Numerical Methods for Deterministic and Stochastic Phase Field Models

The University Of Central Florida Board Of Trustees, Orlando FL

Investigators

Abstract

The project will consider algorithms for moving interface problems that have broad applications in materials science, geometry, and broad physical sciences and engineering. The phase field model is one of the most important models to formulate the moving interface problems, and it arises from many interdisciplinary applications, such as the nucleation and growth processes of polycrystalline materials, chemical reaction, solidification dynamics, and so on. This project proposes several accurate and efficient algorithms to solve the phase field models and to further explore the connections between different areas. In addition, the project will address randomness, which plays an important role in the phase field model applications due to the existence of the impurities, and specifically will consider the design, analysis, and implementation of numerical methods for the stochastic phase field models which incorporate the randomness. Moreover, this project trains graduate students to equip them with necessary skills for their future careers. This research project develops a few accurate and efficient numerical algorithms for both deterministic and stochastic phase field models. The deterministic case discusses the sharpest error bounds for the general phase field models in various spaces and their approximations to geometric flows, the adaptive two-grid algorithms, and some optimization algorithms which are robust to topological changes for the fourth-order phase field models. A package of the phase field models and geometric flows will be developed to better study the theoretical results and to predict other new results. The stochastic case focuses on discrete stability and convergence results for the fourth-order stochastic phase field models, and sharpest discrete stability and convergence results for the second-order stochastic phase field models using stochastic parabolic duality, which may provide theoretical foundation for the approximations to stochastic geometric flows. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.