In this dissertation, I provide a robust, efficient inverse mapping algorithm for use in immersed simulation methods, specifically in the Flex Representation Method. I also explore a structural theory that unifies the theories of solids, shells, beams, and rigid bodies. As part of this, I preform a preliminary exploration of applying the Flex Representation Method to shells. Finally, I explore why higher order elements suffer from small critical time steps in explicit dynamics. I then propose a simple method of remedying this issue by exploiting the properties of U-splines.