Newton-Raphson Procedure Research Articles

Implicit radiation hydrodynamics for 1D-problems

A numerical method is reviewed and described in detail which allows an accurate solution of the equations of radiation hydrodynamics for one-dimensional spherical configurations. Such problems occur in a large variety of astrophysical situations where the interaction of matter with radiation (and vice versa) plays an important role (e.g. stellar structure, stellar pulsations, star and planet formation, radiation driven winds, stellar explosions, etc.). The method relies on a conservative formulation of the physical equations and uses an adaptive grid to resolve and trace any steep features developing within the flow. Due to the wide temporal as well as spatial range of physical time scales the solution is advanced in time by an implicit discretization scheme resulting in a nonlinear system of algebraic equations which has to be solved by a Newton–Raphson procedure. A few examples concerning stellar pulsations of a class of luminous blue stars closes this article.

A general numerical method to solve for dislocation configurations

The shape of a mechanically equilibrated dislocation line is of considerable interest in the study of plastic deformation of metals and alloys. A general numerical method for finding such configurations in arbitrary stress fields has been developed. Analogous to the finite-element method (FEM), a general dislocation line is approximated by a series of straight segments (elements) bounded by nodes. The equilibrium configuration is found by minimizing the system energy with respect to nodal positions using a Newton-Raphson procedure. This approach, termed the finite-segment method (FSM), confers several advantages relative to segment-based, explicit formulations. The utility, generality, and robustness of the FSM is demonstrated by analyzing the Orowan bypass mechanism and a model of dislocation generation and equilibration at misfitting particles. Energy differences from previous analytical methods based on simple loop shapes are significant, up to 80 pct. Explicit expressions for the coordinate transformations, energies, and forces required for numerical implementation are presented.

An algorithm for determining cloud and shadow curves using continuous thermodynamics

A computer algorithm has been developed to determine the cloud and shadow point curves of polydisperse polymer/solvent systems using continuous thermodynamics to represent the polymer. The algorithm is based upon the work of Michelsen [M.L. Michelsen, Fluid Phase Equilibria, 4 (1980), pp. 1–10] and Koak [N. Koak, Polymer Solution Phase Behaviour, PhD Thesis, University of Calgary, Calgary, 1997] where a Newton–Raphson procedure is used to solve the equations defining phase equilibrium for fixed phase segment fractions between 0 and 1. The equations have been developed with the Sanchez–Lacombe equation of state which allows the continuous system to be defined by a set of five equations in five unknowns. Numerical integration is used to determine the values of the indefinite integrals involved. The cloud and shadow curves for a polydisperse polyethylene/ n-hexane system were found at 6 bar using both a log-normal distribution and a Schulz–Flory distribution to represent the polyethylene. Compositions of the cloud point curve covered the entire composition range showing the model's behavior in high temperature regions, a hypothetical liquid–vapour critical point and a three-phase point when the log-normal distribution was used. The cloud and shadow point curves of a polyethylene/ethylene system were also found and indicated a high pressure critical point.

Nonlinear analysis of underconstrained structures

Underconstrained (kinematically indeterminate) assemblies of bars and pin-joints possess a specific kind of nonlinearity which appears even for small displacements and linear elasticity. Various approaches to nonlinear analysis based on the Newton–Raphson procedure are considered. The subspace Newton–Raphson technique is proposed. Theoretical considerations are accompanied by numerical examples of plane and space underconstrained assemblies.

Kinematics of Tripode Transmissions. A New Approach

We present the kinematics of a two-tripode-joint transmission consisting of a sliding joint and a fixed joint. Beginning with a simplified joint having the output shaft in a fixed direction, the kinematic equations are derived from the closure relationships in a formal way. Due to the threefold symmetry of the tripode, the intrinsic parameters of the tulip-to-tripode matrix transformation are obtained; in particular, with this new approach, it is shown why the sum of the roller-to-tripode centre distances depend on the radius of the pot and the joint angle only. The triple frequency of the offset and the double frequency of the rollers on the tripode legs with respect to the driving shaft are also demonstrated, while the roller-to-tulip distances keep the same frequency. A new geometric approach based on the properties of the 120° angle is introduced and the above properties are demonstrated with the aid of geometric tools. For the whole transmission with the two real joints, the kinematic equations are solved numerically through a Newton–Raphson procedure and the results are compared to the simplified joint calculations. The phase angle between the two joints is also introduced and its influence on the kinematics is discussed through the roller-to-tulip distances.

Post-buckling analysis of a strut with general initial imperfection

Computational procedure has been presented for the post-buckling analysis of a strut with initial imperfection when subjected to progressive end shortening. Polynomial expression in very general form is used to simulate the actual initial imperfection, while the deformations are expressed by suitable trigonometric series. Geometric non-linearity is introduced into the strain–displacement relations in a manner that is consistent with the von Karman assumptions. Non-linear equilibrium equations are solved by a Newton–Raphson procedure. Results are presented for cases with symmetric and non-symmetric initial imperfection. Good comparison with other solutions and experimental results is obtained in all applications. © 1998 John Wiley & Sons, Ltd.

Numerical modelling of large deformation in metal powder forming

In this paper, the transient dynamic analysis of metal powder during the cold compaction process is simulated by the finite element method based on a ‘Total’ and ‘Updated’ Lagrangian formulation. Since the compaction process involves a very large reduction in volume, the behaviour of the powders is assumed to be that of a rate-independent elasto-plastic material. The process is therefore described by a large displacement finite element formulation for the spatial discretization. A generalized Newmark scheme is used for the time domain discretization and then the final nonlinear equations are solved by a Newton-Raphson procedure. A combination of a Mohr-Coulomb and elliptical cap yield model is utilised as a constitutive model to describe the nonlinear behaviour of powder materials. An incremental elasto-plastic material model is used to simulate the compaction process and a plasticity theory for friction is employed in the treatment of the powder-tooling interface. The interfacial behaviour between the die and powder is modelled by using an ‘interface’ element mesh. Finally, the powder behaviour during the compaction of a plane bush, a cutting tool and a rotational flanged component is analysed numerically. The predictive compaction forces at different displacements, the variation with time of the displacement, relative density and stress contours are obtained. It is shown that the proposed large displacement elasto-plastic finite element approach is capable of simulating the metal powder during compaction.

A computational model for elasto-viscoplastic solids at finite strain with reference to thin shell applications

This work extends a previously developed methodology for computational plasticity at finite strains that is based on the exponential map and logarithmic stretches to the context of isotropic elasto-viscoplastic solids. A particular form of the strain-energy function, given in terms of its principal values is employed. It is noticeable that within the proposed framework, the small strain integration algorithms, and the corresponding consistent tangent operators, automatically extend to the finite strain regime. Central to the effort of this formulation is the derivation of the closed form of a tangent modulus obtained by linearization of incremental non-linear problem. This ensures asymptotically quadratic rates of convergence of the Newton–Raphson procedure in the implicit finite element solution. To illustrate the performance of the presented formulation, several numerical examples, involving failure by strain localization and finite deformations, are given. © 1998 John Wiley & Sons, Ltd.

A generalized McGhee–von Hippel method for the cooperative binding of different competing ligands to an infinite one-dimensional lattice

Two novel algorithms have been derived for competitive binding of different ligands to an infinite lattice of binding sites. The algorithms are general with no limitation regarding the number of possible interacting ligand species or to lattice or ligand polarity. The first algorithm provides a generalization of the equations of McGhee and von Hippel for the calculation of free ligand concentrations as a function of binding densities. The second algorithm, a Newton–Raphson procedure with analytical derivatives, calculates the binding isotherm as a function of the total concentrations of ligands and binding sites. These new methods, based on efficient matrix power iterations, should in many cases eliminate the need for Monte Carlo simulations in the treatment of complex binding systems.

Nonlinear Behavior of Externally Prestressed Composite Beams: Analytical Model

The analysis and design of composite beams prestressed by external tendons involves difficulties mainly related to the cable slip at the saddle points and to the construction sequences. The paper presents a model for analyzing such structural systems under general conditions. Generic nonlinear constitutive laws are considered. No restrictions are assumed for tendon profiles and static scheme. Usual construction sequences can be considered. Kinematics and the compatibility equations refer to a beam model preserving the cross section planarity, and the tendon is free to slip at the saddle points. The characteristics of the problem make it convenient to obtain the balance condition from the virtual work principle. The numerical solution was performed by variational approximating methods, and the nonlinear equations were solved by the Newton-Raphson procedure. The paper presents a tool for the analysis of externally prestressed composite beams up to failure in order to investigate the influence of typical design parameters on the ultimate capacity. The effectiveness of the presented model is shown in a numerical example.

Layered flow in slider bearings

A procedure for solving the Navier-Stokes equations for the steady, one-dimensional flow of a binary water-based lubricant within an infinite slider bearing is described. The method uses an iterative process (Newton-Raphson procedure) to obtain the interface layer elevation between the liquid layers. The implementation of the Simpson rule integration technique into the equation set allows pressures along the bearing, drag on the moving bearing face, total flow into the bearing and the load-bearing capacity for a binary lubricant to be traced. Results are obtained for a range of non-linearity factors and lead to the conclusion that all the important indicators of bearing performance can be determined using the technique described.

A validation of the quadriphasic mixture theory for intervertebral disc tissue

The swelling and shrinking behaviour of soft biological tissues is described by a quadriphasic mixture model. In this model four phases are distinguished: a charged solid, a fluid, cations and anions. A description of the set of coupled differential equations of this quadriphasic mixture model is given. These equations are solved by the finite element method using a weighted residual approach. The resulting non-linear integral equations are linearized and solved by the Newton-Raphson iteration procedure. We performed some confined swelling and compression experiments on intervertebral disc tissue. These experiments are simulated by a one-dimensional finite element implementation of this quadriphasic mixture model. In contrast to a triphasic mixture model, physically realistic diffusion coefficients can be used to fit the experiments when the fixed charge density is relatively large, because in the quadriphasic mixture model electrical phenomena are not neglected.

The monte carlo newton-raphson algorithm

It is shown that the Monte Carlo Newton-Raphson algorithm is a viable alternative to the Monte Carlo EM algorithm for finding maximum likelihood estimates based on incomplete data. Both Monte Carlo procedures require simulations from the conditional distribution of the missing data given the observed data with the aid of methods like Gibbs sampling and rejective sampling. The Newton-Raphson algorithm is computationally more efficient than the EM algorithm as it converges faster. We further refine the procedure to make it more stable numerically. Our stopping criterion is based on a chi-square test for zero gradient. We control the type II error by working out the number of Monte Carlo replications required to make the non-centrality parameter sufficiently large. The procedure is validated and illustrated using three examples involving binary, survival and count data. In the last example, the Monte Carlo Newton-Raphson procedure is eight times faster than a modified version of the Monte Carlo EM algorithm.

Application of the finite element method for localized failure analysis in dynamic loading

In this paper, we will mainly discuss the localization phenomena for dynamic problems in which the behaviour of the material is incompressible. The finite element method is applied to the mixed formulation for the spatial discretization and then a generalized Newmark scheme is used for the time domain discretization. A Newton-Raphson procedure is finally used to solve the nonlinear equations. This paper will focus on the performance of mixed u-π triangular and quadrilateral elements to study their efficiency in indicating localization for various mesh refinements. It is shown that if a correct approximation is used then both the uniform and nonuniform mesh refinements will converge to the correct answer and clearly indicate the localization phenomenon.

The influence of temperature on the galvanic corrosion of a cast iron-stainless steel couple (prediction by boundary element method)

The general and galvanic corrosion of carbon steel (SS400) and stainless steel (SUS304) in CO 2 + NaCl aqueous solution at elevated temperatures has been analyzed using electrochemical measurements and computational simulation. The potential and current density distributions on the galvanic elements are described by the Laplace equation, and for numerical calculation the boundary element method has been used. The experimentally determined polarization curves have been used in order to define the nonlinear boundary conditions. This boundary value problem has been solved using the Newton-Raphson iterative numerical procedure. Additional thermodynamic and kinetic analysis was conducted to explain the observed corrosion behaviour.

Algebraic expressions for effective potential characteristic parameters in heavy ion scattering

In this paper, we obtain reliable expressions to calculate the barrier and pocket positions of the real part of the effective phenomenological optical potential having Woods-Saxon form factor, for different partial waves. The comparison of the results obtained from these formulae, when compared with the numerical results obtained using Newton-Raphson iterative procedure are found to be quite accurate, with error less than 1%. We also obtain algebraic expressions for estimatinglpoc, the angular momentum at which the potential pocket vanishes, the accuracy of which is tested with the exact calculations, using self-consistent iterative procedures. These and other expressions deduced in this paper provide simple and useful methods for calculating critical parameters of heavy ion effective potentials like barrier and pocket positions, curvatures at the barrier and pocket positions,lpoc and the grazing angular momentumlg to carry out the analysis of heavy ion scattering.

A computationally efficient alternative to the Buckingham potential for molecular mechanics calculations.

This paper describes a (6-8) variant of the Lennard-Jones (6-12) potential, for computing the energy of non-bonded interactions in molecular mechanics calculations, which combines the overall precision of the Buckingham (6-exp) potential with the computational efficiency of the standard Lennard-Jones (6-12) potential. There is also a note on the radius of convergence of the full matrix Newton-Raphson optimization procedure.

Effect of Aspect Ratio and Number of Meshes on Convergence of Steady-State Flow Calculation Using Newton-Raphson Iterative Procedure.

Effect of Aspect Ratio and Number of Meshes on Convergence of Steady-State Flow Calculation Using Newton-Raphson Iterative Procedure.

Analysis of ferroresonant circuits in the time and frequency domains

The steady-state responses of a 2nd order ferroresonant circuit have been determined and the ferroresonance characteristics of typical circuits have been presented. An efficient, hybrid (frequency-time domain) technique has been proposed for the analysis of the nonlinear circuits. The characteristics obtained with this technique agree well with those obtained with the time-domain Newton-Raphson procedure.

Newton–Raphson approach for the analysis of ferroresonant circuits

During switching operations in a power system there is a possibility that an unloaded transformer may be energised by a voltage source through an intervening capacitor. The capacitor could be a stray capacitance between an energised transmission line and the transformer terminals, a capacitance between the open contacts of a circuit breaker or a capacitive divider of a capacitive voltage transformer. The series connection of an alternating voltage source, a linear capacitor and a saturable inductor is generally known as a ferroresonant circuit. Determination of the steady-state periodic responses of a ferroresonant circuit is described. The responses were obtained using a time-domain Newton-Raphson procedure and also using a hybrid technique. A continuation procedure is described for determining the multiple solutions of the ferroresonant circuit and the bifurcating solution branches.