Newton-Raphson Iteration Research Articles

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.

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.

Frequency Transition from Diffusion to Capacitive Response in the Blocked-Diffusion Warburg Impedance for EIS Analysis in Modern Batteries

The use of the Blocked-diffusion Warburg (BDW) impedance within electrochemical impedance spectroscopy (EIS) measurements can unveil diffusion properties of the electroactive material of modern batteries at different states-of-charge. The impedance response of the BDW comprises a diffusion response of charge carriers through a short-diffusion distance (e.g. the solid-phase in electroactive material of battery electrodes) and a capacitive response due to accumulation of charge carriers in a blocked-interface (e.g. impermeable current collector of a thin film electrode). This study has developed a mathematical expression based on the Newton-Raphson iteration method to calculate the frequency and time constant during the transition from diffusion to capacitive response in the BDW impedance. The mathematical procedure to calculate the frequency during the diffusion-capacitive transition response in the BDW has been written in a script in Matlab® software and is applied to BDW impedance responses reported in previous studies and extracted from EIS measurements in Li-ion and NiMH batteries. This study demonstrates that the time constant during the transition from diffusion to capacitive response in the BDW differs from the characteristic time constant commonly represented in the BDW mathematical expression. The characteristic time constant represented in the BDW mathematical expression is related to the rate of accumulation of charge carriers in the blocked-interface of the electrode. On the other hand, the time constant during the transition from diffusion to capacitance responses in the BDW impedance can be related to diffusion properties in solid-phase particles with heterogeneous size distribution in the electroactive material of modern battery electrodes.

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.

Theoretical solution for multi-span continuous cable structures considering sliding

This paper proposes a theoretical solution for multi-span continuous sliding cable structures, which were usually analyzed by numerical simulations. Firstly, the analytical equation for the unstressed length of single-span cable based on the undeformed configuration is derived, in which the tension forces at the ends can be accurately calculated. Then, according to the tension balance condition at sliding joints and the assumption of invariant total unstressed cable length (ITUL method), multivariate transcendental equations for the sliding analysis of multi-span cable structures with and without considering friction are established and solved by multiple Newton-Raphson iterations. Moreover, a theoretical method based on the assumption of invariant total cable length (ITL method) is also proposed and the applicability of the ITL method is investigated. Finally, four typical examples are analyzed by the ITUL method, and the accuracy of the ITL method as well as some existing numerical simulations is evaluated. The result shows that the ITUL method proposed in this paper is more accurate and reliable than the ITL method. The ITUL method can provide a theoretical basis for the design and analysis of multi-span sliding cable structures.

Assessment of simultaneous and nested conservative augmented Lagrangian schemes for constrained multibody dynamics

Two alternative schemes used for the solution of multibody systems are reported and compared between them to evaluate their performance. Within the context of kinematic constraints imposed via augmented Lagrangian technique, one scheme proposes a simultaneous iterative solution for the computation of the Lagrangian multipliers and the nonlinear Newton-Raphson iterations. The second scheme uses a more classical Uzawa approach to compute the Lagrangian multipliers, and the nonlinear iterations are nested into the multiplier update loop. The performance of the methodology is tested by solving numerical experiments: simple, double, triple, and 10 bars pendulums with revolute joints. Moreover, 2D and 3D analyses are performed. The computed results using both the simultaneous iterative, and the nested iterative techniques, are reported to evaluate: kinematic responses, energy conservation, constraints verification, and iterations reduction.

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.

Geometrically Nonlinear Analysis of Axially Functionally Graded Beams by Using Finite Element Method

The aim of this paper is to investigate geometrically nonlinear static analysis of axially functionally graded cantilever beam subjected to transversal non follower load. The considered problem is solved by finite element method with total Lagrangian kinematic approach. The material properties of the beam vary along the longitudinal direction according to the power law function. The finite element model of the beam is considered in the three dimensional continuum approximation for an eight-node quadratic element. The geometrically nonlinear problem is solved by Newton-Raphson iteration method. In the numerical results, the effects of the material distribution on the geometrically nonlinear static displacements of the axially functionally graded beam are investigated. Also, the differences between of material distributions are investigated in geometrically analysis.

Influence of dynamic temperature variation and inplane varying loads over post-buckling and free vibration analysis of sandwich composite beam

In this study, nonlinear post-buckling and free vibration analysis of shape memory polymer sandwich composite (SMPSC) under dynamic temperature variation is performed. For the analysis, simplified Co continuity based on higher-order shear deformation theory (HSDT) has been adopted to perform finite element analysis (FEA). Numerical solutions are obtained by iterative Newton–Raphson method considering Von-Karman nonlinear kinematics. Material properties of SMPSC, with Shape Memory Polymer (SMP) as matrix and carbon fiber as reinforcements, have been calculated by theory of volume averaging. The effect of dynamic temperature variation and axial variable inplane loadings (AVIL) on SMPC and SMPSC has been evaluated for various parameters such as beam thickness ratio, layer variation, boundary conditions (BCs), position of core, thickness of core in sandwich structures for the first time. Apart from these, this study also clearly reveals the difference in magnitude of buckling and free vibration parameters between the shape memory polymer composites (SMPC) and SMPSC, before and after glass transition region of material.

Topology Identification and Line Parameter Estimation for Non-PMU Distribution Network: A Numerical Method

The energy management system becomes increasingly indispensable with the extensive penetration of new players in the distribution networks, such as renewable energy, storage, and controllable load. Also, the operation optimization of the active distribution system requires information on operation state monitoring. Smart measuring equipment enables the topology identification and branch line parameters estimation from a data-driven perspective. Nevertheless, many current methods require the nodal voltage angles measured by phasor measurement units (PMUs), which might be unrealistic for conventional distribution networks. This paper proposes a numerical method to identify the topology and estimate line parameters without the information of voltage angles. We propose a two-step framework: the first step applies a data-driven regression method to provide a preliminary estimation on the topology and line parameter; the second step utilizes a joint data-and-model-driven method, i.e., a specialized Newton-Raphson iteration and power flow equations, to calculate the line parameter, recover voltage angle and further correct the topology. We test the method on IEEE 33 and 123-bus looped networks with load data from 1000 users in Ireland. The results demonstrate that the proposed method can provide an accurate estimation of the topology and line parameters based on limited samples of measurement without voltage angles.

Modified Newton integration neural algorithm for solving the multi-linear [formula omitted]-tensor equation

This paper attends to solve the multi-linear equations with special structure, e.g., the multi-linear M-tensor equation, which frequently appears in engineering applications such as deep learning and hypergraph. For its critical and promising role, there are numbers of resolving schemes devoting to obtain a high-performing solution of the multi-linear M-tensor equation. However, few investigations are discovered with noise-suppression ability till now. To be proper with digital devices and further improve the solving effectiveness, it is desirable to design a discrete-time computational algorithm with noise-suppression ability and high-performing property. Inspired by the aforementioned requirements, this paper proposes a modified Newton integration (MNI) neural algorithm for solving the multi-linear M-tensor equation with noise-suppression ability. Additionally, the corresponding robustness analyses on the proposed MNI neural algorithm are provided. Simultaneously, computer simulative experiments are generated to explain the capabilities and availabilities of the MNI neural algorithm in noise suppression. As a result, in terms of noise suppression, the proposed MNI neural algorithm is superior to other related algorithms, such as Newton–Raphson iterative (NRI) algorithm (Ding and Wei, 2016), discrete time neural network (DTNN) algorithm (Wang et al., 2019), and sufficient descent nonlinear conjugate gradient (SDNCG) algorithm (Liu et al., 2020).

An efficient multi-threaded Newton–Raphson algorithm for strong coupling modeling of multi-physics problems

The Finite Element Gaussian Belief Propagation (FGaBP) method is an iterative algorithm with abundant parallelism making it an alternative for the traditional Finite Element Method (FEM), especially for large multi-physics problems. In this paper, we extend the FGaBP method to solve the coupled electrical–thermal problem that emerges in the modeling of radiofrequency ablation (RFA) of hepatic tumors. The strongest form of coupling algorithms, which is the Newton–Raphson (NR) method, is implemented in parallel using the localized computations of FGaBP. The parallel scalability of the FGaBP method is retained in the proposed algorithm by calculating local Jacobian matrices for each element and then updating the solutions for both electrical and thermal problems accordingly at each NR iteration.

Generalized linear mixed quantile regression with panel data.

A new generalized linear mixed quantile model for panel data is proposed. This proposed approach applies GEE with smoothed estimating functions, which leads to asymptotically equivalent estimation of the regression coefficients. Random effects are predicted by using the best linear unbiased predictors (BLUP) based on the Tweedie exponential dispersion distributions which cover a wide range of distributions, including those widely used ones, such as the normal distribution, Poisson distribution and gamma distribution. A Taylor expansion of the quantile estimating function is used to linearize the random effects in the quantile process. The parameter estimation is based on the Newton-Raphson iteration method. Our proposed quantile mixed model gives consistent estimates that have asymptotic normal distributions. Simulation studies are carried out to investigate the small sample performance of the proposed approach. As an illustration, the proposed method is applied to analyze the epilepsy data.

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.

Improving the initial guess for the Newton-Raphson protocol in time-dependent simulations

A general linearisation procedure for the consistent tangent of a small-strain visco-plastic material model is presented in this note. The procedure is based on multi-variable linearisation around a so-called 'reference state'. In particular, the linerarisation of the time integration scheme is found to yield an extra term compared to classical expressions, which only appears because the material response is time-dependent. It has the effect of yielding a very accurate initial guess for the Newton-Raphson protocol based on the ongoing viscous flow. It is shown, using a modern variational FFT-based solver, that the extra term reduces both the CPU time and the number of Newton-Raphson iterations by around a factor two.