The Numerical Analysis and Application of High Dimensional Higher Index DAEs

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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.