ABSTRACT
In this work, the concept of high-frequency homogenization is extended to the case of one-dimensional periodic media with imperfect interfaces of the spring-mass type. In other words, when considering the propagation of elastic waves in such media, displacement and stress discontinuities are allowed across the borders of the periodic cell. As is customary in high-frequency homogenization, the homogenization is carried out about the periodic and antiperiodic solutions corresponding to the edges of the Brillouin zone. Asymptotic approximations are provided for both the higher branches of the dispersion diagram (second-order) and the resulting wave field (leading-order). The special case of two branches of the dispersion diagram intersecting with a non-zero slope at an edge of the Brillouin zone (occurrence of a so-called Dirac point) is also considered in detail, resulting in an approximation of the dispersion diagram (first-order) and the wave field (zeroth-order) near these points. Finally, a uniform approximation valid for both Dirac and non-Dirac points is provided. Numerical comparisons are made with the exact solutions obtained by the Bloch-Floquet approach for the particular examples of monolayered and bilayered materials. In these two cases, convergence measurements are carried out to validate the approach, and we show that the uniform approximation remains a very good approximation even far from the edges of the Brillouin zone.
ABSTRACT
The investigation of quickly-evolving flow patterns in high-pressure and high-temperature flow rigs requires the use of a high-speed and non-intrusive imaging technique. Electrical Impedance Tomography (EIT) allows reconstructing the admittivity distribution characterizing a flow from the knowledge of currents and voltages on its periphery. The need for images at high frame rates leads to the strategy of simultaneous multi-frequency voltage excitations and simultaneous current measurements, which are discriminated using fast Fourier transforms. The present study introduces the theory for a 16-electrode simultaneous EIT system, which is then built based on a field programmable gate array data acquisition system. An analysis of the propagation of uncertainties through the measurement process is investigated, and experimental results with fifteen simultaneous signals are presented. It is shown that the signals are successfully retrieved experimentally at a rate of 1953 frames per second. The associated signal-to-noise ratio varies from 59.6-69.1 dB, depending on the generated frequency. These preliminary results confirm the relevance and the feasibility of simultaneous multi-frequency excitations and measurements in EIT as a means to significantly increase the imaging rate.
