Interface gradient approximation

This page is currently a WORK IN PROGRESS. Here I will be posting incremental results as I progress my work developing a Reduced Order Method (ROM) that approximates the spatio-temporal gradient of an FSI problem’s interface. In smoothly evolving FSI problems, such as those involving elastic structures submerged in a laminar flow, the power spectrum of the individually examined nodal trajectories (when sampled over a single period of oscillation for example) is shown to be extremely bi-modal with very low variance around its spectrum’s dominant frequencies. As such, by strategically selected only a handful of Fourier modes we can obtain (on a rolling basis) a continuous function that describes each node’s local behavior reasonably well. With this array of continuous functions in hand we can then compute both the approximate location of the interface’s nodes at the forward time step as well as the RO Jacobian matrix that describes the interface’s spatially dependent local rate of change. While not sufficient on its own to ensure problem convergence for any particular set of FSI problems; other than those that may already be implicitly convergent (in which case we will likely see an acceleration in the convergence rate); this technique may prove itself valuable at **imrpoving** the performance of traditional quasi-Newton coupling techniques. That is, while those methods require a minimum of 2 fixed point iterations to appriximate the current interfacial Jacobian this novel approach provides an approximate gradient implicitly. As such, we can better stabilize the first 2 iterations of traditional coupling methods (minimize the degree of solution divergence) and thereby provide a better conditioned solution for these techniques to then progress more efficiently.