Chaotic Process Simulation

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The first step in chemical process design is simulation of the systems, whether they be individual units (such as reactors, heat exchangers, distillation columns, etc.), groups of units, or sometimes the whole plant. These simulations yield equations which are usually nonlinear, and numerical methods of varying complexity have been devised to solve them. Such methods usually begin with some initial educated guess of the unknown parameters and then some iterative method using the equations of the physical system is used to improve these values, such as direct substitution, accelerated direct substitution, various Newton-like methods, etc. These are called fixed-point methods. This research is concerned with building an understanding of: (1) the fundamentals of the chaotic (such as period doubling, period tripling, strange attractors, etc.) behavior of fixed-point methods for steady- state process simulation, and (2) the relationship between the chaotic behavior of fixed-point methods for the steady-state simulation of processes and the chaotic behavior of processes as nonlinear, dynamical systems. Specific tasks under these two areas of work are as follows. To build a better understanding of chaos the PIs will: (a) conduct a numerical investigation and study of the practical implication of the geometric (or topological) structure (i.e., basins of attraction, basin boundaries, Julia sets, the Mandelbrot set, etc.) in the complex domain, (b) study the qualitative effects of stabilization procedures (i.e., step bounding, trust regions, continuation) on geometric structure, and auxiliary implications regarding automatic initialization procedures, and (c) use established and/or develop new analysis tools (from differential topology and geometry) to build a fundamental theoretical understanding and classification of routes to chaos that are relevant to process simulation. To determine if there is an observable signature(s) of the chaotic evolution of unsteady-state processes that manifests itself in the steady-state equation solving task they will: (a) conduct parallel steady- and unsteady-state numerical experiments in the complex domain to determine if signatures of chaotic process dynamics are present in the associated steady-state equation-solving task, and if so, determine the intrinsic properties of those signatures, (b) study the relationship between the geometric structure (i.e., basins of attraction, basin boundaries, Julia sets, etc.) associated with the chaotic behavior of fixed- point methods for steady-state simulation and the geometric structure associated with the chaotic evolution of unsteady- state chemical processes, and (c) build a fundamental theoretical understanding of the signature process in an effort to characterize the chaotic time evolution of chemical processes directly from steady-state models. All of their calculations will be in the complex domain.