In this work we present multi-dimensional mesoscale simulations of detonation initiation in energetic materials. We solve the reactive Euler equations, with the energy equation augmented by a power deposition term. The reaction rate at the mesoscale is modelled using density-based kinetics, while the deposition term is based on simulations of void collapse at the microscale, modelled at the mesoscale as hot spots. We carry out two- and three-dimensional mesoscale simulations of random packs of HMX crystals in a binder, and show that transition between no-detonation and detonation depends on the number density of the hot spots, the packing fraction, and the post-shock pressure of an imposed shock. In particular, we show that, for a fixed post-shock pressure, there exists a critical value of the number density of hot spots, such that when the number density is below this value a detonation wave will not develop. We highlight the importance of morphology to initiation by comparing with a homogeneous counterpart, and we compare relevant length scales by examining their corresponding power spectra. We also examine the effect of packing fraction and show that at low post-shock pressures there is significant variation in the initiation times, but that this variation disappears as the post-shock pressure is increased. Finally, we compare three-dimensional simulations with the experimental data, and show that the model is capable of qualitatively reproducing the trends shown in the data. [ABSTRACT FROM AUTHOR]