General haptic interaction with solid models requires an underlying physically-based model that can generate, in real-time, the forces and deformations to be rendered as a result of user interaction. In order to allow for a rich set of interactions, the physical model must support real-time topological modifications including the embedding of new elements in the model, and the introduction of cuts in the geometry. In this paper, we describe and demonstrate a physically-based framework for real-time interaction with 3D solid models discretized by finite elements. We present a model formulation that allows for fast progressive updates to be used in modeling the addition of new elements as well as dynamic inter- and intra-element changes in model connectivity. Our motivating applications have been in the area of open suturing simulations where cutting through skin and tissue, undermining skin to separate it from the underlying soft tissue, addition of sutures to close wounds, and manipulation using multiple surgical instruments simultaneously, are all tasks that must be supported. We show a new surgical simulator we recently developed to demonstrate the framework.