Successive Iteration Method Research Articles

The influence of the variability of the support of the mortar base plate on the quality of the results obtained in the process of its numerical design

Due to the high costs associated with the purchase of ammunition and firing in certified training ground centers, tests of retaining plate deformations are increasingly replaced by computer simulations using numerical models. Computer programs usually use a single-parameter subsoil model (Winkler-Zimmermann) for calculations, which requires providing the subgrade susceptibility coefficient. The subgrade compliance coefficient is intended to determine the mutual reaction of the subgrade and the structure due to the pressure exerted on the soil by the retaining slab, which settles. When designing slabs in computer programs, it is assumed that the substrate compliance coefficient is constant. Determining the impact of the soil on the retaining slab is important when analyzing its deformations. The subject of the work was the analysis of the influence of ground support on the results obtained during modeling of the retaining slab. In order to obtain data for FEM analysis and validation, the actual strains occurring on the thrust plate were measured using strain gauge rosettes. The plate deformations were measured during field shooting tests. In order to vary the influence of supporting the slab on the ground and obtain reliable stress values on the slab surface, a method of successive iterations was proposed. Calculations are performed using this method until the error is smaller than the assumed one.

Distributed Reactive Power Optimization for Flexible Distribution Networks With Successive Relaxation Iteration Method

Distributed Reactive Power Optimization for Flexible Distribution Networks With Successive Relaxation Iteration Method

A topology optimization framework for 3D phononic crystals via the method of successive iteration of analysis and design

Topology optimization has become a useful tool for designing Phononic Crystals (PnCs) to achieve desired elastic wave propagation properties. However, topological design of three-dimensional (3D) PnCs remains a significant challenge due to extremely high computational cost required for the repeated solution of the nested double-loop problem of design optimization and eigenvalue analysis to determine the band structure during the optimization iteration process. This paper presents a more efficient topology optimization method based on the concept of successive iteration of analysis and design. The method employs a sequential approximation of the eigenpairs through an inverse iteration-like procedure to avoid solving the expensive eigenvalue problems in each design iteration. To further improve the efficiency of the design iteration, the eigenmodes of the intermediate PnCs designs are first computed on a relatively coarse finite element mesh. Then they are mapped onto the fine mesh and used as the initial trial eigenmodes in the eigenpair analysis and design sensitivity analysis. The high efficiency of the proposed method, as compared with conventional nested double-loop optimization approaches, is demonstrated by 3D numerical examples with over one million degrees of freedom. It is shown that this approach can achieve simultaneous convergence of the eigenvectors along with the design evolution of the unit cell, therefore substantially reduces the computational burden for high-resolution topological design of 3D PnCs.

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

MODELING OF NONLINEAR MASS FLOWS DURING WATER SOFTENING IN A MECHANICAL FILTER

A mathematical model of water softening in a mechanical filter based on the balance relations for the mass components of the thermodynamical system is constructed. The initial-boundary value problem of convective diffusion accompanied by a chemical reaction is formulated for one-dimensional case of spatial coordinates and reduced to the equivalent system of integro-differential equations. The solution of this system is constructed by the method of successive iterations by decomposing the functions of the mass concentrations of impurity in water solution and in the body skeleton into series around the solutions of the homogeneous linear initial-boundary value problem. Based on the expression for the impurity concentration in the aqueous solution the mass flows of impurity are obtained and investigated. The nonlinearity of steady-state regime, which is caused by the transfer of mass simultaneously by diffusion and convection mechanisms, is shown.

Adapting Integral Transforms to Create Solitary Solutions for Partial Differential Equations via a New Approach

In this article, a new effective technique is implemented to solve families of nonlinear partial differential equations (NLPDEs). The proposed method combines the double ARA-Sumudu transform with the numerical iterative method to get the exact solutions of NLPDEs. The successive iterative method was used to find the solution of nonlinear terms of these equations. In order to show the efficiency and applicability of the presented method, some physical applications are analyzed and illustrated, and to defend our results, some numerical examples and figures are discussed.

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.

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.

Meshfree methods for nonlinear equilibrium radiation diffusion equation

The radiation diffusion problem is usually applied to astrophysics and inertial confinement fusion. It seems that the traditional numerical methods such as finite difference method and finite element method is not efficiency to solve the problem because of its nonlinearity and complex computational domain. In this paper, some fast meshfree methods based on radial basis function are proposed to solve the equilibrium radiation diffusion equation. Because the modeling problem has two nonlinear terms, so we use the successive permutation iteration and linearization methods to approximate the nonlinear terms. Firstly, diffusion terms of the one-dimensional (1D) and two-dimensional (2D) equilibrium radiation diffusion equations are linearized by constructing a successive permutation iteration method for fully-implicit discretization. Then, three kinds of different linearization methods (factorization iteration, Picard–Newton iteration and Richtmyer iteration) are proposed and used to linearization of other nonlinear terms. Finally, the linearized equation is solved numerically by Kansa’s method with compactly supported radial basis function. The reliability, accuracy and effectiveness of new methods are verified by 1D and 2D numerical experiments. The advantages of presented method are apparent. They require neither domain nor boundary discretization, and no domain integration is required. New methods are easy for implementation and coding. They not only avoid the complexity of generation mesh, but also solve the equilibrium radiation diffusion equation with high accuracy.

Inner-estimating domains of attraction for discrete-time non-polynomial systems with piecewise difference inclusions

ABSTRACT In this paper, we propose two successive heuristic iterative methods for inner-estimating domains of attraction for discrete-time non-polynomial systems. Firstly, we utilise a discrete-time fuzzy model to derive a piecewise difference inclusion for the discrete-time non-polynomial system. Then, an initial heuristic iterative method is presented to inner-estimate the domain of attraction as large as possible by first iteratively calculating common Lyapunov-like functions and then heuristically updating piecewise difference inclusions. Afterwards, the initial heuristic iterative method is further improved by piecewise Lyapunov-like functions, arriving at a refined heuristic iterative method. Especially, we introduce homotopy continuation method to decrease the computational complexity. At last, our methods are applied to three discrete-time non-polynomial examples with a brief discussion.

Control Model and Optimization Study of Temperature Distribution Applied in Thermite Plugging and Abandonment Technology

Abstract With the continuous exploration and development of oil wells, we must pay attention to the risk of leakage from abandoned wells. Therefore, it is necessary to plug and abandon the abandoned well. However, there are many limitations in the traditional plugging and abandonment (P&A) operation, for example, cement's bearing capacity, the cement's corrosion resistance, the problem of the extended operation time, and high cost. To overcome the aforementioned issues, a thermite plugging and abandonment (TP&A) technology is proposed in this field. The technology uses the aluminothermic reaction to melt the original or set materials for P&A operation. To promote the phase transformation of more materials in the well to form a plug with good plugging performance, the temperature distribution in the TP&A system was optimized. Based on the heat conduction theory and successive overrelaxation iterative method, a heat conduction model based on the temperature release law of aluminothermic reaction is established and solved. The temperature change law under different combinations of the downhole environment is studied. The optimized model can maintain the high-efficiency transfer of energy, fluid–structure interaction, and the interaction between fluids. The material after the phase change can be cooled to form a plug with good plugging performance.

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.

On the Application of Successive Over-relaxation Algorithmic and Block Numerical Iterative Solutions for the Stationary Distribution in Markov Chain

The evolution of a system is represented by transitions from one state to the next, and the system's physical or mathematical behavior can also be depicted by defining all of the numerous states it can be in and demonstrating how it moves between them. In this study, the iterative solution methods for the stationary distribution of Markov chains were investigated, which start with an initial estimate of the solution vector and then alter it in such a way that it gets closer and closer to the genuine solution with each step or iteration., and also involved matrices operations such as multiplication with one or more vectors, which leaves the transition matrices unchanged and saves time. Our goal is to use Successive Overrelaxation Algorithmic and Block Numerical Iterative Solution Methods to compute the solutions. With the help of some existing Markov chain laws, theorems, and formulas, the normalization principle and matric operations such as lower, upper, and diagonal matrices are used. The stationary distribution vector’s π^((k+1) )={■(π_1^((k+1) )&π_2^((k+1) )&■(π_3^((k+1) )&π_4^((k+1) )&π_5^((k+1) ) )),k=0,1,2,…,n} are obtained for the illustrative examples, taken the initial stationary solution to be π^((0) )=(■(0.2&0.2&■(0.2&0.2&0.2)))^T and it was observed that all subsequent iterations yield exactly the same result as π^((1) ), and this shows that, the block iterative method requires only a single iteration to obtain the solution to full machine precision.

Half-Sweep Refinement of SOR Iterative Method via Linear Rational Finite Difference Approximation for Second-Order Linear Fredholm Integro-Differential Equations

The numerical solutions of the second-order linear Fredholm integro-differential equations have been considered and discussed based on several discretization schemes. In this paper, the new schemes are developed derived on the hybrid of the three-point half-sweep linear rational finite difference (3HSLRFD) approaches with the half-sweep composite trapezoidal (HSCT) approach. The main advantage of the established schemes is that they discretize the differential terms and integral term of second-order linear Fredholm integro-differential equations into the algebraic equations and generate the corresponding linear system. Furthermore, the half-sweep (HS) concept is combined with the refinement of the successive over-relaxation (RSOR) iterative method to create the new half-sweep successive over-relaxation (HSRSOR) iterative method, which is implemented to get the numerical solution of a system of linear algebraic equations. Apart from that, the classical or full-sweep Gauss-Seidel (FSGS) and full-sweep successive over-relaxation iterative (FSSOR) methods are presented, which serve as the control method in this paper. In the end, we employed FSGS, FSRSOR and HSRSOR methods to obtain numerical solutions of three examples and make a detailed comparison from three aspects of the number of iterations, elapsed time and maximum absolute error. Numerical results demonstrate that FSRSOR and HSRSOR methods have lesser iterations, faster elapsed time, and are more accurate than FSGS. In addition, HSRSOR is the most effective of the three methods. To sum up, this paper has successfully proposed the applicability and superiority of the new HSRSOR method based on 3HSLRFD-HSCT schemes.

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.

Double Sumudu Transform Iterative Method for One-Dimensional Nonlinear Coupled Sine-Gordon Equation

In this paper, the combined double Sumudu transform with iterative method is successfully implemented to obtain the approximate analytical solution of the one-dimensional coupled nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions which cannot be solved by applying double Sumudu transform only. The solution of the nonlinear part of this equation was solved by a successive iterative method, the proposed technique has the advantage of producing an exact solution, and it is easily applied to the given problems analytically. Two test problems from mathematical physics were taken to show the liability, accuracy, convergence, and efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other types of systems of NLPDEs.

Existence theory and numerical solution of leptospirosis disease model via exponential decay law

<abstract><p>We investigated the leptospirosis epidemic model by using Caputo and Fabrizio fractional derivatives. Picard's successive iterative method and Sumudu transform are taken into consideration for developing the iterative solutions for the leptospirosis disease. Employing nonlinear functional analysis, the stability and uniqueness of the proposed model are established. Sensitivity analysis is taken into account to highlight the most sensitive parameters corresponding to the basic reproductive number. Various solutions to the proposed system have been interpolated by graphs with the application of Matlab software.</p></abstract>

On self-similar solutions of time and space fractional sub-diffusion equations

In this paper, we have considered two different sub-diffusion equations involving Hilfer, hyper-Bessel and Erdelyi-Kober fractional derivatives. Using a special transformation, we equivalently reduce the considered boundary value problems for fractional partial differential equation to the corresponding problem for ordinary differential equation. An essential role is played by certain properties of Erd\'elyi-Kober integral and differential operators. We have applied also successive iteration method to obtain self-similar solutions in an explicit form. The obtained self-similar solutions are represented by generalized Wright type function. We have to note that the usage of imposed conditions is important to present self-similar solutions via given data.

Solutions of Two-Dimensional Nonlinear Sine-Gordon Equation via Triple Laplace Transform Coupled with Iterative Method

This article presents triple Laplace transform coupled with iterative method to obtain the exact solution of two-dimensional nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions. The noise term in this equation is vanished by successive iterative method. The proposed technique has the advantage of producing exact solution, and it is easily applied to the given problems analytically. Four test problems from mathematical physics are taken to show the accuracy, convergence, and the efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.

QSSOR and cubic non-polynomial spline method for the solution of two-point boundary value problems

Two-point boundary value problems are commonly used as a numerical test in developing an efficient numerical method. Several researchers studied the application of a cubic non-polynomial spline method to solve the two-point boundary value problems. A preliminary study found that a cubic non-polynomial spline method is better than a standard finite difference method in terms of the accuracy of the solution. Therefore, this paper aims to examine the performance of a cubic non-polynomial spline method through the combination with the full-, half-, and quarter-sweep iterations. The performance was evaluated in terms of the number of iterations, the execution time and the maximum absolute error by varying the iterations from full-, half- to quarter-sweep. A successive over-relaxation iterative method was implemented to solve the large and sparse linear system. The numerical result showed that the newly derived QSSOR method, based on a cubic non-polynomial spline, performed better than the tested FSSOR and HSSOR methods.