Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 7838

Search results for: Newton’s method

7838 New Newton's Method with Third-order Convergence for Solving Nonlinear Equations

Authors: Osama Yusuf Ababneh

Abstract:

For the last years, the variants of the Newton-s method with cubic convergence have become popular iterative methods to find approximate solutions to the roots of non-linear equations. These methods both enjoy cubic convergence at simple roots and do not require the evaluation of second order derivatives. In this paper, we present a new Newton-s method based on contra harmonic mean with cubically convergent. Numerical examples show that the new method can compete with the classical Newton's method.

Keywords: Third-order convergence, non-linear equations, root finding, iterative method.

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7837 A New Method to Solve a Non Linear Differential System

Authors: Seifedine Kadry

Abstract:

In this article, our objective is the analysis of the resolution of non-linear differential systems by combining Newton and Continuation (N-C) method. The iterative numerical methods converge where the initial condition is chosen close to the exact solution. The question of choosing the initial condition is answered by N-C method.

Keywords: Continuation Method, Newton Method, Finite Difference Method, Numerical Analysis and Non-Linear partial Differential Equation.

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7836 An Efficient Method for Solving Multipoint Equation Boundary Value Problems

Authors: Ampon Dhamacharoen, Kanittha Chompuvised

Abstract:

In this work, we solve multipoint boundary value problems where the boundary value conditions are equations using the Newton-Broyden Shooting method (NBSM).The proposed method is tested upon several problems from the literature and the results are compared with the available exact solution. The experiments are given to illustrate the efficiency and implementation of the method.

Keywords: Boundary value problem; Multipoint equation boundary value problems, Shooting Method, Newton-Broyden method.

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7835 A Contractor Iteration Method Using Eigenpairs for Positive Solutions of Nonlinear Elliptic Equation

Authors: Hailong Zhu, Zhaoxiang Li, Kejun Zhuang

Abstract:

By means of Contractor Iteration Method, we solve and visualize the Lane-Emden(-Fowler) equation Δu + up = 0, in Ω, u = 0, on ∂Ω. It is shown that the present method converges quadratically as Newton’s method and the computation of Contractor Iteration Method is cheaper than the Newton’s method.

Keywords: Positive solutions, newton's method, contractor iteration method, Eigenpairs.

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7834 The Application of Homotopy Method In Solving Electrical Circuit Design Problem

Authors: Talib Hashim Hasan

Abstract:

This paper describes simple implementation of homotopy (also called continuation) algorithm for determining the proper resistance of the resistor to dissipate energy at a specified rate of an electric circuit. Homotopy algorithm can be considered as a developing of the classical methods in numerical computing such as Newton-Raphson and fixed point methods. In homoptopy methods, an embedding parameter is used to control the convergence. The method purposed in this work utilizes a special homotopy called Newton homotopy. Numerical example solved in MATLAB is given to show the effectiveness of the purposed method

Keywords: electrical circuit homotopy, methods, MATLAB, Newton homotopy

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7833 Comparison of Newton Raphson and Gauss Seidel Methods for Power Flow Analysis

Authors: H. Abaali, T. Talbi, R.Skouri

Abstract:

This paper presents a comparative study of the Gauss Seidel and Newton-Raphson polar coordinates methods for power flow analysis. The effectiveness of these methods are evaluated and tested through a different IEEE bus test system on the basis of number of iteration, computational time, tolerance value and convergence.

Keywords: Convergence time, Gauss-Seidel Method, Newton-Raphson Method, number of iteration, power flow analysis.

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7832 A Fully Implicit Finite-Difference Solution to One Dimensional Coupled Nonlinear Burgers’ Equations

Authors: Vineet K. Srivastava, Mukesh K. Awasthi, Mohammad Tamsir

Abstract:

A fully implicit finite-difference method has been proposed for the numerical solutions of one dimensional coupled nonlinear Burgers’ equations on the uniform mesh points. The method forms a system of nonlinear difference equations which is to be solved at each iteration. Newton’s iterative method has been implemented to solve this nonlinear assembled system of equations. The linear system has been solved by Gauss elimination method with partial pivoting algorithm at each iteration of Newton’s method. Three test examples have been carried out to illustrate the accuracy of the method. Computed solutions obtained by proposed scheme have been compared with analytical solutions and those already available in the literature by finding L2 and L∞ errors.

Keywords: Burgers’ equation, Implicit Finite-difference method, Newton’s method, Gauss elimination with partial pivoting.

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7831 Newton-Raphson State Estimation Solution Employing Systematically Constructed Jacobian Matrix

Authors: Nursyarizal Mohd Nor, Ramiah Jegatheesan, Perumal Nallagownden

Abstract:

Newton-Raphson State Estimation method using bus admittance matrix remains as an efficient and most popular method to estimate the state variables. Elements of Jacobian matrix are computed from standard expressions which lack physical significance. In this paper, elements of the state estimation Jacobian matrix are obtained considering the power flow measurements in the network elements. These elements are processed one-by-one and the Jacobian matrix H is updated suitably in a simple manner. The constructed Jacobian matrix H is integrated with Weight Least Square method to estimate the state variables. The suggested procedure is successfully tested on IEEE standard systems.

Keywords: State Estimation (SE), Weight Least Square (WLS), Newton-Raphson State Estimation (NRSE), Jacobian matrix H.

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7830 Fast and Efficient Algorithms for Evaluating Uniform and Nonuniform Lagrange and Newton Curves

Authors: Taweechai Nuntawisuttiwong, Natasha Dejdumrong

Abstract:

Newton-Lagrange Interpolations are widely used in numerical analysis. However, it requires a quadratic computational time for their constructions. In computer aided geometric design (CAGD), there are some polynomial curves: Wang-Ball, DP and Dejdumrong curves, which have linear time complexity algorithms. Thus, the computational time for Newton-Lagrange Interpolations can be reduced by applying the algorithms of Wang-Ball, DP and Dejdumrong curves. In order to use Wang-Ball, DP and Dejdumrong algorithms, first, it is necessary to convert Newton-Lagrange polynomials into Wang-Ball, DP or Dejdumrong polynomials. In this work, the algorithms for converting from both uniform and non-uniform Newton-Lagrange polynomials into Wang-Ball, DP and Dejdumrong polynomials are investigated. Thus, the computational time for representing Newton-Lagrange polynomials can be reduced into linear complexity. In addition, the other utilizations of using CAGD curves to modify the Newton-Lagrange curves can be taken.

Keywords: Newton interpolation, Lagrange interpolation, linear complexity.

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7829 An Implicit Methodology for the Numerical Modeling of Locally Inextensible Membranes

Authors: Aymen Laadhari

Abstract:

We present in this paper a fully implicit finite element method tailored for the numerical modeling of inextensible fluidic membranes in a surrounding Newtonian fluid. We consider a highly simplified version of the Canham-Helfrich model for phospholipid membranes, in which the bending force and spontaneous curvature are disregarded. The coupled problem is formulated in a fully Eulerian framework and the membrane motion is tracked using the level set method. The resulting nonlinear problem is solved by a Newton-Raphson strategy, featuring a quadratic convergence behavior. A monolithic solver is implemented, and we report several numerical experiments aimed at model validation and illustrating the accuracy of the proposed method. We show that stability is maintained for significantly larger time steps with respect to an explicit decoupling method.

Keywords: Finite element method, Newton method, level set, Navier-Stokes, inextensible membrane, liquid drop.

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7828 Numerical Study of a Class of Nonlinear Partial Differential Equations

Authors: Kholod M. Abu-Alnaja

Abstract:

In this work, we derive two numerical schemes for solving a class of nonlinear partial differential equations. The first method is of second order accuracy in space and time directions, the scheme is unconditionally stable using Von Neumann stability analysis, the scheme produced a nonlinear block system where Newton-s method is used to solve it. The second method is of fourth order accuracy in space and second order in time. The method is unconditionally stable and Newton's method is used to solve the nonlinear block system obtained. The exact single soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the schemes. The interaction of two solitary waves for different parameters are also discussed.

Keywords: Crank-Nicolson Scheme, Douglas Scheme, Partial Differential Equations

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7827 Numerical Analysis of Thermal Conductivity of Non-Charring Material Ablation Carbon-Carbon and Graphite with Considering Chemical Reaction Effects, Mass Transfer and Surface Heat Transfer

Authors: H. Mohammadiun, A. Kianifar, A. Kargar

Abstract:

Nowadays, there is little information, concerning the heat shield systems, and this information is not completely reliable to use in so many cases. for example, the precise calculation cannot be done for various materials. In addition, the real scale test has two disadvantages: high cost and low flexibility, and for each case we must perform a new test. Hence, using numerical modeling program that calculates the surface recession rate and interior temperature distribution is necessary. Also, numerical solution of governing equation for non-charring material ablation is presented in order to anticipate the recession rate and the heat response of non-charring heat shields. the governing equation is nonlinear and the Newton- Rafson method along with TDMA algorithm is used to solve this nonlinear equation system. Using Newton- Rafson method for solving the governing equation is one of the advantages of the solving method because this method is simple and it can be easily generalized to more difficult problems. The obtained results compared with reliable sources in order to examine the accuracy of compiling code.

Keywords: Ablation rate, surface recession, interior temperaturedistribution, non charring material ablation, Newton Rafson method.

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7826 Spline Collocation for Solving System of Fredholm and Volterra Integral Equations

Authors: N. Ebrahimi, J. Rashidinia

Abstract:

In this paper, numerical solution of system of Fredholm and Volterra integral equations by means of the Spline collocation method is considered. This approximation reduces the system of integral equations to an explicit system of algebraic equations. The solution is collocated by cubic B-spline and the integrand is approximated by the Newton-Cotes formula. The error analysis of proposed numerical method is studied theoretically. The results are compared with the results obtained by other methods to illustrate the accuracy and the implementation of our method.

Keywords: Convergence analysis, Cubic B-spline, Newton- Cotes formula, System of Fredholm and Volterra integral equations.

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7825 Iteration Acceleration for Nonlinear Coupled Parabolic-Hyperbolic System

Authors: Xia Cui, Guang-wei Yuan, Jing-yan Yue

Abstract:

A Picard-Newton iteration method is studied to accelerate the numerical solution procedure of a class of two-dimensional nonlinear coupled parabolic-hyperbolic system. The Picard-Newton iteration is designed by adding higher-order terms of small quantity to an existing Picard iteration. The discrete functional analysis and inductive hypothesis reasoning techniques are used to overcome difficulties coming from nonlinearity and coupling, and theoretical analysis is made for the convergence and approximation properties of the iteration scheme. The Picard-Newton iteration has a quadratic convergent ratio, and its solution has second order spatial approximation and first order temporal approximation to the exact solution of the original problem. Numerical tests verify the results of the theoretical analysis, and show the Picard-Newton iteration is more efficient than the Picard iteration.

Keywords: Nonlinearity, iterative acceleration, coupled parabolic hyperbolic system, quadratic convergence, numerical analysis.

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7824 Implicit Eulerian Fluid-Structure Interaction Method for the Modeling of Highly Deformable Elastic Membranes

Authors: Aymen Laadhari, Gábor Székely

Abstract:

This paper is concerned with the development of a fully implicit and purely Eulerian fluid-structure interaction method tailored for the modeling of the large deformations of elastic membranes in a surrounding Newtonian fluid. We consider a simplified model for the mechanical properties of the membrane, in which the surface strain energy depends on the membrane stretching. The fully Eulerian description is based on the advection of a modified surface tension tensor, and the deformations of the membrane are tracked using a level set strategy. The resulting nonlinear problem is solved by a Newton-Raphson method, featuring a quadratic convergence behavior. A monolithic solver is implemented, and we report several numerical experiments aimed at model validation and illustrating the accuracy of the presented method. We show that stability is maintained for significantly larger time steps.

Keywords: Fluid-membrane interaction, stretching, Eulerian, finite element method, Newton, implicit.

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7823 Cantor Interpolating Spline to Design Electronic Mail Boxes

Authors: Adil Al-Rammahi

Abstract:

Electronic mail is very important in present time. Many researchers work for designing, improving, securing, fasting, goodness and others fields in electronic mail. This paper introduced new algorithm to use Cantor sets and cubic spline interpolating function in the electronic mail design. Cantor sets used as the area (or domain) of the mail, while spline function used for designing formula. The roots of spline function versus Cantor sets used as the controller admin. The roots calculated by the numerical Newton – Raphson's method. The result of this algorithm was promised.

Keywords: Cantor sets, spline, electronic mail design, Newton – Raphson's method.

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7822 New Laguerre-s Type Method for Solving of a Polynomial Equations Systems

Authors: Oleksandr Poliakov, Yevgen Pashkov, Marina Kolesova, Olena Chepenyuk, Mykhaylo Kalinin, Vadym Kramar

Abstract:

In this paper we present a substantiation of a new Laguerre-s type iterative method for solving of a nonlinear polynomial equations systems with real coefficients. The problems of its implementation, including relating to the structural choice of initial approximations, were considered. Test examples demonstrate the effectiveness of the method at the solving of many practical problems solving.

Keywords: Iterative method, Laguerre's method, Newton's method, polynomial equation, system of equations

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7821 A Special Algorithm to Approximate the Square Root of Positive Integer

Authors: Hsian Ming Goo

Abstract:

The paper concerns a special approximate algorithm of the square root of the specific positive integer, which is built by the use of the property of positive integer solution of the Pell’s equation, together with using some elementary theorems of matrices, and then takes it to compare with general used the Newton’s method and give a practical numerical example and error analysis; it is unexpected to find its special property: the significant figure of the approximation value of the square root of positive integer will increase one digit by one. It is well useful in some occasions.

Keywords: Special approximate algorithm, square root, Pell’s equation, Newton’s method, error analysis.

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7820 A Modification on Newton's Method for Solving Systems of Nonlinear Equations

Authors: Jafar Biazar, Behzad Ghanbari

Abstract:

In this paper, we are concerned with the further study for system of nonlinear equations. Since systems with inaccurate function values or problems with high computational cost arise frequently in science and engineering, recently such systems have attracted researcher-s interest. In this work we present a new method which is independent of function evolutions and has a quadratic convergence. This method can be viewed as a extension of some recent methods for solving mentioned systems of nonlinear equations. Numerical results of applying this method to some test problems show the efficiently and reliability of method.

Keywords: System of nonlinear equations.

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7819 Power Flow Control with UPFC in Power Transmission System

Authors: Samina Elyas Mubeen, R. K. Nema, Gayatri Agnihotri

Abstract:

In this paper the performance of unified power flow controller is investigated in controlling the flow of po wer over the transmission line. Voltage sources model is utilized to study the behaviour of the UPFC in regulating the active, reactive power and voltage profile. This model is incorporated in Newton Raphson algorithm for load flow studies. Simultaneous method is employed in which equations of UPFC and the power balance equations of network are combined in to one set of non-linear algebraic equations. It is solved according to the Newton raphson algorithm. Case studies are carried on standard 5 bus network. Simulation is done in Matlab. The result of network with and without using UPFC are compared in terms of active and reactive power flows in the line and active and reactive power flows at the bus to analyze the performance of UPFC.

Keywords: Newton-Raphson algorithm, Load flow, Unified power flow controller, Voltage source model.

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7818 A TFETI Domain Decompositon Solver for Von Mises Elastoplasticity Model with Combination of Linear Isotropic-Kinematic Hardening

Authors: Martin Cermak, Stanislav Sysala

Abstract:

In this paper we present the efficient parallel implementation of elastoplastic problems based on the TFETI (Total Finite Element Tearing and Interconnecting) domain decomposition method. This approach allow us to use parallel solution and compute this nonlinear problem on the supercomputers and decrease the solution time and compute problems with millions of DOFs. In our approach we consider an associated elastoplastic model with the von Mises plastic criterion and the combination of linear isotropic-kinematic hardening law. This model is discretized by the implicit Euler method in time and by the finite element method in space. We consider the system of nonlinear equations with a strongly semismooth and strongly monotone operator. The semismooth Newton method is applied to solve this nonlinear system. Corresponding linearized problems arising in the Newton iterations are solved in parallel by the above mentioned TFETI. The implementation of this problem is realized in our in-house MatSol packages developed in MatLab.

Keywords: Isotropic-kinematic hardening, TFETI, domain decomposition, parallel solution.

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7817 Bond Graph Modeling of Mechanical Dynamics of an Excavator for Hydraulic System Analysis and Design

Authors: Mutuku Muvengei, John Kihiu

Abstract:

This paper focuses on the development of bond graph dynamic model of the mechanical dynamics of an excavating mechanism previously designed to be used with small tractors, which are fabricated in the Engineering Workshops of Jomo Kenyatta University of Agriculture and Technology. To develop a mechanical dynamics model of the manipulator, forward recursive equations similar to those applied in iterative Newton-Euler method were used to obtain kinematic relationships between the time rates of joint variables and the generalized cartesian velocities for the centroids of the links. Representing the obtained kinematic relationships in bondgraphic form, while considering the link weights and momenta as the elements led to a detailed bond graph model of the manipulator. The bond graph method was found to reduce significantly the number of recursive computations performed on a 3 DOF manipulator for a mechanical dynamic model to result, hence indicating that bond graph method is more computationally efficient than the Newton-Euler method in developing dynamic models of 3 DOF planar manipulators. The model was verified by comparing the joint torque expressions of a two link planar manipulator to those obtained using Newton- Euler and Lagrangian methods as analyzed in robotic textbooks. The expressions were found to agree indicating that the model captures the aspects of rigid body dynamics of the manipulator. Based on the model developed, actuator sizing and valve sizing methodologies were developed and used to obtain the optimal sizes of the pistons and spool valve ports respectively. It was found that using the pump with the sized flow rate capacity, the engine of the tractor is able to power the excavating mechanism in digging a sandy-loom soil.

Keywords: Actuators, bond graphs, inverse dynamics, recursive equations, quintic polynomial trajectory.

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7816 Power Flow Analysis for Radial Distribution System Using Backward/Forward Sweep Method

Authors: J. A. Michline Rupa, S. Ganesh

Abstract:

This paper proposes a backward/forward sweep method to analyze the power flow in radial distribution systems. The distribution system has radial structure and high R/X ratios. So the newton-raphson and fast decoupled methods are failed with distribution system. The proposed method presents a load flow study using backward/forward sweep method, which is one of the most effective methods for the load-flow analysis of the radial distribution system. By using this method, power losses for each bus branch and voltage magnitudes for each bus node are determined. This method has been tested on IEEE 33-bus radial distribution system and effective results are obtained using MATLAB.

Keywords: Backward/Forward sweep method, Distribution system, Load flow analysis.

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7815 Evaluation of Mixed-Mode Stress Intensity Factor by Digital Image Correlation and Intelligent Hybrid Method

Authors: K. Machida, H. Yamada

Abstract:

Displacement measurement was conducted on compact normal and shear specimens made of acrylic homogeneous material subjected to mixed-mode loading by digital image correlation. The intelligent hybrid method proposed by Nishioka et al. was applied to the stress-strain analysis near the crack tip. The accuracy of stress-intensity factor at the free surface was discussed from the viewpoint of both the experiment and 3-D finite element analysis. The surface images before and after deformation were taken by a CMOS camera, and we developed the system which enabled the real time stress analysis based on digital image correlation and inverse problem analysis. The great portion of processing time of this system was spent on displacement analysis. Then, we tried improvement in speed of this portion. In the case of cracked body, it is also possible to evaluate fracture mechanics parameters such as the J integral, the strain energy release rate, and the stress-intensity factor of mixed-mode. The 9-points elliptic paraboloid approximation could not analyze the displacement of submicron order with high accuracy. The analysis accuracy of displacement was improved considerably by introducing the Newton-Raphson method in consideration of deformation of a subset. The stress-intensity factor was evaluated with high accuracy of less than 1% of the error.

Keywords: Digital image correlation, mixed mode, Newton-Raphson method, stress intensity factor.

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7814 Time/Temperature-Dependent Finite Element Model of Laminated Glass Beams

Authors: Alena Zemanová, Jan Zeman, Michal Šejnoha

Abstract:

The polymer foil used for manufacturing of laminated glass members behaves in a viscoelastic manner with temperature dependance. This contribution aims at incorporating the time/temperature-dependent behavior of interlayer to our earlier elastic finite element model for laminated glass beams. The model is based on a refined beam theory: each layer behaves according to the finite-strain shear deformable formulation by Reissner and the adjacent layers are connected via the Lagrange multipliers ensuring the inter-layer compatibility of a laminated unit. The time/temperature-dependent behavior of the interlayer is accounted for by the generalized Maxwell model and by the time-temperature superposition principle due to the Williams, Landel, and Ferry. The resulting system is solved by the Newton method with consistent linearization and the viscoelastic response is determined incrementally by the exponential algorithm. By comparing the model predictions against available experimental data, we demonstrate that the proposed formulation is reliable and accurately reproduces the behavior of the laminated glass units.

Keywords: Laminated glass, finite element method, finite-strain Reissner model, Lagrange multipliers, generalized Maxwell model, Williams-Landel-Ferry equation, Newton method.

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7813 Forward Kinematics Analysis of a 3-PRS Parallel Manipulator

Authors: Ghasem Abbasnejad, Soheil Zarkandi, Misagh Imani

Abstract:

In this article the homotopy continuation method (HCM) to solve the forward kinematic problem of the 3-PRS parallel manipulator is used. Since there are many difficulties in solving the system of nonlinear equations in kinematics of manipulators, the numerical solutions like Newton-Raphson are inevitably used. When dealing with any numerical solution, there are two troublesome problems. One is that good initial guesses are not easy to detect and another is related to whether the used method will converge to useful solutions. Results of this paper reveal that the homotopy continuation method can alleviate the drawbacks of traditional numerical techniques.

Keywords: Forward kinematics, Homotopy continuationmethod, Parallel manipulators, Rotation matrix

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7812 High Performance Computing Using Out-of- Core Sparse Direct Solvers

Authors: Mandhapati P. Raju, Siddhartha Khaitan

Abstract:

In-core memory requirement is a bottleneck in solving large three dimensional Navier-Stokes finite element problem formulations using sparse direct solvers. Out-of-core solution strategy is a viable alternative to reduce the in-core memory requirements while solving large scale problems. This study evaluates the performance of various out-of-core sequential solvers based on multifrontal or supernodal techniques in the context of finite element formulations for three dimensional problems on a Windows platform. Here three different solvers, HSL_MA78, MUMPS and PARDISO are compared. The performance of these solvers is evaluated on a 64-bit machine with 16GB RAM for finite element formulation of flow through a rectangular channel. It is observed that using out-of-core PARDISO solver, relatively large problems can be solved. The implementation of Newton and modified Newton's iteration is also discussed.

Keywords: Out-of-core, PARDISO, MUMPS, Newton.

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7811 Load Flow Analysis: An Overview

Authors: P. S. Bhowmik, D. V. Rajan, S. P. Bose

Abstract:

The load flow study in a power system constitutes a study of paramount importance. The study reveals the electrical performance and power flows (real and reactive) for specified condition when the system is operating under steady state. This paper gives an overview of different techniques used for load flow study under different specified conditions.

Keywords: Load Flow Studies, Y-matrix and Z-matrix iteration, Newton-Raphson method, Fast Decoupled method, Fuzzy logic, Artificial Neural Network.

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7810 Reentry Trajectory Optimization Based on Differential Evolution

Authors: Songtao Chang, Yongji Wang, Lei Liu, Dangjun Zhao

Abstract:

Reentry trajectory optimization is a multi-constraints optimal control problem which is hard to solve. To tackle it, we proposed a new algorithm named CDEN(Constrained Differential Evolution Newton-Raphson Algorithm) based on Differential Evolution( DE) and Newton-Raphson.We transform the infinite dimensional optimal control problem to parameter optimization which is finite dimensional by discretize control parameter. In order to simplify the problem, we figure out the control parameter-s scope by process constraints. To handle constraints, we proposed a parameterless constraints handle process. Through comprehensive analyze the problem, we use a new algorithm integrated by DE and Newton-Raphson to solve it. It is validated by a reentry vehicle X-33, simulation results indicated that the algorithm is effective and robust.

Keywords: reentry vehicle, trajectory optimization, constraint optimal, differential evolution.

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7809 Postbuckling Analysis of End Supported Rods under Self-Weight Using Intrinsic Coordinate Finite Elements

Authors: C. Juntarasaid, T. Pulngern, S. Chucheepsakul

Abstract:

A formulation of postbuckling analysis of end supported rods under self-weight has been presented by the variational method. The variational formulation involving the strain energy due to bending and the potential energy of the self-weight, are expressed in terms of the intrinsic coordinates. The variational formulation is accomplished by introducing the Lagrange multiplier technique to impose the boundary conditions. The finite element method is used to derive a system of nonlinear equations resulting from the stationary of the total potential energy and then Newton-Raphson iterative procedure is applied to solve this system of equations. The numerical results demonstrate the postbluckled configurations of end supported rods under self-weight. This finite element method based on variational formulation expressed in term of intrinsic coordinate is highly recommended for postbuckling analysis of end-supported rods under self-weight.

Keywords: Variational method, postbuckling, finite element method, intrinsic coordinate.

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