Newton Iteration Method Research Articles

Simulation of Drying‐Rewetting Processes in Numerical Groundwater Models Using a New Picard Iteration‐Based Method

AbstractWhen simulating groundwater flow in unconfined and convertible aquifers using a groundwater model with the block‐centered finite‐difference approach, such as MODFLOW, it frequently encounters drying and rewetting of cells. Although many drying and rewetting simulation methods have been proposed in the past, balancing simulation accuracy and convergence capability all at once is difficult. MODFLOW‐2005, which has second‐order accuracy, employs a trial‐and‐error method, but it suffers from computational instability when large quantities of grid cells are dried. MODFLOW‐NWT adopts the upstream‐weighting approach and Newton iteration method to ensure the stability of the drying and rewetting simulations. However, the upstream‐weighting approach has only first‐order accuracy, and the Newton iteration method is complex to implement because it necessitates the establishment of an additional Jacobian matrix. The methods employed by MODFLOW‐NWT are also available in MODFLOW 6, therefore it inherits both the strengths and weaknesses of MODFLOW‐NWT. In this study, a new method, Picard iteration‐based always active cell (PAAC), is proposed. Similar to MODFLOW‐NWT, the PAAC method also uses dry cells as active cells. The PAAC method, however, does not use the upstream‐weighting approach and has second‐order accuracy. Moreover, it ensures good convergence stability even under the Picard iteration method. In addition to discussing the algorithm, five cases were used to comprehensively compare the simulation effects of the PAAC method with MODFLOW‐2005 and MODFLOW‐NWT, including an analytical solution, repeated drying‐rewetting of multi‐layer grids, pumping well problem, perched aquifer problem and a nearly dry single‐layer grid, which verified the practicability of the PACC method.

Pricing European call options with interval-valued volatility and interest rate

We propose a novel approach to pricing European call options when both of the volatility of the underlying asset and interest are uncertain. In this approach, we formulate the option pricing problem with uncertain parameters as a partial-differential inequality constrained interval optimization problem. An interior penalty method is then developed for the numerical solution of the finite-dimensional optimization problem arising from the discretization of the continuous pricing problem by a finite difference scheme. A convergence theory for the penalty method is established. An algorithm based on Newton's iterative method is also proposed for solving the penalty equation. Numerical results are presented to demonstrate the effectiveness and usefulness of this approach and the numerical methods.

A model of erosion rate prediction for component with complex geometry based on numerical simulation

The structural components prone to erosion damage often exhibit irregular and complex shapes. Predicting erosion rates for such irregular geometries is a field that needs more attention, since it can be used to evaluate the performance of the component. This study aims to develop an evaluation method for predicting erosion rates of components with complex geometry. Standard erosion tests were performed on small plates to investigate the effects of different impact angles and particle velocities on the erosion rate of the target material. Based on these test results, the Newton iteration method was employed to determine the parameters of the Tabakoff and Grant erosion model. Moreover, these parameters were used in computational fluid dynamics simulation of components with complex geometries. Besides, a model was developed to convert the erosion rate density into erosion rate in numerical simulation, allowing quantitative comparison of simulation results with experimental data. The calibration of the erosion model parameters was conducted by using 1060 aluminum alloy specimens. The predicted erosion rate changes with impact angle showed consistency good agreement with the testing data. Subsequently, the erosion rate of a simplified similar structure for aircraft engine blade made of 1060 aluminum alloy was predicted and verified.

Remarks on Solving Methods of Nonlinear Equations

Abstract: In the field of mechanical engineering, many practical problems can be converted into nonlinear problems, such as the meshing problem of mechanical transmission. So the solution of nonlinear equations has important theoretical research and practical application significance. Whether the traditional Newton iteration method or the intelligent optimization algorithm after the popularization of computers, both them have been greatly enriched and developed through the continuous in-depth research of scholars at home and abroad, and a series of improved algorithms have emerged. This paper mainly reviews the research status of solving nonlinear equations from two aspects of traditional iterative method and intelligent optimization algorithm, systematically reviews the research achievements of domestic and foreign scholars, and puts forward prospects for future research directions.

Pump-stopping-induced hydraulic oscillations in long-distance district heating system: Modelling and a comprehensive analysis of critical factors

To achieve low-carbon goals, long-distance district heating (LDH) systems are widely applied, which utilize the waste heats from renewable energy power plants and waste incineration plants. The hydraulic oscillations induced by pump-stopping in LDH systems may give rise to critical incidents, such as pipeline ruptures and heating disruptions. The systematic analysis of hydraulic oscillations induced by pump-stopping and the subsequent revelation of critical factors that mitigate their intensity are fundamental prerequisites for ensuring the safety and reliability of LDH system. In this paper, firstly, a pump-stopping hydraulic transient model of the LDH system is established based on the distributed parameter method. Then, combining the method of characteristics (MOC) and the Newton iteration method, the numerical solution scheme for this model is proposed. Finally, taking a 20 km LDH system as an example, the characteristics of hydraulic oscillations induced by pump-stopping is analyzed, and the key factors influencing the intensity of these hydraulic oscillations are investigated. In addition, the differences between hydraulic oscillations induced by pump-stopping and those induced by sudden faults on pipelines are also analyzed. The results indicate that an optimal pump inertia moment value and a properly sized bypass pipe at the pump significantly mitigate hydraulic oscillations, while the influence of constant pressure at the heat source inlet on hydraulic oscillations can be negligible. Our study has important reference value for improving the safe and reliable operation of LDH systems.

Dimensionless Analysis of the Spatial–Temporal Coupling Characteristics of the Surrounding Rock Temperature Field in High Geothermal Roadway Realized by Gauss–Newton Iteration Method

Understanding the time–space coupling characteristics of the surrounding rock temperature field in high geothermal roadways is essential for controlling heat damage in mines. However, current research primarily focuses on individually analyzing the temperature changes in the surrounding rock of roadways, either over time or space. Therefore, the Gauss–Newton iteration method is employed to model the coupling relationship between temperature, time, and space. The results demonstrate that the dual coupling function describing the temperature field of the surrounding rock in both time and space provides a more comprehensive characterization of the temperature variations. Over time, as ventilation duration increases, the fitting degree of the characteristic curve steadily rises, and the characteristic curve descends overall. In the spatial dimension, the fitting degree of the characteristic curve gradually decreases with the rise of the dimensionless radius, and the characteristic curve ascends overall. Additionally, as thermal conductivity increases, the fitting degree of the characteristic curve steadily rises.

Hybrid technique of conformal mapping and Chebyshev collocation method for solving time–space fractional order wave equation

AbstractThis work presents a numerical approach for solving the time–space fractional‐order wave equation. The time‐fractional derivative is described in the conformal sense, whereas the space‐fractional derivative is given in the Caputo sense. The investigated technique is based on the third‐kind of shifted Chebyshev polynomials. The main problem is converted into a system of ordinary differential equations using conformal mapping, Caputo derivatives, and the properties of the third‐kind shifted Chebyshev polynomials. Then, the Chebyshev collocation method and the non‐standard finite difference method will be used to convert this system into a system of algebraic equations. Finally, this system can be solved numerically via Newton's iteration method. In the end, physics numerical examples and comparison results are provided to confirm the accuracy, applicability, and efficiency of the suggested approach.

Searching nontrivial magnetic equilibria using the deflated Newton method

Nonlinear systems that model physical experiments often have many equilibrium configurations, and the number of these static solutions grows with the number of degrees of freedom and the presence of symmetries. It is impossible to know a priori how many equilibria exist and which ones are stable or relevant, therefore from the modeler’s perspective, an exhaustive search and symmetry classification in the space of solutions are necessary. With this purpose in mind, the method of deflation (introduced by Farrell as a modification of the classic Newton iterative method) offers a systematic way of finding every possible solution of a set of equations. In this contribution we apply deflated Newton and deflated continuation methods to a model of macroscopic magnetic rotors, and find hundreds of new equilibria that can be classified according to their symmetry. We assess the benefits and limitations of the method for finding branches of solutions in the presence of a symmetry group, and explore the high-dimensional basins of attraction of the method in selected 2-dimensional sections, illustrating the effect of deflation on the convergence.

Prestress analysis and geometry optimization for conical cable domes with zero Gaussian curvature

The conical cable dome with a rotating cone and zero Gaussian curvature is a new type of cable dome with the generatrix that follows a straight line. The paper studies the prestress design and mechanical properties of such cable domes. First, on the basis of the section and vertical diagrams of the proposed cone, a simplified method is developed to calculate feasible prestresses of cables and struts using the principle of nodal balance. Second, prestress distribution, load-bearing capacity, and prestress of the conical cable domes with 15 different combinations of rise-span ratio and inner triangular slope under both prestress and load states are studied. Additionally, the paper proposes five load conditions to assess the performance of the conical cable domes. Finally, the total strain energy, basic frequency, radial reaction, and gravity are optimized through single-criterion and multi-criterion based on the combination of a genetic algorithm and the Newton iteration method. To keep the outer appearance of the conical cable dome, the work selects the strut length as the optimization variable. An optimized conical cable dome is recommended with a low prestress uniformity index and high uniformity index. The results indicate that the conical cable dome is sensitive to the asymmetrical loads, especially the wind loads. For the single-criterion optimization, the structure with the basic frequency under loads as the objective function has the best load-bearing capacity. After optimization, the maximum nodal displacement is 32% and 43% of the original one under LC4 and LC5. Besides, to get a synthetic optimization result, a multi-criterion optimization is suggested.

An inexact relaxed generalized Newton iterative method for solving generalized absolute value equations

An inexact relaxed generalized Newton iterative method for solving generalized absolute value equations

EFFECT ON INCLINED STRETCHING PLATE UNDER SLIP CONDITION DUE TO NON-NEWTONIAN FLUID IN POROUS MEDIUM FOR MAGNETOHYDRODYNAMIC FLOW OF UNSTEADY NATURE

An analytical probe is executed for evaluating the impact of unsteady flow pertaining to non-Newtonian fluid of magnetohydrodynamic nature on a stretching inclined porous plate, built in porous medium. Electrically conducting unsteady flow facing applied magnetic force with applied slip condition are the other factors utilized for the study of convective heat transfer. For converting governing equations to nonlinear differential equations, the similarity transformation method has been deployed. MATLAB was also used. Ordinary differential equations were solved numerically with the help of the Newton iterative method as well as the shooting method. Results for velocity and temperature were derived for a wide range of unsteady and other parameters including magnetic, source/sink, Grashof number, Prandtl number, Deborah number, and angle of inclination. Graphs and tables were used for compiling and computing values of skin friction and heat transfer rate. Results derived in the present analysis have been compared to those available in the previous literature, and parity has been formed to ascertain the accuracy of the current analytical exercise.

A variational-difference method for numerical simulation of equilibrium capillary surfaces

Objectives. A variational-difference method for numerical simulation of equilibrium capillary surfaces based on the minimization of the energy functional is proposed. As a test task a well-known axisymmetric hydrostatic problem on equilibrium shapes of a drop adjacent to a horizontal rotating plane under gravity is considered. The mathematical model of the problem is built on the basis of the variational principle: the shape of the drop satisfies the minimum total energy for a given volume. The problem of the functional minimization is reduced to a system of nonlinear equations using the finite element method. To solve the system a Newton's iterative method is applied.Methods. The variational-difference approach (the finite element method) is used. The finite linear functions are chosen as basic functions.Results. Equilibrium shapes of a drop on a rotating plane are constructed by the finite element method in a wide range of defining parameters: Bond number, rotational Weber number and wetting angle. The influence of these parameters on the shape of a drop is investigated. The numerical results are matched with the results obtained using the iterative-difference approach over the entire range of physical stability with respect to axisymmetric perturbations.Conclusion. The finite element method responds to the loss of stability of a drop with respect to axisymmetric perturbations. Therefore it can be used to study the stability of the equilibrium of axisymmetric capillary surfaces.

A modified Newton-based matrix splitting iteration method for generalized absolute value equations

Many problems in scientific computing and engineering fields lead to the solution of generalized absolute value equations (GAVE). In this paper, by simultaneously splitting both coefficient matrices in the differential and non-differential parts of the GAVE, a modified Newton-based matrix splitting (MNMS) iteration method is proposed. The MNMS iteration method not only covers the well-known generalized Newton iteration method as well as the recent proposed modified Newton-based iteration method and the Newton-based matrix splitting iteration method for solving the GAVE, but also results in a series of relaxation versions that are very flexible in real applications. Convergence properties of the MNMS iteration method are studied in detail when the coefficient matrices are positive definite matrices and H+-matrices. Finally, three numerical examples are presented to show the feasibility and effectiveness of the proposed MNMS iteration method.

Generalized Legendre Polynomial Configuration Method for Solving Numerical Solutions of Fractional Pantograph Delay Differential Equations

This paper develops a numerical approach for solving fractional pantograph delay differential equations using generalized Legendre polynomials. These polynomials are derived from generalized Taylor bases, which facilitate the approximation of the underlying analytical solutions, leading to the formulation of numerical solutions. The fractional pantograph delay differential equation is then transformed into a finite set of nonlinear algebraic equations using collocation points. Following this step, Newton's iterative method is applied to the resultant set of nonlinear algebraic equations to compute their numerical solutions. An error analysis for this methodology is subsequently presented, accompanied by numerical examples demonstrating its accuracy and efficiency. Overall, this study contributes a more streamlined and productive tool for determining the numerical solution of fractional differential equations.

A novel efficient estimator for three‐dimensional bearings‐only source localisation with unknown sensor altitude and systematic measurement errors

AbstractA novel two‐stage profile maximum likelihood estimator is proposed to estimate the source location and the systematic errors jointly, with the aim of addressing the problem of source localisation using angle‐only measurements from a single sensor with unknown sensor altitude and systematic measurement errors. The proposed two‐stage profile maximum likelihood estimator algorithm is capable of decoupling the azimuth and elevation angle measurements while transforming the original maximum likelihood optimisation problem into two sub‐problems, that is, two‐dimensional‐projected maximum likelihood estimator and relative altitude maximum likelihood estimator. In terms of the two‐dimensional‐projected maximum likelihood estimator, an algorithm combining pseudo‐linear estimating and Kalman filtering is proposed to generate an initial estimate. Subsequently, a Gauss–Newton iterative method is developed to estimate the two‐dimensional‐projected target location and the azimuth systematic error jointly. The relative height maximum likelihood estimator is initialised using a pseudo‐linear estimator. Next, a Gauss–Newton iterative algorithm is adopted to estimate the elevation systematic error and the relative height. As indicated by the result of the simulation studies, the proposed algorithm exhibits an estimation performance close to the Cramér–Rao lower bound with unknown sensor altitude and systematic measurement errors.

A Two-Dimensional Variant of Newton’s Method and a Three-Point Hermite Interpolation: Fourth- and Eighth-Order Optimal Iterative Schemes

A nonlinear equation f(x)=0 is mathematically transformed to a coupled system of quasi-linear equations in the two-dimensional space. Then, a linearized approximation renders a fractional iterative scheme xn+1=xn−f(xn)/[a+bf(xn)], which requires one evaluation of the given function per iteration. A local convergence analysis is adopted to determine the optimal values of a and b. Moreover, upon combining the fractional iterative scheme to the generalized quadrature methods, the fourth-order optimal iterative schemes are derived. The finite differences based on three data are used to estimate the optimal values of a and b. We recast the Newton iterative method to two types of derivative-free iterative schemes by using the finite difference technique. A three-point generalized Hermite interpolation technique is developed, which includes the weight functions with certain constraints. Inserting the derived interpolation formulas into the triple Newton method, the eighth-order optimal iterative schemes are constructed, of which four evaluations of functions per iteration are required.

A Semi-Analytical Model for Production Prediction of Deep CBM Wells Considering Gas-Water Two-Phase Flow

The productivity prediction of deep coalbed methane (CBM) wells is significantly influenced by gas-water two-phase flow characteristics and seepage parameters of the fracture network. While numerical simulations offer a comprehensive approach, analytical models are favored for their faster and broader applicability. However, conventional analytical models often oversimplify the complex problem of two-phase seepage equations, leading to substantial errors in dynamic analysis outcomes. Addressing this shortcoming, we establish a gas-water two-phase productivity prediction model for deep CBM reservoirs. This model takes into account the two-phase flow characteristics within the reservoir and fracture network, as well as the stress sensitivity of the reservoir and fractures. Additionally, a modified trilinear flow model characterizes the fractured modification body. By integrating the flowing material balance equation with the Newton Iteration method, we gradually update the seepage model’s nonlinear parameters using the average formation pressure. We also linearize the gas-water two-phase model through successive iterations to derive a semi-analytical solution. The accuracy of the model was verified through comparison with commercial numerical simulation software results and field application. The model also enabled us to scrutinize the influence of reservoir and fracture network parameters on productivity. Our research findings suggest that the semi-analytical solution approach can efficiently address the nonlinear seepage problem of gas-water two-phase flow, enabling quick and accurate prediction of deep CBM well productivity. Moreover, appropriate fracture network parameters are paramount for enhancing the productivity of deep CBM wells. Lastly, during the development of deep CBM reservoirs, it is crucial to control the production pressure difference appropriately to minimize the stress sensitivity impact on production capacity.

Collocation method with Lagrange polynomials for variable‐order time‐fractional advection–diffusion problems

In this study, we proposed a numerical technique to solve a class of variable‐order time‐fractional advection–diffusion equations (VOTFADEs) by applying an operational matrix of differentiation based on fractional‐order Lagrange polynomials (FOLPs). The variable‐order fractional derivative is assumed to be Caputo's derivative. Using the operational matrix and collocation method, the advection–diffusion equation can be reduced to an algebraic system of equations that can be solved using Newton's iterative method. Error analysis also has been carried out for the proposed method. The current approach is simple to use and computer oriented and provides highly accurate approximate solutions. The effectiveness and accuracy of the proposed method are demonstrated using a few numerical examples.

A novel algorithm for evaluating cement azimuthal density based on perturbation theory in horizontal well

Cement density monitoring plays a vital role in evaluating the quality of cementing projects, which is of great significance to the development of oil and gas. However, the presence of inhomogeneous cement distribution and casing eccentricity in horizontal wells often complicates the accurate evaluation of cement azimuthal density. In this regard, this paper proposes an algorithm to calculate the cement azimuthal density in horizontal wells using a multi-detector gamma-ray detection system. The spatial dynamic response functions are simulated to obtain the influence of cement density on gamma-ray counts by the perturbation theory, and the contribution of cement density in six sectors to the gamma-ray recorded by different detectors is obtained by integrating the spatial dynamic response functions. Combined with the relationship between gamma-ray counts and cement density, a multi-parameter calculation equation system is established, and the regularized Newton iteration method is employed to invert casing eccentricity and cement azimuthal density. This approach ensures the stability of the inversion process while simultaneously achieving an accuracy of 0.05 g/cm3 for the cement azimuthal density. This accuracy level is ten times higher compared to density accuracy calculated using calibration equations. Overall, this algorithm enhances the accuracy of cement azimuthal density evaluation, provides valuable technical support for the monitoring of cement azimuthal density in the oil and gas industry.

Finite‐horizon optimal trajectory control of near space hypersonic vehicle with multi‐constraints

AbstractIn this article, a finite‐horizon optimal trajectory control strategy is developed for near space hypersonic vehicle (NSHV) longitudinal model with multi‐constraints including external disturbance, system modeling error, and input saturation. The whole control process has two parts: inner‐loop attitude control and outer‐loop trajectory control. First, the feedback linearization method is applied to design a tracking controller for outer‐loop system, and reference signals for inner‐loop attitude control can be obtained using Newton iteration method. Second, for the inner‐loop attitude system with multi‐constraints, a finite‐horizon optimal tracking control scheme consists of feedforward control input and adaptive dynamic programming based optimal feedback controller is designed. In this way, not only the adverse effects of above multi‐constraints are eliminated, but also the optimally tracking performances are guaranteed. Finally, the Lyapunov analysis method is utilized to ensure the stability of the entire closed‐loop control system, and simulation tests with respect to NSHV longitudinal trajectory tracking are supplied to verify the availability of the proposed strategy.