  The standard cold dark matter (CDM) model predicts too many and too dense
small structures. We consider an alternative model that the dark matter
undergoes two-body decays with cosmological lifetime $\tau$ into only one type
of massive daughters with non-relativistic recoil velocity $V_k$. This decaying
dark matter model (DDM) can suppress the structure formation below its
free-streaming scale at time scale comparable to $\tau$. Comparing with warm
dark matter (WDM), DDM can better reduce the small structures while being
consistent with high redshfit observations. We study the cosmological structure
formation in DDM by performing self-consistent N-body simulations and point out
that cosmological simulations are necessary to understand the DDM structures
especially on non-linear scales. We propose empirical fitting functions for the
DDM suppression of the mass function and the mass-concentration relation, which
depend on the decay parameters lifetime $\tau$ and recoil velocity $V_k$, and
redshift. The fitting functions lead to accurate reconstruction of the the
non-linear power transfer function of DDM to CDM in the framework of halo
model. Using these results, we set constraints on the DDM parameter space by
demanding that DDM does not induce larger suppression than the Lyman-$\alpha$
constrained WDM models. We further generalize and constrain the DDM models to
initial conditions with non-trivial mother fractions and show that the halo
model predictions are still valid after considering a global decayed fraction.
Finally, we point out that the DDM is unlikely to resolve the disagreement on
cluster numbers between the Planck primary CMB prediction and the
Sunyaev-Zeldovich (SZ) effect number count for $\tau \sim H_{0}^{-1}$.
