This paper presents a numerically stable method for rigid body simulation of unstructured meshes undergoing forceful contact, such as in robot locomotion and manipulation. The key contribution is a new contact generation method that treats the geometry as having a thin virtual boundary layer around the underlying meshes. Unlike existing methods, it produces contact estimates that are stable with respect to small displacements, which helps avoid jitter or divergence in the simulator caused by oscillatory discontinuities. Its advantages are particularly apparent on non-watertight meshes and can easily simulate interaction with partially-sensed and noisy objects, such as those that emerge from low-cost 3D scanners. The simulator is tested on a variety of robot locomotion and manipulation examples, and results closely match theoretical predictions and experimental data.