Deformation Methods for Grid Adaptation

科学和工程问题数值模拟成功的一个关键因素是物理域网格的生成。该项目的目标是开发一种可用于各种数值格式（如有限元、有限差分等）的自适应网格生成方法。我们的方法是基于复杂的数学，它的数值实现是直接的。变形法是根据被模拟的物理变量重新分配网格点的方法。在误差较大的情况下，计算一个较小的监控函数。然后，变形法根据监测功能移动每个网格点，以便将新网格集中在需要更好分辨率的地方，以减少误差。流体动力学和其他重要应用中的模型问题将在这种移动网格上进行模拟。我们将证明该方法能够生成适应物理变量（如冲击波、移动锋面和边界层等）的精细特征的网格。该项目旨在大大提高我们模拟对国家战略利益至关重要的大规模计算问题的能力。计算机技术的进步使我们能够模拟一些现实世界的问题，而无需进行昂贵的实验。但是对精确和高效模拟的需求远远超过了当今最强大的计算机。必须进行大量的简化和升级，以保持在当今计算能力的限制之内。固定网格是计算资源的浪费。我们的研究将使软件能够在物理变量快速变化的地方自动添加更多的点，而无需添加新的点。这对于高性能计算和材料科学、环境问题、制造过程和其他需要网格的领域的数值模拟非常有用。通过参与项目，数学和工程专业的学生将获得应用数学解决工程问题的第一手经验。在完成课程后，对项目有显著贡献的研究生将被授予高级学位。

DMS-9732742Guojun LiaoA key factor to the success of numerical simulation of scientific and engineering problems is that of grid generation of the physical domains. The objective of the project is to develop an adaptive grid generation method that can be used with various numerical schemes such as finite element, finite difference, etc. Our method is based on sophisticated mathematics and its numerical implementation is straightforward. It is the deformation method which redistributes the grid points according to the physical variables that are being simulated. A monitor function is calculated which is small where the error is large. The deformation method then moves each grid point according to the monitor function so that the new grid will be concentrated where the need for better resolution exists to reduce the error. Model problems from fluid dynamics and other important applications will be simulated on such moving grids. We will demonstrate that the method is capable of generating grids that are adapted to the fine features of the physical variables such as shock waves, moving fronts, and boundary layers, etc.This project is aimed to greatly enhance our ability to simulate large scale computational problems that are crucial to the national strategic interests. The advance of computing technology allows us to simulate some real world problem without conducting costly experiments. But the needs for accurate and efficient simulation far exceed the most powerful computers of today. A lot of simplification and upgrading have to be made to stay within the limit of today's computing capacity. Fixed grids are wasteful in computing resource. Our research will lead to software that can automatically put more points to the regions where and when the physical variables change rapidly without adding new points. This would be very useful for high performance computing and the numerical simulation of material science, environment problems, manufacturing process and other fields that require a grid. By working in the project, mathematics and engineering students will gain first hand experience on the application of mathematics to the solution of engineering problem. Advanced degrees will be granted to graduate students who make significant contributions to the project after their completion of course work.