Applications of guided wave propagation on waveguides with irregular cross-section

Guided waves are interesting for Non-destructive Testing (NDT) since they offer the potential for rapid inspections of a large variety of structures. Analytical methods are well known for predicting properties of guided waves such as mode shapes and dispersion curves on regular geometries, e.g. plain plates or cylindrical structures. However these methods cannot be used to study guided wave propagation in waveguides having irregular cross-sectional geometries, such as railway lines, T-shape beams or stiffened plates. This thesis applies and develops a Semi-Analytical Finite Element (SAFE) method, which uses finite elements to represent the cross-section of the waveguide and a harmonic description along the propagation direction, to investigate the modal properties of structures with irregular cross-section. Two attractive applications have been investigated with the SAFE method, and the results are encouraging. The first application relates to fluid characterization. Guided torsional waves in a bar with a non-circular cross-section have been exploited by previous researchers to measure the density of fluids. However, due to the complexity of the wave behavior in the non-circular cross-sectional shape, the previous theory can only provide an approximate prediction; thus the accuracy of the measurement has been compromised. The SAFE method is developed to model accurately the propagation velocity and leakage of guided waves along an immersed waveguide with arbitrary non-circular cross-section. An accurate inverse model is then provided to measure the density of the fluid by measuring the change of the torsional wave speed. The model also enables the optimization of the dipstick sensor by changing the material of the dipstick and the geometry of the cross-section. Experimental results obtained with a rectangular bar in a range of fluids show very good agreement with the theoretical predictions. The second application relates to the inspection of large areas of complex structures. An experimental observation on a large welded plate found that the weld can concentrate and guide the energy of a guided wave traveling along the direction of the weld. This is attractive for NDE since it offers the potential to quickly inspect for defects such as cracking or corrosion along long lengths of welds. The SAFE method is applied to provide a modal study of the elastic waves which are guided by the welded joint in a plate. This brings understanding to the compression wave which was previously observed in the experiment. However, during the study, a shear weld-guided mode, which is non-leaky and almost non-dispersive has also been discovered. Its characteristics are particularly attractive for NDT, so this is a significant new finding. The properties for both the compression and the shear mode are discussed and compared, and the physical reason for the energy trapping phenomena is explained. Experiments have been undertaken to validate the existence of the shear weld-guided mode and the accuracy of the FE model, showing very good results. The sensitivity of compression and shear weld-guided modes to different types of defects close to the weld is investigated, by both finite element simulations and experiments. Due to similar reasons for energy trapping, the feature guiding phenomena also exists in a wide range of geometries. This thesis finally discusses feature guided waves on lap joints, stiffened plates and interconnected heat exchanger tube plates, and their potential applications

Simulating the magnetic fields generated by piezoelectric devices using FEM software: Beyond the quasistatic approximation

A method for simulating coupled electromagnetic and mechanical vibrations on arbitrarily shaped piezoelectric structures is presented. This method is based on weak forms and can be implemented in any finite-element-method software, allowing editable access to their definitions. No quasi-static approximation is imposed, meaning that magnetic fields generated by displacement currents within piezoelectric materials are captured, enabling the flow of electromagnetic energy inside and around structures containing such material to be accurately simulated. The method is particularly relevant to the design of piezoelectric antennas, resonators, and waveguides exploiting either bulk or surface-acoustic waves. The accuracy and capabilities of the method are demonstrated by simulating, in COMSOL Multiphysics, (i) a Rayleigh mode on the surface of Z-cut lithium niobate crystal and (ii) a torsional mode of a cylinder of lead zirconium titanate (PZT-5H) ceramic functioning as a micro-antenna

Transverse oscillations of coronal loops

On 14 July 1998 TRACE observed transverse oscillations of a coronal loop generated by an external disturbance most probably caused by a solar flare. These oscillations were interpreted as standing fast kink waves in a magnetic flux tube. Firstly, in this review we embark on the discussion of the theory of waves and oscillations in a homogeneous straight magnetic cylinder with the particular emphasis on fast kink waves. Next, we consider the effects of stratification, loop expansion, loop curvature, non-circular cross-section, loop shape and magnetic twist. An important property of observed transverse coronal loop oscillations is their fast damping. We briefly review the different mechanisms suggested for explaining the rapid damping phenomenon. After that we concentrate on damping due to resonant absorption. We describe the latest analytical results obtained with the use of thin transition layer approximation, and then compare these results with numerical findings obtained for arbitrary density variation inside the flux tube. Very often collective oscillations of an array of coronal magnetic loops are observed. It is natural to start studying this phenomenon from the system of two coronal loops. We describe very recent analytical and numerical results of studying collective oscillations of two parallel homogeneous coronal loops. The implication of the theoretical results for coronal seismology is briefly discussed. We describe the estimates of magnetic field magnitude obtained from the observed fundamental frequency of oscillations, and the estimates of the coronal scale height obtained using the simultaneous observations of the fundamental frequency and the frequency of the first overtone of kink oscillations. In the last part of the review we summarise the most outstanding and acute problems in the theory of the coronal loop transverse oscillations

Short Timescale Core Dynamics: TheoryandObservations

Fluid motions in the Earth's core produce changes in the geomagnetic field (secular variation) and are also an important ingredient in the planet's rotational dynamics. In this article we review current understanding of core dynamics focusing on short timescales of years to centuries. We describe both theoretical models and what may be inferred from geomagnetic and geodetic observations. The kinematic concepts of frozen flux and magnetic diffusion are discussed along with relevant dynamical regimes of magnetostrophic balance, tangential geostrophy, and quasi-geostrophy. An introduction is given to free modes and waves that are expected to be present in Earth's core including axisymmetric torsional oscillations and non-axisymmetric Magnetic-Coriolis waves. We focus on important recent developments and promising directions for future investigation

Modelling borehole wave signatures in elastic and poroelastic media with spectral method

Borehole sonic measurements are an important tool to characterize formation and completion properties of hydrocarbon or water reservoirs. Such measurements can provide direct information about rock physical parameters such as permeability or elastic moduli. These properties are obtained from guided waves propagating along boreholes. The so called tube wave or Stoneley wave is a symmetric mode which compresses the fluid column leading to a piston like motion. If the medium around the borehole wall is permeable, the radial expansion of the fluid column will result in fluid flow across the borehole wall. This results in a sensitivity of the tube wave signature to the permeability of the surrounding formation which manifests itself in a characteristic dispersion and attenuation of the tube wave. Information about the permeability of the surrounding formation provides essential knowledge for reservoir characterization.In addition to the traditional method of using tube wave signatures for formation permeability estimations, the same approach may be used for production monitoring. In sand reservoirs a complicated borehole completion is installed during the production phase for the purpose of controlling sand production. In such a setup highly permeable layers such as a sand screen or a gravel pack are used to prevent sand production.The problem with such completions is that they are very expensive to install and susceptible to plugging or corrosion. No permanent surveillance tool exists to date which allows diagnosis of problems in sand-screened deepwater completions. However, the recently proposed Real-Time Completion Monitoring (RTCM) uses the signature of tube waves to identify permeability changes: the increase of the tube wave velocity can indicate a decrease of permeability and vice versa. Therefore, RTCM has potential to identify problems in sand-screened deepwater completions.In order to understand the acoustic response of such deepwater completions, the dispersion and attenuation of tube waves in this complicated setup needs to be studied. To this end I have developed a modelling algorithm based on a spectral method. The developed algorithm computes the dispersion and attenuation of borehole modes propagating in a cylindrically layered structure with an arbitrary number of fluid, elastic and poroelastic layers. The numerical algorithm discretizes the medium along the radial axis using Chebyshev interpolation points derived from Chebyshev polynomials. The differential operators are discretized using spectral differentiation matrices. Thus, for any number of layers, the corresponding equations can be expressed as a generalized algebraic eigenvalue problem. For a given frequency, the eigenvalues correspond to the wavenumbers of different modes. The eigenvectors, computed along with the eigenvalues, correspond to the displacement potentials. They can be used to obtain the variation of displacement and stress components along the radius of the structure.In this thesis the spectral method was first developed for structures with an arbitrary number of fluid and elastic layers. Subsequently, the algorithm was extended for poroelasticity. The results produced by the modelling program are benchmarked against analytical solutions. Such analytical solutions are known for elastic and poroelastic cylinders as well as fluid filled tubes. The tube wave dispersion in a fluid-filled borehole surrounded by an elastic or poroelastic formation obtained with the spectral method was compared to the analytical low-frequency solution.I obtained the dispersion of the two tube waves propagating in a four layer completion model: fluid – permeable sand-screen – fluid – elastic casing. Varying the permeability of the sand-screen layer allowed me to account for the effect of fluid flow across this layer. Being able to obtain the acoustic response can help to identify broken fluid communication which increases the tube wave velocity. A corroded sand-screen has an extremely attenuated tube wave signature.Furthermore, I have implemented the more complex model of a borehole surrounded by an altered zone in the algorithm. Due to drilling damage the altered zone is an area of reduced permeability. In order to account for the effect of the altered zone on the tube wave signature, up to ten layers were used with stepwise increase of permeability from the borehole towards the formation. Overall, the spectral method proved to be a valuable algorithm to model wave propagation in cylindrical structures.Using borehole modes to evaluate the physical properties of the formation or completions is an important application. However, in borehole seismic modelling, such as crosshole or VSP, it is also important to account for the effect of boreholes and the associated modes. Since the borehole radius is a thousand times smaller then the investigated volume it would require a prohibitively small grid size to explicitly model the borehole. However, it is possible to effectively represent a borehole as a superposition of point sources. This mimics the presence of borehole modes. In order to implement this technique for poroelasticity, it is necessary to model source signatures in poroelastic media. To this end I have analyzed the radiation characteristics and moment tensor solutions for various source types. Together with the spectral method these point source representations can be used to model the effect of boreholes. This will pave the way for more efficient poroelastic seismic modelling in various fluid-filled boreholes and completions

On the application of finite element analysis to wave motion in one-dimensional waveguides

This thesis considers issues concerning the application of the wave finite element (WFE) method to the free and forced vibrations of one-dimensional waveguides. A short section of the waveguide is modelled using conventional finite element (FE) methods. A periodicity condition is applied and the resulting mass and stiffness matrices are post-processed to yield the dispersion relations and so on. First, numerical issues are discussed and methods to reduce the errors are proposed. FE discretisation errors and errors due to round-off of inertia terms are described. A method using concatenated elements is proposed to reduce those round-off errors. Conditioning of the eigenvalue problem is discussed. An application of singular value decomposition is proposed to reduce errors in numerically determining eigenvectors together with Zhong’s formulation of the eigenvalue problem. Effects of the modelling of the cross-section on conditioning are shown. Three methods for numerically determining the group velocity are compared and the power and energy relationship is seen to be reliable. The WFE method is then applied to complicated structures and its accuracy evaluated. Dispersion curves are shown including purely real, purely imaginary and complex wavenumbers. Free wave propagation in a plate strip with free edges, a ring and a cylindrical strip is predicted and the results compared with analytical or numerical solutions to the analytical dispersion equations. In particular, dispersion curves for freely propagating flexural waves, including attenuating waves, are presented. Complicated phenomena such as curve veering, non-zero cut-on phenomena and bifurcations are observed as results of wave coupling in the wave domain. A method of decomposition of the power is proposed to reduce the size of the system matrices and to investigate the wave characteristics of each wave mode. The wave approach is then used to predict the forced response. A well-conditioned formulation for determining the amplitudes of directly excited waves is proposed. The forced response is determined by considering wave propagation and subsequent reflection at boundaries. Numerical examples of a beam, a plate and a cylinder are shown. Inclusion of rapidly decaying waves is discussed. As a practical application, free and forced vibrations of a tyre are analysed. The complicated cross-section of a tyre is modelled using a commercial FE package. Frequency dependent material properties of rubber are included. Free wave propagation is shown including attenuating waves and predicted responses are compared with experiment. Effects of the size of the excited region are discussed

An Experimental and Theoretical Investigation fo Axially Symmetric Wave Propagation In Thick Cylindrical Waveguides

Solid circular cylinders as wavaguides for the propagation of longitudinal elastic waves are used pximarily as buffer rods in high temperature nondestructive evaluation (NDE), and are also found in the split Hopkinson pressure bar (SHPB). Experiments are typically designed so that only the nondispersive range of the first mode propagates. Design constraints sometimes require larger wavcguides and higher ficquencies that propagate multiple dispersive modes, which can add considerable con1plexity to the signal. This thesis presents an analytical modcl for multiple mode wave propagation in a finite solid cylindrical waveguide as a means of interpreting the complex signals and possibly removing the complexity. The model uses the phase velocities and normal stresses of the axially symmetric modes calculated by the Pochhammer-Chree equations to calculate atransfer hnction for each of the propagating modes. The sum of the tranxfcr functions of the propag:,ting modes is the transfer function of the waveguide, which can be used to predict the change of a signal in the waveguide. The ability of the model to accurately capture the general physics of multiple mode wave propagation is demonstrated in the time, frequency and joint time-frequency domain. In the time-reverral domain the calculated dispersed signal for a dispersive multi-mode waveguide is shown to producc a s i p a l with compact support in the time domain. A range of diameter to wavelength ratios is considered for these comparisons, which show the limitations of the model for wavelengths less than th.e raditls. The transfer functions generated by the model indicate which modes are dominant over a particular range of frequencies and which modes have a much smaller magnitude. The transfer functions further indicate that broadband signals are composed of multiple modes. It is found that observed trailing pulses contain energy from multiple propagating modes, and it is the superposition of the modes that creates the trailing pulses. The information from the transfer functions is also used to show the conditions for a sufficiently narrow band signal to excite a single higher order mode with little dispersion