The freeform optical system plays a key role in illumination engineering, and several methods have been reported to manage the design of such system. In this paper, an approach to generate the polar-grids based flux transportation mapping for an arbitrarily-shaped target is proposed based on the conventional variable separation method. The source emitting grid is divided along the azimuth angle and the zenith angle respectively under the spherical coordinate system. Then, the target grid is achieved by solving the flux integral equations in polar coordinates using separation of variables method. When establishing the target grid along the polar radius, a strategy based on uniformly scaling down the external contour of the target is introduced. According to the mapping, a smooth freeform surface is then generated using the geometric construction method according to Snell's law. Finally, an iterative feedback process is adopted to compensate the deterioration of the target distribution caused by surface construction errors and the extension of a real source. Based on this method, a series of freeform lenses are designed for a 1 × 1 mm(2) LED source to generate uniform, Gaussian and multiple-rings illumination distributions within different target regions. High-performance optical systems with the light utilization efficiency η over 0.8 and the relative standard deviation (RSD) of the simulated illumination distribution less than 0.1 are obtained simultaneously for all the cases.