# Newton methods for solving fully monolithic phase-field quasi-static brittle fracture propagation

## Prof. Thomas Wick

**Oct. 10, 2017, 3:30 p.m. S2 416-1**

In this presentation, the purpose is on

the development of a fully monolithic solution

algorithm for quasi-static phase-field fracture propagation. Phase-field

fracture

is a variational approach to fracture and

consists of two coupled partial differential equations and it is well known

that the underlying energy functional is non-convex and

sophisticated algorithms are required.

The incremental, spatially-discretized problem is treated with adaptive

finite elements and predictor-corrector mesh adaptivity that

allows for a very small regularization parameter in the crack region.

The nonlinear problem is solved with adaptive modified Newton algorithms,

which work as inner loop within an inexact augmented

Lagrangian iteration for relaxing

the crack irreversibility

constraint.

Specifically, the fully monolithic approach is compared to a

quasi-monolithic

approach in which phase-field is approximated through extrapolation

in the displacement equation. These comparisons are done in terms of

certain quantities of interest and computational cost.

Moreover, features such as crack nucleation,

joining, branching and fracture networks are addressed.

All findings are critically commented pointing to open questions

and future improvements.