Newton Iteration Technique Research Articles

Nonlinear flow phenomenon of a power-law non-Newtonian fluid falling down a cylinder surface

In this paper, we present a comprehensive study of the fingering phenomenon of a power-law non-Newtonian fluid falling down a cylinder surface. A theoretical analysis is firstly carried out and the governing equation describing the film thickness is established for the non-Newtonian fluid denoted by a power-law index n. Using the lubrication theory with dimensionless variables, the partial differential equation for the film thickness is derived, and both two-dimensional flow and three-dimensional flow are investigated. The traveling wave characteristic of the two-dimensional flow is displayed by using the finite difference scheme associated with a Newton iteration technique. The effect of changing the radius of the cylinder, precursor-layer thickness and power-law index is then considered through examining the variation of the capillary waves. Furthermore in three-dimensional complex flow, the fingering patterns for different parameters are simulated. The linear stability analysis based on the traveling wave solutions is given to elucidate the influence of different physical parameters, explaining the physical mechanism of nonlinear flow behaviors in two dimension and three dimension. Results from linear stability analysis show that the power-law index reinforces the fingering instability whether the fluid is shear-thinning or shear-thickening. Moreover, the numerical results illustrate that increasing the radii of the cylinder expands the unstable region. The flow profiles for different power-law coefficients are consistent with the result of the linear stability analysis, proving the unstable role of the power-law coefficient.

Shape reconstruction for an inverse biharmonic problem from partial Cauchy data

In this paper, we address the Robin inverse problem for the biharmonic equation in a simply connected domain, to reconstruct the geometric shape of a non‐accessible part of the boundary from a single measurement of Riquier–Neumann data on the accessible part of that boundary. Our approach extends the nonlinear boundary integral equation, to recover the shape of the boundary. We propose the Newton iterative technique based on the Fréchet derivatives to linearize the system and then establish an injectivity of the linearized system for certain Robin coefficients, as well as the iteration scheme to describe the inverse algorithm for recovering the shape with respect to the unknowns. The mathematical spirit of the proposed method will be presented, and to illustrate its feasibility, some numerical examples will be provided.

Significance of Cattaneo-Christov double diffusion and induced magnetic field on Maxwell ternary nanofluid flow with magnetic response boundary

The present study delves into the significance of Cattaneo-Christov double diffusion and the induced magnetic field (IMF) on a stagnation-point flow of Maxwell ternary nanofluids. A new boundary condition with the magnetic response is designed to study superior heat and mass transfer of Maxwell ternary nanofluid combined with double diffusion and IMF. The governing equations are formulated by mathematically modeling of the current flow in a Cartesian coordinate system and solved using an improved shooting method and the Runge-Kutta method. Through the application of similarity transformations, the governing equations are transformed into a system of initial boundary value ordinary differential equations. These equations are further transformed into linear equations with initial value problems using the shooting method and subsequently solved using the Runge-Kutta method and Newton's iterative techniques. The numerical results and correlation analysis vividly demonstrate the profound influence of double diffusion and IMF on thermal and concentration patterns via graphical representations. It is found that the double diffusion and the induced magnetic field always helps to achieve lower temperature and concentration. The magnetic response boundary leads to higher heat and mass transfer efficiency for the suction case. The magnetically responsive boundary is verified to be effective for regulating the heat and mass transfer of ternary nanofluid.

Fibonacci Wavelet Method for the Numerical Solution of Nonlinear Reaction-Diffusion Equations of Fisher-Type

This article aims to propose an efficient Fibonacci wavelet-based collocation method for solving the nonlinear reaction-diffusion equation of Fisher-type. The underlying numerical scheme starts by formulating operational matrices of integration corresponding to the Fibonacci wavelets. Besides, we study the error analysis and convergence theorem of the proposed technique. Subsequently, a set of algebraic equations are formed corresponding to the given problem, which could be handled via any conventional method, for instance, the Newton iteration technique. To demonstrate the efficiency of the proposed wavelet-based numerical method, we compare the obtained absolute,L∞,L2, and root mean square (RMS) error norms with the existing Lie symmetry method and cubic trigonometricB-spline (CTB) differential quadrature method in tabular form. From the numerical outcomes, it is ascertained that the proposed numerical technique is computationally more effective and yields precise outcomes in comparison to the existing ones.

Reduced Space Optimization‐Based Evidence Theory Method for Response Analysis of Space‐Coiled Acoustic Metamaterials with Epistemic Uncertainty

Acoustic metamaterials have been widely concerned by researchers because of their excellent sound absorption properties. Traditional uncertainty analysis methods for response analysis of acoustic metamaterials with evidence variables have limitations. The excessive time spent on repetitive extreme analysis of focal elements severely impedes the practical application of evidence theory. To reduce the computational costs of uncertainty quantification for acoustic metamaterials under the evidence theory, a reduced space optimization‐based evidence theory method (RSO‐ETM) is proposed. In RSO‐ETM, a simplified surrogate model is first constructed by a modified adaptive arbitrary orthogonal polynomial (MAAOP) expansion method, and then, the monotonicity of the response surface is examined to reduce the space. Subsequently, the Newton iteration technique and a transformation boundary method are used to obtain the extreme points in the reduced space, through which the boundary of the response over each focal element can be readily obtained. By using RSO‐ETM, the optimization for each focal element can be avoided, and correspondingly the computational costs are reduced. Two mathematical examples and acoustic problems are employed to demonstrate the practicality of the methods.

Atangana–Baleanu derivative-based fractional model of COVID-19 dynamics in Ethiopia

ABSTRACT The paper's main aim is to investigate the 2019 coronavirus disease in Ethiopia using a fractional-order mathematical model. It would also focus on the importance of fractional-order derivatives that may help us in modelling the system and understanding the effect of model parameters and fractional derivative orders on the approximate solutions of our model. A SELAIQHCR model is constructed using nonlinear differential equations in the Atangana–Baleanu non-integer operator in the Caputo sense. After that, the Chebyshev fourth kind spectral collocation method is used to change a fractional system to an algebraic system. Newton iterative technique is used to solve the converted system. The next-generation matrix technique is used to obtain the effective reproduction number. The COVID-19-free equilibrium point and endemic equilibrium point, solution positivity and boundedness, and their stability are all carefully done. The sensitivity of the effective reproduction value with respect to the key model parameters is discussed. The beginning values provided for our system were obtained using reports from the Ethiopian Public Health Institute from 29 February 2021 to 7 June 2021. The fundamental reproduction number is obtained with . The model's numerical solutions are represented graphically.

An improved inverse kinematics solution for 6-DOF robot manipulators with offset wrists

AbstractEfficiently solving inverse kinematics (IK) of robot manipulators with offset wrists remains a challenge in robotics due to noncompliance with Pieper criteria. In this paper, an improved method to solve the IK for 6-DOF robot manipulators with offset wrists is proposed. This method is based on the Newton iteration technique, but it does not require a selection of initial estimation of joint variables. The solution is divided into two parts: the first part is to reconstruct a simplified structure with analytical IK solution, and the second part is to obtain a numerical solution by iteration. Further, a robot manipulator HSR-BR606 with an offset wrist is used as an example to specifically elaborate the mathematical procedure of the method and to investigate the algorithm in terms of accuracy, efficiency, and application of motion planning. A comparative experiment is conducted with a typical IK algorithm, which demonstrates a higher accuracy and shorter calculation time of the proposed method. The mean calculation time for a single IK solution required for this algorithm is only 4% of the comparison algorithm.

A Fréchet derivative‐based novel approach to option pricing models in illiquid markets

Nonlinear option pricing models have been increasingly concerning in financial industries since they build more accurate values by regarding more realistic assumptions such as transaction cost, market liquidity, or uncertain volatility. This study defines a nonclassical numerical method to effectively capture the behavior of the nonlinear option pricing model in illiquid markets where the implementation of a dynamic hedging strategy affects the price of the underlying asset. Unlike the conventional numerical approaches, this study describes a numerical scheme based on the Newton iteration technique and the Fréchet derivative for linearization of the model. The linearized time‐dependent PDE is then discretized by a sixth‐order finite difference scheme in space and a second‐order trapezoidal rule in time. The computations revealed that the current approach appears to be somewhat more effective to some extent and at the same time economical for illustrative examples compared to the existing competitors. In addition, this method helps to prevent considering the convergence issues of the Newton approach applied to the nonlinear algebraic system.

Novel dual vector quaternions based adaptive extended two-step filter for pose and inertial parameters estimation of a free-floating tumbling space target

Estimating the parameters of an uncooperative space target is essential to the on-orbit service missions. A good parameter estimation can provide sufficient prior knowledge for the further operations. This paper proposes a novel dual vector quaternions based adaptive extended two-step filter (DVQ-AETSF) to estimate the pose and inertial parameters of a free-floating tumbling space target. Firstly, both of the rotational and translational motions are modeled by the dual vector quaternions (DVQ). Then, by using the DVQ-based system model, the DVQ-AETSF is designed. The proposed DVQ-AETSF mainly consists of a traditional Kalman filter prediction procedure in the first step and an adaptive regularized Newton iteration technique in the second step. The new proposed two-step filter aims to deal with the high nonlinearities in the measurements equations. By using the proposed DVQ-AETSF, both of the pose and initial parameters of a free floating tumbling space target under large errors of initial guesses and high measurement noise can be well estimated. Finally, the proposed DVQ-AETSF is validated by mathematical simulations to show its performance.

A MASS LUMPING AND DISTRIBUTING FINITE ELEMENT ALGORITHM FOR MODELING FLOW IN VARIABLY SATURATED POROUS MEDIA

The Richards equation for water movement in unsaturated soil is highly nonlinear partial differential equations which are not solvable analytically unless unrealistic and oversimplifying assumptions are made regarding the attributes, dynamics, and properties of the physical systems. Therefore, conventionally, numerical solutions are the only feasible procedures to model flow in partially saturated porous media. The standard Finite element numerical technique is usually coupled with an Euler time discretizations scheme. Except for the fully explicit forward method, any other Euler time-marching algorithm generates nonlinear algebraic equations which should be solved using iterative procedures such as Newton and Picard iterations. In this study, lumped mass and distributed mass in the frame of Picard and Newton iterative techniques were evaluated to determine the most efficient method to solve the Richards equation with finite element model. The accuracy and computational efficiency of the scheme and of the Picard and Newton models are assessed for three test problems simulating one-dimensional flow processes in unsaturated porous media. Results demonstrated that, the conventional mass distributed finite element method suffers from numerical oscillations at the wetting front, especially for very dry initial conditions. Even though small mesh sizes are applied for all the test problems, it is shown that the traditional mass-distributed scheme can still generate an incorrect response due to the highly nonlinear properties of water flow in unsaturated soil and cause numerical oscillation. On the other hand, non oscillatory solutions are obtained and non-physics solutions for these problems are evaded by using the mass-lumped finite element method.

Projection on the intersection of convex sets

In this paper, we give a solution of the problem of projecting a point onto the intersection of several closed convex sets, when a projection on each individual convex set is known. The existing solution methods for this problem are sequential in nature. Here, we propose a highly parallelizable method. The key idea in our approach is the reformulation of the original problem as a system of semi-smooth equations. The benefits of the proposed reformulation are twofold: (a) a fast semi-smooth Newton iterative technique based on Clarke's generalized gradients becomes applicable and (b) the mechanics of the iterative technique is such that an almost decentralized solution method emerges. We proved that the corresponding semi-smooth Newton algorithm converges near the optimal point (quadratically). These features make the overall method attractive for distributed computing platforms, e.g. sensor networks.

CONSEQUENCE OF BACKWARD EULER AND CRANK-NICOLSOM TECHNIQUES IN THE FINITE ELEMENT MODEL FOR THE NUMERICAL SOLUTION OF VARIABLY SATURATED FLOW PROBLEMS

Modeling water flow in variably saturated, porous media is important in many branches of science and engineering. Highly nonlinear relationships between water content and hydraulic conductivity and soil-water pressure result in very steep wetting fronts causing numerical problems. These include poor efficiency when modeling water infiltration into very dry porous media, and numerical oscillation near a steep wetting front. A one-dimensional finite element formulation is developed for the numerical simulation of variably saturated flow systems. First order backward Euler implicit and second order Crank?Nicolson time discretization schemes are adopted as a solution strategy in this formulation based on Picard and Newton iterative techniques. Five examples are used to investigate the numerical performance of two approaches and the different factors are highlighted that can affect their convergence and efficiency. The first test case deals with sharp moisture front that infiltrates into the soil column. It shows the capability of providing a mass-conservative behavior. Saturated conditions are not developed in the second test case. Involving of dry initial condition and steep wetting front are the main numerical complexity of the third test example. Fourth test case is a rapid infiltration of water from the surface, followed by a period of redistribution of the water due to the dynamic boundary condition. The last one-dimensional test case involves flow into a layered soil with variable initial conditions. The numerical results indicate that the Crank?Nicolson scheme is inefficient compared to fully implicit backward Euler scheme for the layered soil problem but offers same accuracy for the other homogeneous soil cases.

Spacecraft intercept using minimum control energy and wait time

A new approach to the minimum energy impulse intercept problem for spacecraft in orbit is explored. The types of orbits investigated in this paper are not restricted to a particular one. The constrained optimization technique is formulated with the universal variable, which is used to describe orbit information with sufficient accuracy for general types of orbits. Two optimization problems are posed. First, a problem for minimum velocity change and time of flight for intercept are investigated with the constraint on the final position of two satellites. Next, the so-called wait time is also added as an additional parameter to be determined. Although a closed-form solution is not obtained, the Newton iteration technique is successfully applicable. Finally, by numerically comparing the proposed solution to the Hohmann transfer, the suggested approach is demonstrated to be a feasible technique applicable to a broad class of orbit transfer problems.

Acceleration techniques for the iterative resolution of the Richards equation by the finite volume method

Groundwater flow in variably saturated soils is described by the nonlinear Richards' equation. The mass-conservative finite volume discretization recently proposed in (Adv. Water Resour. 2004; 27:1199–1215) produces a nonlinear algebraic problem, whose resolution demands for the application of an appropriate iterative strategy, such as the Picard or the Newton scheme. The Picard iterative technique results in a robust fixed-point method, which is globally convergent at a linear or sub-linear rate. On the other hand, the Newton iterative technique shows better convergence rates, but the computational cost of the full calculation is still comparable to the one of the Picard schemes, due to the additional calculation of the Jacobian matrix and its formal inversion. In this work, we investigate two acceleration techniques to reduce the cost of Newton iterations. These techniques are respectively based on a linearization of the nonlinear Jacobian matrix in accordance with the quasi-Newton approach and on a Broyden-type rank-one correction for a fixed number of sub-iterations. Numerical experiments on a set of two-dimensional test case modeling the uptake of plant roots in a highly heterogeneous soil show the performance and the effectiveness of both acceleration schemes in comparison with the original Picard and Newton methods. Copyright © 2009 John Wiley & Sons, Ltd.

Simulation of soft smart hydrogels responsive to pH stimulus: Ionic strength effect and case studies

In this paper, the influence of the ionic strength of surrounding bath solution is presented and discussed on the responsive performance of soft pH-sensitive hydrogels to environmental change in solution pH. The numerical simulations are based on a chemo-electro-mechanical formulation previously termed the multi-effect-coupling pH-stimulus (MECpH) model. It is improved further in this paper via incorporation of the finite deformation theory into the mechanical equilibrium governing equation. The MECpH model is composed of coupled nonlinear partial differential equations and solved by a meshless numerical approach called the Hermite-cloud method with the modified Newton iteration technique. After examination of the improved MECpH model by comparing the simulation results with experiments available in literature, several case studies are carried out to discuss the effects of the ionic strength of surrounding solution on the distributive variations of the diffusive ion concentrations and electric potential as well as the deformation of the pH-stimulus-responsive hydrogels, when the hydrogels are immersed in buffered bath solutions.

Modeling for analysis of the effect of Young’s modulus on soft active hydrogels subject topH stimulus

Modeling is conducted in this paper for analysis of the influence of Young’s moduluson the response of soft active hydrogels to environmental solution pH changes. Achemo–electro–mechanical formulation termed the multi-effect-coupling pH-stimulus(MECpH) model, which was developed previously according to linear elastic theory forsmall deformation description, is improved in this paper through incorporation of the finitedeformation formulation into the mechanical equilibrium equation. The model is expressedby coupled nonlinear partial differential equations and solved via the meshlessHermite-cloud method with the modified Newton iteration technique. The improvedMECpH model is examined by comparison between the computational and publishedexperimental results. Numerical studies are then done on the influence of Young’s moduluson the distributive variations of the diffusive ion concentrations and electric potential, andon the deformation variations of the pH-stimulus-responsive hydrogels within differentbuffered solutions.

Lake eutrophication management modeling using dynamic programming

Lake eutrophication problems have received considerable attention in Taiwan, especially because they relate to the quality of drinking water. In this study, steady-state river water quality and lake eutrophication models are solved using dynamic programming algorithms to find the nutrient removal rates for eutrophication control during dry season. The kinetic cycle of chlorophyll-a, phosphorus and nitrogen for a complete-mixed lake is considered in the optimization framework. The Newton-iterative technique is adopted to solve the nonlinear equations for the steady-state lake eutrophication model. The optimization framework is applied to Cheng-Ching Lake in southern Taiwan. Several nutrient loading scenarios for eutrophication control are studied. Optimization results for nutrient removal rates and corresponding wastewater treatment capacities of each reach of the Kao-Ping River define the least cost approach to lake eutrophication control. A natural purification method, structural free water surface wetland, is also suggested to save more investment and improve river water quality at the same time.

Particle swarm optimization: an efficient method for tracing periodic orbits in three-dimensional galactic potentials

We propose particle swarm optimization (PSO) as an alternative method for locating periodic orbits in a three-dimensional (3D) model of barred galaxies. We develop an appropriate scheme that transforms the problem of finding periodic orbits into the problem of detecting global minimizers of a function, which is defined on the Poincare surface section of the Hamiltonian system. By combining the PSO method with deflection techniques, we succeeded in tracing systematically several periodic orbits of the system. The method succeeded in tracing the initial conditions of periodic orbits in cases where Newton iterative techniques had difficulties. In particular, we found families of two- and three-dimensional periodic orbits associated with the inner 8:1 to 12:1 resonances, between the radial 4:1 and corotation resonances of our 3D Ferrers bar model. The main advantages of the proposed algorithm are its simplicity, its ability to work using function values solely, and its ability to locate many periodic orbits per run at a given Jacobian constant. Ke yw ords: methods: numerical - galaxies: kinematics and dynamics - galaxies: structure.

Analysis of fluctuations in semiconductor devices through self-consistent Poisson-Schrödinger computations

The effects of random doping and random oxide thickness fluctuations in metal-oxide-semiconductor field-effect transistors are analyzed by using self-consistent Poisson-Schrödinger computations. The Poisson and Schrödinger equations are solved by using the Newton iteration technique in which the Jacobian matrix is computed through first-order perturbation theory in quantum mechanics. A very fast technique based on linearization of the transport equations is presented for the computation of threshold voltage fluctuations. This technique is computationally much more efficient than the traditional Monte Carlo approach and it yields information on the sensitivity of threshold voltage fluctuations to the locations of doping and oxide thickness fluctuations. Hence, it can be used in the design of fluctuation resistant structures of semiconductor devices. Sample simulation results obtained by using this linearization technique are reported and compared with those obtained by using the Monte Carlo technique.

Real-Time Measurement of Mean Frequency in Two-Machine System During Power Swings

This paper presents an algorithm for measuring mean frequency in two-machine system during power swings, which applies to sampled current signals in transmission lines with two sinusoidal components. The algorithm is based on rigorous mathematical deducing and recombines itself with Newton iterative technique and Taylor series expansion. The response of the algorithm is provided and the effects of key parameters, that affect the performance of the algorithm, are discussed. Computation requirements are modest and the technique has been implemented on a modern digital signal processor. The proposed technique has also been extensively tested using current signals obtained from dynamic frequency sources. Some test results are presented in the paper.