  Vortex penetration and flux relaxation phenomenon carry the information about
the pinning ability, and consequently current-carrying ability, of a type-II
superconductor. However, the theoretical descriptions to these phenomena are
currently limited to the cases with special initial conditions. A
generalization to the recently developed infinite series models is presented
here. It is shown that one can convert a vortex penetration process with a
non-zero initial internal field into a process with a zero initial internal
field by introducing some time parameters. Similarly, one can also convert a
flux relaxation process starting with an arbitrary internal field into a
process starting with a melting internal field by introducing a virtual time
interval. Therefore, one can predict the melting internal field (or critical
current density) from a flux relaxation process starting with a lower internal
field. Finally, it is shown that the vortex penetration process in an ideal
superconductor is strongly time dependent because of the surface barrier and
internal field repulsive force. But the flux relaxation process does not occur
in the ideal superconductor.
