Chaotic Process Simulation

化工过程设计的第一步是对系统进行模拟，无论它们是单个单元(如反应器、热交换器、蒸馏塔等)、单元组，有时是整个工厂。这些模拟产生了通常是非线性的方程，并设计了各种复杂的数值方法来求解它们。这种方法通常从对未知参数的一些初始的有教育的猜测开始，然后用一些利用物理系统方程的迭代方法来改善这些值，如直接代换、加速直接代换、各种类牛顿方法等，这些方法被称为不动点方法。这项研究致力于建立对以下方面的理解：(1)混沌的基本原理(如周期加倍、周期三倍、奇怪吸引子等)。稳态过程仿真不动点方法的混沌行为；(2)过程稳态仿真不动点方法的混沌行为与过程作为非线性动态系统的混沌行为之间的关系。这两个工作领域的具体任务如下。为了更好地理解混沌，PI将：(A)对几何(或拓扑)结构(即吸引盆地、盆地边界、Julia集、曼德尔布洛特集等。)在复数域中，(B)研究稳定化过程(即步长边界、信赖域、连续性)对几何结构的定性影响，以及关于自动初始化过程的辅助含义，和(C)使用已建立的和/或开发新的分析工具(从微分拓扑学和几何学)，以建立与过程模拟相关的通向混沌的路径的基本理论理解和分类。为了确定在稳态方程求解任务中是否存在非稳态过程混沌演化的可观测特征(S)，他们将：(A)在复数域中进行并行的稳态和非稳态数值实验，以确定在相关联的稳态方程求解任务中是否存在混沌过程动力学的特征，如果是，则确定这些签名的内在属性，(B)研究几何结构(即，吸引盆地，盆地边界，Julia Sets等)与稳态模拟的不动点方法的混沌行为和与非稳态化学过程的混沌演化有关的几何结构相关联，以及(C)建立对签名过程的基本理论理解，以努力直接从稳态模型表征化学过程的混沌时间演化。他们的所有计算都将在复数域中进行。

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.