  We carry out three-dimensional hydrodynamic simulations of the supernova
remnants (SNRs) produced inside molecular clouds (MCs) near their surface using
the HLL code (Harten et al. 1983). We explore the dynamical evolution and the
X-ray morphology of SNRs after breaking through the MC surface for ranges of
the explosion depths below the surface and the density ratios of the clouds to
the intercloud media (ICM). We find that if an SNR breaks out through an MC
surface in its Sedov stage, the outermost dense shell of the remnant is divided
into several layers. The divided layers are subject to the Rayleigh-Taylor
instability and fragmented. On the other hand, if an SNR breaks through an MC
after the remnant enters the snowplow phase, the radiative shell is not divided
to layers. We also compare the predictions of previous analytic solutions for
the expansion of SNRs in stratified media with our onedimensional simulations.
Moreover, we produce synthetic X-ray surface brightness in order to research
the center-bright X-ray morphology shown in thermal composite SNRs. In the late
stages, a breakout SNR shows the center-bright X-ray morphology inside an MC in
our results. We apply our model to the observational results of the X-ray
morphology of the thermal composite SNR 3C 391.
