Isogeometric collocation for phase-field fracture models

Phase-field models based on the variational formulation for brittle fracture have recently been shown capable of accurately and robustly predicting complex crack behavior. Their numerical implementation requires costly operations at the quadrature point level, which may include finding eigenvalues and forming tensor projection operators. We explore the application of isogeometric collocation methods for the discretization of second-order and fourth-order phase-field fracture models. We show that a switch from isogeometric Galerkin to isogeometric collocation methods has the potential to significantly speed up phase-field fracture computations due to a reduction of point evaluations. We advocate a hybrid collocation-Galerkin formulation that provides a consistent way of weakly enforcing Neumann boundary conditions and multi-patch interface constraints, is able to handle the multiple boundary integral terms that arise from the weighted residual formulation, and offers the flexibility to adaptively improve the crack resolution in the fracture zone. We present numerical examples in one and two dimensions that illustrate the advantages of our approach.