Newton-Raphson Iteration Research Articles

Novel calculation method to predict gas–water two-phase production for the fractured tight-gas horizontal well

The prediction of production capacity in tight gas wells is greatly influenced by the characteristics of gas–water two-phase flow and the fracture network permeability parameters. However, traditional analytical models simplify the nonlinear problems of two-phase flow equations to a large extent, resulting in significant errors in dynamic analysis results. To address this issue, this study considers the characteristics of gas–water two-phase flow in the reservoir and fracture network, utilizes a trilinear flow model to characterize the effects of hydraulic fracturing, and takes into account the stress sensitivity of the reservoir and fractures. A predictive model for gas–water two-phase production in tight fractured horizontal wells is established. By combining the mass balance equation with the Newton–Raphson iteration method, the nonlinear parameters of the flow model are updated step by step using the average reservoir pressure. The accuracy of the model is validated through comparisons with results from commercial numerical simulation software and field case applications. The research results demonstrate that the established semi-analytical solution method efficiently handles the nonlinear two-phase flow problems, allowing for the rapid and accurate prediction of production capacity in tight gas wells. Water production significantly affects gas well productivity, and appropriate fracture network parameters are crucial for improving gas well productivity. The findings of this work could provide more clear understanding of the gas production performance from the fractured tight-gas horizontal well.

Applying Potra-Pták Iterative Cycle for Solving Highly Nonlinear Structural Problems

When solving nonlinear algebraic equations that arise from discretization using the finite element method (FEM), it is often observed that the standard Newton-Raphson (N-R) iteration either fails to converge or necessitates a large number of iterations in the vicinity of critical points. This work proposes an additional numerical strategy, known as the Potra-Pták iterative cycle, to improve the efficiency of solving highly nonlinear structural problems. Therefore, the focus here is on making the nonlinear solver more robust and efficient, allowing the analysis of more complex nonlinear structures. In the Potra-Pták iterative cycle, two corrections of the objective function (energy function) are performed. The introduction of a second correction in the iterative cycle makes the Potra-Pták strategy more efficient than the standard or modified N-R iterations. This numerical strategy was implemented in the homemade Computational System for Advanced Structural Analysis (CS-ASA) program. The program is based on the FEM and is capable of performing static and dynamic nonlinear analysis of steel, concrete, and composite structures, and its efficiency is then verified through the analysis of slender frames and arches. The algorithm details for solving the nonlinear structural problem, characterized by the Potra-Pták scheme, are provided.

Divisions and Square Roots with Tight Error Analysis from Newton–Raphson Iteration in Secure Fixed-Point Arithmetic

In this paper, we present new variants of Newton–Raphson-based protocols for the secure computation of the reciprocal and the (reciprocal) square root. The protocols rely on secure fixed-point arithmetic with arbitrary precision parameterized by the total bit length of the fixed-point numbers and the bit length of the fractional part. We perform a rigorous error analysis aiming for tight accuracy claims while minimizing the overall cost of the protocols. Due to the nature of secure fixed-point arithmetic, we perform the analysis in terms of absolute errors. Whenever possible, we allow for stochastic (or probabilistic) rounding as an efficient alternative to deterministic rounding. We also present a new protocol for secure integer division based on our protocol for secure fixed-point reciprocals. The resulting protocol is parameterized by the bit length of the inputs and yields exact results for the integral quotient and remainder. The protocol is very efficient, minimizing the number of secure comparisons. Similarly, we present a new protocol for integer square roots based on our protocol for secure fixed-point square roots. The quadratic convergence of the Newton–Raphson method implies a logarithmic number of iterations as a function of the required precision (independent of the input value). The standard error analysis of the Newton–Raphson method focuses on the termination condition for attaining the required precision, assuming sufficiently precise floating-point arithmetic. We perform an intricate error analysis assuming fixed-point arithmetic of minimal precision throughout and minimizing the number of iterations in the worst case.

Thermodynamic calculations using reverse Monte Carlo: Simultaneously tuning multiple short-range order parameters for 2D lattice adsorption problem.

Lattice simulations are an important class of problems in crystalline solids, surface science, alloys, adsorption, absorption, separation, catalysis, to name a few. We describe a fast computational method for performing lattice thermodynamic calculations that is based on the use of the reverse Monte Carlo (RMC) technique and multiple short-range order (SRO) parameters. The approach is comparable in accuracy to the Metropolis Monte Carlo (MC) method. The equilibrium configuration is determined in 5-10 Newton-Raphson iterations by solving a system of coupled nonlinear algebraic SRO growth rate equations. This makes the RMC-based method computationally more efficient than MC, given that MC typically requires sampling of millions of configurations. The technique is applied to the interacting 2D adsorption problem. Unlike grand canonical MC, RMC is found to be adept at tackling geometric frustration, as it is able to quickly and correctly provide the ordered c(2 × 2) adlayer configuration for Cl adsorbed on a Cu (100) surface.

Nonlinear dynamic behaviors of angular contact ball bearing with waviness based on a synthetical multi-degree-of-freedom mathematical modelling

This paper proposes a synthetical multi-degree-of-freedom mathematical modelling for analyzing the dynamic characteristics of angular contact ball bearing with waviness subjected to the external load working conditions. The above multi-degree-of-freedom mathematical modelling is established based on nonlinear elastic Hertz contact theory and inner raceway control theory by employing improved Newton–Raphson iteration method; the validation of the established mathematical modelling is verified by comparing the presented results with the existing literature results. The above mathematical modelling has considered raceway waviness, ball waviness, bearing clearance, centrifugal force, gyroscopic moment, external loads, and rotating speed comprehensively. Firstly, the effects of waviness order and waviness amplitude on time-varying contact and stiffness characteristics of angular contact ball bearing are investigated systematically. Then, the influences of bearing clearance, rotating speed, axial force, radial force, and torque on the dynamic contact and stiffness characteristics of angular contact ball bearing with waviness are analyzed thoroughly. The main results show that the solution results of mathematical modelling have converged, respectively, when raceway waviness order increases to 25 and ball waviness order increases to 5; The variation tendencies of time-varying contact angle, contact force, and stiffness of angular contact ball bearing with waviness are aperiodic; The amplitudes of time-varying contact angle, contact force, and stiffness of angular contact ball bearing increase evidently with waviness amplitude increased; The bearing clearance has important influence on the contact angle and contact force of outer raceway and inner raceway, respectively; The effects of radial force and torque on contact and stiffness characteristics are consistent basically in the corresponding direction; The influences of axial force and rotating speed on the contact and stiffness characteristics are more remarkable compared with other factors. The investigations of contact and stiffness characteristics of angular contact ball bearing with waviness are helpful for the designation and manufacture of high precision angular contact ball bearing.

Efficient numerical implementation for nonlinear analysis of pile-supported structures during soil liquefaction through adaptive pile element formulations

The structural behavior of pile-supported structures during soil liquefaction is sometimes insufficiently investigated properly due to the inefficiency in the existing line finite element method (LFEM), which is widely used in practice. The traditional LFEM to simulate pile response requires dense element mesh for accurate results, as it simplifies the continuous surrounding soil as discrete spring elements, resulting in high computational costs, especially for pile-supported structures with a significant number of piles. Additionally, analyzing pile-supported structures in different stages of soil liquefaction requires repetitive updating of p-y curves and tedious starting-over iteration processes. To address these issues, an innovative line element formulation named the pile element formulation is employed and improved to adaptively simulate the surrounding soil conditions at different stages during liquefaction within the element formulation, eliminating the need to model discrete soil springs and significantly improving computational efficiency. The adaptive pile element formulation is then implemented within a refined Newton-Raphson iteration procedure that is proposed for a more effective nonlinear analysis process to consider soil liquefaction without repetitive iteration processes from starting over. Through several examples, the computational efficiency of the proposed method is validated, which requires only 20% of the computational cost compared to traditional LFEM while still providing accurate results, highlighting the application potential of the present study.

Asymmetric Dynamic Buckling of Isotropic/ Anisotropic Spherical Caps

This paper deals with nonlinear asymmetric dynamic buckling of clamped isotropic/anisotropic spherical shells under suddenly applied pressure loads. The formulation is based on first-order shear deformation theory and Lagrange's equation of motion. The nonlinearity due to finite deformation of the shell considering von Karman's assumptions is included in the formulation. The buckling loads are obtained through dynamic response history using Newmark's numerical integration scheme coupled with a Newton-Raphson iteration technique. An axisymmetric curved shell element is used to investigate the dynamic characteristics of the spherical caps. The pressure value beyond which the maximum average displacement response shows significant growth rate in the time history of the shell structure is considered as critical dynamic load. Detailed numerical results are presented to highlight the influences of shell geometric parameter, orthotropicity, ply-angle, number of layers and asymmetric mode on the critical load of spherical caps.

Thermal Postbuckling Characteristics Of Angle-ply Laminated Truncated Circular Conical Shells

Here, the nonlinear thermoelastic pre- and post-buckling characteristics of angle-ply laminated composite conical shells subjected to uniform temperature rise are studied using semi-analytical finite element approach. The finite element formulation is based on first-order shear deformation theory and field consistency principle. The nonlinear governing equations, considering geometric nonlinearity based on von Karman’s assumption for moderately large deformation, are solved using Newton-Raphson iteration procedure coupled with displace- ment control method to trace the prebuckling followed by postbuckling equilibrium path. The presence of asymmetric perturbation in the form of small magnitude load spatially proportional to the linear buckling mode shape is assumed to initiate the bifurcation of the shell deformation. The study is carried out to highlight the influences of semi-cone angle, ply-angles, number of layers and number of circumferential waves on the nonlinear prebuckling/postbuckling thermoelastic response of the laminated circular conical shells. The participation of axisymmetric and asymmetric modes in the total response of the shells is brought out through the deformation shape analysis.

Postbuckling of Laminated Composite Circular Conical Shells Under Combined Thermo-Mechanical Loads

In this paper, the postbuckling characteristics of cross- and angle-ply laminated composite conical shells subjected to combined thermo-mechanical loading are studied using shear deformable semi-analytical finite element. The nonlinear governing equations, considering geometric nonlinearity based on Sanders type of kinematic approximations, are solved using Newton-Raphson iteration procedure coupled with adaptive displacement control method to trace the postbuckling equilibrium path. It is brought out from the study that the inward displacement in the postbuckling region is significantly higher as compared to the outward one. The ratio of the lowest load in the postbuckling path and the bifurcation load increases with the increase in the pressure loading proportion and decreases with the increase in the axial loading proportions. The relative load carrying capacity of shells under different set of loading combinations changes with the increase in the maximum postbuckling displacement. It is also observed that the nonlinearity in the load interaction relation corresponding to lowest load in the postbuckling path is less compared to that for bifurcation load.

An XFEM-based hydro-mechanical model to consider the non-Darcy flow in complex fracture networks

Geotechnical instability and failure are closely related to the hydro-mechanical coupled behavior of fractured rocks, and the existing studies often regard the fracture seepage as linear Darcy flow. Due to the neglect of non-Darcy effect caused by rough rock wall and high flow velocity, the estimated rock deformation and seepage flow can lead to incorrect assessment of geotechnical risks. Hence, this study proposed an efficient and robust hydro-mechanical (HM) coupled model to consider non-Darcy flow in complex fracture networks. The fractured rock mass's mechanical response, including the fracture surface's frictional contact, was calculated using the XFEM solver, and the widely used Forchheimer's law was adopted to describe the non-Darcy flow along the fractures. The fracture segments cut by mechanical element edges are considered one-dimensional flow elements, and the Newton-Raphson iteration method is adopted to solve the Forchheimer equation directly. To consider the influence of fracture deformation on the conductivity of fractures, the Barton-Bandis model was adopted to determine the hydraulic aperture of the fractures when the fracture was closed. Subsequently, a staggering Newton-Raphson method can be developed to decouple the fracture deformation and fluid flow process. Experimental results were adopted to validate the accuracy of the numerical model, and several numerical cases were used to illustrate the efficiency and robustness of the proposed model. The effects of inlet pressure, confining stress, and non-Darcy effect on the HM-coupled behavior of fractured rock were investigated. Numerical results suggested that the distribution of fluid pressure and failure process of rock mass based on Darcy assumption could be dramatically changed after considering the nonlinear seepage behavior.

Experimental study on the influence of explosive stress wave on the dynamic crack propagation characteristics under biaxial confining pressure

The effect of biaxial confining pressure on dynamic crack and the interaction between explosive stress wave and dynamic crack are studied based on dynamic photoelastic method in this experiment. The experiment consists of five groups, primarily focusing on the angle at which the stress wave encounters the crack. The experimental results reveal two main fractures: mode I fracture and mixed-mode fracture. The influence of biaxial confining pressure on stress waves and dynamic cracks differs between these two fracture modes. Additionally, the Newton-Raphson iteration method is employed to analyze and calculate the fringe pattern of the crack tip at various stages. Subsequently, the calculation results are used to reconstruct the photoelastic fringe. To validate the accuracy of the calculation method, the reconstructed photoelastic fringe pattern is compared with the experimental results. The results show that the presence of confining biaxial pressure enhances the crack initiation toughness, propagation toughness and propagation speed. It also improves the crack's resistance to the impact of the incident stress wave from all directions. When the stress wave interacts with the crack, the photoelastic fringe patterns at the crack tip can be categorized into two main types. One is primarily influenced by the stress wave, while the other is primarily influenced by the stress field at the crack tip. To extract accurate crack tip information, it is essential to analyze the fringe pattern dominated by the stress field at the crack tip.

Reducing Effects of Initial Imperfection by Investment in the Orthotropic Characteristics of Laminated Composite Plate

The target of this study is to reduce the impact of initial imperfection on the nonlinear dynamical performance of laminated composite plates by taking advantage of the orthotropic characteristics of laminated composite plates by changing carbon fiber sawing in the mass matrix and fiber orientation with different patterns and studying the effect of this optimization without and with initial imperfection (Wo) and different aspect ratios (W/L) and various boundary conditions through analyzing the load-displacement responses for plates under axial in-plane compressive loads by using the FORTRAN 94 programming language. Von-Karman's assumptions are utilized to include geometric nonlinearity for nine node isoperimetric quadrilateral components with five degrees of freedom into the structural model, which is based on first-order shear deformation theory. The Newmark’s implicit time integration method and Newton-Raphson iteration concurrently are employed to solve the nonlinear governing equation in conjunction. The study proved the effectiveness of the carbon fiber's varying geometric distribution and the difference in its directions in reducing the negative effects of the initial imperfection on the large elastic-plastic displacement and critical buckling. To highlight the veracity of the results, some of them have been validated against those found in the literature review. Doi: 10.28991/CEJ-2023-09-07-03 Full Text: PDF

A Closed-Loop Dynamic Controller for Active Vibration Isolation Working on A Parallel Wheel-Legged Robot

Serving the Stewart mechanism as a wheel-legged structure, the most outstanding superiority of the proposed wheel-legged hybrid robot (WLHR) is the active vibration isolation function during rolling on rugged terrain. However, it is difficult to obtain its precise dynamic model, because of the nonlinearity and uncertainty of the heavy robot. This paper presents a dynamic control framework with a decentralized structure for single wheel-leg, position tracking based on model predictive control (MPC) and adaptive impedance module from inside to outside. Through the Newton-Euler dynamic model of the Stewart mechanism, the controller first creates a predictive model by combining Newton-Raphson iteration of forward kinematic and inverse kinematic calculation of Stewart. The actuating force naturally enables each strut to stretch and retract, thereby realizing six degrees-of-freedom (6-DOFs) position-tracking for Stewart wheel-leg. The adaptive impedance control in the outermost loop adjusts environmental impedance parameters by current position and force feedback of wheel-leg along Z-axis. This adjustment allows the robot to adequately control the desired support force tracking, isolating the robot body from vibration that is generated from unknown terrain. The availability of the proposed control methodology on a physical prototype is demonstrated by tracking a Bezier curve and active vibration isolation while the robot is rolling on decelerate strips. By comparing the proportional and integral (PI) and constant impedance controllers, better performance of the proposed algorithm was operated and evaluated through displacement and force sensors internally-installed in each cylinder, as well as an inertial measurement unit (IMU) mounted on the robot body. The proposed algorithm structure significantly enhances the control accuracy and vibration isolation capacity of parallel wheel-legged robot.

Self-similar solutions to the hypoviscous Burgers and SQG equations at criticality

After reviewing the source-type solution of the Burgers equation with standard dissipativity, we study the hypoviscous counterpart of the Burgers equation. (1) We determine an equation that governs the near-identity transformation underlying its self-similar solution. (2) We develop its approximation scheme and construct the first-order approximation. (3) We obtain the source-type solution numerically by the Newton–Raphson iteration scheme and find it to agree well with the first-order approximation. Implications of the source-type solution are given, regarding the possibility of linearisation of the hypoviscous Burgers equation. Finally we address the problems of the incompressible fluid equations in two dimensions, centering on the surface quasi-geostrophic equation with standard and hypoviscous dissipativity.

A full 6 degree-of-freedom visual estimation with laser assisted

Efficient and accurate measurement in unstructured environment is a challenging issue and also research interest in the industrial field. The accuracy of vision is very high in the close-range scene and the stability of laser measurement is better, with a large achievable range. A 6-DOF pose measurement method based on vision and assisted by laser is proposed in this paper. By combining their respective advantages, the method can be immune to ambient light. The optimization objective is established by using the projection coordinates of laser spots in different coordinate systems and solved by Newton–Raphson iteration. The initial value is estimated by Kronecker product. The Monte Carlo method is further used to simulate. The causes of measurement errors and the influence factors affecting the accuracy are analyzed. On this basis, an experimental prototype is constructed. The accuracy and repeatability accuracy experiments are conducted to verify the correctness of the proposed method.

Buckling and Post-Buckling of Bidirectional Porous Beam Under Bidirectional Hygrothermal Environment

In this paper, buckling and nonlinear post-buckling behaviors of a bidirectional porous (BDP) beam are investigated under bidirectional hygrothermal environment. Euler–Bernoulli beam theory with the von Kármán nonlinearity is employed to derive the nonlinear variable coefficient governing differential equations based on Rayleigh quotient method. Analytical solutions of critical buckling load and load–deflection equilibrium path in post-buckling are deduced for the single directional varying (SDV) porous beam. The general numerical solutions for bidirectional varying (BDV) porous beam are obtained by differential quadrature finite element method (DQFEM) with Newton–Raphson iteration method based on the variation principle. The high accuracy of the present numerical method with higher computing efficiency is verified by comparison with published reports and the analytical results in this work. Parametric analysis on effects of the porosity bidirectional distributions, porosity coefficients, distributions of hygrothermal environment and boundary conditions on buckling load and post-buckling response is carried out to enhance the buckling and deformation resistances in design, manufacture and usage of porous structures. The results show that the bidirectional porosity pattern, linear and nonlinear hygrothermal distribution and boundary conditions play a significant role on buckling critical external load and critical hygrothermal increments, buckling form and post-buckling path.

Parametric study for modified couple stress theory on postbuckling of size-dependent FG saturated porous mindlin microplates

Using modified couple stress and Mindlin plate theory, the present study is a parametric study for revealing the effect of length scale parameter on postbuckling of saturated poroelastic rectangular microplates with pore distribution being functionally graded across thickness. The microplate postbuckling is a nonlinear eigenvalue problem with coupled partial differential equations of equilibrium. In order to find the mode shapes, the eigenfunctions have been approximated by some complete polynomial series expansions in accordance with the conventional Ritz approximation technique. After discretizing the governing equations throughout the microplate grid points, the assembled difference equations construct a nonlinear eigenvalue problem with coupled algebraic equations. The nontrivial coefficients of the expansions were calculated such that the eigenfunctions extremize the microplate potential energy for satisfying the equilibrium homogeneous equations. Variation of the potential energy have been derived with respect to the expansions coefficients from the vantage of using Maple software. By gradual reduction in the initial end shortening of the movable supports and solving the assembled difference equations via Newton-Raphson iterations, the critical postbuckling displacement imposed to the movable supports are found along with the nontrivial coefficients regarding eigenfunctions. The results of the parametric study have revealed how geometric parameters, pore distribution, the Skempton’s coefficient, porosity, and different Supports modify the influence of length scale parameter on the microplate postbuckling.

Research on a high-precision real-time improvement method for aero-engine component-level model

Abstract In order to improve the real-time performance of the aero-engine Component-Level Model (CLM) while ensuring accuracy, a method for the Calculation of Thermodynamic Parameters of Working Fluids (CTPWF) based on a Neural Network and Newton Raphson (NN-NR) is proposed. In this method, the enthalpy or entropy under different fuel-air ratio and humidity conditions is mapped to temperature by a neural network, and the mapping output is used as the initial solution of Newton Raphson (NR) iteration. Then, a high-precision solution can be obtained through a few iterations, which avoids the shortcoming that the traditional method uses a fixed initial solution that leads to too many iterative steps. This effectively reduces the number of iterative steps and improves the calculation efficiency. This method is applied to the aero-thermodynamic calculation of each component of an engine CLM, which improves the accuracy and real-time performance of the CLM. The simulation results show that, compared to the traditional method, the proposed method improves the accuracy of the CTPWF and can reduces the single aero-thermodynamic calculation time by 25 % when humidity is not considered and by 47 % when humidity is considered. This effectively improves the real-time performance of the CLM.

The Influence of GPL Reinforcements on the Post-Buckling Behavior of FG Porous Rings Subjected to an External Pressure

The work focuses on the post- buckling behavior of functionally graded graphene platelet (FG-GPL)-reinforced porous thick rings with open-cell internal cavities under a uniform external pressure. The generalized rule of mixture and the modified Halpin–Tsai model are here used to evaluate the effective mechanical properties of the ring. Three types of porosity patterns are assumed together with five different GPL distributions as reinforcement across the ring thickness. The theoretical formulation relies on a 2D-plane stress linear elasticity theory and Green strain field in conjunction a virtual work principle to derive the nonlinear governing equations of the post-buckling problem. Unlike the simple ring models, 2D elasticity considers the thickness stretching. The finite element model combined with an iterative Newton–Raphson algorithm is used to obtain the post-buckling path of the ring up to the collapse. A systematic investigation evaluates the effect of the weight fraction of nanofillers, the coefficient of porosity, porosity distribution, and the GPLs distribution on the deep post-buckling path of the ring. Based on the results, it is found that the buckling value and post-buckling strength increase considerably (by approximately 80%) by increasing the weight fraction of the nanofiller of about 1%.

Static and dynamic analysis of the postbuckling of axially moving spin beams

This paper aims to investigate the static and dynamic behaviors of the postbuckling of axially moving spin beams (bi-gyro system). The Hamilton’s principle is used to derive the nonlinear governing equations of bi-gyro systems, which are based on the Euler-Bernoulli beam model and account for von Karman geometric nonlinearity. After neglecting the time-dependent terms, the static equations are discretized using the differential quadrature method and then solved as a nonlinear algebraic system via the Newton-Raphson iteration method. Subsequently, the linear vibration problem is accurately solved for axially moving spinning beams in their first buckled configuration. Finally, numerical calculations are conducted to investigate the impact of axial and spinning velocities on the steady-state dynamic responses of bi-gyro systems. The findings are as follows: (a) the critical axial velocity where trivial equilibrium is unstable remains unaffected by the presence of spinning velocity; (b) the bifurcation point of the axial velocity corresponding to amplitude is reduced by spinning velocity, resulting from centrifugal force; (c) the vibration amplitude of the bi-gyro system will diverge under a weaker displacement constraint spring; and (d) stable buckled and straight configurations can coexist under certain axial and spinning velocities.