Predictor-corrector Iterative Method Research Articles

Efficient FSI solvers for multiple-degrees-of-freedom flow-induced vibration of a rigid body

Flow-induced vibration (FIV) of a bluff body is a complex fluid-structure interaction (FSI) problem, for which analytical solutions generally don’t exist. Therefore, such problems generally need to be examined with experimental methods or computational simulations. Researchers have developed various numerical methodologies to solve FSI problems using either conforming or non-conforming grid methods. However, these methods are not always ideal for flow-induced vibration problems, as they can be computationally expensive or generate low accuracy solutions, especially for the cases with large body displacements. FIV problems often require long simulation times because, for a given parameter set, the transient flow solution time can be long and generally at least 10 oscillation cycles are required to simulate the representative long term behaviour. Thus, to gain a clear understanding and enhance knowledge of FIV of a bluff body, it is advantageous to have an efficient numerical methodology. We have developed two efficient fully-coupled FSI solvers to accurately predict the FIV of an elastically mounted bluff body and a tethered bluff body. These FSI solvers were developed based on the widely used open-source CFD package OpenFOAM. In these solvers, the fluid flow was modelled in a reference frame attached to the centre of mass of the solid body, so that a non-deforming grid can be employed. A predictor-corrector iterative method was used to enable strong coupling between the solid motion and the fluid flow. Each of the FSI solvers was validated against previously reported investigations. While efficient, the limitation of these FSI solvers is that they can only be used to examine the nature of FIV of a single, rigid bluff body.

A predictor-corrector iterative method for solving linear least squares problems and perturbation error analysis

The motivation of the present work concerns two objectives. Firstly, a predictor-corrector iterative method of convergence order p=45 requiring 10 matrix by matrix multiplications per iteration is proposed for computing the Moore–Penrose inverse of a nonzero matrix of operatorname{rank}=r. Convergence and a priori error analysis of the proposed method are given. Secondly, the numerical solution to the general linear least squares problems by an algorithm using the proposed method and the perturbation error analysis are provided. Furthermore, experiments are conducted on the ill-posed problem of one-dimensional image restoration and on some test problems from Harwell–Boeing collection. Obtained numerical results show the applicability, stability, and the estimated order of convergence of the proposed method.

New Predictor-Corrector Iterative Methods with Twelfth-Order Convergence for Solving Nonlinear Equations

In this paper, we propose and analyze new two efficient iterative methods for finding the simple roots of nonlinear equations. These methods based on a Jarratt's method, Householder's method and Chun&Kim's method by using a predictor-corrector technique. The error equations are given theoretically to show that the proposed methods have twelfth-order convergence. Several numerical examples are given to illustrate the efficiency and robustness of the proposed methods. Comparison with other well-known iterative methods is made.

Higher-order iterative methods by using Householder's method for solving certain nonlinear equations

In this paper, we suggest and analyze some new higher-order iterative methods by using Householder's method free from second derivative for solving nonlinear equations. Here we use new and different technique for implementation of higher-order derivative of the function and derive new higher- order predictor-corrector iterative methods free from second derivative. The efficiency index equals to

On Steffensen’s method on Banach spaces

We present a modification of Steffensen’s method as a predictor–corrector iterative method, so that we can use Steffensen’s method to approximate a solution of a nonlinear equation in Banach spaces from the same starting points from which Newton’s method converges. We study the semilocal convergence of the predictor–corrector method by using the majorant principle. We illustrate the method with an application to a discrete problem.

Improving the domain of starting points for secant-like methods

From the simplified secant method and a family of secant-like methods, we construct a family of predictor–corrector iterative methods for solving a nonlinear equation in Banach spaces.

A third-order convergent iterative method for solving non-linear equations

In this paper, we present and analyse a new predictor-corrector iterative method for solving non-linear single variable equations. The convergence analysis establishes that the new method is cubically convergent. Numerical tests show that the method is comparable with the well-known existing methods and in many cases gives better results.

Modified Efficient Families of Two and Three-Step Predictor-Corrector Iterative Methods for Solving Nonlinear Equations

In this paper, we present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting near the root. The main advantage of our methods is that they perform better and moreover, have the same efficiency indices as that of existing multipoint iterative methods. Furthermore, the convergence analysis of the new methods is discussed and several examples are given to illustrate their efficiency.

Some variants of Newton's method with fifth-order and four-order convergence for solving nonlinear equations

Chun's two-step predictor-corrector type iterative method for solving nonlinear equations (Iterative Methods Improving Newton's Method by the Decomposition Method, J . Comput. Math. Appl.50, 1559-1568 ) is improved and a kind of new iterative method is presented. Some present methods derived from Adomian Decomposition method are unified in a form and a series of new methods with high convergence order and the value of EFF are obtained by introducing parameter also. Those methods can be considered as a significant improvement of the Newton's method and its variant forms.

A plastic nonlocal damage model

In the present paper, a plastic nonlocal damage model is proposed for studying the mechanical response of structural elements made of cementitious materials. A new isotropic damage model, which is able to describe the behavior of a wide class of cementitious materials, is presented. A regularization technique, based on the introduction of the damage Laplacian in the damage limit function, is adopted to overcome the analytical and computational problems induced by the softening constitutive law. A Drucker–Prager type of plastic limit function is proposed considering isotropic hardening. A numerical procedure, based on an implicit `backward-Euler' technique for the time integration of the plastic and damage evolution equations, is presented. To solve each nonlinear step, a predictor–corrector iterative method is developed within the splitting method. In particular, the damage evolution is determined solving a constrained minimization problem of a convex functional. The proposed algorithm is implemented in a finite element code and it is used to study the structural behavior of elements made of masonry materials.