Numerical implementation and comparison study on simulating thermo-elastic fracture using adaptive phase-field method combined with BFGS algorithm and AM algorithm

In this paper, the computational performance of alternate minimization (AM) and Broyden–Fletcher–Goldfarb–Shanno (BFGS) to solve the coupled partial differential equations arising within the framework of the phase field method is numerically studied. Both the numerical approaches are employed on an adaptive phase field method. In the case of BFGS, a double-loop recursive algorithm in combination with a line search is implemented. The local refinement is based on a pre-defined threshold on the damage variable and element size. The incompatible nodes due to adaptive refinement are handled by variable-node elements. The performance of the solution algorithm is numerically studied for a few problems with different loading conditions, viz., pure mechanical, pure thermal and combined thermo-mechanical. From the study, it is opined that the BFGS algorithm is 2$\sim$5 times faster than the AM approach, which is typically used for adaptive phase field framework. In addition, the detailed numerical implementation is presented.