We have performed global simulations of heat transfer during the complete directional solidification (DS) process of multi-crystalline silicon (mc-Si) ingots under a traveling magnetic field (TMF) to investigate the thermal and flow fields in the silicon melt. The melt convection pattern, thermal field, and meltcrystal (m-c) interface shape at different DS stages were compared without a TMF, with an upward TMF, and with a downward TMF. We found that the distribution and the magnitude of the Lorentz force are similar for different melt heights. The melt is mainly occupied by a large vortex, which flows in opposite directions for an upward TMF and a downward TMF. The TMF has a beneficial effect on the melt mixing. The upward TMF makes the m-c interface more concave with respect to the melt, while the downward TMU makes it less concave or convex at the same solidification stage. The interface becomes less concave or convex with increasing solidification fraction under the downward IMF. The amplitude of imposed electric current was adjusted to successfully obtain a slightly convex interface. These results provide important information for optimizing the complete DS process for mc-Si ingots with a IMF. (C) 2014 Elsevier B.V. All rights reserved.