Final Report Summary - MIDAS (Microstructures in Dynamic and Anisotropic Systems)
The project Microstructures in Dynamic and Anisotropic Systems (MIDAS) supported an eighteen-month research fellowship for Nadia Ansini (Sapienza, Rome) at the University of Bath.
Microstructures occur in a wide range of materials, including steel and shape memory materials. Over the past decades, experimental and computational advances have revolutionised our understanding of matter on the atomistic (microscopic) scale. The computation of multiscale problems, such as those exhibiting microstructures, however, is difficult and costly at best, and often impossible if very disparate scales have to be resolved simultaneously. Thus scale-bridging has become a central activity in physics, chemistry and biology; a rich mathematical theory underpinning this topic has been developed over the last two decades in particular. This project studied a range of problems with multiple scales. This includes evolutionary problems with “wiggly” energies, where the energy exhibits oscillations on a fine scale, and the evolution seeks to minimize the energy as much as possible – that is, evolve according to an associated gradient flow. Here a new approach of local minimization and variational evolution in connection to the analytic tool of Gamma-convergence was used to understand in detail the intricate range of evolutions depending on an underlying ration of spatial and temporal (discretization) scales.
One other aspect of the work undertaken concerns degenerate parabolic equations. Here the motivation is the description of the grain size evolution in steel. Modern analytic tools from the theory of Wasserstein gradient flows could be used to analyse the aforementioned class of parabolic equations with degeneracies.
Microstructures occur in a wide range of materials, including steel and shape memory materials. Over the past decades, experimental and computational advances have revolutionised our understanding of matter on the atomistic (microscopic) scale. The computation of multiscale problems, such as those exhibiting microstructures, however, is difficult and costly at best, and often impossible if very disparate scales have to be resolved simultaneously. Thus scale-bridging has become a central activity in physics, chemistry and biology; a rich mathematical theory underpinning this topic has been developed over the last two decades in particular. This project studied a range of problems with multiple scales. This includes evolutionary problems with “wiggly” energies, where the energy exhibits oscillations on a fine scale, and the evolution seeks to minimize the energy as much as possible – that is, evolve according to an associated gradient flow. Here a new approach of local minimization and variational evolution in connection to the analytic tool of Gamma-convergence was used to understand in detail the intricate range of evolutions depending on an underlying ration of spatial and temporal (discretization) scales.
One other aspect of the work undertaken concerns degenerate parabolic equations. Here the motivation is the description of the grain size evolution in steel. Modern analytic tools from the theory of Wasserstein gradient flows could be used to analyse the aforementioned class of parabolic equations with degeneracies.