We present a method for allowing explicit, Lagrangian meshes to undergo topological changes in an automatic way. We employ a method for detecting when topological changes are imminent. When a change occurs, we use a level set method to guide the change of topology of the domain mesh. This is then followed by an optimization step, using a variational formulation, that seeks to improve boundary mesh conformity to the zero level contour of the level set function. The advantage of this method is that it directly allows for using a variational formulation of the physics being modeled and simulated, including the ability to account for important geometric information in the model (such as for surface tension driven flow). Furthermore, the level set update and optimization step are only needed during a topological change. Hence, our method does not significantly affect computational cost.