Deformation Methods for Grid Adaptation

DMS-9732742国军辽宁科学与工程问题数值模拟成功的一个关键因素是物理域的网格生成。该项目的目标是开发一种自适应网格生成方法，可用于各种数值格式，如有限元、有限差分等。我们的方法基于复杂的数学，其数值实现是直接的。它是变形方法，它根据正在模拟的物理变量重新分布网格点。在误差较大的情况下，计算小的监视函数。然后，形变方法根据监测功能移动每个网格点，使新的网格点集中在需要更高分辨率的地方，以减少误差。流体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。我们将演示该方法能够生成适应激波、运动锋面和边界层等物理变量的精细特征的网格。该项目旨在极大地提高我们模拟关系到国家战略利益的大规模计算问题的能力。计算技术的进步使我们能够在不进行昂贵的实验的情况下模拟一些真实世界的问题。但对准确和高效的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，才能保持在今天的计算能力范围内。固定网格是计算资源的浪费。我们的研究将导致软件可以自动将更多的点放到物理变量快速变化的区域，而不需要添加新的点。这将对高性能计算和材料科学、环境问题、制造过程等需要网格的领域的数值模拟非常有用。通过参与该项目的工作，数学和工程专业的学生将获得将数学应用于解决工程问题的第一手经验。高级学位将授予在完成课程作业后对项目做出重大贡献的研究生。

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.