Newton-Raphson Procedure Research Articles

NONLINEAR THERMOELASTIC ANALYSIS OF FUNCTIONALLY GRADED PLATES USING THE THIRD-ORDER SHEAR DEFORMATION THEORY

Linear and nonlinear thermomechanical response of the functionally graded (metal-ceramic) plates subjected to static and dynamic loads is studied. The third-order plate theory of Reddy with the von Karman geometric nonlinearity is used for the kinematic and kinetic descriptions, and two constitutent power-law distribution through the plate thickness is employed. A displacement finite element model is developed with the Newmark time integration scheme, and the Newton-Raphson iterative procedure is used for the solution of nonlinear algebraic equations. While the model is general enough to be used for any boundary conditions, the simply supported and clamped boundary conditions are used to study the parametric effect of the power-law index and surface temperatures and mechanical loads on the thermomechanical response. The most significant finding of this study is that the deflection response does not fall intermediate to those of the metal or ceramic plates. This is due to the nonlinear coupling of mechanical and thermal contributions.

Using finite element method for the determination of elastic moduli by resonant ultrasound spectroscopy

Resonant ultrasound spectroscopy is a recent experimental/numerical method for the determination of moduli of elastic materials. Generally, all 21 elastic components of the elastic tensor can be determined by the numerical procedure based on the knowledge of a mechanical spectrum of a specimen. This involves the solution of a demanding inverse problem. Presently, Levenberg–Marquardt’s (LM) algorithm with the Ritz method for the solution of eigenfrequencies is usually applied. The LM method is based on the modified Newton–Raphson procedure, where all the eigenfrequencies and eigenvectors must be known for the computation of relevant gradients. Finite element method offers an analogous but more general optimization. The tested specimen, for instance, can be made of a composite material consisting of several layers with different material properties; the form of the specimen can be of a more complex shape, etc. In the present work, the elastic moduli are optimized by the fixed point iteration method, which requires only the lowest few eigenfrequencies to be known. The proposed algorithm is verified by test examples and suitable convergence criteria are established.

Wavelet algorithm for solving integral equations of molecular liquids. A test for the reference interaction site model.

A new efficient method is developed for solving integral equations based on the reference interaction site model (RISM) of molecular liquids. The method proposes the expansion of site-site correlation functions into the wavelet series and further calculations of the approximating coefficients. To solve the integral equations we have applied the hybrid scheme in which the coarse part of the solution is calculated by wavelets with the use of the Newton-Raphson procedure, while the fine part is evaluated by the direct iterations. The Coifman 2 basis set is employed for the wavelet treatment of the coarse solution. This wavelet basis set provides compact and accurate approximation of site-site correlation functions so that the number of basis functions and the amplitude of the fine part of solution decrease sufficiently with respect to those obtained by the conventional scheme. The efficiency of the method is tested by calculations of SPC/E model of water. The results indicated that the total CPU time to obtain solution by the proposed procedure reduces to 20% of that required for the conventional hybrid method.

Analytic evaluation of nonadiabatic coupling terms at the MR-CI level. II. Minima on the crossing seam: formaldehyde and the photodimerization of ethylene.

The method for the analytic calculation of the nonadiabatic coupling vector at the multireference configuration-interaction (MR-CI) level and its program implementation into the COLUMBUS program system described in the preceding paper [Lischka et al., J. Chem. Phys. 120, 7322 (2004)] has been combined with automatic searches for minima on the crossing seam (MXS). Based on a perturbative description of the vicinity of a conical intersection, a Lagrange formalism for the determination of MXS has been derived. Geometry optimization by direct inversion in the iterative subspace extrapolation is used to improve the convergence properties of the corresponding Newton-Raphson procedure. Three examples have been investigated: the crossing between the 1(1)B1/2(1)A1 valence states in formaldehyde, the crossing between the 2(1)A1/3(1)A1 pi-pi* valence and ny-3py Rydberg states in formaldehyde, and three crossings in the case of the photodimerization of ethylene. The methods developed allow MXS searches of significantly larger systems at the MR-CI level than have been possible before and significantly more accurate calculations as compared to previous complete-active space self-consistent field approaches.

Electrophoresis of a Concentrated Dispersion of Spherical Particles in a Non-Newtonian Fluid

Electrophoresis is one of the most widely used analytical tools for the quantification of the charged conditions on the surface of fine particles including biological entities. Although it has been studied extensively in the past, relevant results for the case when the dispersion medium is non-Newtonian are very limited. This may occur, for example, when the concentration of the dispersed phase is not low, which is not uncommon in practice. Here, the electrophoresis of a concentrated spherical dispersion in a Carreau fluid is analyzed theoretically under the conditions of low electric potential and weak external applied electrical field. A pseudospectral method coupled with a Newton-Raphson iteration procedure is used to solve the electrokinetic equations describing the phenomenon under consideration. We conclude that the more significant the shear thinning effect of the fluid, the larger the mobility, and this phenomenon is pronounced for the case when the double layer surrounding a particle is thin. We show that if the double layer is thin and the effect of shear thinning is significant, a second vortex can be observed in the neighborhood of a particle.

A super-linear minimization scheme for the nudged elastic band method

In this article, we present a superlinear minimization scheme for the nudged elastic band (NEB) method, which determines a minimum-energy path (MEP) of a reaction via connecting intermediate “replicas” between the reactant and the product. The minimization scheme is based on a quasi-Newton method: the adopted basis Newton–Raphson (ABNR) minimization scheme. In each step of ABNR minimization, the Newton–Raphson procedure is performed in a subspace of a user-defined dimension. The tangent directions of the path at a new Newton–Raphson step are determined self-consistently in the subspace. The acceleration of the proposed scheme over the quenched molecular-dynamic minimization, the current practice for minimizing a path using NEB, is demonstrated in three nontrivial test cases: isomerization of an alanine dipeptide, α-helix to π-helix transition of an alanine decapeptide, and oxidation of dimethyl sulfide. New features are also added such that the distances between replicas can be defined in the root of mean squared (RMS) best-fit space with flexible weighting options. This offers a way to incorporate the effects of a mobile solvent in the process with a finite number of replicas. MEPs obtained from various initial structures can be used to investigate different proposed reaction mechanisms, and the speedup of minimizing a path enhances the applicability of the NEB method. The combination of NEB force projection procedures, the flexible distance definition in the RMS best fit space with arbitrary weighting options, and the superlinear minimization scheme provides a framework to aid in the study of transition processes of biological molecules as such proteins.

Excited state geometry calculations and the resonance Raman spectrum of hexamethylpyrromethene

Abstract A Newton-Raphson procedure was developed for the geometry optimisation of excited states by time-dependent density functional theory (TD-DFT). The algorithm is formulated in internal coordinates and is based on finite differences to approximate the first- and second-order energy derivatives. Using this approach, the geometry of the first excited singlet state of hexamethylpyrromethene (HMPM) was partially optimised with respect to 21 selected internal coordinates at the TD-DFT level with the B3LYP functional and the 6-31G* basis set. The excited state displacements with respect to the ground state geometry were used to calculate the resonance Raman (RR) intensities of the normal modes by means of the transform theory. In general, the predicted enhancement pattern, illustrated by the comparison with the calculated non-resonant Raman spectrum, is in satisfactory agreement with the experimental RR and Raman spectra that were measured at 413 and 1064 nm excitation, respectively. However, relative RR intensities sensitively depend on subtle changes of the excited state geometry as documented by the RR spectra calculated at different steps of the geometry optimisation, which, due to the close proximity of the S1 and S2 potential energy surfaces in HMPM, only yields an approximate energy minimum structure. This specific property of HMPM is most likely the origin for the deviations between the calculated and experimental RR intensities noted for some of the normal modes. Nevertheless, TD-DFT provides a much better reproduction of the experimental data than the Hartree–Fock single-configuration interaction approximation.

Optimisation of continuous casting process using genetic algorithms: studies of spray and radiation cooling regions

A detailed optimisation study has been carried out of the continuous casting process, starting at the exit of the mould region and continuing up to the end of the radiation cooling zone. Biologically inspired genetic algorithms (GAs) have been used to determine the maximum casting speed and solidified shell thickness at the mould exit. Heat transfer models are proposed for both the spray and radiation cooling regions and the cooling water requirements are optimised using GAs. A number of spray bank configurations, studied earlier by experimental researchers in this field, are analysed separately. The extent of surface reheating below the sprays is predicted using a Newton-Raphson procedure. The results are compared with data existing in the literature and found to be in good agreement.

소형 왕복동 압축기의 동적 거동 및 윤활특성 해석

In this paper, a study on the dynamic behavior and lubrication characteristics of a reciprocating compression mechanism used in small refrigeration compressor is performed. In the problem formulation of the compressor dynamics, the viscous frictional force between piston and cylinder wall is considered in order to determine the coupled dynamic behaviors of piston and crankshaft. The solutions of the equations of motion of the reciprocating mechanism along with the time dependent Reynolds equations for the lubricating film between piston and cylinder wall and oil films of the journal bearings are obtained simultaneously. The hydrodynamic forces of journal bearings are calculated using finite bearing model and Gumbel boundary condition. And, a Newton-Raphson procedure was employed in solving the nonlinear equations of piston and crankshaft. The results explored the effects of design parameters on the stability and lubrication characteristics of the compression mechanism.

Harmonic domain periodic steady state modeling of power electronics apparatus: SVC and TCSC

The paper presents a general model for the representation of a thyristor-controlled reactor (TCR) in the frequency domain. This model underlies the implementation of a Newton-Raphson procedure to determine the periodic steady state of the power system with fast convergence and high accuracy. The TCR model is used as a building block to represent more sophisticated power electronics apparatus such as the static VAr compensator (SVC) and the thyristor-controlled series capacitor (TCSC).

Dynamic analysis of a reciprocating compression mechanism considering hydrodynamic forces

In this paper, a dynamic analysis of the reciprocating compression mechanism of a small refrigeration compressor is performed. In the problem formulation of the mechanism dynamics, the viscous frictional force between the piston and the cylinder wall is considered in order to determine the coupled dynamic behaviors of the piston and the crankshaft. Simultaneous solutions are obtained for the equations of motion of the reciprocating mechanism and the time-dependent Reynolds equations for the lubricating film between the piston and the cylinder wall and for the oil films on the journal bearings. The hydrodynamic forces of the journal bearings are calculated by using a finite bearing model along with the Gumbel boundary condition. A Newton-Raphson procedure is employed in solving the nonlinear equations for the piston and crankshaft. The developed computer program can be used to calculate the complete trajectories of the piston and the crankshaft as functions of the crank angle under compressor-running conditions. The results explored the effects of the radial clearance of the piston, oil viscosity, and mass and mass moment of inertia of the piston and connecting rod on the stability of the compression mechanism.

Evaluation of Various Surface Irrigation Numerical Simulation Models

In this study, a zero-inertia model (ZIMOD) is presented to simulate all phases of border and furrow irrigation systems. The governing equations were discretized using a control volume of moving cells, by procedures reported in the literature for borders and for furrows. Later, the discretized equations were linearized by applying a Taylor series expansion according to the Newton-Raphson procedure, and then the algebraic linearized equations were iteratively solved using the Gaussian elimination technique. The ZIMOD was verified against several field experimental data and output of various numerical models in the SIRMOD package. Computational times, errors in predicting advance and recession trajectories, and estimated runoff and infiltrated volumes were compared. All models used in the comparative analysis predicted the advance and recession times and the infiltrated and runoff volumes satisfactory. However, models in the SIRMOD package were computationally faster than ZIMOD.

Two gradient plasticity theories discretized with the element-free Galerkin method

In this paper, the element-free Galerkin method is exploited to analyze gradient plasticity theories. The use of this discretization method has the advantage that higher-order continuity, which can be required for the plastic multiplier field, is provided for. In passing, we show that the regularization properties of the higher-order gradients are necessary, since, similar to finite element methods, a severe discretization sensitivity is encountered otherwise. The mesh-free discretization method is first applied to an established, stress-space gradient plasticity theory. Next, a strain-space gradient plasticity theory, which employs an implicit averaging of an invariant strain measure, is proposed and elaborated. This theory is insensitive to the order of the interpolation polynomials and, as a result, the convergence of the Newton–Raphson procedure is better. The performance of both gradient plasticity models is examined for the mesh-free discretization scheme. A comparative study is carried out for a one-dimensional bar in tension, while the influence of the discretization in biaxial compression is studied next, including that of the numerical parameters of the discretization scheme and that of the internal length scale contained in the gradient plasticity theories.

On an improved numerical method to solve the equilibrium problems of solids with bounded tensile strength that are subjected to thermal strain

In this paper we recall briefly the constitutive equations for solids subjected to thermal strain taking in account the bounded tensile stress of the material. In view to solve the equilibrium problem via the finite element method using the Newton Raphson procedure, we show that the tangent elasticity tensor is semi-definite positive. Therefore, in order to obtain a convergent numerical method, the constitutive equation needs to be modified. Specifically, the dependency of the stress by the anelastic deformation is made explicit by means of a parameter <TEX>${\delta}$</TEX>, varying from 0 to 1, that factorizes the elastic tensor. This parameterization, for <TEX>${\delta}$</TEX> near to 0, assures the positiveness of the tangent elasticity tensor and enforces the convergence of the numerical method. Some numerical examples are illustrated.

Numerical analysis of the piston secondary dynamics in reciprocating compressors

In this study, a numerical analysis for the piston secondary dynamics of small refrigeration reciprocating compressors is performed. In general, the length of cylinder in this class of compressors is shortened to diminish the frictional losses of the piston-cylinder system. So, the contacting length between piston and cylinder wall is in variable with the rotating crank angle around the BDC of the reciprocating piston. In the problem formulation of the piston dynamics, the variation in bearing length of the piston and all corresponding forces and moments are considered in order to determine the piston trajectory, velocity and acceleration at each step. A Newton-Raphson procedure was employed in solving the secondary dynamic equations of the piston. The developed computer program can be used to calculate the entire piston trajectory and the lubrication characteristics as functions of crank angle under compressor running conditions. The results explored the effects of some design parameters and operating conditions on the stability of the piston, the oil leakage, and friction losses.

Computation of Optimum in Average Designs for Experiments with Finite Design Space

For Generalized Linear Models (GLM) optimum designs generally depend on the true but unknown parameter values. If a prior distribution for the parameters is available, it is possible to use a design that is optimum in average. If, in particular, the prior is uniform, the corresponding optimum design is termed a Laplace design. The purpose of this article is to indicate a Newton-Raphson procedure for computation of optimum in average designs for inference about parameters in a GLM when the design space is finite and to study the efficiency properties of Laplace designs in comparison with designs that use a uniform allocation of observations. Three numerical examples are presented, viz. two control group experiments and a Latin square experiment. The efficiency comparisons in these examples indicate that the Laplace designs are likely to be more efficient, when the prior information about the parameters is correct.

A robust algorithmic phase equilibria protocol and its application to a multipurpose compositional simulator

This study presents a new multipurpose compositional model for petroleum reservoirs. The proposed formulation allows the simulation of the isothermal and non-isothermal secondary or tertiary recovery processes, as well as condensate reservoirs where mass transfer of components between phases is considered to be significant. Although the current study limits the number of phases to four (4), the formulation presented in this study does not impose any limitation to the number of phases to be considered. The present compositional model utilizes either Soave–Redlich–Kwong (1972) or Peng and Robinson [Ind. Eng. Chem. Fundam. 15 (1976) 59; Can. J. Chem. Eng. (1976) 595] equations of state to describe the phase behavior of the system in lieu of predetermined equilibrium constants. The phase equilibrium package is capable of handling co-existing liquid phases and is equipped with a robust critical point determination technique, which is used in identifying the phases obtained under the existing temperature and pressure conditions. A linear programming algorithmic protocol utilized to determine the interaction coefficients and/or C 7+ component(s) critical properties is also presented. The partial differential equations describing the transport phenomena in porous media are approximated using the finite-difference technique and are coupled with the flow equations. The resulting system of equations is solved simultaneously using the generalized Newton–Raphson procedure. The simulator has been tested using published data for several options. Results are found to be in good agreement. Also, various sensitivity checks were conducted by varying different parameters in order to test the performance of the model under different conditions. The fully implicit nature of the proposed model is found to be an advantage so that no stability problems have been encountered when large time-step sizes are utilized.

Harmonic Domain Periodic Steady-State Modeling of Power Electronics Apparatus: SVC and TCSC

The paper presents a general model for the representation of a thyristor-controlled reactor (TCR) in the frequency domain. This model underlies the implementation of a Newton-Raphson procedure to determine the periodic steady state of the power system with fast convergence and high accuracy. The TCR model is used as a building block to represent more sophisticated power electronics apparatus such as the static VAr compensator (SVC) and the thyristor-controlled series capacitor (TCSC).

Computation of Steady-State Response in Power Electronic Circuits

A new simulator capable of “direct” steady-state time-domain simulation of circuits with periodic inputs is described. A Newton-Raphson iterative procedure is used to compute the state variables such that, after integrating for one period, they return to their original values. Several Power Electronic circuit examples are discussed. It is shown that the direct method is computationally far superior than the conventional method in which transient simulation is carried out until all non-periodic transients have decayed completely. The new simulator is a public-domain program and can be obtained from the authors.

Higher-Order Finite Strip Method for Postbuckling Analysis of Imperfect Composite Plates

Postbuckling analysis is essential to predict the capacity of composite plates carrying considerable additional load before the ultimate load is reached, and manufacturing-induced geometric imperfections often reduce the load-carrying capacity of composite structures. A higher-order finite strip method based on the higher-order shear deformation plate theory is developed for postbuckling analysis of laminated composite plates with initial geometric imperfection subjected to progressive end shortening. The arbitrary nature of initial geometric imperfection induced during manufacturing is accounted for in the analysis. Nonlinear equilibrium equations are solved by a Newton-Raphson procedure. Examples of postbuckling analyses of unsymmetric cross-ply, angle-ply, and arbitrary laminates are presented, and the accuracy and performance of the method are examined. The numerical higher-order finite strip method presented can be used as an accurate and efficient tool for postbuckling analysis of imperfect composite plates.