Guofeng Zhang, Yuxin Zhang and Hengfei Ding
The purpose of this paper is to present a new family of iterative methods with eighth‐order convergence for solving the nonlinear equation.
Abstract
Purpose
The purpose of this paper is to present a new family of iterative methods with eighth‐order convergence for solving the nonlinear equation.
Design/methodology/approach
The paper uses a family of eighth‐order iterative methods for solving nonlinear equation based on Kou's seventh‐order method.
Findings
This family of methods is preferable to Ostrowski's, Grau's and Kou's methods in high‐precision computations.
Research limitations/implications
This paper only deals with the nonlinear equations.
Practical implications
This paper is concerned with the iterative methods for finding a simple root of the nonlinear equation f(x)=0. One of the reasons for discussing the solution of nonlinear equation is that many methods for high‐dimensional optimization problems involve solving a sub‐problem which is a one‐dimensional search problem. And the nonlinear finite element problem, the boundary‐value problems appearing in Kinetic theory of gases, elasticity and other applied areas are also reduced to solving such an equation.
Originality/value
New methods of this family require three evaluations of the function and one evaluation of its first derivative and without using the second derivatives per iteration. This new family of methods as a new example agrees with Kung‐Traub's conjecture for n=4 and achieves its optimal convergence order 2n−1.
Details
Keywords
Shiyuan Yang, Debiao Meng, Andrés Díaz, Hengfei Yang, Xiaoyan Su and Abilio M.P. de Jesus
Transporting hydrogen through natural gas pipelines in blended compositions has been proven to be a highly feasible solution in the short term. However, under hydrogen-rich…
Abstract
Purpose
Transporting hydrogen through natural gas pipelines in blended compositions has been proven to be a highly feasible solution in the short term. However, under hydrogen-rich environments, steel structures are prone to hydrogen-induced damage (HID). Additionally, uncertainties in various parameters can significantly impact the performance evaluation of hydrogen pipelines. Efficient reliability and sensitivity analyses of medium- to high-strength steel pipelines considering HID have become a challenge. Therefore, the primary aim of this study is to address this issue.
Design/methodology/approach
This study first establishes reliability analysis models for medium- to high-strength steels, represented by X65 and X80. In these models, the effect of HID is expressed by reduced stress, and its statistical parameters are calculated. Then, a highly efficient enhanced first order reliability method (FORM) is proposed for pipeline reliability analysis. This method overcomes the oscillation and convergence issues of traditional FORM when dealing with certain problems and can compute negative reliability indices. The proposed reliability analysis method is applied to solve the constructed reliability models. Finally, a reliability sensitivity analysis is conducted on the models to identify the key variables affecting the reliability of medium- to high-strength steel pipelines under HID.
Findings
First, two reliability analysis examples are used to validate the effectiveness of the proposed enhanced FORM. Then, using this method to solve the constructed reliability models for X65 and X80 steel pipelines under HID reveals that, for both types of steel, the reliability indices decrease significantly when considering HID compared to cases without HID. The decline is more pronounced for X80 steel than for X65 steel. As internal pressure increases, the reliability of both steels drops sharply, showing a concave parabolic trend. Moreover, the reliability sensitivity analysis shows that at a pressure of 10 MPa, for both X80 and X65, internal pressure, pipeline wall thickness and model error are the top three factors influencing reliability. As internal pressure increases, its influence becomes stronger, while the impact of other variables diminishes. Notably, for X80 steel, the presence of hydrogen amplifies the effect of internal pressure on pipeline reliability compared to when HID is not considered, but for X65, this trend is reversed.
Originality/value
Given the urgent need for safety evaluation studies on hydrogen transport through natural gas pipelines, this research provides new insights by constructing reliability models for X65 and X80 pipeline steels under HID and introducing an enhanced FORM method. The results of the reliability and sensitivity analyses of the models offer valuable insights and serve as a reference for engineering design.