Adaptive Anisotropic Mesh Generation

这个项目是关于各向异性网格的自适应生成，用于偏微分方程的数值解。目标是在对插值误差进行严格的数学分析的基础上，研究各向异性网格及其质量度量，并最终开发出二维和三维网格生成的高效可靠的算法和计算机软件。在科学和工程数学建模的核心是一个系统的偏微分方程。数值模拟已成为求解偏微分方程和模拟物理过程不可缺少的工具。网格及其自适应生成对于提高数值模拟的精度和效率起着至关重要的作用。网格自适应的关键是能够同时控制网格元素的形状和大小。到目前为止，研究主要集中在传统的或各向同性的网格自适应，其中单元尺寸适应于一些误差估计，而形状保持接近等边。在求解误差较大的区域，各向同性网格容易集中过多的节点。通过适当选择各向异性网格，允许元素具有较大的纵横比，但与解对齐，可以克服这一缺点，并大大提高数值模拟的效率和准确性。该项目的研究将集中于发展插值误差的一般各向异性估计，并研究它们在构建新的算法和计算机代码中的应用，以生成二维和三维的自适应各向异性网格。网格评估策略和几何、对齐和适应方面的质量措施将得到发展。这些方法将应用于地下水流动系统的数值模拟，其中化学污染和其他物质会导致流动密度发生很大的变化。该项目的成功完成将大大提高我们对网格自适应中的数学的理解，其结果将为进一步研究和开发网格自适应算法和软件提供基础。所开发的网格划分策略可以与许多现有的数值方法和其他科学和工程领域的计算机代码相结合，大大提高了它们的效率和精度。教育将是这个项目的重要组成部分。在项目期间，学生将积极参与并接受专业和跨学科的培训。

This project is concerned with the adaptive generation of anisotropic meshes for use in the numerical solution of partial differential equations. The goal is to study anisotropic meshes and their quality measures based upon rigorous mathematical analysis of the interpolation error and ultimately to develop efficient and reliable algorithms and computer software for their generation in two and three dimensions. At the core of mathematical modeling in science and engineering is a system of partial differential equations. Numerical simulation has become an indispensable tool for solving partial differential equations and therefore simulating physical processes. Meshes and their adaptive generation play a vital role in improving the accuracy and efficiency of numerical simulations. The key to mesh adaptation is the ability to simultaneously control the shape and the size of mesh elements. Research has so far concentrated on traditional or isotropic mesh adaptation where the element size is adapted to some error estimate while the shape is kept close to being equilateral. It is known that isotropic meshes tend to concentrate too many nodes in the regions of large solution error. This drawback can be overcome and the efficiency and accuracy of numerical simulations can be dramatically enhanced by properly choosing an anisotropic mesh where the elements are allowed to have a large aspect ratio but aligned with the solution. The research of this project will focus on the development of general anisotropic estimates for interpolation error and the study of their use in construction of new algorithms and computer codes for generating adaptive anisotropic meshes in two and three dimensions. Mesh assessment strategies and quality measures in geometry, alignment, and adaptation will be developed. The methods will be applied to the numerical simulation of groundwater flow systems where chemical contamination and other substances can cause the flow density to vary considerably. The successful completion of this project will greatly improve our understanding of the mathematics within mesh adaptation, and the outcome will serve as building blocks for further studies and development of mesh adaptation algorithms and software. The developed meshing strategy can be combined with many existing numerical methods and computer codes in other areas of science and engineering to greatly enhance their efficiency and accuracy. Education will be important part of this project. Students will be actively involved and receive their professional and interdisciplinary training during the term of the project.