The Numerical Analysis and Application of High Dimensional Higher Index DAEs

高维高指数DAE的数值分析及应用

• 批准号: 9802259
• 负责人: Stephen Campbell
• 金额: $ 20万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9802259&HistoricalAwards=false
• 关键词: Numerical; Analysis; Application; Dimensional; Higher

9802259 Campbell A differential algebraic equation (DAE) is a system of equations relating various quantities with some of their derivatives. However, unlike with an ordinary differential equation (ODE), the equations do not directly provide all the derivatives of the state variables. A nonnegative integer called the index is one measure of how far a DAE is from an ODE which is index zero. The higher the index the greater the difficulty in working with the system. Many physical problems are most naturally initially modeled as a DAE particularly those that are analyzed and simulated using computer generated mathematical models. Because of the importance of DAEs in applications a variety of numerical techniques have been developed in recent years to simulate and analyze DAEs. These techniques, while very useful, are limited to systems with special structure and low index (no more than three and often limited to one or two) This project is to investigate the analysis, simulation, and application of high dimensional, higher index DAEs of unknown and nonstandard structure. Two applications, of considerable independent interest, will be considered to serve as both application of, and guide for, the theoretical, analytic, and numerical work. The first application is the optimization of the path of a trim tool. A trim tool is a machine used for cutting a part from a piece of metal. Improved performance of trim tools is of great importance for manufacturing and industrial competitiveness. The second application is the simulation of densely packed high frequency circuits for which current industrial simulation packages are inadequate. The robust solution of these types of problems requires additional fundamental theoretical results and algorithms for high index high dimensional DAEs. This project will develop theory and computational algorithms for high index high dimensional DAEs and apply them to the two applications. For both of these applications the high fidelity models needed are mixed systems of partial differential equations, DAEs, and constraints. Their solution will require examining the interplay between type of model, type of approximation process, and resulting types of DAEs.

9802259坎贝尔微分代数方程(DAE)是将各种量与它们的一些导数联系起来的一组方程。然而，与常微分方程(ODE)不同的是，该方程不直接提供状态变量的所有导数。称为索引的非负整数是衡量DAE与索引为零的ODE之间距离的一种指标。指数越高，使用该系统的难度就越大。许多物理问题最初最自然地被建模为DAE，特别是那些使用计算机生成的数学模型进行分析和模拟的问题。由于DAE在应用中的重要性，近年来发展了各种数值技术来模拟和分析DAE。这些技术虽然非常有用，但仅限于具有特殊结构和低指标的系统(不超过三个，通常限于一到两个)。本项目研究的是未知和非标准结构的高维、高指标DAE的分析、模拟和应用。两个具有相当独立兴趣的应用将被认为既是理论工作、分析工作和数值工作的应用，也是指导工作。第一个应用是修剪工具路径的优化。修边工具是用来从一块金属上切割零件的机器。提高修整工具的性能对制造业和工业竞争力具有重要意义。第二个应用是密集封装的高频电路的仿真，目前的工业仿真包不能满足这些应用。这类问题的稳健解决需要更多的高指标高维DAE的基本理论结果和算法。本项目将发展高指数高维DAE的理论和计算算法，并将其应用于这两个应用。对于这两种应用，所需的高保真模型是偏微分方程组、DAE和约束的混合系统。他们的解决方案将需要检查模型类型、近似过程类型和所产生的DAE类型之间的相互作用。