the MN-GSOR method still works well.

On the GSOR iteration method for image restoration

<p style='text-indent:20px;'>In this study, we present a generalization of the successive overrelaxation (GSOR) iteration method to find the solution of the image restoration problem. Moreover, an improved version of the GSOR (IGSOR) method is also given to solve the proposed problem. Convergence of the GSOR and IGSOR methods are investigated. Three numerical examples are given to illustrate the effectiveness and accuracy of the methods.</p>

Lubrication Characteristics of Water-Lubricated Rubber Bearings With Partial Wear

Abstract Water-lubricated rubber bearings are widely used in the propulsion shafting of military craft and ships. These bearings may wear down after a period of service, and consequently, their lubrication characteristics will change, affecting the operation of the shaft. Hydrodynamic lubrication characteristics of water-lubricated rubber bearings with partial wear at the bottom are studied by finite difference method. The steady-state characteristics of water-lubricated rubber bearings with wear and elastic deformation of rubber liner considered are solved with successive over-relaxation iteration method, and the dynamic characteristics are further calculated with finite perturbation method. The results show that water film thickness and distribution of water film pressure is significantly changed by wear. With the same eccentricity ratio, the maximum water film pressure, load capacity, attitude angle, and friction force are reduced by wear, but the friction coefficient is increased by wear. Under the same load, the minimum water film thickness and the maximum water film pressure are slightly affected by wear, the eccentricity ratio increases with the increase of wear, and the attitude angle and the friction coefficient decrease with the increase of wear. For large eccentricity ratios, the direct stiffness coefficients and direct damping coefficients are decreased by wear. The maximum allowable wear depth is approximately 1/6 of the bearing clearance.

Grünwald Implicit Solution for Solving One-Dimensional Time-Fractional Parabolic Equations Using SOR Iteration

The aim of this paper is to examine the effectiveness of Successive Over-Relaxation (SOR) iterative method for solving one-dimensional time-fractional parabolic equations. The Grünwald fractional derivative operator and implicit finite difference scheme have been used to discretize the proposed linear time-fractional equations to construct system of Grünwald implicit approximation equation. The basic formulation and application of the SOR iterative method are also presented. To investigate the effectiveness of the proposed iterative method, numerical experiments and comparison are made based on the iteration numbers, time execution, and maximum absolute error. Based on numerical results, the accuracy of Grünwald implicit solution obtained by proposed iterative method is in excellent agreement, and it can be concluded that the proposed iterative method requires less number of iterations and execution time as compared to the Gauss-Seidel (GS) iterative method.

A preconditioned SSOR iteration method for solving complex symmetric system of linear equations

We present a preconditioned version of the symmetric successive overrelaxation (SSOR) iteration method for a class of complex symmetric linear systems. The convergence results of the proposed method are established and conditions under which the spectral radius of the iteration matrix of the method is smaller than that of the SSOR method are analyzed. Numerical experiments illustrate the theoretical results and depict the efficiency of the new iteration method.

Numerical modelling of natural convection in a square cavity: effect of nanofluid volume fraction and inclination

In this paper, the phenomenon of natural convection in a square cavity filled with copper (Cu) -based nanofluid (Nf) has been studied. The influences of the volume fraction (ф) on the convective flow, as well as the effect of the inclination (γ) of the cavity on thermal performances were examined. The dimensionless governing equations formulated using stream function, vorticity, and temperatures were solved by finite difference methods, where the Successive Over-Relaxation (SOR) iteration and the upwind scheme were adopted. The developed code served as a verification of results where it showed a good agreement with previous works. The calculation code was then used to visualize the influence of the inclination on the thermal enhancement and showed that the transfer becomes more efficient with an inclination of π/6.

Accurate and efficient numerical solutions for elliptic obstacle problems

Elliptic obstacle problems are formulated to find either superharmonic solutions or minimal surfaces that lie on or over the obstacles, by incorporating inequality constraints. In order to solve such problems effectively using finite difference (FD) methods, the article investigates simple iterative algorithms based on the successive over-relaxation (SOR) method. It introduces subgrid FD methods to reduce the accuracy deterioration occurring near the free boundary when the mesh grid does not match with the free boundary. For nonlinear obstacle problems, a method of gradient-weighting is introduced to solve the problem more conveniently and efficiently. The iterative algorithm is analyzed for convergence for both linear and nonlinear obstacle problems. An effective strategy is also suggested to find the optimal relaxation parameter. It has been numerically verified that the resulting obstacle SOR iteration with the optimal parameter converges about one order faster than state-of-the-art methods and the subgrid FD methods reduce numerical errors by one order of magnitude, for most cases. Various numerical examples are given to verify the claim.