Chaotic Process Simulation

化学工艺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）、单元组还是整个装置。这些模拟产生的方程通常是非线性的，并且设计了不同复杂性的数值方法来求解它们。这类方法通常是从对未知参数进行初步的有根据的猜测开始，然后利用物理系统的方程进行一些迭代的方法来改进这些值，如直接代换、加速直接代换、各种类牛顿法等。这些被称为定点方法。本研究旨在建立对以下方面的理解：(1)稳态过程模拟的不动点方法的混沌（如周期加倍、周期三倍、奇异吸引子等）行为的基本原理；(2)稳态过程模拟的不动点方法的混沌行为与非线性过程的混沌行为之间的关系；动力系统。这两方面工作的具体任务如下：为了更好地理解混乱，pi将：(a)对复域中几何（或拓扑）结构（即吸引盆地、盆地边界、Julia集、Mandelbrot集等）的实际含义进行数值调查和研究；(b)研究稳定过程的定性影响(即：步骤边界，信任区域，几何结构的延拓，以及自动初始化过程的辅助含义；和(c)使用建立和/或开发新的分析工具（从微分拓扑和几何），以建立一个基本的理论认识和路线的分类，以混乱是相关的过程模拟。为了确定在稳态方程求解任务中是否存在可观察到的非稳态过程混沌演化特征，他们将：(a)在复杂域中进行稳态和非稳态并行数值实验，确定在相关的稳态方程求解任务中是否存在混沌过程动力学特征，如果存在；确定这些特征的内在性质；(b)研究几何结构(即吸引盆地、盆地边界、Julia集、等)与稳态模拟的不动点方法的混沌行为有关，与非定常化学过程的混沌演化有关的几何结构；(c)建立对特征过程的基本理论理解，以直接从稳态模型描述化学过程的混沌时间演化。所有的计算都在复定义域。

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.