In this paper, Part II of our study, we extend our previous analysis (J Crystal Growth 240 (2002) 267) of the linear evolution of non-spherical growing crystals in three dimensions into the nonlinear regime characterized by large shape perturbations. We focus on a solid crystal growing in an undercooled liquid with isotropic surface tension and interface kinetics. We use a new, adaptive boundary integral method to simulate the morphological evolution of the growing crystals. Our simulations reveal that when the far-field heat flux into the system is prescribed by appropriately varying the undercooling in the far field, the Mullins-Sekerka instability that would arise under constant undercooling can be suppressed. In particular, we demonstrate that there exist critical conditions of flux at which self-similar or nearly self-similar nonlinear evolution occurs and the shape is dominated by a given mode leading to non-spherical, nearly shape invariant growing crystals. This result was predicted by our previous analysis (see Ref. Cristini and Lowengrub) and suggests that our theory is applicable to real physical systems. We provide a simulation of a physical experiment that might be able to be carried out in a laboratory in which a desired shape of a crystal is achieved and maintained during growth by appropriately prescribing the far-field heat flux. (C) 2004 Published by Elsevier B.V.