  The Incremental Potential Contact (IPC) method enables robust complex
simulations of deformable objects with contact and friction. The key to IPC's
robustness is its strict adherence to geometric constraints, avoiding
intersections, which are a common cause of robustness issues in contact
mechanics. A key element of the IPC approach to contact is a geometric barrier
function, which is defined directly in the discrete setting. While IPC achieves
its main goal of providing guarantees for contact constraints, its parameters
need to be chosen carefully to avoid significant simulation artifacts and
inaccuracies. We present a systematic derivation of an IPC-like continuum
potential defined for smooth and piecewise smooth surfaces, starting from
identifying a set of natural requirements for contact potentials, including the
barrier property, locality, differentiable dependence of shape, and absence of
forces in rest configurations, based on the idea of candidate sets. Our
potential is formulated in a way independent of surface discretization.
  This new potential is suitable for piecewise-linear surfaces and its
efficiency is similar to standard IPC. We demonstrate its behavior and compare
it to IPC on a range of challenging contact examples.
