We present a novel physics-based approach to facial animation. Contrary to commonly used generative methods, our solution computes facial expressions by minimizing a set of non-linear potential energies that model the physical interaction of passive flesh, active muscles, and rigid bone structures. By integrating collision and contact handling into the simulation, our algorithm avoids inconsistent poses commonly observed in generative methods such as blendshape rigs. A novel muscle activation model leads to a robust optimization that faithfully reproduces complex facial articulations. We show how person-specific simulation models can be built from a few expression scans with a minimal data acquisition process and an almost entirely automated processing pipeline. Our method supports temporal dynamics due to inertia or external forces, incorporates skin sliding to avoid unnatural stretching, and offers full control of the simulation parameters, which enables a variety of advanced animation effects. For example, slimming or fattening the face is achieved by simply scaling the volume of the soft tissue elements. We show a series of application demos, including artistic editing of the animation model, simulation of corrective facial surgery, or dynamic interaction with external forces and objects.