Newton-Raphson Method Research Articles

Design and analysis of one-pad gas foil bearing with three-part gas film produced by adjusting the length of sliding beams

The one-pad sliding-beam gas foil bearing is studied in this paper. The size of the multiple sliding beams in the bearing is adjusted, which results in dividing the gas film pressure into three parts. The finite element mesh of the bottom foil is established based on rectangle and triangle elements, and the finite element model of sliding beams is established based on the plate element with five degrees of freedom. The separation and friction of the foils are considered, and the model of the foils is established based on minimum potential energy theorem and Newton–Raphson method. The mechanical model of the bearing is finally established by fluid-solid interaction. The calculation scheme is programmed by MATLAB, and the reliability of the model is verified by experiments. The numerical results show that with the increase of preload or the decrease of nominal clearance, the gas film pressure is divided into three parts gradually. Compared with the three-pad structure, the bearing in this paper has better bearing capacity. With nominal clearance being 30 µm, the maximum bearing capacity is obtained with a preload of 10 µm, and the maximum direct damping is obtained with a preload of 20 µm. A proper layout of sliding beams can improve the bearing capacity, but it could also weaken the dynamic performance. The variation of the initial bending stiffness distribution of sliding beams has little impact on the dynamic performance, but it can significantly affect the bearing capacity. When the stiffness of sliding beams of axial middle face is larger than that of the end face, the bearing capacity is obviously improved.

Accurate Dynamic Analysis Method of Cable-Damper System Based on Dynamic Stiffness Method

To suppress large vibrations of the cable in cable-stayed bridges, it is common to install transverse dampers near the end of the cable. This paper focuses on the cable-damper system; based on the dynamic stiffness method, an accurate dynamic analysis method considering cable parameters, damper parameters, and cable forces is proposed. First, a mechanical analysis model is established which is closer to the cable with a transverse damper installed in the bridge. The model considers the cable bending stiffness, sag, inclination angle, cable force, damper stiffness, damping coefficient, and damper installation height. Then, the characteristic frequency equation of the cable-damper system is established, and a solution method that combines the initial value method and Newton–Raphson method is proposed. This method is confirmed to provide more accurate frequency analysis for the cable-damper system. Finally, using this method, the effect of the damper parameters on the dynamic characteristics of the system is investigated.

Optimal approximation of the 1/ x function using Chebyshev polynomials and magic constants

In this paper we analyze low-cost accurate approximation of the function \(1/x\) using Chebyshev polynomials of the first kind and minimizing number of elementary operations in computer codes (in particular, by using the so called magic constants). It is shown that Newton-Raphson iterative method is not optimal and a new approach is proposed. We prove that optimal Chebyshev polynomials can be factorized in terms of Chebyshev polynomials of lower order which leads to new -optimal- iteration schemes. We also construct a family of new algorithms by dividing the considered interval into sub-intervals where different magic constants and multiplicative factors are used (in order to increase the accuracy). Theoretical considerations and proofs are completed with numerical tests on three types of microcontroller processors.

Numerical investigation of existing RC exterior beam-column joints under cyclic loading: case study of 1971 algerian building

Many reinforced concrete (RC) buildings constructed in Algeria before the 1980s may have serious structural deficiencies and are considered substandard according to the Algerian seismic code. Specifically, the failure of the beam-column joints has been the cause of building collapse in many earthquakes in the past: Alasnam October 10, 1980 and Boumerdes May 21, 2003. In this work, the behavior of RC exterior beam-column joints under quasi-static cyclic loading is studied by 3D nonlinear finite element analysis using ANSYS software. To perform the non-linear analysis, the load was applied step by step and the modified Newton-Raphson method was used for the solution. In order to prove that the program successfully captures the necessary response parameters without any modifications to the structure details, some available experimental works were modeled and non-linearly analyzed using ANSYS. Moment-rotation and load-displacement relationships were compared with the experimental data. It was observed that the results are quite similar to each other. Thereafter, a typical RC exterior beam-column joint belonging to a building constructed for residential use in Algeria in 1971 was modeled using ANSYS. The joint was subjected to quasi-static cyclic loading, and its performance was examined in terms of load resistance and failure. The moment-rotation hysteresis curve has been established. The hysteretic loops of the joint, representing the existing structure, showed stiffness degradation and strength deterioration.

Fired Heaters Optimization by Estimating Real-Time Combustion Products Using Numerical Methods

Fired heaters upstream of distillation towers, despite their optimal thermal efficiency, often suffer from performance decline due to fluctuations in fuel composition and unpredictable operational parameters. These heaters have high energy consumption, as fuel properties vary depending on the source of the crude oil. This study aims to optimize the combustion process of a three-gas mixture, mainly refinery gas, by incorporating more stable fuels such as natural gas and liquefied petroleum gas (LPG) to improve energy efficiency and reduce LPG consumption. Using real-time gas chromatography-mass spectrometry (GC-MS) data, we accurately calculate the mass fractions of individual compounds, allowing for more precise burner flow rate determinations. Thermochemical data are used to calculate equilibrium constants as a function of temperature, with the least squares method, while the Newton–Raphson method solves the resulting nonlinear equations. Four key variables (X4,X6,X8, and X11), representing H2,CO,O2, and N2, respectively, are defined, and a Jacobian matrix is constructed to ensure convergence within a tolerance of 1 ×10−6 over a maximum of 200 iterations, implemented via Python 3.10.4 and the scipy.optimize library. The optimization resulted in a reduction in LPG consumption by over 50%. By tailoring the fuel supply to the specific thermal needs of each processing unit, we achieved substantial energy savings. For instance, furnaces in the hydrocracking unit, which handle cleaner subproducts and benefit from hydrogen’s adiabatic reactions, require much less energy than those in the primary distillation unit, where high-impurity crude oil is processed.

Inference on modified Weibull type distribution and its application to Competing risks data using MCMC

For lifetime data analysis, failure time analysis, or survival analysis, the Weibull distribution is commonly used due to its various hazard functions, which can be increasing, decreasing, or constant. We have extended the traditional twoparameter Weibull distribution to accommodate hazard functions that are increasing, decreasing, constant and bathtubshaped. Utilizing a competing risks approach, we applied this modified Weibull distribution to both simulated data and an observed mice dataset. Our findings indicate that the modified Weibull distribution provides a better fit to the mice dataset compared to the traditional Weibull distribution. We estimated the parameters of the modified Weibull distribution using Maximum likelihood estimation (MLE) and Bayesian methods. For MLE, we employed the Newton-Raphson numerical method, while for the Bayesian approach, we used the Metropolis-Hastings algorithm, an MCMC method. Additionally, we plotted hazard curves for both the simulated and mice datasets. The Kaplan-Meier survival curves were plotted along with the survival curve of the modified Weibull distribution.. KEYWORDS :Modified weibull distribution, Competing risks, MCMC, Information criterion, MLE.

Temporal asynchrony analysis for dynamic operation of hydraulic-thermal-electricity multiple energy networks based on holomorphic embedding method

Analyzing the operational states of multiple energy networks (MEN) in multi-energy systems is crucial for ensuring system stability. The dynamic operational characteristics of different energy flows pose challenges for computational analysis. Traditional steady-state methods are inadequate for addressing the dynamics of MEN, especially when dealing with temporal discrepancies between hydraulic and thermal flows in thermal networks (TN) and the heterogeneity between TN and electrical networks. Therefore, this paper proposes a novel holomorphic embedding method (HEM) based on multi-stage decomposition method. The developed HEM constructs a time coefficient matrix and utilize inner-outer loop recursion to handle the time lag between thermal flow and hydraulic flow in the TN. Additionally, we reconstruct a holomorphic matrix, integrating hydraulic flow to bridge thermal and electric power flows, thereby improving the operational heterogeneity among different networks. Real-case simulations show that when the Taylor expansion order in HEM is equal to 4, the proposed method achieves a mere 1% discrepancy from actual operational data, enhancing computational efficiency by 60% compared to the Newton-Raphson method. Moreover, in this real-case, the TN exhibits a maximum delay response time of 180s compared to electrical networks. Exploiting this delay time effectively increases renewable energy generation within multi-energy systems by 961.58kW per day.

Dual phase‐field model for immiscible fluid flow through fractured unsaturated porous media

AbstractThe immiscible multiphase fluid flow in intact and fractured porous media is relevant to numerous engineering sectors, including industrial processes, geo‐engineering, and civil engineering. These encompass crucial applications that pose significant challenges in modeling, such as carbon dioxide () storage in underground reservoirs, contaminant transport, and desiccation‐induced hydraulic fracturing in unsaturated soils. In this work, a macroscopic framework is developed to describe the transport of immiscible and multiphase fluids in intact and fractured porous materials. The modeling approach utilizes the embedding of two phase‐field models within the continuum Theory of Porous Media (TPM). The case of negligible capillary pressure in the macroscopic scale is associated with many numerical challenges in solving this nonlinear DAEs system, which forms the major subject of the current work. Hence, the problem of the neglected capillary pressure is defined as a target problem. To deal with this, a thermodynamically consistent modified problem is defined, which incorporates an additional artificial macroscopic capillary pressure based on the Cahn–Hilliard phase‐field approach. To increase the robustness and stability of the numerical solution, the Newton–Raphson method is replaced by a homotopy method that allows a gradual transformation of a simpler, solvable problem into the original target problem that usually converges poorly. The sharp crack topology in the fracture regions is approximated by another phase‐field method, which forms a diffusive transition zone across the crack edges of the deformable porous material. For the solid matrix, we consider a small strain assumption with a heterogeneous distribution of the permeability parameter. The inclusion of heterogeneity allows for a more realistic modeling of crack propagation and fluid‐fluid interaction. For the numerical framework, the coupled DAE system is approximated by the finite element method (FEM), implemented in the open‐source project FEniCSx. Appropriate stabilization techniques are also discussed.

Estimating the Lifetime Parameters of the Odd-Generalized-Exponential–Inverse-Weibull Distribution Using Progressive First-Failure Censoring: A Methodology with an Application

This paper investigates statistical methods for estimating unknown lifetime parameters using a progressive first-failure censoring dataset. The failure mode’s lifetime distribution is modeled by the odd-generalized-exponential–inverse-Weibull distribution. Maximum-likelihood estimators for the model parameters, including the survival, hazard, and inverse hazard rate functions, are obtained, though they lack closed-form expressions. The Newton–Raphson method is used to compute these estimations. Confidence intervals for the parameters are approximated via the normal distribution of the maximum-likelihood estimation. The Fisher information matrix is derived using the missing information principle, and the delta method is applied to approximate the confidence intervals for the survival, hazard rate, and inverse hazard rate functions. Bayes estimators were calculated with the squared error, linear exponential, and general entropy loss functions, utilizing independent gamma distributions for informative priors. Markov-chain Monte Carlo sampling provides the highest-posterior-density credible intervals and Bayesian point estimates for the parameters and reliability characteristics. This study evaluates these methods through Monte Carlo simulations, comparing Bayes and maximum-likelihood estimates based on mean squared errors for point estimates, average interval widths, and coverage probabilities for interval estimators. A real dataset is also analyzed to illustrate the proposed methods.

Asymptotic Numerical Method for dynamic buckling of shell structures with follower pressure

Asymptotic Numerical Method (ANM) is applied to non-linear dynamics of thin-shells subjected to conservative and non-conservative loads such as follower pressure. ANM is decomposed into several stages: the finite element discretization of the non-linear equations of motion of the shell dynamics, a homotopy transformation of the semi-discrete non-linear equations, a perturbation technique to expand the quantities into Taylor series according to the homotopy parameter and the time integration scheme to solve the series of linear problems resulting from the perturbation technique. ANM is applied here with the 7-parameter shell elements thanks to the Enhanced Assumed Strain (EAS) concept and implicit Newmark integration. In the case of non-conservative force, follower pressure also requires to be decomposed in either Taylor series or rational Padé approximants. The academic case of the cylindrical roof with dynamic snap-through phenomenon is investigated for the purpose of comparing ANM strategies and the classical Newton–Raphson (NR) method. Two engineering cases including an I-shaped thin-walled beam and a closed thin-shell cylinder under dynamic external follower pressure are also investigated. ANM turns out to be accurate, robust and efficient in terms of computation time, providing an alternative method to the well-established Newton–Raphson method.

Space-time shape optimization of rotating electric machines

This paper is devoted to the shape optimization of the internal structure of an electric motor, and more precisely of the arrangement of air and ferromagnetic material inside the rotor part with the aim to increase the torque of the machine. The governing physical problem is the time-dependent, nonlinear magneto-quasi-static version of Maxwell’s equations. This multiphase problem can be reformulated on a 2D section of the real cylindrical 3D configuration; however, due to the rotation of the machine, the geometry of the various material phases at play (the ferromagnetic material, the permanent magnets, air, etc.) undergoes a prescribed motion over the considered time period. This original setting raises a number of issues. From the theoretical viewpoint, we prove the well-posedness of this unusual nonlinear evolution problem featuring a moving geometry. We then calculate the shape derivative of a performance criterion depending on the shape of the ferromagnetic phase via the corresponding magneto-quasi-static potential. Our numerical framework to address this problem is based on a shape gradient algorithm. The nonlinear time periodic evolution problems for the magneto-quasi-static potential is solved in the time domain, with a Newton–Raphson method. The discretization features a space-time finite element method, applied on a precise, meshed representation of the space-time region of interest, which encloses a body-fitted representation of the various material phases of the motor at all the considered stages of the time period. After appraising the efficiency of our numerical framework on an academic problem, we present a quite realistic example of optimal design of the ferromagnetic phase of the rotor of an electric machine.

A novel method for calculating the ultimate bearing capacity of in-service RC arch bridges using sectional constitutive relation

Ultimate bearing capacity is the main basis for evaluating the safety state of in-service reinforced concrete (RC) arch bridges. However, current evaluation methods require a lot of cost to handle the material nonlinearity problem, thereby failing to efficiently addressing such a tough issue. To this end, a novel method for calculating the ultimate bearing capacity of in-service RC arch bridges is proposed. This method accelerates the material nonlinear analysis by the sectional constitutive relation, which avoids the computational intensiveness in conventional methods and may provide a quick solution for structural safety assessment. Firstly, considering the arch axis deviation, temperature change and arch rib damage, the geometric nonlinear differential equation is established, and the real internal force and deformation of the structure are obtained by spline function and Newton–Raphson method. On this basis, the sectional stiffness identification method of in-service arch bridges is developed, in which the identified stiffness is taken as the input parameter. Then, a practical formula of the sectional constitutive relation is proposed to accelerate the material nonlinear analysis based on the fiber model. Finally, the practical formula of the sectional constitutive relation is combined into the geometric nonlinear analysis of arch bridges, by which the double nonlinear analysis of in-service arch RC bridges is fast realized. The effectiveness and superiority of the proposed method is verified experimentally and numerically, and the main factors affecting the service safety of arch bridges are revealed through parameter analysis. The results show that the proposed method can improve the calculation efficiency significantly while maintaining the same accuracy as the traditional method. In numerical validation, the relative calculation error is only 2.08 %, and the total calculation time is only 1/25 of the finite element method.

Linear stability of Rayleigh-Bénard-Poiseuille flow of water near 4°C in a channel bounded by slip walls

The onset of mixed convection of cold water in a horizontal channel is studied using linear stability analysis. The two walls of the channel are modeled using slip conditions and are maintained at different constant temperatures. A model with a parabolic relationship is used to predict the variation of density with temperature in the range around its maximum density. The finite difference method is used to solve numerically the linearized equations, and the critical eigenvalue is obtained using the iterative Newton-Raphson method. The effects of the dimensionless temperature ratio γ and the slip parameter Ω on the onset of thermal instabilities are studied. The obtained results showed that these two parameters induced modifications in the critical Rayleigh number and the corresponding critical wavenumber. The results also indicated that beyond a certain value of dimensionless temperature ratio the slip parameter of the upper wall no longer influences the critical threshold for the onset of mixed convection.

Wind-driven and buoyancy effects for modeling natural ventilation in buildings at urban scale

This work proposes a new model to evaluate the air changes per hour (ach) due to natural infiltrations in buildings. This modeling already exists at building scale, but the new model will implement the hourly ventilation load in a physical-based modeling for space heating and cooling in buildings at urban scale. The proposed improvement considers the wind and buoyancy effects in the calculation of hourly achs in a high-density urban context. A three-zone air flow lumped modeling is applied to describe the air flow in buildings; the air flow rate due to infiltrations is calculated depending only on leakages’ characteristics and pressure variations in various climate conditions. The non-linear equations system of mass and energy conservation is solved by an iterative procedure using the Newton-Raphson numerical method. Besides, two different methodologies are compared to evaluate the external dynamic and static pressure conditions on building façades: experimental values (pressure coefficients Cp) and CFD simulations. For the latter, the air flow field in the urban canyons is described by the windy conditions and by imposing a temperature gradient due to solar irradiation between the windward and leeward facades. This methodology is applied to three urban canyons in Turin, with typical aspect ratios and orientations for some local climate conditions considering both heating and cooling seasons. Comparing the results of hourly ach obtained from the Cp method, the CFD technique allows to modulate the ach considering the impact of the canyon dimension, wind and buoyancy effect of non-isothermal condition, in varying the wind speed on the façades of buildings for different scenarios. It also overcomes the limit of field of applications of Cp, especially in high-density built urban environments. The encouraging results of this work will lead to future developments of the three-zone lumped model and its numerical solution techniques.

Electromagnetic Modeling of Superconducting Bulks in Applied Time-Varying Magnetic Field

An integrodifferential model formulated in terms of the electric vector potential is developed for the 3D numerical modeling of the electromagnetic field in superconducting bulks, for AC losses evaluation. The Newton Raphson method is applied to accelerate the convergence. The model is validated on a benchmark. The comparison results show the accuracy of the model and its performances in terms of computation time compared to classical approaches.

Renewable energy integration impact on power quality of supply of transmission system

Electrical energy is known as an essential part of our day-to-day lives. Renewable energy resources can be regenerated through the natural method within a reasonably short time and can be used to bridge the gap in extended power outages. Achieving more renewable energy (RE) than the low levels typically found in today’s energy supply network will entail continuous additional integration efforts into the future. This study examined the impacts of integrating renewable energy on the power quality of transmission networks. This work considered majorly two prominent renewable technologies (solar photovoltaic and wind energy). To examine the effects, IEEE 9-bus (a transmission network) was used. The transmission network and renewable sources (solar photovoltaic and wind energy technologies) were modelled with MATLAB/SIMULINK®. The Newton-Raphson iteration method of solution was employed for the solution of the load flow owing to its fast convergence and simplicity. The effects of its integration on the quality of the power supply, especially the voltage profile and harmonic content, were determined. It was discovered that the optimal location, where the voltage profile is improved and harmonic distortion is minimal, was at Bus 8 for the wind energy and then Bus 5 for the solar photovoltaic source.

Nonlinear Meshfree Collocation Method for Solving Double-Diffusive Natural Convection in a Porous Enclosure

A meshfree collocation method formulated by using reproducing kernel shape functions with Newton–Raphson method is presented with application to analyzing a three-phase coupling system in a porous enclosure. To tackle such a highly nonlinear phenomenon caused by double-diffusive natural convection, direct collocation using local approximation in conjunction with a two-step iteration solver is employed to stably and efficiently solve this problem. It has been shown that a determined and well-conditioned system is established under the collocation framework with reproducing kernel approximation. By conducting the parametric study, the numerical results have demonstrated the robustness of the proposed method in analyzing the porous enclosure by considering various geometry, boundary conditions, and governing parameters.

Enhancing Photovoltaic Solar Model Parameter Optimization: WSO-MTBO Hybrid Approach based on Newton-Raphson Method

Enhancing Photovoltaic Solar Model Parameter Optimization: WSO-MTBO Hybrid Approach based on Newton-Raphson Method

Analysis of Reliability for the Electrical Industries Company in Diyala using Modified Extension Weibull Distribution (An Application and Simulation study)

In this paper defining modified extension Weibull distribution, reliability function and hazerd function, we explain the maximum likelihood method under type one censoring sample. The censored samples play an important role in reliability analysis .In this paper we estimate and derived modified extension Weibull distribution whith three parameters under the censored samples type one by maximum likelihood estimation method depend on Newton-Raphson method. Then we applied simulation procedure using Monte-Carlo technique to estimate reliability function under different samples sizes and various initial values for the parameters for all estimated parameters of modified extension Weibull model by used MATLAB program. Finally we find the probability density function f(t), reliability function R(t),and hazard function h(t) for real data which get it from (Diyala province Company) is one of the formations of the Ministry of Industry and Minerals. Paper type: Research paper

Application of simplifying jacobian matrix and introducing iterative threshold to improve newton-raphson method in three-phase power flow calculation

Abstract With the expansion of the scale and complexity of the power system, traditional three-phase power flow calculation methods are facing dual challenges of computational efficiency and accuracy. Power flow calculation is a mathematical method for analyzing the power flow of an electrical system. And the Newton-Raphson (NR) method is an effective algorithm for solving nonlinear problems. However, updating the Jacobian matrix in the NR method can take a lot of time. Based on the asymmetry in three-phase power systems and the time-consuming nature of the NR method, an enhanced version of the NR method is proposed in this paper that improves the efficiency of power flow calculations by simplifying the Jacobian matrix and optimizing the iteration speed. Furthermore, in order to minimize the number of iterations in the NR method, it is proposed to use the PO method to determine the ideal iteration threshold.