Empirical Similitude

该项目的目标是通过经验相似法（ESM）推进对数学预测的基本理解，这是一种考虑非线性和属性变化来估计实际产品性能的新方法。传统的相似方法（TSM）或量纲分析结合了Buckingham's Pi定理，通过分析和测试相应的缩放原型来预测系统的行为。该方法的约束和局限性在于假设系统服从幂律，分析后得到的标度参数在整个应用范围内都是唯一不变的，参数只能反映实际系统。TSM分析主要局限于重点领域的预测。ESM的初步工作是通过使用保角映射、线性代数、矢量微积分和统计学等工具在代数领域实施系统的数值操作来解决各种形式的畸变问题，包括模型畸变，如各向同性和正交异性，几何畸变，包括形状和方向，参数畸变，如尺寸和尺寸，以及特征畸变，如方孔与圆孔。该项目旨在开发一个全面的数学推导，以深入了解ESM的现有方法，并进一步扩展复杂系统的这一过程。研究任务包括对保形映射的扩展，在z空间或虚域对系统进行评估，为集总经验相似法制定实用的定义，并利用循环矩阵方法建立ESM的实际推理。在大规模汽车碰撞测试等应用领域与工业界的合作，将证明这种方法对许多复杂系统问题的重大影响。研究人员还将探索与德克萨斯农工大学（TAMU）的一组教师的合作研究活动。这项合作将调查这项工作与TAMU正在进行的不确定性表征的设计空间概率建模研究之间的可能联系。

The goal of this project is to advance the fundamental understanding of mathematical predication through Empirical Similitude Method (ESM), a novel approach that takes non-linearities and property variations into account to estimate the performance of an actual product. The Traditional Similitude Method (TSM) or Dimensional Analysis incorporates the Buckingham's Pi theorem to predict the behavior of a system by analyzing and testing its corresponding scaled prototype. The constraints and limitations in using this method lie in the assumption that the system follows a power law, the scaling parameters obtained after analysis are unique and constant through the entire range of application, and the parameters are indicative of the actual system only. The TSM analysis is primarily confined to prediction in focused domains. Preliminary work in ESM has been achieved by implementing systematic numerical manipulations in the algebraic domain by using tools including Conformal Mapping, Linear Algebra, Vector Calculus and Statistics to address the concerns of various forms of distortion comprising model distortion like isotropic and orthotropic properties, geometric distortion including shape and orientation, parametric distortion like size and dimensions and feature distortion like square holes vs. round holes. This project aims to develop a comprehensive mathematical derivation that gives insights into the existing methods of ESM and further extends this process for complex systems. Research tasks include extending the Conformal Mapping, evaluating the system in the Z-space or the imaginary domain, developing a pragmatic definition for Lumped Empirical Similitude Method and using the Circulant Matrix approach to establish the practical reasoning of ESM. Collaboration with industry on such applications as scaled automotive crash testing will demonstrate the significant impact this approach can have to numerous complex systems problems. The researchers will also explore a collaborative research activity with a group of faculty at Texas A & M University (TAMU). This collaboration will investigate possible connections between this work and research into probability modeling of design spaces with the characterization of uncertainties being conducted at TAMU.