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We present a detailed study of the ground-state magnetic structure of ultrathin Fe films on the surface of fcc Ir(001). We use the spin-cluster expansion technique in combination with the relativistic disordered local moment scheme to obtain parameters of spin models and then determine the favored magnetic structure of the system by means of a mean-field approach and atomistic spin dynamics simulations. For the case of a single monolayer of Fe, we find that layer relaxations very strongly influence the ground-state spin configurations, whereas Dzyaloshinskii-Moriya (DM) interactions and biquadratic couplings also have remarkable effects. To characterize the latter effect, we introduce and analyze spin collinearity maps of the system. While for two monolayers of Fe we find a single- spin spiral as ground state due to DM interactions, for the case of four monolayers, the system shows a noncollinear spin structure …