Newton-Raphson Method Research Articles

Fast implicit update schemes for Cahn–Hilliard-type gradient flow in the context of Fourier-spectral methods

This work discusses a way of allowing fast implicit update schemes for the temporal discretization of phase-field models for gradient flow problems that employ Fourier-spectral methods for their spatial discretization. Through the repeated application of the Sherman–Morrison formula we provide a rule for approximations of the inverted tangent matrix of the corresponding Newton–Raphson method with a selectable order. Since the representation of this inversion is exact for a sufficiently high approximation order, the proposed scheme is shown to provide a fixed-point-type iterative solver for gradient flow problems that require the solution of linear systems in the context of an implicit time-integration. While the proposed scheme is applicable to general gradient flow phase-field models, we discuss the scheme in the context of the Cahn–Hilliard equation, the Swift–Hohenberg equation, and the phase-field crystal equation for which we demonstrate the performance of the proposed method in comparison with classical solvers.

Riding safety prediction of a high-speed train running on transition zone under foundation settlement

The settlement of piers and subgrade bending deformation are widely recognized as common issues in the transition zones of high-speed railway bridges. This study aims to investigate the settlement behavior within these transition zones and its impact on the dynamic interaction between trains and the track. To achieve this, a vehicle-track-transition zone mapping relationship model is developed to analyze both the settlement behavior and the resulting dynamic response characteristics. The study employs the finite element method and multi-body dynamics to construct the simulation model. Settlement effects are simulated using the Newton-Raphson iterative method, with the additional rail deformation caused by foundation settlement serving as the excitation for the vehicle-track-transition zone dynamic interaction system. In the numerical analysis, the dynamic effects of three key factors—train speed, transition zone length, and the amplitude of foundation settlement—are examined based on the performance of the vehicle-track-transition zone interaction. The time-frequency technique is utilized to comprehensively reveal and clarify the spatial-frequency characteristics of system responses influenced by settlement excitation. Moreover, the relationship between the safety-based settlement threshold and these three factors is calibrated.

Selection of sectionalising switch to address any switching failure in a droop controlled islanded microgrid

This paper presents a heuristic approach to address the restoration of droop-controlled islanded microgrids (DCIMGs) in the event of a sectionalising switch (SS) failure. The proposed approach consists of two-step procedures to ensure the smooth operation of the network. First, a faulty SS is isolated by replacing it with a tie switch (TS) that acts as an SS. Then, the switching configuration is adjusted to improve the performance of DCIMGs. To assess the efficacy of the proposed approach, the 33-bus and 118-bus DCIMG test networks are considered. For fulfilling the load demand in each DCIMG, droop-controlled DGs (DDGs) and renewable DGs (RDGs) are employed. Along with satisfying load demand, DGs decrease the power loss and voltage drop. Additionally, shunt capacitor banks (SCBs) are also incorporated into the DCIMG to improve the voltage profile. In conjunction with the load flow analysis, the power injection of the DDGs are determined using the modified Newton–Raphson method. For determining the power injection of RDGs and the size of SCBs, particle swarm optimisation method is implemented. The paper investigates the effectiveness of the proposed method for resolving SS faults in DCIMGs over a 24 hour period amidst hourly demand and renewable power generation variation. By employing Hong’s 2m+1 point estimate method, the uncertainties associated with load demands and renewable power generation are also considered. The effectiveness of the issue formulation is verified using MatLabR2023b. With the proposed method, the voltage and frequency can be effectively maintained after the network is restored following a fault.

Nonlinear Nonmodal Analysis of Hypersonic Flow over Blunt Cones

The linear amplification of modal disturbances that lead to boundary-layer transition in two-dimensional/axisymmetric hypersonic configurations is strongly reduced by the presence of a blunt nose tip, and the mechanisms underlying the low Mack’s second-mode N-factor values at the observed onset of transition over the cone frustum are currently unknown. Linear nonmodal analysis has shown that both planar and oblique traveling disturbances that peak within the entropy-layer experience appreciable energy amplification for moderate to large nose-tip bluntness. The present study extends the previous linear analysis by including the nonlinear effects. Specifically, the harmonic Navier–Stokes equations (HNSE) are solved with a fully implicit formulation and the Newton–Raphson method. The increased number of degrees of freedom for the nonlinear system presents difficulties for solution strategies based on direct solution of the linearized system. Such difficulties are overcome by using the generalized minimal residual method (GMRES) iterative method with a preconditioner corresponding to a simplified Jacobian without the cross-derivative terms. The HNSE solver is verified by comparing with nonlinear parabolized stability equation results for the nonlinear evolution of planar waves in an incompressible Blasius boundary layer and in a Mach 6 flow over a blunt cone. Finally, nonlinear nonmodal results are presented for planar traveling disturbances over a blunt cone configuration with reduced transition N factor as measured in wind-tunnel experiments. The nonmodal analysis demonstrates that entropy-layer disturbances generated close to the nose tip can seed the amplification of higher-frequency Mack’s second-mode instabilities farther downstream and hence can lead to a reduction in the transition N factor.

An Optimal Strategy for Estimating Weibull distribution Parameters by Using Maximum Likelihood Method

Several methods have been used to estimate the Weibull parameters such as least square method (LSM), weighted least square method (WLSM), method of moments (MOM), and maximum likelihood (MLE). The maximum likelihood method is the most popular method (MLE). Newton-Raphson method has been applied to solve the normal equations of MLE’s in order to estimate the Weibull parameters. The method was used to find the optimal values of the Weibull distribution parameters for which the log-likelihood function is maximized. We tried to find the approximation solution to the normal equations of the MLE’s because there is no close form for get analytical solution. In this work, we tried to carry out a study that show the difference between two strategies to solve the MLE equations using Newton-Raphson algorithm. Both two strategies are provided an optimal solution to estimate the Weibull distribution parameters but which one more easer and which one converges faster. Therefore, we applied both strategies to estimate the Weibull’s shape and scale parameters using two different types of data (Real and simulation). We compared between the results that we got by applying the two strategies. Two studies have been done for comparing and selecting the optimal strategy to estimate Weibull distribution parameters using maximum likelihood method. We used some measurements to compare between the results such as number of steps for convergence (convergence condition), the estimated values for AIC, BIC and the RMSE value. The results show the numerical solution that we got by applying first strategy convergence faster than the solution that we got by applying second strategy. Moreover, the MRSE estimated by applying the first strategy is lower than the MRSE estimated by applying second strategy for the simulation study with different noise levels and different samples size.

How to Perform a Statistical Analysis of Non-Destructive Degradation Data to Study Crack Growth in Wind Blades as a Function of the Number of Cycles

Objective: The aim of this paper is to show the application of the Exponential Regression Model to study the failure time of a system over time. The data was taken from the paper (Wang et al., 2021). Theoretical framework: The linear regression model, where the response is associated with the explanatory variables by means of a linear model, is the best known. To formulate the model, it is necessary to specify a deterministic component and a random (stochastic) component. The exponential model should be used when it is assumed that risk is constant over time. Once the model has been specified, its parameters are estimated. In the absence of normal errors and especially in the presence of censoring, a more appropriate option is the maximum likelihood method. As the equations are non-linear in their parameters and do not have an analytical solution, it is necessary to use the Newton-Raphson numerical method. Due to the simplicity of the exponential regression model, few situations in practice are adequately adjusted by this model. The Weibull regression model is widely used in survival analysis (Bastos Lyra et al., 2008) Method: Data taken from the article (Wang et al., 2021) e foi utilizado a Regressão Exponencial para modelar os dados. Final Considerations: This result indicates that Exponential Regression is a very viable option when the aim is to monitor the number of cracks as a function of time to check whether a system will fail. In this specific Case Study, it is clear that one of the blades is better than the others and that there are some Probability Distribution models that are suitable for the model. Implications of the research: The use cases of Exponential Regression are more restricted in the scientific literature and because of this, this case study is interesting to show that this model is effective for treating data in the area of reliability. Originality/value: Despite being a well-known statistical tool, Exponential Regression has a specific application in monitoring the number of cracks over time in a system.

Enhancing Economic Operation and Planning of Power Systems through Artificial Intelligence: Implementation of Optimal Power Flow in Chennai Utility Bus System

The aim of this paper is to integrate Artificial Intelligence (AI) into Economic Operation and Planning (EOP) methodologies of power systems, specifically by implementing an Optimal Power Flow (OPF) optimization method in the Chennai Utility Bus System. Many traditional approaches to optimize power generation and planning are inherently limited and many of these limitations can be overcome by the use of Artificial Intelligence (AI) techniques. Herein, we have used the AI powered optimization algorithms such as Multi Objective Particle Swarm Optimization (MOPSO) and Multi Objective Genetic Algorithm (MOGA) to increase the efficiency in economical ranking and also grid planning. Implementation of AI techniques in Chennai utility bus system and also evaluating it in real time using Multi Objective Particle Swarm Optimization, Multi Objective Genetic Algorithm, Newton Raphson method and their results are compared. These AI-based methods aim to reduce operation costs, minimize power loss and improve voltage stability as well as minimizing deviation of voltages in order to increase the efficiency. Future work will expand the use of these techniques to more intricate systems, such as the Indian utility 146-bus system to validate their effectiveness, in real world applications.

Eliminating eccentricity error in measuring residual stresses via hole-drilling method using strain gauge rosette with five measuring grids: For thin plates using through-holes

In the measurement of residual stresses (RSs) via the hole-drilling method, the error induced by the eccentricity of the drilled hole is inevitable and non-negligible in some cases. In this study, both the eccentricity coordinates and RS state (two principal stresses and one principal angle) are considered unknowns that must be solved. A set of five equations is required to determine these five unknowns. Therefore, two additional measuring grids are added to the strain gauge comprising three grids, which is typically used in measuring RSs. Consequently, a novel strain gauge rosette with five measuring grids (SGR-5MG) is created. Subsequently, an algorithm associated with the SGR-5MG is developed to solve the five unknowns based on the Newton–Raphson method. The algorithm is elasticity-based and derived according to the layout of the SGR-5MG. Finally, the proposed method is verified numerically and experimentally. The results show that (1) the eccentricity error is up to 35.1% when the eccentricity reaches 0.05 D, where D represents the diameter of the gauge circle; (2) Using the proposed method, the error can be significantly reduced to 7.2%; (3) The elasticity-based proposed method cannot make the predicted results exactly converge to the actual stresses in predicting high stresses because of the plasticity deformation around the hole. The eccentricity error can be reduced significantly by using the proposed method based on the SGR-5MG.

Multi-peak soliton dynamics and decoherence via the attenuation effects and trapping potential based on a fractional nonlinear Schrödinger cubic quintic equation in an optical fiber

Fiber optic research continues to develop due to the need for fast information technology. The input signal in an optical fiber is in the form of a soliton wave that propagates in the fiber with a balanced index of refraction, dispersion, and diffraction. Fiber optics may lose some of the signal over time if they solely use a nonlinear refractive index. This study aims to examine into the dynamics and decoherence of multi-peak solitons caused by electromagnetic signal attenuation and potential trap in the propagation via optical fibers. The methods used start with the stationary solution of the nonlinear Schrödinger Cubic Quintic equation, the Newton-Raphson method, and the fourth-order Runge-Kutta. In stationary solutions, the p (state energy) and ρ (Lèvy index) values affect the number, pulse width, and height of soliton peaks. Increased p and ρ values lead to increased instability, causing the data transmitted into the optical fiber to collapse and become unreadable by the receiver. In the excited state (p > 0), the decoherence phenomenon occurs and causes wave instability and destruction. In term of energy intensity, a stable form of transmission in the ground state has a pattern of continuously weakening, decreasing, and smoothing, while in the excited state there is a spike in the intensity of the electromagnetic wave. Based on this study, multi-peak solitons can only be applied for propagation distances, which tend to be short due to their dynamics and decoherence.

A nonlinear interval finite element method for elastic–plastic problems with spatially uncertain parameters

This paper proposes a nonlinear interval finite element method for elastic–plastic analysis of structures with spatially uncertain parameters. The spatially uncertain parameters are described by the interval field, and the variation bounds of the elastic–plastic structural responses can be calculated effectively. Quantified by the interval field, the spatially uncertain parameters are represented by the interval Karhunen–Loève (K-L) expansion, based on which the nonlinear interval finite element equilibrium equation is formulated. An interval iterative method is then presented to solve the equilibrium equation and obtain an outer solution of the variation bounds of structural responses such as displacement. In this method, the Newton-Raphson iterative method is used to transform the nonlinear problem into a linear one, and then the interval iterative method is introduced to solve the interval linear equations. Three numerical examples are employed to illustrate the feasibility and accuracy of the proposed method.

An Efficient Quadratic Programming Method for Kinematic Control of Redundant Manipulators under Joint Velocity Constraints

This paper presents an efficient inverse kinematics solution for redundant robotic arms. The proposed method combines the principles of continuation methods, improves the instability of the computation time by increasing the convergence of the kinematics function, and improves the efficiency of traditional numerical methods. The effectiveness and efficient performance of the method are demonstrated through comparative experiments. The computational speed of the method is twice as fast as the Newton–Raphson method under joint limit constraints and equal solution accuracy.

A Novel Geothermal Wellbore Model Based on the Drift-Flux Approach

Geothermal energy, being a clean energy source, has immense potential, and accurate wellbore modeling is crucial for optimizing the drilling process and ensuring safety. This paper presents a novel geothermal wellbore model based on the drift-flux approach, tested under three different temperature and pressure well conditions. The proposed model integrates the conservation equations of mass, momentum, and energy, incorporating the gas–liquid two-phase flow drift-flux model and heat transfer model. The key features include handling the heat transfer between the formation and the wellbore, addressing the slip relationship between the gas and liquid phases, and accounting for wellbore friction. The nonlinear equations are discretized using the finite difference method, and the highly nonlinear system is solved using the Newton–Raphson method. The numerical simulation, validation, and comparison with existing models demonstrate the enhanced accuracy of this model. In our tests, the model achieved a high accuracy in calculating the bottom-hole pressure and temperature, with mean relative errors (MREs) significantly lower than those of other models. For example, the MREs for the bottom-hole pressure and temperature of the Rongxi area well in Xiongan, calculated by this model, are 1.491% and 1.323%, respectively. These results offer valuable insights for optimizing drilling parameters and ensuring drilling safety. Comparisons indicate that this approach significantly outperforms others in capturing the complex dynamics of geothermal wellbores, making it a superior tool for geothermal energy development.

Transient Friction Analysis of Pressure Waves Propagating in Power-Law Non-Newtonian Fluids

Modulated pressure waves propagating in the drilling fluids inside the drill string are a reliable real-time communication technology that transmit data from downhole to the surface during oil and gas drilling. In the analysis of pressure waves’ propagation characteristics, the modeling of transient friction in non-Newtonian fluids remains a great challenge. This paper establishes a numerical model for transient pipe flow of power-law non-Newtonian fluids by using the weighted residual collocation method. Then, the Newton–Raphson method is applied to solve the nonlinear equations. The numerical method is validated by using the theoretical solution of Newtonian fluids and is proven to converge reliably with larger time steps. Finally, the influencing factors of the wall shear stress are analyzed using this numerical method. For shear-thinning fluids, the friction loss of periodic flow decreases with the increase in flow rate, which is opposite to the variation law of friction with the flow rate for stable pipe flow. Keeping the amplitude of pressure pulsation unchanged, an increase in frequency leads to a decrease in velocity fluctuations; therefore, the friction loss decreases with the increase in frequency.

Extended Lambert-Based Approach for Ground-Track Adjustment with Specified Overflight Altitude

This paper presents an extended Lambert-based approach for ground-track adjustment under J2 perturbation, aiming to achieve a satellite flyby over a designated ground location at a predetermined altitude. Extended equations are developed in the orbital plane based on Lagrange’s form for the unperturbed Lambert’s problem to address the J2-perturbed two-body motion. By combining them with the out-of-plane flight time equation constraint regarding the ground-track adjustment, a nonlinear system of equations with two unknowns and two equations is established and split into different modes. To solve these modes, a modified Newton–Raphson method is used, and a necessary condition is established to determine the maximum allowable number of revolutions. To enhance computational efficiency, a radius restriction method is proposed to narrow down the allowable range of the number of revolutions. The proposed algorithm is applicable to various scenarios, including single- or multiple-day flights, ascending or descending overflights, coplanar or non-coplanar transfers, and initial or reverse-initial directional transfers. The simulation results demonstrate that the proposed algorithm is effective and accurate for the perturbed adjustment problem.

Haar Wavelet Approach for the Mathematical Model on Hepatitis B Virus

Abstract The Haar wavelet collocation method, a wavelet technique, is discussed in this article to examine the mathematical model of Hepatitis B virus infection. We took into account the HB virus, cytotoxic T lymphocytes (CTL) immune response, birth rate, death rate, and infected and uninfected hepatocytes to identify the dynamics of the hepatitis B virus infection. An ordinary differential equation (ODE) system that is nonlinear makes up this model. Using this method, the Hepatitis B Virus model can be solved by expressing each dependent variable as a Haar wavelet and then converting the system of ordinary differential equations into a system of nonlinear algebraic equations. The unknown coefficient values are thought to be extracted using the collocation procedure and the Newton–Raphson method. Tables and graphs are used to illustrate the characteristics of the Hepatitis B virus. The obtained results show that the current approach outperforms other approaches found in the literature in terms of accuracy. Mathematica software is utilized to obtain numerical results and nature.

Application of the Inflection Point in the Evaluation of the Halley and Newton-Raphson Techniques for Finding the Root of Non-Linear Equations

Numeric Method is one of the methods used to solve nonlinear equation roots. Many methods can be used, both open methods and closed methods. In this case, the method used is closed, namely the Newton-Raphson Method and Halley Method. The research aims to find out the comparison result between the Newton-Raphson Method and the Halley Method. The research used a literature method from a book, journal, and any other literature, where it connected with the topic. The steps used are formulation problem, finding and collecting information, describing and explaining the information, analysis, and conclusion of the result. The conclusion can be explained with a table of data and explanations Based on data analysis, it can be stated that the Halley Method is faster toward convergence compared to the Newton-Raphson Method based on the first case or second case.

Investigating the modal behaviors of a deep beam with a transverse open crack

This paper investigates the modal behaviors of a deep beam with a transverse open crack. Unlike cracked shallow beams with aspect ratio (L/h) > 10, cracked deep beams with aspect ratio < 5 simultaneously exhibit near the crack both axial-bending coupling mode caused by mode I crack and shear (or sliding) mode caused by mode II crack, referred to as axial-bending-shear modal coupling. Therefore, for accurate modal behaviors prediction, appropriate compatibility equations accounting for both modes I and II crack should be employed for deriving a frequency equation. Because the derived frequency equation is nonlinear with respect to modal frequency, the modal frequencies are calculated through the Newton–Raphson method. The corresponding mode shapes are determined by substituting the calculated modal frequencies into the spectral solutions for either side of the crack. The validity of the proposed method was demonstrated through comparison to the finite element analysis and experimental results.

Optimal allocation of solar photovoltaic distributed generation for performance enhancement of electrical distribution networks considering optimal volt‐var regulation under uncertainty and high load variation

AbstractThis article proposes an optimal placement and sizing of photovoltaic (PV) power systems based distributed generation (DG) in radial electrical distribution networks considering the capability of PV inverters to regulate the voltage by optimal injecting and absorbing reactive power at the point of common coupling (PCC) using honey badger algorithm (HBA), as a recent and efficient optimization algorithm to solve the complicated optimal allocation. Several objective functions are achieved for distribution system performance enhancement: minimizing the power loss and the voltage deviation index (VDI) and maximizing the voltage stability index (VSI). Based on historical data and probabilistic models, seasonal hourly solar irradiance, ambient temperature, and load variation curves have been modeled, which simultaneously consider the light, normal, and heavy load demand. The essential components of distribution power systems have been characterized. To investigate the validity of the proposed approach, IEEE 33 and IEEE 69 BUS radial distribution test systems have been considered for power flow (PF) analyses, where Newton's Raphson method has been applied to solve the PF issue. The simulation results of different numerical scenarios have shown the effectiveness and validity of the newly proposed method to solve the optimal allocation problem considering optimal volt‐var regulation control of PV inverters compared to several valid and robust optimization algorithms.

Improved Newton-Raphson method with simplified Jacobian matrix and optimized iteration rate for power flow calculation of power system

With high penetration of renewable energy and novel loads connected to the distribution network, the voltage fluctuation becomes more severe and frequent, which may cause over- and under-voltage. The distribution system operator should calculate the power flow and validate the state to optimize the distribution network. Power flow calculation is the solution to the multivariate nonlinear problem, and the Newton-Raphson method is an effective algorithm for solving nonlinear problems. However, calculating the Jacobian matrix is a crucial process of the Newton-Raphson method, which is time-consuming. Therefore, this paper proposed an improved Newton-Raphson method, which simplifies and decreases the iterations of the calculation process of the Jacobian matrix to improve the calculation rate. To verify the effectiveness of the proposed method, the power flow of the IEEE 33-node power distribution system is calculated by the improved Newton-Raphson method and the conventional Newton-Raphson method.

ZEROS OF PARTITION FUNCTION FOR Z5-SYMMETRIC MODEL ON SQUARE LATTICE-PARTICULAR CASE

We study a statistical mechanics model of multiple-phase transitions called the ZQ-symmetric model on the square lattice for five possible spin directions (Q = 5). This study aims to investigate the existence of phase transition(s) and discuss the distribution of partition function zeros on the complex-temperature plane. Some cases of energy list χ are considered where they are written in an arbitrary arrangement of integer numbers (usually in decreasing value due to the angle of separation). The partition functions are computed using a transfer matrix approach, and their zeros are found numerically using the NewtonRaphson method. A Monte-Carlo Metropolis simulation is then performed to study the critical point of phase transition. The results are compared to the zeros distribution of the partition functions for the considered cases. Studying this generalised model is vital to mimic the system in the real world and to help experimental works use a more sustainable approach.