Quasistatic cohesive fracture with an alternating direction method of multipliers


Fracture modeling has undergone tremendous improvements over the years. The initially developed modeling approaches based on linear elastic fracture mechanics (LEFM) have been generalized to develop plasticity zone at the crack tip. Assuming failure criteria based on J-integral and stress intensity factors, these approaches have been employed in post-processing step based on idealized geometries and further incorporated into finite element analyses. In contrast, cohesive fracture seemingly reflects the reality of most materials as it assumes gradual decohesion before the fracture tip. It also permits the regularization of various aspects of LEFM, allowing the treatment of each material point as cracked or uncracked.

LEFM has been cast as an energy minimization problem in several studies. Attendant numerical solutions produce smeared crack methods employing phase fields, which can be extended to include cohesive fracture. Similarly, cohesive fracture postulating no separation prior to critical stress combination has also been formulated as a non-convex and non-differentiable functional problem. While these approaches employ the concept of continuation and subdifferential methods, it is important noting that these works differ from finite element methods using a weak form.

Although remedies to overcome the time discontinuity problem and similar undesirable behaviors have been proposed, it is possible to cure time discontinuity by applying the energy method to a non-convex and non-differentiable functional. The application of smeared and strong-discontinuity energy approaches has been studied. The main advantage of the smeared method is its ability to track the crack path of the element boundaries independently. In contrast, the discrete method assumes a crack path along the element interfaces. Whereas this drawback can be addressed by randomized or special isoperimetric meshes, the method can also be implemented with XFEM or other embedded strong discontinuity methods. Moreover, crack opening can be represented using the discrete method.

Herein, Dr. Katerina Papoulia, a principal at Tesserae Solutions Engineering and also Adjunct Professor at York University together with Dr. Reza Hirmand and PhD candidate James Petrie from the University of Waterloo designed a new model for the implementation of energy approach to quasistatic cohesive fracture using alternating direction method of multipliers (ADMM). In their approach, the avoidance of explicit fracture criterion resulted in time continuity which is associated with the generation of continuous force vector during crack activation. This approach enhanced the model’s ability to perform implicit calculations. The representative convergence paths and sensitivity of the result to different key discretization and optimization parameters were analyzed and discussed. Their work is currently published in the research journal, Engineering Fracture Mechanics.

The research team demonstrated a significant performance improvement with little impact on the results. The optimization problem successfully bypassed the explicit stress criterion of Newtonian methods, thereby minimizing the interference with iterations and convergence. The ADMM algorithm allowed the simulation of larger problems that could not be previously simulated using existing methods. This was attributed to fast iteration times, significant reduction in the computation time and nearly linear time complexity. For instance, the iteration was 2 – 3 orders of magnitude faster than the existing methods. Furthermore, it is worth noting that the model’s run time increased linearly with the problem size, and it could simulate moderately sized problems in a few seconds.

In summary, Dr. Katerina Papoulia and her colleagues developed a reliable and effective ADMM-based model to simulate cohesive fracture by minimizing non-smooth potential energy function. The authors used different examples to successfully demonstrate the effectiveness and insensitivity of the algorithm to numerical parameters. Generally, the Lagrange multiplier method of ADMM was superior to Nitsche and continuation methods for quasistatic problems. Additionally, closed space minima and their effects on complicated microstructures were identified. This technique can be potentially extended to other quasistatic models. In a statement to Advances in Engineering, Dr. Katerina Papoulia, the corresponding author explained the presented new model is a promising tool for multiple runs in machine learning applications.

About the author

James Petrie is a PhD candidate in Applied Mathematics at the University of Waterloo. Previously, he completed a Master’s in Computational Mathematics at Waterloo and a Bachelor’s in Engineering Physics at UBC. His previous research includes numerical modeling of cohesive fracture, AI for soccer playing robots, and design of digital contact tracing protocols. He is currently an ELBI fellow and after graduating, he will be a visiting scholar at the Oxford Big Data Institute.

About the author

Reza Hirmand is a Mechanics Software Developer at Hexagon Manufacturing Intelligence Canada and Adjunct Assistant Professor at the University of Waterloo. He obtained a Master’s degree in Civil Engineering from Sharif University of Technology and a PhD in Mechanical and Mechatronics Engineering from the University of Waterloo in 2019. Previously, he worked as a Research Scientist with CanmetMATERIALS-Hamilton and as a Post-Doctoral Fellow at the University of Waterloo. He has developed several in-house computational codes on cohesive fracture, solid state deposition of powder particles, and metal sheer forming processes.

About the author

Katerina Papoulia is principal at Tesserae Solutions Engineering and Adjunct Professor at York University. She is a graduate of the National Technical University of Athens, Southampton University, and the University of California at Berkeley (PhD 1991). She has taught at Cornell University and at the University of Waterloo and still teaches occasionally at York University. Her research focuses on the study and computational modeling of large time-dependent deformation and material failure. It has been funded by the European Commission, the U.S. National Science Foundation, NSERC, Cornell University, and NASA.


Petrie, J. I., Hirmand, M. R., & Papoulia, K. D. (2022). Quasistatic cohesive fracture with an alternating direction method of multipliers. Engineering Fracture Mechanics, 264, 108267.

Go To Engineering Fracture Mechanics

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