# Recursive multilevel trust region method with application to fully monolithic phase-field models of brittle fracture

## TL;DR

This paper introduces a recursive multilevel trust region method to efficiently solve the non-convex minimization problems in phase-field models of brittle fracture, improving simulation of crack propagation in complex materials.

## Contribution

It proposes a novel RMTR method with level-dependent objective functions tailored for phase-field fracture models, enhancing convergence and computational efficiency.

## Key findings

- Method demonstrates improved convergence in 3D simulations.
- Level-dependent objectives effectively combine fine and coarse mesh information.
- Numerical examples validate the method's efficiency and robustness.

## Abstract

The simulation of crack initiation and propagation in an elastic material is difficult, as crack paths with complex topologies have to be resolved. Phase-field approach allows to simulate crack behavior by circumventing the need to explicitly model crack paths. However, the underlying mathematical model gives rise to a non-convex constrained minimization problem. In this work, we propose a recursive multilevel trust region (RMTR) method to efficiently solve such a minimization problem. The RMTR method combines the global convergence property of the trust region method and the optimality of the multilevel method. The solution process is accelerated by employing level dependent objective functions, minimization of which provides correction to the original/fine-level problem. In the context of the phase-field fracture approach, it is challenging to design efficient level dependent objective functions as the underlying mathematical model relies on the mesh dependent parameters. We introduce level dependent objective functions that combine fine level description of the crack path with the coarse level discretization. The overall performance and the convergence properties of the proposed RMTR method are investigated by means of several numerical examples in three dimensions.