Newton Raphson Research Articles

Numerical solution of fractional PDEs through wavelet approach

To solve fractional partial differential equations (FPDEs) under various physical conditions, this study developed a novel method known as the Hermite wavelet method employing the functional integration matrix. The method that has been suggested is based on the Hermite wavelet collocation process. To determine the solution of the fractional differential equations, the Caputo fractional derivative operator of order α∈(0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,1]$$\\end{document} is used. With the use of appropriate grid points, this method converts FPDEs into a system of nonlinear algebraic equations. We achieve a solution by solving these nonlinear algebraic equations by the Newton–Raphson method. Tables and graphs show that the suggested method produces superior results. We provide various illustrative examples to establish the effectiveness of the suggested concept, and the outcomes support the applicability of the suggested strategy. Obtained results are numerically expressed in terms of absolute errors. Finally, convergence analyses are discussed as some theorem with proof.

Hydraulic fracturing phase-field model in porous viscoelastic media

The mechanical behavior of shale exhibits pronounced time-dependent nonlinearity. Existing phase-field models are inadequate for evaluating the fluid-driven fracture propagation process in porous viscoelastic media. In this study, a new viscoelastic phase-field model for fluid-driven fracture propagation was introduced based on a thermodynamically consistent framework using the Maxwell–Wiechert model to describe viscoelastic constitutive behavior, with the bulk and shear moduli expanded using the Prony series. Volumetric deviatoric splits were used for both elastic and viscous energies. The fluid flow in both the fracture and the matrix adhered to Darcy's law. Additionally, the phase-field driving force considers the combined influence of rock elastic, viscous, and fluid energies. The numerical calculation iteration format was constructed using finite element discretization and the Newton–Raphson (NR) iterative method. A staggered iteration algorithm was applied to address the displacement, pressure, and phase-field problems. This method not only accurately simulates mechanical cyclic loading but also models the extension of multiple hydraulic fractures in homogeneous reservoirs and hydraulic fracture propagation in fractured reservoirs.

Optimal Allocation of Photovoltaic-Green Distributed Generation for Maximizing the Performance of Electrical Distribution Networks

Renewable energy sources provide an environmentally sustainable solution to meet growing energy demands. Consequently, photovoltaics (PV) is regarded as a promising form of green distributed generation (GDG). The penetration of PV-GDG into distribution networks (DNs) is crucial, presenting a significant opportunity to improve power grid quality and reduce power losses. In this study, a comprehensive investigation was conducted to determine the optimal location, number, and capacity of PV-GDG penetrations with DN to achieve these objectives. Therefore, employing the Newton–Raphson (NR) technique and particle swarm optimization (PSO) approach for case studies, the analysis focused on the IEEE 33 bus test system as a benchmark test and the Iraq–Baghdad DN at 11 kV and 0.416 kV as a real case study. The outcomes revealed that integrating 4 × 1 MW PV-GDG units in a centralized configuration at bus 13 of the 11 kV Rusafa DN in the first scenario significantly reduced power losses and alleviated voltage drops across the network. In contrast, the second scenario entailed the utilization of dispersed PV panels with a capacity of 10 kW installed on rooftops at all 400 consumer load points with a cumulative capacity of 4 MW. This approach exemplified the enhancement of DN performance by significantly maximizing the power loss reduction and minimizing the voltage drops across the buses, exceeding the results achieved in the first scenario. The software applications employed in the practical implementation of this study included the CYMDist 9.0 Rev 04 program, PVsyst 7.2.20 software, and MATLAB R2022b.

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.

Analysis of radiative mixed convective heat transfer for non-Newtonian fluid with heat generation/absorption phenomena

The current work consists of investigating the heat generation/absorption and non-linear thermal radiation effects for two-dimensional mixed convective bi-viscosity fluid flow inside the C-shaped enclosure with uniformly heated left and linearly heated top and bottom walls. The physical flow problem is governed by applying the constitutive laws of bi-viscosity model, conservation of mass, momentum and energy balance equations. The governing non-linear coupled partial differential equations is transformed into the algebraic nonlinear system by using Galerkin's finite-element technique after eradicating the pressure terms via the penalty method, and then the Newton–Raphson technique is used to solve it. It is noted that at a low value of [Formula: see text] (which gives the high plasticity of the bi-viscosity liquid), only the primary circulations with intensity [Formula: see text] appeared, but at a high value of [Formula: see text] decreased the plasticity which caused enhanced buoyancy-driven that generate also the secondary bowls with intensity [Formula: see text] inside the enclosure. The heat transfer rate increased [Formula: see text] by an increase in the Reynolds number from [Formula: see text] to [Formula: see text]. The findings of this study will play a role in improving the thermal performance of microelectronic devices, cooling systems, food processing, solar collectors, and high performance boilers by tracing the energy transport trajectories through heatlines.

Assessment of Axisymmetric Dynamic Snap-Through and Thermally Induced Vibrations in FGM Cylindrical Shells Under Instantaneous Heating

This paper addresses the investigation of axisymmetric thermally induced vibrations in Functionally Graded Material (FGM) cylindrical shells. It considers temperature-dependent (TD) properties and geometric non-linearity (the von Karman effect). The study systematically solves a transient heat conduction equation using finite differences method and the Crank–Nicolson method. During the heating stages, the evaluation of thermal forces and moments takes place. Equations of motion are derived through the application of Hamilton’s principle. Spatial dependencies are discretized using the generalized Ritz method, while temporal dependencies are approximated using the [Formula: see text]-Newmark method with Newton–Raphson linearization. A comparative analysis validates the procedure’s efficiency and precision. Parametric studies explore the influence of parameters, including the temperature-dependency material properties, geometric nonlinearity, and shell’s power-law index, providing valuable insights into FGM shell behavior under thermal shock.

Analysis of thermal density and heat sink on dissipative nanofluid along magnetized sheet and applications in microelectronic cooling systems

The objective of present analysis is to control excessive thermal behavior in microelectronic devices using heat sink and magnetic field. Improving technological innovation demands proper temperature evacuation in microelectronic components. A feasible method for cooling microelectronics is to use nanofluid, a novel heat-transport liquid. To achieve significant cooling, microelectronic devices must be small with specific operating liquids than regular liquids. The present work investigates the fluid motion and heat transmission properties of nanofluids as advanced cooling fluids for microelectronic coolant systems. A numerical technique is used to investigate the cooling mechanism using stretching surface. Computational fluid dynamics and 2D liquid movement simulation are employed to determine the mathematical model. The governing model is transformed with Keller box method and Newton Raphson technique through MATLAB program. The base liquid is water using laminar motion. The influence of nanoparticles volume fraction, heat sink, variable density and viscous dissipation in the base liquid on heat transmission efficiency of coolant devices is investigated. It is noticed that the fluid temperature decreases as heat sink increases. It is found that the excessive heat is reduced by using magnetic layer and heat sink. The present numerical mechanism is significant for the cooling of microelectronic devices.

A finite failure software reliability model using extended log-logistic distribution

Software reliability engineering has been developed to examine software systems over the years. Non-homogeneous Poisson process (NHPP) models are widely employed for analyzing software reliability data, as they have the ability to describe the expected number of failures over time using the mean value function. In addition, these models have an intensity function representing the failure rate over time. This article proposes a new non-homogeneous Poisson process model based on extended log-logistic (NHPP ELL) distribution with three parameters. The model properties were derived and represented graphically. Furthermore, the maximum likelihood estimation method was utilized to estimate the model parameters, which were computed numerically using the Newton–Raphson method. Three real software reliability datasets were utilized to evaluate the performance of the new NHPP ELL model based on three distinct criteria: the mean square error, the coefficient of determination (R2), the Theil statistic, the predictive-ratio risk, and the predictive power. Moreover, a comparative study was conducted between the proposed model and several well-known NHPP models regarding goodness-of-fit and predictive performance. All the evaluation measures indicate that the proposed model outperforms the other models across all the considered datasets. In addition, the graphical plots depicting the actual data and the prediction results for all the fitted models show that the proposed model performs well in all three considered software reliability datasets compared to other models.

Design and testing of a high precision parallel manipulator with hyperbolic–elliptic flexure hinges

A novel six-degree-of-freedom (6-DOF) parallel manipulator (PM) with a 6-PSS configuration is designed as a piston error compensation mechanism in an optical system. To meet the high-precision adjustment requirement, the prismatic hinges are driven by nanometer resolution piezoelectric motors, and hyperbolic–elliptic (H-E) flexure hinges are used as spherical hinges because of their zero clearance and greater precision transmission capability than traditional rigid hinges. The analysis of the closed-loop vector relationship of the kinematic chain of the leg results in the construction of the kinematic equation, which can satisfy the inverse kinematic solution within a certain precision. The Newton–Raphson method is used for the iteration to solve the forward kinematics. A kinematic simulation was performed to verify the accuracy and effectiveness of the kinematic model using the rigid–flexible coupling finite element method. The workspace of the PM was analyzed using inverse kinematics and kinematics simulation. Two test systems were constructed to test the working performance of the PM. The experimental tests demonstrated that the resolution reached the nanometer level and the travel reached the millimeter level.

Semi-analytical solutions of Newtonian fluid-FENE-P fluid core annular flow

Water-lubricated transportation of viscous oil is an important application of core annular flow (CAF), which significantly reduces friction pressure drop and saves pump power. However, the core oil floats up due to the density difference of oil and water, causing instability and even destruction of CAF, which restricts the application and development of the drag reduction technology. The viscoelastic fluid in the annular can inhibit the tendency of the core oil to float up and enhance the stability of the CAF. Nevertheless, theoretical studies related to the viscoelastic fluid CAF are currently missing. To make up for the lack of theoretical research, the solutions of laminar concentric viscous oil-viscoelastic fluid CAF in horizontal and inclined pipes are obtained in this work, and the annular fluid is regarded as viscoelastic fluid conforming to the FENE-P model. Based on the Navier–Stokes equation and FENE-P model, a non-dimensional CAF model is established, and the Newton–Raphson method is used to solve the model. The rheological behavior of annular fluid and the effects of viscoelastic fluid rheology and viscosity ratio on various CAF flow characteristics, including holdup, pressure gradient, slip ratio, and Ledinegg instability, are investigated. The results indicate that the shear-thinning effect of viscoelastic fluid has a significant effect on water holdup and expands the multi-solution region. Different from Newtonian fluid, when the annulus fluid is viscoelastic, the slip ratio can be less than 2. The most significant property is that the shear-thinning effect can transform the hydraulic characteristic curve in the multi-valued region into a single-valued curve, which helps to eliminate Ledinegg instability.

A new graph theoretic analytical method for nonlinear distributed order fractional ordinary differential equations by clique polynomial of cocktail party graph

In this paper, we presented a new analytical method for one of the rapidly emerging branches of fractional calculus, the distributed order fractional differential equations (DFDE). Due to its significant applications in modeling complex physical systems, researchers have shown profound interest in developing various analytical and numerical methods to study DFDEs. With this motivation, we proposed an easy computational technique with the help of graph theoretic polynomials from algebraic graph theory for nonlinear distributed order fractional ordinary differential equations (NDFODE). In the method, we used clique polynomials of the cocktail party graph as an approximation solution. With operational integration and fractional differentiation in the Caputo sense, the NDFODEs transformed into a system of algebraic equations and then solved by Newton–Raphson's method to determine the unknowns in the Clique polynomial approximation. The proficiency of the proposed Clique polynomial collocation method (CCM) is illustrated with four numerical examples. The convergence and error analysis are discussed in tabular and graphical depictions by comparing the CCM results with the results of existing numerical methods.

A Novel Higher-Order Numerical Scheme for System of Nonlinear Load Flow Equations

Power flow problems can be solved in a variety of ways by using the Newton–Raphson approach. The nonlinear power flow equations depend upon voltages Vi and phase angle δ. An electrical power system is obtained by taking the partial derivatives of load flow equations which contain active and reactive powers. In this paper, we present an efficient seventh-order iterative scheme to obtain the solutions of nonlinear system of equations, with only three steps in its formulation. Then, we illustrate the computational cost for different operations such as matrix–matrix multiplication, matrix–vector multiplication, and LU-decomposition, which is then used to calculate the cost of our proposed method and is compared with the cost of already seventh-order methods. Furthermore, we elucidate the applicability of our newly developed scheme in an electrical power system. The two-bus, three-bus, and four-bus power flow problems are then solved by using load flow equations that describe the applicability of the new schemes.

Evaluation of FACTS Contributions Using Branch Flow Model and Newton–Raphson Algorithm

Flexible alternating current transmission systems (FACTSs) have been widely incorporated in electric power systems in order to control system parameters. This paper proposes the modeling of four FACTS devices, using the Branch Flow Model (BF) as an optimization problem to reduce the complexity of the Newton–Raphson (NR) load flow code with FACTS devices. The devices are represented as variable impedances, as a function of a firing angle, and as voltage source converters (VSCs) located on the buses and transmission lines. This proposed model solves the problem associated with the selection of appropriate initial conditions of the parameters of each device that guarantee convergence. The model is validated by evaluating its percentage deviation with respect to the NR method, using the standard test systems, IEEE 5-bus, IEEE 14-bus, IEEE 30-bus, and IEEE 57-bus systems.

Nonlinear Dynamic Analysis of a Curved Sandwich Beam with a Time-Dependent Viscoelastic Core Using the Generalized Differential Quadrature Method (GDQM)

This paper focuses on the geometrically nonlinear dynamic analyses of a three-layered curved sandwich beam with isotropic face layers and a time-dependent viscoelastic core. The boundary conditions and equations of motion governing the forced vibration are derived by using Hamilton’s principle. The first-order shear deformation theory is used to obtain kinematic relations. The spatial discretization of the equations is achieved using the generalized differential quadrature method (GDQM), and the Newmark-Beta algorithm is used to solve the time variation of the equations. The Newton–Raphson method is used to transform nonlinear equations into linear equations. The validation of the proposed model and the GDQM solution’s reliability are provided via comparison with the results that already exist in the literature and finite element method (FEM) analyses using ANSYS. Then, a series of parametric studies are carried out for a curved sandwich beam with aluminum face layers and a time-dependent viscoelastic core. The resonance and cancellation phenomena for the nonlinear moving-load problem of curved sandwich beams with a time-dependent viscoelastic core are performed using the GDQM for the first time, to the best of the authors’ knowledge.

Large Deflection Geometrically Nonlinear Bending of Porous Nanocomposite Cylindrical Panels on Elastic Foundation

Large deflection nonlinear bending of functionally graded (FG) porous cylindrical panels reinforced with graphene platelets (GPLs) on a Pasternak-type elastic foundation is examined by developing a reliable and effective 2D meshfree-based nonlinear numerical method. The large displacement field is express by the first-order shear deformation theory (FSDT) and the von Kármán nonlinearity, and approximated by 2D natural element method (NEM) in conjunction with the stabilized MITC3+ shell concept and the shell surface–rectangular grid geometry transformation. The nonlinear simultaneous equations are solved by a load incremental Newton–Raphson scheme. The developed nonlinear numerical method is justified from by comparing with the reference solutions, and the load–deflection and bending moment of FG-GPLRC porous cylindrical panels on elastic foundation are scrutinizingly examined. Four different symmetric GPL distribution patters (except for FG-Λ) and three different symmetric porosity distributions are considered and their combined effects on the nonlinear bending behavior are investigated, as well as the effects of foundation stiffness and GPL amount. Also, the results are compared with those of FG CNT-reinforced porous cylindrical panels.

Probabilistic relative entropy in elasto‐plasticity using the iterative generalized stochastic stress‐based Finite Element Method

AbstractThe generalized iterative stochastic perturbation approach to the stress‐based Finite Element Method has been proposed in this work. This approach is completed using the complementary energy principle, Taylor expansion of the general order applicable to all random functions and parameters as well as nodal polynomial response bases determined with the use of the Least Squares Method. The main aim of this elaboration is the usage of such a probabilistic approach to determine Bhattacharyya relative entropy for some nonlinear engineering stress analysis with uncertainty. Mathematical apparatus with its numerical implementation has been used to study elastoplastic torsion of some prismatic bar with Gaussian material uncertainty and the corresponding reliability measures. This problem has been solved using the Constant Stress Triangular (CST) plane finite elements, the modified Newton–Raphson algorithm, whereas the first four probabilistic characteristics resulting from the iterative generalized stochastic perturbation method have been contrasted with these obtained with the crude Monte‐Carlo sampling and the semi‐analytical probabilistic approach.

Impact on Long Duration of Overvoltage Due to Back-Feed Restoration of Distribution Network

In power system, power outages can be critical; if the environment and public safety within risk situation. Therefore, restoration power is needed to respond on system distribution to reduce the power restoration time and making the system reliable and efficient. Backfeeding using a nearby feeder in a power system is proposed in this research. When a power outage occurred, the faulted area is isolated from the system and searching available nearby feeders that can restore the outage area without any violation. However, analysing the effects of using a nearby feeder for back feeding is needed to avoid voltage violations. This paper analyses the voltage during the healthy and faulted condition, the impact of using a nearby feeder for backfeeding for a full load, half load and during no-load conditions are examine and also determine the maximum increment limitation of a load during back-feed if a qualified feeder is selected. Newton Raphson method is applied to calculate the power flow in the system and observe the stability of backfeed configuration. The results show the highest voltage supply occur during no load followed by half load and lastly during full load. The maximum voltage increment of load on backfeed is determined to be 0.95pu as an ideal voltage value.

A hybrid search space reduction algorithm and Newton–Raphson based selective harmonic elimination for an asymmetric cascade H-bridge multi-level inverter

Abstract In this paper, a selective harmonic elimination (SHE) method for an asymmetric cascade H-bridge multi-level inverter (ACHB-MLI) has been performed by using hybrid search space reduction and Newton–Raphson (SSR-NR) algorithm. Three H-bridges are cascaded in a single phase configuration, while each of the bridge is fed with V dc , 3V dc and 6V dc respectively. This particular asymmetric combination of dc sources will produce 10 possible switching combinations and 21-levels in the output voltage ranging from −10V dc to +10V dc . Hence, there will be 10 SHE equations corresponding to fundamental and harmonics, and the solution will be the 10 switching angles which satisfy those SHE equations. While traditional gradient-based techniques like the Newton–Raphson method can yield the global optimal solution for the SHE equations, they necessitate an initial guess. Conversely, meta-heuristic methods often face challenges related to sub-optimal convergence. To address these issues, a novel the hybrid SSR-NR algorithm has been introduced in this paper. This method combines stochastic and deterministic elements, offering a promising solution for solving the SHE equations. In the first step, the evolutionary SSR algorithm will be used and the output of the SSR algorithm is given to the NR method as an initial guess to converge to the final solution, this way the proposed hybrid SSR-NR algorithm is able to address the drawbacks of both the algorithms such as initial guess requirement for NR method and sub-optimal convergence characteristics of evolutionary algorithms. The simulations were carried out in Matlab/simulink environment for a 21-level ACHB-MLI. The results obtained showcase the efficacy of the proposed SSR-NR algorithm in solving the SHE equations. For demonstrating the effectiveness of SSR algorithm the results were also compared with other algorithms such as particle swarm optimization and grey wolf optimizer as well. Further, the experimental results on a laboratory prototype using dSPACE DS1104 R&D controller board were also presented in the paper, to validate the proposed methodology.

A multi-mechanism numerical simulation model for CO2-EOR and storage in fractured shale oil reservoirs

Under the policy background and advocacy of carbon capture, utilization, and storage (CCUS), CO2-EOR has become a promising direction in the shale oil reservoir industry. The multi-scale pore structure distribution and fracture structure lead to complex multiphase flow, comprehensively considering multiple mechanisms is crucial for development and CO2 storage in fractured shale reservoirs. In this paper, a multi-mechanism coupled model is developed by MATLAB. Compared to the traditional Eclipse 300 and MATLAB Reservoir Simulation Toolbox (MRST), this model considers the impact of pore structure on fluid phase behavior by the modified Peng–Robinson equation of state (PR-EOS), and the effect simultaneously radiate to Maxwell–Stefan (M–S) diffusion, stress sensitivity, the nano-confinement (N-C) effect. Moreover, a modified embedded discrete fracture model (EDFM) is used to model the complex fractures, which optimizes connection types and half-transmissibility calculation approaches between non-neighboring connections (NNCs). The full implicit equation adopts the finite volume method (FVM) and Newton–Raphson iteration for discretization and solution. The model verification with the Eclipse 300 and MRST is satisfactory. The results show that the interaction between the mechanisms significantly affects the production performance and storage characteristics. The effect of molecular diffusion may be overestimated in oil-dominated (liquid-dominated) shale reservoirs. The well spacing and injection gas rate are the most crucial factors affecting the production by sensitivity analysis. Moreover, the potential gas invasion risk is mentioned. This model provides a reliable theoretical basis for CO2-EOR and sequestration in shale oil reservoirs.

A study on Chlamydia transmission in United States through the Haar wavelet technique

In this study, a novel method known as the Haar wavelet collocation method (HWCM) is used to analyze the mathematical model of Chlamydia transmission in the United States. We use dependent variables with various parameters, such as birth rate, mortality rate, and recovery rate, etc., to explore the nature of the Chlamydia disease in susceptible unvaccinated individuals, susceptible vaccinated individuals, infected, treated, and recovered individuals. The ordinary differential equations (ODEs) in this model are coupled nonlinear. In this method, the Chlamydia model is converted into a system of non-linear algebraic equations using properties of the operational matrix of Haar wavelets. Later, the Newton–Raphson approach is used to extract the unknown coefficients. Mathematica software has been used for all calculations. Tables and graphs contain tabulated results of calculations. As a result, it can be seen that the current approach is more precise than those used in the literature. Also, we discuss the effect of different parameters on susceptible unvaccinated individuals, susceptible vaccinated individuals, infected, treated, and recovered individuals through graphs.