Universal meshes for a branched crack

We introduce an efficient approach to obtain conforming meshes for evolving branched cracks immersed in a fixed background mesh. The proposed approach is built on universal meshes (UM) proposed in Rangarajan et al. [34] which is able to construct conforming triangulations for a propagating simple crack. The UM functions by projecting certain nodes of the background mesh onto the crack and simultaneously relaxing their neighboring nodes to improve the quality of the resulting triangular mesh. The essence of the generalization to a branched crack is to determine which side of each branch to select nodes to move to the crack path. The choice is based on the consideration of minimizing mesh distortion. For the case of multiple junctions, we take into account the constraint that the nodes to be moved to the same crack branch must be on the same side of that branch. The proposed method inherits the main advantages of UM, including small perturbation to a fixed background mesh for a family of evolving cracks with no a priori conformity requirements. This advantage saves computational time compared with a brute-force mesh generation step. Numerical examples with one or multiple triple or quadruple junctions are provided.