  A method for quasistatic cohesive fracture is introduced that uses an
alternating direction method of multipliers (ADMM) to implement an energy
approach to cohesive fracture. The ADMM algorithm minimizes a non-smooth,
non-convex potential functional at each strain increment to predict the
evolution of a cohesive-elastic system. The optimization problem bypasses the
explicit stress criterion of force-based (Newtonian) methods, which interferes
with Newton iterations impeding convergence. The model is extended with an
extrapolation method that significantly reduces the computation time of the
sequence of optimizations. The ADMM algorithm is experimentally shown to have
nearly linear time complexity and fast iteration times, allowing it to simulate
much larger problems than were previously feasible. The effectiveness, as well
as the insensitivity of the algorithm to its numerical parameters is
demonstrated through examples. It is shown that the Lagrange multiplier method
of ADMM is more effective than earlier Nitsche and continuation methods for
quasistatic problems. Close spaced minima are identified in complicated
microstructures and their effect discussed.
