Order Perturbation Technique Research Articles

Exact calculation of the time convolutionless master equation generator: Application to the nonequilibrium resonant level model.

The generalized quantum master equation provides a powerful tool to describe the dynamics in quantum impurity models driven away from equilibrium. Two complementary approaches, one based on Nakajima-Zwanzig-Mori time-convolution (TC) and the other on the Tokuyama-Mori time-convolutionless (TCL) formulations provide a starting point to describe the time-evolution of the reduced density matrix. A key in both approaches is to obtain the so called "memory kernel" or "generator," going beyond second or fourth order perturbation techniques. While numerically converged techniques are available for the TC memory kernel, the canonical approach to obtain the TCL generator is based on inverting a super-operator in the full Hilbert space, which is difficult to perform and thus, nearly all applications of the TCL approach rely on a perturbative scheme of some sort. Here, the TCL generator is expressed using a reduced system propagator which can be obtained from system observables alone and requires the calculation of super-operators and their inverse in the reduced Hilbert space rather than the full one. This makes the formulation amenable to quantum impurity solvers or to diagrammatic techniques, such as the nonequilibrium Green's function. We implement the TCL approach for the resonant level model driven away from equilibrium and compare the time scales for the decay of the generator with that of the memory kernel in the TC approach. Furthermore, the effects of temperature, source-drain bias, and gate potential on the TCL/TC generators are discussed.

Stochastic thermal buckling analysis of laminated plates using perturbation technique

Composites are known to display a considerable amount of scatter in their material properties due to large number of parameters involved with the manufacturing and fabrication processes. This paper is concerned with the effect of random system properties on critical thermal buckling temperature of composite laminated plates with temperature dependent properties using micromechanical approach. System parameters are assumed as independent random variables. In this analysis, based on the classical lamination theory in conjunction with the Hamilton’s principle, the basic formulation of random eigenvalue problem has been deduced. A mean-centered first order perturbation technique is used to compute the second-order statistics (mean and standard deviation) of the critical thermal buckling temperature. The performance of outlined stochastic approach has been validated by comparing the present results with those available in the literature and independent Monte Carlo simulation. The effect of random material properties, thermal expansion coefficients, fiber volume fractions, aspect ratios, laying angels and boundary conditions on the critical thermal buckling temperature are presented.

Stochastic fracture analysis of laminated composite plate with arbitrary cracks using X-FEM

In this paper, second order statistics of mixed mode stress intensity factors (MSIFs) and crack propagation analysis of the symmetric angle ply laminated composite plate with through thickness arbitrary curve cracks subjected to tensile and shear stress is presented. The fracture behaviour is analysed using extended finite element method (X-FEM). The cracks like line, semi elliptical, semi circular and arbitrary curves are considered for the detailed numerical study. The material properties, lamination angle, loading, crack width and crack depth are modelled as independent, combine uncorrelated and correlated input random Gaussian variables. The interaction integral (M-integral) is adopted for calculating the MSIFs. The second order perturbation technique and Monte Carlo simulations are proposed to obtain the mean and coefficient of variance of MSIFs by random change in input system parameters. This work signifies the accurate and realistic evaluation of fracture response by handling the various levels of uncertainties. The effect of crack propagation on MSIFs using tensile and shear stresses using global tracking algorithm is also highlighted.

Stochastic Thermal Post Buckling Response of Elastically Supported Laminated Piezoelectric Composite Plate Using Micromechanical approach

Abstract In this paper, second order statistics of thermally induced post buckling response of elastically supported piezoelectric laminated composite plate using micromechanical approach is examined. A Co finite element has been used for deriving eigenvalue problem using higher order shear deformation theory (HSDT) with von-Karman nonlinearity. The uncertain system properties such as material properties of fiber and matrix of composite and piezoelectric, fiber volume fraction, plate thickness, lamination angle and foundation are modeled as random variables. The temperature field considered to be uniform temperature distributions through the plate thickness. A direct iterative based nonlinear finite element method combined with mean-centered second order perturbation technique (SOPT) is used to find the mean and coefficient of variance of the post buckling temperature. The effects of volume fraction, fiber orientation, and length to thickness ratio, aspect ratios, foundation parameters, position and number of piezoelectric layers, amplitude and boundary conditions with random system properties on the critical temperature are analysed. It is found that small amount of variations of uncertain system parameters of the composite plate significantly affect the initial and post buckling temperature of laminated composite plate. The results have been validated with independent Monte Carlo simulation (MCS) and those available in literature.

Multi-Component Diffusion Coefficients in Nitrogen Plasma Under Thermal Equilibrium and Non-equilibrium Conditions

Multi-component diffusion coefficients are calculated for a seven species model of nitrogen plasma under thermal non-equilibrium following the first order perturbation technique of Chapman and Enskog. Binary, thermal, thermal ambipolar, general and general ambipolar diffusion coefficients are presented over electron temperatures ranging from 300 to 50,000 K and thermal non-equilibrium parameter (Te/Th) ranging from 1 to 5. Considering large volume of data, binary, general and general ambipolar diffusion coefficients are presented only for atmospheric pressure. Thermal and thermal ambipolar diffusion coefficients are presented for pressures ranging from 0.1 to 2 atm. The results are compared with published experimental and theoretical data. Necessary electronic levels, associated transition data and collision integrals are collected from recent literature. Details of behaviour of each of the coefficients are presented.

Analytical modeling of cross-phase modulation in coherent fiber-optic system

An analytical model to calculate the variance of cross-phase modulation (XPM) distortion in a wavelength-division multiplexed (WDM) fiber-optic system is developed. The method is based on the first order perturbation technique and it is applicable for both dispersion managed and dispersion uncompensated systems. For dispersion managed systems, it is shown that the variance of XPM distortion scales as Nx where N is the number of spans and x ∈ [1, 2] depending on the amount of inline-dispersion compensation. The analytical model is found to be in good agreement with simulations in most of the cases.

Stochastic finite element nonlinear free vibration analysis of piezoelectric functionally graded materials beam subjected to thermo-piezoelectric loadings with material uncertainties

In this paper, second order statistics of large amplitude free flexural vibration of shear deformable functionally graded materials (FGMs) beams with surface-bonded piezoelectric layers subjected to thermopiezoelectric loadings with random material properties are studied. The material properties such as Young’s modulus, shear modulus, Poisson’s ratio and thermal expansion coefficients of FGMs and piezoelectric materials with volume fraction exponent are modeled as independent random variables. The temperature field considered is assumed to be uniform and non-uniform distribution over the plate thickness and electric field is assumed to be the transverse components E z only. The mechanical properties are assumed to be temperature dependent (TD) and temperature independent (TID). The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics. A C 0 nonlinear finite element method (FEM) based on direct iterative approach combined with mean centered first order perturbation technique (FOPT) is developed for the solution of random eigenvalue problem. Comparison studies have been carried out with those results available in the literature and Monte Carlo simulation (MCS) through normal Gaussian probability density function.

Transformed perturbation stochastic finite element method for static response analysis of stochastic structures

To obtain the probability density functions and the cumulative distribution functions of static responses of stochastic structures, a hybrid stochastic method named as the transformed perturbation stochastic finite element method (TPSFEM) is proposed. In TPSFEM, the static responses of stochastic structures are approximated as the linear functions of random variables by using the first order perturbation technique. According to the approximated linear relationships between static responses and random variables, the probability density functions of static responses are obtained by the change-of-variable technique. The cumulative distribution functions of static responses are calculated by the numerical integration method. The numerical examples on a thin plate, a six-bar truss structure, a Mindlin plate and a shell structure verify the effectiveness and accuracy of the proposed method. Hence, the proposed method can be considered as an alternative engineering method for the static response analysis of stochastic structures.

Statistical Analysis of Basic Dynamic Characteristics of Large Span Cable-Stayed Bridge Based on High Order Perturbation Stochastic FEM

With the quick development of transportation in China, long span cable-stayed bridges have been widely constructed to cross large rivers and deep canyons attribute to their unique structural forms and mechanical properties. For large-scale cable-stayed bridges, how to determine their basic dynamic characteristics or modal data is an important issue on the dynamic analysis of bridge structures. Noting the uncertainty of structural parameters or geometric shape, it is also of significance to find out the statistic dynamic characteristics of uncertain long span cable-stayed bridges. At the present paper, different from traditional low order perturbation stochastic approach, a high order perturbation stochastic FEM called as recursive stochastic finite element method was employed, which is a good practical approximation method compared to the Monte Carlo simulation method. Taking Edong Yangtze River Bridge as a research objective, the recursive stochastic finite element method was employed to solve the modal data of large span cable-stayed bridge with random parameters. Firstly, in the non-orthogonal polynomial presentation, natural frequencies and mode shapes of the random cable-stayed bridge were expressed. Then coefficients of the non-orthogonal polynomial expansion are able to be determined through high order perturbation technique. On the basis of the coefficients obtained, the probability density functions and statistic characteristics of the modal data were straightforward to implement. In accordance with detailed analyzations and discussions on the basic dynamic characteristics of the cable-stayed bridge, it can be found that the randomness of sectional flexural stiffness of the deck beam and bridge towers is very sensitive to many modes of Edong Yangtze River Bridge. However, the elastic stiffness simulating the pile-soil interaction has less influence on the in-plane modes.

Stochastic Nonlinear Free Vibration Analysis of Functionally Graded Beam Subjected to Thermal Loading with Random Material Properties

This paper deals with the stochastic nonlinear free vibration response of functionally graded materials (FGMs) beam subjected to thermal loadings with uncertain material properties subjected to uniform and nonuniform temperature changes with temperature independent (TID) and dependent (TD) material properties. System properties such as material properties of each constituents material and volume fraction index are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strains using modified C0 continuity. A direct iterative based nonlinear finite element method in conjunction with first order perturbation technique (FOPT) is used for FGMs beam to compute the second order statistics (mean and coefficient of variation) of the nonlinear fundamental frequency. The present outlined approach has been validated with the results available in literatures and independent Monte Carlo simulation (MCS).

Stochastic nonlinear bending response of piezoelectric functionally graded beam subjected to thermoelectromechanical loadings with random material properties

This study deals with the stochastic nonlinear bending response of functionally graded materials (FGMs) beam with surface bonded piezoelectric layers subjected to thermoelectromechanical loadings with uncertain material properties. The random material properties FGMs and piezoelectric modeled as basic random variables. The temperature field considered is assumed to be uniform and non-uniform distribution over the plate thickness and electric field is assumed to be the transverse components Ez only. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematic. A direct iterative based C0 nonlinear finite element method combined with first order perturbation technique (FOPT) is used to compute the second order statistics (mean and coefficient of variation) of the nonlinear transverse central deflection. The effects of temperature dependent and independent material properties; slenderness ratios, volume fraction index, boundary conditions, uniform and nonuniform temperature distribution and surface bonded piezoelectric layers acting top and/or bottom of the FGM beam and thermoelectromechanical loadings with random material properties on the nonlinear transverse central deflection have been presented in detail through parametric studies. The results have been validated with independent Monte Carlo simulation and those available in literature.

Stochastic Nonlinear Free Vibration Analysis of Piezolaminated Composite Conical Shell Panel Subjected to Thermoelectromechanical Loading With Random Material Properties

This paper presents the effect of randomness in material properties on piezolaminated composite geometrically nonlinear conical shell panel subjected to thermoelectromechanical loading acting simultaneously or individually. Material properties such as modulus ratio, Poisson’s ratio, and thermal expansion coefficients are modeled as independent random variables. The temperature field considered is assumed to be a uniform distribution over the shell panel surface and through the shell thickness and the electric field is assumed to be the transverse component Ez only. It is assumed that the mechanical properties do not depend on temperature and electric fields. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinearity. A C0 nonlinear finite element model based on direct iterative approach in conjunction with mean centered first order perturbation technique (FOPT) used by the present author for plate is now extended for conical shell panel to solve a random nonlinear generalized eigenvalue problem. Parametric studies are carried out to examine the effect of amplitude ratios, stacking sequences, cone angles, circumferential length to thickness ratios, piezoelectric layers, applied voltages, change in temperature, types of thermoelectromechanical loadings, and support boundary conditions on the dimensionless mean and coefficient of variance (COV) of laminated conical shell panels. The present outlined approach has been validated with those available results in literature and independent Monte Carlo simulation (MCS).

Post buckling response of functionally graded materials plate subjected to mechanical and thermal loadings with random material properties

In this paper, the second order statistics of post buckling response of functionally graded materials plate (FGM) subjected to mechanical and thermal loading with nonuniform temperature changes subjected to temperature independent (TID) and dependent (TD) material properties is examined. Material properties such as material properties of each constituent’s materials, volume fraction index are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear kinematic using modified C0 continuity. A direct iterative based C0 nonlinear finite element method (FEM) combined with mean centered first order perturbation technique (FOPT) proposed by last two authors for the composite plate is extended for Functionally Graded Materials (FGMs) plate with reasonable accuracy to compute the second order statistics (mean and coefficient of variation) of the post buckling load response of the FGM plates. The effect of random material properties with amplitude ratios, volume fraction index, plate thickness ratios, aspect ratios, boundary conditions and types of loadings subjected to TID and TD material properties are presented through numerical examples. The performance of outlined present approach is validated with the results available in literatures and independent Monte Carlo simulation (MCS).

FEM model for stochastic mechanical and thermal postbuckling response of functionally graded material plates applied to panels with circular and square holes having material randomness

In this study, the second order statistics of postbuckling analysis of functionally graded materials (FGMs) plates subjected to mechanical and thermal loadings without and with square and circular holes at center having random material properties is presented. Material properties of each constituent's materials, volume fraction index, thermal expansion coefficients and thermal conductivities are modeled as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) using modified C0 continuity. A nonlinear finite element method (FEM) based on direct iterative technique combined with mean centered first order perturbation technique (FOPT) developed by the author for composite plate is extended for FGM plates to solve the random nonlinear eigenvalue problem. Typical numerical results are presented to examine the effect of volume fractions index, plate length to thickness ratios, plate aspect ratios, types of loadings, amplitude ratios, support conditions and various shaped and sized holes with random thermomechanical properties. The results obtained by the present solution approach are validated with those available in the literatures and independent Monte Carlo Simulation (MCS). It is observed that the plates with circular and square hole have a significant influence on the postbuckling response under mechanical and thermal loading conditions and some new results are presented to demonstrate the applications of present work.

Stochastic nonlinear failure analysis of laminated composite plates under compressive transverse loading

This paper present the second ordered statistics of first-ply failure response of laminated composite plate with random material properties under random loading. The basic formulation is based on higher order shear deformation plate theory (HSDT) with the geometrically nonlinearity in the von-Karman. The direct iterative based C 0 nonlinear finite element method combined with mean centered first order perturbation technique developed by the authors are extended and successfully applied nonlinearity for failure problem with a reasonable accuracy to predict the second order statistics (standard deviation) of first-ply failure response using Tsai–Wu and Hoffman failure criterion with macroscopic analysis. Typical numerical results for various combinations of boundary conditions, plate thickness ratios, aspect ratios, laminates scheme and layers, elastic modulus ratios have been presented to illustrate the application of developed procedure. Some new results are presented and examined which clearly demonstrated the importance of the randomness in the system parameters in the failure response of the structures subjected to transverse loadings.

Hygrothermal Buckling Response of Laminated Composite Plates with Random Material Properties: Micro-Mechanical Model

The aim of this paper is to find out the randomness in the material properties on the buckling of laminated composite plate needed for the economy, safety and reliability of the structures and components in their operational life especially for sensitive Aerospace Engineering applications in hygrothermal environments. Micromechanical model has been taken for the analysis .The used methodology is a C0 finite element method based on higher-order shear deformation plate theory for deriving the standard eigenvalue problem. A Taylor series based mean-centered first order perturbation technique is used to find out the second order statistics of the hygrothermal buckling loads under different sets of environmental conditions..The numerical results for deterministic parameters are compared and validated with available literature and random parameters with independent Monte Carlo Simulation. The result shows that the plate is significantly affected by the hygrothermal buckling load.

Stochastic nonlinear free vibration analysis of elastically supported functionally graded materials plate with system randomness in thermal environment

This paper presents the stochastic nonlinear free vibration response of elastically supported functionally graded materials (FGMs) plate resting on two parameter Pasternak foundation having Winkler cubic nonlinearity with random system properties subjected to uniform and nonuniform temperature changes with temperature independent (TID) and dependent (TD) material properties. System properties such as material properties of each constituent’s material, volume fraction index and foundation parameters are taken as independent random input variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strains using modified C 0 continuity. A direct iterative based nonlinear finite element method in conjunction with first order perturbation technique (FOPT) developed by last two authors for the composite plate is extended for FGM plate to compute the second order statistics (mean and coefficient of variation) of the nonlinear fundamental frequency. The present outlined approach has been validated with those results available in the literature and independent Monte Carlo simulation (MCS).

Effect of random system properties on bending response of thermo-mechanically loaded laminated composite plates

In this paper, effect of random variation in system properties on bending response of geometrically linear laminated composite plates subjected to transverse uniform lateral pressure and thermal loading is examined. System parameters such as the lamina material properties, expansion of thermal coefficients, lamina plate thickness and lateral load are modeled as basic random variables. The basic formulation is based on higher order shear deformation theory to model the system behavior of the composite plate. A C 0 finite element method in conjunction with the first order perturbation technique procedure developed earlier by authors for the plate subjected to lateral loading is employed to obtain the second order response statistics (mean and variance) of the transverse deflection of the plate. Typical numerical results for the second order statistics of the transverse central deflection of geometrically linear composite plates with temperature independent and dependent material properties subjected to uniform temperature and combination of uniform and linearly varying temperature distribution are obtained for various combinations of geometric parameters, uniform lateral pressures, staking sequences and boundary conditions. The performance of the stochastic laminated composite model is demonstrated through comparison of mean transverse central deflection with those results available in literature and standard deviation of the deflection with an independent Monte Carlo simulation.

General formula for phonon-assisted n-photon absorption in solids

A general formula for phonon-assisted n-photon absorption in solids is obtained by (n + 1)-th order perturbation technique. The complicated calculation process for transition element of n-photon absorption is simply demonstrated by a diagram approach that is proposed in this work. We find that the transition element for the n-photon absorption has a simple form, i.e., it is just the first term of the n-th order fist kind Bessel function.

Modeling the lateral pedestrian force on a rigid floor by a self-sustained oscillator

The main goal of this paper is the definition of a nonlinear single-degree-of-freedom oscillator able to accurately predict the lateral walking force of a pedestrian. The force exerted on the floor corresponds to its restoring force. The rigid floor case is analyzed, leading to an autonomous oscillator. Even though such an oscillator is a simplified representation of the human body, it should be able to reproduce two experimentally observed phenomena: (i) the time-history of lateral force is an approximately periodic signal; (ii) the walking motion is self-sustained, in the sense that the pedestrian/oscillator produces by itself the energy needed to sustain its motion. This implies that such an oscillator must be self-sustained. In addition, the self-sustained character entails that the autonomous oscillation has a natural amplitude and frequency, representing the natural walking amplitude and frequency of the pedestrian. An original model is proposed by modifying the so-called hybrid Van der Pol/Rayleigh oscillator, already used for applications in the field of robotics. A dynamic analysis of this oscillator is then performed through an energetic approach and a perturbation technique in order to get the stable limit cycle. The model parameters are finally identified from the experimental force signals, resulting from a test campaign on a population of 12 pedestrians: the agreement between model and experimental results is very good.