Empirical Similitude

该项目的目标是通过经验相似法（ESM），一种新的方法，考虑到非线性和属性变化来估计实际产品的性能，以促进对数学预测的基本理解。传统的相似方法（TSM）或量纲分析结合了白金汉的π定理，通过分析和测试相应的比例原型来预测系统的行为。使用这种方法的约束和限制在于假设系统遵循幂律，分析后获得的缩放参数在整个应用范围内是唯一的和恒定的，并且参数仅指示实际系统。TSM分析主要局限于重点领域的预测。ESM中的初步工作已经通过使用包括保角映射、线性代数、向量微积分和统计在内的工具在代数域中实施系统的数值操作来实现，以解决各种形式的失真的问题，包括模型失真，如各向同性和正交各向异性特性，几何失真，包括形状和方向，参数失真，如大小和尺寸，以及特征失真，如方孔与圆孔。该项目旨在开发一个全面的数学推导，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，并使用循环矩阵方法建立ESM的实用推理。与工业界在汽车碰撞测试等应用领域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A M大学（TAMU）的一组教师合作研究活动。这项合作将调查这项工作和研究之间的可能联系到设计空间的概率建模与不确定性的表征正在TAMU进行。

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.