Empirical Similitude

这个项目的目标是通过经验相似法(ESM)来促进对数学预测的基本理解，ESM是一种新的方法，它考虑了非线性和属性变化来估计实际产品的性能。传统的相似方法(TSM)或量纲分析结合了白金汉的PI定理，通过分析和测试相应的规模化原型来预测系统的行为。该方法的限制和局限性在于假设系统服从幂定律，分析后得到的标度参数在整个应用范围内是唯一和恒定的，并且这些参数仅代表实际系统。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、矢量微积分和统计等工具在代数域实现系统的数值处理来解决各种形式的失真，包括各向同性和正交各向异性的模型失真、包括形状和取向的几何失真、尺寸和尺寸的参数失真以及方孔和圆孔的特征失真。该项目旨在开发一种全面的数学推导，以深入了解ESM的现有方法，并进一步将这一过程扩展到复杂系统。研究任务包括扩展共形映射，在Z空间或虚域中评估系统，开发集总经验相似方法的实用定义，以及使用循环矩阵方法建立ESM的实用推理。与业界在大规模汽车碰撞测试等应用方面的合作将证明这种方法可以对许多复杂的系统问题产生重大影响。研究人员还将探索与德克萨斯农工大学(TAMU)的一群教职员工开展的合作研究活动。这项合作将调查这项工作与在TAMU进行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.