  Dielectric membranes with exceptional mechanical and optical properties
present one of the most promising platforms in quantum opto-mechanics. The
performance of stressed silicon nitride nanomembranes as mechanical resonators
notoriously depends on how their frame is clamped to the sample mount, which in
practice usually necessitates delicate, and difficult-to-reproduce mounting
solutions. Here, we demonstrate that a phononic bandgap shield integrated in
the membrane's silicon frame eliminates this dependence, by suppressing
dissipation through phonon tunneling. We dry-etch the membrane's frame so that
it assumes the form of a $\mathrm{cm}$-sized bridge featuring a 1-dimensional
periodic pattern, whose phononic density of states is tailored to exhibit one,
or several, full band gaps around the membrane's high-$Q$ modes in the
MHz-range. We quantify the effectiveness of this phononic bandgap shield by
optical interferometry measuring both the suppressed transmission of
vibrations, as well as the influence of frame clamping conditions on the
membrane modes. We find suppressions up to $40~\mathrm{dB}$ and, for three
different realized phononic structures, consistently observe significant
suppression of the dependence of the membrane's modes on sample clamping - if
the mode's frequency lies in the bandgap. As a result, we achieve membrane mode
quality factors of $5\times 10^{6}$ with samples that are tightly bolted to the
$8~\mathrm{K}$-cold finger of a cryostat. $Q\times f$-products of $6\times
10^{12}~\mathrm{Hz}$ at $300~\mathrm{K}$ and $14\times 10^{12}~\mathrm{Hz}$ at
$8~\mathrm{K}$ are observed, satisfying one of the main requirements for
optical cooling of mechanical vibrations to their quantum ground-state.
