The full wave surface integral equation computation of the second harmonic generation (SHG) dynamics for metal spheres and nanorods - presented as multimedia files - is performed to reveal the dynamics of the modes supported by the nanostructure. We demonstrate that the interplay between different modes controls the nonlinear response and that the size-induced redshift of the eigenmodes can be manipulated by adjusting the nanostructure geometry, so that the SHG signal can be boosted at specified frequencies. We show that the SHG radiation is not necessarily quadrupolar in spherical nanoparticles, as it is often assumed. Finally, we introduce an efficient way to reduce the SHG calculation time. (C) 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement