Newton-Raphson Iterative Algorithm Research Articles

Simulation of dynamic pulsing fracking in poroelastic media by a hydro-damage-mechanical coupled cohesive phase field model

This study develops a hydro-damage-mechanical fully coupled numerical method capable of modelling complex fluid-driven transient dynamic crack propagation in quasi-brittle poroelastic media. In this method, the fluid flow in both fractures and porous media is described by a fluid continuity equation with the modified Darcy-Poiseuille law based on the Biot's poroelastic theory. The fluid pressure and the inertial force of solids are coupled by governing equations of the mesh-insensitive phase-field regularized cohesive zone model that can simulate quasi-brittle multi-crack initiation, propagation, branching and merging without remeshing or crack tracking. The resultant displacement-pressure-damage coupled multiphysics system of equations is solved using an alternative minimization Newton-Raphson iterative algorithm with an implicit Newmark integration scheme within the finite element framework. The new method was first validated by a few 2D problems, including crack branching and deflecting in solids, fracking in a concrete cube, and consolidation and stress wave propagation in poroelastic media, subjected to various impulsive loadings. It was then applied to pressure pulsing fracking of a 40 m granite rock reservoir with extensive parametric studies of fluid viscosity and pulsing injection rate, mode and period. It was found that pulsing injection with higher fluid rates and lower fluid viscosities resulted in more developed crack patterns, and in particular, there existed an optimal pulsing injection period that could promote fracking under relatively low injection pressures. 3D horizontal well problems with multiple non-planar crack propagation and bifurcation were also successfully simulated to demonstrate the capacity and potential of the new method for engineering design and optimization of pressure pulsing fracking.

A hydro-thermo-damage-mechanical fully coupled cohesive phase-field model for fracking in quasi-brittle thermo-poroelastic media

A hydro-thermo-damage-mechanical fully coupled cohesive phase-field method is developed for modelling of fracking in quasi-brittle thermo-poroelastic media. A new sigmoid interpolation function is proposed to characterize the transitional behaviour from porous media to cracks for permeability, specific heat capacity, and thermal conductivity. A fluid continuity equation with the Darcy-Poiseuille law and an energy conservation equation with the Fourier’s law are then used to simulate fluid flow and heat transfer in cracking porous media, respectively. The fluid pressure and temperature of fluids and solids are fully incorporated into the governing equation of the phase-field regularized cohesive zone model to efficiently simulate complicated quasi-brittle hydro-thermal fracking, without the need of remeshing, crack tracking or auxiliary fields. The resultant displacement–pressure-temperature-damage coupled multiphysics system of equations is solved using a staggered Newton–Raphson iterative algorithm within the finite element framework. The method is first validated by five two-dimensional thermally or hydro-thermally induced fracture examples with analytical solutions, experimental data, and published numerical solutions. Three-dimensional fracking and fluid circulation problems in a vertical five-spot wellbore system with complex natural fracture networks, are then simulated as an application to energy harvesting in enhanced geothermal systems (EGS). It is found that the present method is capable of accurately and robustly modelling complicated hydraulic fracture propagation and fluid circulation problems and has the potential to be used for fracking design and optimization of EGS.

Lesion detection and classification in dermoscopic images using optimal threshold based on Newton Raphson iterative method

Automatic image thresholding is commonly used in the area of computer vision and pattern recognition for object detection. This research work proposes a new method of lesion detection in dermoscopic images using an optimal threshold based on the Newton-Raphson iteration algorithm for diagnosing melanoma. The proposed method incorporates a new strategy of finding the optimal threshold using first-order and second-order edge derivatives. The first- and second-order derivatives values obtained from the images are applied in the Newton-Raphson method which, in turn, converges to a value after a number of iterations. The converged value is considered the optimal threshold and used as a parameter for the separation of lesion regions from healthy skin in dermoscopic images. To test the proposed system performance, the experiment is carried out on two standard data sets, the ISIC Archive and the PH2, of dermoscopic images using different classifiers such as the Naive Bayes (NB), AdaBoost and Bag of Visual words (BOVW). The proposed segmentation technique segments the lesion regions from skin images and Bag of visual words classifier classifies the dermoscopic images into different classes namely common nevi, atypical nevi and melanoma. The proposed thresholding-based segmentation performance is compared against the Otsu method and Otsu using genetic algorithm. The experimental findings reveal that the proposed Newton-Raphson iterative method surpasses various state-of-the-art and recent methods in relation to computational speed, error rate, sensitivity, and accuracy.

Opening-dependent phase field model of hydraulic fracture evolution in porous medium under seepage-stress coupling

In this paper, a seepage-stress coupling numerical analysis method is developed to simulate complex hydraulic fracturing in porous medium, in which the characteristics of hydraulic fracture evolution in porous medium under seepage-stress coupling are revealed. The classical Biot's poroelasticity theory is applied to couple the stress and fluid fields, and the crack propagation path of hydraulic fracturing is described by an opening-dependent phase field model. Potential energy caused by fluid pressure is also coupled into the governing equation of phase field evolution. The system of fracture opening affecting fluid flow is established. Staggered iterative scheme is applied to solve the above three-field coupled problem based on Newton–Raphson iterative algorithm. Three classical cases are presented to validate the proposed method: the first one is an analytical solution of pressure distribution in a fixed opening crack and the second one is a benchmark case of Sneddon analytical solution for a pressurized pre-crack and the third one is the KGD problem for hydraulic fracture propagation. Moreover, numerical simulations of hydraulic fracturing in shale block, multi-layer and natural fractures are also investigated. The numerical results agree well with the results in published literatures. It is indicated that hydraulic fracture evolution under seepage-stress coupling can be forecasted by the proposed method. The proposed method not only provides a powerful analytical tool for the oil or gas production, but also provides assessment tools for the stability of geotechnical engineering in water rich environments.

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%.

Describing function method with pointwise balancing in two-dimensional regularized time domain for quasi-periodic responses

This paper proposes a new semi-analytical approach, called as describing function method with pointwise balancing (DFPB), to solve quasi-periodic (QP) responses of nonlinear dynamical systems especially those with non-smooth nonlinearities. This method is to describe the QP motion by generalized Fourier series with multiple regularized time scales. By discretizing the solution in the regularized time domain, the original equation is transformed into over-determined algebraic equations governing Fourier series coefficients via pointwise balancing. The coefficients are determined by solving the balance equations via the Newton–Raphson iteration algorithm. In addition, the Tikhonov regularization is utilized to enhance the convergence of the iteration scheme, so that the least squares solution of the over-determined equations can be obtained efficiently. On the one hand, the pointwise balancing avoids repeated transformations of non-smooth functions between time- to frequency-domain. On the other hand, the regularized time scales make it convenient to discretize the system equations regardless of unknown fundamental frequencies. With these procedures, the DFPB is well-designed to solve non-smooth systems for QP response with frequencies to be determined. In the numerical examples, the effectiveness and accuracy are elucidated in detail through both smooth and non-smooth models. The DFPB results are in excellent agreement with those attained by the Runge–Kutta method. In addition, it is worthy of pointing out the DFPB can track both stable and unstable QP responses.

Numerical implementation of BBM in FE package for solving unsaturated soil boundary value problems

Unsaturated soil mechanics is of keen relevance when dealing with soil above the water table orcompacted soils. Barcelona Basic Model (BBM) is one of the most widely used constitutive models for describing unsaturated soil behaviour and is available as a soil model in software like PLAXIS and CODE_BRIGHT (developed by UPC). The present study demonstrates a numerical framework to implement BBM within the ABAQUS package as a user-defined soil model using a user subroutine called UMAT (User Material). The backward Euler integration scheme coupled with Newton Raphson iterative algorithm is used to construct the model. A convergence test is conducted to check the accuracy and stability of the integration. UMAT's inability to manage pore pressure led to proxy diffusion to depict suction variation. The heat equation is used as a proxy for diffusion in three-dimensional space and time. Anoedometer test is simulated in which the soil is allowed to consolidate and then set to saturate at constant vertical stress, which leads to wetting collapse. The results demonstrate a reasonable behaviour of wettinginduced compression of soil subjected to loading with varying saturation level.

Numerical evaluation of thermal and hydrodynamic effects caused by heat production well on geothermal Phlegraean Fields

This study describes the geothermal response of the Phlegraean Fields as well as the impact of changes in its thermal and hydrodynamic properties brought on by a deep borehole heat exchanger (DBHE). For this purpose, we have developed a specialized model based on the Galerkin Method (GM) and the iterative Newton–Raphson algorithm to perform a transient simulation of heat transfer with fluid flow in porous media by solving the related system of coupled non-linear differential equations. A two-dimensional domain characterized with an anisotropic saturated porous media and a non-uniform grid is simulated. Extreme characteristics, such as non-uniformity in the distribution of the thermal source, are implemented as well as the fluid flow boundary conditions. While simulating the undisturbed geothermal reservoir and reaching the steady temperature, stream function, and velocity components, a DBHE is placed into the domain to evaluate its impact on the thermal and fluid flow fields. This research aims to identify and investigate the variables involved in the Phlegraean Fields and provide a numerical approach to accurately simulate the thermodynamic and hydrodynamic effects induced in a reservoir by a DBHE. The results show a maximum temperature change of 107.3°C in 200 years of service in the study area and a 65-year time limit is set for sustainable geothermal energy production.

A hydro-mechanical-damage fully coupled cohesive phase field model for complicated fracking simulations in poroelastic media

A hydro-mechanical-damage fully coupled numerical method is developed for simulations of complicated quasi-brittle fracking in poroelastic media. A unified fluid continuity equation with crack-width dependent permeability, based on the Biot’s poroelastic theory, is used for simultaneous modeling of fluid flow in both fractures and porous media. The fluid pressure is coupled into the governing equations of the phase-field regularized cohesive zone model, which can automatically predict quasi-brittle multi-crack initiation, nucleation, and propagation without remeshing, crack tracking, or auxiliary fields as needed by other methods. An alternate minimization Newton–Raphson iterative algorithm is implemented within the finite element framework to solve the above three-fields coupled problem with nodal degrees of freedom of displacements, fluid pressures, and damages. The method is first validated by three problems with analytical solutions, a problem with experimental results, and a two-crack merging problem with numerical results in published literature, in terms of time evolutions of injected fluid pressures, crack widths and lengths, and final crack paths. Horizontal wellbore fracking problems with parallel hydraulic cracks and random natural fractures are then simulated, with the effects of spacing, number, and angle of perforations investigated in detail. It is found that the developed method is capable of modeling complex multi-crack fracking in both homogeneous media and heterogeneous media with natural fractures, and is thus promising for fracking design optimization of practical exploitation of shale gas and oil.

Optimization of Object Injection Current in the Development of Electrical Impedance Tomography for Bone Fracture Detection

Electrical Impedance Tomography (EIT) nearly has all the parameters of the ideal of a modality tomography. There is one parameter that makes the EIT modality not yet ideal. It is the low spatial resolution of the image. Unfortunately, it is limited in the range of 5 percent to 10 percent of the object diameter. The causal factor is the limited number of measurement data obtained. The focus of research on EIT at now is the development of functional applications. The novel application developed in this research is to detect bone fractures, mainly due to sports injuries. The hardware characteristics are certainly different from other applications. Optimization of the injection current to match the most resistive characteristics of bone compared to all human body tissues has been carried out. The image of bone is reconstructed using the adjacent method of data collection and the Newton-Raphson iterative algorithm with Tikhonov regularization. The optimization results are seen from the high quality and contrast of the reconstructed image for various injection current values. The current injection frequency is selected at 10 kHz. In the results of this research, the injection current that can produce the highest quality and contrast is 0.3 mA.

A Projection Transformation Method for Eliminating Crosstalk in Networked Thermopile Arrays

The networked thermopile array in row-column fashion suffered from the crosstalk caused by parasitic parallel paths. To measure accurate two-dimensional thermal information, it is of great importance to eliminate the crosstalk. In this work, based on the response voltage non-linearity of thermopile pixels as well as the response voltage non-uniformity of array, a projection transformation method that suits for small-scale networked thermopile arrays is proposed. It transforms the crosstalk elimination problem into an overdetermined non-linear least squares problem, and is solved by the Newton-Raphson iterative algorithm. Its reconstruction error is analyzed carefully by Monte Carlo simulation, and its ability to implement non-uniformity and defective pixel correction is also discussed. Its validity is verified experimentally using a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3 \times 3$ </tex-math></inline-formula> networked thermopile array. The experimental result shows that the original IR image can be reconstructed from measured IR images by this method, and the reconstruction error is significantly decreased from 17.03% to 2.10%.

Axisymmetric nonlinear behavior of functionally graded saturated poroelastic circular plates under thermo-mechanical loading

In functionally graded saturated poroelastic circular plates with immovable simply supported and clamped rims, the axisymmetric nonlinear bending under transverse thermo-mechanical loading has been parametrically studied and compared with the axisymmetric postbuckling and nonlinear bending under thermal loading. Based on the classical plate theory, Love–Kirchhoff hypotheses and Sander’s assumptions, the general coupled nonlinear radial and transverse equilibrium equations, central continuity, symmetry and boundary conditions has been derived in ordinary and state-spatial forms. The corresponding difference equations have been achieved by using the generalized differential quadrature method. The equations have been assembled and numerically solved by using the Newton–Raphson iterative algorithm. The effects of the mechanical and thermal loads, pore distribution type, porosity parameter, Skempton’s coefficient, and thickness and boundary condition type on the behavior of the deflection, whether caused by thermo-mechanical bending, thermal postbuckling, or thermal bending, have been investigated in detail. From the parametric study, a novel quantity determining bending behavior has been found. The axisymmetric themo-mechanical nonlinear bending deflection is inversely and nonlinearly proportional to thermal load when the quantity is greater than a critical value and is nonlinearly proportional to thermal load when the quantity is less than a critical value. It was verified that the plate behavior complies with the general rules known for FG saturated poroelastic circular plates and with those known for metal–ceramic functionally graded circular plates whose governing equations are mathematically analogous to those of the current research.

Nonlinear time-history earthquake analysis for steel frames

This study presents a simple, efficient, accurate method for nonlinear time-history earthquake analysis of spatial steel frames. The proposed new fiber plastic hinge method simulating only one element for each member captures the time-history dynamic behavior of steel frames as accurately as sophisticated plastic zone methods. Stability functions and the geometric stiffness matrix are employed for predicting the second-order effects that aim to minimize computational time and computer resources. Residual stresses are considered through two fiber plastic hinges by assigning initial stress values. A numerical integration procedure using Newmark integration combined with the Newton-Raphson balanced iteration algorithm is developed to find a solution to nonlinear dynamic equilibrium equations of the structural system. Shear deformation effects are also considered in the dynamic analysis. The proposed software, named Direct Advanced Analysis and Design (DAAD), is proved to be accurate and reliable as compared with the analysis results generated by expensive commercial Finite Element Analysis (FEA) software packages. The proposed DAAD program can be improved for performance-based direct design and analysis.

The use of the spatial trend surface model to predict the distribution of health cadres from pharmacists in Wasit Governorate

Wasit governorate suffers from a poor distribution of health personnel from pharmacists, their concentration in some aspects, and their lack of others. In order to determine the pattern and distribution of the cadres of pharmacists, some of the spatial inference models were used in the prediction process to reach a fair distribution for pharmacists in the governorate, which is the trend surface model of the first, second and third degrees, reviewing the results using the Geo Data program. We estimated the trend surface model after assuming a certain spatial covariance function. In terms of parameters called covariance components, the components were estimated using the MLE method that adopted the Newton-Raphson iterative algorithm Then we compared these models to determine the best model that represents these data, using the criteria (AIC, BIC R2) and we concluded that the best model is the first-order trend surface model. To find out whether there is fairness in the distribution of pharmacists in the governorate the two statistical indicators were calculated the Lorenz curve and the Gini coefficient for the original and predicted data using the i-xplore program. The Lorenz curve showed the lack of equality in the distribution of pharmacists in the governorate and this was confirmed by the Gini coefficient, either for The data predicted, the Lorenz curve showed that there is fairness in the distribution of pharmacists within the governorate, and that the Gini coefficient proved that there is no discrepancy in the distribution.

Modified Newton Integration Algorithm With Noise Tolerance Applied to Robotics

Currently, the Newton–Raphson iterative algorithm has been extensively employed in the fields of basic research and engineering. However, when noise components exist in a system, its performance is largely affected. To remedy shortcomings that the conventional computing methods have encountered in a noisy workspace, a novel modified Newton integration (MNI) algorithm is proposed in this article. In addition, the steady-state error of the proposed MNI algorithm is smaller than that of the Newton–Raphson algorithm under a noise-free or noisy workspace. To lay the foundations for the corresponding theoretical analyses, the proposed MNI algorithm is first converted into a homogeneous linear equation with a residual term. Then, the related theoretical analyses are carried out, which indicate that the MNI algorithm possesses noise-tolerance ability under various noisy environments. Finally, multiple computer simulations and physical experiments on robot control applications are performed to verify the feasibility and advantage of the proposed MNI algorithm.

Solving the Overhead Transmission Conductor Heat Balance Equation Using the Newton Raphson Algorithm

This paper utilizes the Newton Raphson (NR) iterative algorithm to solve a detailed heat balance equation for overhead transmission conductors, for the purpose of calculating a Dynamic Line Rating or temperature-dependent impedance using real-time or forecasted weather data. Provided ambient weather conditions, conductor parameters and electrical current passing through the conductor, a starting guess is formed and the iterative solution method arrives upon a solution within a preset solution tolerance. A simple method to approximate the Rayleigh Number / Nusselt Number and Reynolds Number / Nusselt Number relationships is proposed to ensure successful convergence. The paper performs a thorough, numeric electrothermal conductor simulation to demonstrate proper solution convergence. This exercise identifies a minimum conductor temperature rise necessary for the solution method to converge successfully. Error related to the minimum starting conductor temperature rise, and error introduced by the approximated air property functions is illustrated. The temperature solution method is compared with several well-known thermal models.

Modified Newton Integration Algorithm With Noise Tolerance for Image Deblurring

This paper addresses image deblurring problem, which consists of recovering a rather high resolution image from its blurred and noisy observation version. Different from the existing solving methods, influence of noise perturbation is considered when modeling inverse solution process. To remedy shortcomings that the traditional and current computing methods have encountered under a noisy environment, a modified Newton integration (MNI) algorithm is presented from the control perspective. Besides, through the corresponding theoretical analyses and proofs, the proposed MNI algorithm is confirmed to handle image deblurring problem with noise-tolerance ability and fast convergence performance. Finally, multiple simulative experiments are performed to verify excellent image deblurring ability and noise-tolerance advantage of proposed MNI algorithm compared among Newton-Raphson iterative (NRI) algorithm, gradient descend dynamic (GDD) algorithm, and discrete time derivative (DTD) algorithm.

Nonlinear Inelastic Earthquake Analysis of 2D Steel Frames

In this work, a new method for nonlinear time-history earthquake analysis of 2D steel frames by a fiber plastic hinge method is presented. The beam-column element based on the displacement-based finite element method is established and formulated in detail using a fiber plastic hinge approach and stability functions. Geometric nonlinearities are taken into accounting by stability functions and the geometric stiffness matrix. A nonlinear dynamic algorithm is established based on the combination of the Newmark integration method and the Newton-Raphson iterative algorithm for solving dynamic equations. The proposed program predicts the nonlinear inelastic responses of 2D steel frames subjected to earthquakes as efficiently and accurately as commercial software. This study also shows that the initial residual stresses of steel should be considered in nonlinear inelastic time-history earthquake analysis of 2D steel frames while SAP2000 does not consider the effects of residual stresses.

RBF collocation approach to calculate numerically the solution of the nonlinear system of qFDEs

In this paper, we have implemented an efficient and high accurate radial basis function (RBF) collocation scheme for solving nonlinear systems of q-fractional differential equations. We firstly convert the problems under investigation into the equivalent systems of weakly singular q-integral equations by some essential results of fractional q-calculus. Secondly, we combine RBF collocation method and Newton–Raphson iterative algorithm to solve the latter systems of weakly singular q-integral equations. More precisely, applying RBF collocation scheme will transform the system of q-integral equations into the associated system of nonlinear algebraic equations that can be solved by iterative methods such as the Newton–Raphson algorithm. Finally, various numerical test problems including linear and nonlinear examples are listed to illustrate the robustness of the proposed global scheme with respect to the at least two recent methods in the literature.

Moving Element Analysis of High-Speed Train-Slab Track System Considering Discrete Rail Pads

This paper presents a study of the dynamic behavior of a coupled train-slab track system considering discrete rail pads. The slab track is modeled as a three-layer Timoshenko beam. The study is carried out using the moving element method (MEM). By introducing a convected coordinate system moving at the same speed as the vehicle, the governing equations of motion of the slab track are formulated in a moving frame-of-reference. By adopting Galerkin’s method, the element stiffness, mass and damping matrices of a truncated slab track in the moving coordinate system are derived. The vehicle is modeled as a multi-body with 10 degrees of freedom. The nonlinear Hertz contact model is used to account for the wheel–rail interaction. The Newmark integration method, in conjunction with a global Newton–Raphson iteration algorithm, is employed to solve the nonlinear dynamic equations of motion of the vehicle–track coupled system. The proposed MEM model of the system is validated through comparison with available results in the literature. Further study is then made to investigate the vehicle–track system accounting for track irregularities modeled as short harmonic wave forms. Results showed that irregularities with short wavelengths have a significant effect on wheel–rail contact force and rail acceleration, and the dynamic response of the track structure does not increase monotonously with the increase of the vehicle speed.