Second-order cone interior-point method for quasistatic and moderate dynamic cohesive fracture

TL;DR

This paper introduces a second-order cone interior-point method for implicit time-stepping in cohesive fracture modeling, effectively handling nondifferentiability and contact constraints in quasistatic and moderate dynamic problems.

Contribution

It develops a novel interior-point approach using cone programming to directly address nondifferentiability in cohesive fracture models, improving upon previous smoothing techniques.

Findings

Method effectively handles contact constraints.

Computational results demonstrate practicality.

Captures nondifferentiability without smoothing.

Abstract

Cohesive fracture is among the few techniques able to model complex fracture nucleation and propagation with a sharp (nonsmeared) representation of the crack. Implicit time-stepping schemes are often favored in mechanics due to their ability to take larger time steps in quasistatic and moderate dynamic problems. Furthermore, initially rigid cohesive models are typically preferred when the location of the crack is not known in advance, since initially elastic models artificially lower the material stiffness. It is challenging to include an initially rigid cohesive model in an implicit scheme because the initiation of fracture corresponds to a nondifferentiability of the underlying potential. In this work, an interior-point method is proposed for implicit time stepping of initially rigid cohesive fracture. It uses techniques developed for convex second-order cone programming for the nonconvex problem at hand. The underlying cohesive model is taken from Papoulia (2017) and is based on a nondifferentiable energy function. That previous work proposed an algorithm based on successive smooth approximations to the nondifferential objective for solving the resulting optimization problem. It is argued herein that cone programming can capture the nondifferentiability without smoothing, and the resulting cone formulation is amenable to interior-point algorithms. A further benefit of the formulation is that other conic inequality constraints are straightforward to incorporate. Computational results are provided showing that certain contact constraints can be easily handled and that the method is practical.