Bounds on the hausdorff dimension of a renormalisation map arising from an excitable reaction-diffusion system on a fractal lattice

A renormalisation approach to investigate travelling wave solutions of an excitable reaction- diusion system on a deterministic fractal structure has recently been derived. The dynamics of a particular class of solutions which are governed by a two dimensional subspace of these renormalisation recursion relationships are discussed in this paper. The bifurcations of this mapping are discussed with reference to the discontinuities which arise at the singularities. The map is chaotic for a bounded region in parameter space and bounds on the Hausdor dimension of the associated invariant hyperbolic set are calculated

The causal differential scattering approach to calculating the effective properties of random composite materials with a particle size distribution

An implementation of the Causal Differential Method (CDM) for modelling the effective properties of a random two-phase composite material is presented. Such materials are commonly used as ultrasonic transducer matching layersor backing layers. The method is extended to incorporate a particle size distribution in the inclusion phase. Numerical issues regarding the implementation and convergence of the method are discussed. It is found that, for a given frequency of excitation, the calculated velocity for the composite has a distribution whose variance increases as the volume fraction of inclusions increases. The model predictions would suggest that to reliably and repeatedly manufacture these composites, with a desired mechanical impedance, a low volume fraction of inclusions should be used

A theoretical investigation of 2-2 composite transducers with high shear attenuation in the passive phase

This paper is about a theoretical investigation of 2-2 composite transducers with high shear attenuation in the passive phase. It was presented at the Étude de la propagation ultrasonore en milieux non homogènes en vue du contrôle non-destructif in 2007

Incorporation of viscoelastic loss into the plane wave expansion approach to modelling composite transducers

The plane wave expansion (PWE) method has been proposed as a theoretical model for periodic composite ultrasonic transducers. This paper extends previous work by importantly including viscoelastic loss in the material parameters. Some of the issues with model formulation, such as ill-conditioning in the large matrices, have been addressed through parameter scaling and Tikhonov regularisation. Identification of each mode of vibration has been carried out by visualising the spatial and temporal profiles of the displacement, electrical potential and Poynting vector. A comparison between the theoretical predictions and experimental data from a piezoelectric composite device is presented. The effect that the elastic properties of the passive phase have on device performance is also investigated. It is found that high shear attenuation in the passive phase gives rise to a large frequency stop band gap around the fundamental thickness mode

A theoretical model of a new electrostatic transducer incorporating fluidic amplification

This article concerns the design of a new electrostatic transducer whose backplate consists of a series of drilled pipes. A new one-dimensional model is derived which considers the interaction of the membrane with the air load, the air cavities, and the drilled pipes in the backplate. Dynamic equations for the impedance in each component of the device are calculated analytically and connected using interface conditions of continuity of pressure and radiation conditions into the air load. The model is able to produce solutions to the mechanical impedance of the device and the displacement of the membrane as a function of the device's design parameters. Model results for the output pressure compare well with previous experimental data. The inverse problem of retrieving the design parameters for a desired output is discussed

An inverse problem of reconstructing the electrical and geometrical parameters characterising airframe structures and connector interfaces

This article is concerned with the detection of environmental ageing in adhesively bonded structures used in the aircraft industry. Using a transmission line approach a forward model for the reflection coefficients is constructed and is shown to have an analytic solution in the case of constant permeability and permittivity. The inverse problem is analysed to determine necessary conditions for a unique recovery. The main thrust of this article then involves modelling the connector and then experimental rigs are built for the case of the air-filled line to enable the connector parameters to be identified and the inverse solver to be tested. Some results are also displayed for the dielectric-filled line

Theoretical modelling of frequency dependent elastic loss in composite piezoelectric transducers

The large number of degrees of freedom in the design of piezoelectric transducers requires a theoretical model that is computationally efficient so that a large number of iterations can be performed in the design optimisation. The materials used are often lossy, and indeed loss can be used to enhance the operational characteristics of these designs. Motivated by these needs, this paper extends the one-dimensional linear systems model to incorporate frequency dependent elastic loss. The reception sensitivity, electrical impedance and electromechanical coupling coefficient of a 1-3 composite transducer, with frequency dependent loss in the polymer filler, are investigated. By plotting these operating characteristics as a function of the volume fraction of piezoelectric ceramic an optimum design is obtained. A device with a non-standard, high shear attenuation polymer is also simulated and this leads to an increase in the electromechanical coupling coefficient. A comparison with finite element simulations is then performed. This shows that the two methods are in reasonable agreement in their electrical impedance profiles in all the cases considered. The plots are almost identical away from the main resonant peak where the frequency location of the peaks are comparable but there is in some cases a 20% discrepancy in the magnitude of the peak value and in its bandwidth. The finite element model also shows that the use of a high shear attenuation polymer filler damps out the unwanted, low frequency modes whilst maintaining a reasonable impedance magnitude

Simulation of deposit formation in particle laden flows

Fatty deposits formed on arterial walls lead to atherosclerosis but it is the interplay between these deposits and the vessel walls which govern the growth of plaque formation. Crucially however the vast majority of acute coronary syndromes such as, myocardial infarction, and sudden ischaemic cardiac death are caused by atherosclerotic plaque rupture and not from a stenosis growing and blocking the blood flow. In fact, atherosclerotic plaques expand into the vessel wall during much of their existence and this can make their detection problematic. However inflammation within the necrotic core of the plaque, can be used to detect which plaques may be vulnerable. Thermal mapping of arterial walls can help identify the most likely sites for plaque rupture. This paper aims to provide a direct link between the geometry of these deposits and their thermal properties in order that non-invasive imaging techniques could be used to spot vulnerable plaques. We will discuss a methodology for estimating the thermal conductivity which utilises self-similarity properties using fractal analysis and renormalisation. The selfsimilar microstructure is captured by a family of random fractals called shuffled Sierpinski carpets (SSC). The thermal conductivity of the SSC can then be predicted both from its box counting fractal dimension and via a generalised real space renormalisation method. This latter approach also affords an analysis of the percolation threshold of two phase fractal media