ON THE SECOND ORDER WAVE DIFFRACTION IN TWO LAYER FLUIDS

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Finite element computation of trapped and leaky elastic waves in open stratified waveguides

Elastic guided waves are of interest for inspecting structures due to their ability to propagate over long distances. In numerous applications, the guiding structure is surrounded by a solid matrix that can be considered as unbounded in the transverse directions. The physics of waves in such an open waveguide significantly differs from a closed waveguide, i.e. for a bounded cross-section. Except for trapped modes, part of the energy is radiated in the surrounding medium, yielding attenuated modes along the axis called leaky modes. These leaky modes have often been considered in non destructive testing applications, which require waves of low attenuation in order to maximize the inspection distance. The main difficulty with numerical modeling of open waveguides lies in the unbounded nature of the geometry in the transverse direction. This difficulty is particularly severe due to the unusual behavior of leaky modes: while attenuating along the axis, such modes exponentially grow along the transverse direction. A simple numerical procedure consists in using absorbing layers of artificially growing viscoelasticity, but large layers may be required. The goal of this paper is to explore another approach for the computation of trapped and leaky modes in open waveguides. The approach combines the so-called semi-analytical finite element method and a perfectly matched layer technique. Such an approach has already been successfully applied in scalar acoustics and electromagnetism. It is extended here to open elastic waveguides, which raises specific difficulties. In this paper, two-dimensional stratified waveguides are considered. As it reveals a rich structure, the numerical eigenvalue spectrum is analyzed in a first step. This allows to clarify the spectral objects calculated with the method, including radiation modes, and their dependency on the perfectly matched layer parameters. In a second step, numerical dispersion curves of trapped and leaky modes are compared to analytical results.     

Localized elastic fields in periodic waveguides with defects

The variational method for determining localized waves (trapped modes) is modified for periodic elastic waveguides with partially clamped surfaces. Two sufficient conditions for the existence of localized fields in waveguides with defects (cavities with positive volume and cracks) are established. In the presence of elastic and geometrical symmetries, localized fields were also found in periodic elastic waveguides with surfaces free of external loads.     

The influence of topographic features and density variations upon the internal tides in shelf edge regions

A brief outline of the development of a three‐dimensional primitive equation baroclinic model for computing the internal tides is presented. A series of calculations are performed to examine the internal tide in the shelf edge region off the west coast of Scotland, with the aim of determining the influence upon the internal tide in the region of topographic features, namely large‐scale features, such as seamounts, and small‐scale features, such as a detailed specification of the shelf slope, which is possible using results from a recent survey rather than topographic data taken from a coarse grid model using depths interpolated from charts. The influence upon the solution of small variations in the density field and seasonal variations is examined. Also, the influence of a well‐mixed bottom boundary layer at the base of the shelf slope, produced possibly by cascading of dense water off the shelf during a major wind event, upon the internal tide is also examined. In the calculations described here the model is used in cross‐shelf form. A comparison of internal tides computed with and without the inclusion of an offshore seamount shows that internal tides are produced along the sides of the seamount, and can propagate towards the shelf edge, and thereby influence the internal tidal signal in that region. A comparison of internal tides computed using previously available bottom topography extracted from charts with those computed using a detailed topographic data set from a recent survey shows the importance of accurate topography in the shelf edge region for internal tide calculations. Simulations show that the internal tide is influenced by variations in the density field, suggesting that an accurately measured density field is essential for internal tide determination. Calculations also show that if a well‐mixed sea bed boundary layer occurs at the bottom of the shelf slope, the internal tide can be reflected by this feature giving rise to significant internal tides in the ocean at larger distances from the shelf edge than in the absence of this water mass. Copyright © 1999 John Wiley & Sons, Ltd.     

Long-wave asymptotic theories: The connection between functionally graded waveguides and periodic media

This article explores the deep connections that exist between the mathematical representations of dynamic phenomena in functionally graded waveguides and those in periodic media. These connections are at their most obvious for low-frequency and long-wave asymptotics where well established theories hold. However, there is also a complementary limit of high-frequency long-wave asymptotics corresponding to various features that arise near cut-off frequencies in waveguides, including trapped modes. Simultaneously, periodic media exhibit standing wave frequencies, and the long-wave asymptotics near these frequencies characterise localised defect modes along with other high-frequency phenomena. The physics associated with waveguides and periodic media are, at first sight, apparently quite different, however the final equations that distill the essential physics are virtually identical. The connection is illustrated by the comparative study of a periodic string and a functionally graded acoustic waveguide.     

Dynamically Equilibrium Shape of Intrusive Vortex Formations in the Ocean

The problem of finding the dynamically equilibrium shape of a rotating mass of liquid with homogeneous density (lens) submerged in a stratified ocean at rest on the rotating Earth is formulated. An equation for the shape of the interface between water masses is derived. An exact solution of the problem for an anticyclonically rotating lens in a linearly stratified ocean in the neighborhood of the lens depth shows that the dynamically equilibrium shape of the interface is a triaxial ellipsoid inclined with respect to the horizon and similar to an ellipsoid of revolution for real parameters of the phenomenon. The limiting values of the latitudes at which these formations can exist are determined. Degeneration of the shape with decrease in the intrinsic lens angular rate is investigated.     

Coupled mode theory for acoustic resonators

We develop the effective non-Hermitian Hamiltonian approach for open systems with Neumann boundary conditions. The approach can be used for calculating the scattering matrix and the scattering function in open resonator–waveguide systems. In higher than one dimension the method represents acoustic coupled mode theory in which the scattering solution within an open resonator is found in the form of expansion over the eigenmodes of the closed resonator decoupled from the waveguides. The problem of finding the transmission spectra is reduced to solving a set of linear equations with a non-Hermitian matrix whose anti-Hermitian term accounts for coupling between the resonator eigenmodes and the scattering channels of the waveguides. Numerical applications to acoustic two-, and three-dimensional resonator–waveguide problems are considered.     

Vibration of a membrane strip with a segment of higher density: analysis of trapped modes

The Helmholtz equation governs the dynamics of membranes and waveguides. Using the mode matching method, trapped modes are found for an infinite strip with a segment of inhomogeneity. The exact frequencies and mode shapes are determined as a function of density ratio and length of the segment. Nonuniqueness and nonexistence are demonstrated.     

Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides

We consider the time-harmonic problem of the diffraction of an incident propagative mode by a localized defect, in an infinite elastic waveguide. We propose several iterative algorithms to compute an approximate solution of the problem, using a classical finite element discretization in a small area around the perturbation, and a modal expansion in the unbounded straight parts of the guide. Each algorithm can be related to a so-called domain decomposition method, with an overlap between the domains. Specific transmission conditions are used, so that at each step of the algorithm only the sparse finite element matrix has to be inverted, the modal expansion being obtained by a simple projection, using a bi-orthogonality relation. The benefit of using an overlap between the finite element domain and the modal domain is emphasized. An original choice of transmission conditions is proposed which enhances the effect of the overlap and allows us to handle arbitrary anisotropic materials. As a by-product, we derive transparent boundary conditions for an arbitrary anisotropic waveguide. The transparency of these new boundary conditions is checked for two- and three-dimensional anisotropic waveguides. Finally, in the isotropic case, numerical validation for two- and three-dimensional waveguides illustrates the efficiency of the new approach, compared to other existing methods, in terms of number of iterations and CPU time.     

Time–transient response for ultrasonic guided waves propagating in damped cylinders

Herein, an enhanced spectral finite element (SFE) formulation to calculate the time–transient response in cylindrical waveguides is proposed. The original aspect over SFE-based formulations consists in the possibility to account for the effect of material absorption, i.e. guided waves attenuation, on the calculation of the time–transient response.First, the damped steady-state response is constructed by a weighted superposition of the waveguide modal properties obtained from the spectral decomposition of the governing wave equation. To this purpose an enhanced spectrally formulated finite element is developed, in which material damping is included allowing for complex stress–strain viscoelastic constitutive relations in force of the correspondence principle. Dispersive modal properties for the damped waveguide (phase velocity, energy velocity, attenuation and wavestructures) follow straightforwardly by simple formulae. Next, the frequency response of the problem is calculated by weighting the modal data and the spectrum of the applied time-dependent force via Cauchy residue theorem. Finally, the inverse Fourier transform of the frequency response leads to the time–transient response for propagative damped guided waves.The approach is not restricted to any anisotropy degree, holds for any linear viscoelastic constitutive relation that can be characterized and formulated in the frequency domain and it can be applied to SFE formulations for arbitrary cross-section waveguides. A study on guided waves propagating in a scheduled 4.in-40 ANSI steel pipe is presented, where the steel is considered first as perfectly elastic and then as an hysteretic viscoelastic medium, in order to show the effect of material absorption on the time–transient response.     

A code-independent technique for computational verification of fluid mechanics and heat transfer problems

The goal of this paper is to present a versatile framework for solution verification of PDE＇s. We first generalize the Richardson Extrapolation technique to an optimized extrapolation solution procedure that constructs the best consistent solution from a set of two or three coarse grid solution in the discrete norm of choice. This technique generalizes the Least Square Extrapolation method introduced by one of the author and W. Shyy. We second establish the conditioning number of the problem in a reduced space that approximates the main feature of the numerical solution thanks to a sensitivity analysis. Overall our method produces an a posteriori error estimation in this reduced space of approximation. The key feature of our method is that our construction does not require an internal knowledge of the software neither the source code that produces the solution to be verified. It can be applied in principle as a postprocessing procedure to off the shelf commercial code. We demonstrate the robustness of our method with two steady problems that are separately an incompressible back step flow test case and a heat transfer problem for a battery. Our error estimate might be ultimately verified with a near by manufactured solution. While our pro- cedure is systematic and requires numerous computation of residuals, one can take advantage of distributed computing to get quickly the error estimate.     

桁架频率优化解存在性及其算法研究

给出桁架在给定固频约束时,动力不优化解存在性的基本理论。指出：桁架固频约束通常是决定其动力学优化解是否存在的“关键约束”。基此,给出了一种工程算法,只用到固频对设计变量的一阶导数,就可很快确定约束是否可能满足。反之,当提出的方法判定给定固频约束不能满足时,可以求出一个解存在的极限频率的狭窄范围及相应的设计变量范围,以供设计人员重新设计时参考。三个由简到繁的算例说明了所提出方法实用、有效。     

On a bound for the frequency of surface waves trapped near a cylinder spanning a channel

There has been substantial effort recently put into proving, for a variety of different geometries, the existence of trapped waves, that is unforced time-harmonic motions which do not radiate energy to large distances. Thus it is known that such motions can exist in a deep channel which includes a cylinder spanning the channel, for various shapes of cylinder.The converse problem of proving the absence of such trapped waves has received much less consideration, and the only relevant uniqueness proof for a channel spanned by a cylinder is that of McIver (1991). In an appendix to that paper, McIver demonstrates that no trapped-wave motions can exist for the case in which the cylinder is surface piercing and is entirely contained between vertical planes through the free-surface intersections. This is exactly the same geometrical condition which John (1950) found would ensure uniqueness in water-wave radiation and scattering problems, in finite or infinite depth. Both John and McIver achieved their uniqueness results by consideration of integrals of the potential along vertical lines down from the free surface.John's work was extended by Simon and Ursell (1984) who established uniqueness for a wider class of two-dimensional radiation and scattering problems by consideration of integrals along nonvertical lines. The work presented in this paper is the corresponding extension of McIver's work; although this extension does not rule out trapped waves at all frequencies for any geometry except that already considered by McIver, it does yield an easy lower bound for the ratio of the trapped-mode frequency to the cut-off frequency, in finite or infinite depth.     

On two-dimensional large-scale primitive equations in oceanic dynamics(Ⅱ)

The initial boundary value problem for the two-dimensional primitive equations of largescale oceanic motion in geophysics is considered sequetially.Here the depth of the ocean is positive but not always a constant.By Faedo-Galerkin method and anisotropic inequalities,the existence and uniqueness of the global weakly strong solution and global strong solution for the problem are obtained.Moreover,by studying the asymptotic behavior of solutions for the above problem,the energy is exponential decay with time is proved.     

On two-dimensional large-scale primitive equations in oceanic dynamics （Ⅱ）

The initial boundary value problem for the two-dimensional primitive equations of largescale oceanic motion in geophysics is considered sequetially. Here the depth of the ocean is positive but not always a constant. By Faedo-Galerkin method and anisotropic inequalities, the existence and uniqueness of the global weakly strong solution and global strong solution for the problem are obtained. Moreover, by studying the asymptotic behavior of solutions for the above problem, the energy is exponential decay with time is proved.     

海洋内波研究现状简介

Ⅰ.引言内波是发生在稳定层化流体中的一种波动,其最大振幅在流体内部,频率介于惯性频率f与浮性频率N之间。在海洋中由于海水的层化,内波是很普遍的现象。它具有很强的随机性,时空尺度分布在相当宽的范围内,典型地有:振幅为几米至几十米乃至百米;水平波长为百米至几十公里;铅垂波长为几十米至几公里;周期为几分钟至几十小时。因而海洋内波是很容易观测到的。      

Two-phase flow in heterogeneous porous media: The method of large-scale averaging

The analysis of two-phase flow in porous media begins with the Stokes equations and an appropriate set of boundary conditions. Local volume averaging can then be used to produce the well known extension of Darcy's law for two-phase flow. In addition, a method of closure exists that can be used to predict the individual permeability tensors for each phase. For a heterogeneous porous medium, the local volume average closure problem becomes exceedingly complex and an alternate theoretical resolution of the problem is necessary. This is provided by the method of large-scale averaging which is used to average the Darcy-scale equations over a region that is large compared to the length scale of the heterogeneities. In this paper we present the derivation of the large-scale averaged continuity and momentum equations, and we develop a method of closure that can be used to predict the large-scale permeability tensors and the large-scale capillary pressure. The closure problem is limited by the principle of local mechanical equilibrium. This means that the local fluid distribution is determined by capillary pressure-saturation relations and is not constrained by the solution of an evolutionary transport equation. Special attention is given to the fact that both fluids can be trapped in regions where the saturation is equal to the irreducible saturation, in addition to being trapped in regions where the saturation is greater than the irreducible saturation. Theoretical results are given for stratified porous media and a two-dimensional model for a heterogeneous porous medium.     

Propagation of Normal Waves Through a Fluid Contained in a Thin-Walled Cylinder

The properties of normal axisymmetric waves propagating through a perfect compressible fluid contained in an elastic thin-walled cylinder are investigated. The problem is solved using the complete system of equations of the dynamic theory of elasticity. The effects of interaction between elastic and fluid waves are studied within a wide frequency range. The numerical results are classified on the basis of data on the properties of partial subsystems. Partial subsystems are those for which the interaction effects are insignificant. For special cases of compound waveguides, the dispersion spectra are constructed and the kinematic and energy characteristics of normal waves are analyzed. Particular attention is given to the lowest normal wave, which has specific properties and participates in the elastic–liquid interaction over a wide frequency range.     

Plasma in a high-frequency electric field with a reflective condition on the boundary

An analytical solution is obtained for the linearized problem of the behavior of a collisional gas plasma in a half-space in an external alternating electric field. It is assumed that the electrons are mirror-reflected from the plasma boundary. The solution is used for finding the screened field. The case of an external field frequency close to the plasma frequency is investigated separately.     

The dynamics of a droplet in a capillary constriction underwave excitation

The frequency dependence of the amplitude of the wave excitation mobilizing a droplet trapped in a capillary constriction is determined. The effect of droplet viscosity is analyzed. The problem of free longitudinal oscillations of a viscous-fluid droplet in a capillary constriction is considered. The influence of the surface tension, the droplet volume, and the constriction shape on the natural frequency of droplet longitudinal oscillations is studied. A formula for calculating the droplet natural frequency in the conical constriction is obtained and analyzed.