Search Results (39)

The increasing penetration of direct-current (DC) technologies in power transmission and distribution systems necessitates efficient and robust tools for steady-state analysis. This paper presents a comparative evaluation of the Chebyshev–Halley (CH) family of multipoint iterative methods against the classical Newton–Raphson (NR) method for power flow calculation in monopolar DC networks. Both methods were implemented in MATLAB and tested on four radial test systems of increasing complexity (10, 21, 33, and 69 nodes) under three distinct initialization scenarios: optimal (flat start), adverse ($V^{\left(0\right)}=0.5$ p.u.), and random ($V^{\left(0\right)}\sim \mathcal{U}[0.8,1.2]$ p.u.). Performance was assessed using key metrics including iteration count, CPU time, solution accuracy, and convergence failure rate. The results demonstrate that the cubic convergence of CH consistently reduces the number of iterations by one when compared to NR across all systems. However, this reduction does not translate into computational savings, as CH exhibits median CPU times 1.36 to 2.44 times higher than those of NR, given its higher cost per iteration, which involves solving two additional linear systems. Under adverse starting conditions, both methods converge for the 10-, 21-, and 33-node systems, but CH fails on the 69-node network due to severe Jacobian ill-conditioning, from which NR recovers through an implicit regularization mechanism. Under random initializations, both methods show high failure rates, reaching 100% in the 69-node network. It is concluded that, while CH offers superior convergence order and final accuracy, NR remains more computationally efficient for small- to medium-scale networks under flat-start conditions. The CH family is best justified in high-precision applications or larger networks where the iteration reduction may offset its per-step overhead. Future work should focus on extending CH to meshed and multi-source DC networks, developing quasi-Newton variants to reduce its computational cost, and designing hybrid NR-CH strategies that combine global robustness with local cubic convergence. Full article

Springback control in Multi-Point Stretch Forming (MPSF) is significantly hindered by the computational intensity of Finite Element Analysis (FEA) and the limited predictive robustness of traditional regression methods. This study develops a hybrid GWO-CatBoost model acting as a data-driven surrogate for MPSF simulations by integrating the Grey Wolf Optimizer (GWO) with the CatBoost algorithm for high-precision springback forecasting. An FEA model of the MPSF process was initially validated through experimental comparison under a representative working condition to assess modeling accuracy. A comprehensive dataset comprising 1200 scenarios was generated via a full factorial design, incorporating key variables: curvature radius, sheet thickness, cushion thickness, and pre-stretching rate. In this study, the GWO was employed to perform automated hyperparameter tuning for CatBoost by optimizing the learning rate, tree depth, and number of iterations, thereby enabling accurate modeling of the complex nonlinear relationship between process inputs and numerical springback values. Numerical evaluations demonstrate that the GWO-CatBoost model outperforms GWO-XGBoost and GWO-Random Forest benchmarks, achieving a Coefficient of Determination (R²) of 0.9293, a root mean square error (RMSE) of 0.0274 mm and mean absolute error (MAE) of 0.0189 mm. Sensitivity analysis identifies sheet thickness as the dominant factor (46% contribution), with cushion thickness as the secondary driver (23%). This predictive framework serves as a computationally efficient auxiliary surrogate, designed to assist iterative finite element analyses and support process optimization in the manufacture of complex-curved panels. Full article

Many problems in science, engineering, and economics require solving of nonlinear equations, often arising from attempts to model natural systems and predict their behavior. In this context, iterative methods provide an effective approach to approximate the roots of nonlinear functions. This work introduces five new parametric families of multipoint iterative methods specifically designed for solving nonlinear equations. Each family is built upon a two-step scheme: the first step applies the classical Newton method, while the second incorporates a convex mean, a weight function, and a frozen derivative (i.e., the same derivative from the previous step). The careful design of the weight function was essential to ensure fourth-order convergence while allowing arbitrary parameter values. The proposed methods are theoretically analyzed and dynamically characterized using tools such as stability surfaces, parameter planes, and dynamical planes on the Riemann sphere. These analyses reveal regions of stability and divergence, helping identify suitable parameter values that guarantee convergence to the root. Moreover, a general result proves that all the proposed optimal parametric families of iterative methods are topologically equivalent, under conjugation. Numerical experiments confirm the robustness and efficiency of the methods, often surpassing classical approaches in terms of convergence speed and accuracy. Overall, the results demonstrate that convex-mean-based parametric methods offer a flexible and stable framework for the reliable numerical solution of nonlinear equations. Full article

In this paper, we study the convergence and complex dynamics of a novel higher-order multipoint iteration scheme to solve nonlinear equations. The approach is based upon utilizing cubic interpolation in the second step of the King–Werner method to improve its convergence order from $2.414$ to 3 and the efficiency index from $1.554$ to $1.732$, which is higher than the efficiency of optimal fourth- and eighth-order iterative schemes. The proposed method is validated through numerical and dynamic experiments concerning the absolute error, approximated computational order, regions of convergence, and CPU time (sec) on the real-world problems, including Kepler’s equation, isentropic supersonic flow, and law of population growth, demonstrating superior performance compared to some existing well-known methods. Commonly, regions of convergence of iterative methods are investigated and compared by plotting attractor basins of iteration schemes in the complex plane on polynomial functions of the type $z^n-1$. However, in this paper, the attractor basins of the proposed method are investigated on diverse nonlinear functions. The proposed scheme creates portraits of basins of attraction faster with wider convergence areas outperforming existing well-known iteration schemes. Full article

Coastal aquifers are critical freshwater resources that face increasing threats from contamination and saltwater intrusion. Traditional approaches for characterizing these aquifers are challenged by complex dynamics, high-dimensional parameter spaces, and significant computational demands. This study presents an innovative method that combines an Auto-Regressive Convolutional Neural Network (AR-CNN) surrogate model with the Iterative Local Updating Ensemble Smoother (ILUES) for the joint inversion of contamination source parameters and hydraulic conductivity fields. The AR-CNN surrogate model, trained on synthetic data generated by the SEAWAT model, effectively approximates the complex input–output relationships of coastal aquifer systems, substantially reducing computational burden. The ILUES framework utilizes observational data to iteratively update model parameters. A case study involving a heterogeneous coastal aquifer with multipoint pollution sources demonstrates the efficacy of the proposed method. The results indicate that AR-CNN-ILUES successfully estimates pollution source strengths and characterizes the hydraulic conductivity field, although some limitations are observed in areas with sparse monitoring points and complex geological structures. Compared to the traditional SEAWAT-ILUES framework, the AR-CNN-ILUES approach reduces the total inversion time from approximately 70.4 h to 16.2 h, improving computational efficiency by about 77%. These findings highlight the potential of the AR-CNN-ILUES framework as a promising tool for efficient and accurate characterization of coastal aquifers. By enhancing computational efficiency without significantly compromising accuracy, this method offers a viable solution for the sustainable management and protection of coastal groundwater resources. Full article

Morphing wing technology is crucial for enhancing the flight performance of aircraft. To address the monitoring challenges of full-scale variable-camber leading edges under flight conditions, this study introduces a ground-based strength testing technique aimed at precisely evaluating the deformation patterns and structural strength during actual operation. Firstly, the motion characteristics of the variable-camber leading edge were analyzed using numerical simulation based on kinematic theory. Secondly, a tracking loading test rig was designed and constructed to simulate the actuated deformation and aerodynamic loads of the leading edge. Next, mechanical boundary numerical simulation was then utilized to predict the motion trajectories of loading points on the upper and lower wing surfaces, and a multi-point coordinated control system was developed to achieve accurate experimental control. Finally, a multi-sensor iterative method was employed to ensure loading precision throughout the testing process. A case study was conducted using a leading edge test piece from a specific commercial aircraft. The results indicated that in the motion test of the variable-camber leading edge, the average error of the deflection angle was 4.59%; in the strength test, the average errors in the magnitude and direction of the applied load were 0.54% and 0.24%, respectively. These findings validate the effectiveness of the proposed technique in simulating the flight conditions of deforming wings and accurately obtaining the leading edge shape change curve, deformation accuracy curve, and strain curves of the upper and lower wing surfaces under deflection angles. Furthermore, this paper compares the deformation accuracy of different testing methods under test conditions, providing scientific evidence and technical support for the testing and evaluation of variable-camber leading edges. Full article

In this paper, we have constructed new families of derivative-free three- and four-parametric methods with and without memory for finding the roots of nonlinear equations. Error analysis verifies that the without-memory methods are optimal as per Kung–Traub’s conjecture, with orders of convergence of 4 and 8, respectively. To further enhance their convergence capabilities, the with-memory methods incorporate accelerating parameters, elevating their convergence orders to 7.5311 and 15.5156, respectively, without introducing extra function evaluations. As such, they exhibit exceptional efficiency indices of 1.9601 and 1.9847, respectively, nearing the maximum efficiency index of 2. The convergence domains are also analysed using the basins of attraction, which exhibit symmetrical patterns and shed light on the fascinating interplay between symmetry, dynamic behaviour, the number of diverging points, and efficient root-finding methods for nonlinear equations. Numerical experiments and comparison with existing methods are carried out on some nonlinear functions, including real-world chemical engineering problems, to demonstrate the effectiveness of the new proposed methods and confirm the theoretical results. Notably, our numerical experiments reveal that the proposed methods outperform their existing counterparts, offering superior precision in computation. Full article

Different from conventional reconfigurable intelligent surfaces (RIS), simultaneous transmitting and reflecting RIS (STAR-RIS) can reflect and transmit signals to the receiver. In this paper, to serve more ground users and increase deployment flexibility, we investigate an unmanned aerial vehicle (UAV) equipped with STAR-RIS (STAR-RIS-UAV)-aided wireless communications for multi-user networks. Energy splitting (ES) and mode switching (MS) protocols are considered to control the reflection and transmission coefficients of STAR-RIS elements. To maximize the sum rate of the STAR-RIS-UAV-aided coordinated multipoint (CoMP) cellular system for multi-user networks, the corresponding beamforming vectors as well as transmitted and reflected coefficient matrices are optimized. Specifically, instead of adopting the alternating optimization, we design an iteration method to optimize all variables for both the ES and MS protocols at the same time. Simulation results reveal that the STAR-RIS-UAV-aided CoMP system has a much higher sum rate than systems with conventional RIS or without RIS. Furthermore, the proposed structure is more flexible than fixed STAR-RIS and could greatly promote the sum rate. Full article

In recent years, some Newton-type schemes with noninteger derivatives have been proposed for solving nonlinear transcendental equations by using fractional derivatives (Caputo and Riemann–Liouville) and conformable derivatives. It has also been shown that the methods with conformable derivatives improve the performance of classical schemes. In this manuscript, we design point-to-point higher-order conformable Newton-type and multipoint procedures for solving nonlinear equations and propose a general technique to deduce the conformable version of any classical iterative method with integer derivatives. A convergence analysis is given and the expected orders of convergence are obtained. As far as we know, these are the first optimal conformable schemes, beyond the conformable Newton procedure, that have been developed. The numerical results support the theory and show that the new schemes improve the performance of the original methods in some aspects. Additionally, the dependence on initial guesses is analyzed, and these schemes show good stability properties. Full article

We developed a new family of optimal eighth-order derivative-free iterative methods for finding simple roots of nonlinear equations based on King’s scheme and Lagrange interpolation. By incorporating four self-accelerating parameters and a weight function in a single variable, we extend the proposed family to an efficient iterative scheme with memory. Without performing additional functional evaluations, the order of convergence is boosted from 8 to $15.51560$, and the efficiency index is raised from $1.6817$ to $1.9847$. To compare the performance of the proposed and existing schemes, some real-world problems are selected, such as the eigenvalue problem, continuous stirred-tank reactor problem, and energy distribution for Planck’s radiation. The stability and regions of convergence of the proposed iterative schemes are investigated through graphical tools, such as 2D symmetric basins of attractions for the case of memory-based schemes and 3D stereographic projections in the case of schemes without memory. The stability analysis demonstrates that our newly developed schemes have wider symmetric regions of convergence than the existing schemes in their respective domains. Full article

Undesirable chatter is one of the key problems that restrict the improvement of robot milling quality and efficiency. The prediction of chatter stability, which is used to guide the selection of process parameters, is an effective method to avoid chatter in robot milling. Due to the weak stiffness of the robot, deformation caused by milling forces becomes an unavoidable problem, which will change the tool–workpiece contact area and affect the stability prediction. However, it is often simplified and neglected. In this paper, a multipoint contact dynamic model of robot milling is established, which considers the influence of force-induced deformation on the regenerative effect and process damping. The tool–workpiece contact area is discretized into a finite number of nodes along the axial direction so that the force and deformation at each node can be calculated separately. The different contact forms of the tool–workpiece under different process parameters are discussed in different cases, and the interaction process between cutting force and force-induced deformation is analyzed in detail. An iterative strategy is used to calculate the deformation of each node and the result of the tool–workpiece contact boundary. Finally, chatter stability of robot milling is predicted by a fully discrete method. Robot milling experiments were carried out to verify the predicted results. The results show that force-induced deformation is an important factor improving the stability prediction accuracy of robot milling, and a more accurate prediction result can be obtained by simultaneously considering force-induced deformation and process damping. Full article

Kappa-angle calibration shows its importance in gaze tracking due to the special structure of the eyeball. In a 3D gaze-tracking system, after the optical axis of the eyeball is reconstructed, the kappa angle is needed to convert the optical axis of the eyeball to the real gaze direction. At present, most of the kappa-angle-calibration methods use explicit user calibration. Before eye-gaze tracking, the user needs to look at some pre-defined calibration points on the screen, thereby providing some corresponding optical and visual axes of the eyeball with which to calculate the kappa angle. Especially when multi-point user calibration is required, the calibration process is relatively complicated. In this paper, a method that can automatically calibrate the kappa angle during screen browsing is proposed. Based on the 3D corneal centers and optical axes of both eyes, the optimal objective function of the kappa angle is established according to the coplanar constraint of the visual axes of the left and right eyes, and the differential evolution algorithm is used to iterate through kappa angles according to the theoretical angular constraint of the kappa angle. The experiments show that the proposed method can make the gaze accuracy reach 1.3${}^{°}$ in the horizontal plane and 1.34${}^{°}$ in the vertical plane, both of which are within the acceptable margins of gaze-estimation error. The demonstration of explicit kappa-angle calibration is of great significance to the realization of the instant use of gaze-tracking systems. Full article

This manuscript is focused on a new parametric class of multi-step iterative procedures to find the solutions of systems of nonlinear equations. Starting from Ostrowski’s scheme, the class is constructed by adding a Newton step with a Jacobian matrix taken from the previous step and employing a divided difference operator, resulting in a triparametric scheme with a convergence order of four. The convergence order of the family can be accelerated to six by setting two parameters, resulting in a uniparametric family. We performed dynamic and numerical development to analyze the stability of the sixth-order family. Previous studies for scalar functions allow us to isolate those elements of the family with stable performance for solving practical problems. In this regard, we present dynamical planes showing the complexity of the family. In addition, the numerical properties of the class are analyzed with several test problems. Full article

This article addresses the numerical computation problem of induced inflow ratio based on the helicopter momentum theory in forward flight. The Glauert inflow formula (equation) is a nonlinear equation usually solved by the Newton–Raphson method in a relatively small number of iterations. However, many high-order convergence multipoint iterative methods have been developed over the last decade. The study examines several selected methods in terms of finding ones that provide a solution in only one iteration with acceptable accuracy. Furthermore, the influence of initial guesses on the accuracy of the obtained solutions has been investigated. In this regard, the practical range of parameters of the Glauert inflow equation for helicopters in forward flight is roughly determined by simplified modeling of a power and stall-flutter limitation. For these purposes, a basic low-fidelity longitudinal trim model of a single-rotor helicopter in steady-level flight is modified and numerically solved by a symbolic transformation of a system of 20+ nonlinear equations into a single nonlinear equation. Full article

In this article, a multiple surrogate-model-based optimization method using the multimodal expected improvement criterion (MSMEIC) is proposed. In MSMEIC, an important region is first identified and used alternately with the whole space. Then, in each iteration, three common surrogate models, kriging, radial basis function (RBF), and quadratic response surface (QRS), are constructed, and a multipoint expected improvement (EI) criterion that selects the highest peak and other peaks of EI is proposed to obtain several potential candidates. Furthermore, the optimal predictions of the three surrogate models are regarded as potential candidates. After deleting redundant candidates, the remaining points are saved as the new sampling points. Finally, several well-known benchmark functions and an engineering application are employed to assess the performance of MSMEIC. The testing results demonstrate that, compared with four recent counterparts, the proposed method can obtain more precise solutions more efficiently and with strong robustness. Full article