Structure-preserving Numerical Methods for Engineering Applications

This dissertation develops a variety of structure-preserving algorithms for mechanical systems with external forcing and also extends those methods to systems that evolve on non-Euclidean manifolds. The dissertation is focused on numerical schemes derived from variational principles – schemes that are general enough to apply to a large class of engineering problems. A theoretical framework that encapsulates variational integration for mechanical systems with external forcing and time-dependence and which supports the extension of these methods to systems that evolve on non-Euclidean manifolds is developed. An adaptive time step, energy-preserving variational integrator is developed for mechanical systems with external forcing. It is shown that these methods track the change in energy more accurately than their fixed time step counterparts. This approach is also extended to rigid body systems evolving on Lie groups where the resulting algorithms preserve the geometry of the configuration space in addition to being symplectic as well as energy and momentum-preserving. The advantages of structure-preservation in the numerical simulation are illustrated by various representative examples from engineering applications, which include limit cycle oscillations of an aeroelastic system, dynamics of a neutrally buoyant underwater vehicle, and optimization for spherical shape correlation and matching.