The ability to cut through meshes in real-time is an essential ingredient in a number of practical interactive simulations. Surgical simulation, cloth design, clay sculpting and many other related VR applications require the ability to introduce arbitrary discontinuities through models to separate, reposition, and reshape various pieces of the model as needed for the target application. In addition, in order to provide the necessary realism for these applications, model deformations must be computed from an underlying physically-based model-most commonly a continuum-based finite element model. In this work, we present a method for representing and computing, at interactive rates, the deformations of a mesh whose topology is being dynamically modified with multiple virtual tools. The method relies on introducing controlled discontinuities in the basis functions used to represent the geometry of deformation, and on fast incremental methods for updating global model deformations. The method can also generate the forces needed for force rendering in a haptic environment. The method is shown to scale well with problem size (linearly in the number of nonzeros of the Cholesky factor) allowing realistic interaction with fairly large models.