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.

The Flex Representation Method (FRM) leverages unique computational advantages of splines to address limitations in the process of building CAE simulation models from CAD geometric models. Central to the approach is the envelope CAD domain that encapsulates a CAD model. An envelope CAD domain can be of arbitrary topological and geometric complexity. Envelope domains are constructed from spline representations, like U-splines, that are analysis-suitable. The envelope CAD domain can be used to approximate none, some, or all of the features in a CAD model. This yields additional simulation modeling options that simplify the model-building process while leveraging the properties of splines to control the accuracy and robustness of computed solutions. Modern integration techniques are adapted to envelope domains to maintain accurate solutions regardless of the CAD envelope chosen. The potential of the method is illustrated through several carefully selected benchmark problems.