A two-dimensional fluid model was used to investigate the characteristics of helium dielectric-barrier discharge (DBD) equipped with double-ring electrodes at atmospheric pressure. Simulation results show that although the temporal evolutions of discharge current and current density at different radial positions exhibit the same or similar characteristics to those in traditional DBD, a distinctive spatial discharge structure is observed during the discharge process. The spatial distribution of electron density at current peak moments exhibits a periodical complementary feature between the center-advantage and the periphery-advantage with the inner ring electrode covering the radial axis from 0.5 to 1.0mm. When the inner ring electrode covers the radial axis from 1.0 to 1.5mm, the spatial distribution of electron density satisfies another periodical complementary behavior between the electrode-center-advantage and the coexistence of periphery-advantage and center-advantage. These complementary discharge features mainly result from non-uniform electric field and surface charge distribution at the end of the previous discharge. The difference of the discharge structure between the two cases is attributed to the fact that Laplacian field distributions are largely dependent on the arrangement of double-ring electrodes. Our observations show that electrode geometry plays an important role in the formation of the discharge structure.