Order Power Series Research Articles

The Analitical Solution of Linear and Non-Linear Differential- Algebraic Equations (DAEs) with Laplace-Padé Series Method

In this paper, we apply Laplace-Padé Series method to solve linear and non-linear differentialalgebraic equations (DAEs). Firstly, The basic properties of the Laplace-Padé Series method are given. Secondly, we calculate the arbitrary order power series of differential-algebraic equations (DAEs), then convert it to the series form Laplace-Padé. Then, the three differential-algebraic equations (DAEs) are solved by Laplace-Padé Series method. It was seen that the method gave effective and fast results. Therefore, the method can be easily applied to linear and non-linear differential-algebraic equations (DAEs) problems in different fields.

Modified Drude model for small gold nanoparticles surface plasmon resonance based on the role of classical confinement

In this paper, we study the effect of restoration force caused by the limited size of a small metallic nanoparticle (MNP) on its linear response to the electric field of incident light. In a semi-classical phenomenological Drude-like model for small MNP, we consider restoration force caused by the displacement of conduction electrons with respect to the ionic positive background taking into account a free coefficient as a function of diameter of nanoparticle (NP) in the force term obtained by the idealistic Thomson model in order to adjust the classical approach. All important mechanisms of the energy dissipation such as electron-electron, electron-phonon and electron-NP surface scatterings and radiation are included in the model. In addition a correction term added to the damping factor of mentioned mechanisms in order to rectify the deficiencies of theoretical approaches. For determining the free parameters of model, the experimental data of extinction cross section of gold NPs with different sizes doped in the glass host medium are used and a good agreement between experimental data and results of our model is observed. It is shown that by decreasing the diameter of NP, the restoration force becomes larger and classical confinement effect becomes more dominant in the interaction. According to experimental data, the best fitted parameter for the coefficient of restoration force is a third order negative powers function of diameter. The fitted function for the correction damping factor is proportional to the inverse squared wavelength and third order power series of NP diameter. Based on the model parameters, the real and imaginary parts of permittivity for different sizes of gold NPs are presented and it is seen that the imaginary part is more sensitive to the diameter variations. Increase in the NP diameter causes increase in the real part of permittivity (which is negative) and decrease in the imaginary part.

Power Series Extender Method for the Solution of Nonlinear Differential Equations

We propose a power series extender method to obtain approximate solutions of nonlinear differential equations. In order to assess the benefits of this proposal, three nonlinear problems of different kind are solved and compared against the power series solution obtained using an approximative method. The problems are homogeneous Lane-Emden equation ofαindex, governing equation of a burning iron particle, and an explicit differential-algebraic equation related to battery model simulations. The results show that PSEM generates highly accurate handy approximations requiring only a few steps. The main advantage of PSEM is to extend the domain of convergence of the power series solutions of approximative methods as Taylor series method, homotopy perturbation method, homotopy analysis method, variational iteration method, differential transform method, and Adomian decomposition method, among many others. From the application of PSEM, it results in handy easy computable expressions that extend the domain of convergence of high order power series solutions.

Comparison of IMD Symmetry and Asymmetry Model in Power Amplifier Based on Multisine Excitation

This paper proposes system level behavior model of a nonlinear power amplifier that exhibit the IMD asymmetry effect based on multisine excitation. For this work, a third order power series coefficients are originatively represented with both the system level circuit features such as a second output intercept point (OIP2), a third output intercept point (OIP3), amplifier gain and its excitation amplitudes. In addition, the memory effect like the asymmetry in lower and upper intermodulation products are distinguished through the analytic description and measurement. The main advantage of the proposed theory is able to analyze the composition of the nonlinear response in power amplifier. This paper also presents the measured results to validate the analytic expressions.

Novel Circuit Breaker Modeling in 275kV Substation

Ferroresonance an electrical phenomenon may cause overvoltages in the electrical power system. In this paper, an overview of available papers is provided, at rst ferroresonance is introduced and then various type of ferroresonance in a voltage transformer (VT) is simulated. Then eect of new model of circuit breaker (CB) on damping ferroresonance oscillation is examined. Finally eect of earth resistance on the stabilizing these oscillations in the case of nonlinear core loss has been studied. Core loss is modelled by third order power series in terms of voltage and includes nonlinearities in core loss. For conrmation, the simulation is done on a one phase voltage transformer rated 100VA, 275kV. The simulation results reveal that novel modeling of circuit breaker exhibits great controlling eect on ferrores- onance overvoltages.

Resistive Ferroresonance Limiter for Potential Transformers

The ferroresonance or nonlinear resonance is a complex phenomenon, which may cause overvoltage in the electrical power system and endangers the system reliability and operation. The ability to predict the ferroresonance in the transformer depends on the accuracy of the transformer model used. In this paper, the effect of the new suggested ferroresonance limiter on the control of the chaotic ferroresonance and duration of chaotic transients in a potential transformer including nonlinear core losses is studied. To study the proposed ferroresonance limiter, a single phase 100 VA, 275 kV potential transformer is simulated. The magnetization characteristic of the potential transformer is modeled by a single-value two-term polynomial. The core losses are modeled by third order power series in terms of voltage and include core nonlinearities. The simulation results show that the ferroresonance limiter has a considerable effect on the ferroresonance overvoltage.

An analytical approximation for the orientation-dependent excluded volume of tangent hard sphere chains of arbitrary chain length and flexibility

Onsager-like theories are commonly used to describe the phase behavior of nematic (only orientationally ordered) liquid crystals. A key ingredient in such theories is the orientation-dependent excluded volume of two molecules. Although for hard convex molecular models this is generally known in analytical form, for more realistic molecular models that incorporate intramolecular flexibility, one has to rely on approximations or on computationally expensive Monte Carlo techniques. In this work, we provide a general correlation for the excluded volume of tangent hard-sphere chains of arbitrary chain length and flexibility. The flexibility is introduced by means of the rod-coil model. The resulting correlation is of simple analytical form and accurately covers a wide range of pure component excluded volume data obtained from Monte Carlo simulations of two-chain molecules. The extension to mixtures follows naturally by applying simple combining rules for the parameters involved. The results for mixtures are also in good agreement with data from Monte Carlo simulations. We have expressed the excluded volume as a second order power series in sin (γ), where γ is the angle between the molecular axes. Such a representation is appealing since the solution of the Onsager Helmholtz energy functional usually involves an expansion of the excluded volume in Legendre coefficients. Both for pure components and mixtures, the correlation reduces to an exact expression in the limit of completely linear chains. The expression for mixtures, as derived in this work, is thereby an exact extension of the pure component result of Williamson and Jackson [Mol. Phys. 86, 819-836 (1995)].

Nonlinear Oscillation in Potential Transformers Connected Controlling Circuit

In this work at first ferroresonance phenomenon is introduced and then various type of ferroresonance overvoltages in a potential transformer is simulated. Then effect of neutral earth resistance on controlling these oscillations in the case of nonlinear core losses has been studied. Core losses in the potential transformer are modelled by third order power series in terms of voltage and include nonlinearities in core losses. It is expected that neutral earth resistance generally can cause ferroresonance 'dropout'. For confirmation this aspect Simulation has been done on a one phase potential transformer rated 100VA, 275kV. The simulation results show that connecting the neutral earth resistance on the system configuration, shows a great controlling effect on ferroresonance oscillation.

Effect of Metal Oxide Arrester on Chaotic Behavior of Power Transformers

This paper investigates the Effect of Metal Oxide Arrester (MOV) on the Chaotic Behaviour of power transformers considering nonlinear model for core loss of transformer. The paper contains two parts. In part (1): effect of nonlinear core on the onset of chaotic ferroresonance in a power transformer is evaluated. The core loss is modeled by a third order power series in voltage. in part (2): Effect of Metal Oxide Arrester(MOV) on results in part (1) will be studied. The results reveal that the presence of the arrester has a mitigating effect on ferroresonant chaotic overvoltages. resulted bifurcation diagrams and phase plane diagrams have also been delivered.

Nonlinear calculating method of pile settlement

To study calculating method of settlement on top of extra-long large-diameter pile, the relevant research results were summarized. The hyperbola model, a nonlinear load transfer function, was introduced to establish the basic differential equation with load transfer method. Assumed that the displacement of pile shaft was the high order power series of buried depth, through merging the same orthometric items and arranging the relevant coefficients, the solution which could take the nonlinear pile-soil interaction and stratum properties of soil into account was solved by power series. On the basis of the solution, by determining the load transfer depth with criterion of settlement on pile tip, the method by making boundary conditions compatible was advised to solve the load— settlement curve of pile. The relevant flow chart and mathematic expressions of boundary conditions were also listed. Lastly, the load transfer methods based on both two-broken-line model and hyperbola model were applied to analyzing a real project. The related coefficients of fitting curves by hyperbola were not less than 0.96, which shows that the hyperbola model is truthfulness, and is propitious to avoid personal error. The calculating value of load—settlement curve agrees well with the measured one, which indicates that it can be applied in engineering practice and making the theory that limits the design bearing capacity by settlement on pile top comes true.

A non-iterative and non-singular perturbation solution for transforming Cartesian to geodetic coordinates

The Cartesian-to-Geodetic coordinate transformation is re-cast as a minimization algorithm for the height of the Satellite above the reference Earth surface. Optimal necessary conditions are obtained that fix the satellite ground track vector components in the Earth surface. The introduction of an artificial perturbation variable yields a rapidly converging second order power series solution. The initial starting guess for the solution provides 3–4 digits of precision. Convergence of the perturbation series expansion is accelerated by replacing the series solution with a Pade approximation. For satellites with heights < 30,000 km the second-order expansions yields ~mm satellite geodetic height errors and ~10−12 rad errors for the geodetic latitude. No quartic or cubic solutions are required: the algorithm is both non-iterative and non-singular. Only two square root and two arc-tan calculations are required for the entire transformation. The proposed algorithm has been measured to be ~41% faster than the celebrated Bowring method. Several numerical examples are provided to demonstrate the effectiveness of the new algorithm.

Augmented higher order global–local theory and refined triangular element for laminated composite plates

Abstract The characteristics of higher order theories for laminated composite plate are that the number of unknowns are independent of the number of layers. However, they are unable to predict accurately the inter-element stresses and are also unsuitable for laminated plates with a large number of layers. Based on the third-order global – 1,2-3 order local higher order theory proposed by Li and Liu [Li XiaoYu and Liu D. Generalized laminate theories based on double superposition hypothesis. Int. J. Numer. Meth. Eng.,1997; 40: 1197–1212], which can predict accurately the interlaminar stresses, we propose an augmented higher order global–local theory for laminated composite plates and using it to estimate the applicability to the range of number of layers. The displacement field is composed of a m th-order (9 > m > 3) polynomial of global coordinate z in the thickness direction and 1,2-3 order power series of local coordinate ζ k in the thickness direction of each layer. This theory can satisfy the free surface conditions and the geometric and stress continuity conditions at interfaces, and the number of unknowns is independent of the layer numbers of the laminate. Based on this higher order theory, a refined three-node triangular element satisfying the requirement of C 1 weak-continuity is presented. Numerical results show that the proposed higher order global–local theory can predict accurately in-plane stresses and transverse shear stresses from the constitutive equations, and it is still effective when the number of layers in laminated plates is more than five and up to 14. It is also shown that the present refined triangular element possesses higher accuracy compared with known elements.

A quadrilateral element based on refined global-local higher-order theory for coupling bending and extension thermo-elastic multilayered plates

In this paper a refined higher-order global-local theory is presented to analyze the laminated plates coupled bending and extension under thermo-mechanical loading. The in-plane displacement fields are composed of a third-order polynomial of global coordinate z in the thickness direction and 1,2–3 order power series of local coordinate ζ k in the thickness direction of each layer, which is identical to the 1,2–3 global-local higher-order theory by Li and Liu [Li, X.Y., Liu, D., 1997. Generalized laminate theories based on double superposition hypothesis. Int. J. Numer. Methods Eng. 40, 1197–1212] Moreover, a second-order polynomial of global coordinate z in the thickness direction is chosen as transverse displacement field. The transverse shear stresses can satisfy continuity at interfaces, and the number of unknowns does not depend on the layer numbers of the laminate. Based on this theory, a quadrilateral laminated plate element satisfying the requirement of C 1 continuity is presented. By solving both bending and thermal expansion problems of laminates, it can be found that the present refined theory is very accurate and obviously superior to the existing 1,2–3 global-local higher-order theory. The most attractive feature of this theory is that the transverse shear stresses can be accurately predicted from direct use of constitutive equations without any post-processing method. It is also shown that the present quadrilateral element possesses higher accuracy.

DISTRIBUTIONAL UNCERTAINTY ANALYSIS OF A BATCH CRYSTALLIZATION PROCESS USING POWER SERIES AND POLYNOMIAL CHAOS EXPANSIONS

Computationally efficient approaches are presented that quantify the influence of parameter uncertainties upon the states and outputs of finite-time control trajectories for nonlinear systems. In the first approach, the worst-case values of the states and outputs due to model parameter uncertainties are computed as a function of time along the control trajectories. The approach uses an efficient contour mapping technique to provide an estimate of the distribution of the states and outputs as a function of time. To increase the estimation accuracy of the shape of the distribution, an approach that uses second order power series expansion in combination with Monte Carlo simulations is proposed. Another approach presented here is based on the approximate representation of the model via polynomial chaos expansion. A quantitative and qualitative assessment of the approaches is performed in comparison to the Monte Carlo simulation technique that uses the nonlinear model. It is shown that the power series and polynomial chaos expansion based approaches require a significantly lower computational burden compared to Monte Carlo approaches, while give good approximation of the shape of the distribution. The techniques are applied to the crystallization of an inorganic chemical with uncertainties in the nucleation and growth parameters.

A study of global–local higher-order theories for laminated composite plates

In this paper the special higher-order global–local theories including the mth-order polynomial of global thickness coordinate z and 1, 2-3 order power series of local thickness coordinate ζ k are studied. These theories satisfy the free surface conditions and the geometric and stress continuity conditions at interfaces. Moreover, the number of unknowns is independent of the layer numbers of the laminate. Main aim of this paper is to study the relationship between the order of the global component in mth-order theory and the accuracy of solution. Numerical solutions show the third-order global–local higher-order theory (namely 1, 2-3 double-superposition theory called by Li and Liu [Li X, Liu D. Generalized laminate theories based on double-superposition hypothesis. Int J Numer Meth Eng 1997;40:1197–212] will encounter difficulties when accurately predicting transverse shear stresses as the number of layers of laminated plates is more than five. In particular, the accuracy of this theory drops when used for non-symmetric multilayered plates. However, other higher-order global–local theories can present reasonable in-plane stresses and transverse shear stresses by the direct constitutive equation approach. Neglecting transverse normal strains in the present global–local theories, the transverse normal stresses can be only predicted from equilibrium equation approach.

Dynamic buckling of flat and curved sandwich panels with transversely compressible core

In the present study, the effect of the transverse compressibility of the core on the transient dynamic response of structural sandwich panels under rapid loading conditions is investigated. The analysis is based on a higher-order sandwich shell theory in an effective multilayer formulation. The model is based on the standard Kirchhoff–Love hypothesis for the face sheets whereas a first/second order power series expansion is employed for the core. Consistent equations of motion and boundary conditions are derived by means of Hamilton’s principle. An analytical solution is obtained by an extended Galerkin procedure. The theory is applied to the dynamic buckling and postbuckling analyses of plane and curved sandwich panels subjected to rapidly applied tangential and transverse loads. It is observed that the transverse compressibility of the core can have distinct effects on both, the frequency and the amplitudes of the resulting free oscillations. Due to interactions between the overall oscillation and a local oscillation in the presence of a face wrinkling instability mode, the dynamic response can be chaotic, resulting in oscillating face wrinkling instability modes with unpredictable amplitudes.

Boley’s method for two-dimensional thermoelastic problems applied to piezoelastic structures

We study a composite piezoelastic plate in cylindrical bending. The plate is composed of perfectly bonded substrate and piezoelastic layers. We assume a plane state of strain; the plate mid-plane deforms into a cylindrical surface perpendicular to, and the electric field vector lies in, the ( x, z)-plane. We utilize Bruno Boley’s method for two-dimensional thermoelastic problems: Boley introduced an expansion of the Airy stress function and a step-by-step solution for each term. Furthermore, he discussed relations to strength-of-material theories. For the piezoelastic problem we apply Boley’s method to the charge equations of electrostatics. The electric potential is expanded and each term is calculated using a step-by-step procedure. Use of Boley’s method is facilitated by the capabilities of modern symbolic computer codes. We solve the problem for an arbitrary distribution of strain first; then we consider the variation of displacements in the form of a third order power series expansion in the z-direction. Strength-of-material theories of different approximation levels are finally extracted, for which the level of approximation for the mechanical and electric field is not independent.

On the behaviour of the ANM continuation in the presence of bifurcations

AbstractThe asymptotic‐numerical method (ANM) is a path following technique which is based on high order power series expansions. In this paper, we analyse its behaviour when it is applied to the continuation of a branch with bifurcation points. We show that when the starting point of the continuation is near a bifurcation, the radius of convergence of the power series is exactly the distance from the starting point to the bifurcation. This leads to an accumulation of small steps around the bifurcation point. This phenomenon is related to the presence of inevitable imperfections in the FE models. We also explain that, depending on the maximal tolerated residual error (out‐of‐balance error), the ANM continuation may continue to follow the fundamental path or it may turn onto the bifurcated path without applying any branch switching technique. Copyright © 2003 John Wiley &amp; Sons, Ltd.

A nonlinear theory for doubly curved anisotropic sandwich shells with transversely compressible core

In the present paper, an advanced geometrically nonlinear shell theory of doubly curved structural sandwich panels with transversely compressible core is presented. The model is based on the adoption of the Kirchhoff theory for the face sheets and a second/third order power series expansion for the core displacements. The theory accounts for dynamic effects as well as for initial geometric imperfections. In the v. Kármán sense, large displacement theory is employed with respect to the transverse direction while the displacement gradients with respect to the tangential directions are assumed to be small. The equations of motion are derived by means of Hamilton’s principle and hold valid for all types of elastic and elastic–plastic material models. The theory is illustrated by an analysis of the elastic buckling and postbuckling behavior of flat and curved sandwich panels using an extended Galerkin scheme. Owing to the assumed transverse flexibility of the core, both the global and the local (face wrinkling) instability modes can be addressed.

Effect of iron core loss nonlinearity on chaotic ferroresonance in power transformers

This paper investigates the effect of iron core loss nonlinearity on the onset of chaotic ferroresonance and duration of transient chaos in a power transformer. The transformer chosen for investigation has a rating of 50 MV A, 635.1 kV, the data for which is given by Dommel et al. (Tutorial Course on Digital Simulation of Transients in Power Systems (Chapter 14), IISc, Bangalore, 1983, pp. 17–38). The magnetisation characteristics of the transformer is modelled by a single-value two-term polynomial. The core loss is modelled by a third order power series in voltage. With nonlinearities in core loss included, three effects are clear: (i) onset of chaos at larger values of open phase voltage, (ii) shorter duration of transient chaos, and (iii) less susceptibility to ‘ jump ’ phenomena.