  Recent progress in theory, experiment and observation challenges the mean
field models using the conventional Skyrme interaction, suggesting that the
extension of the conventional Skyrme interaction is necessary. In this work, by
fitting the experimental data of a number of finite nuclei together with a few
additional constraints on nuclear matter using the simulated annealing method,
we construct three Skyrme interaction parameter sets, namely, eMSL07, eMSL08
and eMSL09, based on an extended Skyrme interaction which includes additional
momentum and density dependent two-body forces to effectively simulate the
momentum dependence of the three-body force. The three new interactions can
reasonably describe the ground-state properties and the isoscalar giant
monopole resonance energies of various spherical nuclei used in the fit as well
as the ground-state properties of many other spherical nuclei, nicely conform
to the current knowledge on the equation of state of asymmetric nuclear matter,
eliminate the notorious unphysical instabilities of symmetric nuclear matter
and pure neutron matter up to a very high density of $1.2$ fm$^{-3}$, and
simultaneously support heavier neutron stars with mass larger than two times
solar mass. One important difference of the three new interactions is about the
prediction of the symmetry energy at supra-saturation densities, and these new
interactions are thus potentially useful for the determination of the largely
uncertain high-density symmetry energy in future. In addition, a comparison is
made for the predictions of nuclear matter, finite nuclei and neutron stars
with the three new interactions versus those with three typical interactions
BSk22, BSk24 and BSk26 from Brussels group.
