Mapped phase field method for brittle fracture

Phase field method has proven capable of producing complex crack patterns in solids. It introduces a continuous phase field to regularize the sharp crack discontinuities. However, the applicability of this method to engineering problems is hindered by its computational costs. In this work, we proposed a mapped phase field method as a possible route to resolve this issue. The core of this method is a map that connects the physical domain to a parametric domain, which is essentially a local reparametrization of the physical domain where large gradients are expected. By the use of this map the strongly varying fields can be approximated by a much smoother function. The reparametrized solution is solved via standard finite element in the parametric domain and then mapped back to the physical domain. A simple analysis shows that, with a properly defined map, such method consumes much less computational resources compared with conventional phase field method, without loss in accuracy. The map can also be easily manipulated to adapt to the evolution of cracks, and thus providing a flexible framework to simulate complex crack patterns without any knowledge of the crack path in advance. The advantages of our proposed method are further shown by four different numerical examples: single edge crack under symmetric and anti-symmetric loading, three-point bending, and L-shape panel under cyclic loading.