The Numerical Analysis and Application of High Dimensional Higher Index DAEs

9802259坎贝尔 微分代数方程（英语：Differential algebraic equation，DAE）是一个将各种量与它们的一些导数联系起来的方程组。 然而，与常微分方程（ODE）不同，方程并不直接提供状态变量的所有导数。 称为索引的非负整数是DAE与索引为零的ODE之间距离的一种度量。 指数越高，使用系统的难度就越大。 许多物理问题最初最自然地建模为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，近年来已经开发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数（不超过三个，往往限于一个或两个）的系统 本计画主要研究高维、高指标未知及非标准结构的微分代数系统的分析、模拟与应用。 两个应用程序，相当独立的利益，将被认为是作为两个应用程序，并指导，理论，分析和数值计算工作。 第一个应用是优化修剪工具的路径。 修边工具是一种用于从一块金属上切下一个零件的机器。 改进的修整工具的性能对于制造业和工业竞争力是非常重要的。 第二个应用是模拟密集封装的高频电路，目前的工业仿真软件包是不够的。 这些类型的问题的鲁棒解决方案需要额外的基本理论结果和算法的高指标高维DAE。 本计画将发展高指标高维微分代数系统的理论与演算法，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、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、近似过程类型和生成的DAE类型之间的相互作用。

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.