Scattering by a thin and flat elliptic object: How the conformal mapping approach allows addressing numerical issues

Abstract

In this work, we are interested in modeling the propagation of an ultrasonic field in the complex trabecular bone structure. For this purpose, we use a simplified formulation, based on a representation of the bone tissue as thin and flat elliptic scatterers. With an ultrasonic inspection frequency of 1 MHz, the largest dimension of the ellipse is about half a wavelength, while the smallest dimension is 7–10 times smaller. The computation of the scattered field is based on a modal decomposition, generalized to the case of scatterers of arbitrary geometry (Chati et al., “Modal theory applied to the acoustic scattering by elastic cylinders of arbitrary cross section,” J. Acoust. Soc. Am. 116, 2004). Due to the very thin and flat geometry, this approach suffers huge numerical difficulties that require the use of enhanced precision and very long computation times. In this paper, we illustrate these issues. An alternative has been proposed (Liu et al., “Conformal mapping for the Helmholtz equation: Acoustic wave scattering by a two dimensional inclusion with irregular shape in an ideal fluid,” J. Acoust. Soc. Am. 131, 2012), based on a conformal mapping of the ellipse. We compare the two formulations and show how the conformal mapping allows reducing drastically the numerical issues resulting from the standard approach.