Raphson Iteration Research Articles

Enabling accurate simulations of the nonlinear magnetic hysteresis behavior in ferromagnetic materials

In this work, a stable node - based smoothed finite - element method algorithm is developed to address the lower accurate issues in the simulation of the magnetic hysteresis behavior in ferromagnetic materials, in which the inverse vector Jiles - Atherton model is intimately coupled. The improvement of the accuracy for hysteresis issues is achieved by linking the stable node - based smoothed finite - element method in the framework of the Newton - Raphson iteration and the Jiles - Atherton model. The optimal relaxation coefficient method is introduced to address the natural strong nonlinearity of the Jiles - Atherton model and to ensure the convergence of the model. The algorithm is validated against the experimental results, and several examples are presented for simulations of the ferromagnetic hysteresis issues to illustrate both the accuracy and expandability in the practical scenarios of ferromagnetic materials.

Trajectory planning and automatic docking of LNG five-axis loading arm

PurposeIn response to the challenges posed by the conventional manual flange docking method in the LNG (Liquefied Natural Gas) loading process, such as low positioning accuracy, constraints on production efficiency and safety hazards, this study analyzed the LNG five-axis loading arm’s main functions and structural characteristics.Design/methodology/approachAn automated solution for the joints of the LNG loading arm was designed. The forward kinematic model of the LNG loading arm was established using the Denavit–Hartenberg (D-H) parameter method, and its workspace was analyzed. The Newton–Raphson iteration method was employed to solve the inverse kinematics of the LNG loading arm, facilitating trajectory planning. The relationship between the target position and the joint variables was established to verify the stability of the arm’s motion. Flange center identification was achieved using the Hough transform function. Based on the ROS platform, combined with Gazebo and Rviz, an experimental simulation of automatic docking of the LNG loading arm was conducted.FindingsThe docking errors in the XYZ directions were all less than 0.8 mm, meeting the required docking accuracy. Moreover, the motion performance of the loading arm during docking was smooth and free of abrupt changes, validating its capability to accomplish the automatic docking task.Originality/valueThe proposed trajectory planning and automatic docking scheme can be used for the rapid filling of LNG filling arms and LNG tankers to improve the efficiency of LNG transportation. In guiding the docking, the proposed automatic docking scheme is an accurate and efficient way to improve safety.

Optimization of cantilever beams with multiple flexoelectric actuators based on Newton–Raphson iteration

Optimization of cantilever beams with multiple flexoelectric actuators based on Newton–Raphson iteration

Dynamic contact and stiffness characteristic analysis of angular contact ball bearings under combined external loads

A nonlinear dynamic model of angular contact ball bearing under combined external loads is established based on the nonlinear elastic Hertz contact theory and raceless control theory. The established dynamic model is solved by employing variable step size Newton Raphson iteration method and the validation of model is verified by comparing the proposed results with the corresponding results come from the existing literatures. The solution of the presented model has high accuracy compared with the existing experimental results. In addition, the computational efficiency of the proposed model is improved significantly by introducing iteration step size adjustment factor. According to the proposed model, the dynamic contact and stiffness characteristics of angular contact ball are studied systematically by investigating the effects of combined external load working conditions on the contact parameters including contact angle and contact force and the stiffness parameters including diagonal stiffness and off-diagonal stiffness. This research can provide the theoretical basis and technique guidance for the designation and manufacture of angular contact ball bearing under combined external loads.

Novel multi-degree-of-freedom force–displacement mixed control framework for low loading rate substructure hybrid testing

In substructure hybrid tests, imposing realistic multi-degree-of-freedom (MDOF) displacement and force boundary conditions on a specimen involves controlling several actuators to emulate the MDOF boundary conditions of the experimental substructure. This setup presents significant challenges for decoupling control in MDOF systems. Due to the typically high stiffness of the specimen in the axial direction, force control is preferred over displacement control, which has limited accuracy in this direction. Typically, force control in the axial direction is combined with simultaneous displacement control in the softer direction, forming an MDOF force-displacement mixed control scheme. This scheme transforms force control into displacement control in the inner loop in the axial direction to ensure safe loading. The scheme involves addressing two major challenges: the nonlinear force–displacement transformation and the nonlinear coordinate transformation from the local actuator space (LAS) to the global Cartesian coordinate (GCC) system. To address these challenges, the Modified Newton–Raphson iteration integrated with proportional integral plus feed-forward (MNR-PIFF) and incremental kinematic transformation integrated with proportional integral plus feed-forward (IKT-PIFF) methods are proposed. Based on these methods, a novel MDOF force–displacement mixed control (NMFDC) framework is developed for substructure hybrid testing. A 6-DOF cyclic test simulation using the NMFDC framework was conducted to demonstrate its superiority. Furthermore, a 6-DOF cyclic test and substructure hybrid test at low loading rates were performed to validate the NMFDC framework because of the loading equipment limit. Experimental results and numerical simulations indicate that the NMFDC framework achieves swifter response with lower error levels. For applications in fast and real-time hybrid testing, dynamic loading equipment need to be used, the PIFF controller in the NMFDC framework should be replaced by adaptive time delay compensation in future developments.

Theoretical Evaluation of Lubrication Performance of Thrust-Type Foil Bearings in Liquid Nitrogen

The development of reusable liquid rocket turbopumps has gradually highlighted the disadvantages of rolling bearings, particularly the contradiction between long service life and high rotational speed. It is critical to explore a feasible bearing scheme offering a long wear life and high stability to replace the existing rolling bearings. In this study, liquid nitrogen is adopted to simulate the ultra-low temperature environment of liquid rocket turbopumps, and theoretical evaluations of the lubrication performance of thrust-type foil bearings in liquid nitrogen are conducted. A link-spring model for the bump foil structure and a thin-plate finite element model for the top foil structure are established. The static and dynamic characteristics of the bearings are analyzed using methods including the finite difference method, the Newton–Raphson iteration method, and the finite element method. Detailed analysis includes the effects of factors such as rotational speed, fluid film thickness, thrust disk tilt angle, and the friction coefficient of the bump foil interface on the static and dynamic characteristics of thrust-type foil bearings. The research results indicate that thrust-type foil bearings have a good load-carrying capacity and low frictional power consumption. The adaptive deformation of the foil structure increases the fluid film thickness, preventing dry friction due to direct contact between the rotor journal and the bearing surface. When faced with thrust disk tilt, the direct translational stiffness and damping coefficient of the bearing do not undergo significant changes, ensuring system stability. Based on the results of this study, the exceptional performance characteristics of thrust-type foil bearings make them a promising alternative to rolling bearings for the development of reusable liquid rocket turbopumps.

A Numerical Time Integration Procedure for Secondary Dendrite Arm Spacing in Hyper-Peritectic Steel Alloys

AbstractA numerical time integration procedure for the calculation of the secondary dendrite arm spacing (SDAS) in FeC hyper-peritectic alloys is presented, a preferred group of low-carbon casting steel. The procedure incorporates a three-stage thin arm dissolution model, which is solved at each time step using Newton–Raphson iteration. This is coupled to a coarsening model based on the integral forms of the dissolution model, which are solved using Gaussian quadrature, as well as a growth model for solid fraction evolution. The procedure can easily be embedded into numerical models for solidification, in which the space–time evolution of the SDAS is required for determining the dynamic permeability in the mushy zone. Higher order approximations for both growth and solute concentration evolution can easily be incorporated. Temperature dependence of thermophysical parameters is taken into account using inter-dendritic solidification empirical models, and an alloy-specific peritectic reaction constant is used to determine the isothermal peritectic holding time. The procedure is validated against experimental data presented in literature. Various cases of SDAS as a function of local solidification time, cooling rate and carbon composition are investigated. The method is compared to experimental results of SDAS obtained from test castings of a hyper-peritectic steel alloy and can be used to iteratively determine the alloy-specific peritectic reaction constant by comparing the solid fraction evolution during the peritectic reaction with that found from the experimental cooling curve.

A multi-mechanism numerical simulation model for CO2-EOR and storage in fractured shale oil reservoirs

Under the policy background and advocacy of carbon capture, utilization, and storage (CCUS), CO2-EOR has become a promising direction in the shale oil reservoir industry. The multi-scale pore structure distribution and fracture structure lead to complex multiphase flow, comprehensively considering multiple mechanisms is crucial for development and CO2 storage in fractured shale reservoirs. In this paper, a multi-mechanism coupled model is developed by MATLAB. Compared to the traditional Eclipse 300 and MATLAB Reservoir Simulation Toolbox (MRST), this model considers the impact of pore structure on fluid phase behavior by the modified Peng–Robinson equation of state (PR-EOS), and the effect simultaneously radiate to Maxwell–Stefan (M–S) diffusion, stress sensitivity, the nano-confinement (N-C) effect. Moreover, a modified embedded discrete fracture model (EDFM) is used to model the complex fractures, which optimizes connection types and half-transmissibility calculation approaches between non-neighboring connections (NNCs). The full implicit equation adopts the finite volume method (FVM) and Newton–Raphson iteration for discretization and solution. The model verification with the Eclipse 300 and MRST is satisfactory. The results show that the interaction between the mechanisms significantly affects the production performance and storage characteristics. The effect of molecular diffusion may be overestimated in oil-dominated (liquid-dominated) shale reservoirs. The well spacing and injection gas rate are the most crucial factors affecting the production by sensitivity analysis. Moreover, the potential gas invasion risk is mentioned. This model provides a reliable theoretical basis for CO2-EOR and sequestration in shale oil reservoirs.

Impedance-Based AC/DC Terminal Modeling and Analysis of the MMC-BTB System

Impedance-based small-signal stability analysis is widely applied in practical engineering with modular multilevel converters (MMCs). However, the deficiencies of existing impedance models (IMs) and the idealized extension for the single MMC influence the analyses in multiterminal systems. In this paper, these gaps are filled by focusing on an MMC-based back-to-back system. To obtain the steady-state harmonics of the system, a numerical method is initially proposed based on Newton–Raphson iteration in the frequency domain. Then, by substituting the shared terminal dynamics with active or passive devices, theoretical AC/DC IMs based on typical control loops with pure time delays are directly established via multiharmonic linearization. Further aided by the derived IMs, two neglected aspects in the current literature are investigated, including the influence of power transformers on low-frequency impedance characteristics and the rationality of using simplified IMs for high-frequency resonance studies. The stability of interlinking systems needs to be comprehensively analyzed at both AC and DC terminals. The analyses help to position the instability source, obtain the stability margin, and guide the supplementary control strategy. All IMs and analyses are verified via frequency scans and simulations in PSCAD.

A computational modeling on two-dimensional laminar flow and thermal characteristics through a strongly bent square channel

In order to have a precise knowledge on how pressure gradients and buoyancy force affect fluid flow and energy distribution in a bending channel, it is important to perform a comprehensive study on flow characteristics and heat transfer mechanisms that trigger out the transition of fluids into a turbulent state, subject to a sustained pressure gradient. The present paper explores a computational modeling on two-dimensional fluid flow and thermal characteristics in a bent square channel of strong curvature. The Newton–Raphson (N-R) iteration method is applied to obtain a bifurcation structure depending on the pressure-driven force, the Dean number (De), covering 0 < De ≤ 5000. As a consequence, four branches of asymmetric steady solutions are identified for each of the cases of the Grashof number, Gn (=1000, 1500, and 2000), where only the first branch is found to exhibit asymmetric two-vortex solutions while the remaining branches encompass two- to four-vortex solutions. The similarity and disparity in the branching structure are also demonstrated. Then, adopting the Adam–Bashforth (A-B) method together with Crank–Nicholson (C-N) formula, the unsteady solutions (US) have been explored, validated by power spectrum density (PSD) and phase space Within the realm of US, two- and three-vortex solutions are found and these solutions exhibit transitions from steady to chaotic behavior profoundly. Effects of the Grashof number with convective heat transfer (CHT) are also compared. By analyzing the Nusselt number (Nu), it is observed that in case of highly chaotic flow, CHT experiences substantial enhancement. This intensified CHT arises from increased turbulence and mixing, facilitating more efficient thermal energy exchange under such chaotic flow conditions.

Numerical study of solution structure and nonlinear behavior of Dean flow with vortex structure in a bending square duct

Fluid flow and heat transfer in bending channels are topics of much research interest because of increasing demand in various fields, particularly in the medical and industrial arenas. This paper presents a numerical study of fluid flow and heat transfer in a bending channel with a square cross section. Numerical calculations using spectral methods were performed for a curvature of 0.001 and a Dean number (Dn) in the range of 0<Dn≤5000. A temperature difference was maintained between the horizontal walls for a Grashof number of Gr = 1000, with the bottom wall heated and the outer and inner walls thermally insulated. Applying Newton–Raphson iteration and path continuation, two branching structures of steady solutions with two to eight vortices were obtained. The first branch comprises multi-vortex up to eight and it occurs at Dn≥3500 while the second branch comprises to multi-vortex up to a maximum of four. The first branch forms symmetric solution while the second branch for its complexity forms asymmetric solutions. For unsteady solutions, time-evolution calculations were performed to investigate the nonlinear behavior, and it was found that with increasing Dn, the unsteady flow experienced various flow instabilities. The time evolution was plotted in phase space to clarify the unsteady flow characteristics. Distinctive contours of the secondary flow patterns, streamwise velocity distributions, and isotherms were also obtained, and the unsteady flow was found to comprise two to six vortices. Finally, convective heat transfer was explored by obtaining temperature contours, and the secondary flow was found to magnify the convective heat transfer significantly. Because of the increase of several secondary vortices in the chaotic solutions, heat transfer occurred markedly in the flow.

Modeling of additive manufacturing processes with time‐dependent material properties using physics‐informed neural networks

AbstractRecently, physics‐informed neural networks (PINNs) have been effectively utilized in a wide range of problems within the domains of applied mathematics and engineering. In PINNs, the governing physical equations are directly incorporated into the loss function of the network and a conventional labeled dataset is not required for its training. In order to successfully simulate the additive manufacturing processes with concrete, a novel process‐based FE‐simulation has been developed where the Drucker–Prager plasticity model is used as the material model. In this work, we will examine the deployment of a PINN to substitute the Newton–Raphson iterations that occur in the return‐mapping algorithm of the Drucker–Prager plasticity model.

On the stability of mixed polygonal finite element formulations in nonlinear analysis

AbstractThis article discusses the accuracy and stability of the pressure field in nonlinear mixed displacement‐pressure finite element formulations in solid mechanics. We focus on two‐dimensional mixed polygonal finite element formulations with linear displacement and constant pressure approximations in particular. The inf‐sup stability of these formulations is assessed and compared with classical mixed finite element formulations. An analytical proof is presented, which concludes that the occurrence of spurious pressure modes depends on the chosen meshing strategy. It is shown that these spurious modes are successfully suppressed on any Voronoi mesh in both linear elasticity and nonlinear hyperelasticity without the need for any kind of stabilization. Several linear and nonlinear nearly‐incompressible examples with different discretization strategies and boundary conditions are considered to validate the analytical proof. A mixed polygonal finite element formulation based on the scaled boundary parameterization is used to approximate the field variables, however, the derivations presented herein hold for any lowest‐order mixed polygonal finite element formulation. The nonexistence of checkerboard modes on linear elastic Voronoi discretizations is shown graphically. By evaluating the incremental pressure in each Newton–Raphson iteration, the stabilization effect of the Voronoi discretization is demonstrated for nonlinear problems. In addition, the analytical proof is validated by the numerical (generalized) inf‐sup test.

Novel calculation method to predict gas–water two-phase production for the fractured tight-gas horizontal well

The prediction of production capacity in tight gas wells is greatly influenced by the characteristics of gas–water two-phase flow and the fracture network permeability parameters. However, traditional analytical models simplify the nonlinear problems of two-phase flow equations to a large extent, resulting in significant errors in dynamic analysis results. To address this issue, this study considers the characteristics of gas–water two-phase flow in the reservoir and fracture network, utilizes a trilinear flow model to characterize the effects of hydraulic fracturing, and takes into account the stress sensitivity of the reservoir and fractures. A predictive model for gas–water two-phase production in tight fractured horizontal wells is established. By combining the mass balance equation with the Newton–Raphson iteration method, the nonlinear parameters of the flow model are updated step by step using the average reservoir pressure. The accuracy of the model is validated through comparisons with results from commercial numerical simulation software and field case applications. The research results demonstrate that the established semi-analytical solution method efficiently handles the nonlinear two-phase flow problems, allowing for the rapid and accurate prediction of production capacity in tight gas wells. Water production significantly affects gas well productivity, and appropriate fracture network parameters are crucial for improving gas well productivity. The findings of this work could provide more clear understanding of the gas production performance from the fractured tight-gas horizontal well.

Divisions and Square Roots with Tight Error Analysis from Newton–Raphson Iteration in Secure Fixed-Point Arithmetic

In this paper, we present new variants of Newton–Raphson-based protocols for the secure computation of the reciprocal and the (reciprocal) square root. The protocols rely on secure fixed-point arithmetic with arbitrary precision parameterized by the total bit length of the fixed-point numbers and the bit length of the fractional part. We perform a rigorous error analysis aiming for tight accuracy claims while minimizing the overall cost of the protocols. Due to the nature of secure fixed-point arithmetic, we perform the analysis in terms of absolute errors. Whenever possible, we allow for stochastic (or probabilistic) rounding as an efficient alternative to deterministic rounding. We also present a new protocol for secure integer division based on our protocol for secure fixed-point reciprocals. The resulting protocol is parameterized by the bit length of the inputs and yields exact results for the integral quotient and remainder. The protocol is very efficient, minimizing the number of secure comparisons. Similarly, we present a new protocol for integer square roots based on our protocol for secure fixed-point square roots. The quadratic convergence of the Newton–Raphson method implies a logarithmic number of iterations as a function of the required precision (independent of the input value). The standard error analysis of the Newton–Raphson method focuses on the termination condition for attaining the required precision, assuming sufficiently precise floating-point arithmetic. We perform an intricate error analysis assuming fixed-point arithmetic of minimal precision throughout and minimizing the number of iterations in the worst case.

Nonlinear dynamic behaviors of angular contact ball bearing with waviness based on a synthetical multi-degree-of-freedom mathematical modelling

This paper proposes a synthetical multi-degree-of-freedom mathematical modelling for analyzing the dynamic characteristics of angular contact ball bearing with waviness subjected to the external load working conditions. The above multi-degree-of-freedom mathematical modelling is established based on nonlinear elastic Hertz contact theory and inner raceway control theory by employing improved Newton–Raphson iteration method; the validation of the established mathematical modelling is verified by comparing the presented results with the existing literature results. The above mathematical modelling has considered raceway waviness, ball waviness, bearing clearance, centrifugal force, gyroscopic moment, external loads, and rotating speed comprehensively. Firstly, the effects of waviness order and waviness amplitude on time-varying contact and stiffness characteristics of angular contact ball bearing are investigated systematically. Then, the influences of bearing clearance, rotating speed, axial force, radial force, and torque on the dynamic contact and stiffness characteristics of angular contact ball bearing with waviness are analyzed thoroughly. The main results show that the solution results of mathematical modelling have converged, respectively, when raceway waviness order increases to 25 and ball waviness order increases to 5; The variation tendencies of time-varying contact angle, contact force, and stiffness of angular contact ball bearing with waviness are aperiodic; The amplitudes of time-varying contact angle, contact force, and stiffness of angular contact ball bearing increase evidently with waviness amplitude increased; The bearing clearance has important influence on the contact angle and contact force of outer raceway and inner raceway, respectively; The effects of radial force and torque on contact and stiffness characteristics are consistent basically in the corresponding direction; The influences of axial force and rotating speed on the contact and stiffness characteristics are more remarkable compared with other factors. The investigations of contact and stiffness characteristics of angular contact ball bearing with waviness are helpful for the designation and manufacture of high precision angular contact ball bearing.

Self-similar solutions to the hypoviscous Burgers and SQG equations at criticality

After reviewing the source-type solution of the Burgers equation with standard dissipativity, we study the hypoviscous counterpart of the Burgers equation. (1) We determine an equation that governs the near-identity transformation underlying its self-similar solution. (2) We develop its approximation scheme and construct the first-order approximation. (3) We obtain the source-type solution numerically by the Newton–Raphson iteration scheme and find it to agree well with the first-order approximation. Implications of the source-type solution are given, regarding the possibility of linearisation of the hypoviscous Burgers equation. Finally we address the problems of the incompressible fluid equations in two dimensions, centering on the surface quasi-geostrophic equation with standard and hypoviscous dissipativity.

Buckling and Post-Buckling of Bidirectional Porous Beam Under Bidirectional Hygrothermal Environment

In this paper, buckling and nonlinear post-buckling behaviors of a bidirectional porous (BDP) beam are investigated under bidirectional hygrothermal environment. Euler–Bernoulli beam theory with the von Kármán nonlinearity is employed to derive the nonlinear variable coefficient governing differential equations based on Rayleigh quotient method. Analytical solutions of critical buckling load and load–deflection equilibrium path in post-buckling are deduced for the single directional varying (SDV) porous beam. The general numerical solutions for bidirectional varying (BDV) porous beam are obtained by differential quadrature finite element method (DQFEM) with Newton–Raphson iteration method based on the variation principle. The high accuracy of the present numerical method with higher computing efficiency is verified by comparison with published reports and the analytical results in this work. Parametric analysis on effects of the porosity bidirectional distributions, porosity coefficients, distributions of hygrothermal environment and boundary conditions on buckling load and post-buckling response is carried out to enhance the buckling and deformation resistances in design, manufacture and usage of porous structures. The results show that the bidirectional porosity pattern, linear and nonlinear hygrothermal distribution and boundary conditions play a significant role on buckling critical external load and critical hygrothermal increments, buckling form and post-buckling path.

A full 6 degree-of-freedom visual estimation with laser assisted

Efficient and accurate measurement in unstructured environment is a challenging issue and also research interest in the industrial field. The accuracy of vision is very high in the close-range scene and the stability of laser measurement is better, with a large achievable range. A 6-DOF pose measurement method based on vision and assisted by laser is proposed in this paper. By combining their respective advantages, the method can be immune to ambient light. The optimization objective is established by using the projection coordinates of laser spots in different coordinate systems and solved by Newton–Raphson iteration. The initial value is estimated by Kronecker product. The Monte Carlo method is further used to simulate. The causes of measurement errors and the influence factors affecting the accuracy are analyzed. On this basis, an experimental prototype is constructed. The accuracy and repeatability accuracy experiments are conducted to verify the correctness of the proposed method.

Research on a high-precision real-time improvement method for aero-engine component-level model

Abstract In order to improve the real-time performance of the aero-engine Component-Level Model (CLM) while ensuring accuracy, a method for the Calculation of Thermodynamic Parameters of Working Fluids (CTPWF) based on a Neural Network and Newton Raphson (NN-NR) is proposed. In this method, the enthalpy or entropy under different fuel-air ratio and humidity conditions is mapped to temperature by a neural network, and the mapping output is used as the initial solution of Newton Raphson (NR) iteration. Then, a high-precision solution can be obtained through a few iterations, which avoids the shortcoming that the traditional method uses a fixed initial solution that leads to too many iterative steps. This effectively reduces the number of iterative steps and improves the calculation efficiency. This method is applied to the aero-thermodynamic calculation of each component of an engine CLM, which improves the accuracy and real-time performance of the CLM. The simulation results show that, compared to the traditional method, the proposed method improves the accuracy of the CTPWF and can reduces the single aero-thermodynamic calculation time by 25 % when humidity is not considered and by 47 % when humidity is considered. This effectively improves the real-time performance of the CLM.