Analytic Perturbation Research Articles

A Fast and Accurate Method for dq Impedance Modeling of Power Electronics Systems Based on Finite Differences

This paper presents a finite-difference-based method for numerically deriving the DQ impedance model of power electronics-based power systems, specifically tailored for stability analysis. The proposed method offers a computationally efficient alternative to traditional approaches by directly applying finite-difference approximations to the large-signal dynamic system, without relying on repetitive time-domain simulations or small-signal analytical models. This method eliminates the need for additional models or complex procedures to compute the steady-state solution, streamlining the impedance modeling process. The accuracy, efficiency, and precision of the proposed method are evaluated through comparative studies with analytical and time-domain perturbation methods. Results demonstrate that the proposed approach provides accuracy comparable to analytical models while significantly reducing computational effort, outperforming perturbation methods in both speed and precision. These findings highlight the practical value of the proposed method for real-time and large-scale system analysis, making it a robust tool for power systems stability assessment.

On thermal conduction in the solar atmosphere: An analytical solution for nonlinear diffusivity without compact support

Context. The scientific community employs complicated multiphysics simulations to understand the physics in solar, stellar, and interstellar media. These must be tested against known solutions to ensure their validity. Several well-known tests exist, such as the Sod shock tube test. However, a test for nonlinear diffusivity is missing. This problem is highly relevant in the solar atmosphere, where various events release energy that subsequently diffuses by Spitzer thermal conductivity. Aims. The aim is to derive an analytical solution for nonlinear diffusivity in 1D, 2D, and 3D, which allows for a nonzero background value. The solution is used to design a test for numerical solvers and study Spitzer conductivity in the solar atmosphere. Methods. There exists an ideal solution assuming zero background value. We performed an analytical first-order perturbation of this solution. The first-order solution was first tested against a dedicated nonlinear diffusion solver, whereupon it was used to benchmark the single- and multifluid radiative magnetohydrodynamics code Ebysus, used to study the Sun. The theory and numerical modeling were used to investigate the role of Spitzer conductivity in the transport of energy released in a nanoflare. Results. The derived analytical solution models nonlinear diffusivity accurately within its region of validity and approximately beyond. Various numerical schemes implemented in the Ebysus code is found to model Spitzer conductivity correctly. The energy from a representative nanoflare is found to diffuse 9 Mm within the first second of its lifetime due to Spitzer conductivity alone, strongly dependent on the electron density. Conclusions. The analytical first-order solution is a step forward in ensuring the physical validity of intricate simulations of the Sun. Additionally, since the derivation and argumentation are general, they can easily be followed to treat other nonlinear diffusion problems.

Intuitive understanding of extinction of small particles in absorbing and active host media within the MLWA

In an absorbing or an active host medium characterized by a complex refractive index n2=n2′+in2′′, our previously developed modified dipole long-wave approximation (MLWA) is shown to essentially overlie with the exact Mie theory results for localized surface plasmon resonance of spherical nanoparticles with radius a≲25nm (a≲20nm) in the case of Ag and Au (Al and Mg) nanoparticles. The agreement for Au and Ag (Al and Mg) nanoparticles, slightly better in the case of Au than Ag, continues to be acceptable up to a∼50nm (a∼40nm), and can be used, at least qualitatively, up to a∼70nm (a∼50nm) correspondingly. A first order analytic perturbation theory (PT) in a normalized extinction coefficient, κ¯=n2′′/n2′, around a nonabsorbing host is developed within the dipole MLWA and its properties are investigated. It is shown that, in a suitable parameter range, the PT can reliably isolate and capture the effect of host absorption or host gain on the overall extinction efficiency of various plasmonic nanoparticles.

Invariant subspaces of analytic perturbations

Analytic perturbations are understood here as shifts of the form M z + F M_z + F , where M z M_z is the unilateral shift and F F is a finite rank operator on the Hardy space over the open unit disk. Here the term “a shift” refers to the multiplication operator M z M_z on some analytic reproducing kernel Hilbert space. In this paper, first, a natural class of finite rank operators is isolated for which the corresponding perturbations are analytic, and then a complete classification of invariant subspaces of those analytic perturbations is presented. Some instructive examples and several distinctive properties (like cyclicity, essential normality, hyponormality, etc.) of analytic perturbations are also described.

Investigation of time-dependent mechanical loading on the start-stop process of functionally graded rotating disks under thermal conditions

Rotating disks are used in industries, such as aerospace, automotive, and power plants. These disks are under time-dependent mechanical loading when the machine starts or stops working conditions. Thermal stresses caused by temperature changes with this time-dependent mechanical load create dangerous conditions. Therefore, the estimation of the elastic limit angular velocity and acceleration as a criterion for the initiation of plastic deformations has special importance in the start-stop process. An analytical Homotopy perturbation method is used to solve the heat transfer equation and the governing Navier equations in both radial and tangential directions of functionally graded rotating disks with non-uniform thickness. The Tamura-Tomoto-Ozawa model is implemented to calculate the yield stress at a different radius of the disk. The obtained results will be verified with the higher-order finite difference method. The effect of thickness parameter, type of thermal and boundary conditions on the angular velocity and acceleration, and also the starting radius of the plastic deformations are investigated by numerical examples. It is shown that by defining the appropriate temperature gradient on the outer surface of the disk and the parameters in the time-dependent angular velocity function, the level of stresses can be controlled and optimized in all working processes.

Analytical Inverse QCD Coupling Constant Approach and Its Result for αs

We propose a model for the QCD running coupling constant based on the analytical inverse QCD coupling constant concept with an additional regularization in the low momentum region. Analyticity in the q2-complex plane, where q is the four-momentum transfer, is imposed by methods of the Analytic Perturbation Theory. The model incorporates a peculiar low-momentum behavior for αs(q2) as a divergence at q2=0 to retrieve color confinement, without spoiling its correct high-momentum behavior. This was achieved by means of a two-parameter regularization function, for which we considered three possible analytic expressions. In fact, within the framework of the Analytic Perturbation Theory, αs(q2) assumes a finite value for q2=0, at all perturbative orders (infrared stability), hence the infrared divergence cannot be implemented. For this reason, we found it more straightforward to work with its reciprocal, namely, εs(q2)=1/αs(q2), imposing its vanishing at the origin of the q2-complex plane via the multiplication of the aforementioned regularizing functions and the spectral density. Once the two free parameters of the regularization functions were settled by fitting to the experimental values of αs(q2) at the momenta where these data were available and reliable, the model could reproduce the QCD running coupling constant at any other momentum transferred.

Analytical perturbations of relativistic images in Kerr space-time

Light rays passing very close to black holes may wind several times before escaping. For any given electromagnetic source around the black hole, a distant observer would thus observe two infinite sequences of images on either side of the black hole. These images are generated by light rays performing an increasing numbers of loops. The strong deflection limit provides a simple analytic formalism to describe such higher order images for spherically symmetric metrics, while for axially symmetric black holes one typically resorts to numerical approaches. Here we present the leading order perturbation to higher order images when the black hole spin is turned on. We show that the images slide around the black hole shadow as an effect of space-time dragging. We derive analytical formulae for their shifts and the perturbation of their time delays. We also discuss how such simple analytical formulae for images by Kerr black holes can be of great help in many applications.

MODELING THE DYNAMICS OF DEFORMABLE OBJECTS BASED ON VOLUMETRIC PATCHES OF FREE FORMS

A method for solving nonlinear implicit numerical integration problems based on volumetric patches of free forms for modeling deformable objects with high-speed dynamics is proposed. This method improves the accuracy, consistency and controllability of deformation modeling and animation. The method is characterized by speed, accuracy, stability and uses optimization with region decomposition. The method is well suited for modeling deformable bodies with a large time step, in a wide range of deformation dynamics. We propose decomposed optimization, an optimization method based on free-form patches with region decomposition to minimize incremental potentials at each time step. The method uses quadratic matrix decomposition to combine non-overlapping subregions. The Hessian is evaluated once at the beginning of the time step. The advantages of the method are as follows. Geometric primitives and their mathematical models are proposed, which allow the reasonable application of these primitives to solve problems of volume-oriented modeling. Such requirements are met by volumetric patches of free forms based on analytical perturbation functions relative to the base triangles. The decomposed Hessian is constructed for each subregion and calculated using a set of vertices taken from a complete non-decomposed grid. The weights add the missing second-order Hessian data to the vertices of the subregions from the neighbors along the decomposition boundaries. Thanks to this, the descent is performed along the grid coordinates. There is no need to add gradients. During the descent, the gradient is determined. The Hessians under the region are calculated and factorized in parallel once per time step. They are used as an initializer at each iteration. Then the results are mixed together. This ensures stable and continuous high-quality modeling. An automated and reliable optimization method adapted for modeling nonlinear materials, high-speed dynamics and large deformations is proposed.

Rayleigh Scattering from a Sphere Located Near a Planar Rigid Boundary

Rayleigh scattering from a spherical object located near a planar rigid boundary at distances smaller than the wavelength is calculated. Low frequency analysis reduces a scattering problem to a sequence of potential problems. An analytical solution based on expansion in spherical solid harmonics and the use of addition theorem is presented. Analytical perturbation approach is validated by comparison with numerical calculations. The velocity of the center of the particle and the scattering amplitude are determined. In the lowest order in wavenumber, the scattering amplitude is expressed in terms of the monopole and dipole components. In contrast to the behavior of a bubble, under the same conditions, dipole oscillations of the particle in the direction normal to the boundary are not excited and the monopole component of the scattering amplitude does not depend on the location of the particle relative to the boundary.

On Analytic Perturbations of Linear Equations in the Case of Incomplete Generalized Jordan Set

On Analytic Perturbations of Linear Equations in the Case of Incomplete Generalized Jordan Set

Analysis of clamped-clamped piezoelectric energy harvester under vortex induced vibration considering the stretching effect

This paper investigates the middle layer stretching effect on the dynamic behavior of an energy harvester clamped-clamped beam. Two foam cylinders that are attached together as a dumbbell are mounted on the middle part of the beam for vortex-induced vibrations. The piezoelectric patch collects the electrical energy produced by vortex induced vibration. In this analysis the coupled differential equations governing on the structure oscillation, harvested voltage and fluid lift force are established applying the Hamilton’s principle, Gauss law and wake oscillator model. The obtained differential equations are discretized using Galerkin method, and solved by both numerical and analytical perturbation methods. The results demonstrate that middle layer stretching effect has a significant effect on the lock-in domain, and the output electric voltage must be evaluated by considering this effect. It has been shown that for large values of cylinder diameter the difference between numerical and theoretical results increases due to increasing the middle layer stretching effect.

Analytic Osculating Frozen Orbits Under Perturbation

This work provides a set of closed-form analytical expressions to define osculating frozen orbits under the perturbation effects of the oblateness of the main celestial body. To this end, an analytical perturbation method based on osculating elements is proposed to characterize, define, and study the three existing families of frozen orbits in closed form: the two families of frozen orbits close to the critical inclination and the family of frozen orbits appearing at low eccentricity values. As such, this work aims to complement other analytical approaches based on mean elements by providing an alternative methodology based on the more natural osculating elements that is able to generate closed-form expressions for all known frozen conditions in the main satellite problem. Additionally, this work includes the first- and second-order approximate solutions of the proposed perturbation method, including their applications to the analytical definition of frozen orbits, repeating ground-track orbits, and sun-synchronous orbits under this perturbation. Examples of applications are also provided to show the expected error performance of the proposed approach.

The localised response and filtering performance of centrifugal pendulum vibration absorbers allowing a rotational mobility

Rotating machines are often subjected to fluctuating torques, which causes rotor vibrations, early wear and noise pollution. These vibrations can be reduced using centrifugal pendulum vibration absorbers (CPVAs), which are passive devices made of several bodies (pendulums) oscillating along a given path and rotating relatively to a rotor. Previous studies showed that the dynamics of these devices is subjected to instabilities leading to a localisation of the motion of the pendulums. In this paper, the localised behaviour of a CPVA made of two pendulums allowed to rotate about their centre of mass is investigated. To this aim, a dynamical model based on an analytic perturbation method is established. The aim of this model is to highlight some special features of the localised response, such as the appearance of quasi-periodic regimes. The case studies showed that in some special cases, localisation can improve the filtering efficiency as compared to a unison motion. The validity of the model was confirmed through a comparison with numerical resolutions of the system’s dynamics.

Cosmological perturbations engendered by discrete relativistic species

Within the extension of the varLambda CDM model, allowing for the presence of neutrinos or warm dark matter, we develop the analytical cosmological perturbation theory. It covers all spatial scales where the weak gravitational field regime represents a valid approximation. Discrete particles – the sources of the inhomogeneous gravitational field – may be relativistic. Similarly to the previously investigated case of nonrelativistic matter, the Yukawa interaction range is naturally incorporated into the first-order scalar metric corrections.

Dispersion curves of acoustoelastic borehole waves: the perturbation method with the correct formulation of the stresses around the borehole

SUMMARY The acoustoelastic model has been widely used to investigate the influence of formation stresses on the dispersion curves of borehole waves. The analytical perturbation method (PM), the finite-difference time-domain (FDTD) and the semi-analytical finite element (SAFE) are three common-used methods to calculate the dispersion curves. However, due to different interpretations of the PM and plane strain assumptions, the obtained dispersion curves are incompatible among existing PMs, which may misguide the interpretation of formation stresses. It is therefore necessary to untangle the applicability and limitations of PM. Considering that the conventional PMs are usually inaccurate at the low frequency or inconsistent with Hamilton’s principle, we develop a revised PM to obtain the dispersion curves of borehole waves propagating along a borehole surrounded by the triaxially stressed formation assumed as a monoclinic medium. The revised PM is more accurate, reasonable and logical than existing PMs. When the formation is subjected to low stresses, our finding is of great benefit for quickly computing dispersion curves, since the revised PM is much more efficient than the FDTD method; and there are small discrepancies between the flexural dispersions obtained by the revised PM and those obtained by the FDTD method. Nevertheless, the revised PM has two limitations. The first limitation is that the revised PM cannot be used to compute the Stoneley dispersion curves, which have been validated by comparison with SAFE and FDTD methods. The second limitation is that flexural dispersion curves show significant discrepancies in the high-frequency domain when the low-stress assumption does not hold, as compared to those obtained by the FDTD method.

Analytic Transformation Between Osculating and Mean Elements in the J2 Problem

This work presents an analytical perturbation method to study the dynamics of an orbiting object subject to the term from the gravitational potential of the main celestial body. In particular, this paper focuses on the generation of the analytical transformations between osculating and mean elements under this perturbation. This is done using a power series expansion in the perturbation constant on all the variables of the system, and a time regularization based on the argument of latitude of the orbit. This enables the generation of analytic approximate solutions without the need to control the perturbed frequency of the system. The resultant approximations provide the osculating behavior of the problem as well as the transformations between osculating and mean elements for orbits at any eccentricity, including near-circular, elliptic, parabolic, and hyperbolic orbits. Several examples of application are presented to show the accuracy of the perturbation approach and their related transformations.

Entropy generation analysis on forced and free convection flow in a vertical porous channel with aligned magnetic field and Navier slip

AbstractThe entropy generation rate in a vertical porous channel with injection and suction walls and a uniform magnetic field provided at an angle to the flow direction is studied using an analytical perturbation technique in the presence of Navier slip and buoyancy force. The momentum and energy equations were solved for exemplary values of fluid convection's physical characteristics. Important parameters' impacts on adequate numbers are graphically portrayed and conveyed.

Analytical solutions for chemical dissolution‐front instability problems involving radially divergent flow within fluid‐saturated porous media

AbstractThrough using rigorously mathematical deductions, this article derives analytical solutions for chemical dissolution‐front instability (CDFI) problems, in which radially divergent flow is involved within fluid‐saturated porous media. Since the acid injection‐well to be considered is a circle of nonzero radius in the horizontal plane, a polar coordinate system is used to describe the coupled mathematical governing equations of the CDFI problem. To facilitate deriving analytical solutions, an appropriate mathematical transform is used to convert the coupled mathematical governing equations from a dimensional form into a dimensionless one. This enables the generalized linear stability approach to be utilized for deriving both analytical base and perturbation solutions for the CDFI problem involving radially divergent flow within a fluid‐saturated porous medium. Consequently, a theoretical criterion, which is usable for assessing the CDFI in the chemical dissolution system (CDS), is further established from the corresponding analytical perturbation solutions. The related theoretical results have demonstrated that: (1) the proposed theoretical criterion is useful and applicable for assessing the CDFI associated with radially divergent flow within fluid‐saturated porous media; (2) the permeability ratio between the downstream and upstream regions can affect significantly the critical Peclet number of a CDS associated with radially divergent flow; (3) the initial porosity can have remarkable effects on the critical Peclet number of a CDS associated with radially divergent flow; and (4) the difference between the final and initial porosities can also affect significantly the critical Peclet number of a CDS associated with radially divergent flow in the fluid‐saturated porous medium.

On Analytic Perturbations of a Non-Self-Adjoint Anharmonic Oscillator

On Analytic Perturbations of a Non-Self-Adjoint Anharmonic Oscillator

Laplace eigenvalues of ellipsoids obtained as analytic perturbations of the unit sphere

We study the eigenvalues of the Laplacian on ellipsoids that are obtained as perturbations of the standard Euclidean unit sphere in dimension two. A comparison of these eigenvalues with those of the standard Euclidean unit sphere is obtained under a Gaussian curvature condition, in line with the Lichnerowicz theorem on the first positive eigenvalue on a compact Riemannian manifold.