  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.
