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

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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.