  Micro- and nanoscale mechanical resonators have recently emerged as
ubiquitous devices for use in advanced technological applications, for example
in mobile communications and inertial sensors, and as novel tools for
fundamental scientific endeavors. Their performance is in many cases limited by
the deleterious effects of mechanical damping. Here, we report a significant
advancement towards understanding and controlling support-induced losses in
generic mechanical resonators. We begin by introducing an efficient numerical
solver, based on the "phonon-tunneling" approach, capable of predicting the
design-limited damping of high-quality mechanical resonators. Further, through
careful device engineering, we isolate support-induced losses and perform the
first rigorous experimental test of the strong geometric dependence of this
loss mechanism. Our results are in excellent agreement with theory,
demonstrating the predictive power of our approach. In combination with recent
progress on complementary dissipation mechanisms, our phonon-tunneling solver
represents a major step towards accurate prediction of the mechanical quality
factor.
