  Because phononic topological insulators have primarily been studied in
discrete, graphene-like structures with C$_{6}$ or C$_{3}$ hexagonal symmetry,
an open question is how to systematically achieve double Dirac cones and
topologically non-trivial structures using continuous, non-hexagonal unit
cells. Here, we address this challenge by presenting a novel computational
methodology for the inverse design of continuous two-dimensional square
phononic metamaterials exhibiting C$_{4v}$ and C$_{2v}$ symmetry. This leads to
the systematic design of square unit cell topologies exhibiting a double Dirac
degeneracy, which enables topologically-protected interface propagation based
on the quantum spin Hall effect (QSHE). Numerical simulations prove that
helical edge states emerge at the interface between two topologically distinct
square phononic metamaterials, which opens the possibility of QSHE-based
pseudospin-dependent transport beyond hexagonal lattices.
