# Undergraduate Research Projects

## Mathematical Modelling of Ring Rolling

Ring rolling is an important industrial metal forming process. It is used to make parts for a variety of purposes, from oil platform legs to aircraft engine parts. Surprisingly little is known about the mechanics of ring rolling, and setting up a stable rolling process is largely done by informed trial and error. At present, only circular rings with a rectangular cross-section can be made. With recent interest in using computer control to make more diverse shapes with more exotic metals using less energy, simple quick-to-compute models of the process are needed. This project would use basic mathematical and computational techniques to derive a simple mechanical model for the basic ring rolling process.

## Making Waves with Finite Differences

Waves are all around us. Whether it is understanding wifi coverage, designing quiet vacuum cleaners, or mitigating airport noise, theoretical and computational models of waves are a useful scientific and design tool. Unfortunately, accurate computer simulations of waves are hard: they are finely balanced between being unstable and being too dissipative, and push the bounds of what is computationally possible. The purpose of this project is to investigate bringing together two clever theoretical ideas: wavenumber optimized implicit finite difference schemes (implicit DRP schemes); and summation by parts finite difference schemes (SBP schemes). Don't worry if you don't know what that means yet. Their combination should allow for provably stable wave simulations needing few points per wavelength.

## Theory of Summation By Parts Finite Differences (MA916)

Computational Aero-Acoustics (CAA) involves using computer simulations to model sound generated by and propagating through air flows. Applications range from hand driers and computer cooling fans to aircraft engines and wind turbines. These simulations push the limits of modern computers, and are (or should be) perfectly balanced between being too dissipative and being unstable. The simulations therefore rely on a significant body of mathematical theory. This project concerns recent developments in the theory of finite differences, commonly used in computational aero-acoustics simulations, and possible underlying theoretical links with finite elements.