First-order Perturbation Expansion Research Articles

NURBS-based Isogeometric Analysis and Refined Plate Theory Application on a Functionally Graded Plate Subjected to Random Loads

In structural design standards, loads are often assumed to be random quantities to give load factors. This study deals with the Stochastic Isogeometric Analysis (SIGA) for a Functionally Graded Plate (FGP) subjected to random distribution loads. The spatial random variation of distribution loads is modeled as a homogeneous Gaussian random field in the plane of the functionally graded plate. The governing equation of the functional grade plate is derived using the NURBS-based isogeometric analysis and the refined plate theory. SIGA is developed based on standard NURBS-based isogeometric analysis in conjunction with the first-order perturbation expansions of random loads. This approach was verified with Monte Carlo simulation, and the numerical results showed the effect of random loads on the variation of displacements and stresses of the functionally graded plate.

Perturbation expansions and error bounds for the truncated singular value decomposition

Truncated singular value decomposition is a reduced version of the singular value decomposition in which only a few largest singular values are retained. This paper presents a novel perturbation analysis for the truncated singular value decomposition for real matrices. First, we describe perturbation expansions for the singular value truncation of order r. We extend perturbation results for the singular subspace decomposition to derive the first-order perturbation expansion of the truncated operator about a matrix with rank greater than or equal to r. Observing that the first-order expansion can be greatly simplified when the matrix has exact rank r, we further show that the singular value truncation admits a simple second-order perturbation expansion about a rank-r matrix. Second, we introduce the first-known error bound on the linear approximation of the truncated singular value decomposition of a perturbed rank-r matrix. Our bound only depends on the least singular value of the unperturbed matrix and the norm of the perturbation matrix. Intriguingly, while the singular subspaces are known to be extremely sensitive to additive noises, the newly established error bound holds universally for perturbations with arbitrary magnitude. Finally, we demonstrate an application of our results to the analysis of the mean squared error associated with the TSVD-based matrix denoising solution.

Perturbation and numerical solutions of non‐Newtonian fluid bounded within in a porous channel: Applications of pseudo‐spectral collocation method

In the current study, the effects of fluidic parameters with entropy generation properties on velocity, temperature, and entropy numbers of non-Newtonian fluid flowing through the porous channel are investigated. The complex system of fluid equations is handled with analytically and numerically. The first-order perturbation expansion is employed on both velocity and temperature to obtain the approximate analytical solution. A comparison of the analytical solution is made with numerical results that are obtained by discretizing the system of boundary value problems. The pseudo-spectral collocation method was used for the discretization, and the Newton method was to get the solutions to the complex differential equations. In the Newton method, the finite difference approximation of Jacobian is utilized. The pseudo-spectral solutions are in good agreement with the analytical findings. The order of accuracy in temperature and velocity profiles is of order 10−6 which will be compared in the future with the experimental results of given non-Newtonian fluid.

Adjoint-based parametric sensitivity analysis for swirling M-flames

A linear numerical study is conducted to quantify the effect of swirl on the response behaviour of premixed lean flames to general harmonic excitation in the inlet, upstream of combustion. This study considers axisymmetric M-flames and is based on the linearised compressible Navier–Stokes equations augmented by a simple one-step irreversible chemical reaction. Optimal frequency response gains for both axisymmetric and non-axisymmetric perturbations are computed via a direct–adjoint methodology and singular value decompositions. The high-dimensional parameter space, containing perturbation and base-flow parameters, is explored by taking advantage of generic sensitivity information gained from the adjoint solutions. This information is then tailored to specific parametric sensitivities by first-order perturbation expansions of the singular triplets about the respective parameters. Valuable flow information, at a negligible computational cost, is gained by simple weighted scalar products between direct and adjoint solutions. We find that for non-swirling flows, a mode with azimuthal wavenumber $m=2$ is the most efficiently driven structure. The structural mechanism underlying the optimal gains is shown to be the Orr mechanism for $m=0$ and a blend of Orr and other mechanisms, such as lift-up, for other azimuthal wavenumbers. Further to this, velocity and pressure perturbations are shown to make up the optimal input and output showing that the thermoacoustic mechanism is crucial in large energy amplifications. For $m=0$ these velocity perturbations are mainly longitudinal, but for higher wavenumbers azimuthal velocity fluctuations become prominent, especially in the non-swirling case. Sensitivity analyses are carried out with respect to the Mach number, Reynolds number and swirl number, and the accuracy of parametric gradients of the frequency response curve is assessed. The sensitivity analysis reveals that increases in Reynolds and Mach numbers yield higher gains, through a decrease in temperature diffusion. A rise in mean-flow swirl is shown to diminish the gain, with increased damping for higher azimuthal wavenumbers. This leads to a reordering of the most effectively amplified mode, with the axisymmetric ($m=0$) mode becoming the dominant structure at moderate swirl numbers.

A simple velocity random-walk model for macrodispersion in mildly to highly heterogeneous subsurface formations

Abstract Recently, we have analytically derived a temporal velocity random-walk model for macrodispersion or transport uncertainty quantification based on velocity statistics from classical first-order perturbation expansions. The applicability of these expansions is limited to mildly heterogeneous formations with log-conductivity variances σ Y 2 1 . In this work, we reformulate our model in order to account for velocity skewness and inter-component correlation that emerge as key drivers of non-Fickian dispersion at elevated heterogeneity levels σY. Eventually, we arrive at a light-weight parametrization of macrodispersion that is consistent with our earlier formulation, but at the same time is applicable for formations with multi-Gaussian log-conductivity characterization of variable heterogeneity, i.e. σY from 0 to 1 and beyond.

Generalized Least Squares for ESPRIT-Type Direction of Arrival Estimation

The key task in ESPRIT-based parameter estimation is finding the solution to the shift invariance equation (SIE), which is often an overdetermined, linear system of equations. Additional structure is imposed if the two selection matrices, applied to an estimate of the signal subspace, overlap such that the subspace estimation errors on both sides of the SIE are highly correlated. In this letter, we propose a novel SIE solution for Standard ESPRIT and Unitary ESPRIT based on generalized least squares (GLS), assuming a uniform linear array (ULA) and maximum subarray overlap. GLS directly incorporates the statistics of the subspace estimation error via its covariance matrix, which is found analytically by a first-order perturbation expansion. As the subspace error covariance matrix is not invertible, we introduce a regularization with a clever choice of the regularization parameter. The resulting GLS-based Standard ESPRIT and Unitary ESPRIT algorithms achieve a superior performance over existing ESPRIT-type methods and almost attain the Cramer-Rao bound (CRB).

Relating recent random walk models with classical perturbation theory for dispersion predictions in the heterogeneous porous subsurface

Recently developed stochastic macro-dispersion models enable computationally inexpensive flow and transport predictions for highly heterogeneous formations with statistically non-stationary conductivity and flow statistics. So far, the random processes at the heart of these models have been calibrated numerically based on velocity statistics from Monte Carlo simulation studies. In this work, we provide a more rigorous foundation to some of these models by analytically relating the process definition to given velocity statistics from classical first-order perturbation expansions.

Highly responsive ac scalar atomic magnetometer with long relaxation time

We propose an operating condition for an optically pumped scalar atomic magnetometer with a unique band-pass frequency response. Under this operating condition, the signal response of the magnetometer is increased threefold around a nonzero resonance frequency. The resonance frequency is equal to the detuning between the Larmor frequency and the frequency of the oscillating magnetic field used for excitation. The resonance is explained in terms of the characteristics of the first-order perturbation expansion of the Bloch equation for a modulated bias magnetic field. The scalar magnetometer provides an enhanced magnetic response around its resonance frequency even beyond the cutoff frequency, which is defined as the inverse of the phase relaxation time of the spin polarization.

Sensitivity analysis for the multivariate eigenvalue problem

This paper concerns with the sensitivity analysis for the multivariate eigenvalue problem (MEP). The concept of a simple multivariate eigenvalue of a matrix is generalized to the MEP and the first-order perturbation expansions of a simple multivariate eigenvalue and the corresponding multivariate eigenvector are presented. The explicit expressions of condition numbers, perturbation upper bounds and backward errors for multivariate eigenpairs are derived.

Perturbation, extraction and refinement of invariant pairs for matrix polynomials

Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefits can be expected for polynomial eigenvalue problems, for which the concept of an invariant subspace needs to be replaced by the concept of an invariant pair. Little has been known so far about numerical aspects of such invariant pairs. The aim of this paper is to fill this gap. The behavior of invariant pairs under perturbations of the matrix polynomial is studied and a first-order perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe efficient refinement procedures directly based on the polynomial formulation. Numerical experiments with matrix polynomials from a number of applications demonstrate the effectiveness of our extraction and refinement procedures.

Regular Perturbation Approach to Nonsteady Drift of the Cox-Ingersoll-Ross Model

The nonsteady drift of the Cox-Ingersoll-Ross model is obtained by noticing that the affine term-structures for small volatility is a perturbation of the Hull-White affine term-structure. Fast mean reversion ensures that a first-order perturbation expansion is sufficient when the volatility is not small.

Perturbation Hamiltonians in heterogeneous anisotropic weakly dissipative media

SUMMARY Perturbation Hamiltonians simplify considerably the solution of various traveltime perturbation problems of seismic body waves propagating in heterogeneous, isotropic or anisotropic, weakly dissipative media. In addition to the phase-space coordinates, representing the Cartesian coordinates and Cartesian components of the slowness vector, the perturbation Hamiltonians also depend on one or more dimensionless perturbation parameters, which can be chosen in different ways. General traveltime perturbation procedures, based on the perturbation Hamiltonians, were proposed by Klimes. The evaluation of the perturbation expansions for the traveltime and for its derivatives does not require the computation of perturbed rays in these procedures. It consists in dynamic ray tracing along the reference ray, in the determination of the partial derivatives of the perturbation Hamiltonian with respect to the perturbation parameters, and in additional quadratures along a reference ray, computed in the reference medium. Different forms of the perturbation Hamiltonian yield different perturbation expansions with different accuracy and numerical efficiency of computations. In this paper, the special case of the perturbation Hamiltonian, herein referred to as the linear perturbation Hamiltonian, is used to compute the second-order perturbation expansion of the complex-valued traveltime and the first-order perturbation expansion of the complex-valued traveltime gradient. The advantage of the linear perturbation Hamiltonian is that it is linear with respect to perturbation parameters. Heterogeneous, perfectly elastic, isotropic or anisotropic, reference media are considered, in which reference rays can be calculated without problems by well-known methods. The derived perturbation expansions are valid very generally. For the isotropic reference model, the perturbed medium may be isotropic perfectly elastic, isotropic weakly dissipative, weakly anisotropic perfectly elastic or weakly anisotropic weakly dissipative. For the anisotropic reference model, the perturbed medium may be anisotropic perfectly elastic, or anisotropic weakly dissipative. Expressions for the attenuation vector, the dissipation filter, the global absorption factor, the direction-dependent scalar local quality factor and the attenuation coefficient are also derived for seismic body waves propagating in heterogeneous anisotropic weakly dissipative media.

Strongly coupled overdamped pendulums

It is shown, by a first-order perturbation expansion, that the dimensionality of the dynamical equations for the angular variables of two strongly coupled identical overdamped pendulums can be reduced from two to one. The resulting dynamical equation is seen to be similar to the one of a single pendulum with an additional fictitious torque characterized by a second harmonic contribution.

Numerical comparison of the isotropic-common-ray and anisotropic-common-ray approximations of the coupling ray theory

In the common-ray approximation of the coupling ray theory, the S-wave traveltimes in the coupling equations are approximated by the first-order perturbation expansion from the common reference ray. The common-ray approximation eliminates problems with ray tracing through S-wave singularities and also considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in traveltimes due to the perturbation from the common reference ray. The errors of S-wave traveltimes due to the common-ray approximation may be approximated by the second-order terms in the perturbation expansion. The first-order and second-order terms in the perturbation expansion can be calculated by simple numerical quadratures along the common reference ray. The common reference ray may be represented by the isotropic common ray or by the anisotropic common ray. The anisotropic-common-ray approximation of the coupling ray theory is more accurate than the isotropic-common-ray approximation. The equations for the first-order and second-order perturbation expansions of traveltime from both the isotropic and anisotropic common rays to the anisotropic-ray-theory rays have already been derived. We numerically test the equations in three smooth 1-D velocity models of differing degrees of anisotropy. We first compare the second-order perturbation expansions of traveltime from the isotropic common rays to the anisotropic common rays and vice versa. This comparison not only demonstrates the accuracy of the perturbation expansions, but also allows us to estimate the order of magnitude of the third-order, fourth-order and fifth-order terms in the perturbation expansions. The first-order and second-order perturbation expansions of traveltime from the isotropic common rays and from the anisotropic common rays to the anisotropic-ray-theory rays are then compared with the correct anisotropic-ray-theory traveltimes in order to illustrate the accuracy of the perturbation expansions and reliability of the error estimates. Finally, the synthetic seismograms obtained by the isotropic-common-ray and anisotropic-common-ray approximations of the coupling ray theory are compared with the more accurate coupling-ray-theory synthetic seismograms simulated by the second-order perturbation expansion of traveltime from the anisotropic common rays. In the numerical examples, the errors of the anisotropic-common-ray approximation of the coupling ray theory are considerably smaller than the errors of the isotropic-common-ray approximation.

Numerical comparison of the isotropic-common-ray and anisotropic-common-ray approximations of the coupling ray theory

SUMMARY In the common-ray approximation of the coupling ray theory, the S-wave traveltimes in the coupling equations are approximated by the first-order perturbation expansion from the common reference ray. The common-ray approximation eliminates problems with ray tracing through S-wave singularities and also considerably simplifies the numerical algorithm of the coupling ray theory for S waves, but may introduce errors in traveltimes due to the perturbation from the common reference ray. The errors of S-wave traveltimes due to the common-ray approximation may be approximated by the second-order terms in the perturbation expansion. The first-order and second-order terms in the perturbation expansion can be calculated by simple numerical quadratures along the common reference ray. The common reference ray may be represented by the isotropic common ray or by the anisotropic common ray. The anisotropic-common-ray approximation of the coupling ray theory is more accurate than the isotropic-common-ray approximation. The equations for the first-order and second-order perturbation expansions of traveltime from both the isotropic and anisotropic common rays to the anisotropic-ray-theory rays have already been derived. We numerically test the equations in three smooth 1-D velocity models of differing degrees of anisotropy. We first compare the second-order perturbation expansions of traveltime from the isotropic common rays to the anisotropic common rays and vice versa. This comparison not only demonstrates the accuracy of the perturbation expansions, but also allows us to estimate the order of magnitude of the third-order, fourth-order and fifth-order terms in the perturbation expansions. The first-order and second-order perturbation expansions of traveltime from the isotropic common rays and from the anisotropic common rays to the anisotropic-ray-theory rays are then compared with the correct anisotropic-ray-theory traveltimes in order to illustrate the accuracy of the perturbation expansions and reliability of the error estimates. Finally, the synthetic seismograms obtained by the isotropic-common-ray and anisotropiccommon-ray approximations of the coupling ray theory are compared with the more accurate coupling-ray-theory synthetic seismograms simulated by the second-order perturbation expansion of traveltime from the anisotropic common rays. In the numerical examples, the errors of the anisotropic-common-ray approximation of the coupling ray theory are considerably smaller than the errors of the isotropic-common-ray approximation.

STOCHASTIC POST-BUCKLING OF FRAMES USING KOITER'S METHOD

The objective of this paper is to explore the impact of stochastic inputs on the buckling and post-buckling response of structural frames. In particular, we examine the impact of random member stiffness on the buckling load, and the initial slope and curvature of the post-buckling response of three example frames. A finite element implementation of Koiter's perturbation method is employed to efficiently examine the post-buckling response. Monte Carlo simulations where the member stiffness is treated as a random variable, as well as correlated and uncorrelated random fields, are completed. The efficiency of Koiter's perturbation method is the key to the feasibility of applying Monte Carlo simulation techniques, which typically requires a large number of sample simulations. In an attempt to curtail the need for multiple sample calculations, an alternative first-order perturbation expansion is proposed for approximating the mean and variance of the post-buckling behavior. However, the limitations of this first-order perturbation approximation are demonstrated to be significant. The simulations indicate that deterministic characteristics of the post-buckling response can be inadequate in the face of input randomness. In one case, a frame that is stable symmetric in the deterministic case is found to be asymmetric when randomness in the input is incorporated; therefore, this frame has real potential for imperfection sensitivity. The importance of random field models for the member stiffness as opposed to random variable models is highlighted. The simulations indicate that the post-buckling response can magnify input randomness, as variability in the post-buckling parameters can be greater than the variability in the input parameters.

Super-Poissonian noise, negative differential conductance, and relaxation effects in transport through molecules, quantum dots, and nanotubes

We consider charge transport through a nanoscopic object, e.g. single molecules, short nanotubes, or quantum dots, that is weakly coupled to metallic electrodes. We account for several levels of the molecule/quantum dot with level-dependent coupling strengths, and allow for relaxation of the excited states. The current-voltage characteristics as well as the current noise are calculated within first-order perturbation expansion in the coupling strengths. For the case of asymmetric coupling to the leads we predict negative-differential-conductance accompanied with super-poissonian noise. Both effects are destroyed by fast relaxation processes. The non-monotonic behavior of the shot noise as a function of bias and relaxation rate reflects the details of the electronic structure and level-dependent coupling strengths.

Transfer-matrix approach based on modal analysis for modeling corrugated long-period fiber gratings

A transfer-matrix method is developed for modeling a corrugated long-period fiber grating. Cladding-mode resonance in such a corrugated structure can be controlled by the applied tensile stress based on the photoelastic effect. A first-order vectorial perturbation expansion is used to derive the mode fields of the two basic regions under the strain-induced index perturbation. Because the etched cladding radius is much smaller than the unetched radius, the effect of the corrugated structure on cladding modes cannot be treated as a small perturbation. Thus the conventional coupled-mode theory is inadequate for the modeling of such a structure. Based on a self-consistent mode-matching technique, mode coupling within the corrugated structure can be described by a set of transfer matrices. We apply the formulation to the calculation of the transmission spectra of a corrugated long-period grating and compare the calculated with the experimental results. The transfer-matrix approach is found to account well for the features of the transmission spectra of the corrugated long-period gratings.

Propagation and diffraction of locally excited surface plasmons.

With the use of optical near-field techniques, it is now possible to excite or observe surface plasmons with high lateral resolution. A theoretical study is presented of surface plasmon excitation by near-field optical probes and the influence of well-defined structures on surface plasmon propagation and surface plasmon detection in the far field. The generation and the diffraction of the surface plasmon is calculated by using a theoretical scheme founded upon a first-order perturbation expansion of the Rayleigh-Fano method. A very good agreement is obtained between numerical and experimental results. The theoretical tools used should prove a useful guideline for future experiments of nanooptics with surface plasmons.

An efficient probabilistic finite element method for stochastic groundwater flow

We present an efficient numerical method for solving stochastic porous media flow problems. Single-phase flow with a random conductivity field is considered in a standard first-order perturbation expansion framework. The numerical scheme, based on finite element techniques, is computationally more efficient than traditional approaches because one can work with a much coarser finite element mesh. This is achieved by avoiding the common finite element representation of the conductivity field. Computations with the random conductivity field only arise in integrals of the log conductivity covariance function. The method is demonstrated in several two- and three-dimensional flow situations and compared to analytical solutions and Monte Carlo simulations. Provided that the integrals involving the covariance of the log conductivity are computed by higher-order Gaussian quadrature rules, excellent results can be obtained with characteristic element sizes equal to about five correlation lengths of the log conductivity field. Investigations of the validity of the proposed first-order method are performed by comparing nonlinear Monte Carlo results with linear solutions. In box-shaped domains the log conductivity standard deviation σ Y may be as large as 1.5, while the head variance is considerably influenced by nonlinear effects as σ Y approaches unity in more general domains.