Successive Overrelaxation Research Articles

Influence of electric field, blood velocity, and pharmacokinetics on electrochemotherapy efficiency

The convective delivery of chemotherapeutic drugs in cancerous tissues is directly proportional to the blood perfusion rate, which in turns can be transiently reduced by the application of high-voltage and short-duration electric pulses due to vessel vasoconstriction. However, electric pulses can also increase vessel wall and cell membrane permeabilities, boosting the extravasation and cell internalization of drug. These opposite effects, as well as possible adverse impacts on the viability of tissues and endothelial cells, suggest the importance of conducting in silico studies about the influence of physical parameters involved in electric-mediated drug transport. In the present work, the global method of approximate particular solutions for axisymmetric domains, together with two solution schemes (Gauss-Seidel iterative and linearization+successive over-relaxation), is applied for the simulation of drug transport in electroporated cancer tissues, using a continuum tumor cord approach and considering both the electropermeabilization and vasoconstriction phenomena. The developed global method of approximate particular solutions algorithm is validated with numerical and experimental results previously published, obtaining a satisfactory accuracy and convergence. Then, a parametric study about the influence of electric field magnitude and inlet blood velocity on the internalization efficacy, drug distribution uniformity, and cell-kill capacity of the treatment, as expressed by the number of internalized moles into viable cells, homogeneity of exposure to bound intracellular drug, and cell survival fraction, respectively, is analyzed for three pharmacokinetic profiles, namely one-short tri-exponential, mono-exponential, and uniform. According to numerical results, the trade-off between vasoconstriction and electropermeabilization effects and, consequently, the influence of electric field magnitude and inlet blood velocity on the assessment parameters considered here (efficacy, uniformity, and cell-kill capacity) is different for each pharmacokinetic profile deemed.

Efficient Iterative Detection Based on Conjugate Gradient and Successive Over-Relaxation Methods for Uplink Massive MIMO Systems

Being a crucial aspect of fifth-generation (5G) mobile communications systems, massively multiple-input multiple-output (mMIMO) architectures are expected to help achieve the highest key performance indicators. However, the huge numbers of antennas used in such systems make it difficult to determine the inversion of the signal channel matrix relied upon by several detection methods, hence posing a problem with accurate estimation of the symbols sent. In this paper, conjugate gradient (CG) and successive over-relaxation (SOR) methods are selected to construct a new iterative approach that avoids the matrix inversion computation issue. This suggested approach for uplink mMIMO detection is based on a joint cascade structure of both iterative methods. The CG method is first applied and adjusted for the initial solution, followed by the SOR method in the final iterations for terminal computations, resulting in an algorithm with robust performance and low computational complexity. Furthermore, the new hybrid scheme operates based on the relaxation parameter, whose value has a great impact on error performance and, whose optimal determination is necessary. Numerical simulations reveal that the proposed scheme is capable of significantly improving signal detection accuracy with minimum complexity. The simulation results indicated that the proposed detector outperforms CG and SOR detectors, achieves close to optimal performance, requires fewer iterations, and reduces complexity.

Optical waveguide analysis using alternative direction implicit (ADI) method in combination with successive over-relaxation (SOR) algorithm

Optical waveguide analysis using alternative direction implicit (ADI) method in combination with successive over-relaxation (SOR) algorithm

Multigrid Incomplete Factorization Methods in Krylov Subspaces

The paper studies multigrid methods for solving systems of linear algebraic equations resulting from the seven-point discretization of the three-dimensional Dirichlet problem for an elliptic differential equation of the second order in a parallepipedal domain on a regular grid. The algorithms suggested are presented as special iteration processes of incomplete factorization in Krylov subspaces with a hierarchical recursive vector structure that corresponds to a sequence of embedded grids and gives rise to a block tridiagonal recursive representation of the coefficient matrix of the original linear algebraic system. The convergence of iterations is enhanced by using the principle of compensation of the row sums and also the symmetric successive block overrelaxation. An arbitrary m-grid method is defined recursively, based on the two-grid method. For simplicity, the algorithms are considered for linear systems with Stieltjes coefficient matrices. Issues related to generalization of the algorithms to larger classes of problems and, in particular, those with unsymmetric matrices are discussed.

MHD mixed convection of nanofluid in a cavity with isothermal local heater under an influence of velocity modulation of upper cold wall

Flow and heat transfer dynamics are extremely complicated phenomena in various enclosures. The mechanism of heat transmission and its enhancement due to the presence of nanoscale particles is an important research area. Owing to the broad significance of convection fluid flows with nanoparticles in different configurations and limitations, we have addressed in this study the mixed convection of aluminum oxide (Al2O3)-water nanosuspension in a closed chamber having a moveable upper boundary and a semicircular heat source of constant temperature. The simulation has been carried out with the impacts of thermal radiation and uniform horizontal magnetic field. Mass, momentum, and energy conservation laws have been used to develop the mathematical model for assessing the effectiveness regarding the thermal performance of nanoparticles. Objective of this analysis is to study the factors that improve the effectiveness of thermal transport in heat transfer machinery and heat recovery units used in engineering and industrial processes. Firstly, appropriate transformations are employed to alter the governing system into non-dimensional forms. The dimensionless system was then numerically solved for the wide range of emerging parameters. The system of governing equations has been solved on the basis of the finite difference method. Parabolic equations have been solved using the locally one-dimensional Samarskii scheme and then using the Thomas algorithm. An elliptic equation has been approximated by central differences and then solved by the successive over-relaxation method. The obtained distributions characterize the consequences of parameters on the fluid flow, establishment time of the quasi-stationary regime, and efficiency of heat transfer in the investigated cavity. Thus, it has been revealed that the effective energy removal can be obtained for ξ = 0.01 under an influence of magnetic field and nanoparticles addition.

A parallel unstructured multi-color SOR solver for 3D Navier–Stokes equations on graphics processing units

In this paper, we develop a GPU parallelization technique to accelerate the pressure Poisson solver. We apply it in three-dimensional Navier–Stokes simulations with arbitrary geometries. The numerical method is based on a projection method, and the discretization is obtained by combining a second-order unstructured finite-volume method and a σ transformation. The pressure solution is responsible for this formulation’s most time-consuming computation tasks. This is because the Poisson equation must be approximated. This work proposes a GPU-based Multi-Color Successive Over-Relaxation (MC-SOR) method to solve the resulting sparse linear system. The proposed parallel finite-volume code is verified and validated by solving the Poisson problem. We employ analytical solutions and well-known benchmark problems with complex flows. These include the Taylor–Green vortex problem and the problem of turbulent flow around a circular cylinder.

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 novel high-dimensional and multi-physics modeling approach of proton exchange membrane fuel cell for real-time simulation

Fuel cell model provides state observation and quantitative analysis for Hardware-In-the-Loop (HIL) test platform. For a high-dimensional multi-physics model, the observed distribution of physical properties is meaningful for fuel cell stack optimization, manufacturing process and model-based control development. However, traditional modeling approach is difficult to consider both the calculation accuracy and efficiency. To solve this problem, this paper proposes a high-dimensional multi-physics real-time modeling approach for fuel cells based on partial differential equations (PDEs)-tridiagonal matrix equation transformation (TMET) and symmetric successive overrelaxation (SSOR) algorithm. In the proposed method, a generalized framework of TMET and a novel staggered grid technique is developed. In this case, both the convection and diffusion phenomena can be accurately described. For different fuel cells or parameters change, supplementary model modification or development is not required, only the global variables need to be re-assigned. Moreover, SSOR with bi-directional iteration mechanism is used to increase the probability of computational convergence and thus improve the calculation efficiency. Compared with commercial COMSOL model, by using the proposed modeling approach, the mean absolute percentage error (MAPE) of the simulated polarization output characteristics is 1.77%, the MAPE of different distribution characteristics can remain within 10%, and the calculation time can be shortened from hours even days to seconds.

A novel flow update policy in solving traffic assignment problems: Successive over relaxation iteration method

This paper presents a novel flow update policy, namely the successive over relaxation (SOR) iteration method, which can be implemented in traffic assignment algorithms. Most existing solution algorithms for the user equilibrium traffic assignment problem (UE-TAP) mainly use two flow update policies: Jacobi and Gauss-Seidel iteration methods. The proposed flow update policy SOR can be a more efficient replacement. Following the path-based gradient projection (GP) algorithm, we developed a new method GP-SOR for the UE-TAP. This study first provides the complete procedure of applying the GP-SOR algorithm to solve the UE-TAP. Subsequently, a few properties of the proposed method are rigorously proven. However, empirical tests of the GP-SOR algorithm demonstrate serious oscillations and poor convergence. To cope with this problem, the Armijo Rule is employed to determine the relaxation factor, which substantially improves the convergence of GP-SOR algorithm. The preliminary numerical examples show that the GP-SOR algorithm has speedier convergence compared with the known alternatives, which is reflected by the evident reduction of the computing time and the number of iterations.

An Analytical Study on Two High-Order Hybrid Methods to Solve Systems of Nonlinear Equations

In order to solve systems of nonlinear equations, two novel iterative methods are presented. The successive over-relaxation method and the Chebyshev-like iterative methods to solve systems of nonlinear equations have combined to obtain the new algorithms. By this combination, two powerful hybrid methods are obtained. Necessary conditions for convergence of these methods are presented. Furthermore, the stability analysis of both algorithms is investigated. These algorithms are applied for solving two real stiff systems of ordinary differential equations. These systems arise from an HIV spreading model and an SIR model of an epidemic which formulates the spread of a nonfatal disease in a certain population. Numerical results show promising convergence and stability for both new hybrid methods.

Comparison study of different forward modeling approaches by using the finite-element method for 3D magnetotelluric data in anisotropic media

We have examined three approaches for analyzing 3D magnetotelluric (MT) data with general anisotropy. These included the direct electromagnetic (EM) approach, which solved the electric field Helmholtz equation directly, and the Coulomb-gauged and Ungauged approaches. The three approaches are discretized using finite-element (FE) methods. First, the correctness of the three approaches is validated using a typical isotropic 3D model and a 2D anisotropic model. The FE solutions of each approach obtained by the direct solver MUMPS are compared with the analytical solutions to demonstrate that all approaches generated accurate results. Subsequently, an anisotropic 3D model is designed to investigate the forward characteristics of the different approaches. The FE solutions are obtained using the quasi-minimum residual (QMR) solver with a symmetric successive overrelaxation (SSOR) preconditioner in this model. The forward modeling results of each approach were compared in terms of stiffness matrices, the convergence of the iterative solver, and apparent resistivity and phase. We find that the FE solution of the Coulomb-gauged and Ungauged approaches produces larger stiffness matrices, but the proportion of non-zero elements in the Coulomb-gauged approach is the least. The Ungauged approach can yield faster solutions over a wide range of frequencies. Finally, a complex numerical example is provided to demonstrate the good convergence behavior of the Ungauged approach.

Generation of Two-Dimensional Structured Meshes for Simply and Doubly Connected Regions

We present in this work Thompson’s equations, with spacing factors P and Q, and we use them to computationally generate two-dimensional structured meshes for simply and doubly connected regions. For the internal boundary of doubly connected regions we selected some wing profiles of an aircraft. We discretized the equations using second-order centered finite differences and we used the SOR method to numerically solve the system of linear equations resulting from discretization. In SOR method, we test optimal values for the relaxation parameter

Multi-phase Flow Phenomena under Alternating Magnetic Field: Numerical Analysis

This article outlines the fundamentals of electric conducting fluid analysis and its numerical model for an electrically conductive droplet when alternating magnetic field is applied. The level-set method was applied for interfacial capture and the electromagnetic potential was introduced for electromagnetic field analysis. The pressure and electromagnetic potential were computed by the Red-Black SOR method using GPU (Graphics processing unit). It is exhibited that the solid, liquid and gas phases were able to be computed simultaneously and that an electromagnetic levitation phenomenon was successfully simulated.

Effect of lubricant inertia on steady-state characteristics of textured hydrodynamic journal bearing

PurposeThis paper aims to investigate the effects of inertia of the lubricant on the performance parameters of journal bearings textured with spherical-shaped textures.Design/methodology/approachThe lubricant is assumed to be Newtonian, and the flow is considered laminar. Considering the lubricant inertia effects, the modified Reynolds equation is discretized using the finite difference method and solved with the Gauss–Seidel successive over-relaxation scheme using the progressive mesh densification method.FindingsThe results from this numerical study indicate that the lubricant inertia improves textured journal bearing performance characteristics significantly. The improvement is more significant in the case of heavily loaded bearings. Furthermore, it is observed that protruded texturing in journal bearings shows better results compared to dimple textured journal bearings.Originality/valueUnderstanding the effect of lubricant inertia is essential for efficiently designing textured journal bearings. Thus, the results shown here would be helpful for the researchers and the bearing designers.

Preconditioned successive over relaxation iterative method via semi-approximate approach for Burgers’ equation

<span lang="EN-US">This paper proposes the combination of a preconditioner applied with successive over relaxation (SOR) iterative method for solving a sparse and huge scale linear system (LS) in which its coefficient matrix is a tridiagonal matrix. The purpose for applying the preconditioner is to enhance the convergence rate of SOR iterative method. Hence, in order to examine the feasibility of the proposed iterative method which is preconditioner SOR (PSOR) iterative method, first we need to derive the approximation equation of one-dimensional (1D) Burgers’ equation through the discretization process in which the second-order implicit finite difference (SIFD) scheme together with semi-approximate (SA) approach have been applied to the proposed problem. Then, the generated LS is modified into preconditioned linear system (PLS) to construct the formulation of PSOR iterative method. Furthemore, to analyze the feasibility of PSOR iterative method compared with other point iterative methods, three examples of 1D Burgers’ equation are considered. In conclusion, the PSOR iterative method is superior than PGS iterative method. The simulation results showed that our proposed iterative method has low iteration numbers and execution time.</span>

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

Agent Navigation based on Boundary Value Problem using Iterative Methods

This paper presents the simulation of numerical solutions to the navigational problem of an agent traveling safely in its environment. The approach is based on the numeric solutions of the boundary value problem (BVP) that generate harmonic potential fields through a differential equation whose gradient represents navigation routes to the destination. Two methods, namely KSOR and KAOR, were tested to solve the BVP. KSOR and KAOR are variants of the standard SOR and AOR methods, respectively. In this work, the KSOR and KAOR methods were used to solve the BVP by applying Laplace's equation to obtain harmonic functions. The generated harmonic functions are then utilized by the searching algorithm to find a smooth navigational route for an agent to travel in its environment without colliding with any obstacles. The numerical results from the solutions of BVP demonstrate that the KAOR provides a faster execution time with fewer iterations compared to the KSOR method.

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

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

Comparison of Linear Iteration Schemes to Improve the Convergence of Iterative Physical Optics for an Impedance Scatterer

The conventional iterative physical optics (IPO) method updates the surface current based on the Jacobi iteration scheme, which typically diverges for large objects. To control the convergence property of the IPO method, other iteration schemes, such as Gauss–Seidel and successive over-relaxation, can be used. In this study, we compare the convergence properties of three iteration schemes for scatterings by five scatterers comprising electrically perfect or imperfect conductors modeled with an impedance material. The accuracy of the IPO method is compared with that of the multi-level fast multipole method.

Comparison of Basic Iterative Methods Used to Solve of Heat and Fluid Flow Problems

In this research, a comparison of the convergence rate of different basic methods is made in order to solve the Poisson equation, which is similar to some of the resulting equations in computational fluid mechanics. The Jacobi method, the point Gauss-Seidel method, the successive over-relaxation method (SOR), the line Gauss method (TDMA), the ADI method and Strongly Implicit (SIP) method are among the iterative approaches investigated, which are then compared to find the most optimal method. The selection criteria included the number of iterations and the time needed to reach convergence. In both selection criteria, the SIP approach has been shown to be the most efficient.