Phase field crystal (PFC) models are an efficient methodology to simulate phases and phase transitions in many physical systems. In this talk, a class of gradient based approaches, which is the so-called adaptive accelerated Bregman proximal gradient (AA-BPG) methods, will be presented and the convergence property is established without the global Lipschitz constant requirements. One key feature of our algorithms is that the energy dissipation and mass conservation properties hold during the iteration process. Moreover, we develop a hybrid acceleration framework to accelerate the AA-BPG methods and most of existing approaches through coupling with the practical Newton method. Extensive numerical experiments demonstrate that our approaches lead to a significant acceleration over many existing methods, especially in computing complex periodic crystals and quasicrystals. We also apply our approaches to simulate a class of PFC models, and find some new physical phenomena.