Rootfinding and Optimization Techniques for Solving Nonlinear Systems of Equations Arising from Cohesive Zone Models

TL;DR

This paper evaluates various rootfinding and optimization methods for solving nonlinear equations in cohesive zone models, highlighting their performance differences and proposing techniques for improvement in computational fracture modeling.

Contribution

It systematically compares established nonlinear solution methods on cohesive zone models and discusses strategies to enhance their effectiveness for complex problems.

Findings

Established methods vary in performance depending on problem complexity

Higher degrees of freedom impact solution method effectiveness

Proposed techniques can improve convergence and robustness

Abstract

While approaches to model the progression of fracture have received significant attention, methods to find the solution to the associated nonlinear equations have not. In general, nonlinear solution methods and optimization methods have a rich body of work spanning back to at least the first century, providing the opportunity for advancement in the field of computational discrete damage modeling. In this paper, we explore the performance of established methods when applied to problems involving cohesive zone models to identify promising methods for further improvement in this specialized application. We first use a simple 1D example problem with low degrees of freedom (DoF) to compare nonlinear solution methods, thereby allowing for both straightforward and intuitive visualization of the residual space and reasoning about the cause for each method's performance. We then explore the impact of higher DoF discretizations of the same problem on the performance of the solution methods. Finally, we discuss techniques to improve performance or to overcome limitations of the various methods.