The project deals generally with the analysis and development of reliable and suitable approximation schemes for mathematical models of complex physical phenomena such as damped elastic structures, processes with delays, coupled structures (including models of smart materials), etc. A standard approach to analyzing and solving problems for such systems is to instead consider an approximate system, which ideally is both finite dimensional and rich enough to retain properties of the original (infinite dimensional) system. But it is well known that some approximate systems do not retain important properties of the original system, such as growth rate, stability robustness, etc. (In fact, it is well known that some models of elastic structures with boundary damping have an energy decay rate that is not preserved by standard "finite element" or "finite difference" approximate systems). This project focuses on how and why certain properties are retained or lost under approximation, and on developing a systematic method for constructing "property-preserving" approximation schemes. The key is analyzing the geometry (inner product) under which the original system is projected onto the approximating system, and one thrust is that changing geometries can improve approximations so that important properties are better preserved. This is a recent idea on which the PI has demonstrated successful implementation for some preliminary examples. This project will extend in significant ways the method and the applications.
There are many practical applications in science and engineering that involve elastic structures, such as large antennae, fluid flow over an airfoil, vibrating buildings, etc. In the presence of evolving "smart materials" and actuator technologies, these lead to complex mathematical models that cannot be fully resolved on even the most powerful computers. A standard strategy for overcoming this difficulty is to instead consider a simpler model (an "approximation scheme") which is a reasonably good approximation of the original complex model, and which can also be solved on a computer. There are typically many approximation schemes for any given model, and since something is always lost in the approximation, some important questions are - how good is a 'reasonably good approximation'? and - which important properties are lost or preserved? Further, since these computer models are used to better understand and control these complex systems, high-fidelity approximation schemes are essential. This project will focus on these questions, with the goal of developing a systematic methodology for a broad range of applications, and of developing software based on the theory.
