[en] In this paper, we present a three-dimensional model for self-consistently modeling ion beam formation from plasma ion sources and transporting in low energy beam transport systems. A multi-section overlapped computational domain has been used to break the original transport system into a number of weakly coupled subsystems. Within each subsystem, macro-particle tracking is used to obtain the charge density distribution in this subdomain. The three-dimensional Poisson equation is solved within the subdomain after each particle tracking to obtain the self-consistent space-charge forces and the particle tracking is repeated until the solution converges. Two new Poisson solvers based on a combination of the spectral method and the finite difference multigrid method have been developed to solve the Poisson equation in cylindrical coordinates for the straight beam transport section and in Frenet-Serret coordinates for the bending magnet section. This model can have important application in design and optimization of the low energy beam line optics of the proposed Rare Isotope Accelerator (RIA) front end