  Barrier functions are crucial for maintaining an intersection and inversion
free simulation trajectory but existing methods which directly use distance can
restrict implementation design and performance. We present an approach to
rewriting the barrier function for arriving at an efficient and robust
approximation of its Hessian. The key idea is to formulate a simplicial
geometric measure of contact using mesh boundary elements, from which analytic
eigensystems are derived and enhanced with filtering and stiffening terms that
ensure robustness with respect to the convergence of a Project-Newton solver. A
further advantage of our rewriting of the barrier function is that it naturally
caters to the notorious case of nearly-parallel edge-edge contacts for which we
also present a novel analytic eigensystem. Our approach is thus well suited for
standard second order unconstrained optimization strategies for resolving
contacts, minimizing nonlinear nonconvex functions where the Hessian may be
indefinite. The efficiency of our eigensystems alone yields a 3x speedup over
the standard IPC barrier formulation. We further apply our analytic proxy
eigensystems to produce an entirely GPU-based implementation of IPC with
significant further acceleration.
