Algorithms for continuum processes on complex, moving surfaces

连续过程的偏微分方程（PDE）模型出现在整个应用科学和自然科学中。在一维、二维或三维空间中，许多重要的偏微分方程的数值方法已经得到了很大的发展。然而，很多问题都涉及到曲面上的微分方程。在曲面上求解偏微分方程的数值方法比在标准笛卡尔坐标空间R2或R3中求解要复杂得多。求解此类偏微分方程的一类常用方法是嵌入方法。在最近的研究中，我们引入了最近点法，这是一种将曲面几何与底层微分算子解耦的嵌入方法。这项研究是获得2011年格蒙德·达奎斯特奖的工作的一部分。*******最近点法使用嵌入空间中的标准数值PDE方法，以精确的方式在光滑、静止的表面上求解相当一般的PDE。然而，许多实际问题涉及到不光滑的移动表面。在这项资助中，将引入对最近点方法的一些几何增强：(1)将研究在具有褶皱、结和其他非光滑特征的表面上的一般偏微分方程的算法，(2)将开发在复杂、不断变化的表面上逼近偏微分方程的鲁棒方法，(3)也将研究在点云表面上的一般偏微分方程的精确方法。除了这些几何改进之外，还将使用域分解和优化的时间步进方法来提高效率。还将进行分析研究，为该方法的收敛性提供新的见解，特别是在其性能出乎意料地准确的情况下。在整个过程中，高素质的人才将参与数学软件的开发，数值算法的分析，以及求解表面偏微分方程的数值方法的设计。这为学术机构和技术相关行业的广泛职业提供了宝贵的经验。*******提出的研究将寻求精确和高效的算法和软件，但在某种意义上来说，它们在充分利用现有的3D技术和软件的同时，尽可能统一地计算不同连续体模型的解决方案。预计在这项资助下获得的方法将使研究人员和最终用户能够以极高的精度对复杂运动表面上的一般连续过程的现实模型进行数值研究。**** *****

***Partial differential equation (PDE) models for continuum processes arise throughout the applied and natural sciences. There has been a great effort made to develop numerical methods for many important classes of PDEs in one, two or three spatial dimensions. However, a remarkable variety of problems involve differential equations on curved surfaces. Numerical methods for PDEs on curved surfaces are much more complicated than in the standard Cartesian coordinate spaces R2 or R3. A popular class of methods for solving such PDEs are the embedding methods. In recent research, we introduced the closest point method, which is an embedding method that decouples surface geometry from the underlying differential operators. This research is part of the work that led to the 2011 Germund Dahquist Prize. *******The closest point method solves rather general PDEs on smooth, stationary surfaces in an accurate manner, using standard numerical PDE methods in the embedding space. However, many practical problems involve nonsmooth, moving surfaces. In this grant, a number of geometric enhancements to the closest point method will be introduced: (1) Research will be conducted on algorithms for general PDEs on surfaces with folds, junctions and other nonsmooth features, (2) robust methods for approximating PDEs on complex, evolving surfaces will be developed, (3) research on accurate methods for general PDEs on point cloud surfaces will also be carried out. Alongside these geometric improvements, efficiency enhancements will be developed using domain decomposition and optimized time-stepping methods. Analytical research will also be conducted to provide new insight into the convergence of the method, especially in situations where its performance is unexpectedly accurate. Throughout, highly qualified personnel will participate in the development of mathematical software, the analysis of numerical algorithms, and the design of numerical methods to solve PDEs on surfaces. This provides valuable experience for a broad range of careers in academic institutions and technology-related industries.*******The proposed research will seek algorithms and software that are accurate and efficient, yet are simple in the sense that they compute solutions to different continuum models as uniformly as possible while making strong use of existing techniques and software in 3D. It is anticipated that the methods obtained under this grant will enable researchers and end-users to numerically investigate realistic models of general continuum processes on complex moving surfaces with great accuracy.**** *****