  We study the quantum phase transitions in two-dimensional arrays of
Josephson-couples junctions with short range Josephson couplings (given by the
Josephson energy) and the charging energy. We map the problem onto the solvable
quantum generalization of the spherical model that improves over the mean-field
theory method. The arrays are placed on the top of a two-dimensional electron
gas separated by an insulator. We include effects of the local dissipation in
the presence of an external magnetic flux f in square lattice for several
rational fluxes f=0,1/2,1/3,1/4 and 1/6. We also have examined the T=0
superconducting-insulator phase boundary as function of a dissipation alpha for
two different geometry of the lattice: square and triangular. We have found
critical value of the dissipation parameter independent on geometry of the
lattice and presence magnetic field.
