# Moving mesh finite element simulation for phase-field modeling of brittle fracture and convergence of newton's iteration

## TL;DR

This paper introduces a moving mesh finite element method combined with regularization techniques to improve the numerical simulation and convergence of phase-field models for brittle fracture, effectively capturing complex crack propagation.

## Contribution

The study develops a novel moving mesh finite element approach with regularization methods to enhance convergence and accuracy in phase-field modeling of brittle fracture.

## Key findings

- Regularization improves Newton's iteration convergence for small parameters.
- Moving mesh adapts to complex crack systems and propagating cracks.
- Methods maintain solution accuracy while enhancing numerical stability.

## Abstract

A moving mesh finite element method is studied for the numerical solution of a phase-field model for brittle fracture. The moving mesh partial differential equation approach is employed to dynamically track crack propagation. Meanwhile, the decomposition of the strain tensor into tensile and compressive components is essential for the success of the phase-field modeling of brittle fracture but results in a non-smooth elastic energy and stronger nonlinearity in the governing equation. This makes the governing equation much more difficult to solve and, in particular, Newton's iteration often to fail to converge. Three regularization methods are proposed to smooth out the decomposition of the strain tensor. Numerical examples of fracture propagation under quasi-static load demonstrate that all of the methods can effectively improve the convergence of Newton's iteration for relatively small values of the regularization parameter but without comprising the accuracy of the numerical solution. They also show that the moving mesh finite element method is able to adaptively concentrate the mesh elements around propagating cracks and handle multiple and complex crack systems.