Designing a push-button with desired sensation and performance is challenging because the mechanical construction must have the right response characteristics. Physical simulation of a button's force-displacement (FD) response has been studied to facilitate prototyping; however, the simulations' scope and realism have been limited. In this paper, we extend FD modeling to include vibration (V) and velocity-dependence characteristics (V). The resulting FDVV models better capture tactility characteristics of buttons, including snap. They increase the range of simulated buttons and the perceived realism relative to FD models. The paper also demonstrates methods for obtaining these models, editing them, and simulating accordingly. This end-to-end approach enables the analysis, prototyping, and optimization of buttons, and supports exploring designs that would be hard to implement mechanically.