Newton-Raphson Iterative Algorithm Research Articles

Nonlinear Inelastic Earthquake Analysis of 2D Steel Frames

In this work, a new method for nonlinear time-history earthquake analysis of 2D steel frames by a fiber plastic hinge method is presented. The beam-column element based on the displacement-based finite element method is established and formulated in detail using a fiber plastic hinge approach and stability functions. Geometric nonlinearities are taken into accounting by stability functions and the geometric stiffness matrix. A nonlinear dynamic algorithm is established based on the combination of the Newmark integration method and the Newton-Raphson iterative algorithm for solving dynamic equations. The proposed program predicts the nonlinear inelastic responses of 2D steel frames subjected to earthquakes as efficiently and accurately as commercial software. This study also shows that the initial residual stresses of steel should be considered in nonlinear inelastic time-history earthquake analysis of 2D steel frames while SAP2000 does not consider the effects of residual stresses.

RBF collocation approach to calculate numerically the solution of the nonlinear system of qFDEs

In this paper, we have implemented an efficient and high accurate radial basis function (RBF) collocation scheme for solving nonlinear systems of q-fractional differential equations. We firstly convert the problems under investigation into the equivalent systems of weakly singular q-integral equations by some essential results of fractional q-calculus. Secondly, we combine RBF collocation method and Newton–Raphson iterative algorithm to solve the latter systems of weakly singular q-integral equations. More precisely, applying RBF collocation scheme will transform the system of q-integral equations into the associated system of nonlinear algebraic equations that can be solved by iterative methods such as the Newton–Raphson algorithm. Finally, various numerical test problems including linear and nonlinear examples are listed to illustrate the robustness of the proposed global scheme with respect to the at least two recent methods in the literature.

Moving Element Analysis of High-Speed Train-Slab Track System Considering Discrete Rail Pads

This paper presents a study of the dynamic behavior of a coupled train-slab track system considering discrete rail pads. The slab track is modeled as a three-layer Timoshenko beam. The study is carried out using the moving element method (MEM). By introducing a convected coordinate system moving at the same speed as the vehicle, the governing equations of motion of the slab track are formulated in a moving frame-of-reference. By adopting Galerkin’s method, the element stiffness, mass and damping matrices of a truncated slab track in the moving coordinate system are derived. The vehicle is modeled as a multi-body with 10 degrees of freedom. The nonlinear Hertz contact model is used to account for the wheel–rail interaction. The Newmark integration method, in conjunction with a global Newton–Raphson iteration algorithm, is employed to solve the nonlinear dynamic equations of motion of the vehicle–track coupled system. The proposed MEM model of the system is validated through comparison with available results in the literature. Further study is then made to investigate the vehicle–track system accounting for track irregularities modeled as short harmonic wave forms. Results showed that irregularities with short wavelengths have a significant effect on wheel–rail contact force and rail acceleration, and the dynamic response of the track structure does not increase monotonously with the increase of the vehicle speed.

A noise-suppressing Newton-Raphson iteration algorithm for solving the time-varying Lyapunov equation and robotic tracking problems

The Newton-Raphson iteration (NRI) algorithm is a prevalent computational methodology in many fields and can be used to solve the time-varying Lyapunov equation. However, the performance of the traditional NRI (TNRI) algorithm is severely degraded under noisy conditions. To overcome this weakness, a noise-suppressing NRI (NSNRI) algorithm is proposed in this paper. By utilizing the Kronecker product theorem, the time-varying Lyapunov equation can be transformed into a linear matrix equation. Based on that linear matrix equation, the related theoretical analyses of the convergence and the noise-suppressing property of the NSNRI algorithm under various noise conditions are provided. To verify the theoretical analyses, numerical simulations for solving the time-varying Lyapunov equation and an application to manipulator motion tracking are presented. For comparison purpose, the TNRI and two zeroing neural network (ZNN) algorithms are also introduced in these simulations. As indicated by the simulation results, the NSNRI algorithm is superior in terms of the convergence accuracy and the robustness to noise.

Affine-Type Model of Variable Renewable Energy and Its Application in the Uncertain Power Flow

The intermittent variable renewable energy (VRE) results in uncertainty in distribution networks, and the existing affine iterative algorithm cannot consider the characteristics of VRE. To deal with such issues, an affine-type model of VRE considering its regulation characteristics was proposed in this paper. Furthermore, an alternating iterative algorithm based on Newton-Raphson iterative algorithm was proposed, which can consider both power flow equations and the VRE models. Finally, the effectiveness of the affine-type model of VRE and the affine iterative algorithm were evaluated on the modified IEEE 33-bus system and PG&E 69-bus system. The simulation results showed that the proposed method has sufficient completeness and slight conservativeness. Moreover, the proposed method with VRE models is superior over Monte Carlo method, particularly in terms of calculation burden.

Postbuckling analysis of multi-directional perforated FGM plates using NURBS-based IGA and FCM

Geometric modeling and numerical analysis of multi-directional FGM (Functionally Graded Material) plate, whose material properties grade continuously both in its thickness and in-plane directions, are increasingly required. In this work, postbuckling behavior of this type of plates with multiple cutouts is, for the first time, numerically investigated through the combination of NURBS-based IGA (IsoGeonetric Analysis) and FCM (Finite Cell Method). The nonlinear deformation of plate is determined by TSDT (Third-order Shear Deformation Theory) and von Kármán nonlinear assumptions without the requirement of SCFs (shear correction factors). Besides, the higher continuity advantage of NURBS basis functions can easily meet the C1-continuous requirement of the displacement field. The main contribution is introducing the FCM to deal with the influence of complex cutouts on the postbuckling characteristics. The geometric interfaces of the cutouts are approached and approximated by adaptive quadrature procedure in the distinguished cut elements. The advantage of this implementation is that the previously tricky process of representing the geometry of perforated plate with multiple NURBS patches can be eliminated, which naturally avoids the imposition of C1-continuity condition across the patch boundaries. The cylinder arc-length method combined with modified Newton–Raphson iteration algorithm, which takes into account of the initial geometric imperfections, is applied to implement geometrically nonlinear stability analysis and track the postbuckling paths. The effectiveness and reliability of the presented method are verified with available solutions of isotropic and conventional perfect FGM plates. Subsequently, a series of factors, including material volume fraction, length-to-thickness ratio, boundary condition, cutout size, etc., affecting the postbuckling responses of multi-directional perforated FGM plates are considered and investigated.

A High-Performance Computing of Internal Rate of Return using a Centroid-based Newton-Raphson Iterative Algorithm

A High-Performance Computing of Internal Rate of Return using a Centroid-based Newton-Raphson Iterative Algorithm

Slender metal web stability analysis: Impact of initial imperfection on the mode of buckling

The post buckling of a rectangular slender web in compression has been analyzed. Shapes of a buckling area obtained from the nonlinear analysis have been compared with buckling modes from the linearized problem for various aspect ratios. Effects of initial shape imperfections upon the analysis have been investigated using nonlinear approach. To trace the complete nonlinear equilibrium curves, specialized code based on FEM was created. The Newton-Raphson iteration algorithm was used, load versus displacement control was changed during the process of calculation. Obtained results were verified using Ansys system, in this case arc-length method was activated for overcoming critical points.

Design of wake-adapted contra-rotating propellers for high-speed underwater vehicles

Based on the lifting-surface vortex lattice model, a numerical design method of wake-adapted contra-rotating propellers (CRPs) for high-speed underwater vehicles is proposed. According to the given radial circulation distribution, the method can use prescribed camber line shapes to design maximum cambers and pitches of blade sections by controlling circulation at the leading edge, which makes the chordwise distribution of blade loading similar to that of NACA a = 0.8. It also can be performed under prescribed chordwise circulation distributions, where camber line shape and blade section pitch are designed. The Newton–Raphson iterative algorithm is utilised in the design of the pitch and camber. The radial circulation distribution of a set of CRPs for an underwater vehicle is used to redesign CRPs by the proposed method, and the design results are then validated via numerical simulations by solving the Reynolds-averaged Navier-Stokes equations. The results indicate that the proposed method is suitable for the design of CRPs with tapered hubs and skewed blades, and it also exhibits good mesh convergence. The CRPs designed with the given camber line shape and the given chordwise loading distribution both have relatively uniform pressure distributions, with the latter being superior.

Time-Domain Impulse Response With the TD-VSIE Field-Circuit Coupling Algorithm for Nonlinear Analysis of Microwave Amplifier

In this letter, the time-domain impulse response with the time-domain volume–surface integral equation field-circuit coupling algorithm is presented. This method uses the time-domain impulse response technique to separate the calculation of the linear electromagnetic field structure from the nonlinear equivalent circuit in the microwave circuit. First, the time-domain impulse response sources of the linear microwave structure are extracted at the location of ports. Then, the nonlinear system equations are built by the time-domain impulse response sources and the equivalent circuit equations. Meanwhile, the Newton–Raphson iteration algorithm is utilized to solve the nonlinear system equations. Finally, the simulation results illustrate that the proposed method is more accurate and effective than the conventional method.

Basins of convergence of equilibrium points in the generalized Hénon–Heiles system

We numerically explore the Newton–Raphson basins of convergence, related to the libration points (which act as attractors of the convergence process), in the generalized Hénon-Heiles system (GHH). The evolution of the position as well as of the linear stability of the equilibrium points is determined, as a function of the value of the perturbation parameter. The attracting regions, on the configuration (x,y) plane, are revealed by using the multivariate version of the classical Newton–Raphson iterative algorithm. We perform a systematic investigation in an attempt to understand how the perturbation parameter affects the geometry as well as of the basin entropy of the attracting domains. The convergence regions are also related with the required number of iterations.

An enriched–FEM technique for numerical simulation of interacting discontinuities in naturally fractured porous media

In this paper, an extended finite element method is presented for simulation of interaction between hydraulic fracturing and natural fractures in saturated porous media. The well-known u−p formulation is employed in order to obtain the fully coupled set of governing equations. Natural faults are modeled for both opening and closure modes where the fluid inflow and contact conditions are considered at the interface, respectively. The Darcy law is employed in conjunction with an aperture dependent permeability for the fracture channel to describe the interfacial inflow. The contact constraints of both the solid and fluid phases are imposed using the Penalty method. The Heaviside and modified ramp enrichment functions are introduced to model the required strong and weak discontinuities in the displacement and pressure fields, respectively. Furthermore, new enrichment functions are introduced to model the junction and/or intersection of the discontinuities. The system of nonlinear equations is solved using the Newton–Raphson iterative algorithm with an implicit time stepping scheme. Finally, several numerical examples are chosen to illustrate the robustness of the proposed formulation as well as to study the interaction of hydraulic fracturing with natural faults.

Analyzing of Wedge Splitting Test on Asphalt Pavement Using Optical Measurements

Abstract This paper dealt with the characterization of opening mode fracture using Digital Image Correlation, coupled with a kinematic approach. Tests were carried out using a Wedge Splitting sample made in asphalt concrete AC12. The asphalt concrete was provided in the RILEM-SIB. The samples were a bi-layered asphalt concrete with or without a carbon fiber grid at interface. During experimentation, the sample deformation was recorded by using Digital Image Correlation. Based on the optical measurements coupled with the analytical approach, the energy release rate was performed by means the crack relative displacement and stress intensity factors. Using an adjustment procedure, based on an iterative Newton–Raphson algorithm, the crack relative displacement factor was calculated from optical measurements. the stress intensity factor was estimated from an analytical approach. Furthermore, the crack opening displacement was measured from the mark tracking method. This approach allowed the assessment of fracture parameters for the real structures to be considered.

Geometrically nonlinear isogeometric analysis of functionally graded microplates with the modified couple stress theory

In this study, a new and efficient computational approach based on isogeometric analysis (IGA) and refined plate theory (RPT) is proposed for the geometrically nonlinear analysis of functionally graded (FG) microplates. While the microplates’ size-dependent effects are efficiently captured by a simple modified couple stress theory (MCST) with only one length scale parameter, the four-unknown RPT is employed to establish the displacement fields which are eventually used to derive the nonlinear von Kámán strains. The NURBS-based isogeometric analysis is used to construct high-continuity elements, which is essentially required in the modified couple stress and refined plate theories, before the iterative Newton-Raphson algorithm is employed to solve the nonlinear problems. The successful convergence and comparison studies as well as benchmark results of the nonlinear analysis of FG microplates ascertain the validity and reliability of the proposed approach. In addition, a number of studies have been carried out to investigate the effects of material length scale, material and geometrical parameters on the nonlinear bending behaviours of microplates.

Advanced analysis for planar steel frames with semi-rigid connections using plastic-zone method

This paper presents a displacement-based finite element procedure for second-order distributed plasticity analysis of planar steel frames with semi-rigid beam-to-column connections under static loadings. A partially strain-hardening elastic-plastic beam-column element, which directly takes into account geometric nonlinearity, gradual yielding of material, and flexibility of semi-rigid connections, is proposed. The second-order effects and distributed plasticity are considered by dividing the member into several sub-elements and meshing the cross-section into several fibers. A new nonlinear solution procedure based on the combination of the Newton-Raphson equilibrium iterative algorithm and the constant work method for adjusting the incremental load factor is proposed for solving nonlinear equilibrium equations. The nonlinear inelastic behavior predicted by the proposed program compares well with previous studies. Coupling effects of three primary sources of nonlinearity, geometric imperfections, and residual stress are investigated and discussed in this paper.

Further investigation of wheel climb initiation: Three-point contact

In previous research, a set of nonlinear algebraic kinematic constraint equations were developed that describe the configuration of a wheelset in contact with a track at two distinct points. In such a case of two points of contact, a simplified wheelset model that has the lateral displacement and angle of attack as the independent variables can be developed. In the current investigation, this approach is extended to the new case of a wheelset in contact with a tangent track at three distinct points. The solution of this three-point contact problem requires specifying the wheelset angle of attack only. This wheelset configuration is significant in derailment investigations because it is a possible configuration at the initiation of a wheel climb derailment. In order to study this wheel climb initiation configuration, a set of nonlinear kinematic constraint equations is developed as a function of the wheelset angle of attack and solved for the unknown system coordinates and contact surface parameters using an iterative Newton–Raphson algorithm. The wheelset angle of attack during wheel climb derailments can be determined forensically at the derailment site, making this approach of practical significance. It is shown in this investigation that the system configuration can be fully defined for wheel climb derailment initiation, which allows for the investigation of various derailment parameters such as the wheel–rail contact angle. It is then reinforced in this study that the wheelset flange angle, which is the angle between the tangent to the wheel surface at the contact point and the wheelset axle, is not representative of the wheel–rail contact angle, which is the angle between the tangent to the contact surfaces and the lateral common tangent to the two railheads; this distinction can only be demonstrated through full definition of the system configuration that accounts for the wheelset roll angle. This investigation therefore calls into question the Nadal [Formula: see text] derailment limit as well as any investigation that chooses to neglect the wheelset orientation or the effect of such orientation on the wheel/rail contact geometry. This new formulation is validated and supported using a three-dimensional fully nonlinear unconstrained multibody system wheel climb derailment model recently proposed in a previous study. Using this model, new results demonstrate that initiation of the wheel climb motion is correctly predicted using proper geometry definitions in the derailment criteria, whereas previous results demonstrated such motion was not correctly predicted using the geometry definitions used by Nadal. This investigation is not intended as a derailment criteria proposal, but rather as support and rationalization for the use of correct contact geometry in derailment investigations. This investigation reiterates the important result that the Nadal [Formula: see text] derailment limit is not conservative, and demonstrates that, with proper formulation, more accurate and justifiable derailment criteria can be developed.

Feedrate optimization of complex surface milling based on predictive model of cutting force

In terms of the low machining efficiency and bad surface quality using constant feedrate in complex surface milling, a feedrate optimization method is proposed based on cutting force model. Firstly, by the analysis and calculation of the tool path, the geometry of complex surface and tool, tool-workpiece engagement zone and uncut chip thickness, a cutting force model is derived in multi-axis maching of complex surface and the mapping relation is obtained between feedrate and cutting force. Then the objective function about the absolute value of difference between the reference cutting force and the maximum one calculated in each tool position is put forward. Then the method adopts Newton-Raphson iteration algorithm to alter the feedrate in the original tool path file, thus ensures the constant cutting forces and enhance steadiness of machining process. The experiment is carried out about model propeller, the results show that the proposed optimization method ensures the machining time of the blade reduced by more than 25%, the surface residual stresses and micro-hardness machining accuracy to be increased, which has a clear effect on increasing the machining efficiency and improving the machining surface quality of the blade. This thus validates the effectiveness of the proposed optimization method.

Stability Analysis of an Imperfect Slender Web Subjected to the Shear Load

The stability analysis of an imperfect slender web subjected to the shearing load is presented, a specialized code based on FEM has been created. The nonlinear finite element method equations are derived from the variational principle of minimum of total potential energy. To obtain the nonlinear equilibrium paths, the Newton-Raphson iteration algorithm is used. Corresponding levels of the total potential energy are defined. The peculiarities of the effects of the initial imperfections are investigated. Special attention is paid to the influence of imperfections on the post-critical buckling mode. Obtained results are compared with those gained using ANSYS system.

Dynamic Gust Load Analysis for Rotors

Dynamic load of helicopter rotors due to gust directly affects the structural stress and flight performance for helicopters. Based on a large deflection beam theory, an aeroelastic model for isolated helicopter rotors in the time domain is constructed. The dynamic response and structural load for a rotor under the impulse gust and slope-shape gust are calculated, respectively. First, a nonlinear Euler beam model with 36 degrees-of-freedoms per element is applied to depict the structural dynamics for an isolated rotor. The generalized dynamic wake model and Leishman-Beddoes dynamic stall model are applied to calculate the nonlinear unsteady aerodynamic forces on rotors. Then, we transformed the differential aeroelastic governing equation to an algebraic one. Hence, the widely used Newton-Raphson iteration algorithm is employed to simulate the dynamic gust load. An isolated helicopter rotor with four blades is studied to validate the structural model and the aeroelastic model. The modal frequencies based on the Euler beam model agree well with published ones by CAMRAD. The flap deflection due to impulse gust with the speed of 2m/s increases twice to the one without gust. In this numerical example, results indicate that the bending moment at the blade root is alleviated due to elastic effect.

Optimal air route flight conflict resolution based on receding horizon control

To deal with the air route flight conflicts caused by abnormal situations during four-dimensional based air traffic operation, the air route flight conflict resolution problem for two aircrafts was addressed. Based on the optimized static single heading angle or ground speed adjustment strategy, we proposed an optimized dynamic mixed conflict resolution strategy based on Receding Horizon Control (RHC), which considered the possibility of one aircraft's ground speed variation. In particular, the wind speed vector disturbance during conflict resolution might lead to a model mismatch, so the maximum likelihood estimation method and the Newton–Raphson iterative algorithm were employed to identify the wind speed vector using the aircraft true airspeed inputs and ground based trajectory measurements. Moreover, we considered the convergence of conflict resolution method based on RHC and proposed the pre-condition that local and global optimization resolution can be achieved. We compared three situations, i.e., static single strategy optimization, dynamic mixed strategy optimization using RHC with one aircraft's ground speed variation, and dynamic mixed strategy optimization using RHC with the wind speed vector disturbance. We demonstrated that the dynamic mixed strategy responds to one aircraft's ground speed disturbance quickly, and resolved conflict by track angle and ground speed adjustments in an effective manner after the wind speed vector being identified accurately.