Abstract: Phase-field models are a popular choice in computational physics to describe complex dynamics of substances with multiple phases, and are widely used in various applications including solidification or fracture mechanics. While these models provide a flexible framework and avoid explicit tracking of the interface, the underlying models governed by local differential operators (e.g., Allen-Cahn or Cahn-Hilliard) always lead to diffuse interfaces, which, in the case of thin interfaces, requires very fine meshes and high computational cost. In contrast, we analyze models where the interface evolution is represented by a nonlocal operator, which allows to capture long-range interactions between the particles in the substances and recovers the classical (local) operator for vanishing nonlocal interactions. We demonstrate that under certain conditions and with a careful choice of the nonlocal operator we can obtain a model that allows for a sharp interface in the solution and can be simulated on coarse meshes.
First, I will present different nonlocal models of Cahn-Hilliard and Allen-Cahn type involving a nonsmooth obstacle double-well potential. We prove the well-posedness and regularity properties of the solutions and present efficient space-time discretizations that can handle these sharp interfaces. Second, I will discuss the ongoing development of extensions to more complex models arising in the context of solidification, where the nonlocal phase-field model is coupled to an additional temperature equation. We provide an analysis of the model, discuss discretization schemes, and supplement our findings with several numerical experiments.
This is a joint work with Steve DeWitt (ORNL), Max Gunzburger (FSU, UT Austin) and Balasubramaniam Radhakrishnan (ORNL).
Speaker’s Bio: Olena Burkovska is currently a Householder Fellow in the Multiscale Methods and Dynamics Group. Prior to joining ORNL, Olena was a Postdoctoral Researcher at the Florida State University and a visiting researcher at the University of Colorado Boulder. She received her Ph.D. in Numerical Mathematics from Technical University of Munich in 2016. Olena’s current research interests lie in the area of numerical analysis, model order reduction, nonlocal and fractional models, and variational inequalities.
Last Updated: July 26, 2021 - 7:45 am