  The transmission of acoustic phonons is an important element in the design
and performance of nano-mechanical devices operating in the mesoscopic limit.
Analytic expressions for the power transmission coefficient, T, exist only in
the low-frequency (quasi-static) limit described by thin-plate elastic theory,
and for well-defined elastic wave-guiding geometries. We compare two numerical
techniques based on finite-element computations to determine the frequency
dependence of T for arbitrary phonon scattering structures. Both methods take
into account acoustic mode conversion to acoustic and optical modes. In one
case, phase and amplitude of complex-valued reflected waves are determined and
related to transmission through a Fresnel equation, while in the other the
magnitude of the transmitted mechanical power is directly calculated. The
numerical robustness of these methods is demonstrated by considering the
transmission across an abrupt junction in a rectangular elastic beam, a
well-known problem of considerable importance in mesoscopic device physics. The
simulations presented extend the standard results for acoustic phonon
transmission at an abrupt junction, and are in good agreement with analytic
predictions from thin-plate elastic theory in the long-wavelength limit. More
generally, the numerical methods developed provide an effective tool for
calculating acoustic mode energy loss in nano-mechanical resonators through
mode conversion and heat transfer in arbitrary mesoscopic structures.
