Secant Method Research Articles

Study on a curvature compensation method for improving the end face matching of large-segment steel box girders during full-span erection

As precast segments are increasingly preferred for bridge construction owing to their efficiency and environmentally benign attributes, this study aims to enhance the end face matching technology for large-segment steel box girders. The curvature compensation method, which is a form of extension of the elevation compensation for pre-camber setting, is proposed based on the unstressed geometry of the steel box girder. It is found that the curvature compensation is equivalent to the end face angle compensation of segments if the construction technology of “combining several straight lines into a curve” is adopted. Three practical formulas for calculating curvature compensation angles are derived, namely, the moment method, intersection angle secant method, and incremental moment method, and theoretical values of the errors caused by using matching line shape calculation are provided. Furthermore, considering the specific continuous bridge with large-segment steel box girders in the Shenzhen-Zhongshan Channel as a case study, the calculation results of manufacturing parameters, matching line shapes, and end matching are presented. The findings indicate that the curvature compensation angle's maximum error reaches 55.7 % when calculated conservatively using erection geometry, while the manufacturing geometry's maximum error reaches 4.84 cm if the curvature compensation angle is disregarded. The results of on-site monitoring show that the curvature compensation method proposed in this study is able to provide the continuous fabrication geometry and achieve the perfect matching of end faces of neighbouring large-segment steel box girders.

Behavior of circular concrete-filled steel tubular columns with different end moment ratios: Modelling and discussion

This paper develops a nonlinear model for the analysis of circular CFST columns under equal and unequal end moments using fiber beam element (FBE) method. An iterative process adopting Newton’s method combined with secant method was proposed to improve the calculation efficiency. A test database was collected to assess the accuracy of the proposed model regarding the prediction of axial load-bearing capacity, deflection curves and load–deflection curves. The verified model was then further used to investigate the behavior of CFST columns under different end moments. It was found that the normalized sectional locations of maximum deflection of CFST columns are only significantly affected by the end moment ratio, whereas those of critical sections are also influenced by the axial load. The specimen with a larger end moment ratio, steel strength, L/D ratio and/or a smaller e1/D ratio, D/t ratio experiences more significant second order effect. The bending stiffness of CFST columns is hardly affected by the end moment ratio. Moreover, the current codes, i.e., EC4 and AISC 360, may underestimate the capacities of CFST columns under unequal end moments.

Secant-inexact projection algorithms for solving a new class of constrained mixed generalized equations problems

In this paper, a new version of a secant-type method for solving constrained mixed generalized equations is addressed. The method is a combination of the secant method applied to generalized equations with the conditional gradient method. We use the contraction mapping principle to establish the convergence results. Moreover, by assuming the Lipschitz condition on the gradient and the metric regularity property, we show that the sequence generated by the proposed algorithm is well-defined and locally convergent for a solution with linear or superlinear rate.

An Analytical Method for Mechanical Analysis of Offshore Pipelines during Lifting Operation.

The lifting operation of offshore pipelines is an important step in ocean pipeline engineering. An effective analytical method is developed for investigating the mechanical properties of the pipeline based on mechanical, physical, and geometric relationships. By using the shooting and the secant methods to transform the boundary value problem into an initial value one and then solving them with the Runge-Kutta method, the deformation and mechanical properties of the pipeline are calculated. Furthermore, based on the Det Norske Veritas (DNV) offshore standard, the mechanical properties of the pipeline are checked. The finite element method (FEM) by Orcaflex is employed to verify the accuracy of the analytical model. The effects of some factors such as the current velocity and lifting point position on the mechanical properties of the pipeline are analyzed based on the analytical model. The results indicate that the change in current velocity during the lifting process has a minimal effect on the pipeline, but the change in lifting point position significantly affects the deformation and mechanical properties of the pipeline.

Reduced basis stochastic Galerkin methods for partial differential equations with random inputs

We present a reduced basis stochastic Galerkin method for partial differential equations with random inputs. In this method, the reduced basis methodology is integrated into the stochastic Galerkin method, resulting in a significant reduction in the cost of solving the Galerkin system. To reduce the main cost of matrix-vector manipulation involved in our reduced basis stochastic Galerkin approach, the secant method is applied to identify the number of reduced basis functions. We present a general mathematical framework of the methodology, validate its accuracy and demonstrate its efficiency with numerical experiments.

Optimal design of a Low-Cost SAE JA2954 compliant WPT system using NSGA-II

Wireless Power Transfer (WPT) systems for electric vehicle charging are one of the most promising methods that, given the advantages they bring, will help the desired deployment of electric vehicles. This paper presents a mathematical optimisation method applied to the design of an 11 kW S-S system that complies with the SAE J2954 standard. A proposal is made to calculate the electrical parameters of the circuit based on equations that are compared with the results obtained by simulation with finite elements and experimental measurements, achieving very tight results with a reduced computational time. The NSGA-II multi-objective genetic algorithm is then applied together with the secant method, defining three different scenarios: minimisation of the primary copper volume, minimisation of the secondary copper volume and a compromise solution optimising the total primary and secondary copper volume. The result is a set of Pareto optimal solutions, from which the one that meets the standard can be extracted that suits the designer’s needs.

Newton’s forward interpolation method for solving nonlinear algebraic equation

As we know, the interpolation is one of the most basic and most useful numerical techniques in Mathematics. Newton’s forward interpolation method is one of most important of these methods. Its most important task in numerical analysis to find roots of nonlinear equations, several methods already exist to find roots. But in this paper, we introduce the interpolation technique for this purpose. The proposed method derived from the newton forward interpolation method and we compared the results with another existing method (Bisection Method (BM), Regula-Falsi Method (RFM), Secant Method (SM), Newton Raphson Method (NRM)) and the method proposed by J. Sanaullah (SJM). It’s observed that the proposed method has fast convergence but it has same order of convergence of the method (SJM). Maple software is used to solve problems by different methods.

Analysis of entropy generation in Carreau ternary hybrid nanofluid flow over a stretching sheet

In this study, we consider a steady-state viscous,laminar Carreau ternary-hybrid nanofluid flow over a stretching sheet with first-order slip and Ohmic dissipation.Further, we consider the effects of suction/injection, oblique Lorentz force, and a temperature jump boundary condition. We investigate here how to transport a Carreau ternary hybrid nano-fluid while minimizing the entropy it generates. We analyze in this article the flow of molybdenum disulfide (MoS2), silver (Ag), and multi-wall carbon nanotubes (MWCNT). Carboxymethyl cellulose (CMC-water) is the observed fluid of choice. Several industries could benefit from this research, including those dealing with the equations of the model are converted to a dimensionless form by appropriate similarity transformations. Using a shooting strategy with Runge-Kutta-Fehlberg and Secant methods the velocity and the temperature fields are obtained. The physical quantities such as the fluid flow rate, the temperature, the skin friction, the Nusselt number, the entropy minimization, and the Bejan number are presented graphically and analyzed. In our Carreau ternary hybrid nanofluid model it is found that an increase in the Weissenberg number is to decrease the velocity field. We also noticed a significant effect of the Brinkmann number which has a deleterious effect on the entropy-minimization and Bejan-number profiles.

A Modified Secant Method for Entropic Lattice Boltzmann Equations

Stability of lattice Boltzmann equations is governed by a parameter that is responsible for the relaxation time of the nonequilibrium system which, in turn, affects the viscosity of the flow under examination. In the entropic approach, the relaxation time is evaluated from the entropy balance equation in such a way that the entropy does not decrease at each time and spatial point. In this paper, a technique for solving the entropy balance equation using a modified secant method is proposed. It is shown that this approach provides high accuracy. As an application of the proposed method, numerical solutions of the two-dimensional double shear problem are considered. The simulation results are compared with the results obtained by other entropic methods.

Graphical User Interface for Solving Non-Linear Equations for Undergraduate Students

Solving non-linear equations problems for root-finding is more challenging rather than linear equations problems. Generally, this is because it cannot be solved by using the standard formula. The numerical method is one of the ways that can be applied, consisting of an iterative procedure. A simulator named Root Calculator to Solve Non-Linear Equations was developed, where it is a graphical user interface (GUI) system that can be used to approximate the root based on the equation and interval given. Four methods were chosen which are the Graphical Method, the Bisection Method, Newton’s Method, and the Secant Method to produce the approximation root. App Designer in MATLAB R2020b software was used for this development.

Steady state population balance modelling of precipitation processes: Nucleation, growth and size-dependent agglomeration

In this work a numerical methodology to solve the steady state Population Balance Equation (PBE) is developed. Three crystallisation mechanisms are included, namely: nucleation, size-independent growth and size-dependent loose agglomeration. The numerical method is based on the discretisation of the crystal size as distributed variable. In order to describe the loose agglomeration, the numerical methodology solves two PBE: one including the nucleation and growth mechanisms and one accounting for the agglomeration process. From the first PBE, liquid phase composition, supersaturation, developed crystal surface and Crystallite Size Distribution (CSD) are obtained. Similarly, the second PBE leads to the Agglomerate Size Distribution (ASD). The study of the size-dependant agglomeration kernel induces an additional numerical difficulty due to the dependency of both PBE and agglomeration kernel on the particle size. An accelerated fixed point algorithm based on the crossed secant method is adapted to overcome the difficulty and accurately solve the agglomeration PBE. The oxalic precipitation of uranium is simulated using this numerical methodology. First, the experimental results of a reference case are compared with the numerical predictions in terms of particle size distribution, mean size, mass fraction and moments. Then, the operating conditions are varied in order to test the algorithm robustness and performances. In all cases, the crossed secant method ensures the size-dependent agglomeration PBE solution and properly predicts the ASD. The developed numerical methodology predicts the mean particle size under the experimental uncertainty in a reasonable computation time and number of iterations.

Extrapolation methods as nonlinear Krylov subspace methods

When applied to linear vector sequences, extrapolation methods are equivalent to Krylov subspace methods. Both types of methods can be expressed as particular cases of the multisecant equations, the secant method generalized to higher dimensions. Through these equations, there is also equivalence with a variety of quasi-Newton methods. This paper presents a framework to connect these various methods.

The development of Newton’s method by enhancing the starting point

In general, nonlinear problems cannot be solved analytically. A special theory or method is needed to simplify calculations. Many problems that are too complex, an exact solution is needed to support numerical solution. There are many numerical methods that can be used to solve nonlinear problems, including the Bisection, Secant and Newton methods, also known as the Newton-Raphson method. However, these methods cannot be used for large-scale of nonlinear programming problems; for example the Newton-Raphson method which does not always converge if it takes the wrong initial value. The Newton-Raphson method is widely used to find approximations to the roots of real functions. However, the Newton-Raphson method does not always converge if it takes the wrong initial value. Therefore, it is necessary to develop the Newton-Raphson method without using other methods in order to have a higher convergence. This research is a literature study compiled based on literature references with the initial step of understanding problems that appear from the use of Newton's method, it is base on the problem of divergence or oscillation. Newtonian method was developed without modification of other methods, but took two starting points. Then prove the super-quadratic convergence of the proposed method by extending the Taylor expansion and giving or assuming the error rate. After that, the stability test of the proposed model is carried out and provides an example of the application by solving the root search using Newton's method and the proposed method can be seen as a comparison.

Numerical solution of an extra-wide angle parabolic equation through diagonalization of a 1-D indefinite Schrödinger operator with a piecewise constant potential

We present a numerical method for computing the solution of a partial differential equation (PDE) for modeling acoustic pressure, known as an extra-wide angle parabolic equation, that features the square root of a differential operator. The differential operator is the negative of an indefinite Schrödinger operator with a piecewise constant potential. This work primarily deals with the 3-piece case; however, a generalization is made the case of an arbitrary number of pieces. Through restriction to a judiciously chosen lower-dimensional subspace, approximate eigenfunctions are used to obtain estimates for the eigenvalues of the operator. Then, the estimated eigenvalues are used as initial guesses for the Secant Method to find the exact eigenvalues, up to roundoff error. An eigenfunction expansion of the solution is then constructed. The computational expense of obtaining each eigenpair is independent of the grid size. The accuracy, efficiency, and scalability of this method is shown through numerical experiments and comparisons with other methods.

Estimation of A Partial Linear Model with Instrumental Variable for the Longitudinal Data

A partial linear model with instrumental variables was developed for longitudinal data. In the partially linear model, the explanatory variable is an endogenous variable, which is correlated with the error term. The endogenous variable was expressed by an instrumental variable and an error item. The endogenous variable was estimated by the instrumental variable through the least square method. B-spline regression combined with QR decomposition was used to approximate the nonparametric function. For the estimation of parametric, the Quadratic inference function and Secant method were applied. Under some conditions, the estimator was consistent and asymptotic normality. Some simulation was conducted to prove the finite sample behavior of the estimator.

Unified Convergence Criteria of Derivative-Free Iterative Methods for Solving Nonlinear Equations

A local and semi-local convergence is developed of a class of iterative methods without derivatives for solving nonlinear Banach space valued operator equations under the classical Lipschitz conditions for first-order divided differences. Special cases of this method are well-known iterative algorithms, in particular, the Secant, Kurchatov, and Steffensen methods as well as the Newton method. For the semi-local convergence analysis, we use a technique of recurrent functions and majorizing scalar sequences. First, the convergence of the scalar sequence is proved and its limit is determined. It is then shown that the sequence obtained by the proposed method is bounded by this scalar sequence. In the local convergence analysis, a computable radius of convergence is determined. Finally, the results of the numerical experiments are given that confirm obtained theoretical estimates.

An improved method for computing acoustic ray incident angle based on secant method

Constant gradient acoustic ray tracking is usually used in high-precision underwater acoustic positioning. However, it needs to calculate the incident angle of the acoustic ray, and the traditional incident angle computation method is very inefficient. To resolve this problem, we propose an improved acoustic ray incident angle iterative computation method. It firstly constructs the mathematical model for incident angle by using the function between horizontal propagation distance and incident angle based on secant method. The initial value of the incident angle is then determined by the incident angle calculated from the previous epoch and the last iteration. Meanwhile, the root-containing interval of the incident angle is reduced according to experimental parameters and coordinate correction values. Finally, the incident angle calculated in the last iteration is directly used for acoustic ray tracking and positioning solution until the end, when the coordinate correction values reach a threshold. Experiment conducted on the simulation cases demonstrates that the positioning calculation time of the proposed method is 77.7%, 77.1% and 88.1% lower than traditional methods in order of the dichotomy, secant method, and Newton method, respectively. As for the real experiment of the South China Sea, the computation efficiency improvement of the proposed method is about 62.9%, 67.6% and 66.1% compared with the dichotomy, secant method, and Newton method respectively. These results suggest that the proposed method can significantly improve the positioning calculation efficiency without loss of positioning accuracy compared with conventional methods.

Modified Secant Method for Reduction in Number of Iterations

Modified Secant Method for Reduction in Number of Iterations

On the set of initial guesses for the secant method

The secant method is the most used iterative method to solve an operator equation where the operator involved is nondifferentiable. A known problem that arises when applying this method is its accessibility. Then, we try to improve it by using the technique of decomposition of the operator involved, so that the operator is in turn the sum of two operators, one differentiable and the other continuous but not differentiable. For this, we use a family of Newton‐secant‐type iterative methods that arises from Newton's method and from a well‐known family of secant‐type iterative methods. We study the accessibility of the new methods in two different ways: From the convergence balls of the methods, obtained from a local study of the convergence, and from a dynamic study of the methods. Some examples related to chemistry are also presented to prove the theoretical results.

A new multipoint symmetric secant method with a dense initial matrix

In large-scale optimization, when either forming or storing Hessian matrices are prohibitively expensive, quasi-Newton methods are often used in lieu of Newton's method because they only require first-order information to approximate the true Hessian. Multipoint symmetric secant (MSS) methods can be thought of as generalizations of quasi-Newton methods in that they attempt to impose additional requirements on their approximation of the Hessian. Given an initial Hessian approximation, MSS methods generate a sequence of possibly-indefinite matrices using rank-2 updates to solve nonconvex unconstrained optimization problems. For practical reasons, up to now, the initialization has been a constant multiple of the identity matrix. In this paper, we propose a new limited-memory MSS method for large-scale nonconvex optimization that allows for dense initializations. Numerical results on the CUTEst test problems suggest that the MSS method using a dense initialization outperforms the standard initialization. Numerical results also suggest that this approach is competitive with both a basic L-SR1 trust-region method and an L-PSB method.