  We solve exactly a class of Frenkel-Kontorova models with piecewise parabolic
potential, which has $d$ sub-wells in a period. With careful analysis, we show
that the phase diagram of the minimum enthalpy configurations exhibits the
structure of a complete $d$-dimensional devil's staircase. The winding number
of a minimum enthalpy configuration is locked to rational values, while the
fraction of atoms in each sub-well is locked to values which are
sub-commensurable with the winding number.
