Effects ranging from very small to very large scales govern physical phenomena such as the evolution of a storm system or the structural deformation of an automobile during an accident. Accurately predicting these by resolving the finest scales in a computer simulation is prohibitively expensive. The investigators are studying how fine scale information impacts coarse scale behavior and vice versa. In effect, "summarizing" these relationships allow us to model coarse scale effects accurately and efficiently without the need to explicitly resolve the finest scales in a computation. A key to this study lies in the careful transfer of structures present in the mathematical models of these phenomena (which in essence have infinite resolution) to the computational realm with its finite resolution and finite computational resources. The methods being developed will allow rapid assessment of overall effects with the ability "to drill down" computationally where additional detail is required.
Physical systems are typically described by a set of continuous equations using tools from geometric mechanics and differential geometry to analyze and capture their properties. For purposes of computation one must derive discrete (in space and time) representations of the underlying equations. Theories which are discrete from the start (rather than discretized after the fact), with key geometric properties built in, can more readily yield robust numerical simulations which are true to the underlying continuous systems: they exactly preserve invariants of the continuous systems in the discrete computational realm. So far these methods have not accounted for effects across scales. Yet both physics and numerical computation require such multi-resolution strategies. This research project is developing a multi-resolution theory for discrete variational methods and discrete differential geometry with applications to thin-shell and fluid modeling. The principal scientific innovation lies in techniques to conserve symmetries across computational scales.
