Order Perturbation Solution Research Articles

A perturbative picture of cubic tensors in dually flat spaces

Some effects of cubic tensors in divergences, which characterize dually flat spaces, are investigated. Focusing on the small area around a specific point, how tangent space approximation is corrected by the cubic tensor at the point, is described. The cubic tensor in divergence is treated as a perturbation, supposing that the coordinates in tangent space are unperturbed solutions. A first order perturbation solution is presented as a main result. On the other hand, the unperturbed solution is yet another coordinate different from the usual affine coordinate. The divergences are shown to be written in a simple form if it is written in terms of the unperturbed solution. Furthermore, a relation which transforms similarities to the divergences is shown. Consequently, the asymmetry in the divergences between the tangent point and the other point is shown to be attributed to the difference between the squared norms of the two kinds of coordinates.

Internal resonances for heave, roll and pitch modes of a spar platform considering wave and vortex-induced loads in the main roll resonance

We present a study of the nonlinear coupling internal resonance for the heave roll and pitch performance of a spar platform under the wave and vortex-induced loads when the ratio of the frequencies of heave, roll and pitch are approximately 2:1:1. In consideration of varying wet surface, the three DOFs nonlinear coupled equations are established for the spar platform under the effect of the first-order wave loads in the heave and pitch, and vortexinduced loads in the roll. By utilizing the method of multi-scales when the vortex-induced frequency is close to the natural roll frequency, the first-order perturbation solution is obtained analytically and further validated by the numerical integration. Sensitivity analysis is performed to understand the influence of the damping and the internal detuning parameter. Two cases with internal resonance are shown. The first case is that no saturation phenomenon exists under small vortex-induced loads. The first order perturbation solution illustrates that only the vortex-induced frequency motion in roll and the super-harmonic frequency motion in heave are excited. The second case is that the vortex-induced loads are large enough to excite the pitch and a saturation phenomenon in the heave mode follows. The results show that there is no steady response occurrence for some cases. For these cases chaos occurs and large amplitudes response can be induced by the vortex-induced excitation.

Fifth Order Perturbation Solution to DSGE Models

This paper derives a fifth-order perturbation solution to DSGE models. Solutions of this order are required in models with rare disasters (Barro 2006), where lower order solutions fail to approximate accurately the true solution. The paper develops a new notation that is simpler than the standard notation of Schmitt-Grohe and Uribe (2004). Specifically, the perturbation parameter is treated as a state variable and the model is presented as a composite function. This yields a notational gain that rises exponentially with the perturbation order. Further notational gain is obtained by developing a compact matrix notation for high order chain rules. Finally, memory constraints are relaxed by exploiting symmetry and sparsity of the model derivatives.

A simplified perturbation solution of Michaelis-Menten kinetics equations in a “total” framework

A uniformly valid first order perturbation solution of the Michaelis-Menten kinetics equations in a total quasi-steady state setting is derived. This approximate solution has a simpler form than that obtained by Dell’Acqua and Bersani (J Math Chem 50:1136–1148, 2012). We also indicate an inconsistency in the perturbation equations obtained by these authors.

Darcy-Brinkman Flow Through a Bumpy Channel

The forced flow through a channel with bumpy walls which sandwich a porous medium is studied. The problem models micro-fluidics where, due to the small size of the channel width, the surface roughness of the walls is amplified. The Darcy-Brinkman equation is solved analytically through small perturbations on the ratio of bump amplitude to the half width of the channel. The first- order perturbation solutions give the three-dimensional velocity effects of the bumpiness and the second-order perturbation solutions give the increased resistance due to roughness. The problem depends heavily on the non-dimensional porous medium parameter \(k\) which represents the importance of length scale to the square root of permeability. Our solutions reduce to the clear fluid limit when \(k\) is zero and to the Darcy limit when \(k\) approaches infinity.

An Efficient Method of Computing Higher-Order Bond Price Perturbation Approximations

This paper develops a fast method of computing arbitrary order perturbation approximations to bond prices in DSGE models. The procedure is implemented to third order where it can shorten the approximation process by more than one hundred times. In a consumption-based endowment model with habits, it is further shown that a third order perturbation solution is more accurate than the log-normal method and a procedure using consol bonds. In addition, we present MATLAB codes that implement the suggested method to third order.

A finite extensibility nonlinear oscillator

The dynamics of a finite extensibility nonlinear oscillator (FENO) is studied analytically by means of two different approaches: a generalized decomposition method (GDM) and a linearized harmonic balance procedure (LHB). From both approaches, analytical approximations to the frequency of oscillation and periodic solutions are obtained, which are valid for a large range of amplitudes of oscillation. Within the generalized decomposition method, two different versions are presented, which provide different kinds of approximate analytical solutions. In the first version, it is shown that the truncation of the perturbation solution up to the third order provides a remarkable degree of accuracy for almost the whole range of amplitudes. The second version, which expands the nonlinear term in Taylor’s series around the equilibrium point, exhibits a little lower degree of accuracy, but it supplies an infinite series as the approximate solution. On the other hand, a linearized harmonic balance method is also employed, and the comparison between the approximate period and the exact one (numerically calculated) is slightly better than that obtained by both versions of the GDM. In general, the agreement between the results obtained by the three methods and the exact solution (numerically integrated) for amplitudes ( A) between 0 < A ⩽ 0.9 is very good both for the period and the amplitude of oscillation. For the rest of the amplitude range (0.9 < A < 1), an exponentially large L 2 error demonstrates that all three approximations do not represent a good description for the FENO, and higher order perturbation solutions are needed instead. As a complement, very accurate asymptotic representations of the period are provided for the whole range of amplitudes of oscillation.

An explicit solution of the Alekseevski–Tate penetration equations

Abstract The Alekseevski–Tate equations are typically used to predict the penetration, penetration velocity, rod velocity, and rod length of long rod penetrators and similar projectiles impacting targets. These nonlinear equations were originally solved numerically and more recently by the exact analytical solution of Walters and Segletes. However, due to the nonlinear nature of the equations, the penetration was obtained implicitly as a function of time, so that an explicit functional dependence of the penetration on material properties was not obtained. The current paper obtains the velocities, length, and penetration as an explicit function of time by employing a perturbation solution of the nondimensional Alekseevski–Tate equations. Simple (algebraic) analytical equations are given. Perturbation solutions of the Alekseevski–Tate equations were first undertaken by Forrestal et al., up to the first–order, and good agreement with the exact solutions was shown for relatively short times. In retrospect, this solution was only valid for penetrators impacting weak targets. The current study obtains a third-order perturbation solution and includes both penetrator and target strength terms, and is applicable for strong targets. The paper compares the exact solution to the perturbation solutions, and a typical comparison between the exact and approximate solutions for a tungsten rod impacting a semi-infinite steel armor target is shown. Also, alternate ways are investigated to normalize the governing equations in order to obtain an optimum perturbation parameter. In most cases, the third-order perturbation solution shows good agreement with the exact solution of the Alekseevski–Tate equations. In addition, simple equations based on the first–order perturbation solution are presented, which are accurate for the perforation of finite thickness (short penetration time) targets.

Solving, Estimating and Selecting Nonlinear Dynamic Economic Models without the Curse of Dimensionality

A welfare analysis of policies for risk averse preferences is handicapped within linear or linearized models and their certainty equivalence property. I calculate the optimal policy in a nonlinear stochastic dynamic general equilibrium model. Then I estimate the posterior density of structural parameters and the marginal likelihood within a nonlinear state space model. The computational challenges are considerable and I concentrate on the numerics and statistics for a simple model of dynamic consumption and labor choice. It is estimated on simulated data in order to test the routines with known true parameters. The code is written for a general model class. The policy function is approximated by Smolyak Chebyshev polynomials and the rational expectation integral by Smolyak Gaussian quadrature. The Smolyak operator is used to extend univariate approximation and integration operators to many dimensions. It is also called boolean method, hyperbolic cross points, discrete or sparse grids, fully symmetrical integration rules or complete polynomials. It reduces the curse of dimensionality from exponential to polynomial growth. The likelihood integrals are evaluated by a Gaussian quadrature and Gaussian quadrature particle filter. The bootstrap or sequential importance resampling particle filter, based on a Monte Carlo integration, is used as an accuracy benchmark. The posterior is estimated by the Gaussian filter and a Metropolis-Hastings algorithm. I propose a genetic extension of the standard Metropolis-Hastings algorithm by parallel random walk sequences. The candidate parameter vectors are constructed by innovation from the parallel sequences in addition to the usual random walk shocks. This improves the robustness of start values and the global maximization properties. Moreover it simplifies a cluster implementation and the random walk variances decision is reduced to only two parameters so that almost no trial sequences are needed. Finally the marginal likelihood is calculated as a criterion for nonnested and quasi-true models in order to select between the nonlinear estimates and a first order perturbation solution combined with the Kalman filter.

Plane stress crack growing steadily in metal foams

In this paper a continuum constitutive model proposed by Deshpande and Fleck is developed to solve the crack problem of metal foams in which the effects of cells can be concluded as compressible plasticity. The asymptotic analysis of near tip field for plane stress Mode-I crack growing steadily in metal foams has been performed by using singular perturbation expansion, and the lowest order perturbation solution of the stress and deformation velocity fields is obtained. The work presents an analytic approach for calculating slowly extending crack-tip fields in metal foams.

Dynamic Coefficient Prediction in Multi-lobe Journal Bearings Using a Mass Conservation Algorithm

A numerical procedure incorporating cavitation modeling in the predication of dynamic coefficients for four multi-lobe bearings is presented. A first order perturbation solution of the density (or fractional film content) is obtained by using the Elrod universal equation. The Elrod algorithm conserves mass throughout the bearing and automatically predicts the full film and cavitation regions using a switch function. In this work, the cavitation boundary predicted under steady state is considered for calculation of four unsteady pressure gradients. Results of stiffness and damping coefficients are presented for four multi-lobe bearing configurations, viz., two-axial groove, elliptical, three-lobe and offset cylindrical bearing for various L/D and eccentricity ratios. The results of bearing dynamic coefficients obtained using the cavitation algorithm agree well with the published work based on Reynolds boundary on the conditions as the film rupture boundary is predicted based on the continuity of fluid flow in the journal bearings. Presented at the 58th Annual Meeting in New York City April 28–May 1, 2003

SUPER AND COMBINATORIAL HARMONIC RESPONSE OF FLEXIBLE ELASTIC CABLES WITH SMALL SAG

The paper deals with the analysis of cables in stayed bridges and TV-towers, where the excitation is caused by harmonically varying in-plane motions of the upper support point with the amplitude U. Such cables are characterized by a sag-to-chord-length ratio below &0uml;02, which means that the lowest circular eigenfrequencies for in-plane and out-of-plane eigenvibrations, ω1and ω2, are closely separated. The dynamic analysis is performed by a two-degree-of-freedom modal decomposition in the lowest in-plane and out-of-plane eigenmodes. Modal parameters are evaluated based on the eigenmodes for the parabolic approximation to the equilibrium suspension. Superharmonic components of the ordern , supported by the parametric terms of the excitation and the non-linear coupling terms, are registered in the response for circular frequency ω≃ω1/n. At moderate U, the cable response takes place entirely in the static equilibrium plane. At larger amplitudes the in-plane response becomes unstable and a coupled whirling superharmonic component occurs. In the paper a first order perturbation solution to the superharmonic response is performed based on the averaging method. For ω≃(m/n)ω1, m<n, the geometrical non-linear restoring forces gives rise to a substantial combinatorial harmonic component with the circular frequency (n/m)ω. Both entirely in-plane and coupled in-plane and out-of-plane responses occur. Based on an initial frequency analysis of the response, an analytical model for these vibrations is formulated with emphasis on superharmonics of the order n=3 and combinatorial harmonics of the order (n, m)=(3,2). All analytical solutions have been verified by direct numerical integration of the modal equations of motion.

High‐Pressure Annular Seal Leakage and Rotordynamics with Application to Turbomachinery

The paper presents theoretical investigations that deal with the steady state mass flow rate, labyrinth seal-influence coefficients, and unbalance response of a cryogenic turbopump. A numerical method for calculating the leakage flow rates, stiffness, and damping coefficients of liquid hydrogen‐ lubricated seals is presented. Rotation and seal surface roughness effects are also included for leakage and dynamic force calculation. A finite element solution of a Reynolds equation is obtained for steady state as well as dynamic pressure distributions produced by a small amplitude whirl of the seal journal center (a first order perturbation solution). Film rupture is taken into account with the boundary to the ruptured film zone determined by an iterative procedure. A brief procedure is presented for modeling of the rotor/seals system. This work extends the previous theory for plain centered seals to large eccentricities using a perturbation analysis, which includes surface roughness effects. Investigations have been carried out for seal breadth-to-diameter ratio (BDR) of 0.25 to 0.5. The performance of a convergent and three-stepped seal is estimated. Numerical data for these seals for an eccentricity ratio up to 0.8 has been obtained.

Theoretical Study of Acoustic Multipole Logging in Anisotropic Two-Phase Medium Formation

Anisotropic formation is modeled as a transversely isotropic two-phase medium whose symmetric principal axis is perpendicular to the borehole axis. An effective analytical perturbation method is presented for the multipole acoustic logging in this formationborehole model. The anisotropic formation is regarded as the perturbations added to the relevant isotropic two-phase medium. Five perturbation quantities about moduli deviated from the isotropic formation are introduced. The acoustic multipole logging in anisotropy is theoretically studied and analyzed. The zero- and first- order perturbation solutions of the multipole acoustic fields inside and outside the borehole are derived. It is found that not only the nth multipole acoustic field can be excited by the nth multipole source, but the multipole fields whose orders are higher and lower than n can also be excited in such a formation-borehole system.

A stochastic wavefield solution of the acoustic wave equation based on the random Fourier–Stieltjes increments

In this paper we investigate the problem of compressional wave seismic propagation in random media. This problem is important because almost all geologic media is spatially heterogeneous by nature, consisting of a random agglomerate of many-sized rocks, soil and strata. In our formulation, a plane-harmonic seismic wave propagates in a medium having random material properties in the vertical direction. The random field representation is introduced through the intrinsic rock physical properties of the elastic medium. Each of these intrinsic properties is assumed to have a log-normal probability density function, and the random field representation is expressed in terms of these log-normal probability density functions. The constitutive law, the mass balance, and the moment balance equations are written in the Fourier–Stieltjes representation using random Lamé coefficients and random mass density. The stochastic wave equation is developed by introducing a perturbation approach based on an infinite series expansion of both random coefficients and the displacement solution in terms of σ-parameters (standard deviations of the random material properties). The method yields an integral representation of the displacement wavefield based on the Green's function. This representation is expressed in terms of the random rock physical properties of the medium. The key feature of this paper is that we have expressed the solution as a function of statistical parameters of 1D random medium, including the second order moments. Contrary to most previous derivations, the solutions can also simulate the coda and can be easily extended to simulate waves propagating in 2D and 3D random media. To test the displacement wave solution, synthetic seismograms and dispersion due to scattering effects were calculated for stiffness and density fluctuations of the random medium. This paper is the underlying foundation for the development of the effective propagation vector of acoustic waves in randomly heterogeneous media. This development is presented in a companion paper. In this case, an analytical expression is obtained using a second order perturbation solution. The solution is obtained in terms of the standard deviations of the density and the Young's modulus, respectively, as well as the cross-correlation coefficient and an integral that includes the spectral density and a kernel. In addition, this paper introduces practical expressions for the calculation of the effective attenuation and phase velocity of waves in randomly heterogeneous media. In this companion paper the solution is applied to interpret phase velocity curves that were obtained from interwell acoustic data recorded at Buckhorn test site, Illinois. The objective in this case is to be able to simulate the effect of scattering and intrinsic attenuation associated with acoustic waves in randomly heterogeneous media.

A finite-element simulation method for a system of growing cracks in a heterogeneous material

In this paper a numerical simulation method is proposed for the growth of interacting non-collinear cracks in a finite, heterogeneous, and two-dimensional elastic solid. The stress analysis is based on a step-by-step finite element method using the method of superposition of analytical and finite-element solutions accompanied by the quadtree automatic mesh generation algorithm. An incremental crack growth pattern is predicted considering the interaction of all growing cracks by the Schwartz–Neumann alternating procedure using a first order perturbation solution of slightly kinked and curved extension of all growing cracks, which analytically gives the mixed mode stress intensity factors at extending crack tips. The local symmetry criterion is used as the crack path criterion at the extending crack tips. Several numerical examples, including the avoidance of collinear edge cracks and cracks initiated at the ends of fibers show the validity of the present simulation method.

Mixed boundary–value problems of two–dimensional anisotropic elasticity with perturbed boundaries

The mixed boundary–value problems of two–dimensional anisotropic elasticity are considered in this paper. The boundary treated may be the one perturbed by a straight line or an ellipse. A general solution up to the first–order perturbation has been found by using Stroh9s formalism, analytical continuation method, conformal mapping function, and perturbation technique. As to higher–order perturbation solutions, general procedure is depicted in this paper. In order to illustrate the use of general solution, two typical examples are solved completely. One is a cosine wave–shaped surface indented by a rigid flat–ended punch, the other is a triangular hole boundary indented by a rigid stamp.

Capillary Performance of Evaporating Flow in Micro Grooves: An Analytical Approach for Very Small Tilt Angles

The axial flow of an evaporating thin film through a V-shaped micro channel was investigated for the case of a grooved plate inclined with a very small tilt angle. For this problem, the addition of the gravity term caused by the tilt angle alters the form of the governing equation from linear to nonlinear. Because the effect of the tilt angle is very small, a perturbation method was applied to obtain a first order perturbation solution of R = R0 + εR1, where R0 is the undisturbed solution caused by the tilt angle. A generalized graph of the nondimensional radius of curvature as a function of the nondimensionalized axial length was obtained. The perturbation parameter, ε, was expressed as a function of five nondimensional parameters. The results indicate that when the geometry is fixed, the effects of the Bond number and Capillary number are dominant. In addition, it was found that ε is inversely proportional to the square root of the Capillary number. This indicates that the perturbation method is applicable and accurate, even for the case of somewhat larger tilt angles (where the initial applicability of perturbation appears to be violated), is a sufficiently large heat flux is applied.

A CONSISTENT CRACKED BAR VIBRATION THEORY

A consistent continuous cracked bar vibration theory is developed. The stress and displacement field about the crack was used to modify the stress and displacement field throughout the bar, and reduction to one spatial dimension was achieved by integrating the stress and displacement fields throughout the bar cross-sections so that the total displacement would be exact. The resulting linear differential equation with variable coefficients has the modified displacement field due to the crack imbedded in it. Any number of cracks can be introduced into the differential equation as modifications of the displacement field. A numerical solution and a first order perturbation solution are presented for the prediction of changes in longitudinal vibration natural frequencies of a fixed-free bar with a single open-edge transverse crack. To assess the validity of the assumptions made experiments on aluminium bars with fatigue cracks were performed. The analytical results correlate very closely with experimental results with better correlation than the local flexibility solution.

A second comparison of two higher order perturbation schemes

The order three superharmonic resonance in the harmonically excited Duffing oscillator with hardening non-linearity is considered. The second order perturbation solutions determined by the multiple scales and frequency-damping expansion methods are compared with each other, and with the exact numerical results. By using the example of third superharmonic resonance in the Duffing oscillator, this work establishes the following: (1) the approximate periodic solutions determined by the frequency-damping expansion procedure are based on the fast time scale only; (2) these solutions provide an approximation for the steady state response only; (3) the second order approximate solutions for the third superharmonic resonance in the Duffing equation determined by these methods can uncover a second type of frequency response curve; (4) both second order approximations predict extraneous solutions; and (5) the extension, achieved by the second order approximation, in the range of the system parameters for which the perturbation solution provides a reasonable approximation to the actual system response, may depend on the problem on hand and, in this regard, performance of the frequency-damping expansion procedure may not be substantially superior to the performance of the multiple scales-reconstitution procedure.