  We numerically study imbalanced two component Fermi gases with attractive
interactions in highly elongated harmonic traps. An accurate parametrization
formula for the ground state energy is presented for a spin-polarized
attractive Gaudin-Yang model. Our studies are based on an accurate microscopic
spin-density-functional theory through the Kohn-Sham scheme which employs the
one-dimensional homogeneous Gaudin-Yang model with Luther-Emery-liquid
ground-state correlation as a reference system. A Thomas-Fermi approximation is
examined incorporating the exchange-correlation interaction. By studying the
charge and spin density profiles of the system based on these methods, we gain
a quantitative understanding of the role of attractive interactions and
polarization on the formation of the two-shell structure, with the coexisted
Fulde-Ferrell-Larkin-Ovchinnikov-type phase in the center of the trap and
either the BCS superfluid phase or the normal phase at the edges of the trap.
Our results are in good agreement with the recent theoretical consequences.
