Algorithms for Multiple Phases

该项目将开发严格理解，高效和准确的计算工具（算法及其软件实现，基于坚实的理论基础），用于模拟几何运动下曲面网络的运动，如加权平均曲率运动和表面扩散运动。该方法基于Merriman， Bence和Osher的阈值动力学（也称为扩散生成运动）的一种新的变分公式，使其能够正确地扩展原始算法，例如具有不等和各向异性表面张力的多相加权平均曲率流。本项目将研究的几何运动在材料科学中发挥核心作用，它们描述了多晶材料在常见工业过程（如退火（热处理））下的微观结构演变。多晶材料非常常见：大多数金属和陶瓷都属于这一类。它们是由微小的单晶片组成的，这些单晶片被称为颗粒，沿着它们的表面连接在一起。这些材料的物理性能，如导电性和屈服强度，与它们的微观结构（构成材料的颗粒的形状和大小）密切相关。该项目将提高我们模拟这些材料微观结构变化的能力，这对材料的计算设计很重要，其中计算机模拟而不是昂贵的实验将确定实现所需特性的最有利的加工参数。

This project will develop rigorously understood, highly efficient and accurate computational tools (algorithms and their software implementation, based on a solid theoretical foundation) for simulating the motion of networks of surfaces under geometric motions such as motion by weighted mean curvature and motion by surface diffusion. The approach is based on a new, variational formulation of threshold dynamics (also known as diffusion generated motion) of Merriman, Bence, and Osher that makes it possible to correctly extend the original algorithm to e.g. weighted mean curvature flow of multiple phases with unequal and anisotropic surface tensions.The geometric motions that will be studied in this project play a central role in materials science, where they describe the microstructural evolution of polycrystalline materials under common industrial processes such as annealing (heat treatment). Polycrystalline materials are very common: most metals and ceramics belong to this category. They are composed of tiny single-crystal pieces, known as grains, that are joined together along their faces. The physical properties of these materials, such as conductivity and yield strength, depend intimately on their microstructure (the shapes and sizes of the grains that make up the material). This project will advance our ability to simulate how the microstructure of these materials change, which is important for computational design of materials, where computer simulations rather than costly experiments would identify the most favorable processing parameters for achieving desired characteristics.