  Current massive star evolution grids are not able to simultaneously reproduce
the empirical upper luminosity limit of red supergiants, the Humphrey-Davidson
(HD) limit at high and low metallicity. In this study, we provide a better
understanding of what drives massive star evolution to blue and red supergiant
phases, with the ultimate aim of reproducing the HD limit at varied
metallicities (Z). For solar, LMC, and SMC Z, we develop eight grids of MESA
models for the mass range 20-60M to probe the effect of semiconvection and
overshooting. We compare rotating and non-rotating models with efficient
(alpha_semi = 100) and inefficient semi-convection (alpha_semi = 0.1), with
high and low core overshooting (alpha_ov of 0.1 or 0.5). The red and blue
supergiant evolutionary phases are investigated by comparing the fraction of
core He-burning lifetimes spent in each phase. We find that the extension of
the convective core by overshooting alpha_ov = 0.5 has an effect on the post-MS
evolution which can disable semiconvection leading to more RSGs, but a lack of
BSGs. We therefore implement alpha_ov = 0.1 which switches on semiconvective
mixing, though for standard alpha_semi = 1, would result in an HD limit which
is higher than observed at low Z. Therefore, we need to implement very
efficient semiconvection of alpha_semi = 100 which reproduces the HD limit at
log L ~ 5.5 for the Magellanic Clouds while simultaneously reproducing the
Galactic HD limit of log L ~ 5.8 naturally. The effect of semiconvection is not
active at high Z due to the depletion of the envelope structure by strong mass
loss such that semiconvective regions could not form. Z-dependent mass loss
plays an indirect, yet decisive role in setting the HD limit as a function of
Z. For a combination of efficient semiconvection and low overshooting with
standard Z-dependent mass loss, we find a natural HD limit at all
metallicities.
