Numerical Analysis and Methods for Simulating Moving Interfaces and Controlling Shape

了解物理规律并开发数学模型进行预测，可以推动现代技术和工业进程的进步。此外，这些工艺的优化可以实现更好、更高效的设计。对社会的直接影响通常是节省成本、更强大的基础设施和更好的生活质量。该研究项目(见技术说明)的目标是针对涉及移动边界或界面的一组特定的物理/工业过程(例如，水中融化的冰块的表面是固体“冰”相和液体“水”相之间的界面)。例如，从液体熔体中生长出所需形状的固体相(受控晶体生长)，模拟液晶(显示技术)，通过电场控制微小液体液滴的微流体(在生物医学领域有用)，以及粒子和聚合物链在可变形的膜表面上的自组装(材料设计)。这项研究将使这些系统能够更好地设计、优化和控制。此外，该项目将创建新的方法来创建复杂形状的计算机模型，从而有效地捕捉移动边界。这被称为网格生成，这是许多领域使用的基本工具，从计算机图形(用于创建逼真的动画)一直到商业产品设计(桥梁等结构的工程设计)。就工时和金钱而言，电网发电仍然是一项昂贵的任务。这项研究开发的任何计算工具都将通过在线资源向公众提供。此外，本科生和研究生将受益于一门由PI开发的关于计算自由边界问题的特别课程。最后，在本次研究的激励下，PI正在引导中学生参与科学博览会项目。这项拟议的研究将开发数值分析工具和方法，用于模拟移动界面和多物理问题，并对自由边界进行最优控制。我们将在以下项目中利用变分/有限元方法、稳定性/能量估计、形状微分和自动网格技术：(A)发展和分析二维和三维Stefan问题(相变/凝固)的新的离散格式，并探索与时间相关的界面运动的适定性问题(在有限元方法中)；(B)发展一种新的具有可证明的稳定性/收敛性质的液晶有限元方法，并探索动力学和平衡构型；(C)发展基于PDE的最优控制技术和数值方法来指导和控制形状；(D)开发稳定和精确的数值方法(FEM)来模拟服从无自穿透条件的几何物体；以及(E)创建新的网格生成方法，以稳健和自动的方式处理三维问题，并注意并行实现问题。本项目将通过研究含时区域变形问题的适定性和形状的最优控制来推进自由边界问题的基本理论。这将进一步发展耦合到非线性界面物理的多物理问题的方法，如各向异性表面张力。它将创建新的数值分析和有效的方法来模拟动态曲线和曲面，这些曲线和曲面尊重*无自渗透*，即强制执行排除的体积约束。

Understanding physical laws and developing mathematical models to make predictions enables advancement of modern technology and industrial processes. Moreover, optimization of these processes allows for better and more efficient designs. The direct impacts to society are in general cost savings, more robust infrastructure, and better quality of life. The goal of this research project (see technical description) is to target a specific set of physical/industrial processes that involve moving boundaries or interfaces (e.g. the surface of a melting ice-cube in water is an interface between the solid "ice" phase and the liquid "water" phase). Examples include growing solid phases in a desired shape from a liquid melt (controlled crystal growth), modeling liquid crystals (display technology), micro-fluidics that control small droplets of liquid by electric fields (useful in the bio-medical field), and self-assembly of particles and polymer chains on deformable membrane surfaces (design of materials). The research will enable better design, optimization, and control of these systems. In addition, the project will create new methods for creating computer models of complex shapes that efficiently capture moving boundaries. This is known as grid generation, which is a basic tool used in many areas, from computer graphics (for creating life-like animations) all the way to commercial product design (engineering design of structures such as bridges). Grid generation is still an expensive task in terms of man-hours and money. Any computational tools developed by this research will be made available to the public through online resources. Furthermore, undergraduate and graduate students will benefit from a special course developed by the PI on computational free boundary problems. Lastly, the PI is guiding middle school students in science fair projects motivated by this research. The proposed research will develop numerical analysis tools and methods for simulating moving interface and multi-physics problems and for optimal control of free boundaries. We will take advantage of variational/finite element methods, stability/energy estimates, shape differentiation, and automatic meshing technology in the following projects: (A) develop and analyze new discrete formulations of the Stefan problem (phase change/solidification) in 2-D and 3-D and explore well posedness questions of the time-dependent interface motion (in a finite element method (FEM) setting); (B) develop a new FEM for liquid crystals with provable stability/convergence properties and explore dynamics and equilibrium configurations; (C) develop PDE-based optimal control techniques and numerics for directing and controlling shape; (D) develop stable and accurate numerical methods (FEM) for simulating geometric objects that obey the no self-penetration condition; and (E) create new mesh generation methods to handle 3-D problems in a robust and automatic way with attention given to parallel implementation issues. This project will advance the fundamental theory of free boundary problems by investigating well-posedness of time dependent domain-deforming problems and optimal control of shape. It will further the development of methods for multi-physics problems with coupling to non-linear interfacial physics, such as anisotropic surface tension. It will create novel numerical analysis and efficient methods for simulating dynamic curves and surfaces that respect *no self-penetration*, i.e. that enforce the excluded volume constraint.