Successive Iteration Method Research Articles

Numerical computation of successive approximations method and variational iteration method for solving Klein-Gordon Schrödinger equation

This paper is devoted to investigating and comparing the Successive Approximations Method (SAM) and Variational Iteration Method (VIM) for solving Klein-Gordon Schrodinger (KGS) Equation. Furthermore, the approximate solutions that obtained by both methods have been represented numerically and graphically.

Successive Approximation, Variational Iteration, and Multistage-Analytical Methods for a SEIR Model of Infectious Disease Involving Vaccination Strategy

We consider a SEIR model for the spread (transmission) of an infectious disease. The model has played an important role due to world pandemic disease spread cases. Our contributions in this paper are three folds. Our first contribution is to provide successive approximation and variational iteration methods to obtain analytical approximate solutions to the SEIR model. Our second contribution is to prove that for solving the SEIR model, the variational iteration and successive approximation methods are identical when we have some particular values of Lagrange multipliers in the variational iteration formulation. Third, we propose a new multistage-analytical method for solving the SEIR model. Computational experiments show that the successive approximation and variational iteration methods are accurate for small size of time domain. In contrast, our proposed multistage-analytical method is successful to solve the SEIR model very accurately for large size of time domain. Furthermore, the order of accuracy of the multistage-analytical method can be made higher simply by taking more number of successive iterations in the multistage evolution.

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.

Grid impedance estimation based on improved firefly algorithm

Grid impedance has a great influence on the performance of grid-connected inverters in distributed power generation systems, so accurate grid impedance measurement is a key technique to achieve high-performance adaptive control of grid-connected inverters in weak grid situations. Firefly algorithm has the advantages of global search capability and fast convergence speed, and it has been widely used in the field of parameter estimation. In order to estimate grid impedance accurately and quickly, a grid impedance estimation algorithm based on the improved firefly algorithm is proposed. The grid impedance parameters are taken as the individual of the firefly algorithm, and the simulation model is built in the Simulink environment to compare and analyze the grid impedance estimation effect between the firefly algorithm and the successive iteration method. The simulation results show that the improved firefly algorithm has a higher accuracy for grid impedance estimation.

Numerical computation of successive approximations method and variational iteration method for solving Klein-Gordon Schrödinger equation

This paper is devoted to investigating and comparing the Successive Approximations Method (SAM) and Variational Iteration Method (VIM) for solving Klein-Gordon Schrödinger (KGS) Equation. Furthermore, the approximate solutions that obtained by both methods have been represented numerically and graphically.

Existence and Uniqueness of Uncertain Fractional Backward Difference Equations of Riemann–Liouville Type

In this article, we consider the analytic solutions of the uncertain fractional backward difference equations in the sense of Riemann–Liouville fractional operators which are solved by using the Picard successive iteration method. Also, we consider the existence and uniqueness theorem of the solution to an uncertain fractional backward difference equation via the Banach contraction fixed-point theorem under the conditions of Lipschitz constant and linear combination growth. Finally, we point out some examples to confirm the validity of the existence and uniqueness of the solution.

Efficient and robust phase-split computations in the internal energy, volume, and moles (UVN) space

Phase-split computations in an isolated system, which in general may be defined as one where the total internal energy (U), volume (V), and the number of moles (N=N1,N2,…,Nn) of the components are fixed at some specified set of values, involves the determination of the temperature (T), pressure (P), plus the amount and composition of the various phases that constitute the system. A simpler, but analogous problem is one where T, P, and N are specified instead. In TPN space, one may first perform a stability analysis to determine whether the system is stable, meaning whether at equilibrium it will split up into multiple phases or remain in single phase. If the single-phase state is unstable, the stability analysis reliably provides a good set of initial guesses in the subsequent phase-split computations. In UVN space, however, we demonstrate that the stability analysis (which is the main subject of our earlier study [1]) may not in general provide good enough initial guesses; we offer alternative strategies for setting up good initial guesses. Furthermore, we show that a combination of successive substitution iteration (SSI) and Newton's method---two iterative methods that are prevalent in the literature in TPN space---facilitates a robust and efficient algorithm for phase-split computations in isolated systems. This combination has so far not been applied in UVN space.

Vapor-liquid equilibrium calculations at specified composition, density and temperature with the perturbed chain statistical associating fluid theory (PC-SAFT) equation of state

In this study, the PC-SAFT equation of state is used for vapor-liquid equilibrium calculations using as independent variables the mixture composition, density and temperature. The method is based on unconstrained minimisation of the Helmholtz Free energy via a combination of the successive substitution iteration and Newton-Raphson minimisation methods with line-search; the positive definiteness of the Hessian is guaranteed by a modified Cholesky decomposition. The algorithm consists of two stages; initially, the mixture is assumed to be a single-phase and its stability is assessed; in case of being found unstable, a second stage of phase splitting (flash) takes place, in which the pressure of the fluid and compositions of both the liquid and vapor phases are calculated. The reliability of two different methods presented in the existing literature, (i) using mole numbers and (ii) using the logarithm of the equilibrium constants as iterative variables, is evaluated in terms of both iterations and computational time needed to reach convergence, for seven test cases. These include both single and multicomponent Diesel fuel surrogates, known to give incomplete density information when using pressure and temperature as independent variables. Results show that iterating with the logarithm of the equilibrium constants also reproduces this issue, while it requires a smaller number of iterations than using with mole numbers as independent variables. However, the total computational time needed for the latter case is vastly inferior. Pressure and vapor volume fraction fields are discussed for a range of temperatures and densities, apart from the number of iterations needed during the flash calculation stage. A performance comparison is obtained against the Peng-Robinson equation of state, showing similar number of iterations required but a computational time increasing with the number of components. While for a single component PC-SAFT needs around 3 times more CPU time, for 4 components it is 6 times and for a mixture of 8 components it increases up to 14 times. Finally, the method is demonstrated to converge unconditionally for all conditions tested.

Theory Analysis of Nonlinear Excitation of Second Harmonic THz Gyrotron for DNP-NMR

As an important parameter, start current reveals the excitation condition of cavity modes. Normally, the method of successive iterations with one mode associated in the calculation is used to get the start current, which just reflects the self-excitation conditions in the presence of single mode. Nevertheless, in many cases, the prior-excited mode may affect the start current of the subsequent ones. In this article, we re-derived electron motion equations with two modes involved in the calculation and calculated the start current numerically based on those equations. We analyzed start currents of operation mode TE26+ and parasitic modes of a 394-GHz gyrotron with the two methods. In the calculation using the conventional method, TE26+ satisfies self-excitation condition while TE26− does not. In the calculation using the latter method, the start current of TE26− decreases under the influence of TE26+. This nonlinear oscillation of TE26− is further observed in time domain multimode simulation. In practical application, the beam velocity spread, ohmic loss, and high-frequency loss are inevitable. We studied their effects on mode interaction and then optimized the operation parameters. When the equivalent conductivity is $1.7e^{7}~\text {s/m}$ and beam velocity spread is 0.03, the optimized output power is 149.2 W.

Cyclic viscoplastic behavior of thick spherical vessels under strain controlled loads based on the Chaboche unified model

The purpose of this paper is to investigate the effect of stress classification on cyclic viscoplastic response of a structure (here, a thick spherical vessel which has an application in pressure vessel technology). The response is either reverse plasticity and shakedown or ratcheting behavior of a structure. The viscoplastic behavior of thick spherical vessels subjected to thermo-mechanical loads, using the rate-dependent Chaboche unified viscoplastic model with combined kinematic and isotropic hardening theory of plasticity, is evaluated. The material properties of the structure is assumed to be temperature dependent. A combination of generalized differential quadrature method (GDQ) and a precise numerical scheme, using the successive approximation iterative method, is employed to solve the resulting nonlinear differential equations. It is concluded that the Chaboche viscoplastic hardening model, as well as the plastic hardening model, predicts reverse plasticity under thermal cyclic loading with and without the creep consideration (a strain controlled cyclic loading). It is also shown that the rates of loading and unloading do not have significant influence on the cyclic viscoplastic behavior of the vessel, although they have influence on the strain and stress distributions.

Solutions of reaction-diffusion equations using similarity reduction and HSSOR iteration

<p>Similarity method is used in finding the solutions of partial differential equation (PDE) in reduction to the corresponding ordinary differential equation (ODE) which are not easily integrable in terms of elementary or tabulated functions. Then, the Half-Sweep Successive Over-Relaxation (HSSOR) iterative method is applied in solving the sparse linear system which is generated from the discretization process of the corresponding second order ODEs with Dirichlet boundary conditions. Basically, this ODEs has been constructed from one-dimensional reaction-diffusion equations by using wave variable transformation. Having a large-scale and sparse linear system, we conduct the performances analysis of three iterative methods such as Full-sweep Gauss-Seidel (FSGS), Full-sweep Successive Over-Relaxation (FSSOR) and HSSOR iterative methods to examine the effectiveness of their computational cost. Therefore, four examples of these problems were tested to observe the performance of the proposed iterative methods. Throughout implementation of numerical experiments, three parameters have been considered which are number of iterations, execution time and maximum absolute error. According to the numerical results, the HSSOR method is the most efficient iterative method in solving the proposed problem with the least number of iterations and execution time followed by FSSOR and FSGS iterative methods.</p>

Long-term optimal operation of cascade hydropower stations based on the utility function of the carryover potential energy

Challenge remains to find the optimal carryover storage to balance the immediate and carryover utilities for long-term hydropower reservoir operation due to low forecast accuracy, limited foresight, and complex hydrologic, hydraulic and electric connections between cascade hydropower reservoirs. Thus, this paper proposes a long-term optimal operation model to dynamically control the optimal carryover storages for cascade hydropower reservoirs. First, the carryover utility of cascade hydropower stations is expressed as a function of the carryover potential energy rather than the carryover storage, thus decreasing the function parameters and using the limited prior knowledge. Then, the utility function of the carryover potential energy (UFCPE) is fitted by a quadratic function based on a successive iteration method, based on which a two-stage optimal operation model (TSOOM) is developed to guide the long-term optimal operation of cascade hydropower stations. The case study with five cascade hydropower stations in the Yalong River Basin shows that: 1) UFCPE can be well fitted by a quadratic function, and it increases linearly in the dry season, but monotonically at a decreasing rate in the flood season resulting from a large quantity of surplus water; and 2) The performance of UFCPE-based TSOOM is much superior to those of conventional rules and conventional optimization methods and quite close to that of dynamic programming successive approximation in guiding the long-term cascade hydropower reservoirs operation.

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.

A Generalized Uncertain Fractional Forward Difference Equations of Riemann-Liouville Type

In this paper, we firstly recall the definition of an uncertain fractional forward difference equation with Riemann-Liouvillelike forward difference. After that analytic solutions to a generalized uncertain fractional difference equations are solved by using the Picard successive iteration method. Moreover, the existence and uniqueness theorem of the solutions are proved by applying Banach contraction mapping theorem. Finally, two examples are presented to illustrate the validity of the existence and uniqueness theorem.

Two-stage stochastic optimal operation model for hydropower station based on the approximate utility function of the carryover stage

Challenge remains to find the optimal carryover storage to balance the immediate and carryover utilities for long-term hydropower reservoir operation due to high uncertainties of long-term forecasts. Thus, this paper develops a two-stage stochastic optimal operation model to dynamically decide the optimal carryover storage. First, a successive iteration method based on periodic Markov characteristics of reservoir operation is proposed to obtain the approximate utility function of the carryover stage. Then, three two-stage stochastic optimal operation models based on different forecast accuracy (no forecasts, perfect forecasts, and uncertainty forecasts) are developed to guide the long-term hydropower reservoir operation. The applications shows that: 1) the back propagation neural network can approximate the utility function of the carryover stage with a high accuracy and avoid the need to predetermine the function type; 2) the approximate utility function of the carryover stage increases with the carryover storage and current inflow, and it changes gradually from a nearly linear surface to an approximate concave surface with the shift from the dry season to the flood season; 3) two-stage stochastic optimal operation models outperform the conventional operating rules and conventional optimization method in guiding the long-term hydropower operation.

Kinematic analysis and dimensional synthesis of a three-degrees-of-freedom hybrid-drive parallel mechanism

With the increasing demand for the attitude adjustment and vibration isolation of the precision instruments, this paper proposes a three-degrees-of-freedom hybrid-drive parallel mechanism and carries out the kinematic analysis and dimensional synthesis for it. First, the advantages of the hybrid-drive unit are introduced by comparison with the common driving methods. Then, the degrees of freedom of the hybrid-drive parallel mechanism is analyzed by constrained screw method, and the displacement mapping relationship between the generalized input and the moving platform is obtained by a successive approximation iteration method. Furthermore, the first-order and the second-order influence coefficients of the hybrid-drive mechanism are established on the basis of screw theory and the direct method. Besides, the operational dexterity index, the transfer performance index and the stiffness performance index of the hybrid-drive mechanism are introduced, and each single-objective dimensional synthesis of the hybrid-drive mechanism is carried out by the interior point method. Moreover, the multi-objective dimensional synthesis of the hybrid-drive mechanism is proposed by combining the genetic algorithm and the Pareto-optimal solution. Finally, taking the integer numbers of No.23 Pareto set as the numerical example, the kinematics formulas of the hybrid-drive mechanism are verified by the simulation and the experiment.

Effect of a Starting Model on the Solution of a Travel Time Seismic Tomography Problem

In the problems of three-dimensional (3D) travel time seismic tomography where the data are travel times of diving waves and the starting model is a system of plane layers where the velocity is a function of depth alone, the solution turns out to strongly depend on the selection of the starting model. This is due to the fact that in the different starting models, the rays between the same points can intersect different layers, which makes the tomography problem fundamentally nonlinear. This effect is demonstrated by the model example. Based on the same example, it is shown how the starting model should be selected to ensure a solution close to the true velocity distribution. The starting model (the average dependence of the seismic velocity on depth) should be determined by the method of successive iterations at each step of which the horizontal velocity variations in the layers are determined by solving the two-dimensional tomography problem. An example illustrating the application of this technique to the P-wave travel time data in the region of the Black Sea basin is presented.

A GPU based iteration approach to efficiently evaluate radiation symmetry for laser driven inertial confinement fusion

• Efficient computing approach presented for laser driven inertial confinement fusion • Guaranteed symmetry, strictly diagonally dominant and positive definite properties. • GPU based Preconditioned Conjugate gradient iteration approach. • The radiation models are efficiently solved and validated with two examples. Radiation computation is very important for high energy density experiments design in the laser-driven Inertial Confinement Fusion. The view-factor based models are often used to calculate the radiation on the capsule inside a hohlraum. However, it usually takes much time to solve them when the number of equations is very large. In this paper, an efficient iteration approach GPU is presented. The core idea is: (1) guaranteed symmetry, strictly diagonally dominant, and positive definite properties underlying the models are described, (2) a preconditioned conjugate gradient iteration approach is presented to compute the radiation based on such guaranteed properties, and (3) such approach is then parallelized and implemented for GPU so that the large scale models, especially for the non-linear model, can be efficiently solved in reasonable time. Finally, two experimental targets for Shenguang laser facilities built in China are demonstrated and compared to validate the efficiency of the presented approach. The results show that, the models’ computation (1) can be speeded up with successive over-relax iteration method by eight times as compared with Cholesky factorization based direct approach, (2) can be accelerated more with the preconditioned conjugate gradient iteration approach by almost eight times, and (3) can be further accelerated about 2 to 4 times as it parallelized and run on the GPU, which enables the large scale models, can be efficiently solved in reasonable time on the usual desktop computers.

BOUNDEDNESS OF SOLUTIONS FOR IMPULSIVE DIFFERENTIAL EQUATIONS WITH INTEGRAL JUMP CONDITIONS

The boundedness of solutions for certain nonlinear impulsive differential equations are obtained, the jumping conditions at discontinuous points are related to the integral of the past states, rather than a left hand limit at the discontinuous points. These results are obtained by new built impulsive integral inequalities with integral jumping conditions using the method of successive iteration.