Successive Overrelaxation Research Articles

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.

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.

On cost-efficient parallel iterative solvers for 3D frequency-domain seismic multisource viscoelastic anisotropic wave modeling

Solving large sparse linear systems in 3D frequency-domain seismic wave modeling, especially in viscoelastic anisotropic media, poses significant challenges due to the increasing number of discrete moduli and nonzero elements in the linear system matrix. The computational load surpasses that of acoustic or viscoacoustic media, making it even more challenging when dealing with multisource problems. Popular scientific tools for solving a linear system, such as multifrontal massively parallel sparse direct solver (MUMPS), structured matrix package (STRUMPACK), and portable extensible toolkit for scientific computation (PETSc), can be used, but their applicability to our specific problem has not been comprehensively evaluated. Our study aims to tackle the challenges in solving large, sparse, complex-valued symmetric linear systems with multiple right-hand side vectors for 3D frequency-domain seismic wave modeling. We have adopted the preconditioned conjugate gradient iterative algorithms as the foundation for our research, introducing two highly cost-effective parallel iterative solvers: the parallel symmetric successive overrelaxation conjugate gradient (P-SSORCG) and the parallel incomplete Cholesky conjugate gradient (P-ICCG). These novel solvers are subjected to a comprehensive comparative analysis against well-established scientific tools, such as MUMPS, STRUMPACK, and PETSc, in the context of 3D frequency-domain seismic wave modeling. We show their promising performances in a practical 3D SEG/EAGE overthrust model and demonstrate that the grouped P-SSORCG offers an efficient alternative to parallel direct solvers, particularly in situations wherein computational resources are limited.

Novel Discounted Adaptive Critic Control Designs With Accelerated Learning Formulation.

Inspired by the successive relaxation method, a novel discounted iterative adaptive dynamic programming framework is developed, in which the iterative value function sequence possesses an adjustable convergence rate. The different convergence properties of the value function sequence and the stability of the closed-loop systems under the new discounted value iteration (VI) are investigated. Based on the properties of the given VI scheme, an accelerated learning algorithm with convergence guarantee is presented. Moreover, the implementations of the new VI scheme and its accelerated learning design are elaborated, which involve value function approximation and policy improvement. A nonlinear fourth-order ball-and-beam balancing plant is used to verify the performance of the developed approaches. Compared with the traditional VI, the present discounted iterative adaptive critic designs greatly accelerate the convergence rate of the value function and reduce the computational cost simultaneously.

Entropy optimization of lid-driven micropolar hybrid nanofluid flow in a partially porous hexagonal-shaped cavity with relevance to energy efficient storage processes

Hydromagnetically associated heat convection can greatly enhance the performance of high-efficiency thermal appliances and renewable energy sources through an optimized design. This investigation examines the production of thermodynamic irreversibility and heat convection for a double lid-driven flow within a partially porous stratified hexagonal enclosure. The top and bottom-wall are moving in the opposite direction with an equal velocity U0. The top-wall and the bottom-wall are kept at temperature Tc and Th (Th >\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$>$$\\end{document} Tc) while the slanted walls are assumed to be thermally insulated. A constant magnetic field is employed in the horizontal x-direction. The hexagonal cavity was filled with a micropolar hybrid nanofluid Ag-MgO/water. The system of dimensionless equations was solved by the finite difference method (FDM) associated with successive over-relaxation (SOR), successive under-relaxation (SUR), and Gauss–Seidel iteration tactics and required results are computed with problem specific program in MATLAB code. The results indicate that the Ra and the thickness of the porous layer (Xp) significantly influences heat convection and thermal irreversibility processes. The Nuavg and STotal rises 6.299% and 3.373% as ‘ϕhnf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\upphi }_{{\ ext{hnf}}}$$\\end{document}’ enhances from 0 to 4%, respectively. Furthermore, as the values of Ra, Ha, K0, and ϕhnf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\upphi }_{{\ ext{hnf}}}$$\\end{document} increase, Beavg experiences a decline of 53.73%, 11.04%, 38.36%, and 0.09% respectively. Also, movement of wall has a significant impact on heat transfer rates and entropy production. The present study may be extended in numerous areas to mimic the problems like—(1) onset of thermo-mechanical process for solid–fluid interaction in a conduit. (2) Thermos-chemical process with extraction of ions in two-phase fluid for double layer plating on a continuously moving sheet, as region of porous stratum saturated with a class of fluid and region without porous medium occupied with other fluid.

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/

Optimizing heat transfer efficiency with injection in ventilated cavity in the presence of internal thermal effects

Venturing into unexplored territory, the study explores the dynamics of mixed convection flow within a square cavity, influenced by a ventilated channel and an injection port. By conducting numerical simulations, the impact of internal heat generation/absorption and viscous dissipation on heat transfer efficiency is thoroughly examined. Employing a finite difference method alongside an alternate direction implicit scheme and iterative successive under-relaxation (SUR) technique, the study systematically solves the governing equations. The effect of physical parameters, namely the Reynolds number ( 50 ≤ Re ≤ 500 ), Richardson number ( 0.01 ≤ Ri ≤ 10 ), volumetric internal heat generation ( △ * > 0 )/absorption ( △ * < 0 ), Eckert number ( 0 ≤ Ec ≤ 0.3 ), and especially variation of injection strength, on the streamlines, isotherms, and average Nusselt number are discussed graphically. The findings demonstrate a significant improvement in heat transfer efficiency, ranging from 17% to 73% with injection, underscoring its substantial implications for advancing thermal management across diverse engineering domains.

The Influence of Short Video on Consumer Consumption Behavior under the Perspective of Impulsive Consumption

Nowadays, the rapid development of mobile media technology causes short videos to become one of the most useful media. Short videos not only provide interesting daily life sharing but also contain lots of product merchandising. This type of merchandising always makes people appear the impulse consumption behavior. Therefore, we started researching the effect short videos have on consumers' consumption behavior under the impulsive consumption perspective. Based on this article, we give a detailed concept explanation of impulsive consumption, merchandising, and the SOR model we used in the following research. Besides, we also define the short videos. Then, we analyze the reasons short videos stimulate people's impulse consumption behavior using the SOR method, the factors that affect consumers consumption behavior, and the situational factors that affect consumers behavior. We compare the difference between offline stores promoting strategies and online promoting strategies in short videos, and analyze the cause of those factors using the SOR model. As a consequence, we find out that those factors we listed can explain the influence short videos bring to consumer consumption behavior.

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.

Optimization of the Non-Linear Diffussion Equations

Partial Differential Equations are used in smoothening of images. Under partial differential equations an image is termed as a function; f(x, y), XÎR&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt;. The pixel flux is referred to as an edge stopping function since it ensures that diffusion occurs within the image region but zero at the boundaries; u&amp;lt;sub&amp;gt;x&amp;lt;/sub&amp;gt;(0, y, t) = u&amp;lt;sub&amp;gt;x&amp;lt;/sub&amp;gt;(p, y, t) = u&amp;lt;sub&amp;gt;y&amp;lt;/sub&amp;gt;(x, 0, t) = u&amp;lt;sub&amp;gt;y&amp;lt;/sub&amp;gt;(x, q, t). Nonlinear PDEs tend to adjust the quality of the image, thus giving images desirable outlooks. In the digital world there is need for images to be smoothened for broadcast purposes, medical display of internal organs i.e MRI (Magnetic Resonance Imaging), study of the galaxy, CCTV (Closed Circuit Television) among others. This model inputs optimization in the smoothening of images. The solutions of the diffusion equations were obtained using iterative algorithms i.e. Alternating Direction Implicit (ADI) method, Two-point Explicit Group Successive Over-Relaxation (2-EGSOR) and a successive implementation of these two approaches. These schemes were executed in MATLAB (Matrix Laboratory) subject to an initial condition of a noisy images characterized by pepper noise, Gaussian noise, Brownian noise, Poisson noise etc. As the algorithms were implemented in MATLAB, the smoothing effect reduced at places with possibilities of being boundaries, the parameters C&amp;lt;sub&amp;gt;v&amp;lt;/sub&amp;gt; (pixel flux), C&amp;lt;sub&amp;gt;f&amp;lt;/sub&amp;gt; (coefficient of the forcing term), b (the threshold parameter) alongside time t were estimated through optimization. Parameter b maintained the highest value, while C&amp;lt;sub&amp;gt;v&amp;lt;/sub&amp;gt; exhibited the lowest value implying that diffusion of pixels within the various images i.e. CCTV, MRI &amp; Galaxy was limited to enhance smoothening. On the other hand the threshold parameter (b) took an escalated value across the images translating to a high level of the force responsible for smoothening.

Higher order phase-field modeling of brittle fracture via isogeometric analysis

AbstractThe evolution of brittle fracture in a material can be conveniently investigated by means of the phase-field technique introducing a smooth crack density functional. Following Borden et al. (2014), two distinct types of phase-field functional are considered: (i) a second-order model and (ii) a fourth-order one. The latter approach involves the bi-Laplacian of the phase field and therefore the resulting Galerkin form requires continuously differentiable basis functions: a condition we easily fulfill via Isogeometric Analysis. In this work, we provide an extensive comparison of the considered formulations performing several tests that progressively increase the complexity of the crack patterns. To measure the fracture length necessary in our accuracy evaluations, we propose an image-based algorithm that features an automatic skeletonization technique able to track complex fracture patterns. In all numerical results, damage irreversibility is handled in a straightforward and rigorous manner using the Projected Successive Over-Relaxation algorithm that is suitable to be adopted for both phase-field formulations since it can be used in combination with higher continuity isogeometric discretizations. Based on our results, the fourth-order approach provides higher rates of convergence and a greater accuracy. Moreover, we observe that fourth- and second-order models exhibit a comparable accuracy when the former methods employ a mesh-size approximately two times larger, entailing a substantial reduction of the computational effort.

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.

Improve refinement approach iterative method for solution linear equition of sparse matrices

In this paper, systems of linear equations on sparse matrices investigated through modified improve method using Gauss-Seidel and successive overrelaxation (SOR) approach. Taking into adapted convergence rate on the Improve refinement Gauss-seidel outperformed the prior two Gauss-Seidel methods in terms of rate of convergence and number of iterations required to solve the problem by applying a modified version of the Gauss-Seidel approach. to observe the effectiveness of this method, the numerical example is given. The main findings in this study, that Gauss seidel improvement refinement gives optimum spectral radius and convergence rate. Similarly, the SOR improved refinement method gives. Considering their performance, using parameters such as time to converge, number of iterations required to converge and spectral radius level of accuracy. However, SOR works with relaxation values so that it greatly affects the convergence rate and spectral radius results if given greater than 1.

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.

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.

NON-NESTED MULTILEVEL ACCELERATION OF MESHLESS SOLUTION OF HEAT CONDUCTION EQUATION IN COMPLEX DOMAINS

In this paper, we developed and demonstrated a non-nested multilevel procedure for solving the heat conduction equation in complex domains using a meshless discretization method. Previous multilevel methods for solving elliptic partial differential equations in complex domains have mostly used one of the four approaches: nested refinement, agglomeration, additive corrections, and algebraic coarsening. Each of these techniques has some issues of generality, robustness, and speed of acceleration. In this paper, we developed a generally applicable multilevel algorithm for partial differential equations discretized on complex domains using unstructured finite volume, finite element, and meshless methods. We applied this multilevel method to accelerate convergence of the set of discrete equations obtained by a meshless technique. The heat conduction equation is discretized at scattered points using a polyharmonic spine (PHS) radial basis function (RBF) interpolation with appended polynomials to achieve exponential convergence of discretization errors. The RBF interpolations are performed over clouds of points, and the partial differential equation is collocated at the scattered points. The multilevel algorithm to solve the set of linear equations utilizes multiple independently generated coarser sets of points. Restriction of residuals and prolongation of the corrections are also performed using the RBF interpolations. The fast convergence of the algorithm is demonstrated for solution of the heat conduction equation in three model complex domains with manufactured solutions. A simple successive over-relaxation point solver is used as the relaxation scheme.

Novel Hybrid SOR- and AOR-Based Multi-User Detection for Uplink M-MIMO B5G Systems

The Internet of Things (IoT) is one of the most important wireless sensor network (WSN) applications in 5G systems and requires a large amount of wireless data transmission. Therefore, massive multiple-input multiple-output (M-MIMO) has become a crucial type of technology and trend in the future of beyond fifth-generation (B5G) wireless network communication systems. However, as the number of antennas increases, this also causes a significant increase in complexity at the receiving end. This is a challenge that must be overcome. To reduce the BER, confine the computational complexity, and produce a form of detection suitable for 4G and B5G environments simultaneously, we propose a novel multi-user detection (MUD) scheme for the uplink of M-MIMO orthogonal frequency division multiplexing (OFDM) and universal filtered multi-carrier (UFMC) systems that combines the merits of successive over-relaxation (SOR) and accelerated over-relaxation (AOR) named mixed over-relaxation (MOR). Herein, we divide MOR into the initial and collaboration stages. The former will produce the appropriate initial parameters to improve feasibility and divergence risk. Then, the latter achieves rapid convergence and refinement performance through alternating iterations. The conducted simulations show that our proposed form of detection, compared with the BER performance of traditional SOR and AOR, can achieve 99.999% and 99.998% improvement, respectively, and keep the complexity at O(N2). It balances BER performance and complexity with fewer iterations.

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.

Flutter Phenomenon and Safety Implications in Transonic Flow

As advancements in aircraft manufacturing technology persistently elevate the boundaries of flight speed, challenges emerge when aircraft velocities verge on sonic speeds. Notably, empirical observations have highlighted instances of pronounced vibrations and, in extreme cases, structural failures, casting shadows over flight stability and safety. This research endeavors to dissect the ramifications of shock waves and flutter phenomena, precipitated by fluid discontinuities during supersonic flight, on the overarching safety of aircraft. Methodologically, the study delineates a series of simulation protocols, commencing with the derivation of simplified functions representing the aircraft's 2D contour, followed by error function computations, and finally, using the finite difference method in conjunction with the successive over-relaxation (SOR) iterative technique to model the aircraft's velocity distribution. Preliminary findings indicate that for β values spanning 0.1, 0.15, and 0.175, the surrounding airflow retains its continuity and stability, with the Mach number's rate of change exhibiting a decelerating trend. Conversely, at a β threshold of 0.2, the airflow descends into turbulence, manifesting in erratic Mach number fluctuations. Such fluidic discontinuities underscore potential threats to the aircraft's flight safety, necessitating further exploration and mitigation strategies.