Secant Method Research Articles

A numerical strain-softening solution of a circular opening in nonlinear yield rock masses

Strain-softening behavior occurs when the surrounding rock reaches its peak load, complicating the prediction of stress and displacement. The linear Mohr-Coulomb yield criterion and semi-linear Hoek-Brown yield criterion are widely used in elastoplastic analysis. However, the established criteria struggle to universally describe the nonlinear behavior of various rock types. This study aims to overcome the difficulty and propose the solution of surrounding rock in a circular tunnel following any nonlinear yield criterion and plastic potential function. The derivation of the equivalent cohesive strength and frictional angle for an arbitrary nonlinear yield criterion in the form of σ1=fσ3 using the secant method, along with the determination of the equivalent dilatancy angle for any nonlinear plastic potential functions, was conducted. Concerning the elastoplastic solution of a thick-cylinder, a numerical strain-softening solution for a circular tunnel in nonlinear yield rock masses was derived. Verification of the proposed solution was performed against the established analytical and/or numerical elasto-(brittle-)plastic and strain-softening solutions in rock masses following the Mohr-Coulomb, (generalized) Hoek-Brown, and unified strength criteria. The impact of critical plastic shear strain on displacement, plastic radius, and residual radius was further analyzed. The predictions of displacement and residual region for the Jinping II project also meet the field test results.

The influence of mineral inclusion on the effective strength of rock-like geomaterials

The influences of microstructure on the macroscopic mechanical behavior of a composite with a porous matrix reinforced by mineral inclusions are investigated in the present work by both numerical and theoretical methods. The mineral inclusions are embedded at the mesoscopic scale and much bigger than the pores which are located at the microscopic scale. In order to consider the properties of the studied rock-like geomaterials, such as the dissymmetry between tension and compression, the solid phase at the microscopic scale is assumed to obey to a Drucker–Prager criterion. Based on the studied microstructure, the Fast Fourier Transform (FFT) based numerical method is firstly adopted to investigate the macroscopic plastic yield stress of the studied composite. Different microstructure having different volume fraction of inclusion, micro-porosity and frictional coefficient of the solid phase are considered. Based on these numerical results, the existing theoretical yield criterion is estimated. One finds that it should be improved for the case of a microstructure having a high inclusion content. Then, a new macroscopic yield criterion is constructed in the present work by using the modified secant method. This criterion ameliorates fundamentally the one proposed in Shen et al. (2013), specially for the case of a deviatoric loading. It is then estimated and validated by comparing with the FFT based numerical results which were carried out in this work with different volume fractions of heterogeneous phase.

Numerical Procedure for Solving the Nonlinear Behaviour of a Spherical Absorber

The aim of the paper is to perform numerical simulations for a system of nonlinear differential equations that describe the behaviour of a spherical absorber placed in a support bowl and to describe the applied techniques. The motion of the sphere is constrained to a plane problem. The derived system of equations is numerically solved using the continuation method and the modified secant method. The absorber's response to different harmonic excitation forces is simulated to demonstrate the applicability of these techniques in its analysis. The dependence of the response amplitude on the angular frequency of excitation is presented graphically. The results also include a response stability analysis using the Routh-Hurwitz criteria.

PERBANDINGAN ESTIMASI VOLATILITAS HARGA OPSI BELI SAHAM APPLE INC. (AAPL) DENGAN METODE BISECTION DAN SECANT

Stock price volatility is a measure of how far a stock price moves in a given Stock price volatility is a measure of how far a stock price moves in a given time. The theory developed by Black-Scholes states that every option price with the same 'underlying asset' and the same time to maturity but with different exercise values will have the same Implied Volatility value. However, this is not always the case in the market. Therefore, it is necessary to estimate volatility known as Implied Volatility, which is considered an appropriate method in estimating volatility values. This study compares the Bisection and Secant methods to estimate the volatility of Apple Inc. (Aapl) stock. This study uses data on the closing price of the stock in the period September 29, 2022 to September 29, 2023. Volatility estimation for the Bisection and Secant methods by determining the initial approximation and limiting it to a maximum of 100 simulations and iteration stops if it has produced a relative error smaller than = . The is an error tolerance limit, the smaller the error tolerance, the more accurate it is. According to the research results, the Bisection method produces an estimated volatility value of 0.498212 at the 9th iteration, while the Secant method produces an estimated value of 0.498590 at the 10th iteration. The Secant method produces a smaller relative error value of 0.000096, indicating that the Secant method is more accurate than the Bisection method.

A global springback compensation method for manufacturing metallic seal ring with complex cross-section and submillimeter precision

The fabrication of complex thin-walled components with irregular cross-sectional geometries often necessitates multi-pass forming processes. The springback phenomena at the microscale level contribute significantly difficulty to the precise control of geometrical accuracy for such components. To achieve submillimeter precision control of micro features in metal seal rings, this research proposed a global springback compensation method to facilitate the attainment of the optimized surfaces of dies. Through the analysis of changes in cross-sectional shape, stress distribution and bending moment in the multiple stages, several partition points were set up to divide the cross-section into four zones. For a partitional control of the manufactured surface, the compensation magnitudes for each segment are determined by iteratively refining the compensation factors using the secant method. The iteration of compensation factors is based on the deviation of dimensions measured by finite element simulation, which greatly saves experimental costs. The experimental results indicate that the predictive dimensional alterations by the global compensation agree with the experimental ones exhibiting a maximum error of less than 3 %. Furthermore, the springback behavior during the unloading and trimming stages coupling caused a dimensional error of 14.9 % for the wave width and 8.3 % for the peak width. Following the application of coupled compensation through two iterations, a distinct decrease in the relative average errors of springback values for both part width and peak width, achieving reductions to 2.4 % and 0.6 %, respectively. Compared to the traditional single springback compensation technique, the global compensation strategy effectively mitigates the out-of-tolerance issues induced by the trimming process.

Numerical Methods for Statistical Data Generation

A comprehensive comparison was undertaken between two numerical methods, namely bisection and secant methods, to generate random variable data for distribution. two methodologies were suggested to acquire the initial guesses for secant and interval for the bisection method. The results of 100 cases for the choices of the parameters are shown and a comparison was made according to the number of iterations between the methods that was conducted.

On Spectral Dai-Yuan Structured Technique for Solving Nonlinear Least-Squares Optimisation

Handling nonlinear least squares (NLS) problems as unconstrained optimisation problems has led to the development and adaptation of numerous numerical approaches. Calculating the Hessian matrix expensively for each iteration is one of the problems with the current numerical approaches for solving NLS problems. Some methods designed to rely solely on first-derivative information while ignoring second-order details tend to underperform when dealing with problems involving non-zero residuals. This paper suggests a modification to the Dai-Yuan type (DY) formula by deriving a spectral parameter for the proposed search direction. The sufficient descent results of the new formula are established using the standard assumptions under the strong Wolfe line search. Furthermore, experiments are conducted on some benchmark functions to demonstrate the computational efficiency of the proposed technique. The results highlight the effectiveness of the algorithm compared to existing methods.Keywords: Conjugate gradient, convergence theory, Hessian matrix, optimisation, Quasi-Newton, spectral, structured secant method

Comparison of the Number of Iterations of Roots of Non-Linear Equations in the Regula Falsi Method, Secant Method, and Bisection Method using Python

Comparison of the Number of Iterations of Roots of Non-Linear Equations in the Regula Falsi Method, Secant Method, and Bisection Method using Python

PUSHOVER ANALYSIS OF EARTHQUAKE RESISTANCE RCC STRUCTURE USING STAAD.PRO SOFTWARE

RCC structure has become crucial to the construction industry in recent decades. A structure must be designed with strong seismic performance in mind. A multi-story reinforced concrete building's seismic performance is planned in compliance with the current Indian code (IS 456 -2000). The study of each structural members performance is conducted using nonlinear static analysis. This paper study about a popular method to ascertain the performance level of a building is through pushover analysis, which yields results that indicate whether collapse occurs at the member or structure level. Performance-based design is used to evaluate the performance level of buildings subjected to earthquakes. Various approaches in nonlinear static analysis were developed to study the performance of a building. This assessment has taken into account a non-existing solid Rcc structure that is assumed to be located in Zone IV, as defined by IS 1893:2000, a classification of seismic zones in India. During the entire inquiry, brick work infills have been considered to be non-structural individuals. Systems for inelastic static static study include the Secant Method, the Displacement Coefficient Method, and the Capacity Spectrum Method. Pushover Analysis, a non-direct static approach, was used to examine the structure's weak points, building on static testing. The structure was recreated in SeismoStruct Version 5.2.2 using M20 cement and Fe415 steel support, as predicted in STAAD Pro v8i. After that, the research is carried out for 150% of the predetermined objective displacing for the structure to track individual yielding and the sufficiency of the core quality. The degree of damage suffered by the structure during the objective displacing is thought to represent the damage the structure would experience under configuration level ground shaking. Key Words: Pushover analysis, RCC structure, Pushover curve, drift velocity, STAAD.PRO, Earthquake structure.

A numerical method to consider the interaction between multiple fractures in frozen rocks based on XFEM

The initiation, propagation, and coalescence of fractures in frozen rocks plays an important role in determining the deterioration and stability of the geotechnical engineering structure in cold regions. This study proposes a numerical method to consider the interaction of multiple fractures in frozen rock based on the volumetric expansion theory and extended finite element method (XFEM). The effective expansion coefficient is adopted in this paper to describe the volume change of ice-water mixture. To accurately determine the magnitude of frost-heaving pressure during the deformation and propagation of fractures, two iterative methods have been proposed. The correctness of the numerical method is verified by the analytical solution of frost-heaving pressure within a single fracture. In addition, the secant method shows the higher efficiency in solving the problems considering the fracture interaction. The evolution of frost-heaving pressure and cracking behavior under different geometric conditions were studied by the proposed method. The numerical results suggest that stress interference may promote the propagation of fractures and accelerate the deterioration of rock mass when the fractures are located in the favorable zone. After considering the multi-fracture interaction, the configuration of two fractures has a significant influence on the evolution of the frost-heaving pressure.

Calculation and Analysis of Equilibrium Position of Aerostatic Bearings Based on Bivariate Interpolation Method

The solution of equilibrium positions is a critical component in the calculation of the dynamic characteristic coefficients of aerostatic bearings. The movement of the rotor in one direction leads to bidirectional variations in the air film force, resulting in low efficiency when using conventional calculation methods. It can even lead to iterative divergence if the initial value is improperly selected. This study concentrates on the orifice throttling aerostatic bearings and proposes a novel method called the bivariate interpolation method (BIM) to calculate the equilibrium position. The equilibrium equation for the rotor under the combined influence of air film forces, gravity, and external loads is established. A calculation program based on the finite difference method is developed to determine the equilibrium position. The process of solving the equilibrium position and the convergence is compared with the secant method and the search method. Furthermore, the variation trend of the equilibrium position and stiffness when the external loads changes are studied based on the BIM. Finally, the correctness of the BIM to solve the equilibrium position is proved by comparing it with the experiment results. The calculation results indicate that the BIM successfully resolves the problem of initial value selection and exhibits superior computational efficiency and accuracy. The equilibrium position initially moves away from the direction of the external load as the load increases, and then this gradually approaches the load direction. The main stiffness increases with increases in the external load, while the variation in cross stiffness depends on the direction of the external load.

On the application of numerical differentiation methods to the determination of the fatigue crack growth rate

Using a sample of test results from 68 compact eccentric tensile specimens made of titanium alloys, nickel alloys and steel, the effect of the choice of numerical differentiation method (secant method and the method of differential polynomials on three, five and seven points) used to calculate the fatigue crack growth rate on characteristics of the linear section of the kinetic diagram of the fatigue failure. The purpose of the study is to determine the advantages, disadvantages and consistent patterns of the considered methods. The coefficient of determination R 2, integral criterion χ which characterizes the difference between the predicted and actual number of cycles corresponding to the section of stable crack growth, and correlation between the logarithms of the Paris constants for alloys of the same class were used as criteria for the correct choice of the method of numerical differentiation. The main results and conclusions of the study: the use of the method of differential polynomials over three points compared to the secant method slightly increases the correlation between the logarithms of the fatigue crack growth rate and the range of the stress intensity factor (an increase in R 2) and increases the difference between the calculated and experimental number of cycles corresponding to stable crack growth (an increase in χ). However, when determining the fatigue crack growth rate by the method of differential polynomials for five and seven points, a more significant smoothing of the experimental data is observed, accompanied by a significant increase in R 2 and a decrease in χ; proximity to zero of the integral accuracy parameter χ is a necessary but not sufficient criterion for good agreement between the test result and the mathematical model that describes it, while the combination of parameters χ and R 2 uniquely forms this criterion; the choice of the method of numerical differentiation does not affect the correlation of the logarithms of the constants of the Paris equation.

The optimal scaling factor based on Secant Method for image watermarking using the hybrid DTCWT-DCT domain

This paper proposes an implementation of the secant optimization method on the watermarkembedding in DTCWT domain. An objective function combining PSNR and NC (The Normalized Correlation)values calculated for a set of common attacks is used in the optimization process. The aim is to enhance the tradeoff between robustness and imperceptibility against the various image processing attacks via an automatic andcareful selection of the scaling factor using optimization. The watermark is inserted in the high energy bands ofDCT-transformed sub-vectors, which were generated from the DTCWT coefficient decomposition of the hostimage. Experimental results performed on a set of 512×512 greyscale images, show better results than theexisting schemes for most common image attacks such as compression, low-pass filtering and noise addition.

Convergence and Stability Improvement of Quasi-Newton Methods by Full-Rank Update of the Jacobian Approximates

A system of simultaneous multi-variable nonlinear equations can be solved by Newton’s method with local q-quadratic convergence if the Jacobian is analytically available. If this is not the case, then quasi-Newton methods with local q-superlinear convergence give solutions by approximating the Jacobian in some way. Unfortunately, the quasi-Newton condition (Secant equation) does not completely specify the Jacobian approximate in multi-dimensional cases, so its full-rank update is not possible with classic variants of the method. The suggested new iteration strategy (“T-Secant”) allows for a full-rank update of the Jacobian approximate in each iteration by determining two independent approximates for the solution. They are used to generate a set of new independent trial approximates; then, the Jacobian approximate can be fully updated. It is shown that the T-Secant approximate is in the vicinity of the classic quasi-Newton approximate, providing that the solution is evenly surrounded by the new trial approximates. The suggested procedure increases the superlinear convergence of the Secant method φS=1.618… to super-quadratic φT=φS+1=2.618… and the quadratic convergence of the Newton method φN=2 to cubic φT=φN+1=3 in one-dimensional cases. In multi-dimensional cases, the Broyden-type efficiency (mean convergence rate) of the suggested method is an order higher than the efficiency of other classic low-rank-update quasi-Newton methods, as shown by numerical examples on a Rosenbrock-type test function with up to 1000 variables. The geometrical representation (hyperbolic approximation) in single-variable cases helps explain the basic operations, and a vector-space description is also given in multi-variable cases.

Fuel consumption optimization of a multimachines microgrid by secant method combined with IPPD table

The objective consists in optimizing the fuel consumption of a power station of small cogeneration multi machines while minimizing generated pollution. At first, a study carried out on the strategies of Economic Dispatch , was used to us to solve the unit commitment problem, whose main aim is to determine the optimal schedule of online generating units so as to meet the power demand at minimum operating cost under various system and operating constraints. This work was carried out by Secant method combined with Improved Pre-prepared Power Demand (IPPD) table which obtains the unit status information and then the optimal solution is achieved by Secant method at each power demand for 24 hours. The comparison of results of the proposed method with the results of classical method, by Matlab/Simulink, shows that the proposed approach provides qualitative solution.

Adaptation of nanofluids with magnetohydrodynamic Williamson fluid to enhance the thermal and solutal flow analysis with viscous dissipation: A numerical study

The Williamson nanofluid flow along with the thermal and solutal transport effects like enthalpy and viscous dissipation, has lots of applications in numerous engineering fields such as heat exchangers, automotive engineering, cooling systems, aerospace, and biomedical applications. The temperature-dependent viscosity has implications in the polymer processing, chemical processing, microfluidic devices, environmental fluid dynamics, and oil and gas industries. The current research deals with the numerical study of the magnetohydrodynamic flow of 2-dimensional Williamson nanofluid flowing on a stretched cylinder. The thermophysical property (viscosity) is assumed to depend on temperature for this flow system. The transmission analysis of solutal and heat processes, along with enthalpy and viscous dissipation, is also considered here. The boundary layer approach is used to address the main leading equations, and ODEs are generated from the main equations by assuming similarity transformations. The main novelty of this problem is the use of a multistep technique, which is the Adams-Bashforth Predictor and Corrector method, along with the RK-4 scheme and the secant method. The key parameters are used to illustrate the graphical and numerical behavior of the velocity, temperature, and concentration regions. The variations in the Williamson fluid coefficient cause the fluid velocity to increase, whereas alterations in the curvature parameter cause the velocity region to decrease. The temperature in a given area declines due to the impact of the Prandtl number. The fluids temperature shows augmented values in the occurrence of variation in the thermopherosis coefficient, Brownian motion coefficient, curvature parameter, and heat generation parameter. The curvature parameter, thermopherosis coefficient, and activation energy lead to an increase in the fluid concentration region, whereas a decrease in concentration is noted due to the Brownian motion coefficient, Damkholer number and Schmidt number.

Implementing the multistep adams-bashforth numerical approach on magnetohydrodynamic radiated casson fluid with darcy forchheimer, dissipation and activation energy

The main goal behind the conduction of this work is to use the Adams-Bashforth method in the field of fluid dynamics to solve the Navier–Stokes equations based on the boundary layer flow of magnetohydrodynamic Casson fluid passing on the upper horizontal parabolic surface with variable fluid characteristics and Darcy-Forchheimer effect. The temperature flow rate and mass rate are also investigated with the help of thermal radiation, viscous dissipation, heat generation and activation energy. The Adams-Bashforth technique is widely used in numerical analysis, scientific computing, fluid dynamics, chemical kinetics, astrophysics and is particularly useful in solving the stiff ODEs, where other methods may fail to converge. It is particularly useful for simulating the flow of fluids around complex geometries, such as aircraft wings or turbine blade. The equations of basic governing laws (Momentum, temperature and concentration) are achieved in the form of PDE’s. These PDEs are then transmuted into ODEs via similarity variables and the numerical behaviors of these ODEs are obtained with Adam-Bashforth (Predictor-Corrector) along with RK-4 technique on the Matlab software. The initial guesses are modified by using secant method. Graphical analyses are used to explain how different parameters affect the temperature, velocity and mass transport rate. The finding of results revealed that the inertia coefficient, Hartmann number and fluid parameter make the decrement in the velocity field. The amplification in temperature of fluid is noted for varying the radiation coefficient, dissipation and thermal generation coefficient. The solutal field drops due to activation energy and reaction coefficient. The findings are useful to observe the flow attributes, behaviors and helpful in the designing of thermal equipment in the thermal industries.

Numerical analysis of the frequency-dependent Jiles-Atherton hysteresis model using the example of Terfenol-D material

<p>The Jiles-Atherton model has been widely used in describing the hysteretic property of a magnetic material or device. However, the calculation errors are not so easily discovered. With a complex expression, the frequency-dependent Jiles-Atherton model should be solved numerically with appropriate settings. This paper proposes an effective solving method for this model and describes some necessary analysis built on the numerical results. In the numerical method proposed in this manuscript, the anhysteretic magnetization was calculated by the secant method, and the trapezoidal rule was utilized to form the implicit function, which can be calculated by the fixed-point iteration. Compared to the other common methods, the proposed one has a friendly expression and fast computation speed. The Terfenol-D material was taken as an example for the numerical analysis. The feasible region was determined and the commonly used approximation that neglects the term of the magnetic field when calculating the magnetic induction intensity was tested. At last, the required number of sampling points per period was reached to guarantee high precision from analyzing its influence on the computation precision. The proposed numerical method is helpful for high-precision solutions of the frequency-dependent Jiles-Atherton model. The results from the numerical analysis can also help users avoid some incorrect calculations when employing this hysteresis model.</p>

CORRECTION OF THE CYLINDRICAL HELICAL-GEAR WHEEL PAIR FOR WEAR DECREASE OF TEETH’ INVOLUTE SURFACES

Purpose. It is to create a method for determination of profile shift coefficients, such that during the operation of a cylindrical helical-gear pair the sum of the wear rates of involute surfaces on the contacting sections at the highest and lowest active points of teeth’ profiles of driving and driven wheels will be a minimum.
 Research methods are based on integrated application of the involute gear theory and tribological laws. The secant method that is known in computational mathematics is used to find a root of nonlinear equation.
 Results. The method is created for determination of rational values of correction parameters for cylindrical helical-gear wheels, which maximize the service life of this gear pair until the maximum sum of worn-layer thicknesses on the contacting sections of the teeth’ active surfaces reaches limiting permissible value. An example of application of this method in design calculation of the cylindrical helical-gear pair is given.
 Scientific novelty. Calculation model is elaborated for determination of the sum of worn-layer thicknesses on the most wearing contacting sections of the teeth’ active surfaces of a cylindrical helical-gear wheel pair. It is derived an equation of the linkage between such values of profile shift coefficients for the driving and driven helical-gear wheels which minimize the greatest sum of wear rates on mentioned sections of teeth’ surfaces. Solution of this equation by numerical method for the helical-gear pair having standard distance between axes allows determination of rational values of correction parameters for these wheels.
 Practical value. Application of the created method for determination of profile shift coefficients is expedient in design of cylindrical helical-gear pairs for machines and equipment, which operate in conditions when it is impossible to protect the engagement region from ingress of abrasive particles.

Investigation on voltage stability evaluation indicators and algorithms for power systems based on neural network algorithms

Abstract In order to avoid the occurrence of unstable phenomena in the Power Grid (PG), it is necessary to have a thorough understanding of the current operating status of the PG. Research on voltage stability evaluation indicators and algorithms in the power system can assist power system operators in making effective decisions. In this paper, an improved continuous power flow scheme based on Lagrange quadratic interpolation was adopted. Secant method and Lagrange quadratic interpolation method were used for prediction comparison, and Newton method was used to correct the predicted results. This article utilized an online voltage safety assessment and warning model to ensure the safe operation of the PG and reduce the occurrence of faults. To ensure the integrity and accuracy of network dynamic information, a least squares fitting method based on orthogonal functions was adopted to improve the fitting accuracy without increasing communication volume. As the step size increases from 2 to 3, the number of power flow solutions decreased and the solution time increased. The effectiveness of the voltage stability evaluation algorithm in this paper was verified through simulation of two node systems: New England 39 and IEEE118.