Goal oriented error estimation for space-time adaptivity in phase-field fracture

                                
==Abstract==

This work focuses on temporal adaptivity for phase-field fracture problems. The methodology requires a space-time formulation and utilizes a space-time Galerkin finite element discretization for the governing phase-field equations. Then, goal functionals (i.e., quantities of interest) are introduced. The computational implementation of goal-oriented error control employs the dual-weighted residual method in which an adjoint problem must be solved. As the analysis is quasi-static, without a temporal derivative, the adjoint problem of the quasistatic primal problem decouples in time. Nonetheless, time-averaged goal functionals can also be considered. The temporal errors are localized using a partition of unity, which allows one to adaptively refine and coarsen the time intervals in the space-time cylinder. Numerical tests are performed on a single edge notched tensile test to investigate the quality of the proposed error estimator.