Feasibility in Chemical Process Design and Simulation

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Abstract - Lucia - 9409158 Limits of feasibility for mathematical models of many problems in science and engineering are characterized by smooth turning or bifurcation points, which are the singular points of some associated Jacobian matrix. These turning and bifurcation points, as well as the solutions themselves, can be real or complex-valued and many numerical methods for obtaining them can experience periodic, aperiodic or divergent behavior. Accordingly, the PI plans to build a better understanding of the role and utility of singularity in defining and determining limits of feasibility in chemical process design and simulation by: 1. Conducting numerical studies that revolve around the use of (a) nonsingular implicit composite equation systems to define turning and bifurcation points, (b) trust region methods in the complex domain to compute these turning and bifurcation points, and (c) hybrid methods consisting of approximating some of the first partial derivatives of the implicit equations. 2. Developing the necessary mathematical framework for understanding the convergence and rate of convergence of complex domain or extended trust region methods for calculating limits of feasibility. Problem specifications in science and engineering applications always give rise to either feasible or infeasible sets of model equations. Feasible specifications always give rise to zero-valued minima in the two-norm, while infeasible ones correspond to nonzero-valued minima and singular points in the norm. Information regarding feasibility or infeasiblity, whether a priori or concomitant, is important to equation-solving tasks in design and simulation. The PI plans to: 1. Conduct numerical investigations on the use of (a) extended trust region methods for identifying and calculating conditions of feasibility or infeasibility and (b) accelerating convergence to singular points using complex domain quasi-Newton updates to build a quadratic model in the null space. 2. Study t he convergence and rates of convergence of the methods developed above.