Quasistatic cohesive fracture with an alternating direction method of multipliers

TL;DR

This paper introduces an ADMM-based method for quasistatic cohesive fracture that efficiently predicts crack evolution, surpassing traditional force-based methods in speed and problem size, with demonstrated effectiveness in complex microstructures.

Contribution

The paper presents a novel ADMM algorithm for cohesive fracture modeling that improves computational efficiency and robustness over existing methods.

Findings

Nearly linear time complexity observed.

Significant reduction in computation time with extrapolation.

Effective in complex microstructure simulations.

Abstract

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.