Virtual Node Algorithms for Simulating and Cutting Deformable Solids

Physics based simulations often require solving differential equations on domains with irregular geometry. A popular approach to their numerical approximation is the finite element method with unstructured meshes that conform to the domain geometries. However, in many applications such as shape optimization for elastic material and virtual surgery, the simulation domains frequently change shape. Under these cases, continuous remeshing can be a daunting task, and often produces ill-conditioned elements, which will reduce the accuracy and stability of the simulations. Virtual node algorithms(VNA) were invented to overcome this difficulty by embedding the simulation domains on Cartesian grids or structured meshes that do not conform to their geometry. Solutions are then computed on the embedding domains to take advantage of their regularity. The term `virtual nodes" refers to the degrees of freedom that lie outside the actual domains, reflecting the fact that there is no material associated with them. In this dissertation we present two novel virtual node algorithms