Mathematical Science: General Methods for the Numerical Solution of Nonlinear DAEs

微分代数方程由微分关系和代数关系组成。DAE最初最自然地描述了许多物理问题。DAE出现在各种物理问题中，包括飞行控制、机器人、化学过程控制以及力学中的接触或约束问题。此外，在用直线法求解偏微分方程组时，以及在许多电路模型中都会出现偏微分方程组。计算机生成的物理过程的数学模型通常是DAE。现有的数字代码虽然非常有用，但需要特殊的问题结构，并且不能处理许多感兴趣的问题。此外，这些方法中的许多方法不仅要求潜在用户认识问题的数学结构，而且还要求明确地利用这种结构。这既限制了用户数量，也限制了可以解决的问题的复杂性。该项目将开发一般非线性DAE的数值方法。理论分析、计算实现和应用都将被考虑。拟议研究的长期目标是让科学家和工程师能够直接使用原始的隐式模型，从而减少设计和模拟时间，并有助于考虑更复杂的问题。物理原型的构建既耗时又昂贵，并限制了可以试验的想法的数量。计算机模拟对于设计和评估都变得越来越重要。在许多技术领域，计算机必须求解的方程式最容易建立为微分方程式和代数方程式或DAE的混合物。DAE不是经典的数值方法和数学理论所设计的那种方程。将DAE转换为另一种类型的数学问题通常既困难又耗时。有时，这种转换需要考虑一个更简单的数学方程，它不能很好地预测真实的物理系统将会做什么。这导致了世界范围内的努力，以开发求解DAE的数值方法。已经取得了显著的成功，其中一些努力已经导致了许多行业和实验室使用的软件，如DASSL。然而，许多重要的DAE类仍然很难或不可能使用现有软件直接解决。该项目将开发可用于求解重要的附加类型的DAE的数值方法。虽然在许多领域出现了DAE，但该项目将专注于受约束的机械系统，包括机器人操作手的模拟和控制，车辆系统的模拟和控制，以及化学制造过程的模拟、控制和优化。

A differential algebraic equation (DAE) is composed of both differential and algebraic relationships. Many physical problems are most naturally initially described by a DAE. DAEs arise in a variety of physical problems including flight control, robotics, chemical process control, and contact or constrained problems in mechanics. In addition, DAEs occur in the solution of partial differential equations by the method of lines and in many circuit models. Computer generated mathematical models for physical processes are often DAEs. Existing numerical codes, while very useful, require special problem structure and do not handle many problems of interest. In addition, many of these methods require the potential user not only to recognize the mathematical structure of the problem, but also to explicitly exploit this structure. This restricts both the number of users and the complexity of the problems that can be solved. This project will develop numerical methods for general nonlinear DAEs. Theoretical analysis, computational implementation, and applications will all be considered. The long range goal of the proposed research is for scientists and engineers to be able to work directly with the original implicit model thus reducing design and simulation time as well as facilitating the consideration of more complex problems. The building of physical prototypes is time consuming, expensive, and restricts the number of ideas that can be tried out. Computer simulation is becoming increasingly essential for both design and evaluation. In many areas of technology, the equations that the computer must solve are most easily set up as a mixture of differential and algebraic equations or a DAE. DAEs are not the kinds of equations that classical numerical methods and mathematical theory were designed for. Converting a DAE into another type of mathematical problem is often difficult and time consuming. Sometimes this conversion requires considering a simpler mathematical equation which does not do as good a job of predicting what the real physical system will do. This has led to a world wide effort to develop numerical methods for solving DAEs. There have been notable successes and some of this effort has resulted in software, such as DASSL, that is used in many industries and laboratories. However, many important classes of DAEs are still difficult, or impossible, to solve directly with existing software. This project will develop numerical methods that can be used to solve important additional types of DAEs. While DAEs arise in many areas, this project will focus on constrained mechanical systems including the simulation and control of robotic manipulators, simulation and control of vehicular systems, and the simulation, control, and optimization of chemical manufacturing processes.