  We use high-resolution N-body simulations to study the galaxy-cluster
cross-sections and the abundance of giant arcs in the $\Lambda$CDM model.
Clusters are selected from the simulations using the friends-of-friends method,
and their cross-sections for forming giant arcs are analyzed. The background
sources are assumed to follow a uniform ellipticity distribution from 0 to 0.5
and to have an area identical to a circular source with diameter $1\arcsec$. We
find that the optical depth scales as the source redshift approximately as
$\tau_{1''} = 2.25 \times 10^{-6}/[1+(\zs/3.14)^{-3.42}]$ ($0.6<\zs<7$). The
amplitude is about 50% higher for an effective source diameter of $0.5\arcsec$.
The optimal lens redshift for giant arcs with the length-to-width ratio ($L/W$)
larger than 10 increases from 0.3 for $\zs=1$, to 0.5 for $\zs=2$, and to
0.7-0.8 for $\zs>3$. The optical depth is sensitive to the source redshift, in
qualitative agreement with Wambsganss et al. (2004). However, our overall
optical depth appears to be only $\sim$ 10% to 70% of those from previous
studies. The differences can be mostly explained by different power spectrum
normalizations ($\sigma_8$) used and different ways of determining the $L/W$
ratio. Finite source size and ellipticity have modest effects on the optical
depth. We also found that the number of highly magnified (with magnification
$|\mu|>10$) and ``undistorted'' images (with $L/W<3$) is comparable to the
number of giant arcs with $|\mu|>10$ and $L/W>10$. We conclude that our
predicted rate of giant arcs may be lower than the observed rate, although the
precise `discrepancy' is still unclear due to uncertainties both in theory and
observations.
