Moir\'e and frustration physics of dipolar supersolids under periodic confinement

We study the ground-state phases of a two-dimensional dipolar supersolid subjected to external periodic confinement by numerically solving the extended Gross--Pitaevskii equation. Focusing on a regime in which the unconfined system forms an intrinsic triangular droplet crystal, we consider triangular, honeycomb, and square optical lattices and classify them into isostructural and heterostructural settings relative to the spontaneous supersolid order. We map out the stationary states as functions of the lattice depth $V_0$ and the commensurability ratio between the intrinsic droplet spacing and the external lattice period. For triangular and honeycomb confinements, the competition between the soft self-organized supersolid lattice and the rigid external potential can generate long-wavelength moir\'e superstructures in the weak- to intermediate-lattice regime, together with a sequence of reconstructed states including ring-like clusters and stripe-segment configurations. By contrast, the square lattice introduces strong symmetry mismatch between the intrinsic $C_6$ order and the imposed $C_4$ geometry, leading to frustration-induced anisotropic states and symmetry-reduced cluster arrangements. Our results establish dipolar supersolids under periodic confinement as an unconventional route to exploring moir\'e physics, where moir\'e superstructures arise from the competition between a self-organized soft lattice and an externally imposed rigid one.