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We show that every smooth Schubert variety of affine type $\tilde{A}$ is aniterated fibre bundle of Grassmannians, extending an analogous result by Ryanand Wolper for Schubert varieties of finite type $A$. As a consequence, wefinish a conjecture of Billey-Crites that a Schubert variety in affine type$\tilde{A}$ is smooth if and only if the corresponding affine permutationavoids the patterns $4231$ and $3412$. Using this iterated fibre bundlestructure, we compute the generating function for the number of smooth Schubertvarieties of affine type $\tilde{A}$.

Smooth Schubert varieties in the affine flag variety of type $\tilde{A}$