Raphson Iteration Research Articles

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.

Calculation of Flight Data Fusion Coefficients Using Newton–Raphson Iterations

The problem of plotting the flight path of an aircraft based on flight data containing numerous measurement errors is investigated. A theoretical (continuous) model of the flight data fusion problem is proposed in the form of a boundary value problem for a system of differential equations with unknown coefficients. The application of the Newton–Raphson iteration method for calculating the sought-for coefficients is described.

Parameter Estimation and Hypothesis Testing of Multivariate Poisson Inverse Gaussian Regression

Multivariate Poisson regression is used in order to model two or more count response variables. The Poisson regression has a strict assumption, that is the mean and the variance of response variables are equal (equidispersion). Practically, the variance can be larger than the mean (overdispersion). Thus, a suitable method for modelling these kind of data needs to be developed. One alternative model to overcome the overdispersion issue in the multi-count response variables is the Multivariate Poisson Inverse Gaussian Regression (MPIGR) model, which is extended with an exposure variable. Additionally, a modification of Bessel function that contain factorial functions is proposed in this work to make it computable. The objective of this study is to develop the parameter estimation and hypothesis testing of the MPIGR model. The parameter estimation uses the Maximum Likelihood Estimation (MLE) method, followed by the Newton–Raphson iteration. The hypothesis testing is constructed using the Maximum Likelihood Ratio Test (MLRT) method. The MPIGR model that has been developed is then applied to regress three response variables, i.e., the number of infant mortality, the number of under-five children mortality, and the number of maternal mortality on eight predictors. The unit observation is the cities and municipalities in Java Island, Indonesia. The empirical results show that three response variables that are previously mentioned are significantly affected by all predictors.

Point Inversion for triparametric NURBS

In this paper a Point Inversion algorithm for NURBS volumes is presented. This algorithm is an extension to 3D of the classic Newton–Raphson iteration, thus implying the calculation of the NURBS volume partial derivatives. Explicit formulas for the derivatives are deduced and reported, in turn requiring the evaluation of the corresponding derivatives of the rational basis functions. Also these derivatives are inferred and shown together with some examples of applications. The method, applicable to any free-form shape NURBS volume is moreover compared with Fast Mapping, a novel approach presented here that, once applied to simple NURBS boxes give performance 20–50 times higher.

Nonlinear modeling and analysis of rotors supported by magnetorheological squeeze film journal bearings

The driver coupled to a driven system through mechanical couplings is very common in rotating machinery. These couplings can present angular and parallel misalignments with more or less degree due to manufacturing tolerances or maintenance proceedings. Theoretical and experimental analyses have been published demonstrating the effects of rotor misalignment and the vibration stability of rotor systems. In this work the nonlinear formulation of a magnetorheological fluid journal bearing is included in the finite element model that evaluates the nonlinear responses of a complete rotor system subject to rigid coupling misalignment. The modified Reynolds equations for Bingham viscoplastic materials are implemented in the finite element procedures to evaluate the nonlinear hydrodynamic reaction forces acting on the bearing positions. The finite element formulation for the shaft-line and mechanical couplings is based on the Timoshenko beam theory. Misalignment forces are calculated and included in the equations of motion. The nonlinear dynamic responses are calculated by the modified Newmark method incorporating the Newton–Raphson iteration method to find the equilibrium position at each time step. Bifurcation analysis demonstrates the influence of misalignment to obtain periodic and period-doubling orbit for the center position of the rotor. Results are demonstrated through displacements versus time and frequency responses.

On the Abramov approach for the approximation of whispering gallery modes in prolate spheroids

In this paper, we present the Abramov approach for the numerical simulation of the whispering gallery modes in prolate spheroids. The main idea of this approach is the Newton–Raphson technique combined with the quasi-time marching. In the first step, a solution of a simpler problem, as an initial guess for the Newton–Raphson iterations, is provided. Then, step-by-step, this simpler problem is converted into the original problem, while the quasi-time parameter τ runs from τ=0 to τ=1. While following the involved imaginary path two numerical approaches are realized, the first is based on the Prüfer angle technique, the second on high order finite difference schemes.

DESC: A stellarator equilibrium solver

The new code DESC is presented to solve for fixed-boundary ideal magnetohydrodynamic equilibria in stellarators. The approach directly solves the equilibrium force balance as a system of nonlinear equations in the form f(x)=0. The independent variables x represent nested magnetic flux surfaces expressed in the inverse representation with toroidal flux coordinates, and the equations f(x) quantify equilibrium force balance errors at discrete points in real space. Discretizing with global Fourier–Zernike basis functions properly treats the magnetic axis and minimizes the number of coefficients needed to describe the flux surfaces. The pseudospectral method provides great flexibility in where the errors are evaluated, and the system of equations is efficiently solved with a Newton–Raphson iteration. Equilibria are computed and compared against VMEC for both axisymmetric and non-axisymmetric examples. The results show fast convergence rates and solutions with low errors throughout the plasma volume.

Nonlinear finite element analysis of temperature-dependent functionally graded porous micro-plates under thermal and mechanical loads

In the present study, a displacement based nonlinear finite element model for functionally graded porous micro-plates is developed based on the general third-order shear deformation plate theory and the modified couple stress theory. The developed finite element model accounts for von Kármán nonlinear strains, a power-law variation of two material constituents through the plate thickness, and different distributions of porosity with a constant volume of voids on static bending of micro-plates are analyzed. The length scale dependency is captured using a single parameter of the modified couple stress theory. A power-law distribution is assumed to model the variation of the two material constituents, while the porosity distributions vary according to cosine functions. The temperature-dependent properties are obtained using a cubic-spline interpolation from experimental data. A one-dimensional steady-state heat conduction problem is solved using the effective thermal conductivity of the porous material based on the Maxwell–Eucken model to obtain temperature distribution through the plate thickness. The Newton–Raphson iteration scheme is used to solve the nonlinear system of equations. A parametric study is conducted to demonstrate the effects of material and porosity parameters, temperature and length scale dependencies, and boundary conditions on the deflections and stress distributions.

On Nonlinear Bending Study of a Piezo-Flexomagnetic Nanobeam Based on an Analytical-Numerical Solution.

Among various magneto-elastic phenomena, flexomagnetic (FM) coupling can be defined as a dependence between strain gradient and magnetic polarization and, contrariwise, elastic strain and magnetic field gradient. This feature is a higher-order one than piezomagnetic, which is the magnetic response to strain. At the nanoscale, where large strain gradients are expected, the FM effect is significant and could be even dominant. In this article, we develop a model of a simultaneously coupled piezomagnetic–flexomagnetic nanosized Euler–Bernoulli beam and solve the corresponding problems. In order to evaluate the FM on the nanoscale, the well-known nonlocal model of strain gradient (NSGT) is implemented, by which the nanosize beam can be transferred into a continuum framework. To access the equations of nonlinear bending, we use the variational formulation. Converting the nonlinear system of differential equations into algebraic ones makes the solution simpler. This is performed by the Galerkin weighted residual method (GWRM) for three conditions of ends, that is to say clamp, free, and pinned (simply supported). Then, the system of nonlinear algebraic equations is solved on the basis of the Newton–Raphson iteration technique (NRT) which brings about numerical values of nonlinear deflections. We discovered that the FM effect causes the reduction in deflections in the piezo-flexomagnetic nanobeam.

New methods for projecting a 3D object onto a free-form surface

PurposeThis paper aims to provide a series of new methods for projecting a three-dimensional (3D) object onto a free-form surface. The projection algorithms presented can be divided into three types, namely, orthogonal, perspective and parallel projection.Design/methodology/approachFor parametric surfaces, the computing strategy of the algorithm is to obtain an approximate solution by using a geometric algorithm, then improve the accuracy of the approximate solution using the Newton–Raphson iteration. For perspective projection and parallel projection on an implicit surface, the strategy replaces Newton–Raphson iteration by multi-segment tracing. The implementation takes two mesh objects as an example of calculating an image projected onto parametric and implicit surfaces. Moreover, a comparison is made for orthogonal projections with Hu’s and Liu’s methods.FindingsThe results show that the new method can solve the 3D objects projection problem in an effective manner. For orthogonal projection, the time taken by the new method is substantially less than that required for Hu’s method. The new method is also more accurate and faster than Liu’s approach, particularly when the 3D object has a large number of points.Originality/valueThe algorithms presented in this paper can be applied in many industrial applications such as computer aided design, computer graphics and computer vision.

Fast algorithms for the quantile regression process

The widespread use of quantile regression methods depends crucially on the existence of fast algorithms. Despite numerous algorithmic improvements, the computation time is still non-negligible because researchers often estimate many quantile regressions and use the bootstrap for inference. We suggest two new fast algorithms for the estimation of a sequence of quantile regressions at many quantile indexes. The first algorithm applies the preprocessing idea of Portnoy and Koenker (Stat Sci 12(4):279–300, 1997) but exploits a previously estimated quantile regression to guess the sign of the residuals. This step allows for a reduction in the effective sample size. The second algorithm starts from a previously estimated quantile regression at a similar quantile index and updates it using a single Newton–Raphson iteration. The first algorithm is exact, while the second is only asymptotically equivalent to the traditional quantile regression estimator. We also apply the preprocessing idea to the bootstrap by using the sample estimates to guess the sign of the residuals in the bootstrap sample. Simulations show that our new algorithms provide very large improvements in computation time without significant (if any) cost in the quality of the estimates. For instance, we divide by 100 the time required to estimate 99 quantile regressions with 20 regressors and 50,000 observations.

Accurate iteration-free mixed-stabilised formulation for laminar incompressible Navier–Stokes: Applications to fluid–structure interaction

Stabilised mixed velocity–pressure formulations are one of the widely-used finite element schemes for computing the numerical solutions of laminar incompressible Navier–Stokes. In these formulations, the Newton–Raphson scheme is employed to solve the nonlinearity in the convection term. One fundamental issue with this approach is the computational cost incurred in the Newton–Raphson iterations at every load/time step. In this paper, we present an iteration-free mixed finite element formulation for incompressible Navier–Stokes that preserves second-order temporal accuracy of the generalised-alpha and related schemes for both velocity and pressure fields. First, we demonstrate the second-order temporal accuracy using numerical convergence studies for an example with a manufactured solution. Later, we assess the accuracy and the computational benefits of the proposed scheme by studying the benchmark example of flow past a fixed circular cylinder. Towards showcasing the applicability of the proposed technique in a wider context, the inf–sup stable P2–P1 pair for the formulation without stabilisation is also considered. Finally, the resulting benefits of using the proposed scheme for fluid–structure interaction problems are illustrated using two benchmark examples in fluid-flexible structure interaction.

Nonlinear supersonic flutter study of porous 2D curved panels including graphene platelets reinforcement effect using trigonometric shear deformable finite element

The nonlinear panel flutter behaviour of two-dimensional porous curved panel reinforced by graphene platelets exposed to a supersonic flow is investigated. A curved beam element developed based on the trigonometric shear deformation theory is employed. The formulation integrates the geometric nonlinearity with von Karman’s approximation. The effort to model the fluid–structure interaction is reduced by implementing the first-order form of piston theory aerodynamics to describe the flow and accounting for the influence of static aerodynamic load due to the inherent geometric curvature of the panel. The nonlinear governing equations are formulated adopting the Lagrangian formulation. The panel deflection under the static aerodynamic load is evaluated using the Newton–Raphson iteration method. The flutter behaviour is analysed with reference to the large deflection equilibrium state through an eigenvalue analysis and by tracing the complex eigenvalues and identifying the first coalescence of any two vibratory modes. The flutter dynamic pressure is also predicted iteratively using the eigenvalue approach for the selected range of limit cycle amplitudes. The influence of static aerodynamic load and vibration amplitude on the flutter characteristics is brought out for both isotropic and graphene reinforced composite panels with different boundary conditions. The pre-flutter static deflection shape of the panel is also examined. The material parameters such as porosity level in the metal foam and the graphene platelet content are assessed on the nonlinear flutter features of 2D panels.

Geometrically nonlinear isogeometric analysis of a partly wrinkled membrane structure

An isogeometric membrane element based on the non-uniform rational basis spline (NURBS) model is presented that accounts for membrane wrinkling based on tension-field theory. First, the element is validated by means of a benchmark problem involving a partly wrinkled membrane. It is then applied to the large deformation of a thin membrane structure using a two-stage procedure that combines dynamic relaxation and Newton–Raphson iteration. A simple technique is introduced that takes advantage of the geometrical symmetry of an isogeometric analysis model by using a “one-sided” open knot vector to treat the continuity of the membrane surface with respect to the symmetry plane. Because NURBS has many suitable features for representing complex geometries, it enriches the function space of the membrane element. Consequently, the characteristic mechanical responses of membranes, such as deep folding, are captured appropriately by the present isogeometric membrane element. In addition, a numerical example demonstrates that the convergence rate of the isogeometric membrane analysis with respect to refinement of the discretization is much better when tension-field theory is introduced in the analysis.

Effects of elastic foundation on the large-amplitude vibration analysis of functionally graded GPL-RC annular sector plates

The main purpose of this research is to investigate the impacts of the two-parameter elastic medium on the geometrically nonlinear free vibration of nanocomposite annular sector plates reinforced with graphene platelets. The modified Halpin–Tsai micromechanical model is implemented to determine the overall mechanical properties. With the aid of Hamilton’s principle and in according the first-order shear deformation plate theory in conjunction with geometric nonlinearity effects, the nonlinear sets of motion equations are derived for functionally graded nanocomposite annular sector plates. The 2D generalized differential quadrature GDQ method, the harmonic balance method and Newton–Raphson iteration technique are utilized to numerically solve the problem and obtain the nonlinear vibration response. Multiple comparative investigations are provided to express the efficiency of the model. The impacts of involved geometrical and material factors are also examined on the nonlinear vibration behavior of functionally graded nanocomposite annular sector plates.

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.

Simulation of Dynamics During Deployment of Foldable Origami Structures

Foldable origami-based structures are a type of deployable structures that are increasingly applied in the space and building industries. When folded, the small size of such structures facilitates transportation and storage. Meanwhile, the properties of their larger deployed state may be of interest to different applications. A stable working condition is established by locking the structure in its deployed state, as in the process of deployment, the driving forces may generate a dynamic effect, thus leading to instability of the system. Hence, the study of dynamic characteristics of such structures, including trajectory, duration, velocity, and acceleration is of paramount importance. In this paper, based on the general dynamic equation and Lagrange’s equations of the first kind, the finite element method is adopted to investigate the dynamic deployment of foldable plate structures in terms of the generalized nodal coordinates. The proposed geometric description of a quadrilateral plate element is based on a folding plate composed of refined triangular elements, which are used to approximate the real shells in the structure. Subsequently, a MATLAB framework is developed on the basis of the element using the Newmark integration and the Newton–Raphson iteration method to simulate the deployment process of the structure. Comparisons between MATLAB results and ADAMS results verify the reliability of the framework in analyzing the dynamic deployment of the foldable origami-based structures with sufficient accuracy.

Modeling Claim Frequency in Indonesia Auto Insurance Using Generalized Poisson-Lindley Linear Model

This paper will discuss the modeling of claim frequency from Indonesian auto insurance using the generalized Poisson-Lindley linear model. This modeling method assumes that the data of claim frequency are from populations that follow generalized Poisson-Lindley distribution. Generalized Poisson-Lindley linear model is an alternative to modeling count data that contains overdispersion. The parameters in the generalized Poisson-Lindley linear model can be estimated using the maximum likelihood estimation method through Newton Raphson's iteration numerical method. The data are the secondary data took from XYZ Company for the 2013 policy which is overdispersed. The data contains policyholder partial loss claims for comprehensive motor vehicle insurance products. From the research conducted it was concluded that the data is suitable to be modeled with generalized Poisson-Lindley linear models and produce better models than ordinary Poisson linear modeling because of produced the smaller AIC value. Of the 3 predictor variables that are modeled on the frequency of claims, 2 variables influenced they are the use variable and vehicle brand variable.

ГОЛОМОРФНОЕ ПОГРУЖЕНИЕ КАК АНАЛИТИЧЕСКИЙ МЕТОД РАСЧЕТА ЭЛЕКТРИЧЕСКИХ СЕТЕЙ НЕФТЯНЫХ И ГАЗОВЫХ МЕСТОРОЖДЕНИЙ

Для того чтобы обеспечить требуемый уровень надежного функционирования объектов нефтедобычи, необходимо обращать внимание на условия эксплуатации энергосистемы. Это важно, если принять во внимание, что для добычи нефти и газа необходимы достаточно мощные источники энергии, которые существенно влияют на режим работы электросети. При этом актуальным является задача расчета установившихся режимов электрической сети, питающей объекты нефтедобычи. Расчеты установившихся режимов имеют большое практическое значение для обеспечения эффективного и безопасного управления режимами работы нефтяных и газовых предприятий, являются важными при проектировании электрических сетей, питающих объекты нефтегазовых предприятий. Однако применение классических итерационных методов расчета установившихся режимов, таких как метод Гаусса–Зайделя и Ньютона–Рафсона, не всегда позволяет найти правильное решение системы нелинейных уравнений, описывающих установившиеся режимы работы сети, так как сходимость данных методов зависит от начальных приближений. В работе предлагается аналитический метод расчета установившихся режимов электрических сетей, описываемых нелинейными уравнениями. Метод основан на аппроксимации Паде и методе возмущений. Приводятся преимущества метода перед известным методом итераций Гаусса–Зайделя и Ньютона–Рафсона. Приводятся примеры решения задач электроэнергетических сетей, и обсуждаются недостатки предлагаемого метода. Рассматриваются проблемы устойчивости. Цель: применить аналитический метод голоморфного погружения для расчета двух и трехузловой энергетической схемы; сравнить возможности метода с другими альтернативными методами; исследовать ограничения метода голоморфного погружения и показать область его работы. Методы: разложение Тейлора, аналитическое продолжение, решения алгебраических уравнений рекуррентным методом, бесконечные дроби. Результаты. Получены примеры использования метода голоморфного погружения для двух и трех PQ узловых схем, и показаны недостатки метода голоморфного погружения. Проведено сравнение метода голоморфного погружения с альтернативными методами. Выводы. Аналитический метод голоморфного погружения обладает рядом преимуществ: физической наглядностью, простотой алгоритмической реализации, заключающейся в рекуррентных соотношениях для коэффициентов разложения искомой функции в ряд Тейлора. Разложенная в ряд функция является голоморфной, что позволяет осуществлять ее аналитическое продолжение и получить желаемую точность решения.

Nonlinear vibration analysis of third-order shear deformable functionally graded beams by a new method based on direct numerical integration technique

This study investigates the nonlinear vibration and dynamic response of a beam made of functionally graded material (FGM) within the framework of the improved third-order shear deformable theory. The beam is subjected to a concentrated moving load, and the included angle of the load direction and axial direction varies with time. Considering the von Karman geometric nonlinearity, the nonlinear formulations of the FGM beam is derived. The Newmark method in conjunction with the Newton–Raphson iteration is adopted to analyze the dynamic response of the beam. A new method, based on a direct numerical integration technique for the matrix form motion equation, is presented to study the nonlinear vibration of the FGM beam. The proposed method overcomes the problem of signal determination for the solution of the eigenvector equation that often leads to nonconvergence or a false result of the nonlinear eigenvalue equation. In relation to the numerical results, the effects of material property distribution, vibration amplitude on the nonlinear dynamic behavior of the FGM beams, and the energy transference phenomenon are discussed in this paper.